{"text": "[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\n\u22a2 p \u2208 lifts f \u2194 \u2203 q, map f q = p\n[PROOFSTEP]\nsimp only [coe_mapRingHom, lifts, RingHom.mem_rangeS]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\n\u22a2 p \u2208 lifts f \u2194 p \u2208 Set.range (map f)\n[PROOFSTEP]\nsimp only [coe_mapRingHom, lifts, Set.mem_range, RingHom.mem_rangeS]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\n\u22a2 p \u2208 lifts f \u2194 p \u2208 RingHom.rangeS (mapRingHom f)\n[PROOFSTEP]\nsimp only [coe_mapRingHom, lifts, Set.mem_range, RingHom.mem_rangeS]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\n\u22a2 p \u2208 lifts f \u2194 \u2200 (n : \u2115), coeff p n \u2208 Set.range \u2191f\n[PROOFSTEP]\nrw [lifts_iff_ringHom_rangeS, mem_map_rangeS f]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\n\u22a2 (\u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f) \u2194 \u2200 (n : \u2115), coeff p n \u2208 Set.range \u2191f\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nr : R\n\u22a2 \u2191(mapRingHom f) (\u2191C r) = \u2191C (\u2191f r)\n[PROOFSTEP]\nsimp only [coe_mapRingHom, map_C, Set.mem_univ, Subsemiring.coe_top, eq_self_iff_true, and_self_iff]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\ns : S\nh : s \u2208 Set.range \u2191f\n\u22a2 \u2191C s \u2208 lifts f\n[PROOFSTEP]\nobtain \u27e8r, rfl\u27e9 := Set.mem_range.1 h\n[GOAL]\ncase intro\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nr : R\nh : \u2191f r \u2208 Set.range \u2191f\n\u22a2 \u2191C (\u2191f r) \u2208 lifts f\n[PROOFSTEP]\nuse C r\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nr : R\nh : \u2191f r \u2208 Set.range \u2191f\n\u22a2 \u2191(mapRingHom f) (\u2191C r) = \u2191C (\u2191f r)\n[PROOFSTEP]\nsimp only [coe_mapRingHom, map_C, Set.mem_univ, Subsemiring.coe_top, eq_self_iff_true, and_self_iff]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\n\u22a2 \u2191(mapRingHom f) X = X\n[PROOFSTEP]\nsimp only [coe_mapRingHom, Set.mem_univ, Subsemiring.coe_top, eq_self_iff_true, map_X, and_self_iff]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nn : \u2115\n\u22a2 \u2191(mapRingHom f) (X ^ n) = X ^ n\n[PROOFSTEP]\nsimp only [coe_mapRingHom, map_pow, Set.mem_univ, Subsemiring.coe_top, eq_self_iff_true, map_X, and_self_iff]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nr : R\nhp : p \u2208 lifts f\n\u22a2 \u2191C (\u2191f r) * p \u2208 lifts f\n[PROOFSTEP]\nsimp only [lifts, RingHom.mem_rangeS] at hp \u22a2\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nr : R\nhp : \u2203 x, \u2191(mapRingHom f) x = p\n\u22a2 \u2203 x, \u2191(mapRingHom f) x = \u2191C (\u2191f r) * p\n[PROOFSTEP]\nobtain \u27e8p\u2081, rfl\u27e9 := hp\n[GOAL]\ncase intro\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nr : R\np\u2081 : R[X]\n\u22a2 \u2203 x, \u2191(mapRingHom f) x = \u2191C (\u2191f r) * \u2191(mapRingHom f) p\u2081\n[PROOFSTEP]\nuse C r * p\u2081\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nr : R\np\u2081 : R[X]\n\u22a2 \u2191(mapRingHom f) (\u2191C r * p\u2081) = \u2191C (\u2191f r) * \u2191(mapRingHom f) p\u2081\n[PROOFSTEP]\nsimp only [coe_mapRingHom, map_C, map_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nh : s \u2208 Set.range \u2191f\n\u22a2 \u2191(monomial n) s \u2208 lifts f\n[PROOFSTEP]\nobtain \u27e8r, rfl\u27e9 := Set.mem_range.1 h\n[GOAL]\ncase intro\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nr : R\nh : \u2191f r \u2208 Set.range \u2191f\n\u22a2 \u2191(monomial n) (\u2191f r) \u2208 lifts f\n[PROOFSTEP]\nuse monomial n r\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nr : R\nh : \u2191f r \u2208 Set.range \u2191f\n\u22a2 \u2191(mapRingHom f) (\u2191(monomial n) r) = \u2191(monomial n) (\u2191f r)\n[PROOFSTEP]\nsimp only [coe_mapRingHom, Set.mem_univ, map_monomial, Subsemiring.coe_top, eq_self_iff_true, and_self_iff]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nn : \u2115\nh : p \u2208 lifts f\n\u22a2 erase n p \u2208 lifts f\n[PROOFSTEP]\nrw [lifts_iff_ringHom_rangeS, mem_map_rangeS] at h \u22a2\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nn : \u2115\nh : \u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f\n\u22a2 \u2200 (n_1 : \u2115), coeff (erase n p) n_1 \u2208 RingHom.rangeS f\n[PROOFSTEP]\nintro k\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nn : \u2115\nh : \u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f\nk : \u2115\n\u22a2 coeff (erase n p) k \u2208 RingHom.rangeS f\n[PROOFSTEP]\nby_cases hk : k = n\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nn : \u2115\nh : \u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f\nk : \u2115\nhk : k = n\n\u22a2 coeff (erase n p) k \u2208 RingHom.rangeS f\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nn : \u2115\nh : \u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f\nk : \u2115\nhk : k = n\n\u22a2 \u2191f 0 = coeff (erase n p) k\n[PROOFSTEP]\nsimp only [hk, RingHom.map_zero, erase_same]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nn : \u2115\nh : \u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f\nk : \u2115\nhk : \u00ack = n\n\u22a2 coeff (erase n p) k \u2208 RingHom.rangeS f\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := h k\n[GOAL]\ncase neg.intro\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nn : \u2115\nh : \u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f\nk : \u2115\nhk : \u00ack = n\ni : R\nhi : \u2191f i = coeff p k\n\u22a2 coeff (erase n p) k \u2208 RingHom.rangeS f\n[PROOFSTEP]\nuse i\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nn : \u2115\nh : \u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f\nk : \u2115\nhk : \u00ack = n\ni : R\nhi : \u2191f i = coeff p k\n\u22a2 \u2191f i = coeff (erase n p) k\n[PROOFSTEP]\nsimp only [hi, hk, erase_ne, Ne.def, not_false_iff]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhl : \u2191(monomial n) s \u2208 lifts f\n\u22a2 \u2203 q, map f q = \u2191(monomial n) s \u2227 degree q = degree (\u2191(monomial n) s)\n[PROOFSTEP]\nby_cases hzero : s = 0\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhl : \u2191(monomial n) s \u2208 lifts f\nhzero : s = 0\n\u22a2 \u2203 q, map f q = \u2191(monomial n) s \u2227 degree q = degree (\u2191(monomial n) s)\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhl : \u2191(monomial n) s \u2208 lifts f\nhzero : s = 0\n\u22a2 map f 0 = \u2191(monomial n) s \u2227 degree 0 = degree (\u2191(monomial n) s)\n[PROOFSTEP]\nsimp only [hzero, degree_zero, eq_self_iff_true, and_self_iff, monomial_zero_right, Polynomial.map_zero]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhl : \u2191(monomial n) s \u2208 lifts f\nhzero : \u00acs = 0\n\u22a2 \u2203 q, map f q = \u2191(monomial n) s \u2227 degree q = degree (\u2191(monomial n) s)\n[PROOFSTEP]\nrw [lifts_iff_set_range] at hl \n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhl : \u2191(monomial n) s \u2208 Set.range (map f)\nhzero : \u00acs = 0\n\u22a2 \u2203 q, map f q = \u2191(monomial n) s \u2227 degree q = degree (\u2191(monomial n) s)\n[PROOFSTEP]\nobtain \u27e8q, hq\u27e9 := hl\n[GOAL]\ncase neg.intro\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhzero : \u00acs = 0\nq : R[X]\nhq : map f q = \u2191(monomial n) s\n\u22a2 \u2203 q, map f q = \u2191(monomial n) s \u2227 degree q = degree (\u2191(monomial n) s)\n[PROOFSTEP]\nreplace hq := (ext_iff.1 hq) n\n[GOAL]\ncase neg.intro\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhzero : \u00acs = 0\nq : R[X]\nhq : coeff (map f q) n = coeff (\u2191(monomial n) s) n\n\u22a2 \u2203 q, map f q = \u2191(monomial n) s \u2227 degree q = degree (\u2191(monomial n) s)\n[PROOFSTEP]\nhave hcoeff : f (q.coeff n) = s := by\n simp [coeff_monomial] at hq \n exact hq\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhzero : \u00acs = 0\nq : R[X]\nhq : coeff (map f q) n = coeff (\u2191(monomial n) s) n\n\u22a2 \u2191f (coeff q n) = s\n[PROOFSTEP]\nsimp [coeff_monomial] at hq \n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhzero : \u00acs = 0\nq : R[X]\nhq : \u2191f (coeff q n) = s\n\u22a2 \u2191f (coeff q n) = s\n[PROOFSTEP]\nexact hq\n[GOAL]\ncase neg.intro\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhzero : \u00acs = 0\nq : R[X]\nhq : coeff (map f q) n = coeff (\u2191(monomial n) s) n\nhcoeff : \u2191f (coeff q n) = s\n\u22a2 \u2203 q, map f q = \u2191(monomial n) s \u2227 degree q = degree (\u2191(monomial n) s)\n[PROOFSTEP]\nuse monomial n (q.coeff n)\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhzero : \u00acs = 0\nq : R[X]\nhq : coeff (map f q) n = coeff (\u2191(monomial n) s) n\nhcoeff : \u2191f (coeff q n) = s\n\u22a2 map f (\u2191(monomial n) (coeff q n)) = \u2191(monomial n) s \u2227 degree (\u2191(monomial n) (coeff q n)) = degree (\u2191(monomial n) s)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhzero : \u00acs = 0\nq : R[X]\nhq : coeff (map f q) n = coeff (\u2191(monomial n) s) n\nhcoeff : \u2191f (coeff q n) = s\n\u22a2 map f (\u2191(monomial n) (coeff q n)) = \u2191(monomial n) s\n[PROOFSTEP]\nsimp only [hcoeff, map_monomial]\n[GOAL]\ncase h.right\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhzero : \u00acs = 0\nq : R[X]\nhq : coeff (map f q) n = coeff (\u2191(monomial n) s) n\nhcoeff : \u2191f (coeff q n) = s\n\u22a2 degree (\u2191(monomial n) (coeff q n)) = degree (\u2191(monomial n) s)\n[PROOFSTEP]\nhave hqzero : q.coeff n \u2260 0 := by\n intro habs\n simp only [habs, RingHom.map_zero] at hcoeff \n exact hzero hcoeff.symm\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhzero : \u00acs = 0\nq : R[X]\nhq : coeff (map f q) n = coeff (\u2191(monomial n) s) n\nhcoeff : \u2191f (coeff q n) = s\n\u22a2 coeff q n \u2260 0\n[PROOFSTEP]\nintro habs\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhzero : \u00acs = 0\nq : R[X]\nhq : coeff (map f q) n = coeff (\u2191(monomial n) s) n\nhcoeff : \u2191f (coeff q n) = s\nhabs : coeff q n = 0\n\u22a2 False\n[PROOFSTEP]\nsimp only [habs, RingHom.map_zero] at hcoeff \n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhzero : \u00acs = 0\nq : R[X]\nhq : coeff (map f q) n = coeff (\u2191(monomial n) s) n\nhabs : coeff q n = 0\nhcoeff : 0 = s\n\u22a2 False\n[PROOFSTEP]\nexact hzero hcoeff.symm\n[GOAL]\ncase h.right\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhzero : \u00acs = 0\nq : R[X]\nhq : coeff (map f q) n = coeff (\u2191(monomial n) s) n\nhcoeff : \u2191f (coeff q n) = s\nhqzero : coeff q n \u2260 0\n\u22a2 degree (\u2191(monomial n) (coeff q n)) = degree (\u2191(monomial n) s)\n[PROOFSTEP]\nrw [\u2190 C_mul_X_pow_eq_monomial]\n[GOAL]\ncase h.right\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhzero : \u00acs = 0\nq : R[X]\nhq : coeff (map f q) n = coeff (\u2191(monomial n) s) n\nhcoeff : \u2191f (coeff q n) = s\nhqzero : coeff q n \u2260 0\n\u22a2 degree (\u2191C (coeff q n) * X ^ n) = degree (\u2191(monomial n) s)\n[PROOFSTEP]\nrw [\u2190 C_mul_X_pow_eq_monomial]\n[GOAL]\ncase h.right\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\nn : \u2115\nhzero : \u00acs = 0\nq : R[X]\nhq : coeff (map f q) n = coeff (\u2191(monomial n) s) n\nhcoeff : \u2191f (coeff q n) = s\nhqzero : coeff q n \u2260 0\n\u22a2 degree (\u2191C (coeff q n) * X ^ n) = degree (\u2191C s * X ^ n)\n[PROOFSTEP]\nsimp only [hzero, hqzero, Ne.def, not_false_iff, degree_C_mul_X_pow]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nhlifts : p \u2208 lifts f\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p\n[PROOFSTEP]\ngeneralize hd : p.natDegree = d\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nhlifts : p \u2208 lifts f\nd : \u2115\nhd : natDegree p = d\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p\n[PROOFSTEP]\nrevert hd p\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nd : \u2115\n\u22a2 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = d \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\n[PROOFSTEP]\ninduction' d using Nat.strong_induction_on with n hn\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\n\u22a2 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = n \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\n[PROOFSTEP]\nintros p hlifts hdeg\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p\n[PROOFSTEP]\nby_cases erase_zero : p.eraseLead = 0\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : eraseLead p = 0\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p\n[PROOFSTEP]\nrw [\u2190 eraseLead_add_monomial_natDegree_leadingCoeff p, erase_zero, zero_add, leadingCoeff]\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : eraseLead p = 0\n\u22a2 \u2203 q,\n map f q = \u2191(monomial (natDegree p)) (coeff p (natDegree p)) \u2227\n degree q = degree (\u2191(monomial (natDegree p)) (coeff p (natDegree p)))\n[PROOFSTEP]\nexact monomial_mem_lifts_and_degree_eq (monomial_mem_lifts p.natDegree ((lifts_iff_coeff_lifts p).1 hlifts p.natDegree))\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p\n[PROOFSTEP]\nhave deg_erase := Or.resolve_right (eraseLead_natDegree_lt_or_eraseLead_eq_zero p) erase_zero\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p\n[PROOFSTEP]\nhave pzero : p \u2260 0 := by\n intro habs\n exfalso\n rw [habs, eraseLead_zero, eq_self_iff_true, not_true] at erase_zero \n exact erase_zero\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\n\u22a2 p \u2260 0\n[PROOFSTEP]\nintro habs\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\nhabs : p = 0\n\u22a2 False\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\nhabs : p = 0\n\u22a2 False\n[PROOFSTEP]\nrw [habs, eraseLead_zero, eq_self_iff_true, not_true] at erase_zero \n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : False\ndeg_erase : natDegree (eraseLead p) < natDegree p\nhabs : p = 0\n\u22a2 False\n[PROOFSTEP]\nexact erase_zero\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p \u2260 0\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p\n[PROOFSTEP]\nhave lead_zero : p.coeff p.natDegree \u2260 0 := by rw [\u2190 leadingCoeff, Ne.def, leadingCoeff_eq_zero]; exact pzero\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p \u2260 0\n\u22a2 coeff p (natDegree p) \u2260 0\n[PROOFSTEP]\nrw [\u2190 leadingCoeff, Ne.def, leadingCoeff_eq_zero]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p \u2260 0\n\u22a2 \u00acp = 0\n[PROOFSTEP]\nexact pzero\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p \u2260 0\nlead_zero : coeff p (natDegree p) \u2260 0\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p\n[PROOFSTEP]\nobtain \u27e8lead, hlead\u27e9 :=\n monomial_mem_lifts_and_degree_eq (monomial_mem_lifts p.natDegree ((lifts_iff_coeff_lifts p).1 hlifts p.natDegree))\n[GOAL]\ncase neg.intro\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p \u2260 0\nlead_zero : coeff p (natDegree p) \u2260 0\nlead : R[X]\nhlead :\n map f lead = \u2191(monomial (natDegree p)) (coeff p (natDegree p)) \u2227\n degree lead = degree (\u2191(monomial (natDegree p)) (coeff p (natDegree p)))\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p\n[PROOFSTEP]\nhave deg_lead : lead.degree = p.natDegree := by\n rw [hlead.2, \u2190 C_mul_X_pow_eq_monomial, degree_C_mul_X_pow p.natDegree lead_zero]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p \u2260 0\nlead_zero : coeff p (natDegree p) \u2260 0\nlead : R[X]\nhlead :\n map f lead = \u2191(monomial (natDegree p)) (coeff p (natDegree p)) \u2227\n degree lead = degree (\u2191(monomial (natDegree p)) (coeff p (natDegree p)))\n\u22a2 degree lead = \u2191(natDegree p)\n[PROOFSTEP]\nrw [hlead.2, \u2190 C_mul_X_pow_eq_monomial, degree_C_mul_X_pow p.natDegree lead_zero]\n[GOAL]\ncase neg.intro\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p \u2260 0\nlead_zero : coeff p (natDegree p) \u2260 0\nlead : R[X]\nhlead :\n map f lead = \u2191(monomial (natDegree p)) (coeff p (natDegree p)) \u2227\n degree lead = degree (\u2191(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = \u2191(natDegree p)\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p\n[PROOFSTEP]\nrw [hdeg] at deg_erase \n[GOAL]\ncase neg.intro\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p \u2260 0\nlead_zero : coeff p (natDegree p) \u2260 0\nlead : R[X]\nhlead :\n map f lead = \u2191(monomial (natDegree p)) (coeff p (natDegree p)) \u2227\n degree lead = degree (\u2191(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = \u2191(natDegree p)\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p\n[PROOFSTEP]\nobtain \u27e8erase, herase\u27e9 :=\n hn p.eraseLead.natDegree deg_erase (erase_mem_lifts p.natDegree hlifts) (refl p.eraseLead.natDegree)\n[GOAL]\ncase neg.intro.intro\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p \u2260 0\nlead_zero : coeff p (natDegree p) \u2260 0\nlead : R[X]\nhlead :\n map f lead = \u2191(monomial (natDegree p)) (coeff p (natDegree p)) \u2227\n degree lead = degree (\u2191(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = \u2191(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p \u2227 degree erase = degree (Polynomial.erase (natDegree p) p)\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p\n[PROOFSTEP]\nuse erase + lead\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p \u2260 0\nlead_zero : coeff p (natDegree p) \u2260 0\nlead : R[X]\nhlead :\n map f lead = \u2191(monomial (natDegree p)) (coeff p (natDegree p)) \u2227\n degree lead = degree (\u2191(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = \u2191(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p \u2227 degree erase = degree (Polynomial.erase (natDegree p) p)\n\u22a2 map f (erase + lead) = p \u2227 degree (erase + lead) = degree p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p \u2260 0\nlead_zero : coeff p (natDegree p) \u2260 0\nlead : R[X]\nhlead :\n map f lead = \u2191(monomial (natDegree p)) (coeff p (natDegree p)) \u2227\n degree lead = degree (\u2191(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = \u2191(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p \u2227 degree erase = degree (Polynomial.erase (natDegree p) p)\n\u22a2 map f (erase + lead) = p\n[PROOFSTEP]\nsimp only [hlead, herase, Polynomial.map_add]\n[GOAL]\ncase h.left\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p \u2260 0\nlead_zero : coeff p (natDegree p) \u2260 0\nlead : R[X]\nhlead :\n map f lead = \u2191(monomial (natDegree p)) (coeff p (natDegree p)) \u2227\n degree lead = degree (\u2191(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = \u2191(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p \u2227 degree erase = degree (Polynomial.erase (natDegree p) p)\n\u22a2 Polynomial.erase (natDegree p) p + \u2191(monomial (natDegree p)) (coeff p (natDegree p)) = p\n[PROOFSTEP]\nrw [\u2190 eraseLead, \u2190 leadingCoeff]\n[GOAL]\ncase h.left\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p \u2260 0\nlead_zero : coeff p (natDegree p) \u2260 0\nlead : R[X]\nhlead :\n map f lead = \u2191(monomial (natDegree p)) (coeff p (natDegree p)) \u2227\n degree lead = degree (\u2191(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = \u2191(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p \u2227 degree erase = degree (Polynomial.erase (natDegree p) p)\n\u22a2 eraseLead p + \u2191(monomial (natDegree p)) (leadingCoeff p) = p\n[PROOFSTEP]\nrw [eraseLead_add_monomial_natDegree_leadingCoeff p]\n[GOAL]\ncase h.right\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p \u2260 0\nlead_zero : coeff p (natDegree p) \u2260 0\nlead : R[X]\nhlead :\n map f lead = \u2191(monomial (natDegree p)) (coeff p (natDegree p)) \u2227\n degree lead = degree (\u2191(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = \u2191(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p \u2227 degree erase = degree (Polynomial.erase (natDegree p) p)\n\u22a2 degree (erase + lead) = degree p\n[PROOFSTEP]\nrw [degree_eq_natDegree pzero, \u2190 deg_lead]\n[GOAL]\ncase h.right\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p \u2260 0\nlead_zero : coeff p (natDegree p) \u2260 0\nlead : R[X]\nhlead :\n map f lead = \u2191(monomial (natDegree p)) (coeff p (natDegree p)) \u2227\n degree lead = degree (\u2191(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = \u2191(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p \u2227 degree erase = degree (Polynomial.erase (natDegree p) p)\n\u22a2 degree (erase + lead) = degree lead\n[PROOFSTEP]\napply degree_add_eq_right_of_degree_lt\n[GOAL]\ncase h.right.h\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p \u2260 0\nlead_zero : coeff p (natDegree p) \u2260 0\nlead : R[X]\nhlead :\n map f lead = \u2191(monomial (natDegree p)) (coeff p (natDegree p)) \u2227\n degree lead = degree (\u2191(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = \u2191(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p \u2227 degree erase = degree (Polynomial.erase (natDegree p) p)\n\u22a2 degree erase < degree lead\n[PROOFSTEP]\nrw [herase.2, deg_lead, \u2190 degree_eq_natDegree pzero]\n[GOAL]\ncase h.right.h\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 \u2200 {p : S[X]}, p \u2208 lifts f \u2192 natDegree p = m \u2192 \u2203 q, map f q = p \u2227 degree q = degree p\np : S[X]\nhlifts : p \u2208 lifts f\nhdeg : natDegree p = n\nerase_zero : \u00aceraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p \u2260 0\nlead_zero : coeff p (natDegree p) \u2260 0\nlead : R[X]\nhlead :\n map f lead = \u2191(monomial (natDegree p)) (coeff p (natDegree p)) \u2227\n degree lead = degree (\u2191(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = \u2191(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p \u2227 degree erase = degree (Polynomial.erase (natDegree p) p)\n\u22a2 degree (Polynomial.erase (natDegree p) p) < degree p\n[PROOFSTEP]\nexact degree_erase_lt pzero\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p \u2227 Monic q\n[PROOFSTEP]\ncases' subsingleton_or_nontrivial R with hR hR\n[GOAL]\ncase inl\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Subsingleton R\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p \u2227 Monic q\n[PROOFSTEP]\nobtain \u27e8q, hq\u27e9 := mem_lifts_and_degree_eq hlifts\n[GOAL]\ncase inl.intro\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Subsingleton R\nq : R[X]\nhq : map f q = p \u2227 degree q = degree p\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p \u2227 Monic q\n[PROOFSTEP]\nexact \u27e8q, hq.1, hq.2, monic_of_subsingleton _\u27e9\n[GOAL]\ncase inr\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial R\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p \u2227 Monic q\n[PROOFSTEP]\nhave H : erase p.natDegree p + X ^ p.natDegree = p := by\n simpa only [hp.leadingCoeff, C_1, one_mul, eraseLead] using eraseLead_add_C_mul_X_pow p\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial R\n\u22a2 erase (natDegree p) p + X ^ natDegree p = p\n[PROOFSTEP]\nsimpa only [hp.leadingCoeff, C_1, one_mul, eraseLead] using eraseLead_add_C_mul_X_pow p\n[GOAL]\ncase inr\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial R\nH : erase (natDegree p) p + X ^ natDegree p = p\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p \u2227 Monic q\n[PROOFSTEP]\nby_cases h0 : erase p.natDegree p = 0\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial R\nH : erase (natDegree p) p + X ^ natDegree p = p\nh0 : erase (natDegree p) p = 0\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p \u2227 Monic q\n[PROOFSTEP]\nrw [\u2190 H, h0, zero_add]\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial R\nH : erase (natDegree p) p + X ^ natDegree p = p\nh0 : erase (natDegree p) p = 0\n\u22a2 \u2203 q, map f q = X ^ natDegree p \u2227 degree q = degree (X ^ natDegree p) \u2227 Monic q\n[PROOFSTEP]\nrefine' \u27e8X ^ p.natDegree, _, _, monic_X_pow p.natDegree\u27e9\n[GOAL]\ncase pos.refine'_1\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial R\nH : erase (natDegree p) p + X ^ natDegree p = p\nh0 : erase (natDegree p) p = 0\n\u22a2 map f (X ^ natDegree p) = X ^ natDegree p\n[PROOFSTEP]\nrw [Polynomial.map_pow, map_X]\n[GOAL]\ncase pos.refine'_2\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial R\nH : erase (natDegree p) p + X ^ natDegree p = p\nh0 : erase (natDegree p) p = 0\n\u22a2 degree (X ^ natDegree p) = degree (X ^ natDegree p)\n[PROOFSTEP]\nrw [degree_X_pow, degree_X_pow]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial R\nH : erase (natDegree p) p + X ^ natDegree p = p\nh0 : \u00acerase (natDegree p) p = 0\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p \u2227 Monic q\n[PROOFSTEP]\nobtain \u27e8q, hq\u27e9 := mem_lifts_and_degree_eq (erase_mem_lifts p.natDegree hlifts)\n[GOAL]\ncase neg.intro\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial R\nH : erase (natDegree p) p + X ^ natDegree p = p\nh0 : \u00acerase (natDegree p) p = 0\nq : R[X]\nhq : map f q = erase (natDegree p) p \u2227 degree q = degree (erase (natDegree p) p)\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p \u2227 Monic q\n[PROOFSTEP]\nhave p_neq_0 : p \u2260 0 := by intro hp; apply h0; rw [hp]; simp only [natDegree_zero, erase_zero]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial R\nH : erase (natDegree p) p + X ^ natDegree p = p\nh0 : \u00acerase (natDegree p) p = 0\nq : R[X]\nhq : map f q = erase (natDegree p) p \u2227 degree q = degree (erase (natDegree p) p)\n\u22a2 p \u2260 0\n[PROOFSTEP]\nintro hp\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp\u271d : Monic p\nhR : Nontrivial R\nH : erase (natDegree p) p + X ^ natDegree p = p\nh0 : \u00acerase (natDegree p) p = 0\nq : R[X]\nhq : map f q = erase (natDegree p) p \u2227 degree q = degree (erase (natDegree p) p)\nhp : p = 0\n\u22a2 False\n[PROOFSTEP]\napply h0\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp\u271d : Monic p\nhR : Nontrivial R\nH : erase (natDegree p) p + X ^ natDegree p = p\nh0 : \u00acerase (natDegree p) p = 0\nq : R[X]\nhq : map f q = erase (natDegree p) p \u2227 degree q = degree (erase (natDegree p) p)\nhp : p = 0\n\u22a2 erase (natDegree p) p = 0\n[PROOFSTEP]\nrw [hp]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp\u271d : Monic p\nhR : Nontrivial R\nH : erase (natDegree p) p + X ^ natDegree p = p\nh0 : \u00acerase (natDegree p) p = 0\nq : R[X]\nhq : map f q = erase (natDegree p) p \u2227 degree q = degree (erase (natDegree p) p)\nhp : p = 0\n\u22a2 erase (natDegree 0) 0 = 0\n[PROOFSTEP]\nsimp only [natDegree_zero, erase_zero]\n[GOAL]\ncase neg.intro\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial R\nH : erase (natDegree p) p + X ^ natDegree p = p\nh0 : \u00acerase (natDegree p) p = 0\nq : R[X]\nhq : map f q = erase (natDegree p) p \u2227 degree q = degree (erase (natDegree p) p)\np_neq_0 : p \u2260 0\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p \u2227 Monic q\n[PROOFSTEP]\nhave hdeg : q.degree < (X ^ p.natDegree).degree :=\n by\n rw [@degree_X_pow R, hq.2, \u2190 degree_eq_natDegree p_neq_0]\n exact degree_erase_lt p_neq_0\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial R\nH : erase (natDegree p) p + X ^ natDegree p = p\nh0 : \u00acerase (natDegree p) p = 0\nq : R[X]\nhq : map f q = erase (natDegree p) p \u2227 degree q = degree (erase (natDegree p) p)\np_neq_0 : p \u2260 0\n\u22a2 degree q < degree (X ^ natDegree p)\n[PROOFSTEP]\nrw [@degree_X_pow R, hq.2, \u2190 degree_eq_natDegree p_neq_0]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial R\nH : erase (natDegree p) p + X ^ natDegree p = p\nh0 : \u00acerase (natDegree p) p = 0\nq : R[X]\nhq : map f q = erase (natDegree p) p \u2227 degree q = degree (erase (natDegree p) p)\np_neq_0 : p \u2260 0\n\u22a2 degree (erase (natDegree p) p) < degree p\n[PROOFSTEP]\nexact degree_erase_lt p_neq_0\n[GOAL]\ncase neg.intro\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial R\nH : erase (natDegree p) p + X ^ natDegree p = p\nh0 : \u00acerase (natDegree p) p = 0\nq : R[X]\nhq : map f q = erase (natDegree p) p \u2227 degree q = degree (erase (natDegree p) p)\np_neq_0 : p \u2260 0\nhdeg : degree q < degree (X ^ natDegree p)\n\u22a2 \u2203 q, map f q = p \u2227 degree q = degree p \u2227 Monic q\n[PROOFSTEP]\nrefine' \u27e8q + X ^ p.natDegree, _, _, (monic_X_pow _).add_of_right hdeg\u27e9\n[GOAL]\ncase neg.intro.refine'_1\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial R\nH : erase (natDegree p) p + X ^ natDegree p = p\nh0 : \u00acerase (natDegree p) p = 0\nq : R[X]\nhq : map f q = erase (natDegree p) p \u2227 degree q = degree (erase (natDegree p) p)\np_neq_0 : p \u2260 0\nhdeg : degree q < degree (X ^ natDegree p)\n\u22a2 map f (q + X ^ natDegree p) = p\n[PROOFSTEP]\nrw [Polynomial.map_add, hq.1, Polynomial.map_pow, map_X, H]\n[GOAL]\ncase neg.intro.refine'_2\nR : Type u\ninst\u271d\u00b2 : Semiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Nontrivial S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial R\nH : erase (natDegree p) p + X ^ natDegree p = p\nh0 : \u00acerase (natDegree p) p = 0\nq : R[X]\nhq : map f q = erase (natDegree p) p \u2227 degree q = degree (erase (natDegree p) p)\np_neq_0 : p \u2260 0\nhdeg : degree q < degree (X ^ natDegree p)\n\u22a2 degree (q + X ^ natDegree p) = degree p\n[PROOFSTEP]\nrw [degree_add_eq_right_of_degree_lt hdeg, degree_X_pow, degree_eq_natDegree hp.ne_zero]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\n\u22a2 \u2203 q, map f q = p \u2227 natDegree q = natDegree p \u2227 Monic q\n[PROOFSTEP]\ncases' subsingleton_or_nontrivial S with hR hR\n[GOAL]\ncase inl\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Subsingleton S\n\u22a2 \u2203 q, map f q = p \u2227 natDegree q = natDegree p \u2227 Monic q\n[PROOFSTEP]\nobtain rfl : p = 1 := Subsingleton.elim _ _\n[GOAL]\ncase inl\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nhR : Subsingleton S\nhlifts : 1 \u2208 lifts f\nhp : Monic 1\n\u22a2 \u2203 q, map f q = 1 \u2227 natDegree q = natDegree 1 \u2227 Monic q\n[PROOFSTEP]\nrefine' \u27e81, Subsingleton.elim _ _, by simp, by simp\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nhR : Subsingleton S\nhlifts : 1 \u2208 lifts f\nhp : Monic 1\n\u22a2 natDegree 1 = natDegree 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\nhR : Subsingleton S\nhlifts : 1 \u2208 lifts f\nhp : Monic 1\n\u22a2 Monic 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial S\n\u22a2 \u2203 q, map f q = p \u2227 natDegree q = natDegree p \u2227 Monic q\n[PROOFSTEP]\nobtain \u27e8p', h\u2081, h\u2082, h\u2083\u27e9 := lifts_and_degree_eq_and_monic hlifts hp\n[GOAL]\ncase inr.intro.intro.intro\nR : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nhlifts : p \u2208 lifts f\nhp : Monic p\nhR : Nontrivial S\np' : R[X]\nh\u2081 : map f p' = p\nh\u2082 : degree p' = degree p\nh\u2083 : Monic p'\n\u22a2 \u2203 q, map f q = p \u2227 natDegree q = natDegree p \u2227 Monic q\n[PROOFSTEP]\nexact \u27e8p', h\u2081, natDegree_eq_of_degree_eq h\u2082, h\u2083\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\nS : Type v\ninst\u271d : Ring S\nf : R \u2192+* S\np : S[X]\n\u22a2 p \u2208 lifts f \u2194 p \u2208 liftsRing f\n[PROOFSTEP]\nsimp only [lifts, liftsRing, RingHom.mem_range, RingHom.mem_rangeS]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Algebra R S\np : R[X]\n\u22a2 \u2191(mapAlg R S) p = map (algebraMap R S) p\n[PROOFSTEP]\nsimp only [mapAlg, aeval_def, eval\u2082_eq_sum, map, algebraMap_apply, RingHom.coe_comp]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Algebra R S\np : R[X]\n\u22a2 (sum p fun e a => \u2191C (\u2191(algebraMap R S) a) * X ^ e) = sum p fun e a => (\u2191C \u2218 \u2191(algebraMap R S)) a * X ^ e\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Algebra R S\np : R[X]\nn\u271d : \u2115\n\u22a2 coeff (sum p fun e a => \u2191C (\u2191(algebraMap R S) a) * X ^ e) n\u271d =\n coeff (sum p fun e a => (\u2191C \u2218 \u2191(algebraMap R S)) a * X ^ e) n\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\nR\u271d : Type u\ninst\u271d\u2075 : CommSemiring R\u271d\nS\u271d : Type v\ninst\u271d\u2074 : Semiring S\u271d\ninst\u271d\u00b3 : Algebra R\u271d S\u271d\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Algebra R S\np : S[X]\n\u22a2 p \u2208 lifts (algebraMap R S) \u2194 p \u2208 AlgHom.range (mapAlg R S)\n[PROOFSTEP]\nsimp only [coe_mapRingHom, lifts, mapAlg_eq_map, AlgHom.mem_range, RingHom.mem_rangeS]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Algebra R S\np : S[X]\nr : R\nhp : p \u2208 lifts (algebraMap R S)\n\u22a2 r \u2022 p \u2208 lifts (algebraMap R S)\n[PROOFSTEP]\nrw [mem_lifts_iff_mem_alg] at hp \u22a2\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Type v\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Algebra R S\np : S[X]\nr : R\nhp : p \u2208 AlgHom.range (mapAlg R S)\n\u22a2 r \u2022 p \u2208 AlgHom.range (mapAlg R S)\n[PROOFSTEP]\nexact Subalgebra.smul_mem (mapAlg R S).range hp r\n", "meta": {"mathlib_filename": "Mathlib.Data.Polynomial.Lifts", "llama_tokens": 20474, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.511716619597144, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.2498627284462005}} {"text": "[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 Discrete PUnit\nX : C\n\u22a2 F.obj X = G.obj X\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 Discrete PUnit\nX : C\n\u22a2 F.obj X = G.obj X\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\n\u22a2 Nonempty (C \u224c Discrete PUnit) \u2194 Nonempty C \u2227 \u2200 (x y : C), Nonempty (Unique (x \u27f6 y))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\ninst\u271d : Category.{v, u} C\n\u22a2 Nonempty (C \u224c Discrete PUnit) \u2192 Nonempty C \u2227 \u2200 (x y : C), Nonempty (Unique (x \u27f6 y))\n[PROOFSTEP]\nrintro \u27e8h\u27e9\n[GOAL]\ncase mp.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\n\u22a2 Nonempty C \u2227 \u2200 (x y : C), Nonempty (Unique (x \u27f6 y))\n[PROOFSTEP]\nrefine' \u27e8\u27e8h.inverse.obj \u27e8\u27e8\u27e9\u27e9\u27e9, fun x y => Nonempty.intro _\u27e9\n[GOAL]\ncase mp.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\nx y : C\n\u22a2 Unique (x \u27f6 y)\n[PROOFSTEP]\nlet f : x \u27f6 y := by\n have hx : x \u27f6 h.inverse.obj \u27e8\u27e8\u27e9\u27e9 := by convert h.unit.app x\n have hy : h.inverse.obj \u27e8\u27e8\u27e9\u27e9 \u27f6 y := by convert h.unitInv.app y\n exact hx \u226b hy\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\nx y : C\n\u22a2 x \u27f6 y\n[PROOFSTEP]\nhave hx : x \u27f6 h.inverse.obj \u27e8\u27e8\u27e9\u27e9 := by convert h.unit.app x\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\nx y : C\n\u22a2 x \u27f6 h.inverse.obj { as := PUnit.unit }\n[PROOFSTEP]\nconvert h.unit.app x\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\nx y : C\nhx : x \u27f6 h.inverse.obj { as := PUnit.unit }\n\u22a2 x \u27f6 y\n[PROOFSTEP]\nhave hy : h.inverse.obj \u27e8\u27e8\u27e9\u27e9 \u27f6 y := by convert h.unitInv.app y\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\nx y : C\nhx : x \u27f6 h.inverse.obj { as := PUnit.unit }\n\u22a2 h.inverse.obj { as := PUnit.unit } \u27f6 y\n[PROOFSTEP]\nconvert h.unitInv.app y\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\nx y : C\nhx : x \u27f6 h.inverse.obj { as := PUnit.unit }\nhy : h.inverse.obj { as := PUnit.unit } \u27f6 y\n\u22a2 x \u27f6 y\n[PROOFSTEP]\nexact hx \u226b hy\n[GOAL]\ncase mp.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\nx y : C\nf : x \u27f6 y :=\n let_fun hx :=\n Eq.mpr (_ : (x \u27f6 h.inverse.obj { as := PUnit.unit }) = ((\ud835\udfed C).obj x \u27f6 (h.functor \u22d9 h.inverse).obj x))\n (id (NatTrans.app (Equivalence.unit h) x));\n let_fun hy :=\n Eq.mpr (_ : (h.inverse.obj { as := PUnit.unit } \u27f6 y) = ((h.functor \u22d9 h.inverse).obj y \u27f6 (\ud835\udfed C).obj y))\n (id (NatTrans.app (Equivalence.unitInv h) y));\n hx \u226b hy\n\u22a2 Unique (x \u27f6 y)\n[PROOFSTEP]\nsuffices sub : Subsingleton (x \u27f6 y) from uniqueOfSubsingleton f\n[GOAL]\ncase mp.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\nx y : C\nf : x \u27f6 y :=\n let_fun hx :=\n Eq.mpr (_ : (x \u27f6 h.inverse.obj { as := PUnit.unit }) = ((\ud835\udfed C).obj x \u27f6 (h.functor \u22d9 h.inverse).obj x))\n (id (NatTrans.app (Equivalence.unit h) x));\n let_fun hy :=\n Eq.mpr (_ : (h.inverse.obj { as := PUnit.unit } \u27f6 y) = ((h.functor \u22d9 h.inverse).obj y \u27f6 (\ud835\udfed C).obj y))\n (id (NatTrans.app (Equivalence.unitInv h) y));\n hx \u226b hy\n\u22a2 Subsingleton (x \u27f6 y)\n[PROOFSTEP]\nhave : \u2200 z, z = h.unit.app x \u226b (h.functor \u22d9 h.inverse).map z \u226b h.unitInv.app y :=\n by\n intro z\n simp [congrArg (\u00b7 \u226b h.unitInv.app y) (h.unit.naturality z)]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\nx y : C\nf : x \u27f6 y :=\n let_fun hx :=\n Eq.mpr (_ : (x \u27f6 h.inverse.obj { as := PUnit.unit }) = ((\ud835\udfed C).obj x \u27f6 (h.functor \u22d9 h.inverse).obj x))\n (id (NatTrans.app (Equivalence.unit h) x));\n let_fun hy :=\n Eq.mpr (_ : (h.inverse.obj { as := PUnit.unit } \u27f6 y) = ((h.functor \u22d9 h.inverse).obj y \u27f6 (\ud835\udfed C).obj y))\n (id (NatTrans.app (Equivalence.unitInv h) y));\n hx \u226b hy\n\u22a2 \u2200 (z : x \u27f6 y),\n z = NatTrans.app (Equivalence.unit h) x \u226b (h.functor \u22d9 h.inverse).map z \u226b NatTrans.app (Equivalence.unitInv h) y\n[PROOFSTEP]\nintro z\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\nx y : C\nf : x \u27f6 y :=\n let_fun hx :=\n Eq.mpr (_ : (x \u27f6 h.inverse.obj { as := PUnit.unit }) = ((\ud835\udfed C).obj x \u27f6 (h.functor \u22d9 h.inverse).obj x))\n (id (NatTrans.app (Equivalence.unit h) x));\n let_fun hy :=\n Eq.mpr (_ : (h.inverse.obj { as := PUnit.unit } \u27f6 y) = ((h.functor \u22d9 h.inverse).obj y \u27f6 (\ud835\udfed C).obj y))\n (id (NatTrans.app (Equivalence.unitInv h) y));\n hx \u226b hy\nz : x \u27f6 y\n\u22a2 z = NatTrans.app (Equivalence.unit h) x \u226b (h.functor \u22d9 h.inverse).map z \u226b NatTrans.app (Equivalence.unitInv h) y\n[PROOFSTEP]\nsimp [congrArg (\u00b7 \u226b h.unitInv.app y) (h.unit.naturality z)]\n[GOAL]\ncase mp.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\nx y : C\nf : x \u27f6 y :=\n let_fun hx :=\n Eq.mpr (_ : (x \u27f6 h.inverse.obj { as := PUnit.unit }) = ((\ud835\udfed C).obj x \u27f6 (h.functor \u22d9 h.inverse).obj x))\n (id (NatTrans.app (Equivalence.unit h) x));\n let_fun hy :=\n Eq.mpr (_ : (h.inverse.obj { as := PUnit.unit } \u27f6 y) = ((h.functor \u22d9 h.inverse).obj y \u27f6 (\ud835\udfed C).obj y))\n (id (NatTrans.app (Equivalence.unitInv h) y));\n hx \u226b hy\nthis :\n \u2200 (z : x \u27f6 y),\n z = NatTrans.app (Equivalence.unit h) x \u226b (h.functor \u22d9 h.inverse).map z \u226b NatTrans.app (Equivalence.unitInv h) y\n\u22a2 Subsingleton (x \u27f6 y)\n[PROOFSTEP]\napply Subsingleton.intro\n[GOAL]\ncase mp.intro.allEq\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\nx y : C\nf : x \u27f6 y :=\n let_fun hx :=\n Eq.mpr (_ : (x \u27f6 h.inverse.obj { as := PUnit.unit }) = ((\ud835\udfed C).obj x \u27f6 (h.functor \u22d9 h.inverse).obj x))\n (id (NatTrans.app (Equivalence.unit h) x));\n let_fun hy :=\n Eq.mpr (_ : (h.inverse.obj { as := PUnit.unit } \u27f6 y) = ((h.functor \u22d9 h.inverse).obj y \u27f6 (\ud835\udfed C).obj y))\n (id (NatTrans.app (Equivalence.unitInv h) y));\n hx \u226b hy\nthis :\n \u2200 (z : x \u27f6 y),\n z = NatTrans.app (Equivalence.unit h) x \u226b (h.functor \u22d9 h.inverse).map z \u226b NatTrans.app (Equivalence.unitInv h) y\n\u22a2 \u2200 (a b : x \u27f6 y), a = b\n[PROOFSTEP]\nintro a b\n[GOAL]\ncase mp.intro.allEq\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\nx y : C\nf : x \u27f6 y :=\n let_fun hx :=\n Eq.mpr (_ : (x \u27f6 h.inverse.obj { as := PUnit.unit }) = ((\ud835\udfed C).obj x \u27f6 (h.functor \u22d9 h.inverse).obj x))\n (id (NatTrans.app (Equivalence.unit h) x));\n let_fun hy :=\n Eq.mpr (_ : (h.inverse.obj { as := PUnit.unit } \u27f6 y) = ((h.functor \u22d9 h.inverse).obj y \u27f6 (\ud835\udfed C).obj y))\n (id (NatTrans.app (Equivalence.unitInv h) y));\n hx \u226b hy\nthis :\n \u2200 (z : x \u27f6 y),\n z = NatTrans.app (Equivalence.unit h) x \u226b (h.functor \u22d9 h.inverse).map z \u226b NatTrans.app (Equivalence.unitInv h) y\na b : x \u27f6 y\n\u22a2 a = b\n[PROOFSTEP]\nrw [this a, this b]\n[GOAL]\ncase mp.intro.allEq\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\nx y : C\nf : x \u27f6 y :=\n let_fun hx :=\n Eq.mpr (_ : (x \u27f6 h.inverse.obj { as := PUnit.unit }) = ((\ud835\udfed C).obj x \u27f6 (h.functor \u22d9 h.inverse).obj x))\n (id (NatTrans.app (Equivalence.unit h) x));\n let_fun hy :=\n Eq.mpr (_ : (h.inverse.obj { as := PUnit.unit } \u27f6 y) = ((h.functor \u22d9 h.inverse).obj y \u27f6 (\ud835\udfed C).obj y))\n (id (NatTrans.app (Equivalence.unitInv h) y));\n hx \u226b hy\nthis :\n \u2200 (z : x \u27f6 y),\n z = NatTrans.app (Equivalence.unit h) x \u226b (h.functor \u22d9 h.inverse).map z \u226b NatTrans.app (Equivalence.unitInv h) y\na b : x \u27f6 y\n\u22a2 NatTrans.app (Equivalence.unit h) x \u226b (h.functor \u22d9 h.inverse).map a \u226b NatTrans.app (Equivalence.unitInv h) y =\n NatTrans.app (Equivalence.unit h) x \u226b (h.functor \u22d9 h.inverse).map b \u226b NatTrans.app (Equivalence.unitInv h) y\n[PROOFSTEP]\nsimp only [Functor.comp_map]\n[GOAL]\ncase mp.intro.allEq\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\nx y : C\nf : x \u27f6 y :=\n let_fun hx :=\n Eq.mpr (_ : (x \u27f6 h.inverse.obj { as := PUnit.unit }) = ((\ud835\udfed C).obj x \u27f6 (h.functor \u22d9 h.inverse).obj x))\n (id (NatTrans.app (Equivalence.unit h) x));\n let_fun hy :=\n Eq.mpr (_ : (h.inverse.obj { as := PUnit.unit } \u27f6 y) = ((h.functor \u22d9 h.inverse).obj y \u27f6 (\ud835\udfed C).obj y))\n (id (NatTrans.app (Equivalence.unitInv h) y));\n hx \u226b hy\nthis :\n \u2200 (z : x \u27f6 y),\n z = NatTrans.app (Equivalence.unit h) x \u226b (h.functor \u22d9 h.inverse).map z \u226b NatTrans.app (Equivalence.unitInv h) y\na b : x \u27f6 y\n\u22a2 NatTrans.app (Equivalence.unit h) x \u226b h.inverse.map (h.functor.map a) \u226b NatTrans.app (Equivalence.unitInv h) y =\n NatTrans.app (Equivalence.unit h) x \u226b h.inverse.map (h.functor.map b) \u226b NatTrans.app (Equivalence.unitInv h) y\n[PROOFSTEP]\ncongr 3\n[GOAL]\ncase mp.intro.allEq.e_a.e_a.e_a\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\nx y : C\nf : x \u27f6 y :=\n let_fun hx :=\n Eq.mpr (_ : (x \u27f6 h.inverse.obj { as := PUnit.unit }) = ((\ud835\udfed C).obj x \u27f6 (h.functor \u22d9 h.inverse).obj x))\n (id (NatTrans.app (Equivalence.unit h) x));\n let_fun hy :=\n Eq.mpr (_ : (h.inverse.obj { as := PUnit.unit } \u27f6 y) = ((h.functor \u22d9 h.inverse).obj y \u27f6 (\ud835\udfed C).obj y))\n (id (NatTrans.app (Equivalence.unitInv h) y));\n hx \u226b hy\nthis :\n \u2200 (z : x \u27f6 y),\n z = NatTrans.app (Equivalence.unit h) x \u226b (h.functor \u22d9 h.inverse).map z \u226b NatTrans.app (Equivalence.unitInv h) y\na b : x \u27f6 y\n\u22a2 h.functor.map a = h.functor.map b\n[PROOFSTEP]\napply ULift.ext\n[GOAL]\ncase mp.intro.allEq.e_a.e_a.e_a.h\nC : Type u\ninst\u271d : Category.{v, u} C\nh : C \u224c Discrete PUnit\nx y : C\nf : x \u27f6 y :=\n let_fun hx :=\n Eq.mpr (_ : (x \u27f6 h.inverse.obj { as := PUnit.unit }) = ((\ud835\udfed C).obj x \u27f6 (h.functor \u22d9 h.inverse).obj x))\n (id (NatTrans.app (Equivalence.unit h) x));\n let_fun hy :=\n Eq.mpr (_ : (h.inverse.obj { as := PUnit.unit } \u27f6 y) = ((h.functor \u22d9 h.inverse).obj y \u27f6 (\ud835\udfed C).obj y))\n (id (NatTrans.app (Equivalence.unitInv h) y));\n hx \u226b hy\nthis :\n \u2200 (z : x \u27f6 y),\n z = NatTrans.app (Equivalence.unit h) x \u226b (h.functor \u22d9 h.inverse).map z \u226b NatTrans.app (Equivalence.unitInv h) y\na b : x \u27f6 y\n\u22a2 (h.functor.map a).down = (h.functor.map b).down\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\n\u22a2 (Nonempty C \u2227 \u2200 (x y : C), Nonempty (Unique (x \u27f6 y))) \u2192 Nonempty (C \u224c Discrete PUnit)\n[PROOFSTEP]\nrintro \u27e8\u27e8p\u27e9, h\u27e9\n[GOAL]\ncase mpr.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nh : \u2200 (x y : C), Nonempty (Unique (x \u27f6 y))\np : C\n\u22a2 Nonempty (C \u224c Discrete PUnit)\n[PROOFSTEP]\nhaveI := fun x y => (h x y).some\n[GOAL]\ncase mpr.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nh : \u2200 (x y : C), Nonempty (Unique (x \u27f6 y))\np : C\nthis : (x y : C) \u2192 Unique (x \u27f6 y)\n\u22a2 Nonempty (C \u224c Discrete PUnit)\n[PROOFSTEP]\nrefine'\n Nonempty.intro\n (CategoryTheory.Equivalence.mk ((Functor.const _).obj \u27e8\u27e8\u27e9\u27e9) ((@Functor.const <| Discrete PUnit).obj p) ?_\n (by apply Functor.punitExt))\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nh : \u2200 (x y : C), Nonempty (Unique (x \u27f6 y))\np : C\nthis : (x y : C) \u2192 Unique (x \u27f6 y)\n\u22a2 (Functor.const (Discrete PUnit)).obj p \u22d9 (Functor.const C).obj { as := PUnit.unit } \u2245 \ud835\udfed (Discrete PUnit)\n[PROOFSTEP]\napply Functor.punitExt\n[GOAL]\ncase mpr.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nh : \u2200 (x y : C), Nonempty (Unique (x \u27f6 y))\np : C\nthis : (x y : C) \u2192 Unique (x \u27f6 y)\n\u22a2 \ud835\udfed C \u2245 (Functor.const C).obj { as := PUnit.unit } \u22d9 (Functor.const (Discrete PUnit)).obj p\n[PROOFSTEP]\nexact\n NatIso.ofComponents fun _ =>\n { hom := default\n inv := default }\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.PUnit", "llama_tokens": 5337, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.45713671682749474, "lm_q1q2_score": 0.2481627389556736}} {"text": "[GOAL]\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsIrrefl \u03b1 r\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : Lex r (a :: l\u2081) (a :: l\u2082)\n\u22a2 Lex r l\u2081 l\u2082\n[PROOFSTEP]\ncases' h with _ _ _ _ _ h _ _ _ _ h\n[GOAL]\ncase cons\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsIrrefl \u03b1 r\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : Lex r l\u2081 l\u2082\n\u22a2 Lex r l\u2081 l\u2082\ncase rel \u03b1 : Type u r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : IsIrrefl \u03b1 r a : \u03b1 l\u2081 l\u2082 : List \u03b1 h : r a a \u22a2 Lex r l\u2081 l\u2082\n[PROOFSTEP]\nexacts [h, (irrefl_of r a h).elim]\n[GOAL]\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : IsOrderConnected \u03b1 r\ninst\u271d : IsTrichotomous \u03b1 r\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 l\u2083 : List \u03b1\nh : Lex r l\u2081 l\u2083\n\u22a2 Lex r (a :: l\u2081) (b :: l\u2082) \u2228 Lex r (b :: l\u2082) (a :: l\u2083)\n[PROOFSTEP]\nrcases trichotomous_of r a b with (ab | rfl | ab)\n[GOAL]\ncase inl\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : IsOrderConnected \u03b1 r\ninst\u271d : IsTrichotomous \u03b1 r\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 l\u2083 : List \u03b1\nh : Lex r l\u2081 l\u2083\nab : r a b\n\u22a2 Lex r (a :: l\u2081) (b :: l\u2082) \u2228 Lex r (b :: l\u2082) (a :: l\u2083)\n[PROOFSTEP]\nexact Or.inl (rel ab)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : IsOrderConnected \u03b1 r\ninst\u271d : IsTrichotomous \u03b1 r\na : \u03b1\nl\u2081 l\u2082 l\u2083 : List \u03b1\nh : Lex r l\u2081 l\u2083\n\u22a2 Lex r (a :: l\u2081) (a :: l\u2082) \u2228 Lex r (a :: l\u2082) (a :: l\u2083)\n[PROOFSTEP]\nexact (aux _ l\u2082 _ h).imp cons cons\n[GOAL]\ncase inr.inr\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : IsOrderConnected \u03b1 r\ninst\u271d : IsTrichotomous \u03b1 r\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 l\u2083 : List \u03b1\nh : Lex r l\u2081 l\u2083\nab : r b a\n\u22a2 Lex r (a :: l\u2081) (b :: l\u2082) \u2228 Lex r (b :: l\u2082) (a :: l\u2083)\n[PROOFSTEP]\nexact Or.inr (rel ab)\n[GOAL]\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsTrichotomous \u03b1 r\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 : List \u03b1\n\u22a2 Lex r (a :: l\u2081) (b :: l\u2082) \u2228 a :: l\u2081 = b :: l\u2082 \u2228 Lex r (b :: l\u2082) (a :: l\u2081)\n[PROOFSTEP]\nrcases trichotomous_of r a b with (ab | rfl | ab)\n[GOAL]\ncase inl\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsTrichotomous \u03b1 r\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 : List \u03b1\nab : r a b\n\u22a2 Lex r (a :: l\u2081) (b :: l\u2082) \u2228 a :: l\u2081 = b :: l\u2082 \u2228 Lex r (b :: l\u2082) (a :: l\u2081)\n[PROOFSTEP]\nexact Or.inl (rel ab)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsTrichotomous \u03b1 r\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 Lex r (a :: l\u2081) (a :: l\u2082) \u2228 a :: l\u2081 = a :: l\u2082 \u2228 Lex r (a :: l\u2082) (a :: l\u2081)\n[PROOFSTEP]\nexact (aux l\u2081 l\u2082).imp cons (Or.imp (congr_arg _) cons)\n[GOAL]\ncase inr.inr\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsTrichotomous \u03b1 r\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 : List \u03b1\nab : r b a\n\u22a2 Lex r (a :: l\u2081) (b :: l\u2082) \u2228 a :: l\u2081 = b :: l\u2082 \u2228 Lex r (b :: l\u2082) (a :: l\u2081)\n[PROOFSTEP]\nexact Or.inr (Or.inr (rel ab))\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : DecidableEq \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl\u2081 : List \u03b1\nh : Lex r l\u2081 []\n\u22a2 False\n[PROOFSTEP]\ncases h\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : DecidableEq \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 : List \u03b1\n\u22a2 Decidable (Lex r (a :: l\u2081) (b :: l\u2082))\n[PROOFSTEP]\nhaveI := decidableRel r l\u2081 l\u2082\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : DecidableEq \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 : List \u03b1\nthis : Decidable (Lex r l\u2081 l\u2082)\n\u22a2 Decidable (Lex r (a :: l\u2081) (b :: l\u2082))\n[PROOFSTEP]\nrefine' decidable_of_iff (r a b \u2228 a = b \u2227 Lex r l\u2081 l\u2082) \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\ninst\u271d\u00b9 : DecidableEq \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 : List \u03b1\nthis : Decidable (Lex r l\u2081 l\u2082)\nh : r a b \u2228 a = b \u2227 Lex r l\u2081 l\u2082\n\u22a2 Lex r (a :: l\u2081) (b :: l\u2082)\n[PROOFSTEP]\nrcases h with (h | \u27e8rfl, h\u27e9)\n[GOAL]\ncase refine'_1.inl\n\u03b1 : Type u\ninst\u271d\u00b9 : DecidableEq \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 : List \u03b1\nthis : Decidable (Lex r l\u2081 l\u2082)\nh : r a b\n\u22a2 Lex r (a :: l\u2081) (b :: l\u2082)\n[PROOFSTEP]\nexact Lex.rel h\n[GOAL]\ncase refine'_1.inr.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : DecidableEq \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nthis : Decidable (Lex r l\u2081 l\u2082)\nh : Lex r l\u2081 l\u2082\n\u22a2 Lex r (a :: l\u2081) (a :: l\u2082)\n[PROOFSTEP]\nexact Lex.cons h\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\ninst\u271d\u00b9 : DecidableEq \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 : List \u03b1\nthis : Decidable (Lex r l\u2081 l\u2082)\nh : Lex r (a :: l\u2081) (b :: l\u2082)\n\u22a2 r a b \u2228 a = b \u2227 Lex r l\u2081 l\u2082\n[PROOFSTEP]\nrcases h with (_ | h | h)\n[GOAL]\ncase refine'_2.cons\n\u03b1 : Type u\ninst\u271d\u00b9 : DecidableEq \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nthis : Decidable (Lex r l\u2081 l\u2082)\nh : Lex r l\u2081 l\u2082\n\u22a2 r a a \u2228 a = a \u2227 Lex r l\u2081 l\u2082\n[PROOFSTEP]\nexact Or.inr \u27e8rfl, h\u27e9\n[GOAL]\ncase refine'_2.rel\n\u03b1 : Type u\ninst\u271d\u00b9 : DecidableEq \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 : List \u03b1\nthis : Decidable (Lex r l\u2081 l\u2082)\nh : r a b\n\u22a2 r a b \u2228 a = b \u2227 Lex r l\u2081 l\u2082\n[PROOFSTEP]\nexact Or.inl h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nH : length l\u2081 \u2264 length l\u2082\nh : l\u2081 \u2260 l\u2082\n\u22a2 Lex (fun x x_1 => x \u2260 x_1) l\u2081 l\u2082\n[PROOFSTEP]\ninduction' l\u2081 with a l\u2081 IH generalizing l\u2082\n[GOAL]\ncase nil\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nH\u271d : length l\u2081 \u2264 length l\u2082\u271d\nh\u271d : l\u2081 \u2260 l\u2082\u271d\nl\u2082 : List \u03b1\nH : length [] \u2264 length l\u2082\nh : [] \u2260 l\u2082\n\u22a2 Lex (fun x x_1 => x \u2260 x_1) [] l\u2082\n[PROOFSTEP]\ncases' l\u2082 with b l\u2082\n[GOAL]\ncase cons\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : length l\u2081\u271d \u2264 length l\u2082\u271d\nh\u271d : l\u2081\u271d \u2260 l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, length l\u2081 \u2264 length l\u2082 \u2192 l\u2081 \u2260 l\u2082 \u2192 Lex (fun x x_1 => x \u2260 x_1) l\u2081 l\u2082\nl\u2082 : List \u03b1\nH : length (a :: l\u2081) \u2264 length l\u2082\nh : a :: l\u2081 \u2260 l\u2082\n\u22a2 Lex (fun x x_1 => x \u2260 x_1) (a :: l\u2081) l\u2082\n[PROOFSTEP]\ncases' l\u2082 with b l\u2082\n[GOAL]\ncase nil.nil\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nH\u271d : length l\u2081 \u2264 length l\u2082\nh\u271d : l\u2081 \u2260 l\u2082\nH : length [] \u2264 length []\nh : [] \u2260 []\n\u22a2 Lex (fun x x_1 => x \u2260 x_1) [] []\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase nil.cons\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nH\u271d : length l\u2081 \u2264 length l\u2082\u271d\nh\u271d : l\u2081 \u2260 l\u2082\u271d\nb : \u03b1\nl\u2082 : List \u03b1\nH : length [] \u2264 length (b :: l\u2082)\nh : [] \u2260 b :: l\u2082\n\u22a2 Lex (fun x x_1 => x \u2260 x_1) [] (b :: l\u2082)\n[PROOFSTEP]\napply nil\n[GOAL]\ncase cons.nil\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2081\u271d l\u2082 : List \u03b1\nH\u271d : length l\u2081\u271d \u2264 length l\u2082\nh\u271d : l\u2081\u271d \u2260 l\u2082\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, length l\u2081 \u2264 length l\u2082 \u2192 l\u2081 \u2260 l\u2082 \u2192 Lex (fun x x_1 => x \u2260 x_1) l\u2081 l\u2082\nH : length (a :: l\u2081) \u2264 length []\nh : a :: l\u2081 \u2260 []\n\u22a2 Lex (fun x x_1 => x \u2260 x_1) (a :: l\u2081) []\n[PROOFSTEP]\nexact (not_lt_of_ge H).elim (succ_pos _)\n[GOAL]\ncase cons.cons\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : length l\u2081\u271d \u2264 length l\u2082\u271d\nh\u271d : l\u2081\u271d \u2260 l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, length l\u2081 \u2264 length l\u2082 \u2192 l\u2081 \u2260 l\u2082 \u2192 Lex (fun x x_1 => x \u2260 x_1) l\u2081 l\u2082\nb : \u03b1\nl\u2082 : List \u03b1\nH : length (a :: l\u2081) \u2264 length (b :: l\u2082)\nh : a :: l\u2081 \u2260 b :: l\u2082\n\u22a2 Lex (fun x x_1 => x \u2260 x_1) (a :: l\u2081) (b :: l\u2082)\n[PROOFSTEP]\nby_cases ab : a = b\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : length l\u2081\u271d \u2264 length l\u2082\u271d\nh\u271d : l\u2081\u271d \u2260 l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, length l\u2081 \u2264 length l\u2082 \u2192 l\u2081 \u2260 l\u2082 \u2192 Lex (fun x x_1 => x \u2260 x_1) l\u2081 l\u2082\nb : \u03b1\nl\u2082 : List \u03b1\nH : length (a :: l\u2081) \u2264 length (b :: l\u2082)\nh : a :: l\u2081 \u2260 b :: l\u2082\nab : a = b\n\u22a2 Lex (fun x x_1 => x \u2260 x_1) (a :: l\u2081) (b :: l\u2082)\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : length l\u2081\u271d \u2264 length l\u2082\u271d\nh\u271d : l\u2081\u271d \u2260 l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, length l\u2081 \u2264 length l\u2082 \u2192 l\u2081 \u2260 l\u2082 \u2192 Lex (fun x x_1 => x \u2260 x_1) l\u2081 l\u2082\nl\u2082 : List \u03b1\nH : length (a :: l\u2081) \u2264 length (a :: l\u2082)\nh : a :: l\u2081 \u2260 a :: l\u2082\n\u22a2 Lex (fun x x_1 => x \u2260 x_1) (a :: l\u2081) (a :: l\u2082)\n[PROOFSTEP]\napply cons\n[GOAL]\ncase pos.h\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : length l\u2081\u271d \u2264 length l\u2082\u271d\nh\u271d : l\u2081\u271d \u2260 l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, length l\u2081 \u2264 length l\u2082 \u2192 l\u2081 \u2260 l\u2082 \u2192 Lex (fun x x_1 => x \u2260 x_1) l\u2081 l\u2082\nl\u2082 : List \u03b1\nH : length (a :: l\u2081) \u2264 length (a :: l\u2082)\nh : a :: l\u2081 \u2260 a :: l\u2082\n\u22a2 Lex (fun x x_1 => x \u2260 x_1) l\u2081 l\u2082\n[PROOFSTEP]\nexact IH (le_of_succ_le_succ H) (mt (congr_arg _) h)\n[GOAL]\ncase neg\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : length l\u2081\u271d \u2264 length l\u2082\u271d\nh\u271d : l\u2081\u271d \u2260 l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, length l\u2081 \u2264 length l\u2082 \u2192 l\u2081 \u2260 l\u2082 \u2192 Lex (fun x x_1 => x \u2260 x_1) l\u2081 l\u2082\nb : \u03b1\nl\u2082 : List \u03b1\nH : length (a :: l\u2081) \u2264 length (b :: l\u2082)\nh : a :: l\u2081 \u2260 b :: l\u2082\nab : \u00aca = b\n\u22a2 Lex (fun x x_1 => x \u2260 x_1) (a :: l\u2081) (b :: l\u2082)\n[PROOFSTEP]\nexact rel ab\n[GOAL]\n\u03b1 : Type u\nl\u2081 l\u2082 : List \u03b1\nH : length l\u2081 \u2264 length l\u2082\n\u22a2 Lex (fun x x_1 => x \u2260 x_1) l\u2081 l\u2082 \u2194 l\u2081 \u2260 l\u2082\n[PROOFSTEP]\nclassical exact Decidable.List.Lex.ne_iff H\n[GOAL]\n\u03b1 : Type u\nl\u2081 l\u2082 : List \u03b1\nH : length l\u2081 \u2264 length l\u2082\n\u22a2 Lex (fun x x_1 => x \u2260 x_1) l\u2081 l\u2082 \u2194 l\u2081 \u2260 l\u2082\n[PROOFSTEP]\nexact Decidable.List.Lex.ne_iff H\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\n\u22a2 lt l l' \u2194 Lex (fun x x_1 => x < x_1) l l'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\n\u22a2 lt l l' \u2192 Lex (fun x x_1 => x < x_1) l l'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\n\u22a2 Lex (fun x x_1 => x < x_1) l l' \u2192 lt l l'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\nh : lt l l'\n\u22a2 Lex (fun x x_1 => x < x_1) l l'\n[PROOFSTEP]\ninduction h with\n| nil b bs => exact Lex.nil\n| @head a as b bs hab => apply Lex.rel; assumption\n|\n @tail a as b bs hab hba _ ih =>\n have heq : a = b := _root_.le_antisymm (le_of_not_lt hba) (le_of_not_lt hab)\n subst b; apply Lex.cons; assumption\n[GOAL]\ncase mp\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\nh : lt l l'\n\u22a2 Lex (fun x x_1 => x < x_1) l l'\n[PROOFSTEP]\ninduction h with\n| nil b bs => exact Lex.nil\n| @head a as b bs hab => apply Lex.rel; assumption\n|\n @tail a as b bs hab hba _ ih =>\n have heq : a = b := _root_.le_antisymm (le_of_not_lt hba) (le_of_not_lt hab)\n subst b; apply Lex.cons; assumption\n[GOAL]\ncase mp.nil\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\nb : \u03b1\nbs : List \u03b1\n\u22a2 Lex (fun x x_1 => x < x_1) [] (b :: bs)\n[PROOFSTEP]\n\n| nil b bs => exact Lex.nil\n[GOAL]\ncase mp.nil\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\nb : \u03b1\nbs : List \u03b1\n\u22a2 Lex (fun x x_1 => x < x_1) [] (b :: bs)\n[PROOFSTEP]\nexact Lex.nil\n[GOAL]\ncase mp.head\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas : List \u03b1\nb : \u03b1\nbs : List \u03b1\nhab : a < b\n\u22a2 Lex (fun x x_1 => x < x_1) (a :: as) (b :: bs)\n[PROOFSTEP]\n\n| @head a as b bs hab => apply Lex.rel; assumption\n[GOAL]\ncase mp.head\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas : List \u03b1\nb : \u03b1\nbs : List \u03b1\nhab : a < b\n\u22a2 Lex (fun x x_1 => x < x_1) (a :: as) (b :: bs)\n[PROOFSTEP]\napply Lex.rel\n[GOAL]\ncase mp.head.h\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas : List \u03b1\nb : \u03b1\nbs : List \u03b1\nhab : a < b\n\u22a2 a < b\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mp.tail\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas : List \u03b1\nb : \u03b1\nbs : List \u03b1\nhab : \u00aca < b\nhba : \u00acb < a\na\u271d : lt as bs\nih : Lex (fun x x_1 => x < x_1) as bs\n\u22a2 Lex (fun x x_1 => x < x_1) (a :: as) (b :: bs)\n[PROOFSTEP]\n\n|\n @tail a as b bs hab hba _ ih =>\n have heq : a = b := _root_.le_antisymm (le_of_not_lt hba) (le_of_not_lt hab)\n subst b; apply Lex.cons; assumption\n[GOAL]\ncase mp.tail\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas : List \u03b1\nb : \u03b1\nbs : List \u03b1\nhab : \u00aca < b\nhba : \u00acb < a\na\u271d : lt as bs\nih : Lex (fun x x_1 => x < x_1) as bs\n\u22a2 Lex (fun x x_1 => x < x_1) (a :: as) (b :: bs)\n[PROOFSTEP]\nhave heq : a = b := _root_.le_antisymm (le_of_not_lt hba) (le_of_not_lt hab)\n[GOAL]\ncase mp.tail\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas : List \u03b1\nb : \u03b1\nbs : List \u03b1\nhab : \u00aca < b\nhba : \u00acb < a\na\u271d : lt as bs\nih : Lex (fun x x_1 => x < x_1) as bs\nheq : a = b\n\u22a2 Lex (fun x x_1 => x < x_1) (a :: as) (b :: bs)\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase mp.tail\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas bs : List \u03b1\na\u271d : lt as bs\nih : Lex (fun x x_1 => x < x_1) as bs\nhab hba : \u00aca < a\n\u22a2 Lex (fun x x_1 => x < x_1) (a :: as) (a :: bs)\n[PROOFSTEP]\napply Lex.cons\n[GOAL]\ncase mp.tail.h\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas bs : List \u03b1\na\u271d : lt as bs\nih : Lex (fun x x_1 => x < x_1) as bs\nhab hba : \u00aca < a\n\u22a2 Lex (fun x x_1 => x < x_1) as bs\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mpr\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\nh : Lex (fun x x_1 => x < x_1) l l'\n\u22a2 lt l l'\n[PROOFSTEP]\ninduction h with\n| @nil a as => apply lt.nil\n| @cons a as bs _ ih => apply lt.tail <;> simp [ih]\n| @rel a as b bs h => apply lt.head; assumption\n[GOAL]\ncase mpr\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\nh : Lex (fun x x_1 => x < x_1) l l'\n\u22a2 lt l l'\n[PROOFSTEP]\ninduction h with\n| @nil a as => apply lt.nil\n| @cons a as bs _ ih => apply lt.tail <;> simp [ih]\n| @rel a as b bs h => apply lt.head; assumption\n[GOAL]\ncase mpr.nil\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas : List \u03b1\n\u22a2 lt [] (a :: as)\n[PROOFSTEP]\n\n| @nil a as => apply lt.nil\n[GOAL]\ncase mpr.nil\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas : List \u03b1\n\u22a2 lt [] (a :: as)\n[PROOFSTEP]\napply lt.nil\n[GOAL]\ncase mpr.cons\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas bs : List \u03b1\nh\u271d : Lex (fun x x_1 => x < x_1) as bs\nih : lt as bs\n\u22a2 lt (a :: as) (a :: bs)\n[PROOFSTEP]\n\n| @cons a as bs _ ih => apply lt.tail <;> simp [ih]\n[GOAL]\ncase mpr.cons\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas bs : List \u03b1\nh\u271d : Lex (fun x x_1 => x < x_1) as bs\nih : lt as bs\n\u22a2 lt (a :: as) (a :: bs)\n[PROOFSTEP]\napply lt.tail\n[GOAL]\ncase mpr.cons.a\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas bs : List \u03b1\nh\u271d : Lex (fun x x_1 => x < x_1) as bs\nih : lt as bs\n\u22a2 \u00aca < a\n[PROOFSTEP]\nsimp [ih]\n[GOAL]\ncase mpr.cons.a\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas bs : List \u03b1\nh\u271d : Lex (fun x x_1 => x < x_1) as bs\nih : lt as bs\n\u22a2 \u00aca < a\n[PROOFSTEP]\nsimp [ih]\n[GOAL]\ncase mpr.cons.a\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas bs : List \u03b1\nh\u271d : Lex (fun x x_1 => x < x_1) as bs\nih : lt as bs\n\u22a2 lt as bs\n[PROOFSTEP]\nsimp [ih]\n[GOAL]\ncase mpr.rel\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas : List \u03b1\nb : \u03b1\nbs : List \u03b1\nh : a < b\n\u22a2 lt (a :: as) (b :: bs)\n[PROOFSTEP]\n\n| @rel a as b bs h => apply lt.head; assumption\n[GOAL]\ncase mpr.rel\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas : List \u03b1\nb : \u03b1\nbs : List \u03b1\nh : a < b\n\u22a2 lt (a :: as) (b :: bs)\n[PROOFSTEP]\napply lt.head\n[GOAL]\ncase mpr.rel.a\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\nl l' : List \u03b1\na : \u03b1\nas : List \u03b1\nb : \u03b1\nbs : List \u03b1\nh : a < b\n\u22a2 a < b\n[PROOFSTEP]\nassumption\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Lex", "llama_tokens": 7273, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5813030761371503, "lm_q2_score": 0.4263215925474903, "lm_q1q2_score": 0.2478220531715449}} {"text": "[GOAL]\n\u03b1 : Type ?u.37\n\u03b2 : Type ?u.40\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\n\u22a2 inst\u271d\u00b9 \u2264 OuterMeasure.caratheodory (OuterMeasure.dirac a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type ?u.2205\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\nh : a \u2208 s\n\u22a2 \u2191\u2191(dirac a) s = 1\n[PROOFSTEP]\nhave : \u2200 t : Set \u03b1, a \u2208 t \u2192 t.indicator (1 : \u03b1 \u2192 \u211d\u22650\u221e) a = 1 := fun t ht => indicator_of_mem ht 1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type ?u.2205\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\nh : a \u2208 s\nthis : \u2200 (t : Set \u03b1), a \u2208 t \u2192 indicator t 1 a = 1\n\u22a2 \u2191\u2191(dirac a) s = 1\n[PROOFSTEP]\nrefine' le_antisymm (this univ trivial \u25b8 _) (this s h \u25b8 le_dirac_apply)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type ?u.2205\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\nh : a \u2208 s\nthis : \u2200 (t : Set \u03b1), a \u2208 t \u2192 indicator t 1 a = 1\n\u22a2 \u2191\u2191(dirac a) s \u2264 indicator univ 1 a\n[PROOFSTEP]\nrw [\u2190 dirac_apply' a MeasurableSet.univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type ?u.2205\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\nh : a \u2208 s\nthis : \u2200 (t : Set \u03b1), a \u2208 t \u2192 indicator t 1 a = 1\n\u22a2 \u2191\u2191(dirac a) s \u2264 \u2191\u2191(dirac a) univ\n[PROOFSTEP]\nexact measure_mono (subset_univ s)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type ?u.3101\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1\ns : Set \u03b1\n\u22a2 \u2191\u2191(dirac a) s = indicator s 1 a\n[PROOFSTEP]\nby_cases h : a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.3101\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1\ns : Set \u03b1\nh : a \u2208 s\n\u22a2 \u2191\u2191(dirac a) s = indicator s 1 a\n[PROOFSTEP]\nrw [dirac_apply_of_mem h, indicator_of_mem h, Pi.one_apply]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.3101\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1\ns : Set \u03b1\nh : \u00aca \u2208 s\n\u22a2 \u2191\u2191(dirac a) s = indicator s 1 a\n[PROOFSTEP]\nrw [indicator_of_not_mem h, \u2190 nonpos_iff_eq_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.3101\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1\ns : Set \u03b1\nh : \u00aca \u2208 s\n\u22a2 \u2191\u2191(dirac a) s \u2264 0\n[PROOFSTEP]\ncalc\n dirac a s \u2264 dirac a { a }\u1d9c := measure_mono (subset_compl_comm.1 <| singleton_subset_iff.2 h)\n _ = 0 := by simp [dirac_apply' _ (measurableSet_singleton _).compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type ?u.3101\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1\ns : Set \u03b1\nh : \u00aca \u2208 s\n\u22a2 \u2191\u2191(dirac a) {a}\u1d9c = 0\n[PROOFSTEP]\nsimp [dirac_apply' _ (measurableSet_singleton _).compl]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(map f (dirac a)) s = \u2191\u2191(dirac (f a)) s\n[PROOFSTEP]\nsimp [hs, map_apply hf hs, hf hs, indicator_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7321\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\n\u22a2 restrict \u03bc {a} = \u2191\u2191\u03bc {a} \u2022 dirac a\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7321\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s\n[PROOFSTEP]\nby_cases ha : a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7321\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : a \u2208 s\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s\n[PROOFSTEP]\nhave : s \u2229 { a } = { a } := by simpa\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7321\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : a \u2208 s\n\u22a2 s \u2229 {a} = {a}\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7321\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : a \u2208 s\nthis : s \u2229 {a} = {a}\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7321\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : \u00aca \u2208 s\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s\n[PROOFSTEP]\nhave : s \u2229 { a } = \u2205 := inter_singleton_eq_empty.2 ha\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7321\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : \u00aca \u2208 s\nthis : s \u2229 {a} = \u2205\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns : Set \u03b1\ninst\u271d\u00b9 : Countable \u03b2\ninst\u271d : MeasurableSingletonClass \u03b2\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\n\u22a2 map f \u03bc = sum fun b => \u2191\u2191\u03bc (f \u207b\u00b9' {b}) \u2022 dirac b\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : Countable \u03b2\ninst\u271d : MeasurableSingletonClass \u03b2\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(map f \u03bc) s = \u2191\u2191(sum fun b => \u2191\u2191\u03bc (f \u207b\u00b9' {b}) \u2022 dirac b) s\n[PROOFSTEP]\nhave : \u2200 y \u2208 s, MeasurableSet (f \u207b\u00b9' { y }) := fun y _ => hf (measurableSet_singleton _)\n[GOAL]\ncase h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : Countable \u03b2\ninst\u271d : MeasurableSingletonClass \u03b2\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\nthis : \u2200 (y : \u03b2), y \u2208 s \u2192 MeasurableSet (f \u207b\u00b9' {y})\n\u22a2 \u2191\u2191(map f \u03bc) s = \u2191\u2191(sum fun b => \u2191\u2191\u03bc (f \u207b\u00b9' {b}) \u2022 dirac b) s\n[PROOFSTEP]\nsimp [\u2190 tsum_measure_preimage_singleton (to_countable s) this, *, tsum_subtype s fun b => \u03bc (f \u207b\u00b9' { b }), \u2190\n indicator_mul_right s fun b => \u03bc (f \u207b\u00b9' { b })]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type ?u.17308\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns : Set \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\n\u03bc : Measure \u03b1\n\u22a2 (sum fun a => \u2191\u2191\u03bc {a} \u2022 dirac a) = \u03bc\n[PROOFSTEP]\nsimpa using (map_eq_sum \u03bc id measurable_id).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type ?u.19567\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2211' (x : \u03b1), indicator s (fun x => \u2191\u2191\u03bc {x}) x = \u2191\u2191(sum fun a => \u2191\u2191\u03bc {a} \u2022 dirac a) s\n[PROOFSTEP]\nsimp only [Measure.sum_apply _ hs, Measure.smul_apply, smul_eq_mul, Measure.dirac_apply, Set.indicator_apply, mul_ite,\n Pi.one_apply, mul_one, mul_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type ?u.19567\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(sum fun a => \u2191\u2191\u03bc {a} \u2022 dirac a) s = \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [\u03bc.sum_smul_dirac]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type ?u.22509\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\nhs : MeasurableSet s\n\u22a2 s \u2208 ae (dirac a) \u2194 a \u2208 s\n[PROOFSTEP]\nby_cases a \u2208 s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type ?u.22509\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\nhs : MeasurableSet s\n\u22a2 s \u2208 ae (dirac a) \u2194 a \u2208 s\n[PROOFSTEP]\nby_cases a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.22509\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\nhs : MeasurableSet s\nh : a \u2208 s\n\u22a2 s \u2208 ae (dirac a) \u2194 a \u2208 s\n[PROOFSTEP]\nsimp [mem_ae_iff, dirac_apply', hs.compl, indicator_apply, *]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.22509\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\nhs : MeasurableSet s\nh : \u00aca \u2208 s\n\u22a2 s \u2208 ae (dirac a) \u2194 a \u2208 s\n[PROOFSTEP]\nsimp [mem_ae_iff, dirac_apply', hs.compl, indicator_apply, *]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type ?u.24298\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1\n\u22a2 ae (dirac a) = pure a\n[PROOFSTEP]\next s\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type ?u.24298\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1\ns : Set \u03b1\n\u22a2 s \u2208 ae (dirac a) \u2194 s \u2208 pure a\n[PROOFSTEP]\nsimp [mem_ae_iff, imp_false]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type ?u.26464\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns : Set \u03b1\n\u03b4 : Type u_1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b4\n\u22a2 f =\u1da0[ae (dirac a)] const \u03b1 (f a)\n[PROOFSTEP]\nsimp [Filter.EventuallyEq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27478\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\nhs : MeasurableSet s\ninst\u271d : Decidable (a \u2208 s)\n\u22a2 Measure.restrict (dirac a) s = if a \u2208 s then dirac a else 0\n[PROOFSTEP]\nsplit_ifs with has\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27478\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\nhs : MeasurableSet s\ninst\u271d : Decidable (a \u2208 s)\nhas : a \u2208 s\n\u22a2 Measure.restrict (dirac a) s = dirac a\n[PROOFSTEP]\napply restrict_eq_self_of_ae_mem\n[GOAL]\ncase pos.hs\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27478\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\nhs : MeasurableSet s\ninst\u271d : Decidable (a \u2208 s)\nhas : a \u2208 s\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202dirac a, x \u2208 s\n[PROOFSTEP]\nrw [ae_dirac_iff]\n[GOAL]\ncase pos.hs\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27478\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\nhs : MeasurableSet s\ninst\u271d : Decidable (a \u2208 s)\nhas : a \u2208 s\n\u22a2 a \u2208 s\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.hs\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27478\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\nhs : MeasurableSet s\ninst\u271d : Decidable (a \u2208 s)\nhas : a \u2208 s\n\u22a2 MeasurableSet {x | x \u2208 s}\n[PROOFSTEP]\nassumption\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27478\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\nhs : MeasurableSet s\ninst\u271d : Decidable (a \u2208 s)\nhas : \u00aca \u2208 s\n\u22a2 Measure.restrict (dirac a) s = 0\n[PROOFSTEP]\nrw [restrict_eq_zero, dirac_apply' _ hs, indicator_of_not_mem has]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type ?u.28375\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : Decidable (a \u2208 s)\n\u22a2 Measure.restrict (dirac a) s = if a \u2208 s then dirac a else 0\n[PROOFSTEP]\nsplit_ifs with has\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.28375\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : Decidable (a \u2208 s)\nhas : a \u2208 s\n\u22a2 Measure.restrict (dirac a) s = dirac a\n[PROOFSTEP]\napply restrict_eq_self_of_ae_mem\n[GOAL]\ncase pos.hs\n\u03b1 : Type u_1\n\u03b2 : Type ?u.28375\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : Decidable (a \u2208 s)\nhas : a \u2208 s\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202dirac a, x \u2208 s\n[PROOFSTEP]\nrwa [ae_dirac_eq]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.28375\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : Decidable (a \u2208 s)\nhas : \u00aca \u2208 s\n\u22a2 Measure.restrict (dirac a) s = 0\n[PROOFSTEP]\nrw [restrict_eq_zero, dirac_apply, indicator_of_not_mem has]\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Measure.Dirac", "llama_tokens": 5565, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5156199157230157, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.24774437548176784}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Nontrivial \u03b1\n\u22a2 Disjoint atBot atTop\n[PROOFSTEP]\nrcases exists_pair_ne \u03b1 with \u27e8x, y, hne\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Nontrivial \u03b1\nx y : \u03b1\nhne : x \u2260 y\n\u22a2 Disjoint atBot atTop\n[PROOFSTEP]\nby_cases hle : x \u2264 y\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Nontrivial \u03b1\nx y : \u03b1\nhne : x \u2260 y\nhle : x \u2264 y\n\u22a2 Disjoint atBot atTop\n[PROOFSTEP]\nrefine' disjoint_of_disjoint_of_mem _ (Iic_mem_atBot x) (Ici_mem_atTop y)\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Nontrivial \u03b1\nx y : \u03b1\nhne : x \u2260 y\nhle : x \u2264 y\n\u22a2 Disjoint (Iic x) (Ici y)\n[PROOFSTEP]\nexact Iic_disjoint_Ici.2 (hle.lt_of_ne hne).not_le\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Nontrivial \u03b1\nx y : \u03b1\nhne : x \u2260 y\nhle : \u00acx \u2264 y\n\u22a2 Disjoint atBot atTop\n[PROOFSTEP]\nrefine' disjoint_of_disjoint_of_mem _ (Iic_mem_atBot y) (Ici_mem_atTop x)\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Nontrivial \u03b1\nx y : \u03b1\nhne : x \u2260 y\nhle : \u00acx \u2264 y\n\u22a2 Disjoint (Iic y) (Ici x)\n[PROOFSTEP]\nexact Iic_disjoint_Ici.2 hle\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Preorder \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1da0 (x : \u03b1) in atTop, \u2200 (y : \u03b1), x \u2264 y \u2192 p y) \u2194 \u2200\u1da0 (x : \u03b1) in atTop, p x\n[PROOFSTEP]\nrefine \u27e8fun h \u21a6 h.mono fun x hx \u21a6 hx x le_rfl, fun h \u21a6 ?_\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Preorder \u03b1\np : \u03b1 \u2192 Prop\nh : \u2200\u1da0 (x : \u03b1) in atTop, p x\n\u22a2 \u2200\u1da0 (x : \u03b1) in atTop, \u2200 (y : \u03b1), x \u2264 y \u2192 p y\n[PROOFSTEP]\nrcases(hasBasis_iInf_principal_finite _).eventually_iff.1 h with \u27e8S, hSf, hS\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Preorder \u03b1\np : \u03b1 \u2192 Prop\nh : \u2200\u1da0 (x : \u03b1) in atTop, p x\nS : Set \u03b1\nhSf : Set.Finite S\nhS : \u2200 \u2983x : \u03b1\u2984, x \u2208 \u22c2 (i : \u03b1) (_ : i \u2208 S), Ici i \u2192 p x\n\u22a2 \u2200\u1da0 (x : \u03b1) in atTop, \u2200 (y : \u03b1), x \u2264 y \u2192 p y\n[PROOFSTEP]\nrefine mem_iInf_of_iInter hSf (V := fun x \u21a6 Ici x.1) (fun _ \u21a6 Subset.rfl) fun x hx y hy \u21a6 ?_\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Preorder \u03b1\np : \u03b1 \u2192 Prop\nh : \u2200\u1da0 (x : \u03b1) in atTop, p x\nS : Set \u03b1\nhSf : Set.Finite S\nhS : \u2200 \u2983x : \u03b1\u2984, x \u2208 \u22c2 (i : \u03b1) (_ : i \u2208 S), Ici i \u2192 p x\nx : \u03b1\nhx : x \u2208 \u22c2 (i : \u2191S), (fun x => Ici \u2191x) i\ny : \u03b1\nhy : x \u2264 y\n\u22a2 p y\n[PROOFSTEP]\nsimp only [mem_iInter] at hS hx \n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Preorder \u03b1\np : \u03b1 \u2192 Prop\nh : \u2200\u1da0 (x : \u03b1) in atTop, p x\nS : Set \u03b1\nhSf : Set.Finite S\nx y : \u03b1\nhy : x \u2264 y\nhS : \u2200 \u2983x : \u03b1\u2984, (\u2200 (i : \u03b1), i \u2208 S \u2192 x \u2208 Ici i) \u2192 p x\nhx : \u2200 (i : \u2191S), x \u2208 Ici \u2191i\n\u22a2 p y\n[PROOFSTEP]\nexact hS fun z hz \u21a6 le_trans (hx \u27e8z, hz\u27e9) hy\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2\u271d : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b2 : Type u_7\ninst\u271d : Preorder \u03b2\nl : Filter \u03b1\np : \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nhf : Tendsto f l atTop\nh_evtl : \u2200\u1da0 (x : \u03b2) in atTop, p x\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2200 (y : \u03b2), f x \u2264 y \u2192 p y\n[PROOFSTEP]\nrw [\u2190 Filter.eventually_forall_ge_atTop] at h_evtl \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2\u271d : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b2 : Type u_7\ninst\u271d : Preorder \u03b2\nl : Filter \u03b1\np : \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nhf : Tendsto f l atTop\nh_evtl : \u2200\u1da0 (x : \u03b2) in atTop, \u2200 (y : \u03b2), x \u2264 y \u2192 p y\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2200 (y : \u03b2), f x \u2264 y \u2192 p y\n[PROOFSTEP]\nexact (h_evtl.comap f).filter_mono hf.le_comap\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2\u271d : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b2 : Type u_7\ninst\u271d : Preorder \u03b2\nl : Filter \u03b1\np : \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nhf : Tendsto f l atBot\nh_evtl : \u2200\u1da0 (x : \u03b2) in atBot, p x\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2200 (y : \u03b2), y \u2264 f x \u2192 p y\n[PROOFSTEP]\nrw [\u2190 Filter.eventually_forall_le_atBot] at h_evtl \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2\u271d : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b2 : Type u_7\ninst\u271d : Preorder \u03b2\nl : Filter \u03b1\np : \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nhf : Tendsto f l atBot\nh_evtl : \u2200\u1da0 (x : \u03b2) in atBot, \u2200 (y : \u03b2), y \u2264 x \u2192 p y\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2200 (y : \u03b2), y \u2264 f x \u2192 p y\n[PROOFSTEP]\nexact (h_evtl.comap f).filter_mono hf.le_comap\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b9 : Subsingleton \u03b1\ninst\u271d : Preorder \u03b1\n\u22a2 atTop = \u22a4\n[PROOFSTEP]\nrefine' top_unique fun s hs x => _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b9 : Subsingleton \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhs : s \u2208 atTop\nx : \u03b1\n\u22a2 x \u2208 s\n[PROOFSTEP]\nrw [atTop, ciInf_subsingleton x, mem_principal] at hs \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b9 : Subsingleton \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nx : \u03b1\nhs : Ici x \u2286 s\n\u22a2 x \u2208 s\n[PROOFSTEP]\nexact hs left_mem_Ici\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : Nonempty \u03b1\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\n\u22a2 (\u2203 b, \u2200\u1da0 (a : \u03b1) in atTop, r a b) \u2194 \u2200\u1da0 (a\u2080 : \u03b1) in atTop, \u2203 b, \u2200 (a : \u03b1), a \u2265 a\u2080 \u2192 r a b\n[PROOFSTEP]\nsimp_rw [eventually_atTop, \u2190 exists_swap (\u03b1 := \u03b1)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : Nonempty \u03b1\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\n\u22a2 (\u2203 x y, \u2200 (b : \u03b1), b \u2265 x \u2192 r b y) \u2194 \u2203 a, \u2200 (b : \u03b1), b \u2265 a \u2192 \u2203 b_1, \u2200 (a : \u03b1), a \u2265 b \u2192 r a b_1\n[PROOFSTEP]\nexact exists_congr fun a \u21a6 .symm <| forall_ge_iff <| Monotone.exists fun _ _ _ hb H n hn \u21a6 H n (hb.trans hn)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : Nonempty \u03b1\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\n\u22a2 (\u2203 b, \u2200\u1da0 (a : \u03b1) in atBot, r a b) \u2194 \u2200\u1da0 (a\u2080 : \u03b1) in atBot, \u2203 b, \u2200 (a : \u03b1), a \u2264 a\u2080 \u2192 r a b\n[PROOFSTEP]\nsimp_rw [eventually_atBot, \u2190 exists_swap (\u03b1 := \u03b1)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : Nonempty \u03b1\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\n\u22a2 (\u2203 x y, \u2200 (b : \u03b1), b \u2264 x \u2192 r b y) \u2194 \u2203 a, \u2200 (b : \u03b1), b \u2264 a \u2192 \u2203 b_1, \u2200 (a : \u03b1), a \u2264 b \u2192 r a b_1\n[PROOFSTEP]\nexact exists_congr fun a \u21a6 .symm <| forall_le_iff <| Antitone.exists fun _ _ _ hb H n hn \u21a6 H n (hn.trans hb)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : Nonempty \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200 (i : \u03b1), True \u2192 \u2203 x, x \u2208 Ici i \u2227 p x) \u2194 \u2200 (a : \u03b1), \u2203 b, b \u2265 a \u2227 p b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : NoMaxOrder \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200 (i : \u03b1), True \u2192 \u2203 x, x \u2208 Ioi i \u2227 p x) \u2194 \u2200 (a : \u03b1), \u2203 b, b > a \u2227 p b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Preorder \u03b2\nm : \u03b1 \u2192 \u03b2\nf : Filter \u03b1\n\u22a2 Tendsto m f atTop \u2194 \u2200 (b : \u03b2), \u2200\u1da0 (a : \u03b1) in f, b \u2264 m a\n[PROOFSTEP]\nsimp only [atTop, tendsto_iInf, tendsto_principal, mem_Ici]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Preorder \u03b2\nl : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 \u03b2\nh : f\u2081 \u2264\u1da0[l] f\u2082\nh\u2081 : Tendsto f\u2081 l atTop\nb : \u03b2\n\u22a2 \u2200\u1da0 (a : \u03b1) in l, b \u2264 f\u2082 a\n[PROOFSTEP]\nfilter_upwards [tendsto_atTop.1 h\u2081 b, h] with x using le_trans\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\n\u22a2 comap (\u2191e) atTop = atTop\n[PROOFSTEP]\nsimp [atTop, \u2190 e.surjective.iInf_comp]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\n\u22a2 map (\u2191e) atTop = atTop\n[PROOFSTEP]\nrw [\u2190 e.comap_atTop, map_comap_of_surjective e.surjective]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nl : Filter \u03b3\nf : \u03b3 \u2192 \u03b1\ne : \u03b1 \u2243o \u03b2\n\u22a2 Tendsto (fun x => \u2191e (f x)) l atTop \u2194 Tendsto f l atTop\n[PROOFSTEP]\nrw [\u2190 e.comap_atTop, tendsto_comap_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nl : Filter \u03b3\nf : \u03b3 \u2192 \u03b1\ne : \u03b1 \u2243o \u03b2\n\u22a2 Tendsto (fun x => \u2191e (f x)) l atTop \u2194 Tendsto (\u2191e \u2218 f) l atTop\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : Nonempty \u03b1\nF : Filter \u03b2\nu : \u03b1 \u2192 \u03b2\n\u22a2 NeBot (F \u2293 map u atTop) \u2194 \u2200 (U : Set \u03b2), U \u2208 F \u2192 \u2200 (N : \u03b1), \u2203 n, n \u2265 N \u2227 u n \u2208 U\n[PROOFSTEP]\nsimp_rw [inf_neBot_iff_frequently_left, frequently_map, frequently_atTop]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : Nonempty \u03b1\nF : Filter \u03b2\nu : \u03b1 \u2192 \u03b2\n\u22a2 (\u2200 {p : \u03b2 \u2192 Prop}, (\u2200\u1da0 (x : \u03b2) in F, p x) \u2192 \u2200 (a : \u03b1), \u2203 b, b \u2265 a \u2227 p (u b)) \u2194\n \u2200 (U : Set \u03b2), U \u2208 F \u2192 \u2200 (N : \u03b1), \u2203 n, n \u2265 N \u2227 u n \u2208 U\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 Prop\nh : \u2200 (N : \u2115), \u2203 n, n > N \u2227 P n\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (n : \u2115), P (\u03c6 n)\n[PROOFSTEP]\nchoose u hu hu' using h\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 Prop\nu : \u2115 \u2192 \u2115\nhu : \u2200 (N : \u2115), u N > N\nhu' : \u2200 (N : \u2115), P (u N)\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (n : \u2115), P (\u03c6 n)\n[PROOFSTEP]\nrefine \u27e8fun n => u^[n + 1] 0, strictMono_nat_of_lt_succ fun n => ?_, fun n => ?_\u27e9\n[GOAL]\ncase refine_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 Prop\nu : \u2115 \u2192 \u2115\nhu : \u2200 (N : \u2115), u N > N\nhu' : \u2200 (N : \u2115), P (u N)\nn : \u2115\n\u22a2 u^[n + 1] 0 < u^[n + 1 + 1] 0\n[PROOFSTEP]\nexact Trans.trans (hu _) (Function.iterate_succ_apply' _ _ _).symm\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 Prop\nu : \u2115 \u2192 \u2115\nhu : \u2200 (N : \u2115), u N > N\nhu' : \u2200 (N : \u2115), P (u N)\nn : \u2115\n\u22a2 P ((fun n => u^[n + 1] 0) n)\n[PROOFSTEP]\nsimpa only [Function.iterate_succ_apply'] using hu' _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 Prop\nh : \u2203\u1da0 (n : \u2115) in atTop, P n\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (n : \u2115), P (\u03c6 n)\n[PROOFSTEP]\nrw [frequently_atTop'] at h \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 Prop\nh : \u2200 (a : \u2115), \u2203 b, b > a \u2227 P b\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (n : \u2115), P (\u03c6 n)\n[PROOFSTEP]\nexact extraction_of_frequently_atTop' h\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 \u2115 \u2192 Prop\nh : \u2200 (n : \u2115), \u2203\u1da0 (k : \u2115) in atTop, P n k\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (n : \u2115), P n (\u03c6 n)\n[PROOFSTEP]\nsimp only [frequently_atTop'] at h \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 \u2115 \u2192 Prop\nh : \u2200 (n a : \u2115), \u2203 b, b > a \u2227 P n b\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (n : \u2115), P n (\u03c6 n)\n[PROOFSTEP]\nchoose u hu hu' using h\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 \u2115 \u2192 Prop\nu : \u2115 \u2192 \u2115 \u2192 \u2115\nhu : \u2200 (n a : \u2115), u n a > a\nhu' : \u2200 (n a : \u2115), P n (u n a)\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (n : \u2115), P n (\u03c6 n)\n[PROOFSTEP]\nuse(fun n => Nat.recOn n (u 0 0) fun n v => u (n + 1) v : \u2115 \u2192 \u2115)\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 \u2115 \u2192 Prop\nu : \u2115 \u2192 \u2115 \u2192 \u2115\nhu : \u2200 (n a : \u2115), u n a > a\nhu' : \u2200 (n a : \u2115), P n (u n a)\n\u22a2 (StrictMono fun n => Nat.recOn n (u 0 0) fun n v => u (n + 1) v) \u2227\n \u2200 (n : \u2115), P n (Nat.recOn n (u 0 0) fun n v => u (n + 1) v)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 \u2115 \u2192 Prop\nu : \u2115 \u2192 \u2115 \u2192 \u2115\nhu : \u2200 (n a : \u2115), u n a > a\nhu' : \u2200 (n a : \u2115), P n (u n a)\n\u22a2 StrictMono fun n => Nat.recOn n (u 0 0) fun n v => u (n + 1) v\n[PROOFSTEP]\napply strictMono_nat_of_lt_succ\n[GOAL]\ncase h.left.hf\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 \u2115 \u2192 Prop\nu : \u2115 \u2192 \u2115 \u2192 \u2115\nhu : \u2200 (n a : \u2115), u n a > a\nhu' : \u2200 (n a : \u2115), P n (u n a)\n\u22a2 \u2200 (n : \u2115), (Nat.recOn n (u 0 0) fun n v => u (n + 1) v) < Nat.recOn (n + 1) (u 0 0) fun n v => u (n + 1) v\n[PROOFSTEP]\nintro n\n[GOAL]\ncase h.left.hf\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 \u2115 \u2192 Prop\nu : \u2115 \u2192 \u2115 \u2192 \u2115\nhu : \u2200 (n a : \u2115), u n a > a\nhu' : \u2200 (n a : \u2115), P n (u n a)\nn : \u2115\n\u22a2 (Nat.recOn n (u 0 0) fun n v => u (n + 1) v) < Nat.recOn (n + 1) (u 0 0) fun n v => u (n + 1) v\n[PROOFSTEP]\napply hu\n[GOAL]\ncase h.right\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 \u2115 \u2192 Prop\nu : \u2115 \u2192 \u2115 \u2192 \u2115\nhu : \u2200 (n a : \u2115), u n a > a\nhu' : \u2200 (n a : \u2115), P n (u n a)\n\u22a2 \u2200 (n : \u2115), P n (Nat.recOn n (u 0 0) fun n v => u (n + 1) v)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase h.right\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 \u2115 \u2192 Prop\nu : \u2115 \u2192 \u2115 \u2192 \u2115\nhu : \u2200 (n a : \u2115), u n a > a\nhu' : \u2200 (n a : \u2115), P n (u n a)\nn : \u2115\n\u22a2 P n (Nat.recOn n (u 0 0) fun n v => u (n + 1) v)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase h.right.zero\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 \u2115 \u2192 Prop\nu : \u2115 \u2192 \u2115 \u2192 \u2115\nhu : \u2200 (n a : \u2115), u n a > a\nhu' : \u2200 (n a : \u2115), P n (u n a)\n\u22a2 P Nat.zero (Nat.recOn Nat.zero (u 0 0) fun n v => u (n + 1) v)\n[PROOFSTEP]\nsimp [hu']\n[GOAL]\ncase h.right.succ\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 \u2115 \u2192 Prop\nu : \u2115 \u2192 \u2115 \u2192 \u2115\nhu : \u2200 (n a : \u2115), u n a > a\nhu' : \u2200 (n a : \u2115), P n (u n a)\nn\u271d : \u2115\n\u22a2 P (Nat.succ n\u271d) (Nat.recOn (Nat.succ n\u271d) (u 0 0) fun n v => u (n + 1) v)\n[PROOFSTEP]\nsimp [hu']\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 \u2115 \u2192 Prop\nh : \u2200 (n : \u2115), \u2203 N, \u2200 (k : \u2115), k \u2265 N \u2192 P n k\n\u22a2 \u2200 (n : \u2115), \u2200\u1da0 (k : \u2115) in atTop, P n k\n[PROOFSTEP]\nsimp [eventually_atTop, h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\np : \u2115 \u2192 Prop\nn : \u2115\nhn : n \u2260 0\nhp : \u2200 (k : \u2115), k < n \u2192 \u2200\u1da0 (a : \u2115) in atTop, p (n * a + k)\n\u22a2 \u2200\u1da0 (a : \u2115) in atTop, p a\n[PROOFSTEP]\nsimp only [eventually_atTop] at hp \u22a2\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\np : \u2115 \u2192 Prop\nn : \u2115\nhn : n \u2260 0\nhp : \u2200 (k : \u2115), k < n \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 p (n * b + k)\n\u22a2 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 p b\n[PROOFSTEP]\nchoose! N hN using hp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\np : \u2115 \u2192 Prop\nn : \u2115\nhn : n \u2260 0\nN : \u2115 \u2192 \u2115\nhN : \u2200 (k : \u2115), k < n \u2192 \u2200 (b : \u2115), b \u2265 N k \u2192 p (n * b + k)\n\u22a2 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 p b\n[PROOFSTEP]\nrefine \u27e8(Finset.range n).sup (n * N \u00b7), fun b hb => ?_\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\np : \u2115 \u2192 Prop\nn : \u2115\nhn : n \u2260 0\nN : \u2115 \u2192 \u2115\nhN : \u2200 (k : \u2115), k < n \u2192 \u2200 (b : \u2115), b \u2265 N k \u2192 p (n * b + k)\nb : \u2115\nhb : b \u2265 Finset.sup (Finset.range n) fun x => n * N x\n\u22a2 p b\n[PROOFSTEP]\nrw [\u2190 Nat.div_add_mod b n]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\np : \u2115 \u2192 Prop\nn : \u2115\nhn : n \u2260 0\nN : \u2115 \u2192 \u2115\nhN : \u2200 (k : \u2115), k < n \u2192 \u2200 (b : \u2115), b \u2265 N k \u2192 p (n * b + k)\nb : \u2115\nhb : b \u2265 Finset.sup (Finset.range n) fun x => n * N x\n\u22a2 p (n * (b / n) + b % n)\n[PROOFSTEP]\nhave hlt := Nat.mod_lt b hn.bot_lt\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\np : \u2115 \u2192 Prop\nn : \u2115\nhn : n \u2260 0\nN : \u2115 \u2192 \u2115\nhN : \u2200 (k : \u2115), k < n \u2192 \u2200 (b : \u2115), b \u2265 N k \u2192 p (n * b + k)\nb : \u2115\nhb : b \u2265 Finset.sup (Finset.range n) fun x => n * N x\nhlt : b % n < n\n\u22a2 p (n * (b / n) + b % n)\n[PROOFSTEP]\nrefine hN _ hlt _ ?_\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\np : \u2115 \u2192 Prop\nn : \u2115\nhn : n \u2260 0\nN : \u2115 \u2192 \u2115\nhN : \u2200 (k : \u2115), k < n \u2192 \u2200 (b : \u2115), b \u2265 N k \u2192 p (n * b + k)\nb : \u2115\nhb : b \u2265 Finset.sup (Finset.range n) fun x => n * N x\nhlt : b % n < n\n\u22a2 b / n \u2265 N (b % n)\n[PROOFSTEP]\nrw [ge_iff_le, Nat.le_div_iff_mul_le hn.bot_lt, mul_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\np : \u2115 \u2192 Prop\nn : \u2115\nhn : n \u2260 0\nN : \u2115 \u2192 \u2115\nhN : \u2200 (k : \u2115), k < n \u2192 \u2200 (b : \u2115), b \u2265 N k \u2192 p (n * b + k)\nb : \u2115\nhb : b \u2265 Finset.sup (Finset.range n) fun x => n * N x\nhlt : b % n < n\n\u22a2 n * N (b % n) \u2264 b\n[PROOFSTEP]\nexact (Finset.le_sup (f := (n * N \u00b7)) (Finset.mem_range.2 hlt)).trans hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : Preorder \u03b2\nu : \u03b1 \u2192 \u03b2\nh : Tendsto u atTop atTop\na : \u03b1\nb : \u03b2\n\u22a2 \u2203 a', a' \u2265 a \u2227 b \u2264 u a'\n[PROOFSTEP]\nhave : Nonempty \u03b1 := \u27e8a\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : Preorder \u03b2\nu : \u03b1 \u2192 \u03b2\nh : Tendsto u atTop atTop\na : \u03b1\nb : \u03b2\nthis : Nonempty \u03b1\n\u22a2 \u2203 a', a' \u2265 a \u2227 b \u2264 u a'\n[PROOFSTEP]\nhave : \u2200\u1da0 x in atTop, a \u2264 x \u2227 b \u2264 u x := (eventually_ge_atTop a).and (h.eventually <| eventually_ge_atTop b)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : Preorder \u03b2\nu : \u03b1 \u2192 \u03b2\nh : Tendsto u atTop atTop\na : \u03b1\nb : \u03b2\nthis\u271d : Nonempty \u03b1\nthis : \u2200\u1da0 (x : \u03b1) in atTop, a \u2264 x \u2227 b \u2264 u x\n\u22a2 \u2203 a', a' \u2265 a \u2227 b \u2264 u a'\n[PROOFSTEP]\nexact this.exists\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u03b1 \u2192 \u03b2\nh : Tendsto u atTop atTop\na : \u03b1\nb : \u03b2\n\u22a2 \u2203 a', a' \u2265 a \u2227 b < u a'\n[PROOFSTEP]\ncases' exists_gt b with b' hb'\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u03b1 \u2192 \u03b2\nh : Tendsto u atTop atTop\na : \u03b1\nb b' : \u03b2\nhb' : b < b'\n\u22a2 \u2203 a', a' \u2265 a \u2227 b < u a'\n[PROOFSTEP]\nrcases exists_le_of_tendsto_atTop h a b' with \u27e8a', ha', ha''\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u03b1 \u2192 \u03b2\nh : Tendsto u atTop atTop\na : \u03b1\nb b' : \u03b2\nhb' : b < b'\na' : \u03b1\nha' : a' \u2265 a\nha'' : b' \u2264 u a'\n\u22a2 \u2203 a', a' \u2265 a \u2227 b < u a'\n[PROOFSTEP]\nexact \u27e8a', ha', lt_of_lt_of_le hb' ha''\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u2115 \u2192 \u03b2\nhu : Tendsto u atTop atTop\n\u22a2 \u2200 (N : \u2115), \u2203 n, n \u2265 N \u2227 \u2200 (k : \u2115), k < n \u2192 u k < u n\n[PROOFSTEP]\nintro N\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u2115 \u2192 \u03b2\nhu : Tendsto u atTop atTop\nN : \u2115\n\u22a2 \u2203 n, n \u2265 N \u2227 \u2200 (k : \u2115), k < n \u2192 u k < u n\n[PROOFSTEP]\nobtain \u27e8k : \u2115, - : k \u2264 N, hku : \u2200 l \u2264 N, u l \u2264 u k\u27e9 : \u2203 k \u2264 N, \u2200 l \u2264 N, u l \u2264 u k\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u2115 \u2192 \u03b2\nhu : Tendsto u atTop atTop\nN : \u2115\n\u22a2 \u2203 k, k \u2264 N \u2227 \u2200 (l : \u2115), l \u2264 N \u2192 u l \u2264 u k\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u2115 \u2192 \u03b2\nhu : Tendsto u atTop atTop\nN k : \u2115\nhku : \u2200 (l : \u2115), l \u2264 N \u2192 u l \u2264 u k\n\u22a2 \u2203 n, n \u2265 N \u2227 \u2200 (k : \u2115), k < n \u2192 u k < u n\n[PROOFSTEP]\nexact exists_max_image _ u (finite_le_nat N) \u27e8N, le_refl N\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u2115 \u2192 \u03b2\nhu : Tendsto u atTop atTop\nN k : \u2115\nhku : \u2200 (l : \u2115), l \u2264 N \u2192 u l \u2264 u k\n\u22a2 \u2203 n, n \u2265 N \u2227 \u2200 (k : \u2115), k < n \u2192 u k < u n\n[PROOFSTEP]\nhave ex : \u2203 n \u2265 N, u k < u n := exists_lt_of_tendsto_atTop hu _ _\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u2115 \u2192 \u03b2\nhu : Tendsto u atTop atTop\nN k : \u2115\nhku : \u2200 (l : \u2115), l \u2264 N \u2192 u l \u2264 u k\nex : \u2203 n, n \u2265 N \u2227 u k < u n\n\u22a2 \u2203 n, n \u2265 N \u2227 \u2200 (k : \u2115), k < n \u2192 u k < u n\n[PROOFSTEP]\nobtain \u27e8n : \u2115, hnN : n \u2265 N, hnk : u k < u n, hn_min : \u2200 m, m < n \u2192 N \u2264 m \u2192 u m \u2264 u k\u27e9 :\n \u2203 n \u2265 N, u k < u n \u2227 \u2200 m, m < n \u2192 N \u2264 m \u2192 u m \u2264 u k :=\n by\n rcases Nat.findX ex with \u27e8n, \u27e8hnN, hnk\u27e9, hn_min\u27e9\n push_neg at hn_min \n exact \u27e8n, hnN, hnk, hn_min\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u2115 \u2192 \u03b2\nhu : Tendsto u atTop atTop\nN k : \u2115\nhku : \u2200 (l : \u2115), l \u2264 N \u2192 u l \u2264 u k\nex : \u2203 n, n \u2265 N \u2227 u k < u n\n\u22a2 \u2203 n, n \u2265 N \u2227 u k < u n \u2227 \u2200 (m : \u2115), m < n \u2192 N \u2264 m \u2192 u m \u2264 u k\n[PROOFSTEP]\nrcases Nat.findX ex with \u27e8n, \u27e8hnN, hnk\u27e9, hn_min\u27e9\n[GOAL]\ncase mk.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u2115 \u2192 \u03b2\nhu : Tendsto u atTop atTop\nN k : \u2115\nhku : \u2200 (l : \u2115), l \u2264 N \u2192 u l \u2264 u k\nex : \u2203 n, n \u2265 N \u2227 u k < u n\nn : \u2115\nhn_min : \u2200 (m : \u2115), m < n \u2192 \u00ac(m \u2265 N \u2227 u k < u m)\nhnN : n \u2265 N\nhnk : u k < u n\n\u22a2 \u2203 n, n \u2265 N \u2227 u k < u n \u2227 \u2200 (m : \u2115), m < n \u2192 N \u2264 m \u2192 u m \u2264 u k\n[PROOFSTEP]\npush_neg at hn_min \n[GOAL]\ncase mk.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u2115 \u2192 \u03b2\nhu : Tendsto u atTop atTop\nN k : \u2115\nhku : \u2200 (l : \u2115), l \u2264 N \u2192 u l \u2264 u k\nex : \u2203 n, n \u2265 N \u2227 u k < u n\nn : \u2115\nhnN : n \u2265 N\nhnk : u k < u n\nhn_min : \u2200 (m : \u2115), m < n \u2192 m \u2265 N \u2192 u m \u2264 u k\n\u22a2 \u2203 n, n \u2265 N \u2227 u k < u n \u2227 \u2200 (m : \u2115), m < n \u2192 N \u2264 m \u2192 u m \u2264 u k\n[PROOFSTEP]\nexact \u27e8n, hnN, hnk, hn_min\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u2115 \u2192 \u03b2\nhu : Tendsto u atTop atTop\nN k : \u2115\nhku : \u2200 (l : \u2115), l \u2264 N \u2192 u l \u2264 u k\nex : \u2203 n, n \u2265 N \u2227 u k < u n\nn : \u2115\nhnN : n \u2265 N\nhnk : u k < u n\nhn_min : \u2200 (m : \u2115), m < n \u2192 N \u2264 m \u2192 u m \u2264 u k\n\u22a2 \u2203 n, n \u2265 N \u2227 \u2200 (k : \u2115), k < n \u2192 u k < u n\n[PROOFSTEP]\nuse n, hnN\n[GOAL]\ncase right\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u2115 \u2192 \u03b2\nhu : Tendsto u atTop atTop\nN k : \u2115\nhku : \u2200 (l : \u2115), l \u2264 N \u2192 u l \u2264 u k\nex : \u2203 n, n \u2265 N \u2227 u k < u n\nn : \u2115\nhnN : n \u2265 N\nhnk : u k < u n\nhn_min : \u2200 (m : \u2115), m < n \u2192 N \u2264 m \u2192 u m \u2264 u k\n\u22a2 \u2200 (k : \u2115), k < n \u2192 u k < u n\n[PROOFSTEP]\nrintro (l : \u2115) (hl : l < n)\n[GOAL]\ncase right\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u2115 \u2192 \u03b2\nhu : Tendsto u atTop atTop\nN k : \u2115\nhku : \u2200 (l : \u2115), l \u2264 N \u2192 u l \u2264 u k\nex : \u2203 n, n \u2265 N \u2227 u k < u n\nn : \u2115\nhnN : n \u2265 N\nhnk : u k < u n\nhn_min : \u2200 (m : \u2115), m < n \u2192 N \u2264 m \u2192 u m \u2264 u k\nl : \u2115\nhl : l < n\n\u22a2 u l < u n\n[PROOFSTEP]\nhave hlk : u l \u2264 u k := by\n cases' (le_total l N : l \u2264 N \u2228 N \u2264 l) with H H\n \u00b7 exact hku l H\n \u00b7 exact hn_min l hl H\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u2115 \u2192 \u03b2\nhu : Tendsto u atTop atTop\nN k : \u2115\nhku : \u2200 (l : \u2115), l \u2264 N \u2192 u l \u2264 u k\nex : \u2203 n, n \u2265 N \u2227 u k < u n\nn : \u2115\nhnN : n \u2265 N\nhnk : u k < u n\nhn_min : \u2200 (m : \u2115), m < n \u2192 N \u2264 m \u2192 u m \u2264 u k\nl : \u2115\nhl : l < n\n\u22a2 u l \u2264 u k\n[PROOFSTEP]\ncases' (le_total l N : l \u2264 N \u2228 N \u2264 l) with H H\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u2115 \u2192 \u03b2\nhu : Tendsto u atTop atTop\nN k : \u2115\nhku : \u2200 (l : \u2115), l \u2264 N \u2192 u l \u2264 u k\nex : \u2203 n, n \u2265 N \u2227 u k < u n\nn : \u2115\nhnN : n \u2265 N\nhnk : u k < u n\nhn_min : \u2200 (m : \u2115), m < n \u2192 N \u2264 m \u2192 u m \u2264 u k\nl : \u2115\nhl : l < n\nH : l \u2264 N\n\u22a2 u l \u2264 u k\n[PROOFSTEP]\nexact hku l H\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u2115 \u2192 \u03b2\nhu : Tendsto u atTop atTop\nN k : \u2115\nhku : \u2200 (l : \u2115), l \u2264 N \u2192 u l \u2264 u k\nex : \u2203 n, n \u2265 N \u2227 u k < u n\nn : \u2115\nhnN : n \u2265 N\nhnk : u k < u n\nhn_min : \u2200 (m : \u2115), m < n \u2192 N \u2264 m \u2192 u m \u2264 u k\nl : \u2115\nhl : l < n\nH : N \u2264 l\n\u22a2 u l \u2264 u k\n[PROOFSTEP]\nexact hn_min l hl H\n[GOAL]\ncase right\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u2115 \u2192 \u03b2\nhu : Tendsto u atTop atTop\nN k : \u2115\nhku : \u2200 (l : \u2115), l \u2264 N \u2192 u l \u2264 u k\nex : \u2203 n, n \u2265 N \u2227 u k < u n\nn : \u2115\nhnN : n \u2265 N\nhnk : u k < u n\nhn_min : \u2200 (m : \u2115), m < n \u2192 N \u2264 m \u2192 u m \u2264 u k\nl : \u2115\nhl : l < n\nhlk : u l \u2264 u k\n\u22a2 u l < u n\n[PROOFSTEP]\ncalc\n u l \u2264 u k := hlk\n _ < u n := hnk\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u2115 \u2192 \u03b2\nhu : Tendsto u atTop atTop\n\u22a2 \u2203\u1da0 (n : \u2115) in atTop, \u2200 (k : \u2115), k < n \u2192 u k < u n\n[PROOFSTEP]\nsimpa [frequently_atTop] using high_scores hu\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : OrderedAddCommGroup \u03b2\nl : Filter \u03b1\nf g : \u03b1 \u2192 \u03b2\nC : \u03b2\nhf : \u2200\u1da0 (x : \u03b1) in l, C \u2264 f x\nhg : Tendsto g l atTop\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, (fun x => -f x) x \u2264 -C\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : OrderedAddCommGroup \u03b2\nl : Filter \u03b1\nf g : \u03b1 \u2192 \u03b2\nC : \u03b2\nhf : \u2200\u1da0 (x : \u03b1) in l, C \u2264 f x\nhg : Tendsto g l atTop\n\u22a2 Tendsto (fun x => (fun x => -f x) x + (fun x => f x + g x) x) l atTop\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : OrderedAddCommGroup \u03b2\nl : Filter \u03b1\nf g : \u03b1 \u2192 \u03b2\nC : \u03b2\nhf : Tendsto f l atTop\nhg : \u2200\u1da0 (x : \u03b1) in l, C \u2264 g x\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, (fun x => -g x) x \u2264 -C\n[PROOFSTEP]\nsimp [hg]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : OrderedAddCommGroup \u03b2\nl : Filter \u03b1\nf g : \u03b1 \u2192 \u03b2\nC : \u03b2\nhf : Tendsto f l atTop\nhg : \u2200\u1da0 (x : \u03b1) in l, C \u2264 g x\n\u22a2 Tendsto (fun x => (fun x => f x + g x) x + (fun x => -g x) x) l atTop\n[PROOFSTEP]\nsimp [hf]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : StrictOrderedSemiring \u03b1\nl : Filter \u03b2\nf g : \u03b2 \u2192 \u03b1\nhf : Tendsto f l atTop\nhg : Tendsto g l atTop\n\u22a2 Tendsto (fun x => f x * g x) l atTop\n[PROOFSTEP]\nrefine' tendsto_atTop_mono' _ _ hg\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : StrictOrderedSemiring \u03b1\nl : Filter \u03b2\nf g : \u03b2 \u2192 \u03b1\nhf : Tendsto f l atTop\nhg : Tendsto g l atTop\n\u22a2 g \u2264\u1da0[l] fun x => f x * g x\n[PROOFSTEP]\nfilter_upwards [hg.eventually (eventually_ge_atTop 0), hf.eventually (eventually_ge_atTop 1)] with _ using\n le_mul_of_one_le_left\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : StrictOrderedRing \u03b1\nl : Filter \u03b2\nf g : \u03b2 \u2192 \u03b1\nhf : Tendsto f l atTop\nhg : Tendsto g l atBot\n\u22a2 Tendsto (fun x => f x * g x) l atBot\n[PROOFSTEP]\nhave := hf.atTop_mul_atTop <| tendsto_neg_atBot_atTop.comp hg\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : StrictOrderedRing \u03b1\nl : Filter \u03b2\nf g : \u03b2 \u2192 \u03b1\nhf : Tendsto f l atTop\nhg : Tendsto g l atBot\nthis : Tendsto (fun x => f x * (Neg.neg \u2218 g) x) l atTop\n\u22a2 Tendsto (fun x => f x * g x) l atBot\n[PROOFSTEP]\nsimpa only [(\u00b7 \u2218 \u00b7), neg_mul_eq_mul_neg, neg_neg] using tendsto_neg_atTop_atBot.comp this\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : StrictOrderedRing \u03b1\nl : Filter \u03b2\nf g : \u03b2 \u2192 \u03b1\nhf : Tendsto f l atBot\nhg : Tendsto g l atTop\n\u22a2 Tendsto (fun x => f x * g x) l atBot\n[PROOFSTEP]\nhave : Tendsto (fun x => -f x * g x) l atTop := (tendsto_neg_atBot_atTop.comp hf).atTop_mul_atTop hg\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : StrictOrderedRing \u03b1\nl : Filter \u03b2\nf g : \u03b2 \u2192 \u03b1\nhf : Tendsto f l atBot\nhg : Tendsto g l atTop\nthis : Tendsto (fun x => -f x * g x) l atTop\n\u22a2 Tendsto (fun x => f x * g x) l atBot\n[PROOFSTEP]\nsimpa only [(\u00b7 \u2218 \u00b7), neg_mul_eq_neg_mul, neg_neg] using tendsto_neg_atTop_atBot.comp this\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : StrictOrderedRing \u03b1\nl : Filter \u03b2\nf g : \u03b2 \u2192 \u03b1\nhf : Tendsto f l atBot\nhg : Tendsto g l atBot\n\u22a2 Tendsto (fun x => f x * g x) l atTop\n[PROOFSTEP]\nhave : Tendsto (fun x => -f x * -g x) l atTop :=\n (tendsto_neg_atBot_atTop.comp hf).atTop_mul_atTop (tendsto_neg_atBot_atTop.comp hg)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : StrictOrderedRing \u03b1\nl : Filter \u03b2\nf g : \u03b2 \u2192 \u03b1\nhf : Tendsto f l atBot\nhg : Tendsto g l atBot\nthis : Tendsto (fun x => -f x * -g x) l atTop\n\u22a2 Tendsto (fun x => f x * g x) l atTop\n[PROOFSTEP]\nsimpa only [neg_mul_neg] using this\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedAddCommGroup \u03b1\n\u22a2 comap abs atTop = atBot \u2294 atTop\n[PROOFSTEP]\nrefine'\n le_antisymm (((atTop_basis.comap _).le_basis_iff (atBot_basis.sup atTop_basis)).2 _)\n (sup_le tendsto_abs_atBot_atTop.le_comap tendsto_abs_atTop_atTop.le_comap)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedAddCommGroup \u03b1\n\u22a2 \u2200 (i' : \u03b1 \u00d7 \u03b1), True \u2227 True \u2192 \u2203 i, True \u2227 abs \u207b\u00b9' Ici i \u2286 Iic i'.fst \u222a Ici i'.snd\n[PROOFSTEP]\nrintro \u27e8a, b\u27e9 -\n[GOAL]\ncase mk\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\n\u22a2 \u2203 i, True \u2227 abs \u207b\u00b9' Ici i \u2286 Iic (a, b).fst \u222a Ici (a, b).snd\n[PROOFSTEP]\nrefine' \u27e8max (-a) b, trivial, fun x hx => _\u27e9\n[GOAL]\ncase mk\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b x : \u03b1\nhx : x \u2208 abs \u207b\u00b9' Ici (max (-a) b)\n\u22a2 x \u2208 Iic (a, b).fst \u222a Ici (a, b).snd\n[PROOFSTEP]\nrw [mem_preimage, mem_Ici, le_abs', max_le_iff, \u2190 min_neg_neg, le_min_iff, neg_neg] at hx \n[GOAL]\ncase mk\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b x : \u03b1\nhx : x \u2264 a \u2227 x \u2264 -b \u2228 -a \u2264 x \u2227 b \u2264 x\n\u22a2 x \u2208 Iic (a, b).fst \u222a Ici (a, b).snd\n[PROOFSTEP]\nexact hx.imp And.left And.right\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedSemiring \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nn : \u2115\nh : Tendsto (fun x => x ^ n) atTop atTop\nhn : n = 0\n\u22a2 False\n[PROOFSTEP]\nsimp only [hn, pow_zero, not_tendsto_const_atTop] at h \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedRing \u03b1\n\u22a2 \u00acTendsto (fun x => x ^ 0) atTop atBot\n[PROOFSTEP]\nsimp [not_tendsto_const_atBot]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedSemifield \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr c : \u03b1\nn : \u2115\nhr : 0 < r\nh : Tendsto f l atTop\n\u22a2 Tendsto (fun x => r\u207b\u00b9 * (r * f x)) l atTop\n[PROOFSTEP]\nsimpa only [inv_mul_cancel_left\u2080 hr.ne']\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedSemifield \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr c : \u03b1\nn : \u2115\nhr : 0 < r\n\u22a2 Tendsto (fun x => f x * r) l atTop \u2194 Tendsto f l atTop\n[PROOFSTEP]\nsimpa only [mul_comm] using tendsto_const_mul_atTop_of_pos hr\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr c : \u03b1\nn : \u2115\ninst\u271d : NeBot l\nh : Tendsto f l atTop\n\u22a2 Tendsto (fun x => r * f x) l atTop \u2194 0 < r\n[PROOFSTEP]\nrefine' \u27e8fun hrf => not_le.mp fun hr => _, fun hr => (tendsto_const_mul_atTop_of_pos hr).mpr h\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr c : \u03b1\nn : \u2115\ninst\u271d : NeBot l\nh : Tendsto f l atTop\nhrf : Tendsto (fun x => r * f x) l atTop\nhr : r \u2264 0\n\u22a2 False\n[PROOFSTEP]\nrcases((h.eventually_ge_atTop 0).and (hrf.eventually_gt_atTop 0)).exists with \u27e8x, hx, hrx\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr c : \u03b1\nn : \u2115\ninst\u271d : NeBot l\nh : Tendsto f l atTop\nhrf : Tendsto (fun x => r * f x) l atTop\nhr : r \u2264 0\nx : \u03b2\nhx : 0 \u2264 f x\nhrx : 0 < r * f x\n\u22a2 False\n[PROOFSTEP]\nexact (mul_nonpos_of_nonpos_of_nonneg hr hx).not_lt hrx\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr c : \u03b1\nn : \u2115\ninst\u271d : NeBot l\nh : Tendsto f l atTop\n\u22a2 Tendsto (fun x => f x * r) l atTop \u2194 0 < r\n[PROOFSTEP]\nsimp only [mul_comm _ r, tendsto_const_mul_atTop_iff_pos h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedSemifield \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr c : \u03b1\nn : \u2115\nhr : 0 < r\nhf : Tendsto f l atTop\n\u22a2 Tendsto (fun x => f x / r) l atTop\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using hf.atTop_mul_const (inv_pos.2 hr)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedSemifield \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr c : \u03b1\nn : \u2115\n\u22a2 Tendsto (fun x => c * x ^ n) atTop atTop \u2194 n \u2260 0 \u2227 0 < c\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8_, _\u27e9, fun h => tendsto_const_mul_pow_atTop h.1 h.2\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedSemifield \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr c : \u03b1\nn : \u2115\nh : Tendsto (fun x => c * x ^ n) atTop atTop\n\u22a2 n \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedSemifield \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr c : \u03b1\nh : Tendsto (fun x => c * x ^ 0) atTop atTop\n\u22a2 False\n[PROOFSTEP]\nsimp only [pow_zero, not_tendsto_const_atTop] at h \n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedSemifield \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr c : \u03b1\nn : \u2115\nh : Tendsto (fun x => c * x ^ n) atTop atTop\n\u22a2 0 < c\n[PROOFSTEP]\nrcases((h.eventually_gt_atTop 0).and (eventually_ge_atTop 0)).exists with \u27e8k, hck, hk\u27e9\n[GOAL]\ncase refine'_2.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedSemifield \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr c : \u03b1\nn : \u2115\nh : Tendsto (fun x => c * x ^ n) atTop atTop\nk : \u03b1\nhck : 0 < c * k ^ n\nhk : 0 \u2264 k\n\u22a2 0 < c\n[PROOFSTEP]\nexact pos_of_mul_pos_left hck (pow_nonneg hk _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\nhr : 0 < r\n\u22a2 Tendsto (fun x => r * f x) l atBot \u2194 Tendsto f l atBot\n[PROOFSTEP]\nsimpa only [\u2190 mul_neg, \u2190 tendsto_neg_atTop_iff] using tendsto_const_mul_atTop_of_pos hr\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\nhr : 0 < r\n\u22a2 Tendsto (fun x => f x * r) l atBot \u2194 Tendsto f l atBot\n[PROOFSTEP]\nsimpa only [mul_comm] using tendsto_const_mul_atBot_of_pos hr\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\nhr : r < 0\n\u22a2 Tendsto (fun x => r * f x) l atTop \u2194 Tendsto f l atBot\n[PROOFSTEP]\nsimpa only [neg_mul, tendsto_neg_atBot_iff] using tendsto_const_mul_atBot_of_pos (neg_pos.2 hr)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\nhr : r < 0\n\u22a2 Tendsto (fun x => f x * r) l atTop \u2194 Tendsto f l atBot\n[PROOFSTEP]\nsimpa only [mul_comm] using tendsto_const_mul_atTop_of_neg hr\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\nhr : r < 0\n\u22a2 Tendsto (fun x => r * f x) l atBot \u2194 Tendsto f l atTop\n[PROOFSTEP]\nsimpa only [neg_mul, tendsto_neg_atTop_iff] using tendsto_const_mul_atTop_of_pos (neg_pos.2 hr)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\nhr : r < 0\n\u22a2 Tendsto (fun x => f x * r) l atBot \u2194 Tendsto f l atTop\n[PROOFSTEP]\nsimpa only [mul_comm] using tendsto_const_mul_atBot_of_neg hr\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\ninst\u271d : NeBot l\n\u22a2 Tendsto (fun x => r * f x) l atTop \u2194 0 < r \u2227 Tendsto f l atTop \u2228 r < 0 \u2227 Tendsto f l atBot\n[PROOFSTEP]\nrcases lt_trichotomy r 0 with (hr | rfl | hr)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\ninst\u271d : NeBot l\nhr : r < 0\n\u22a2 Tendsto (fun x => r * f x) l atTop \u2194 0 < r \u2227 Tendsto f l atTop \u2228 r < 0 \u2227 Tendsto f l atBot\n[PROOFSTEP]\nsimp [hr, hr.not_lt, tendsto_const_mul_atTop_of_neg]\n[GOAL]\ncase inr.inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : NeBot l\n\u22a2 Tendsto (fun x => 0 * f x) l atTop \u2194 0 < 0 \u2227 Tendsto f l atTop \u2228 0 < 0 \u2227 Tendsto f l atBot\n[PROOFSTEP]\nsimp [not_tendsto_const_atTop]\n[GOAL]\ncase inr.inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\ninst\u271d : NeBot l\nhr : 0 < r\n\u22a2 Tendsto (fun x => r * f x) l atTop \u2194 0 < r \u2227 Tendsto f l atTop \u2228 r < 0 \u2227 Tendsto f l atBot\n[PROOFSTEP]\nsimp [hr, hr.not_lt, tendsto_const_mul_atTop_of_pos]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\ninst\u271d : NeBot l\n\u22a2 Tendsto (fun x => f x * r) l atTop \u2194 0 < r \u2227 Tendsto f l atTop \u2228 r < 0 \u2227 Tendsto f l atBot\n[PROOFSTEP]\nsimp only [mul_comm _ r, tendsto_const_mul_atTop_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\ninst\u271d : NeBot l\n\u22a2 Tendsto (fun x => r * f x) l atBot \u2194 0 < r \u2227 Tendsto f l atBot \u2228 r < 0 \u2227 Tendsto f l atTop\n[PROOFSTEP]\nsimp only [\u2190 tendsto_neg_atTop_iff, \u2190 mul_neg, tendsto_const_mul_atTop_iff, neg_neg]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\ninst\u271d : NeBot l\n\u22a2 Tendsto (fun x => f x * r) l atBot \u2194 0 < r \u2227 Tendsto f l atBot \u2228 r < 0 \u2227 Tendsto f l atTop\n[PROOFSTEP]\nsimp only [mul_comm _ r, tendsto_const_mul_atBot_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\ninst\u271d : NeBot l\nh : Tendsto f l atBot\n\u22a2 Tendsto (fun x => r * f x) l atTop \u2194 r < 0\n[PROOFSTEP]\nsimp [tendsto_const_mul_atTop_iff, h, h.not_tendsto disjoint_atBot_atTop]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\ninst\u271d : NeBot l\nh : Tendsto f l atBot\n\u22a2 Tendsto (fun x => f x * r) l atTop \u2194 r < 0\n[PROOFSTEP]\nsimp only [mul_comm _ r, tendsto_const_mul_atTop_iff_neg h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\ninst\u271d : NeBot l\nh : Tendsto f l atBot\n\u22a2 Tendsto (fun x => r * f x) l atBot \u2194 0 < r\n[PROOFSTEP]\nsimp [tendsto_const_mul_atBot_iff, h, h.not_tendsto disjoint_atBot_atTop]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\ninst\u271d : NeBot l\nh : Tendsto f l atBot\n\u22a2 Tendsto (fun x => f x * r) l atBot \u2194 0 < r\n[PROOFSTEP]\nsimp only [mul_comm _ r, tendsto_const_mul_atBot_iff_pos h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\ninst\u271d : NeBot l\nh : Tendsto f l atTop\n\u22a2 Tendsto (fun x => r * f x) l atBot \u2194 r < 0\n[PROOFSTEP]\nsimp [tendsto_const_mul_atBot_iff, h, h.not_tendsto disjoint_atTop_atBot]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\ninst\u271d : NeBot l\nh : Tendsto f l atTop\n\u22a2 Tendsto (fun x => f x * r) l atBot \u2194 r < 0\n[PROOFSTEP]\nsimp only [mul_comm _ r, tendsto_const_mul_atBot_iff_neg h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\nhr : 0 < r\nhf : Tendsto f l atBot\n\u22a2 Tendsto (fun x => f x / r) l atBot\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using hf.atBot_mul_const (inv_pos.2 hr)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr c : \u03b1\nn : \u2115\n\u22a2 Tendsto (fun x => c * x ^ n) atTop atBot \u2194 n \u2260 0 \u2227 c < 0\n[PROOFSTEP]\nsimp only [\u2190 tendsto_neg_atTop_iff, \u2190 neg_mul, tendsto_const_mul_pow_atTop_iff, neg_pos]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192 \u03b2\nl : Filter \u03b2\n\u22a2 Tendsto f atTop l \u2194 \u2200 (s : Set \u03b2), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u03b1), b \u2265 a \u2192 f b \u2208 s\n[PROOFSTEP]\nsimp only [tendsto_def, mem_atTop_sets]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192 \u03b2\nl : Filter \u03b2\n\u22a2 (\u2200 (s : Set \u03b2), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u03b1), b \u2265 a \u2192 b \u2208 f \u207b\u00b9' s) \u2194 \u2200 (s : Set \u03b2), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u03b1), b \u2265 a \u2192 f b \u2208 s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b2 \u2192 \u03b1\ns : Set \u03b1\n\u22a2 Tendsto f atTop (\ud835\udcdf s) \u2194 \u2203 N, \u2200 (n : \u03b2), n \u2265 N \u2192 f n \u2208 s\n[PROOFSTEP]\nrw [tendsto_iff_comap, comap_principal, le_principal_iff, mem_atTop_sets]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b2 \u2192 \u03b1\ns : Set \u03b1\n\u22a2 (\u2203 a, \u2200 (b : \u03b2), b \u2265 a \u2192 b \u2208 f \u207b\u00b9' s) \u2194 \u2203 N, \u2200 (n : \u03b2), n \u2265 N \u2192 f n \u2208 s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2\ne : \u03b2 \u2192 \u03b3\nl : Filter \u03b1\nhm : \u2200 (b\u2081 b\u2082 : \u03b2), e b\u2081 \u2264 e b\u2082 \u2194 b\u2081 \u2264 b\u2082\nhu : \u2200 (c : \u03b3), \u2203 b, c \u2264 e b\n\u22a2 Tendsto (e \u2218 f) l atTop \u2194 Tendsto f l atTop\n[PROOFSTEP]\nrw [\u2190 comap_embedding_atTop hm hu, tendsto_comap_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u22a2 atTop = \u2a05 (x : \u03b1), \ud835\udcdf (Ici {x})\n[PROOFSTEP]\nrefine' le_antisymm (le_iInf fun i => le_principal_iff.2 <| mem_atTop { i }) _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u22a2 \u2a05 (x : \u03b1), \ud835\udcdf (Ici {x}) \u2264 atTop\n[PROOFSTEP]\nrefine' le_iInf fun s => le_principal_iff.2 <| mem_iInf_of_iInter s.finite_toSet (fun i => mem_principal_self _) _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns : Finset \u03b1\n\u22a2 \u22c2 (i : \u2191\u2191s), Ici {\u2191i} \u2286 Ici s\n[PROOFSTEP]\nsimp only [subset_def, mem_iInter, SetCoe.forall, mem_Ici, Finset.le_iff_subset, Finset.mem_singleton,\n Finset.subset_iff, forall_eq]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns : Finset \u03b1\n\u22a2 \u2200 (x : Finset \u03b1), (\u2200 (x_1 : \u03b1), x_1 \u2208 \u2191s \u2192 x_1 \u2208 x) \u2192 \u2200 \u2983x_1 : \u03b1\u2984, x_1 \u2208 s \u2192 x_1 \u2208 x\n[PROOFSTEP]\nexact fun t => id\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Preorder \u03b2\nf : \u03b2 \u2192 Finset \u03b1\nh : Monotone f\nh' : \u2200 (x : \u03b1), \u2203 n, x \u2208 f n\n\u22a2 Tendsto f atTop atTop\n[PROOFSTEP]\nsimp only [atTop_finset_eq_iInf, tendsto_iInf, tendsto_principal]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Preorder \u03b2\nf : \u03b2 \u2192 Finset \u03b1\nh : Monotone f\nh' : \u2200 (x : \u03b1), \u2203 n, x \u2208 f n\n\u22a2 \u2200 (i : \u03b1), \u2200\u1da0 (a : \u03b2) in atTop, f a \u2208 Ici {i}\n[PROOFSTEP]\nintro a\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Preorder \u03b2\nf : \u03b2 \u2192 Finset \u03b1\nh : Monotone f\nh' : \u2200 (x : \u03b1), \u2203 n, x \u2208 f n\na : \u03b1\n\u22a2 \u2200\u1da0 (a_1 : \u03b2) in atTop, f a_1 \u2208 Ici {a}\n[PROOFSTEP]\nrcases h' a with \u27e8b, hb\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Preorder \u03b2\nf : \u03b2 \u2192 Finset \u03b1\nh : Monotone f\nh' : \u2200 (x : \u03b1), \u2203 n, x \u2208 f n\na : \u03b1\nb : \u03b2\nhb : a \u2208 f b\n\u22a2 \u2200\u1da0 (a_1 : \u03b2) in atTop, f a_1 \u2208 Ici {a}\n[PROOFSTEP]\nexact (eventually_ge_atTop b).mono fun b' hb' => (Finset.singleton_subset_iff.2 hb).trans (h hb')\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : DecidableEq \u03b2\ni : \u03b2 \u2192 \u03b3\nj : \u03b3 \u2192 \u03b2\nh : Function.LeftInverse j i\na : \u03b2\n\u22a2 a \u2208 Finset.image j {i a}\n[PROOFSTEP]\nsimp only [Finset.image_singleton, h a, Finset.mem_singleton]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\n\u22a2 atTop \u00d7\u02e2 atTop = atTop\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b1\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u271d : IsEmpty \u03b1\n\u22a2 atTop \u00d7\u02e2 atTop = atTop\ncase inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u271d : Nonempty \u03b1\n\u22a2 atTop \u00d7\u02e2 atTop = atTop\n[PROOFSTEP]\nexact Subsingleton.elim _ _\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u271d : Nonempty \u03b1\n\u22a2 atTop \u00d7\u02e2 atTop = atTop\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b2\n[GOAL]\ncase inr.inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u271d\u00b9 : Nonempty \u03b1\nh\u271d : IsEmpty \u03b2\n\u22a2 atTop \u00d7\u02e2 atTop = atTop\ncase inr.inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u271d\u00b9 : Nonempty \u03b1\nh\u271d : Nonempty \u03b2\n\u22a2 atTop \u00d7\u02e2 atTop = atTop\n[PROOFSTEP]\nexact Subsingleton.elim _ _\n[GOAL]\ncase inr.inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u271d\u00b9 : Nonempty \u03b1\nh\u271d : Nonempty \u03b2\n\u22a2 atTop \u00d7\u02e2 atTop = atTop\n[PROOFSTEP]\nsimpa [atTop, prod_iInf_left, prod_iInf_right, iInf_prod] using iInf_comm\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1\u2081 : Type u_6\n\u03b1\u2082 : Type u_7\n\u03b2\u2081 : Type u_8\n\u03b2\u2082 : Type u_9\ninst\u271d\u00b9 : Preorder \u03b2\u2081\ninst\u271d : Preorder \u03b2\u2082\nu\u2081 : \u03b2\u2081 \u2192 \u03b1\u2081\nu\u2082 : \u03b2\u2082 \u2192 \u03b1\u2082\n\u22a2 map u\u2081 atTop \u00d7\u02e2 map u\u2082 atTop = map (Prod.map u\u2081 u\u2082) atTop\n[PROOFSTEP]\nrw [prod_map_map_eq, prod_atTop_atTop_eq, Prod.map_def]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeInf \u03b1\n\u22a2 Tendsto (fun a => (a, a)) atBot atBot\n[PROOFSTEP]\nrw [\u2190 prod_atBot_atBot_eq]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeInf \u03b1\n\u22a2 Tendsto (fun a => (a, a)) atBot (atBot \u00d7\u02e2 atBot)\n[PROOFSTEP]\nexact tendsto_id.prod_mk tendsto_id\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\n\u22a2 Tendsto (fun a => (a, a)) atTop atTop\n[PROOFSTEP]\nrw [\u2190 prod_atTop_atTop_eq]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\n\u22a2 Tendsto (fun a => (a, a)) atTop (atTop \u00d7\u02e2 atTop)\n[PROOFSTEP]\nexact tendsto_id.prod_mk tendsto_id\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeInf \u03b3\nF : Filter \u03b1\nG : Filter \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\nhf : Tendsto f F atBot\nhg : Tendsto g G atBot\n\u22a2 Tendsto (Prod.map f g) (F \u00d7\u02e2 G) atBot\n[PROOFSTEP]\nrw [\u2190 prod_atBot_atBot_eq]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeInf \u03b3\nF : Filter \u03b1\nG : Filter \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\nhf : Tendsto f F atBot\nhg : Tendsto g G atBot\n\u22a2 Tendsto (Prod.map f g) (F \u00d7\u02e2 G) (atBot \u00d7\u02e2 atBot)\n[PROOFSTEP]\nexact hf.prod_map hg\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b3\nF : Filter \u03b1\nG : Filter \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\nhf : Tendsto f F atTop\nhg : Tendsto g G atTop\n\u22a2 Tendsto (Prod.map f g) (F \u00d7\u02e2 G) atTop\n[PROOFSTEP]\nrw [\u2190 prod_atTop_atTop_eq]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b3\nF : Filter \u03b1\nG : Filter \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\nhf : Tendsto f F atTop\nhg : Tendsto g G atTop\n\u22a2 Tendsto (Prod.map f g) (F \u00d7\u02e2 G) (atTop \u00d7\u02e2 atTop)\n[PROOFSTEP]\nexact hf.prod_map hg\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : SemilatticeInf \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : Tendsto f atBot atBot\nhg : Tendsto g atBot atBot\n\u22a2 Tendsto (Prod.map f g) atBot atBot\n[PROOFSTEP]\nrw [\u2190 prod_atBot_atBot_eq]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : SemilatticeInf \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : Tendsto f atBot atBot\nhg : Tendsto g atBot atBot\n\u22a2 Tendsto (Prod.map f g) (atBot \u00d7\u02e2 atBot) atBot\n[PROOFSTEP]\nexact hf.prod_map_prod_atBot hg\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : SemilatticeSup \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : Tendsto f atTop atTop\nhg : Tendsto g atTop atTop\n\u22a2 Tendsto (Prod.map f g) atTop atTop\n[PROOFSTEP]\nrw [\u2190 prod_atTop_atTop_eq]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : SemilatticeSup \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : Tendsto f atTop atTop\nhg : Tendsto g atTop atTop\n\u22a2 Tendsto (Prod.map f g) (atTop \u00d7\u02e2 atTop) atTop\n[PROOFSTEP]\nexact hf.prod_map_prod_atTop hg\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : Nonempty \u03b1\np : \u03b1 \u00d7 \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1da0 (x : \u03b1 \u00d7 \u03b1) in atBot, p x) \u2194 \u2203 a, \u2200 (k l : \u03b1), k \u2264 a \u2192 l \u2264 a \u2192 p (k, l)\n[PROOFSTEP]\nsimp [\u2190 prod_atBot_atBot_eq, (@atBot_basis \u03b1 _ _).prod_self.eventually_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : Nonempty \u03b1\np : \u03b1 \u00d7 \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1da0 (x : \u03b1 \u00d7 \u03b1) in atBot, p x) \u2194 \u2203 a, \u2200 (k : \u03b1), k \u2264 a \u2192 \u2200 (l : \u03b1), l \u2264 a \u2192 p (k, l)\n[PROOFSTEP]\nsimp only [eventually_atBot_prod_self, ball_cond_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : Nonempty \u03b1\np : \u03b1 \u00d7 \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1da0 (x : \u03b1 \u00d7 \u03b1) in atTop, p x) \u2194 \u2203 a, \u2200 (k : \u03b1), k \u2265 a \u2192 \u2200 (l : \u03b1), l \u2265 a \u2192 p (k, l)\n[PROOFSTEP]\nsimp only [eventually_atTop_prod_self, ball_cond_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : SemilatticeSup \u03b2\np : \u03b1 \u00d7 \u03b2 \u2192 Prop\nhp : \u2200\u1da0 (x : \u03b1 \u00d7 \u03b2) in atTop, p x\n\u22a2 \u2200\u1da0 (k : \u03b1) in atTop, \u2200\u1da0 (l : \u03b2) in atTop, p (k, l)\n[PROOFSTEP]\nrw [\u2190 prod_atTop_atTop_eq] at hp \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : SemilatticeSup \u03b2\np : \u03b1 \u00d7 \u03b2 \u2192 Prop\nhp : \u2200\u1da0 (x : \u03b1 \u00d7 \u03b2) in atTop \u00d7\u02e2 atTop, p x\n\u22a2 \u2200\u1da0 (k : \u03b1) in atTop, \u2200\u1da0 (l : \u03b2) in atTop, p (k, l)\n[PROOFSTEP]\nexact hp.curry\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nb' : \u03b2\nhf : Monotone f\ngc : \u2200 (a : \u03b1) (b : \u03b2), b \u2265 b' \u2192 (f a \u2264 b \u2194 a \u2264 g b)\nhgi : \u2200 (b : \u03b2), b \u2265 b' \u2192 b \u2264 f (g b)\n\u22a2 map f atTop = atTop\n[PROOFSTEP]\nrefine' le_antisymm (hf.tendsto_atTop_atTop fun b => \u27e8g (b \u2294 b'), le_sup_left.trans <| hgi _ le_sup_right\u27e9) _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nb' : \u03b2\nhf : Monotone f\ngc : \u2200 (a : \u03b1) (b : \u03b2), b \u2265 b' \u2192 (f a \u2264 b \u2194 a \u2264 g b)\nhgi : \u2200 (b : \u03b2), b \u2265 b' \u2192 b \u2264 f (g b)\n\u22a2 atTop \u2264 map f atTop\n[PROOFSTEP]\nrw [@map_atTop_eq _ _ \u27e8g b'\u27e9]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nb' : \u03b2\nhf : Monotone f\ngc : \u2200 (a : \u03b1) (b : \u03b2), b \u2265 b' \u2192 (f a \u2264 b \u2194 a \u2264 g b)\nhgi : \u2200 (b : \u03b2), b \u2265 b' \u2192 b \u2264 f (g b)\n\u22a2 atTop \u2264 \u2a05 (a : \u03b1), \ud835\udcdf (f '' {a' | a \u2264 a'})\n[PROOFSTEP]\nrefine' le_iInf fun a => iInf_le_of_le (f a \u2294 b') <| principal_mono.2 fun b hb => _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nb' : \u03b2\nhf : Monotone f\ngc : \u2200 (a : \u03b1) (b : \u03b2), b \u2265 b' \u2192 (f a \u2264 b \u2194 a \u2264 g b)\nhgi : \u2200 (b : \u03b2), b \u2265 b' \u2192 b \u2264 f (g b)\na : \u03b1\nb : \u03b2\nhb : b \u2208 Ici (f a \u2294 b')\n\u22a2 b \u2208 f '' {a' | a \u2264 a'}\n[PROOFSTEP]\nrw [mem_Ici, sup_le_iff] at hb \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nb' : \u03b2\nhf : Monotone f\ngc : \u2200 (a : \u03b1) (b : \u03b2), b \u2265 b' \u2192 (f a \u2264 b \u2194 a \u2264 g b)\nhgi : \u2200 (b : \u03b2), b \u2265 b' \u2192 b \u2264 f (g b)\na : \u03b1\nb : \u03b2\nhb : f a \u2264 b \u2227 b' \u2264 b\n\u22a2 b \u2208 f '' {a' | a \u2264 a'}\n[PROOFSTEP]\nexact \u27e8g b, (gc _ _ hb.2).1 hb.1, le_antisymm ((gc _ _ hb.2).2 le_rfl) (hgi _ hb.2)\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\ns : Set \u03b1\nh : Ici a \u2286 s\n\u22a2 map Subtype.val atTop = atTop\n[PROOFSTEP]\nhaveI : Nonempty s := \u27e8\u27e8a, h le_rfl\u27e9\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\ns : Set \u03b1\nh : Ici a \u2286 s\nthis : Nonempty \u2191s\n\u22a2 map Subtype.val atTop = atTop\n[PROOFSTEP]\nhave : Directed (\u00b7 \u2265 \u00b7) fun x : s => \ud835\udcdf (Ici x) := by\n intro x y\n use\u27e8x \u2294 y \u2294 a, h le_sup_right\u27e9\n simp only [ge_iff_le, principal_mono, Ici_subset_Ici, \u2190 Subtype.coe_le_coe, Subtype.coe_mk]\n exact \u27e8le_sup_left.trans le_sup_left, le_sup_right.trans le_sup_left\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\ns : Set \u03b1\nh : Ici a \u2286 s\nthis : Nonempty \u2191s\n\u22a2 Directed (fun x x_1 => x \u2265 x_1) fun x => \ud835\udcdf (Ici x)\n[PROOFSTEP]\nintro x y\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\ns : Set \u03b1\nh : Ici a \u2286 s\nthis : Nonempty \u2191s\nx y : \u2191s\n\u22a2 \u2203 z,\n (fun x x_1 => x \u2265 x_1) ((fun x => \ud835\udcdf (Ici x)) x) ((fun x => \ud835\udcdf (Ici x)) z) \u2227\n (fun x x_1 => x \u2265 x_1) ((fun x => \ud835\udcdf (Ici x)) y) ((fun x => \ud835\udcdf (Ici x)) z)\n[PROOFSTEP]\nuse\u27e8x \u2294 y \u2294 a, h le_sup_right\u27e9\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\ns : Set \u03b1\nh : Ici a \u2286 s\nthis : Nonempty \u2191s\nx y : \u2191s\n\u22a2 (fun x x_1 => x \u2265 x_1) ((fun x => \ud835\udcdf (Ici x)) x)\n ((fun x => \ud835\udcdf (Ici x)) { val := \u2191x \u2294 \u2191y \u2294 a, property := (_ : \u2191x \u2294 \u2191y \u2294 a \u2208 s) }) \u2227\n (fun x x_1 => x \u2265 x_1) ((fun x => \ud835\udcdf (Ici x)) y)\n ((fun x => \ud835\udcdf (Ici x)) { val := \u2191x \u2294 \u2191y \u2294 a, property := (_ : \u2191x \u2294 \u2191y \u2294 a \u2208 s) })\n[PROOFSTEP]\nsimp only [ge_iff_le, principal_mono, Ici_subset_Ici, \u2190 Subtype.coe_le_coe, Subtype.coe_mk]\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\ns : Set \u03b1\nh : Ici a \u2286 s\nthis : Nonempty \u2191s\nx y : \u2191s\n\u22a2 \u2191x \u2264 \u2191x \u2294 \u2191y \u2294 a \u2227 \u2191y \u2264 \u2191x \u2294 \u2191y \u2294 a\n[PROOFSTEP]\nexact \u27e8le_sup_left.trans le_sup_left, le_sup_right.trans le_sup_left\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\ns : Set \u03b1\nh : Ici a \u2286 s\nthis\u271d : Nonempty \u2191s\nthis : Directed (fun x x_1 => x \u2265 x_1) fun x => \ud835\udcdf (Ici x)\n\u22a2 map Subtype.val atTop = atTop\n[PROOFSTEP]\nsimp only [le_antisymm_iff, atTop, le_iInf_iff, le_principal_iff, mem_map, mem_setOf_eq, map_iInf_eq this,\n map_principal]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\ns : Set \u03b1\nh : Ici a \u2286 s\nthis\u271d : Nonempty \u2191s\nthis : Directed (fun x x_1 => x \u2265 x_1) fun x => \ud835\udcdf (Ici x)\n\u22a2 (\u2200 (i : \u03b1), Ici i \u2208 \u2a05 (i : \u2191s), \ud835\udcdf (Subtype.val '' Ici i)) \u2227 \u2200 (i : \u2191s), Subtype.val '' Ici i \u2208 \u2a05 (a : \u03b1), \ud835\udcdf (Ici a)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\ns : Set \u03b1\nh : Ici a \u2286 s\nthis\u271d : Nonempty \u2191s\nthis : Directed (fun x x_1 => x \u2265 x_1) fun x => \ud835\udcdf (Ici x)\n\u22a2 \u2200 (i : \u03b1), Ici i \u2208 \u2a05 (i : \u2191s), \ud835\udcdf (Subtype.val '' Ici i)\n[PROOFSTEP]\nintro x\n[GOAL]\ncase left\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\ns : Set \u03b1\nh : Ici a \u2286 s\nthis\u271d : Nonempty \u2191s\nthis : Directed (fun x x_1 => x \u2265 x_1) fun x => \ud835\udcdf (Ici x)\nx : \u03b1\n\u22a2 Ici x \u2208 \u2a05 (i : \u2191s), \ud835\udcdf (Subtype.val '' Ici i)\n[PROOFSTEP]\nrefine' mem_of_superset (mem_iInf_of_mem \u27e8x \u2294 a, h le_sup_right\u27e9 (mem_principal_self _)) _\n[GOAL]\ncase left\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\ns : Set \u03b1\nh : Ici a \u2286 s\nthis\u271d : Nonempty \u2191s\nthis : Directed (fun x x_1 => x \u2265 x_1) fun x => \ud835\udcdf (Ici x)\nx : \u03b1\n\u22a2 Subtype.val '' Ici { val := x \u2294 a, property := (_ : x \u2294 a \u2208 s) } \u2286 Ici x\n[PROOFSTEP]\nrintro _ \u27e8y, hy, rfl\u27e9\n[GOAL]\ncase left.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\ns : Set \u03b1\nh : Ici a \u2286 s\nthis\u271d : Nonempty \u2191s\nthis : Directed (fun x x_1 => x \u2265 x_1) fun x => \ud835\udcdf (Ici x)\nx : \u03b1\ny : \u2191s\nhy : y \u2208 Ici { val := x \u2294 a, property := (_ : x \u2294 a \u2208 s) }\n\u22a2 \u2191y \u2208 Ici x\n[PROOFSTEP]\nexact le_trans le_sup_left (Subtype.coe_le_coe.2 hy)\n[GOAL]\ncase right\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\ns : Set \u03b1\nh : Ici a \u2286 s\nthis\u271d : Nonempty \u2191s\nthis : Directed (fun x x_1 => x \u2265 x_1) fun x => \ud835\udcdf (Ici x)\n\u22a2 \u2200 (i : \u2191s), Subtype.val '' Ici i \u2208 \u2a05 (a : \u03b1), \ud835\udcdf (Ici a)\n[PROOFSTEP]\nintro x\n[GOAL]\ncase right\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\ns : Set \u03b1\nh : Ici a \u2286 s\nthis\u271d : Nonempty \u2191s\nthis : Directed (fun x x_1 => x \u2265 x_1) fun x => \ud835\udcdf (Ici x)\nx : \u2191s\n\u22a2 Subtype.val '' Ici x \u2208 \u2a05 (a : \u03b1), \ud835\udcdf (Ici a)\n[PROOFSTEP]\nfilter_upwards [mem_atTop (\u2191x \u2294 a)] with b hb\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\ns : Set \u03b1\nh : Ici a \u2286 s\nthis\u271d : Nonempty \u2191s\nthis : Directed (fun x x_1 => x \u2265 x_1) fun x => \ud835\udcdf (Ici x)\nx : \u2191s\nb : \u03b1\nhb : \u2191x \u2294 a \u2264 b\n\u22a2 b \u2208 Subtype.val '' Ici x\n[PROOFSTEP]\nexact \u27e8\u27e8b, h <| le_sup_right.trans hb\u27e9, Subtype.coe_le_coe.1 (le_sup_left.trans hb), rfl\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\n\u22a2 atTop = comap Subtype.val atTop\n[PROOFSTEP]\nrcases isEmpty_or_nonempty (Ioi a) with h | \u27e8\u27e8b, hb\u27e9\u27e9\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\nh : IsEmpty \u2191(Ioi a)\n\u22a2 atTop = comap Subtype.val atTop\n[PROOFSTEP]\nexact Subsingleton.elim _ _\n[GOAL]\ncase inr.intro.mk\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na b : \u03b1\nhb : b \u2208 Ioi a\n\u22a2 atTop = comap Subtype.val atTop\n[PROOFSTEP]\nrw [\u2190 map_val_atTop_of_Ici_subset (Ici_subset_Ioi.2 hb), comap_map Subtype.coe_injective]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\n\u22a2 atTop = comap Subtype.val atTop\n[PROOFSTEP]\nrw [\u2190 map_val_Ici_atTop a, comap_map Subtype.coe_injective]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\nf : \u03b2 \u2192 \u2191(Ioi a)\nl : Filter \u03b2\n\u22a2 Tendsto f l atTop \u2194 Tendsto (fun x => \u2191(f x)) l atTop\n[PROOFSTEP]\nrw [atTop_Ioi_eq, tendsto_comap_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\nf : \u03b2 \u2192 \u2191(Ioi a)\nl : Filter \u03b2\n\u22a2 Tendsto (Subtype.val \u2218 f) l atTop \u2194 Tendsto (fun x => \u2191(f x)) l atTop\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeInf \u03b1\na : \u03b1\nf : \u03b2 \u2192 \u2191(Iio a)\nl : Filter \u03b2\n\u22a2 Tendsto f l atBot \u2194 Tendsto (fun x => \u2191(f x)) l atBot\n[PROOFSTEP]\nrw [atBot_Iio_eq, tendsto_comap_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeInf \u03b1\na : \u03b1\nf : \u03b2 \u2192 \u2191(Iio a)\nl : Filter \u03b2\n\u22a2 Tendsto (Subtype.val \u2218 f) l atBot \u2194 Tendsto (fun x => \u2191(f x)) l atBot\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\nf : \u03b2 \u2192 \u2191(Ici a)\nl : Filter \u03b2\n\u22a2 Tendsto f l atTop \u2194 Tendsto (fun x => \u2191(f x)) l atTop\n[PROOFSTEP]\nrw [atTop_Ici_eq, tendsto_comap_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\nf : \u03b2 \u2192 \u2191(Ici a)\nl : Filter \u03b2\n\u22a2 Tendsto (Subtype.val \u2218 f) l atTop \u2194 Tendsto (fun x => \u2191(f x)) l atTop\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeInf \u03b1\na : \u03b1\nf : \u03b2 \u2192 \u2191(Iic a)\nl : Filter \u03b2\n\u22a2 Tendsto f l atBot \u2194 Tendsto (fun x => \u2191(f x)) l atBot\n[PROOFSTEP]\nrw [atBot_Iic_eq, tendsto_comap_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeInf \u03b1\na : \u03b1\nf : \u03b2 \u2192 \u2191(Iic a)\nl : Filter \u03b2\n\u22a2 Tendsto (Subtype.val \u2218 f) l atBot \u2194 Tendsto (fun x => \u2191(f x)) l atBot\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : NoMaxOrder \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nl : Filter \u03b2\n\u22a2 Tendsto (fun x => f \u2191x) atTop l \u2194 Tendsto f atTop l\n[PROOFSTEP]\nrw [\u2190 map_val_Ioi_atTop a, tendsto_map'_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : NoMaxOrder \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nl : Filter \u03b2\n\u22a2 Tendsto (fun x => f \u2191x) atTop l \u2194 Tendsto (f \u2218 Subtype.val) atTop l\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nl : Filter \u03b2\n\u22a2 Tendsto (fun x => f \u2191x) atTop l \u2194 Tendsto f atTop l\n[PROOFSTEP]\nrw [\u2190 map_val_Ici_atTop a, tendsto_map'_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nl : Filter \u03b2\n\u22a2 Tendsto (fun x => f \u2191x) atTop l \u2194 Tendsto (f \u2218 Subtype.val) atTop l\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : NoMinOrder \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nl : Filter \u03b2\n\u22a2 Tendsto (fun x => f \u2191x) atBot l \u2194 Tendsto f atBot l\n[PROOFSTEP]\nrw [\u2190 map_val_Iio_atBot a, tendsto_map'_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : NoMinOrder \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nl : Filter \u03b2\n\u22a2 Tendsto (fun x => f \u2191x) atBot l \u2194 Tendsto (f \u2218 Subtype.val) atBot l\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeInf \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nl : Filter \u03b2\n\u22a2 Tendsto (fun x => f \u2191x) atBot l \u2194 Tendsto f atBot l\n[PROOFSTEP]\nrw [\u2190 map_val_Iic_atBot a, tendsto_map'_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeInf \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nl : Filter \u03b2\n\u22a2 Tendsto (fun x => f \u2191x) atBot l \u2194 Tendsto (f \u2218 Subtype.val) atBot l\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nk a : \u2115\nh : a \u2265 k\n\u22a2 a \u2264 (fun a => a - k) a + k\n[PROOFSTEP]\nrw [tsub_add_cancel_of_le h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nk b : \u2115\nx\u271d : b \u2265 0\n\u22a2 b \u2264 (fun a => a + k) b - k\n[PROOFSTEP]\nrw [add_tsub_cancel_right]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u2115 \u2192 \u03b1\nl : Filter \u03b1\nk : \u2115\n\u22a2 Tendsto (f \u2218 fun n => n + k) atTop l \u2194 Tendsto f atTop l\n[PROOFSTEP]\nrw [\u2190 tendsto_map'_iff, map_add_atTop_eq_nat]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nk : \u2115\nhk : 0 < k\na b : \u2115\nx\u271d : b \u2265 1\n\u22a2 a / k \u2264 b \u2194 a \u2264 (fun b => b * k + (k - 1)) b\n[PROOFSTEP]\nsimp only [\u2190 Nat.lt_succ_iff, Nat.div_lt_iff_lt_mul hk, Nat.succ_eq_add_one, add_assoc,\n tsub_add_cancel_of_le (Nat.one_le_iff_ne_zero.2 hk.ne'), add_mul, one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nk : \u2115\nhk : 0 < k\nb : \u2115\nx\u271d : b \u2265 1\n\u22a2 b = b * k / k\n[PROOFSTEP]\nrw [Nat.mul_div_cancel b hk]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : LinearOrder \u03b1\nu : \u03b9 \u2192 \u03b1\nh : Monotone u\nH : \u00acBddAbove (range u)\n\u22a2 Tendsto u atTop atTop\n[PROOFSTEP]\napply h.tendsto_atTop_atTop\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : LinearOrder \u03b1\nu : \u03b9 \u2192 \u03b1\nh : Monotone u\nH : \u00acBddAbove (range u)\n\u22a2 \u2200 (b : \u03b1), \u2203 a, b \u2264 u a\n[PROOFSTEP]\nintro b\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : LinearOrder \u03b1\nu : \u03b9 \u2192 \u03b1\nh : Monotone u\nH : \u00acBddAbove (range u)\nb : \u03b1\n\u22a2 \u2203 a, b \u2264 u a\n[PROOFSTEP]\nrcases not_bddAbove_iff.1 H b with \u27e8_, \u27e8N, rfl\u27e9, hN\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : LinearOrder \u03b1\nu : \u03b9 \u2192 \u03b1\nh : Monotone u\nH : \u00acBddAbove (range u)\nb : \u03b1\nN : \u03b9\nhN : b < u N\n\u22a2 \u2203 a, b \u2264 u a\n[PROOFSTEP]\nexact \u27e8N, le_of_lt hN\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b3 : Nonempty \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nh : Tendsto f atTop atTop\n\u22a2 \u00acBddAbove (range f)\n[PROOFSTEP]\nrintro \u27e8M, hM\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b3 : Nonempty \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nh : Tendsto f atTop atTop\nM : \u03b2\nhM : M \u2208 upperBounds (range f)\n\u22a2 False\n[PROOFSTEP]\ncases' mem_atTop_sets.mp (h <| Ioi_mem_atTop M) with a ha\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b3 : Nonempty \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nh : Tendsto f atTop atTop\nM : \u03b2\nhM : M \u2208 upperBounds (range f)\na : \u03b1\nha : \u2200 (b : \u03b1), b \u2265 a \u2192 b \u2208 f \u207b\u00b9' Ioi M\n\u22a2 False\n[PROOFSTEP]\napply lt_irrefl M\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b3 : Nonempty \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMaxOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nh : Tendsto f atTop atTop\nM : \u03b2\nhM : M \u2208 upperBounds (range f)\na : \u03b1\nha : \u2200 (b : \u03b1), b \u2265 a \u2192 b \u2208 f \u207b\u00b9' Ioi M\n\u22a2 M < M\n[PROOFSTEP]\ncalc\n M < f a := ha a le_rfl\n _ \u2264 M := hM (Set.mem_range_self a)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\nf : \u03b2 \u2192 \u03b1\ng : \u03b3 \u2192 \u03b1\nh_eq : \u2200 (u : Finset \u03b3), \u2203 v, \u2200 (v' : Finset \u03b2), v \u2286 v' \u2192 \u2203 u', u \u2286 u' \u2227 \u220f x in u', g x = \u220f b in v', f b\n\u22a2 map (fun s => \u220f b in s, f b) atTop \u2264 map (fun s => \u220f x in s, g x) atTop\n[PROOFSTEP]\nclassical\nrefine' ((atTop_basis.map _).le_basis_iff (atTop_basis.map _)).2 fun b _ => _\nlet \u27e8v, hv\u27e9 := h_eq b\nrefine \u27e8v, trivial, ?_\u27e9\nsimpa [image_subset_iff] using hv\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\nf : \u03b2 \u2192 \u03b1\ng : \u03b3 \u2192 \u03b1\nh_eq : \u2200 (u : Finset \u03b3), \u2203 v, \u2200 (v' : Finset \u03b2), v \u2286 v' \u2192 \u2203 u', u \u2286 u' \u2227 \u220f x in u', g x = \u220f b in v', f b\n\u22a2 map (fun s => \u220f b in s, f b) atTop \u2264 map (fun s => \u220f x in s, g x) atTop\n[PROOFSTEP]\nrefine' ((atTop_basis.map _).le_basis_iff (atTop_basis.map _)).2 fun b _ => _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\nf : \u03b2 \u2192 \u03b1\ng : \u03b3 \u2192 \u03b1\nh_eq : \u2200 (u : Finset \u03b3), \u2203 v, \u2200 (v' : Finset \u03b2), v \u2286 v' \u2192 \u2203 u', u \u2286 u' \u2227 \u220f x in u', g x = \u220f b in v', f b\nb : Finset \u03b3\nx\u271d : True\n\u22a2 \u2203 i, True \u2227 (fun s => \u220f b in s, f b) '' Ici i \u2286 (fun s => \u220f x in s, g x) '' Ici b\n[PROOFSTEP]\nlet \u27e8v, hv\u27e9 := h_eq b\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\nf : \u03b2 \u2192 \u03b1\ng : \u03b3 \u2192 \u03b1\nh_eq : \u2200 (u : Finset \u03b3), \u2203 v, \u2200 (v' : Finset \u03b2), v \u2286 v' \u2192 \u2203 u', u \u2286 u' \u2227 \u220f x in u', g x = \u220f b in v', f b\nb : Finset \u03b3\nx\u271d : True\nv : Finset \u03b2\nhv : \u2200 (v' : Finset \u03b2), v \u2286 v' \u2192 \u2203 u', b \u2286 u' \u2227 \u220f x in u', g x = \u220f b in v', f b\n\u22a2 \u2203 i, True \u2227 (fun s => \u220f b in s, f b) '' Ici i \u2286 (fun s => \u220f x in s, g x) '' Ici b\n[PROOFSTEP]\nrefine \u27e8v, trivial, ?_\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\nf : \u03b2 \u2192 \u03b1\ng : \u03b3 \u2192 \u03b1\nh_eq : \u2200 (u : Finset \u03b3), \u2203 v, \u2200 (v' : Finset \u03b2), v \u2286 v' \u2192 \u2203 u', u \u2286 u' \u2227 \u220f x in u', g x = \u220f b in v', f b\nb : Finset \u03b3\nx\u271d : True\nv : Finset \u03b2\nhv : \u2200 (v' : Finset \u03b2), v \u2286 v' \u2192 \u2203 u', b \u2286 u' \u2227 \u220f x in u', g x = \u220f b in v', f b\n\u22a2 (fun s => \u220f b in s, f b) '' Ici v \u2286 (fun s => \u220f x in s, g x) '' Ici b\n[PROOFSTEP]\nsimpa [image_subset_iff] using hv\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : HasAntitoneBasis f s\nr : \u2115 \u2192 \u2115 \u2192 Prop\nhr : \u2200 (m : \u2115), \u2200\u1da0 (n : \u2115) in atTop, r m n\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 (\u2200 \u2983m n : \u2115\u2984, m < n \u2192 r (\u03c6 m) (\u03c6 n)) \u2227 HasAntitoneBasis f (s \u2218 \u03c6)\n[PROOFSTEP]\nsuffices : \u2203 \u03c6 : \u2115 \u2192 \u2115, StrictMono \u03c6 \u2227 \u2200 m n, m < n \u2192 r (\u03c6 m) (\u03c6 n)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : HasAntitoneBasis f s\nr : \u2115 \u2192 \u2115 \u2192 Prop\nhr : \u2200 (m : \u2115), \u2200\u1da0 (n : \u2115) in atTop, r m n\nthis : \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (m n : \u2115), m < n \u2192 r (\u03c6 m) (\u03c6 n)\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 (\u2200 \u2983m n : \u2115\u2984, m < n \u2192 r (\u03c6 m) (\u03c6 n)) \u2227 HasAntitoneBasis f (s \u2218 \u03c6)\n[PROOFSTEP]\nrcases this with \u27e8\u03c6, h\u03c6, hr\u03c6\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : HasAntitoneBasis f s\nr : \u2115 \u2192 \u2115 \u2192 Prop\nhr : \u2200 (m : \u2115), \u2200\u1da0 (n : \u2115) in atTop, r m n\n\u03c6 : \u2115 \u2192 \u2115\nh\u03c6 : StrictMono \u03c6\nhr\u03c6 : \u2200 (m n : \u2115), m < n \u2192 r (\u03c6 m) (\u03c6 n)\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 (\u2200 \u2983m n : \u2115\u2984, m < n \u2192 r (\u03c6 m) (\u03c6 n)) \u2227 HasAntitoneBasis f (s \u2218 \u03c6)\n[PROOFSTEP]\nexact \u27e8\u03c6, h\u03c6, hr\u03c6, hs.comp_strictMono h\u03c6\u27e9\n[GOAL]\ncase this\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : HasAntitoneBasis f s\nr : \u2115 \u2192 \u2115 \u2192 Prop\nhr : \u2200 (m : \u2115), \u2200\u1da0 (n : \u2115) in atTop, r m n\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (m n : \u2115), m < n \u2192 r (\u03c6 m) (\u03c6 n)\n[PROOFSTEP]\nhave : \u2200 t : Set \u2115, t.Finite \u2192 \u2200\u1da0 n in atTop, \u2200 m \u2208 t, m < n \u2227 r m n := fun t ht =>\n (eventually_all_finite ht).2 fun m _ => (eventually_gt_atTop m).and (hr _)\n[GOAL]\ncase this\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : HasAntitoneBasis f s\nr : \u2115 \u2192 \u2115 \u2192 Prop\nhr : \u2200 (m : \u2115), \u2200\u1da0 (n : \u2115) in atTop, r m n\nthis : \u2200 (t : Set \u2115), Set.Finite t \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (m : \u2115), m \u2208 t \u2192 m < n \u2227 r m n\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (m n : \u2115), m < n \u2192 r (\u03c6 m) (\u03c6 n)\n[PROOFSTEP]\nrcases seq_of_forall_finite_exists fun t ht => (this t ht).exists with \u27e8\u03c6, h\u03c6\u27e9\n[GOAL]\ncase this.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : HasAntitoneBasis f s\nr : \u2115 \u2192 \u2115 \u2192 Prop\nhr : \u2200 (m : \u2115), \u2200\u1da0 (n : \u2115) in atTop, r m n\nthis : \u2200 (t : Set \u2115), Set.Finite t \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (m : \u2115), m \u2208 t \u2192 m < n \u2227 r m n\n\u03c6 : \u2115 \u2192 \u2115\nh\u03c6 : \u2200 (n m : \u2115), m \u2208 \u03c6 '' Iio n \u2192 m < \u03c6 n \u2227 r m (\u03c6 n)\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (m n : \u2115), m < n \u2192 r (\u03c6 m) (\u03c6 n)\n[PROOFSTEP]\nsimp only [ball_image_iff, forall_and, mem_Iio] at h\u03c6 \n[GOAL]\ncase this.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : HasAntitoneBasis f s\nr : \u2115 \u2192 \u2115 \u2192 Prop\nhr : \u2200 (m : \u2115), \u2200\u1da0 (n : \u2115) in atTop, r m n\nthis : \u2200 (t : Set \u2115), Set.Finite t \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (m : \u2115), m \u2208 t \u2192 m < n \u2227 r m n\n\u03c6 : \u2115 \u2192 \u2115\nh\u03c6 : (\u2200 (x x_1 : \u2115), x_1 < x \u2192 \u03c6 x_1 < \u03c6 x) \u2227 \u2200 (x x_1 : \u2115), x_1 < x \u2192 r (\u03c6 x_1) (\u03c6 x)\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (m n : \u2115), m < n \u2192 r (\u03c6 m) (\u03c6 n)\n[PROOFSTEP]\nexact \u27e8\u03c6, forall_swap.2 h\u03c6.1, forall_swap.2 h\u03c6.2\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated f\ninst\u271d : NeBot f\n\u22a2 \u2203 x, Tendsto x atTop f\n[PROOFSTEP]\nobtain \u27e8B, h\u27e9 := f.exists_antitone_basis\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated f\ninst\u271d : NeBot f\nB : \u2115 \u2192 Set \u03b1\nh : HasAntitoneBasis f B\n\u22a2 \u2203 x, Tendsto x atTop f\n[PROOFSTEP]\nchoose x hx using fun n => Filter.nonempty_of_mem (h.mem n)\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated f\ninst\u271d : NeBot f\nB : \u2115 \u2192 Set \u03b1\nh : HasAntitoneBasis f B\nx : \u2115 \u2192 \u03b1\nhx : \u2200 (n : \u2115), x n \u2208 B n\n\u22a2 \u2203 x, Tendsto x atTop f\n[PROOFSTEP]\nexact \u27e8x, h.tendsto hx\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u22a2 \u2203 xs, Monotone xs \u2227 Tendsto xs atTop atTop\n[PROOFSTEP]\nhaveI h_ne_bot : (atTop : Filter \u03b1).NeBot := atTop_neBot\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\n\u22a2 \u2203 xs, Monotone xs \u2227 Tendsto xs atTop atTop\n[PROOFSTEP]\nobtain \u27e8ys, h\u27e9 := exists_seq_tendsto (atTop : Filter \u03b1)\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : Tendsto ys atTop atTop\n\u22a2 \u2203 xs, Monotone xs \u2227 Tendsto xs atTop atTop\n[PROOFSTEP]\nlet xs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : Tendsto ys atTop atTop\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\n\u22a2 \u2203 xs, Monotone xs \u2227 Tendsto xs atTop atTop\n[PROOFSTEP]\nhave h_mono : Monotone xs := by\n intro i j hij\n rw [Finset.sup'_le_iff]\n intro k hk\n refine' Finset.le_sup'_of_le _ _ le_rfl\n rw [Finset.mem_range] at hk \u22a2\n exact hk.trans_le (add_le_add_right hij _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : Tendsto ys atTop atTop\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\n\u22a2 Monotone xs\n[PROOFSTEP]\nintro i j hij\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : Tendsto ys atTop atTop\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\ni j : \u2115\nhij : i \u2264 j\n\u22a2 xs i \u2264 xs j\n[PROOFSTEP]\nrw [Finset.sup'_le_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : Tendsto ys atTop atTop\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\ni j : \u2115\nhij : i \u2264 j\n\u22a2 \u2200 (b : \u2115), b \u2208 Finset.range (i + 1) \u2192 ys b \u2264 xs j\n[PROOFSTEP]\nintro k hk\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : Tendsto ys atTop atTop\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\ni j : \u2115\nhij : i \u2264 j\nk : \u2115\nhk : k \u2208 Finset.range (i + 1)\n\u22a2 ys k \u2264 xs j\n[PROOFSTEP]\nrefine' Finset.le_sup'_of_le _ _ le_rfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : Tendsto ys atTop atTop\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\ni j : \u2115\nhij : i \u2264 j\nk : \u2115\nhk : k \u2208 Finset.range (i + 1)\n\u22a2 k \u2208 Finset.range (j + 1)\n[PROOFSTEP]\nrw [Finset.mem_range] at hk \u22a2\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : Tendsto ys atTop atTop\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\ni j : \u2115\nhij : i \u2264 j\nk : \u2115\nhk : k < i + 1\n\u22a2 k < j + 1\n[PROOFSTEP]\nexact hk.trans_le (add_le_add_right hij _)\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : Tendsto ys atTop atTop\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\nh_mono : Monotone xs\n\u22a2 \u2203 xs, Monotone xs \u2227 Tendsto xs atTop atTop\n[PROOFSTEP]\nrefine' \u27e8xs, h_mono, _\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : Tendsto ys atTop atTop\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\nh_mono : Monotone xs\n\u22a2 Tendsto xs atTop atTop\n[PROOFSTEP]\nrefine' tendsto_atTop_atTop_of_monotone h_mono _\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : Tendsto ys atTop atTop\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\nh_mono : Monotone xs\n\u22a2 \u2200 (b : \u03b1), \u2203 a, b \u2264 xs a\n[PROOFSTEP]\nhave : \u2200 a : \u03b1, \u2203 n : \u2115, a \u2264 ys n := by\n rw [tendsto_atTop_atTop] at h \n intro a\n obtain \u27e8i, hi\u27e9 := h a\n exact \u27e8i, hi i le_rfl\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : Tendsto ys atTop atTop\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\nh_mono : Monotone xs\n\u22a2 \u2200 (a : \u03b1), \u2203 n, a \u2264 ys n\n[PROOFSTEP]\nrw [tendsto_atTop_atTop] at h \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : \u2200 (b : \u03b1), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 ys a\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\nh_mono : Monotone xs\n\u22a2 \u2200 (a : \u03b1), \u2203 n, a \u2264 ys n\n[PROOFSTEP]\nintro a\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : \u2200 (b : \u03b1), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 ys a\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\nh_mono : Monotone xs\na : \u03b1\n\u22a2 \u2203 n, a \u2264 ys n\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := h a\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : \u2200 (b : \u03b1), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 ys a\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\nh_mono : Monotone xs\na : \u03b1\ni : \u2115\nhi : \u2200 (a_1 : \u2115), i \u2264 a_1 \u2192 a \u2264 ys a_1\n\u22a2 \u2203 n, a \u2264 ys n\n[PROOFSTEP]\nexact \u27e8i, hi i le_rfl\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : Tendsto ys atTop atTop\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\nh_mono : Monotone xs\nthis : \u2200 (a : \u03b1), \u2203 n, a \u2264 ys n\n\u22a2 \u2200 (b : \u03b1), \u2203 a, b \u2264 xs a\n[PROOFSTEP]\nintro a\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : Tendsto ys atTop atTop\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\nh_mono : Monotone xs\nthis : \u2200 (a : \u03b1), \u2203 n, a \u2264 ys n\na : \u03b1\n\u22a2 \u2203 a_1, a \u2264 xs a_1\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := this a\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : Tendsto ys atTop atTop\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\nh_mono : Monotone xs\nthis : \u2200 (a : \u03b1), \u2203 n, a \u2264 ys n\na : \u03b1\ni : \u2115\nhi : a \u2264 ys i\n\u22a2 \u2203 a_1, a \u2264 xs a_1\n[PROOFSTEP]\nrefine' \u27e8i, hi.trans _\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : Tendsto ys atTop atTop\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\nh_mono : Monotone xs\nthis : \u2200 (a : \u03b1), \u2203 n, a \u2264 ys n\na : \u03b1\ni : \u2115\nhi : a \u2264 ys i\n\u22a2 ys i \u2264 xs i\n[PROOFSTEP]\nrefine' Finset.le_sup'_of_le _ _ le_rfl\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : IsCountablyGenerated atTop\nh_ne_bot : NeBot atTop\nys : \u2115 \u2192 \u03b1\nh : Tendsto ys atTop atTop\nxs : \u2115 \u2192 \u03b1 := fun n => Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) ys\nh_mono : Monotone xs\nthis : \u2200 (a : \u03b1), \u2203 n, a \u2264 ys n\na : \u03b1\ni : \u2115\nhi : a \u2264 ys i\n\u22a2 i \u2208 Finset.range (i + 1)\n[PROOFSTEP]\nrw [Finset.mem_range_succ_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2\nk : Filter \u03b1\nl : Filter \u03b2\ninst\u271d : IsCountablyGenerated k\n\u22a2 Tendsto f k l \u2194 \u2200 (x : \u2115 \u2192 \u03b1), Tendsto x atTop k \u2192 Tendsto (f \u2218 x) atTop l\n[PROOFSTEP]\nrefine' \u27e8fun h x hx => h.comp hx, fun H s hs => _\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2\nk : Filter \u03b1\nl : Filter \u03b2\ninst\u271d : IsCountablyGenerated k\nH : \u2200 (x : \u2115 \u2192 \u03b1), Tendsto x atTop k \u2192 Tendsto (f \u2218 x) atTop l\ns : Set \u03b2\nhs : s \u2208 l\n\u22a2 s \u2208 map f k\n[PROOFSTEP]\ncontrapose! H\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2\nk : Filter \u03b1\nl : Filter \u03b2\ninst\u271d : IsCountablyGenerated k\ns : Set \u03b2\nhs : s \u2208 l\nH : \u00acs \u2208 map f k\n\u22a2 \u2203 x, Tendsto x atTop k \u2227 \u00acTendsto (f \u2218 x) atTop l\n[PROOFSTEP]\nhave : NeBot (k \u2293 \ud835\udcdf (f \u207b\u00b9' s\u1d9c)) := by simpa [neBot_iff, inf_principal_eq_bot]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2\nk : Filter \u03b1\nl : Filter \u03b2\ninst\u271d : IsCountablyGenerated k\ns : Set \u03b2\nhs : s \u2208 l\nH : \u00acs \u2208 map f k\n\u22a2 NeBot (k \u2293 \ud835\udcdf (f \u207b\u00b9' s\u1d9c))\n[PROOFSTEP]\nsimpa [neBot_iff, inf_principal_eq_bot]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2\nk : Filter \u03b1\nl : Filter \u03b2\ninst\u271d : IsCountablyGenerated k\ns : Set \u03b2\nhs : s \u2208 l\nH : \u00acs \u2208 map f k\nthis : NeBot (k \u2293 \ud835\udcdf (f \u207b\u00b9' s\u1d9c))\n\u22a2 \u2203 x, Tendsto x atTop k \u2227 \u00acTendsto (f \u2218 x) atTop l\n[PROOFSTEP]\nrcases(k \u2293 \ud835\udcdf (f \u207b\u00b9' s\u1d9c)).exists_seq_tendsto with \u27e8x, hx\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2\nk : Filter \u03b1\nl : Filter \u03b2\ninst\u271d : IsCountablyGenerated k\ns : Set \u03b2\nhs : s \u2208 l\nH : \u00acs \u2208 map f k\nthis : NeBot (k \u2293 \ud835\udcdf (f \u207b\u00b9' s\u1d9c))\nx : \u2115 \u2192 \u03b1\nhx : Tendsto x atTop (k \u2293 \ud835\udcdf (f \u207b\u00b9' s\u1d9c))\n\u22a2 \u2203 x, Tendsto x atTop k \u2227 \u00acTendsto (f \u2218 x) atTop l\n[PROOFSTEP]\nrw [tendsto_inf, tendsto_principal] at hx \n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2\nk : Filter \u03b1\nl : Filter \u03b2\ninst\u271d : IsCountablyGenerated k\ns : Set \u03b2\nhs : s \u2208 l\nH : \u00acs \u2208 map f k\nthis : NeBot (k \u2293 \ud835\udcdf (f \u207b\u00b9' s\u1d9c))\nx : \u2115 \u2192 \u03b1\nhx : Tendsto x atTop k \u2227 \u2200\u1da0 (a : \u2115) in atTop, x a \u2208 f \u207b\u00b9' s\u1d9c\n\u22a2 \u2203 x, Tendsto x atTop k \u2227 \u00acTendsto (f \u2218 x) atTop l\n[PROOFSTEP]\nrefine' \u27e8x, hx.1, fun h => _\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2\nk : Filter \u03b1\nl : Filter \u03b2\ninst\u271d : IsCountablyGenerated k\ns : Set \u03b2\nhs : s \u2208 l\nH : \u00acs \u2208 map f k\nthis : NeBot (k \u2293 \ud835\udcdf (f \u207b\u00b9' s\u1d9c))\nx : \u2115 \u2192 \u03b1\nhx : Tendsto x atTop k \u2227 \u2200\u1da0 (a : \u2115) in atTop, x a \u2208 f \u207b\u00b9' s\u1d9c\nh : Tendsto (f \u2218 x) atTop l\n\u22a2 False\n[PROOFSTEP]\nrcases(hx.2.and (h hs)).exists with \u27e8N, hnmem, hmem\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2\nk : Filter \u03b1\nl : Filter \u03b2\ninst\u271d : IsCountablyGenerated k\ns : Set \u03b2\nhs : s \u2208 l\nH : \u00acs \u2208 map f k\nthis : NeBot (k \u2293 \ud835\udcdf (f \u207b\u00b9' s\u1d9c))\nx : \u2115 \u2192 \u03b1\nhx : Tendsto x atTop k \u2227 \u2200\u1da0 (a : \u2115) in atTop, x a \u2208 f \u207b\u00b9' s\u1d9c\nh : Tendsto (f \u2218 x) atTop l\nN : \u2115\nhnmem : x N \u2208 f \u207b\u00b9' s\u1d9c\nhmem : (f \u2218 x) N \u2208 s\n\u22a2 False\n[PROOFSTEP]\nexact hnmem hmem\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nx : \u03b9 \u2192 \u03b1\nf : Filter \u03b1\nl : Filter \u03b9\n\u22a2 \u00acTendsto x l f \u2194 \u2203 s, s \u2208 f \u2227 \u2203\u1da0 (n : \u03b9) in l, \u00acx n \u2208 s\n[PROOFSTEP]\nsimp only [tendsto_iff_forall_eventually_mem, not_forall, exists_prop, Filter.Frequently, not_not]\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\n\u22a2 (\u2203\u1da0 (n : \u03b9) in l, p n) \u2194 \u2203 x, Tendsto x atTop l \u2227 \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n[PROOFSTEP]\nrefine' \u27e8fun h_freq => _, fun h_exists_freq => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nh_freq : \u2203\u1da0 (n : \u03b9) in l, p n\n\u22a2 \u2203 x, Tendsto x atTop l \u2227 \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n[PROOFSTEP]\nhave : NeBot (l \u2293 \ud835\udcdf {x : \u03b9 | p x}) := by simpa [neBot_iff, inf_principal_eq_bot]\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nh_freq : \u2203\u1da0 (n : \u03b9) in l, p n\n\u22a2 NeBot (l \u2293 \ud835\udcdf {x | p x})\n[PROOFSTEP]\nsimpa [neBot_iff, inf_principal_eq_bot]\n[GOAL]\ncase refine'_1\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nh_freq : \u2203\u1da0 (n : \u03b9) in l, p n\nthis : NeBot (l \u2293 \ud835\udcdf {x | p x})\n\u22a2 \u2203 x, Tendsto x atTop l \u2227 \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := exists_seq_tendsto (l \u2293 \ud835\udcdf {x : \u03b9 | p x})\n[GOAL]\ncase refine'_1.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nh_freq : \u2203\u1da0 (n : \u03b9) in l, p n\nthis : NeBot (l \u2293 \ud835\udcdf {x | p x})\nx : \u2115 \u2192 \u03b9\nhx : Tendsto x atTop (l \u2293 \ud835\udcdf {x | p x})\n\u22a2 \u2203 x, Tendsto x atTop l \u2227 \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n[PROOFSTEP]\nrw [tendsto_inf] at hx \n[GOAL]\ncase refine'_1.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nh_freq : \u2203\u1da0 (n : \u03b9) in l, p n\nthis : NeBot (l \u2293 \ud835\udcdf {x | p x})\nx : \u2115 \u2192 \u03b9\nhx : Tendsto x atTop l \u2227 Tendsto x atTop (\ud835\udcdf {x | p x})\n\u22a2 \u2203 x, Tendsto x atTop l \u2227 \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n[PROOFSTEP]\ncases' hx with hx_l hx_p\n[GOAL]\ncase refine'_1.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nh_freq : \u2203\u1da0 (n : \u03b9) in l, p n\nthis : NeBot (l \u2293 \ud835\udcdf {x | p x})\nx : \u2115 \u2192 \u03b9\nhx_l : Tendsto x atTop l\nhx_p : Tendsto x atTop (\ud835\udcdf {x | p x})\n\u22a2 \u2203 x, Tendsto x atTop l \u2227 \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n[PROOFSTEP]\nrefine' \u27e8x, hx_l, _\u27e9\n[GOAL]\ncase refine'_1.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nh_freq : \u2203\u1da0 (n : \u03b9) in l, p n\nthis : NeBot (l \u2293 \ud835\udcdf {x | p x})\nx : \u2115 \u2192 \u03b9\nhx_l : Tendsto x atTop l\nhx_p : Tendsto x atTop (\ud835\udcdf {x | p x})\n\u22a2 \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n[PROOFSTEP]\nrw [tendsto_principal] at hx_p \n[GOAL]\ncase refine'_1.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nh_freq : \u2203\u1da0 (n : \u03b9) in l, p n\nthis : NeBot (l \u2293 \ud835\udcdf {x | p x})\nx : \u2115 \u2192 \u03b9\nhx_l : Tendsto x atTop l\nhx_p : \u2200\u1da0 (a : \u2115) in atTop, x a \u2208 {x | p x}\n\u22a2 \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n[PROOFSTEP]\nexact hx_p.frequently\n[GOAL]\ncase refine'_2\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nh_exists_freq : \u2203 x, Tendsto x atTop l \u2227 \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n\u22a2 \u2203\u1da0 (n : \u03b9) in l, p n\n[PROOFSTEP]\nobtain \u27e8x, hx_tendsto, hx_freq\u27e9 := h_exists_freq\n[GOAL]\ncase refine'_2.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nx : \u2115 \u2192 \u03b9\nhx_tendsto : Tendsto x atTop l\nhx_freq : \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n\u22a2 \u2203\u1da0 (n : \u03b9) in l, p n\n[PROOFSTEP]\nsimp_rw [Filter.Frequently, Filter.Eventually] at hx_freq \u22a2\n[GOAL]\ncase refine'_2.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nx : \u2115 \u2192 \u03b9\nhx_tendsto : Tendsto x atTop l\nhx_freq : \u00ac{x_1 | \u00acp (x x_1)} \u2208 atTop\n\u22a2 \u00ac{x | \u00acp x} \u2208 l\n[PROOFSTEP]\nhave : {n : \u2115 | \u00acp (x n)} = {n | x n \u2208 {y | \u00acp y}} := rfl\n[GOAL]\ncase refine'_2.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nx : \u2115 \u2192 \u03b9\nhx_tendsto : Tendsto x atTop l\nhx_freq : \u00ac{x_1 | \u00acp (x x_1)} \u2208 atTop\nthis : {n | \u00acp (x n)} = {n | x n \u2208 {y | \u00acp y}}\n\u22a2 \u00ac{x | \u00acp x} \u2208 l\n[PROOFSTEP]\nrw [this, \u2190 mem_map'] at hx_freq \n[GOAL]\ncase refine'_2.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nx : \u2115 \u2192 \u03b9\nhx_tendsto : Tendsto x atTop l\nhx_freq : \u00ac{y | \u00acp y} \u2208 map (fun n => x n) atTop\nthis : {n | \u00acp (x n)} = {n | x n \u2208 {y | \u00acp y}}\n\u22a2 \u00ac{x | \u00acp x} \u2208 l\n[PROOFSTEP]\nexact mt (@hx_tendsto _) hx_freq\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\n\u22a2 (\u2200\u1da0 (n : \u03b9) in l, p n) \u2194 \u2200 (x : \u2115 \u2192 \u03b9), Tendsto x atTop l \u2192 \u2200\u1da0 (n : \u2115) in atTop, p (x n)\n[PROOFSTEP]\nhave : (\u2200\u1da0 n in l, p n) \u2194 \u00ac\u2203\u1da0 n in l, \u00acp n := by\n rw [not_frequently]\n simp_rw [not_not]\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\n\u22a2 (\u2200\u1da0 (n : \u03b9) in l, p n) \u2194 \u00ac\u2203\u1da0 (n : \u03b9) in l, \u00acp n\n[PROOFSTEP]\nrw [not_frequently]\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\n\u22a2 (\u2200\u1da0 (n : \u03b9) in l, p n) \u2194 \u2200\u1da0 (x : \u03b9) in l, \u00ac\u00acp x\n[PROOFSTEP]\nsimp_rw [not_not]\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nthis : (\u2200\u1da0 (n : \u03b9) in l, p n) \u2194 \u00ac\u2203\u1da0 (n : \u03b9) in l, \u00acp n\n\u22a2 (\u2200\u1da0 (n : \u03b9) in l, p n) \u2194 \u2200 (x : \u2115 \u2192 \u03b9), Tendsto x atTop l \u2192 \u2200\u1da0 (n : \u2115) in atTop, p (x n)\n[PROOFSTEP]\nrw [this, frequently_iff_seq_frequently]\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nthis : (\u2200\u1da0 (n : \u03b9) in l, p n) \u2194 \u00ac\u2203\u1da0 (n : \u03b9) in l, \u00acp n\n\u22a2 (\u00ac\u2203 x, Tendsto x atTop l \u2227 \u2203\u1da0 (n : \u2115) in atTop, \u00acp (x n)) \u2194\n \u2200 (x : \u2115 \u2192 \u03b9), Tendsto x atTop l \u2192 \u2200\u1da0 (n : \u2115) in atTop, p (x n)\n[PROOFSTEP]\npush_neg\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nthis : (\u2200\u1da0 (n : \u03b9) in l, p n) \u2194 \u00ac\u2203\u1da0 (n : \u03b9) in l, \u00acp n\n\u22a2 (\u2200 (x : \u2115 \u2192 \u03b9), Tendsto x atTop l \u2192 \u00ac\u2203\u1da0 (n : \u2115) in atTop, \u00acp (x n)) \u2194\n \u2200 (x : \u2115 \u2192 \u03b9), Tendsto x atTop l \u2192 \u2200\u1da0 (n : \u2115) in atTop, p (x n)\n[PROOFSTEP]\nsimp_rw [not_frequently, not_not]\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nx : \u2115 \u2192 \u03b9\np : \u03b9 \u2192 Prop\nl : Filter \u03b9\nh_tendsto : Tendsto x atTop l\nh : \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n\u22a2 \u2203 ns, Tendsto (fun n => x (ns n)) atTop l \u2227 \u2200 (n : \u2115), p (x (ns n))\n[PROOFSTEP]\nrw [tendsto_iff_seq_tendsto] at h_tendsto \n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nx : \u2115 \u2192 \u03b9\np : \u03b9 \u2192 Prop\nl : Filter \u03b9\nh_tendsto : \u2200 (x_1 : \u2115 \u2192 \u2115), Tendsto x_1 atTop atTop \u2192 Tendsto (x \u2218 x_1) atTop l\nh : \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n\u22a2 \u2203 ns, Tendsto (fun n => x (ns n)) atTop l \u2227 \u2200 (n : \u2115), p (x (ns n))\n[PROOFSTEP]\nchoose ns hge hns using frequently_atTop.1 h\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nx : \u2115 \u2192 \u03b9\np : \u03b9 \u2192 Prop\nl : Filter \u03b9\nh_tendsto : \u2200 (x_1 : \u2115 \u2192 \u2115), Tendsto x_1 atTop atTop \u2192 Tendsto (x \u2218 x_1) atTop l\nh : \u2203\u1da0 (n : \u2115) in atTop, p (x n)\nns : \u2115 \u2192 \u2115\nhge : \u2200 (a : \u2115), ns a \u2265 a\nhns : \u2200 (a : \u2115), p (x (ns a))\n\u22a2 \u2203 ns, Tendsto (fun n => x (ns n)) atTop l \u2227 \u2200 (n : \u2115), p (x (ns n))\n[PROOFSTEP]\nexact \u27e8ns, h_tendsto ns (tendsto_atTop_mono hge tendsto_id), hns\u27e9\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nh : \u2203\u1da0 (n : \u03b9) in l, p n\n\u22a2 \u2203 ns, Tendsto ns atTop l \u2227 \u2200 (n : \u2115), p (ns n)\n[PROOFSTEP]\nrw [frequently_iff_seq_frequently] at h \n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nh : \u2203 x, Tendsto x atTop l \u2227 \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n\u22a2 \u2203 ns, Tendsto ns atTop l \u2227 \u2200 (n : \u2115), p (ns n)\n[PROOFSTEP]\nobtain \u27e8x, hx_tendsto, hx_freq\u27e9 := h\n[GOAL]\ncase intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nx : \u2115 \u2192 \u03b9\nhx_tendsto : Tendsto x atTop l\nhx_freq : \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n\u22a2 \u2203 ns, Tendsto ns atTop l \u2227 \u2200 (n : \u2115), p (ns n)\n[PROOFSTEP]\nobtain \u27e8n_to_n, h_tendsto, h_freq\u27e9 := subseq_forall_of_frequently hx_tendsto hx_freq\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nx : \u2115 \u2192 \u03b9\nhx_tendsto : Tendsto x atTop l\nhx_freq : \u2203\u1da0 (n : \u2115) in atTop, p (x n)\nn_to_n : \u2115 \u2192 \u2115\nh_tendsto : Tendsto (fun n => x (n_to_n n)) atTop l\nh_freq : \u2200 (n : \u2115), p (x (n_to_n n))\n\u22a2 \u2203 ns, Tendsto ns atTop l \u2227 \u2200 (n : \u2115), p (ns n)\n[PROOFSTEP]\nexact \u27e8x \u2218 n_to_n, h_tendsto, h_freq\u27e9\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b9 : Type u_7\nx : \u03b9 \u2192 \u03b1\nf : Filter \u03b1\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nhxy : \u2200 (ns : \u2115 \u2192 \u03b9), Tendsto ns atTop l \u2192 \u2203 ms, Tendsto (fun n => x (ns (ms n))) atTop f\n\u22a2 Tendsto x l f\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b9 : Type u_7\nx : \u03b9 \u2192 \u03b1\nf : Filter \u03b1\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nhxy : \u2200 (ns : \u2115 \u2192 \u03b9), Tendsto ns atTop l \u2192 \u2203 ms, Tendsto (fun n => x (ns (ms n))) atTop f\nh : \u00acTendsto x l f\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8s, hs, hfreq\u27e9 : \u2203 s \u2208 f, \u2203\u1da0 n in l, x n \u2209 s := by rwa [not_tendsto_iff_exists_frequently_nmem] at h \n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b9 : Type u_7\nx : \u03b9 \u2192 \u03b1\nf : Filter \u03b1\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nhxy : \u2200 (ns : \u2115 \u2192 \u03b9), Tendsto ns atTop l \u2192 \u2203 ms, Tendsto (fun n => x (ns (ms n))) atTop f\nh : \u00acTendsto x l f\n\u22a2 \u2203 s, s \u2208 f \u2227 \u2203\u1da0 (n : \u03b9) in l, \u00acx n \u2208 s\n[PROOFSTEP]\nrwa [not_tendsto_iff_exists_frequently_nmem] at h \n[GOAL]\ncase intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b9 : Type u_7\nx : \u03b9 \u2192 \u03b1\nf : Filter \u03b1\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nhxy : \u2200 (ns : \u2115 \u2192 \u03b9), Tendsto ns atTop l \u2192 \u2203 ms, Tendsto (fun n => x (ns (ms n))) atTop f\nh : \u00acTendsto x l f\ns : Set \u03b1\nhs : s \u2208 f\nhfreq : \u2203\u1da0 (n : \u03b9) in l, \u00acx n \u2208 s\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8y, hy_tendsto, hy_freq\u27e9 := exists_seq_forall_of_frequently hfreq\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b9 : Type u_7\nx : \u03b9 \u2192 \u03b1\nf : Filter \u03b1\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nhxy : \u2200 (ns : \u2115 \u2192 \u03b9), Tendsto ns atTop l \u2192 \u2203 ms, Tendsto (fun n => x (ns (ms n))) atTop f\nh : \u00acTendsto x l f\ns : Set \u03b1\nhs : s \u2208 f\nhfreq : \u2203\u1da0 (n : \u03b9) in l, \u00acx n \u2208 s\ny : \u2115 \u2192 \u03b9\nhy_tendsto : Tendsto y atTop l\nhy_freq : \u2200 (n : \u2115), \u00acx (y n) \u2208 s\n\u22a2 False\n[PROOFSTEP]\nspecialize hxy y hy_tendsto\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b9 : Type u_7\nx : \u03b9 \u2192 \u03b1\nf : Filter \u03b1\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nh : \u00acTendsto x l f\ns : Set \u03b1\nhs : s \u2208 f\nhfreq : \u2203\u1da0 (n : \u03b9) in l, \u00acx n \u2208 s\ny : \u2115 \u2192 \u03b9\nhy_tendsto : Tendsto y atTop l\nhy_freq : \u2200 (n : \u2115), \u00acx (y n) \u2208 s\nhxy : \u2203 ms, Tendsto (fun n => x (y (ms n))) atTop f\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8ms, hms_tendsto\u27e9 := hxy\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b9 : Type u_7\nx : \u03b9 \u2192 \u03b1\nf : Filter \u03b1\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nh : \u00acTendsto x l f\ns : Set \u03b1\nhs : s \u2208 f\nhfreq : \u2203\u1da0 (n : \u03b9) in l, \u00acx n \u2208 s\ny : \u2115 \u2192 \u03b9\nhy_tendsto : Tendsto y atTop l\nhy_freq : \u2200 (n : \u2115), \u00acx (y n) \u2208 s\nms : \u2115 \u2192 \u2115\nhms_tendsto : Tendsto (fun n => x (y (ms n))) atTop f\n\u22a2 False\n[PROOFSTEP]\nspecialize hms_tendsto hs\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b9 : Type u_7\nx : \u03b9 \u2192 \u03b1\nf : Filter \u03b1\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nh : \u00acTendsto x l f\ns : Set \u03b1\nhs : s \u2208 f\nhfreq : \u2203\u1da0 (n : \u03b9) in l, \u00acx n \u2208 s\ny : \u2115 \u2192 \u03b9\nhy_tendsto : Tendsto y atTop l\nhy_freq : \u2200 (n : \u2115), \u00acx (y n) \u2208 s\nms : \u2115 \u2192 \u2115\nhms_tendsto : s \u2208 map (fun n => x (y (ms n))) atTop\n\u22a2 False\n[PROOFSTEP]\nrw [mem_map] at hms_tendsto \n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b9 : Type u_7\nx : \u03b9 \u2192 \u03b1\nf : Filter \u03b1\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nh : \u00acTendsto x l f\ns : Set \u03b1\nhs : s \u2208 f\nhfreq : \u2203\u1da0 (n : \u03b9) in l, \u00acx n \u2208 s\ny : \u2115 \u2192 \u03b9\nhy_tendsto : Tendsto y atTop l\nhy_freq : \u2200 (n : \u2115), \u00acx (y n) \u2208 s\nms : \u2115 \u2192 \u2115\nhms_tendsto : (fun n => x (y (ms n))) \u207b\u00b9' s \u2208 atTop\n\u22a2 False\n[PROOFSTEP]\nhave hms_freq : \u2200 n : \u2115, x (y (ms n)) \u2209 s := fun n => hy_freq (ms n)\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b9 : Type u_7\nx : \u03b9 \u2192 \u03b1\nf : Filter \u03b1\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nh : \u00acTendsto x l f\ns : Set \u03b1\nhs : s \u2208 f\nhfreq : \u2203\u1da0 (n : \u03b9) in l, \u00acx n \u2208 s\ny : \u2115 \u2192 \u03b9\nhy_tendsto : Tendsto y atTop l\nhy_freq : \u2200 (n : \u2115), \u00acx (y n) \u2208 s\nms : \u2115 \u2192 \u2115\nhms_tendsto : (fun n => x (y (ms n))) \u207b\u00b9' s \u2208 atTop\nhms_freq : \u2200 (n : \u2115), \u00acx (y (ms n)) \u2208 s\n\u22a2 False\n[PROOFSTEP]\nhave h_empty : (fun n : \u2115 => x (y (ms n))) \u207b\u00b9' s = \u2205 := by\n ext1 n\n simp only [Set.mem_preimage, Set.mem_empty_iff_false, iff_false_iff]\n exact hms_freq n\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b9 : Type u_7\nx : \u03b9 \u2192 \u03b1\nf : Filter \u03b1\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nh : \u00acTendsto x l f\ns : Set \u03b1\nhs : s \u2208 f\nhfreq : \u2203\u1da0 (n : \u03b9) in l, \u00acx n \u2208 s\ny : \u2115 \u2192 \u03b9\nhy_tendsto : Tendsto y atTop l\nhy_freq : \u2200 (n : \u2115), \u00acx (y n) \u2208 s\nms : \u2115 \u2192 \u2115\nhms_tendsto : (fun n => x (y (ms n))) \u207b\u00b9' s \u2208 atTop\nhms_freq : \u2200 (n : \u2115), \u00acx (y (ms n)) \u2208 s\n\u22a2 (fun n => x (y (ms n))) \u207b\u00b9' s = \u2205\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b9 : Type u_7\nx : \u03b9 \u2192 \u03b1\nf : Filter \u03b1\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nh : \u00acTendsto x l f\ns : Set \u03b1\nhs : s \u2208 f\nhfreq : \u2203\u1da0 (n : \u03b9) in l, \u00acx n \u2208 s\ny : \u2115 \u2192 \u03b9\nhy_tendsto : Tendsto y atTop l\nhy_freq : \u2200 (n : \u2115), \u00acx (y n) \u2208 s\nms : \u2115 \u2192 \u2115\nhms_tendsto : (fun n => x (y (ms n))) \u207b\u00b9' s \u2208 atTop\nhms_freq : \u2200 (n : \u2115), \u00acx (y (ms n)) \u2208 s\nn : \u2115\n\u22a2 n \u2208 (fun n => x (y (ms n))) \u207b\u00b9' s \u2194 n \u2208 \u2205\n[PROOFSTEP]\nsimp only [Set.mem_preimage, Set.mem_empty_iff_false, iff_false_iff]\n[GOAL]\ncase h\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b9 : Type u_7\nx : \u03b9 \u2192 \u03b1\nf : Filter \u03b1\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nh : \u00acTendsto x l f\ns : Set \u03b1\nhs : s \u2208 f\nhfreq : \u2203\u1da0 (n : \u03b9) in l, \u00acx n \u2208 s\ny : \u2115 \u2192 \u03b9\nhy_tendsto : Tendsto y atTop l\nhy_freq : \u2200 (n : \u2115), \u00acx (y n) \u2208 s\nms : \u2115 \u2192 \u2115\nhms_tendsto : (fun n => x (y (ms n))) \u207b\u00b9' s \u2208 atTop\nhms_freq : \u2200 (n : \u2115), \u00acx (y (ms n)) \u2208 s\nn : \u2115\n\u22a2 \u00acx (y (ms n)) \u2208 s\n[PROOFSTEP]\nexact hms_freq n\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b9 : Type u_7\nx : \u03b9 \u2192 \u03b1\nf : Filter \u03b1\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nh : \u00acTendsto x l f\ns : Set \u03b1\nhs : s \u2208 f\nhfreq : \u2203\u1da0 (n : \u03b9) in l, \u00acx n \u2208 s\ny : \u2115 \u2192 \u03b9\nhy_tendsto : Tendsto y atTop l\nhy_freq : \u2200 (n : \u2115), \u00acx (y n) \u2208 s\nms : \u2115 \u2192 \u2115\nhms_tendsto : (fun n => x (y (ms n))) \u207b\u00b9' s \u2208 atTop\nhms_freq : \u2200 (n : \u2115), \u00acx (y (ms n)) \u2208 s\nh_empty : (fun n => x (y (ms n))) \u207b\u00b9' s = \u2205\n\u22a2 False\n[PROOFSTEP]\nrw [h_empty] at hms_tendsto \n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b1 : Type u_6\n\u03b9 : Type u_7\nx : \u03b9 \u2192 \u03b1\nf : Filter \u03b1\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nh : \u00acTendsto x l f\ns : Set \u03b1\nhs : s \u2208 f\nhfreq : \u2203\u1da0 (n : \u03b9) in l, \u00acx n \u2208 s\ny : \u2115 \u2192 \u03b9\nhy_tendsto : Tendsto y atTop l\nhy_freq : \u2200 (n : \u2115), \u00acx (y n) \u2208 s\nms : \u2115 \u2192 \u2115\nhms_tendsto : \u2205 \u2208 atTop\nhms_freq : \u2200 (n : \u2115), \u00acx (y (ms n)) \u2208 s\nh_empty : (fun n => x (y (ms n))) \u207b\u00b9' s = \u2205\n\u22a2 False\n[PROOFSTEP]\nexact empty_not_mem atTop hms_tendsto\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ninst\u271d : IsCountablyGenerated f\nu : \u2115 \u2192 \u03b1\nhx : NeBot (f \u2293 map u atTop)\n\u22a2 \u2203 \u03b8, StrictMono \u03b8 \u2227 Tendsto (u \u2218 \u03b8) atTop f\n[PROOFSTEP]\nobtain \u27e8B, h\u27e9 := f.exists_antitone_basis\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ninst\u271d : IsCountablyGenerated f\nu : \u2115 \u2192 \u03b1\nhx : NeBot (f \u2293 map u atTop)\nB : \u2115 \u2192 Set \u03b1\nh : HasAntitoneBasis f B\n\u22a2 \u2203 \u03b8, StrictMono \u03b8 \u2227 Tendsto (u \u2218 \u03b8) atTop f\n[PROOFSTEP]\nhave : \u2200 N, \u2203 n \u2265 N, u n \u2208 B N := fun N => Filter.inf_map_atTop_neBot_iff.mp hx _ (h.1.mem_of_mem trivial) N\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ninst\u271d : IsCountablyGenerated f\nu : \u2115 \u2192 \u03b1\nhx : NeBot (f \u2293 map u atTop)\nB : \u2115 \u2192 Set \u03b1\nh : HasAntitoneBasis f B\nthis : \u2200 (N : \u2115), \u2203 n, n \u2265 N \u2227 u n \u2208 B N\n\u22a2 \u2203 \u03b8, StrictMono \u03b8 \u2227 Tendsto (u \u2218 \u03b8) atTop f\n[PROOFSTEP]\nchoose \u03c6 h\u03c6 using this\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ninst\u271d : IsCountablyGenerated f\nu : \u2115 \u2192 \u03b1\nhx : NeBot (f \u2293 map u atTop)\nB : \u2115 \u2192 Set \u03b1\nh : HasAntitoneBasis f B\n\u03c6 : \u2115 \u2192 \u2115\nh\u03c6 : \u2200 (N : \u2115), \u03c6 N \u2265 N \u2227 u (\u03c6 N) \u2208 B N\n\u22a2 \u2203 \u03b8, StrictMono \u03b8 \u2227 Tendsto (u \u2218 \u03b8) atTop f\n[PROOFSTEP]\ncases' forall_and.mp h\u03c6 with \u03c6_ge \u03c6_in\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ninst\u271d : IsCountablyGenerated f\nu : \u2115 \u2192 \u03b1\nhx : NeBot (f \u2293 map u atTop)\nB : \u2115 \u2192 Set \u03b1\nh : HasAntitoneBasis f B\n\u03c6 : \u2115 \u2192 \u2115\nh\u03c6 : \u2200 (N : \u2115), \u03c6 N \u2265 N \u2227 u (\u03c6 N) \u2208 B N\n\u03c6_ge : \u2200 (x : \u2115), \u03c6 x \u2265 x\n\u03c6_in : \u2200 (x : \u2115), u (\u03c6 x) \u2208 B x\n\u22a2 \u2203 \u03b8, StrictMono \u03b8 \u2227 Tendsto (u \u2218 \u03b8) atTop f\n[PROOFSTEP]\nhave lim_u\u03c6 : Tendsto (u \u2218 \u03c6) atTop f := h.tendsto \u03c6_in\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ninst\u271d : IsCountablyGenerated f\nu : \u2115 \u2192 \u03b1\nhx : NeBot (f \u2293 map u atTop)\nB : \u2115 \u2192 Set \u03b1\nh : HasAntitoneBasis f B\n\u03c6 : \u2115 \u2192 \u2115\nh\u03c6 : \u2200 (N : \u2115), \u03c6 N \u2265 N \u2227 u (\u03c6 N) \u2208 B N\n\u03c6_ge : \u2200 (x : \u2115), \u03c6 x \u2265 x\n\u03c6_in : \u2200 (x : \u2115), u (\u03c6 x) \u2208 B x\nlim_u\u03c6 : Tendsto (u \u2218 \u03c6) atTop f\n\u22a2 \u2203 \u03b8, StrictMono \u03b8 \u2227 Tendsto (u \u2218 \u03b8) atTop f\n[PROOFSTEP]\nhave lim_\u03c6 : Tendsto \u03c6 atTop atTop := tendsto_atTop_mono \u03c6_ge tendsto_id\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ninst\u271d : IsCountablyGenerated f\nu : \u2115 \u2192 \u03b1\nhx : NeBot (f \u2293 map u atTop)\nB : \u2115 \u2192 Set \u03b1\nh : HasAntitoneBasis f B\n\u03c6 : \u2115 \u2192 \u2115\nh\u03c6 : \u2200 (N : \u2115), \u03c6 N \u2265 N \u2227 u (\u03c6 N) \u2208 B N\n\u03c6_ge : \u2200 (x : \u2115), \u03c6 x \u2265 x\n\u03c6_in : \u2200 (x : \u2115), u (\u03c6 x) \u2208 B x\nlim_u\u03c6 : Tendsto (u \u2218 \u03c6) atTop f\nlim_\u03c6 : Tendsto \u03c6 atTop atTop\n\u22a2 \u2203 \u03b8, StrictMono \u03b8 \u2227 Tendsto (u \u2218 \u03b8) atTop f\n[PROOFSTEP]\nobtain \u27e8\u03c8, h\u03c8, h\u03c8\u03c6\u27e9 : \u2203 \u03c8 : \u2115 \u2192 \u2115, StrictMono \u03c8 \u2227 StrictMono (\u03c6 \u2218 \u03c8)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ninst\u271d : IsCountablyGenerated f\nu : \u2115 \u2192 \u03b1\nhx : NeBot (f \u2293 map u atTop)\nB : \u2115 \u2192 Set \u03b1\nh : HasAntitoneBasis f B\n\u03c6 : \u2115 \u2192 \u2115\nh\u03c6 : \u2200 (N : \u2115), \u03c6 N \u2265 N \u2227 u (\u03c6 N) \u2208 B N\n\u03c6_ge : \u2200 (x : \u2115), \u03c6 x \u2265 x\n\u03c6_in : \u2200 (x : \u2115), u (\u03c6 x) \u2208 B x\nlim_u\u03c6 : Tendsto (u \u2218 \u03c6) atTop f\nlim_\u03c6 : Tendsto \u03c6 atTop atTop\n\u22a2 \u2203 \u03c8, StrictMono \u03c8 \u2227 StrictMono (\u03c6 \u2218 \u03c8)\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ninst\u271d : IsCountablyGenerated f\nu : \u2115 \u2192 \u03b1\nhx : NeBot (f \u2293 map u atTop)\nB : \u2115 \u2192 Set \u03b1\nh : HasAntitoneBasis f B\n\u03c6 : \u2115 \u2192 \u2115\nh\u03c6 : \u2200 (N : \u2115), \u03c6 N \u2265 N \u2227 u (\u03c6 N) \u2208 B N\n\u03c6_ge : \u2200 (x : \u2115), \u03c6 x \u2265 x\n\u03c6_in : \u2200 (x : \u2115), u (\u03c6 x) \u2208 B x\nlim_u\u03c6 : Tendsto (u \u2218 \u03c6) atTop f\nlim_\u03c6 : Tendsto \u03c6 atTop atTop\n\u03c8 : \u2115 \u2192 \u2115\nh\u03c8 : StrictMono \u03c8\nh\u03c8\u03c6 : StrictMono (\u03c6 \u2218 \u03c8)\n\u22a2 \u2203 \u03b8, StrictMono \u03b8 \u2227 Tendsto (u \u2218 \u03b8) atTop f\n[PROOFSTEP]\nexact strictMono_subseq_of_tendsto_atTop lim_\u03c6\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : Filter \u03b1\ninst\u271d : IsCountablyGenerated f\nu : \u2115 \u2192 \u03b1\nhx : NeBot (f \u2293 map u atTop)\nB : \u2115 \u2192 Set \u03b1\nh : HasAntitoneBasis f B\n\u03c6 : \u2115 \u2192 \u2115\nh\u03c6 : \u2200 (N : \u2115), \u03c6 N \u2265 N \u2227 u (\u03c6 N) \u2208 B N\n\u03c6_ge : \u2200 (x : \u2115), \u03c6 x \u2265 x\n\u03c6_in : \u2200 (x : \u2115), u (\u03c6 x) \u2208 B x\nlim_u\u03c6 : Tendsto (u \u2218 \u03c6) atTop f\nlim_\u03c6 : Tendsto \u03c6 atTop atTop\n\u03c8 : \u2115 \u2192 \u2115\nh\u03c8 : StrictMono \u03c8\nh\u03c8\u03c6 : StrictMono (\u03c6 \u2218 \u03c8)\n\u22a2 \u2203 \u03b8, StrictMono \u03b8 \u2227 Tendsto (u \u2218 \u03b8) atTop f\n[PROOFSTEP]\nexact \u27e8\u03c6 \u2218 \u03c8, h\u03c8\u03c6, lim_u\u03c6.comp h\u03c8.tendsto_atTop\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b3 : Type u_5\n\u03b2 : \u03b1 \u2192 Type u_6\ninst\u271d\u00b2 : Preorder \u03b9\ns : \u03b9 \u2192 Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidablePred fun x => x \u2208 s i\ninst\u271d : DecidablePred fun x => x \u2208 \u22c3 (i : \u03b9), s i\nhs : Monotone s\nf g : (a : \u03b1) \u2192 \u03b2 a\na : \u03b1\n\u22a2 \u2200\u1da0 (i : \u03b9) in atTop, Set.piecewise (s i) f g a = Set.piecewise (\u22c3 (i : \u03b9), s i) f g a\n[PROOFSTEP]\nrcases em (\u2203 i, a \u2208 s i) with \u27e8i, hi\u27e9 | ha\n[GOAL]\ncase inl.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b3 : Type u_5\n\u03b2 : \u03b1 \u2192 Type u_6\ninst\u271d\u00b2 : Preorder \u03b9\ns : \u03b9 \u2192 Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidablePred fun x => x \u2208 s i\ninst\u271d : DecidablePred fun x => x \u2208 \u22c3 (i : \u03b9), s i\nhs : Monotone s\nf g : (a : \u03b1) \u2192 \u03b2 a\na : \u03b1\ni : \u03b9\nhi : a \u2208 s i\n\u22a2 \u2200\u1da0 (i : \u03b9) in atTop, Set.piecewise (s i) f g a = Set.piecewise (\u22c3 (i : \u03b9), s i) f g a\n[PROOFSTEP]\nrefine (eventually_ge_atTop i).mono fun j hij \u21a6 ?_\n[GOAL]\ncase inl.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b3 : Type u_5\n\u03b2 : \u03b1 \u2192 Type u_6\ninst\u271d\u00b2 : Preorder \u03b9\ns : \u03b9 \u2192 Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidablePred fun x => x \u2208 s i\ninst\u271d : DecidablePred fun x => x \u2208 \u22c3 (i : \u03b9), s i\nhs : Monotone s\nf g : (a : \u03b1) \u2192 \u03b2 a\na : \u03b1\ni : \u03b9\nhi : a \u2208 s i\nj : \u03b9\nhij : i \u2264 j\n\u22a2 Set.piecewise (s j) f g a = Set.piecewise (\u22c3 (i : \u03b9), s i) f g a\n[PROOFSTEP]\nsimp only [Set.piecewise_eq_of_mem, hs hij hi, subset_iUnion _ _ hi]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b3 : Type u_5\n\u03b2 : \u03b1 \u2192 Type u_6\ninst\u271d\u00b2 : Preorder \u03b9\ns : \u03b9 \u2192 Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidablePred fun x => x \u2208 s i\ninst\u271d : DecidablePred fun x => x \u2208 \u22c3 (i : \u03b9), s i\nhs : Monotone s\nf g : (a : \u03b1) \u2192 \u03b2 a\na : \u03b1\nha : \u00ac\u2203 i, a \u2208 s i\n\u22a2 \u2200\u1da0 (i : \u03b9) in atTop, Set.piecewise (s i) f g a = Set.piecewise (\u22c3 (i : \u03b9), s i) f g a\n[PROOFSTEP]\nrefine eventually_of_forall fun i \u21a6 ?_\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b3 : Type u_5\n\u03b2 : \u03b1 \u2192 Type u_6\ninst\u271d\u00b2 : Preorder \u03b9\ns : \u03b9 \u2192 Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidablePred fun x => x \u2208 s i\ninst\u271d : DecidablePred fun x => x \u2208 \u22c3 (i : \u03b9), s i\nhs : Monotone s\nf g : (a : \u03b1) \u2192 \u03b2 a\na : \u03b1\nha : \u00ac\u2203 i, a \u2208 s i\ni : \u03b9\n\u22a2 Set.piecewise (s i) f g a = Set.piecewise (\u22c3 (i : \u03b9), s i) f g a\n[PROOFSTEP]\nsimp only [Set.piecewise_eq_of_not_mem, not_exists.1 ha i, mt mem_iUnion.1 ha]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b3 : Type u_5\n\u03b2 : \u03b1 \u2192 Type u_6\ninst\u271d\u00b2 : Preorder \u03b9\ns : \u03b9 \u2192 Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidablePred fun x => x \u2208 s i\ninst\u271d : DecidablePred fun x => x \u2208 \u22c2 (i : \u03b9), s i\nhs : Antitone s\nf g : (a : \u03b1) \u2192 \u03b2 a\na : \u03b1\n\u22a2 \u2200\u1da0 (i : \u03b9) in atTop, Set.piecewise (s i) f g a = Set.piecewise (\u22c2 (i : \u03b9), s i) f g a\n[PROOFSTEP]\nclassical\nconvert \u2190 (compl_anti.comp hs).piecewise_eventually_eq_iUnion g f a using 3\n\u00b7 convert congr_fun (Set.piecewise_compl (s _) g f) a\n\u00b7 simp only [(\u00b7 \u2218 \u00b7), \u2190 compl_iInter, Set.piecewise_compl]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b3 : Type u_5\n\u03b2 : \u03b1 \u2192 Type u_6\ninst\u271d\u00b2 : Preorder \u03b9\ns : \u03b9 \u2192 Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidablePred fun x => x \u2208 s i\ninst\u271d : DecidablePred fun x => x \u2208 \u22c2 (i : \u03b9), s i\nhs : Antitone s\nf g : (a : \u03b1) \u2192 \u03b2 a\na : \u03b1\n\u22a2 \u2200\u1da0 (i : \u03b9) in atTop, Set.piecewise (s i) f g a = Set.piecewise (\u22c2 (i : \u03b9), s i) f g a\n[PROOFSTEP]\nconvert \u2190 (compl_anti.comp hs).piecewise_eventually_eq_iUnion g f a using 3\n[GOAL]\ncase h.e'_2.h.h.e'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b3 : Type u_5\n\u03b2 : \u03b1 \u2192 Type u_6\ninst\u271d\u00b2 : Preorder \u03b9\ns : \u03b9 \u2192 Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidablePred fun x => x \u2208 s i\ninst\u271d : DecidablePred fun x => x \u2208 \u22c2 (i : \u03b9), s i\nhs : Antitone s\nf g : (a : \u03b1) \u2192 \u03b2 a\na : \u03b1\nx\u271d : \u03b9\n\u22a2 Set.piecewise ((compl \u2218 s) x\u271d) g f a = Set.piecewise (s x\u271d) f g a\n[PROOFSTEP]\nconvert congr_fun (Set.piecewise_compl (s _) g f) a\n[GOAL]\ncase h.e'_2.h.h.e'_3\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b3 : Type u_5\n\u03b2 : \u03b1 \u2192 Type u_6\ninst\u271d\u00b2 : Preorder \u03b9\ns : \u03b9 \u2192 Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidablePred fun x => x \u2208 s i\ninst\u271d : DecidablePred fun x => x \u2208 \u22c2 (i : \u03b9), s i\nhs : Antitone s\nf g : (a : \u03b1) \u2192 \u03b2 a\na : \u03b1\nx\u271d : \u03b9\n\u22a2 Set.piecewise (\u22c3 (i : \u03b9), (compl \u2218 s) i) g f a = Set.piecewise (\u22c2 (i : \u03b9), s i) f g a\n[PROOFSTEP]\nsimp only [(\u00b7 \u2218 \u00b7), \u2190 compl_iInter, Set.piecewise_compl]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\ng : \u03b3 \u2192 \u03b2\nhg : Injective g\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 (x : \u03b2), \u00acx \u2208 Set.range g \u2192 f x = 1\n\u22a2 Filter.map (fun s => \u220f i in s, f (g i)) atTop = Filter.map (fun s => \u220f i in s, f i) atTop\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b2\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\ng : \u03b3 \u2192 \u03b2\nhg : Injective g\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 (x : \u03b2), \u00acx \u2208 Set.range g \u2192 f x = 1\nthis : DecidableEq \u03b2\n\u22a2 Filter.map (fun s => \u220f i in s, f (g i)) atTop = Filter.map (fun s => \u220f i in s, f i) atTop\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\ng : \u03b3 \u2192 \u03b2\nhg : Injective g\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 (x : \u03b2), \u00acx \u2208 Set.range g \u2192 f x = 1\nthis : DecidableEq \u03b2\n\u22a2 Filter.map (fun s => \u220f i in s, f (g i)) atTop \u2264 Filter.map (fun s => \u220f i in s, f i) atTop\n[PROOFSTEP]\nrefine' map_atTop_finset_prod_le_of_prod_eq fun s => _\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\ng : \u03b3 \u2192 \u03b2\nhg : Injective g\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 (x : \u03b2), \u00acx \u2208 Set.range g \u2192 f x = 1\nthis : DecidableEq \u03b2\n\u22a2 Filter.map (fun s => \u220f i in s, f i) atTop \u2264 Filter.map (fun s => \u220f i in s, f (g i)) atTop\n[PROOFSTEP]\nrefine' map_atTop_finset_prod_le_of_prod_eq fun s => _\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\ng : \u03b3 \u2192 \u03b2\nhg : Injective g\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 (x : \u03b2), \u00acx \u2208 Set.range g \u2192 f x = 1\nthis : DecidableEq \u03b2\ns : Finset \u03b2\n\u22a2 \u2203 v, \u2200 (v' : Finset \u03b3), v \u2286 v' \u2192 \u2203 u', s \u2286 u' \u2227 \u220f x in u', f x = \u220f b in v', f (g b)\n[PROOFSTEP]\nrefine' \u27e8s.preimage g (hg.injOn _), fun t ht => _\u27e9\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\ng : \u03b3 \u2192 \u03b2\nhg : Injective g\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 (x : \u03b2), \u00acx \u2208 Set.range g \u2192 f x = 1\nthis : DecidableEq \u03b2\ns : Finset \u03b2\nt : Finset \u03b3\nht : Finset.preimage s g (_ : InjOn g (g \u207b\u00b9' \u2191s)) \u2286 t\n\u22a2 \u2203 u', s \u2286 u' \u2227 \u220f x in u', f x = \u220f b in t, f (g b)\n[PROOFSTEP]\nrefine' \u27e8t.image g \u222a s, Finset.subset_union_right _ _, _\u27e9\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\ng : \u03b3 \u2192 \u03b2\nhg : Injective g\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 (x : \u03b2), \u00acx \u2208 Set.range g \u2192 f x = 1\nthis : DecidableEq \u03b2\ns : Finset \u03b2\nt : Finset \u03b3\nht : Finset.preimage s g (_ : InjOn g (g \u207b\u00b9' \u2191s)) \u2286 t\n\u22a2 \u220f x in Finset.image g t \u222a s, f x = \u220f b in t, f (g b)\n[PROOFSTEP]\nrw [\u2190 Finset.prod_image (hg.injOn _)]\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\ng : \u03b3 \u2192 \u03b2\nhg : Injective g\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 (x : \u03b2), \u00acx \u2208 Set.range g \u2192 f x = 1\nthis : DecidableEq \u03b2\ns : Finset \u03b2\nt : Finset \u03b3\nht : Finset.preimage s g (_ : InjOn g (g \u207b\u00b9' \u2191s)) \u2286 t\n\u22a2 \u220f x in Finset.image g t \u222a s, f x = \u220f x in Finset.image (fun x => g x) t, f x\n[PROOFSTEP]\nrefine' (prod_subset (subset_union_left _ _) _).symm\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\ng : \u03b3 \u2192 \u03b2\nhg : Injective g\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 (x : \u03b2), \u00acx \u2208 Set.range g \u2192 f x = 1\nthis : DecidableEq \u03b2\ns : Finset \u03b2\nt : Finset \u03b3\nht : Finset.preimage s g (_ : InjOn g (g \u207b\u00b9' \u2191s)) \u2286 t\n\u22a2 \u2200 (x : \u03b2), x \u2208 Finset.image g t \u222a s \u2192 \u00acx \u2208 Finset.image g t \u2192 f x = 1\n[PROOFSTEP]\nsimp only [Finset.mem_union, Finset.mem_image]\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\ng : \u03b3 \u2192 \u03b2\nhg : Injective g\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 (x : \u03b2), \u00acx \u2208 Set.range g \u2192 f x = 1\nthis : DecidableEq \u03b2\ns : Finset \u03b2\nt : Finset \u03b3\nht : Finset.preimage s g (_ : InjOn g (g \u207b\u00b9' \u2191s)) \u2286 t\n\u22a2 \u2200 (x : \u03b2), (\u2203 a, a \u2208 t \u2227 g a = x) \u2228 x \u2208 s \u2192 (\u00ac\u2203 a, a \u2208 t \u2227 g a = x) \u2192 f x = 1\n[PROOFSTEP]\nrefine' fun y hy hyt => hf y (mt _ hyt)\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\ng : \u03b3 \u2192 \u03b2\nhg : Injective g\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 (x : \u03b2), \u00acx \u2208 Set.range g \u2192 f x = 1\nthis : DecidableEq \u03b2\ns : Finset \u03b2\nt : Finset \u03b3\nht : Finset.preimage s g (_ : InjOn g (g \u207b\u00b9' \u2191s)) \u2286 t\ny : \u03b2\nhy : (\u2203 a, a \u2208 t \u2227 g a = y) \u2228 y \u2208 s\nhyt : \u00ac\u2203 a, a \u2208 t \u2227 g a = y\n\u22a2 y \u2208 Set.range g \u2192 \u2203 a, a \u2208 t \u2227 g a = y\n[PROOFSTEP]\nrintro \u27e8x, rfl\u27e9\n[GOAL]\ncase a.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\ng : \u03b3 \u2192 \u03b2\nhg : Injective g\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 (x : \u03b2), \u00acx \u2208 Set.range g \u2192 f x = 1\nthis : DecidableEq \u03b2\ns : Finset \u03b2\nt : Finset \u03b3\nht : Finset.preimage s g (_ : InjOn g (g \u207b\u00b9' \u2191s)) \u2286 t\nx : \u03b3\nhy : (\u2203 a, a \u2208 t \u2227 g a = g x) \u2228 g x \u2208 s\nhyt : \u00ac\u2203 a, a \u2208 t \u2227 g a = g x\n\u22a2 \u2203 a, a \u2208 t \u2227 g a = g x\n[PROOFSTEP]\nexact \u27e8x, ht (Finset.mem_preimage.2 <| hy.resolve_left hyt), rfl\u27e9\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\ng : \u03b3 \u2192 \u03b2\nhg : Injective g\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 (x : \u03b2), \u00acx \u2208 Set.range g \u2192 f x = 1\nthis : DecidableEq \u03b2\ns : Finset \u03b3\n\u22a2 \u2203 v, \u2200 (v' : Finset \u03b2), v \u2286 v' \u2192 \u2203 u', s \u2286 u' \u2227 \u220f x in u', f (g x) = \u220f b in v', f b\n[PROOFSTEP]\nrefine' \u27e8s.image g, fun t ht => _\u27e9\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\ng : \u03b3 \u2192 \u03b2\nhg : Injective g\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 (x : \u03b2), \u00acx \u2208 Set.range g \u2192 f x = 1\nthis : DecidableEq \u03b2\ns : Finset \u03b3\nt : Finset \u03b2\nht : Finset.image g s \u2286 t\n\u22a2 \u2203 u', s \u2286 u' \u2227 \u220f x in u', f (g x) = \u220f b in t, f b\n[PROOFSTEP]\nsimp only [\u2190 prod_preimage _ _ (hg.injOn _) _ fun x _ => hf x]\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b1\ng : \u03b3 \u2192 \u03b2\nhg : Injective g\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 (x : \u03b2), \u00acx \u2208 Set.range g \u2192 f x = 1\nthis : DecidableEq \u03b2\ns : Finset \u03b3\nt : Finset \u03b2\nht : Finset.image g s \u2286 t\n\u22a2 \u2203 u', s \u2286 u' \u2227 \u220f x in u', f (g x) = \u220f x in Finset.preimage t g (_ : InjOn g (g \u207b\u00b9' \u2191t)), f (g x)\n[PROOFSTEP]\nexact \u27e8_, (image_subset_iff_subset_preimage _).1 ht, rfl\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Order.Filter.AtTopBot", "llama_tokens": 62102, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5544704502361149, "lm_q2_score": 0.44552953503957277, "lm_q1q2_score": 0.24703296188687882}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\n\u22a2 HasSum (fun x => 0) 0\n[PROOFSTEP]\nsimp [HasSum, tendsto_const_nhds]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\ninst\u271d : IsEmpty \u03b2\n\u22a2 HasSum f 0\n[PROOFSTEP]\nconvert @hasSum_zero \u03b1 \u03b2 _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\nh : \u00acSummable f\n\u22a2 \u2211' (b : \u03b2), f b = 0\n[PROOFSTEP]\nsimp [tsum_def, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g\u271d : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\ng : \u03b3 \u2192 \u03b2\nhg : Injective g\nhf : \u2200 (x : \u03b2), \u00acx \u2208 Set.range g \u2192 f x = 0\n\u22a2 HasSum (f \u2218 g) a \u2194 HasSum f a\n[PROOFSTEP]\nsimp only [HasSum, Tendsto, comp_apply, hg.map_atTop_finset_sum_eq hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g\u271d : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\ng : \u03b2 \u2192 \u03b3\nhg : Injective g\n\u22a2 HasSum (extend g f 0) a \u2194 HasSum f a\n[PROOFSTEP]\nrw [\u2190 hg.hasSum_iff, extend_comp hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g\u271d : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\ng : \u03b2 \u2192 \u03b3\nhg : Injective g\n\u22a2 \u2200 (x : \u03b3), \u00acx \u2208 Set.range g \u2192 extend g f 0 x = 0\n[PROOFSTEP]\nexact extend_apply' _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns\u271d : Finset \u03b2\ns : Set \u03b2\nhf : support f \u2286 s\n\u22a2 \u2200 (x : \u03b2), (\u00acx \u2208 Set.range fun a => \u2191a) \u2192 f x = 0\n[PROOFSTEP]\nsimpa using support_subset_iff'.1 hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns\u271d : Finset \u03b2\ns : Set \u03b2\n\u22a2 HasSum (f \u2218 Subtype.val) a \u2194 HasSum (Set.indicator s f) a\n[PROOFSTEP]\nrw [\u2190 Set.indicator_range_comp, Subtype.range_coe, hasSum_subtype_iff_of_support_subset Set.support_indicator_subset]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns\u271d s : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 HasSum (f \u2218 Subtype.val) (\u2211 b in s, f b)\n[PROOFSTEP]\nrw [\u2190 sum_attach]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns\u271d s : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 HasSum (f \u2218 Subtype.val) (\u2211 x in attach s, f \u2191x)\n[PROOFSTEP]\nexact hasSum_fintype _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns\u271d : Finset \u03b2\ns : Set \u03b2\nhs : Set.Finite s\nf : \u03b2 \u2192 \u03b1\n\u22a2 Summable (f \u2218 Subtype.val)\n[PROOFSTEP]\nhave := hs.toFinset.summable f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns\u271d : Finset \u03b2\ns : Set \u03b2\nhs : Set.Finite s\nf : \u03b2 \u2192 \u03b1\nthis : Summable (f \u2218 Subtype.val)\n\u22a2 Summable (f \u2218 Subtype.val)\n[PROOFSTEP]\nrwa [hs.coe_toFinset] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\nh : Set.Finite (support f)\n\u22a2 Summable f\n[PROOFSTEP]\napply summable_of_ne_finset_zero (s := h.toFinset)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\nh : Set.Finite (support f)\n\u22a2 \u2200 (b : \u03b2), \u00acb \u2208 Set.Finite.toFinset h \u2192 f b = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\nha : Summable f\n\u22a2 HasSum f (\u2211' (b : \u03b2), f b)\n[PROOFSTEP]\nsimp only [tsum_def, ha, dite_true]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\nha : Summable f\n\u22a2 HasSum f (if Set.Finite (support fun b => f b) then \u2211\u1da0 (b : \u03b2), f b else Classical.choose (_ : Summable fun b => f b))\n[PROOFSTEP]\nby_cases H : (support f).Finite\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\nha : Summable f\nH : Set.Finite (support f)\n\u22a2 HasSum f (if Set.Finite (support fun b => f b) then \u2211\u1da0 (b : \u03b2), f b else Classical.choose (_ : Summable fun b => f b))\n[PROOFSTEP]\nsimp [H, hasSum_sum_of_ne_finset_zero, finsum_eq_sum]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\nha : Summable f\nH : \u00acSet.Finite (support f)\n\u22a2 HasSum f (if Set.Finite (support fun b => f b) then \u2211\u1da0 (b : \u03b2), f b else Classical.choose (_ : Summable fun b => f b))\n[PROOFSTEP]\nsimpa [H] using Classical.choose_spec ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na b\u271d : \u03b1\ns : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : \u2200 (b' : \u03b2), b' \u2260 b \u2192 f b' = 0\n\u22a2 \u2200 (b_1 : \u03b2), \u00acb_1 \u2208 {b} \u2192 f b_1 = 0\n[PROOFSTEP]\nsimpa [hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na b\u271d : \u03b1\ns : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : \u2200 (b' : \u03b2), b' \u2260 b \u2192 f b' = 0\nthis : HasSum f (\u2211 b' in {b}, f b')\n\u22a2 HasSum f (f b)\n[PROOFSTEP]\nsimpa using this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d b\u271d : \u03b1\ns : Finset \u03b2\nb : \u03b2\ninst\u271d : DecidablePred fun x => x = b\na : \u03b1\n\u22a2 HasSum (fun b' => if b' = b then a else 0) a\n[PROOFSTEP]\nconvert @hasSum_single _ _ _ _ (fun b' => if b' = b then a else 0) b (fun b' hb' => if_neg hb')\n[GOAL]\ncase h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d b\u271d : \u03b1\ns : Finset \u03b2\nb : \u03b2\ninst\u271d : DecidablePred fun x => x = b\na : \u03b1\n\u22a2 a = if b = b then a else 0\n[PROOFSTEP]\nexact (if_pos rfl).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d b\u271d : \u03b1\ns : Finset \u03b2\ninst\u271d : DecidableEq \u03b2\nb : \u03b2\na : \u03b1\n\u22a2 HasSum (Pi.single b a) a\n[PROOFSTEP]\nconvert hasSum_ite_eq b a\n[GOAL]\ncase h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d b\u271d : \u03b1\ns : Finset \u03b2\ninst\u271d : DecidableEq \u03b2\nb : \u03b2\na : \u03b1\nx\u271d : \u03b2\n\u22a2 Pi.single b a x\u271d = if x\u271d = b then a else 0\n[PROOFSTEP]\nsimp [Pi.single_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\ne : \u03b3 \u2243 \u03b2\n\u22a2 \u2200 (x : \u03b2), \u00acx \u2208 Set.range \u2191e \u2192 f x = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g\u271d : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\ng : \u03b3 \u2192 \u03b1\ne : \u2191(support f) \u2243 \u2191(support g)\nhe : \u2200 (x : \u2191(support f)), g \u2191(\u2191e x) = f \u2191x\n\u22a2 HasSum f a \u2194 HasSum g a\n[PROOFSTEP]\nhave : (g \u2218 (\u2191)) \u2218 e = f \u2218 (\u2191) := funext he\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g\u271d : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\ng : \u03b3 \u2192 \u03b1\ne : \u2191(support f) \u2243 \u2191(support g)\nhe : \u2200 (x : \u2191(support f)), g \u2191(\u2191e x) = f \u2191x\nthis : (g \u2218 Subtype.val) \u2218 \u2191e = f \u2218 Subtype.val\n\u22a2 HasSum f a \u2194 HasSum g a\n[PROOFSTEP]\nrw [\u2190 hasSum_subtype_support, \u2190 this, e.hasSum_iff, hasSum_subtype_support]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommMonoid \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\nf g\u271d : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\ninst\u271d\u00b3 : AddCommMonoid \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nG : Type u_5\nG' : Type u_6\ninst\u271d\u00b9 : AddMonoidHomClass G \u03b1 \u03b3\ninst\u271d : AddMonoidHomClass G' \u03b3 \u03b1\ng : G\ng' : G'\nhg : Continuous \u2191g\nhg' : Continuous \u2191g'\nhinv : LeftInverse \u2191g' \u2191g\nh : Summable (\u2191g \u2218 f)\n\u22a2 Summable f\n[PROOFSTEP]\nhave := h.map _ hg'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommMonoid \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\nf g\u271d : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\ninst\u271d\u00b3 : AddCommMonoid \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nG : Type u_5\nG' : Type u_6\ninst\u271d\u00b9 : AddMonoidHomClass G \u03b1 \u03b3\ninst\u271d : AddMonoidHomClass G' \u03b3 \u03b1\ng : G\ng' : G'\nhg : Continuous \u2191g\nhg' : Continuous \u2191g'\nhinv : LeftInverse \u2191g' \u2191g\nh : Summable (\u2191g \u2218 f)\nthis : Summable (\u2191g' \u2218 \u2191g \u2218 f)\n\u22a2 Summable f\n[PROOFSTEP]\nrwa [\u2190 Function.comp.assoc, hinv.id] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns : Finset \u03b2\ninst\u271d : T2Space \u03b1\nf : \u2115 \u2192 \u03b1\na : \u03b1\nhf : Summable f\n\u22a2 HasSum f a \u2194 Tendsto (fun n => \u2211 i in range n, f i) atTop (\ud835\udcdd a)\n[PROOFSTEP]\nrefine' \u27e8fun h => h.tendsto_sum_nat, fun h => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns : Finset \u03b2\ninst\u271d : T2Space \u03b1\nf : \u2115 \u2192 \u03b1\na : \u03b1\nhf : Summable f\nh : Tendsto (fun n => \u2211 i in range n, f i) atTop (\ud835\udcdd a)\n\u22a2 HasSum f a\n[PROOFSTEP]\nrw [tendsto_nhds_unique h hf.hasSum.tendsto_sum_nat]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns : Finset \u03b2\ninst\u271d : T2Space \u03b1\nf : \u2115 \u2192 \u03b1\na : \u03b1\nhf : Summable f\nh : Tendsto (fun n => \u2211 i in range n, f i) atTop (\ud835\udcdd a)\n\u22a2 HasSum f (\u2211' (b : \u2115), f b)\n[PROOFSTEP]\nexact hf.hasSum\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\nhf : HasSum f a\nhg : HasSum g b\n\u22a2 HasSum (fun b => f b + g b) (a + b)\n[PROOFSTEP]\ndsimp only [HasSum] at hf hg \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\nhf : Tendsto (fun s => \u2211 b in s, f b) atTop (\ud835\udcdd a)\nhg : Tendsto (fun s => \u2211 b in s, g b) atTop (\ud835\udcdd b)\n\u22a2 Tendsto (fun s => \u2211 b in s, (f b + g b)) atTop (\ud835\udcdd (a + b))\n[PROOFSTEP]\nsimp_rw [sum_add_distrib]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\nhf : Tendsto (fun s => \u2211 b in s, f b) atTop (\ud835\udcdd a)\nhg : Tendsto (fun s => \u2211 b in s, g b) atTop (\ud835\udcdd b)\n\u22a2 Tendsto (fun s => \u2211 x in s, f x + \u2211 x in s, g x) atTop (\ud835\udcdd (a + b))\n[PROOFSTEP]\nexact hf.add hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\na : \u03b3 \u2192 \u03b1\ns : Finset \u03b3\n\u22a2 (\u2200 (i : \u03b3), i \u2208 \u2205 \u2192 HasSum (f i) (a i)) \u2192 HasSum (fun b => \u2211 i in \u2205, f i b) (\u2211 i in \u2205, a i)\n[PROOFSTEP]\nsimp only [hasSum_zero, sum_empty, forall_true_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\na : \u03b3 \u2192 \u03b1\ns : Finset \u03b3\n\u22a2 \u2200 \u2983a_1 : \u03b3\u2984 {s : Finset \u03b3},\n \u00aca_1 \u2208 s \u2192\n ((\u2200 (i : \u03b3), i \u2208 s \u2192 HasSum (f i) (a i)) \u2192 HasSum (fun b => \u2211 i in s, f i b) (\u2211 i in s, a i)) \u2192\n (\u2200 (i : \u03b3), i \u2208 insert a_1 s \u2192 HasSum (f i) (a i)) \u2192\n HasSum (fun b => \u2211 i in insert a_1 s, f i b) (\u2211 i in insert a_1 s, a i)\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [mem_insert, forall_eq_or_imp, not_false_iff, sum_insert, and_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\na : \u03b3 \u2192 \u03b1\ns : Finset \u03b3\n\u22a2 \u2200 \u2983a_1 : \u03b3\u2984 {s : Finset \u03b3},\n \u00aca_1 \u2208 s \u2192\n ((\u2200 (i : \u03b3), i \u2208 s \u2192 HasSum (f i) (a i)) \u2192 HasSum (fun b => \u2211 i in s, f i b) (\u2211 i in s, a i)) \u2192\n HasSum (f a_1) (a a_1) \u2192\n (\u2200 (a_5 : \u03b3), a_5 \u2208 s \u2192 HasSum (f a_5) (a a_5)) \u2192\n HasSum (fun b => f a_1 b + \u2211 i in s, f i b) (a a_1 + \u2211 i in s, a i)\n[PROOFSTEP]\nexact fun x s _ IH hx h \u21a6 hx.add (IH h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\ns t : Set \u03b2\nhs : Disjoint s t\nha : HasSum (f \u2218 Subtype.val) a\nhb : HasSum (f \u2218 Subtype.val) b\n\u22a2 HasSum (f \u2218 Subtype.val) (a + b)\n[PROOFSTEP]\nrw [hasSum_subtype_iff_indicator] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\ns t : Set \u03b2\nhs : Disjoint s t\nha : HasSum (Set.indicator s f) a\nhb : HasSum (Set.indicator t f) b\n\u22a2 HasSum (Set.indicator (s \u222a t) f) (a + b)\n[PROOFSTEP]\nrw [Set.indicator_union_of_disjoint hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\ns t : Set \u03b2\nhs : Disjoint s t\nha : HasSum (Set.indicator s f) a\nhb : HasSum (Set.indicator t f) b\n\u22a2 HasSum (fun a => Set.indicator s f a + Set.indicator t f a) (a + b)\n[PROOFSTEP]\nexact ha.add hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\n\u03b9 : Type u_5\ns : Finset \u03b9\nt : \u03b9 \u2192 Set \u03b2\na : \u03b9 \u2192 \u03b1\nhs : Set.Pairwise (\u2191s) (Disjoint on t)\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 HasSum (f \u2218 Subtype.val) (a i)\n\u22a2 HasSum (f \u2218 Subtype.val) (\u2211 i in s, a i)\n[PROOFSTEP]\nsimp_rw [hasSum_subtype_iff_indicator] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\n\u03b9 : Type u_5\ns : Finset \u03b9\nt : \u03b9 \u2192 Set \u03b2\na : \u03b9 \u2192 \u03b1\nhs : Set.Pairwise (\u2191s) (Disjoint on t)\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 HasSum (Set.indicator (t i) f) (a i)\n\u22a2 HasSum (Set.indicator (\u22c3 (i : \u03b9) (_ : i \u2208 s), t i) f) (\u2211 i in s, a i)\n[PROOFSTEP]\nrw [Set.indicator_finset_biUnion _ _ hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\n\u03b9 : Type u_5\ns : Finset \u03b9\nt : \u03b9 \u2192 Set \u03b2\na : \u03b9 \u2192 \u03b1\nhs : Set.Pairwise (\u2191s) (Disjoint on t)\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 HasSum (Set.indicator (t i) f) (a i)\n\u22a2 HasSum (fun a => \u2211 i in s, Set.indicator (t i) f a) (\u2211 i in s, a i)\n[PROOFSTEP]\nexact hasSum_sum hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\ns t : Set \u03b2\nhs : IsCompl s t\nha : HasSum (f \u2218 Subtype.val) a\nhb : HasSum (f \u2218 Subtype.val) b\n\u22a2 HasSum f (a + b)\n[PROOFSTEP]\nsimpa [\u2190 hs.compl_eq] using (hasSum_subtype_iff_indicator.1 ha).add (hasSum_subtype_iff_indicator.1 hb)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\nf : \u2115 \u2192 \u03b1\nhe : HasSum (fun k => f (2 * k)) a\nho : HasSum (fun k => f (2 * k + 1)) b\n\u22a2 HasSum f (a + b)\n[PROOFSTEP]\nhave := mul_right_injective\u2080 (two_ne_zero' \u2115)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\nf : \u2115 \u2192 \u03b1\nhe : HasSum (fun k => f (2 * k)) a\nho : HasSum (fun k => f (2 * k + 1)) b\nthis : Injective ((fun x x_1 => x * x_1) 2)\n\u22a2 HasSum f (a + b)\n[PROOFSTEP]\nreplace he := this.hasSum_range_iff.2 he\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\nf : \u2115 \u2192 \u03b1\nho : HasSum (fun k => f (2 * k + 1)) b\nthis : Injective ((fun x x_1 => x * x_1) 2)\nhe : HasSum (fun x => f \u2191x) a\n\u22a2 HasSum f (a + b)\n[PROOFSTEP]\nreplace ho := ((add_left_injective 1).comp this).hasSum_range_iff.2 ho\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\nf : \u2115 \u2192 \u03b1\nthis : Injective ((fun x x_1 => x * x_1) 2)\nhe : HasSum (fun x => f \u2191x) a\nho : HasSum (fun x => f \u2191x) b\n\u22a2 HasSum f (a + b)\n[PROOFSTEP]\nrefine' he.add_isCompl _ ho\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\nf : \u2115 \u2192 \u03b1\nthis : Injective ((fun x x_1 => x * x_1) 2)\nhe : HasSum (fun x => f \u2191x) a\nho : HasSum (fun x => f \u2191x) b\n\u22a2 IsCompl (Set.range ((fun x x_1 => x * x_1) 2)) (Set.range ((fun x => x + 1) \u2218 (fun x x_1 => x * x_1) 2))\n[PROOFSTEP]\nsimpa [(\u00b7 \u2218 \u00b7)] using Nat.isCompl_even_odd\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns : Finset \u03b2\ninst\u271d\u00b9 : ContinuousAdd \u03b1\ninst\u271d : RegularSpace \u03b1\n\u03b3 : \u03b2 \u2192 Type u_5\nf : (b : \u03b2) \u00d7 \u03b3 b \u2192 \u03b1\ng : \u03b2 \u2192 \u03b1\na : \u03b1\nha : HasSum f a\nhf : \u2200 (b : \u03b2), HasSum (fun c => f { fst := b, snd := c }) (g b)\n\u22a2 HasSum g a\n[PROOFSTEP]\nrefine' (atTop_basis.tendsto_iff (closed_nhds_basis a)).mpr _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns : Finset \u03b2\ninst\u271d\u00b9 : ContinuousAdd \u03b1\ninst\u271d : RegularSpace \u03b1\n\u03b3 : \u03b2 \u2192 Type u_5\nf : (b : \u03b2) \u00d7 \u03b3 b \u2192 \u03b1\ng : \u03b2 \u2192 \u03b1\na : \u03b1\nha : HasSum f a\nhf : \u2200 (b : \u03b2), HasSum (fun c => f { fst := b, snd := c }) (g b)\n\u22a2 \u2200 (ib : Set \u03b1), ib \u2208 \ud835\udcdd a \u2227 IsClosed ib \u2192 \u2203 ia, True \u2227 \u2200 (x : Finset \u03b2), x \u2208 Set.Ici ia \u2192 \u2211 b in x, g b \u2208 id ib\n[PROOFSTEP]\nrintro s \u27e8hs, hsc\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d\u00b9 : ContinuousAdd \u03b1\ninst\u271d : RegularSpace \u03b1\n\u03b3 : \u03b2 \u2192 Type u_5\nf : (b : \u03b2) \u00d7 \u03b3 b \u2192 \u03b1\ng : \u03b2 \u2192 \u03b1\na : \u03b1\nha : HasSum f a\nhf : \u2200 (b : \u03b2), HasSum (fun c => f { fst := b, snd := c }) (g b)\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd a\nhsc : IsClosed s\n\u22a2 \u2203 ia, True \u2227 \u2200 (x : Finset \u03b2), x \u2208 Set.Ici ia \u2192 \u2211 b in x, g b \u2208 id s\n[PROOFSTEP]\nrcases mem_atTop_sets.mp (ha hs) with \u27e8u, hu\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d\u00b9 : ContinuousAdd \u03b1\ninst\u271d : RegularSpace \u03b1\n\u03b3 : \u03b2 \u2192 Type u_5\nf : (b : \u03b2) \u00d7 \u03b3 b \u2192 \u03b1\ng : \u03b2 \u2192 \u03b1\na : \u03b1\nha : HasSum f a\nhf : \u2200 (b : \u03b2), HasSum (fun c => f { fst := b, snd := c }) (g b)\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd a\nhsc : IsClosed s\nu : Finset ((b : \u03b2) \u00d7 \u03b3 b)\nhu : \u2200 (b : Finset ((b : \u03b2) \u00d7 \u03b3 b)), b \u2265 u \u2192 b \u2208 (fun s => \u2211 b in s, f b) \u207b\u00b9' s\n\u22a2 \u2203 ia, True \u2227 \u2200 (x : Finset \u03b2), x \u2208 Set.Ici ia \u2192 \u2211 b in x, g b \u2208 id s\n[PROOFSTEP]\nuse u.image Sigma.fst, trivial\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d\u00b9 : ContinuousAdd \u03b1\ninst\u271d : RegularSpace \u03b1\n\u03b3 : \u03b2 \u2192 Type u_5\nf : (b : \u03b2) \u00d7 \u03b3 b \u2192 \u03b1\ng : \u03b2 \u2192 \u03b1\na : \u03b1\nha : HasSum f a\nhf : \u2200 (b : \u03b2), HasSum (fun c => f { fst := b, snd := c }) (g b)\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd a\nhsc : IsClosed s\nu : Finset ((b : \u03b2) \u00d7 \u03b3 b)\nhu : \u2200 (b : Finset ((b : \u03b2) \u00d7 \u03b3 b)), b \u2265 u \u2192 b \u2208 (fun s => \u2211 b in s, f b) \u207b\u00b9' s\n\u22a2 \u2200 (x : Finset \u03b2), x \u2208 Set.Ici (image Sigma.fst u) \u2192 \u2211 b in x, g b \u2208 id s\n[PROOFSTEP]\nintro bs hbs\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d\u00b9 : ContinuousAdd \u03b1\ninst\u271d : RegularSpace \u03b1\n\u03b3 : \u03b2 \u2192 Type u_5\nf : (b : \u03b2) \u00d7 \u03b3 b \u2192 \u03b1\ng : \u03b2 \u2192 \u03b1\na : \u03b1\nha : HasSum f a\nhf : \u2200 (b : \u03b2), HasSum (fun c => f { fst := b, snd := c }) (g b)\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd a\nhsc : IsClosed s\nu : Finset ((b : \u03b2) \u00d7 \u03b3 b)\nhu : \u2200 (b : Finset ((b : \u03b2) \u00d7 \u03b3 b)), b \u2265 u \u2192 b \u2208 (fun s => \u2211 b in s, f b) \u207b\u00b9' s\nbs : Finset \u03b2\nhbs : bs \u2208 Set.Ici (image Sigma.fst u)\n\u22a2 \u2211 b in bs, g b \u2208 id s\n[PROOFSTEP]\nsimp only [Set.mem_preimage, ge_iff_le, Finset.le_iff_subset] at hu \n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d\u00b9 : ContinuousAdd \u03b1\ninst\u271d : RegularSpace \u03b1\n\u03b3 : \u03b2 \u2192 Type u_5\nf : (b : \u03b2) \u00d7 \u03b3 b \u2192 \u03b1\ng : \u03b2 \u2192 \u03b1\na : \u03b1\nha : HasSum f a\nhf : \u2200 (b : \u03b2), HasSum (fun c => f { fst := b, snd := c }) (g b)\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd a\nhsc : IsClosed s\nu : Finset ((b : \u03b2) \u00d7 \u03b3 b)\nbs : Finset \u03b2\nhbs : bs \u2208 Set.Ici (image Sigma.fst u)\nhu : \u2200 (b : Finset ((b : \u03b2) \u00d7 \u03b3 b)), u \u2286 b \u2192 \u2211 b in b, f b \u2208 s\n\u22a2 \u2211 b in bs, g b \u2208 id s\n[PROOFSTEP]\nhave : Tendsto (fun t : Finset (\u03a3 b, \u03b3 b) => \u2211 p in t.filter fun p => p.1 \u2208 bs, f p) atTop (\ud835\udcdd <| \u2211 b in bs, g b) :=\n by\n simp only [\u2190 sigma_preimage_mk, sum_sigma]\n refine' tendsto_finset_sum _ fun b _ => _\n change Tendsto (fun t => (fun t => \u2211 s in t, f \u27e8b, s\u27e9) (preimage t (Sigma.mk b) _)) atTop (\ud835\udcdd (g b))\n exact (hf b).comp (tendsto_finset_preimage_atTop_atTop (sigma_mk_injective))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d\u00b9 : ContinuousAdd \u03b1\ninst\u271d : RegularSpace \u03b1\n\u03b3 : \u03b2 \u2192 Type u_5\nf : (b : \u03b2) \u00d7 \u03b3 b \u2192 \u03b1\ng : \u03b2 \u2192 \u03b1\na : \u03b1\nha : HasSum f a\nhf : \u2200 (b : \u03b2), HasSum (fun c => f { fst := b, snd := c }) (g b)\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd a\nhsc : IsClosed s\nu : Finset ((b : \u03b2) \u00d7 \u03b3 b)\nbs : Finset \u03b2\nhbs : bs \u2208 Set.Ici (image Sigma.fst u)\nhu : \u2200 (b : Finset ((b : \u03b2) \u00d7 \u03b3 b)), u \u2286 b \u2192 \u2211 b in b, f b \u2208 s\n\u22a2 Tendsto (fun t => \u2211 p in filter (fun p => p.fst \u2208 bs) t, f p) atTop (\ud835\udcdd (\u2211 b in bs, g b))\n[PROOFSTEP]\nsimp only [\u2190 sigma_preimage_mk, sum_sigma]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d\u00b9 : ContinuousAdd \u03b1\ninst\u271d : RegularSpace \u03b1\n\u03b3 : \u03b2 \u2192 Type u_5\nf : (b : \u03b2) \u00d7 \u03b3 b \u2192 \u03b1\ng : \u03b2 \u2192 \u03b1\na : \u03b1\nha : HasSum f a\nhf : \u2200 (b : \u03b2), HasSum (fun c => f { fst := b, snd := c }) (g b)\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd a\nhsc : IsClosed s\nu : Finset ((b : \u03b2) \u00d7 \u03b3 b)\nbs : Finset \u03b2\nhbs : bs \u2208 Set.Ici (image Sigma.fst u)\nhu : \u2200 (b : Finset ((b : \u03b2) \u00d7 \u03b3 b)), u \u2286 b \u2192 \u2211 b in b, f b \u2208 s\n\u22a2 Tendsto\n (fun t =>\n \u2211 a in bs,\n \u2211 s in preimage t (Sigma.mk a) (_ : Set.InjOn (Sigma.mk a) (Sigma.mk a \u207b\u00b9' \u2191t)), f { fst := a, snd := s })\n atTop (\ud835\udcdd (\u2211 b in bs, g b))\n[PROOFSTEP]\nrefine' tendsto_finset_sum _ fun b _ => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na\u271d b\u271d : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d\u00b9 : ContinuousAdd \u03b1\ninst\u271d : RegularSpace \u03b1\n\u03b3 : \u03b2 \u2192 Type u_5\nf : (b : \u03b2) \u00d7 \u03b3 b \u2192 \u03b1\ng : \u03b2 \u2192 \u03b1\na : \u03b1\nha : HasSum f a\nhf : \u2200 (b : \u03b2), HasSum (fun c => f { fst := b, snd := c }) (g b)\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd a\nhsc : IsClosed s\nu : Finset ((b : \u03b2) \u00d7 \u03b3 b)\nbs : Finset \u03b2\nhbs : bs \u2208 Set.Ici (image Sigma.fst u)\nhu : \u2200 (b : Finset ((b : \u03b2) \u00d7 \u03b3 b)), u \u2286 b \u2192 \u2211 b in b, f b \u2208 s\nb : \u03b2\nx\u271d : b \u2208 bs\n\u22a2 Tendsto\n (fun t => \u2211 s in preimage t (Sigma.mk b) (_ : Set.InjOn (Sigma.mk b) (Sigma.mk b \u207b\u00b9' \u2191t)), f { fst := b, snd := s })\n atTop (\ud835\udcdd (g b))\n[PROOFSTEP]\nchange Tendsto (fun t => (fun t => \u2211 s in t, f \u27e8b, s\u27e9) (preimage t (Sigma.mk b) _)) atTop (\ud835\udcdd (g b))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na\u271d b\u271d : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d\u00b9 : ContinuousAdd \u03b1\ninst\u271d : RegularSpace \u03b1\n\u03b3 : \u03b2 \u2192 Type u_5\nf : (b : \u03b2) \u00d7 \u03b3 b \u2192 \u03b1\ng : \u03b2 \u2192 \u03b1\na : \u03b1\nha : HasSum f a\nhf : \u2200 (b : \u03b2), HasSum (fun c => f { fst := b, snd := c }) (g b)\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd a\nhsc : IsClosed s\nu : Finset ((b : \u03b2) \u00d7 \u03b3 b)\nbs : Finset \u03b2\nhbs : bs \u2208 Set.Ici (image Sigma.fst u)\nhu : \u2200 (b : Finset ((b : \u03b2) \u00d7 \u03b3 b)), u \u2286 b \u2192 \u2211 b in b, f b \u2208 s\nb : \u03b2\nx\u271d : b \u2208 bs\n\u22a2 Tendsto\n (fun t =>\n (fun t => \u2211 s in t, f { fst := b, snd := s })\n (preimage t (Sigma.mk b) (_ : Set.InjOn (Sigma.mk b) (Sigma.mk b \u207b\u00b9' \u2191t))))\n atTop (\ud835\udcdd (g b))\n[PROOFSTEP]\nexact (hf b).comp (tendsto_finset_preimage_atTop_atTop (sigma_mk_injective))\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d\u00b9 : ContinuousAdd \u03b1\ninst\u271d : RegularSpace \u03b1\n\u03b3 : \u03b2 \u2192 Type u_5\nf : (b : \u03b2) \u00d7 \u03b3 b \u2192 \u03b1\ng : \u03b2 \u2192 \u03b1\na : \u03b1\nha : HasSum f a\nhf : \u2200 (b : \u03b2), HasSum (fun c => f { fst := b, snd := c }) (g b)\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd a\nhsc : IsClosed s\nu : Finset ((b : \u03b2) \u00d7 \u03b3 b)\nbs : Finset \u03b2\nhbs : bs \u2208 Set.Ici (image Sigma.fst u)\nhu : \u2200 (b : Finset ((b : \u03b2) \u00d7 \u03b3 b)), u \u2286 b \u2192 \u2211 b in b, f b \u2208 s\nthis : Tendsto (fun t => \u2211 p in filter (fun p => p.fst \u2208 bs) t, f p) atTop (\ud835\udcdd (\u2211 b in bs, g b))\n\u22a2 \u2211 b in bs, g b \u2208 id s\n[PROOFSTEP]\nrefine' hsc.mem_of_tendsto this (eventually_atTop.2 \u27e8u, fun t ht => hu _ fun x hx => _\u27e9)\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d\u00b9 : ContinuousAdd \u03b1\ninst\u271d : RegularSpace \u03b1\n\u03b3 : \u03b2 \u2192 Type u_5\nf : (b : \u03b2) \u00d7 \u03b3 b \u2192 \u03b1\ng : \u03b2 \u2192 \u03b1\na : \u03b1\nha : HasSum f a\nhf : \u2200 (b : \u03b2), HasSum (fun c => f { fst := b, snd := c }) (g b)\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd a\nhsc : IsClosed s\nu : Finset ((b : \u03b2) \u00d7 \u03b3 b)\nbs : Finset \u03b2\nhbs : bs \u2208 Set.Ici (image Sigma.fst u)\nhu : \u2200 (b : Finset ((b : \u03b2) \u00d7 \u03b3 b)), u \u2286 b \u2192 \u2211 b in b, f b \u2208 s\nthis : Tendsto (fun t => \u2211 p in filter (fun p => p.fst \u2208 bs) t, f p) atTop (\ud835\udcdd (\u2211 b in bs, g b))\nt : Finset ((b : \u03b2) \u00d7 \u03b3 b)\nht : t \u2265 u\nx : (b : \u03b2) \u00d7 \u03b3 b\nhx : x \u2208 u\n\u22a2 x \u2208 filter (fun p => p.fst \u2208 bs) t\n[PROOFSTEP]\nexact mem_filter.2 \u27e8ht hx, hbs <| mem_image_of_mem _ hx\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na\u271d b : \u03b1\ns : Finset \u03b2\ninst\u271d\u00b9 : ContinuousAdd \u03b1\ninst\u271d : T3Space \u03b1\n\u03b3 : \u03b2 \u2192 Type u_5\nf : (b : \u03b2) \u00d7 \u03b3 b \u2192 \u03b1\ng : \u03b2 \u2192 \u03b1\na : \u03b1\nha : HasSum g a\nhf : \u2200 (b : \u03b2), HasSum (fun c => f { fst := b, snd := c }) (g b)\nhf' : Summable f\n\u22a2 HasSum f a\n[PROOFSTEP]\nsimpa [(hf'.hasSum.sigma hf).unique ha] using hf'.hasSum\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\n\u22a2 a + x = a' + f b\n[PROOFSTEP]\nhave : \u2200 b', f b' + ite (b' = b) x 0 = update f b x b' + ite (b' = b) (f b) 0 :=\n by\n intro b'\n split_ifs with hb'\n \u00b7 simpa only [Function.update_apply, hb', eq_self_iff_true] using add_comm (f b) x\n \u00b7 simp only [Function.update_apply, hb', if_false]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\n\u22a2 \u2200 (b' : \u03b2), (f b' + if b' = b then x else 0) = update f b x b' + if b' = b then f b else 0\n[PROOFSTEP]\nintro b'\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\nb' : \u03b2\n\u22a2 (f b' + if b' = b then x else 0) = update f b x b' + if b' = b then f b else 0\n[PROOFSTEP]\nsplit_ifs with hb'\n[GOAL]\ncase pos\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\nb' : \u03b2\nhb' : b' = b\n\u22a2 f b' + x = update f b x b' + f b\n[PROOFSTEP]\nsimpa only [Function.update_apply, hb', eq_self_iff_true] using add_comm (f b) x\n[GOAL]\ncase neg\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\nb' : \u03b2\nhb' : \u00acb' = b\n\u22a2 f b' + 0 = update f b x b' + 0\n[PROOFSTEP]\nsimp only [Function.update_apply, hb', if_false]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\nthis : \u2200 (b' : \u03b2), (f b' + if b' = b then x else 0) = update f b x b' + if b' = b then f b else 0\n\u22a2 a + x = a' + f b\n[PROOFSTEP]\nhave h := hf.add (hasSum_ite_eq b x)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\nthis : \u2200 (b' : \u03b2), (f b' + if b' = b then x else 0) = update f b x b' + if b' = b then f b else 0\nh : HasSum (fun b_1 => f b_1 + if b_1 = b then x else 0) (a + x)\n\u22a2 a + x = a' + f b\n[PROOFSTEP]\nsimp_rw [this] at h \n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na a' : \u03b1\nhf : HasSum f a\nb : \u03b2\nx : \u03b1\nhf' : HasSum (update f b x) a'\nthis : \u2200 (b' : \u03b2), (f b' + if b' = b then x else 0) = update f b x b' + if b' = b then f b else 0\nh : HasSum (fun b_1 => update f b x b_1 + if b_1 = b then f b else 0) (a + x)\n\u22a2 a + x = a' + f b\n[PROOFSTEP]\nexact HasSum.unique h (hf'.add (hasSum_ite_eq b (f b)))\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nhf : HasSum f a\nb : \u03b2\na' : \u03b1\nhf' : HasSum (fun n => if n = b then 0 else f n) a'\n\u22a2 a = a' + f b\n[PROOFSTEP]\nrefine' (add_zero a).symm.trans (hf.update' b 0 _)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nhf : HasSum f a\nb : \u03b2\na' : \u03b1\nhf' : HasSum (fun n => if n = b then 0 else f n) a'\n\u22a2 HasSum (update f b 0) a'\n[PROOFSTEP]\nconvert hf'\n[GOAL]\ncase h.e'_5.h\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommMonoid \u03b1\u271d\ninst\u271d\u2075 : TopologicalSpace \u03b1\u271d\nf\u271d g : \u03b2\u271d \u2192 \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\ns : Finset \u03b2\u271d\ninst\u271d\u2074 : ContinuousAdd \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nhf : HasSum f a\nb : \u03b2\na' : \u03b1\nhf' : HasSum (fun n => if n = b then 0 else f n) a'\nx\u271d : \u03b2\n\u22a2 update f b 0 x\u271d = if x\u271d = b then 0 else f x\u271d\n[PROOFSTEP]\napply update_apply\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\ns t : Set \u03b2\nh : s = t\n\u22a2 \u2211' (x : \u2191s), f \u2191x = \u2211' (x : \u2191t), f \u2191x\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nhf : Set.Finite (support f)\n\u22a2 \u2211' (b : \u03b2), f b = \u2211\u1da0 (b : \u03b2), f b\n[PROOFSTEP]\nsimp [tsum_def, summable_of_finite_support hf, hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ns : Finset \u03b2\nhf : \u2200 (b : \u03b2), \u00acb \u2208 s \u2192 f b = 0\n\u22a2 \u2211' (b : \u03b2), f b = \u2211 b in s, f b\n[PROOFSTEP]\nhave I : support f \u2286 s := by\n intros x hx\n contrapose! hx\n rw [nmem_support]\n exact hf _ hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ns : Finset \u03b2\nhf : \u2200 (b : \u03b2), \u00acb \u2208 s \u2192 f b = 0\n\u22a2 support f \u2286 \u2191s\n[PROOFSTEP]\nintros x hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ns : Finset \u03b2\nhf : \u2200 (b : \u03b2), \u00acb \u2208 s \u2192 f b = 0\nx : \u03b2\nhx : x \u2208 support f\n\u22a2 x \u2208 \u2191s\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ns : Finset \u03b2\nhf : \u2200 (b : \u03b2), \u00acb \u2208 s \u2192 f b = 0\nx : \u03b2\nhx : \u00acx \u2208 \u2191s\n\u22a2 \u00acx \u2208 support f\n[PROOFSTEP]\nrw [nmem_support]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ns : Finset \u03b2\nhf : \u2200 (b : \u03b2), \u00acb \u2208 s \u2192 f b = 0\nx : \u03b2\nhx : \u00acx \u2208 \u2191s\n\u22a2 f x = 0\n[PROOFSTEP]\nexact hf _ hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ns : Finset \u03b2\nhf : \u2200 (b : \u03b2), \u00acb \u2208 s \u2192 f b = 0\nI : support f \u2286 \u2191s\n\u22a2 \u2211' (b : \u03b2), f b = \u2211 b in s, f b\n[PROOFSTEP]\nsimp [tsum_def, summable_of_ne_finset_zero hf, Set.Finite.subset (finite_toSet s) I]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ns : Finset \u03b2\nhf : \u2200 (b : \u03b2), \u00acb \u2208 s \u2192 f b = 0\nI : support f \u2286 \u2191s\n\u22a2 \u2211\u1da0 (b : \u03b2), f b = \u2211 b in s, f b\n[PROOFSTEP]\nexact finsum_eq_sum_of_support_subset f I\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\n\u22a2 \u2211' (x : \u03b2), 0 = 0\n[PROOFSTEP]\nrw [tsum_eq_finsum]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\n\u22a2 \u2211\u1da0 (b : \u03b2), 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\n\u22a2 Set.Finite (support fun x => 0)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : IsEmpty \u03b2\n\u22a2 \u2211' (b : \u03b2), f b = 0\n[PROOFSTEP]\nrw [tsum_eq_sum (s := (\u2205 : Finset \u03b2))]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : IsEmpty \u03b2\n\u22a2 \u2211 b in \u2205, f b = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : IsEmpty \u03b2\n\u22a2 \u2200 (b : \u03b2), \u00acb \u2208 \u2205 \u2192 f b = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : Fintype \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 \u2211' (b : \u03b2), f b = \u2211 b : \u03b2, f b\n[PROOFSTEP]\napply tsum_eq_sum\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : Fintype \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 \u2200 (b : \u03b2), \u00acb \u2208 univ \u2192 f b = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : Bool \u2192 \u03b1\n\u22a2 \u2211' (i : Bool), f i = f (decide False) + f (decide True)\n[PROOFSTEP]\nrw [tsum_fintype, Finset.sum_eq_add]\n[GOAL]\ncase hn\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : Bool \u2192 \u03b1\n\u22a2 decide False \u2260 decide True\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2080\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : Bool \u2192 \u03b1\n\u22a2 \u2200 (c : Bool), c \u2208 univ \u2192 c \u2260 decide False \u2227 c \u2260 decide True \u2192 f c = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase ha\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : Bool \u2192 \u03b1\n\u22a2 \u00acdecide False \u2208 univ \u2192 f (decide False) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hb\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : Bool \u2192 \u03b1\n\u22a2 \u00acdecide True \u2208 univ \u2192 f (decide True) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : \u2200 (b' : \u03b2), b' \u2260 b \u2192 f b' = 0\n\u22a2 \u2211' (b : \u03b2), f b = f b\n[PROOFSTEP]\nrw [tsum_eq_sum (s := { b }), sum_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : \u2200 (b' : \u03b2), b' \u2260 b \u2192 f b' = 0\n\u22a2 \u2200 (b_1 : \u03b2), \u00acb_1 \u2208 {b} \u2192 f b_1 = 0\n[PROOFSTEP]\nexact fun b' hb' \u21a6 hf b' (by simpa using hb')\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : \u2200 (b' : \u03b2), b' \u2260 b \u2192 f b' = 0\nb' : \u03b2\nhb' : \u00acb' \u2208 {b}\n\u22a2 b' \u2260 b\n[PROOFSTEP]\nsimpa using hb'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nb : \u03b2\ninst\u271d : DecidablePred fun x => x = b\na : \u03b1\n\u22a2 (\u2211' (b' : \u03b2), if b' = b then a else 0) = a\n[PROOFSTEP]\nrw [tsum_eq_single b]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nb : \u03b2\ninst\u271d : DecidablePred fun x => x = b\na : \u03b1\n\u22a2 (if b = b then a else 0) = a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nb : \u03b2\ninst\u271d : DecidablePred fun x => x = b\na : \u03b1\n\u22a2 \u2200 (b' : \u03b2), b' \u2260 b \u2192 (if b' = b then a else 0) = 0\n[PROOFSTEP]\nintro b' hb'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nb : \u03b2\ninst\u271d : DecidablePred fun x => x = b\na : \u03b1\nb' : \u03b2\nhb' : b' \u2260 b\n\u22a2 (if b' = b then a else 0) = 0\n[PROOFSTEP]\nsimp [hb']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\ninst\u271d : DecidableEq \u03b2\nb : \u03b2\na : \u03b1\n\u22a2 \u2211' (b' : \u03b2), Pi.single b a b' = a\n[PROOFSTEP]\nrw [tsum_eq_single b]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\ninst\u271d : DecidableEq \u03b2\nb : \u03b2\na : \u03b1\n\u22a2 Pi.single b a b = a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\ninst\u271d : DecidableEq \u03b2\nb : \u03b2\na : \u03b1\n\u22a2 \u2200 (b' : \u03b2), b' \u2260 b \u2192 Pi.single b a b' = 0\n[PROOFSTEP]\nintro b' hb'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\ninst\u271d : DecidableEq \u03b2\nb : \u03b2\na : \u03b1\nb' : \u03b2\nhb' : b' \u2260 b\n\u22a2 Pi.single b a b' = 0\n[PROOFSTEP]\nsimp [hb']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ns : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 \u2211' (x : { x // x \u2208 s }), f \u2191x = \u2211 x in s, f x\n[PROOFSTEP]\nrw [\u2190 sum_attach]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ns : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 \u2211' (x : { x // x \u2208 s }), f \u2191x = \u2211 x in attach s, f \u2191x\n[PROOFSTEP]\nexact tsum_fintype _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ns : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 \u2211' (x : \u2191\u2191s), f \u2191x = \u2211 x in s, f x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nb : \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 \u2211' (x : \u2191{b}), f \u2191x = f b\n[PROOFSTEP]\nrw [\u2190 coe_singleton, Finset.tsum_subtype', sum_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\nP : Prop\ninst\u271d : Decidable P\nx : \u03b2 \u2192 \u00acP \u2192 \u03b1\n\u22a2 (\u2211' (b : \u03b2), if h : P then 0 else x b h) = if h : P then 0 else \u2211' (b : \u03b2), x b h\n[PROOFSTEP]\nby_cases hP : P\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\nP : Prop\ninst\u271d : Decidable P\nx : \u03b2 \u2192 \u00acP \u2192 \u03b1\nhP : P\n\u22a2 (\u2211' (b : \u03b2), if h : P then 0 else x b h) = if h : P then 0 else \u2211' (b : \u03b2), x b h\n[PROOFSTEP]\nsimp [hP]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\nP : Prop\ninst\u271d : Decidable P\nx : \u03b2 \u2192 \u00acP \u2192 \u03b1\nhP : \u00acP\n\u22a2 (\u2211' (b : \u03b2), if h : P then 0 else x b h) = if h : P then 0 else \u2211' (b : \u03b2), x b h\n[PROOFSTEP]\nsimp [hP]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\nP : Prop\ninst\u271d : Decidable P\nx : \u03b2 \u2192 P \u2192 \u03b1\n\u22a2 (\u2211' (b : \u03b2), if h : P then x b h else 0) = if h : P then \u2211' (b : \u03b2), x b h else 0\n[PROOFSTEP]\nby_cases hP : P\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\nP : Prop\ninst\u271d : Decidable P\nx : \u03b2 \u2192 P \u2192 \u03b1\nhP : P\n\u22a2 (\u2211' (b : \u03b2), if h : P then x b h else 0) = if h : P then \u2211' (b : \u03b2), x b h else 0\n[PROOFSTEP]\nsimp [hP]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\nP : Prop\ninst\u271d : Decidable P\nx : \u03b2 \u2192 P \u2192 \u03b1\nhP : \u00acP\n\u22a2 (\u2211' (b : \u03b2), if h : P then x b h else 0) = if h : P then \u2211' (b : \u03b2), x b h else 0\n[PROOFSTEP]\nsimp [hP]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\n\u03b1' : Type u_5\ninst\u271d\u00b9 : AddCommMonoid \u03b1'\ninst\u271d : TopologicalSpace \u03b1'\ne : \u03b1' \u2192 \u03b1\nhes : Surjective e\nh0 : e 0 = 0\nf : \u03b2 \u2192 \u03b1\ng : \u03b3 \u2192 \u03b1'\nh : \u2200 {a : \u03b1'}, HasSum f (e a) \u2194 HasSum g a\nhg : \u00acSummable g\n\u22a2 \u2211' (b : \u03b2), f b = e (\u2211' (c : \u03b3), g c)\n[PROOFSTEP]\nhave hf : \u00acSummable f := mt (hes.summable_iff_of_hasSum_iff @h).1 hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\n\u03b1' : Type u_5\ninst\u271d\u00b9 : AddCommMonoid \u03b1'\ninst\u271d : TopologicalSpace \u03b1'\ne : \u03b1' \u2192 \u03b1\nhes : Surjective e\nh0 : e 0 = 0\nf : \u03b2 \u2192 \u03b1\ng : \u03b3 \u2192 \u03b1'\nh : \u2200 {a : \u03b1'}, HasSum f (e a) \u2194 HasSum g a\nhg : \u00acSummable g\nhf : \u00acSummable f\n\u22a2 \u2211' (b : \u03b2), f b = e (\u2211' (c : \u03b3), g c)\n[PROOFSTEP]\nsimp [tsum_def, hf, hg, h0]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\ng : \u03b3 \u2192 \u03b2\nf : \u03b2 \u2192 \u03b1\nhg : Injective g\n\u22a2 \u2211' (x : \u2191(Set.range g)), f \u2191x = \u2211' (x : \u03b3), f (g x)\n[PROOFSTEP]\nrw [\u2190 Set.image_univ, tsum_image f (hg.injOn _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\ng : \u03b3 \u2192 \u03b2\nf : \u03b2 \u2192 \u03b1\nhg : Injective g\n\u22a2 \u2211' (x : \u2191Set.univ), f (g \u2191x) = \u2211' (x : \u03b3), f (g x)\n[PROOFSTEP]\nsimp_rw [\u2190 comp_apply (g := g), tsum_univ (f \u2218 g)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : Summable (update f b 0)\nn : \u03b2\n\u22a2 f n = (if n = b then f n else 0) + update f b 0 n\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : Summable (update f b 0)\nn : \u03b2\nh : n = b\n\u22a2 f n = f n + update f b 0 n\n[PROOFSTEP]\nsimp [update_apply, h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : Summable (update f b 0)\nn : \u03b2\nh : \u00acn = b\n\u22a2 f n = 0 + update f b 0 n\n[PROOFSTEP]\nsimp [update_apply, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : Summable (update f b 0)\n\u22a2 (\u2211' (x : \u03b2), if x = b then f x else 0) + \u2211' (x : \u03b2), update f b 0 x =\n (if b = b then f b else 0) + \u2211' (x : \u03b2), update f b 0 x\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : Summable (update f b 0)\n\u22a2 (\u2211' (x : \u03b2), if x = b then f x else 0) = if b = b then f b else 0\n[PROOFSTEP]\nexact tsum_eq_single b fun b' hb' => if_neg hb'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : ContinuousAdd \u03b1\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : Summable (update f b 0)\n\u22a2 (if b = b then f b else 0) + \u2211' (x : \u03b2), update f b 0 x = f b + \u2211' (x : \u03b2), if x = b then 0 else f x\n[PROOFSTEP]\nsimp only [update, eq_self_iff_true, if_true, eq_rec_constant, dite_eq_ite]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2077 : AddCommMonoid \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u2075 : T2Space \u03b1\ninst\u271d\u2074 : ContinuousAdd \u03b1\ninst\u271d\u00b3 : AddCommMonoid \u03b4\ninst\u271d\u00b2 : TopologicalSpace \u03b4\ninst\u271d\u00b9 : T3Space \u03b4\ninst\u271d : ContinuousAdd \u03b4\nf : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nh : Summable (uncurry f)\nh\u2081 : \u2200 (b : \u03b2), Summable (f b)\nh\u2082 : \u2200 (c : \u03b3), Summable fun b => f b c\n\u22a2 \u2211' (c : \u03b3) (b : \u03b2), f b c = \u2211' (b : \u03b2) (c : \u03b3), f b c\n[PROOFSTEP]\nerw [\u2190 tsum_prod' h h\u2081, \u2190 tsum_prod' h.prod_symm h\u2082, \u2190 (Equiv.prodComm \u03b3 \u03b2).tsum_eq (uncurry f)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2077 : AddCommMonoid \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u2075 : T2Space \u03b1\ninst\u271d\u2074 : ContinuousAdd \u03b1\ninst\u271d\u00b3 : AddCommMonoid \u03b4\ninst\u271d\u00b2 : TopologicalSpace \u03b4\ninst\u271d\u00b9 : T3Space \u03b4\ninst\u271d : ContinuousAdd \u03b4\nf : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nh : Summable (uncurry f)\nh\u2081 : \u2200 (b : \u03b2), Summable (f b)\nh\u2082 : \u2200 (c : \u03b3), Summable fun b => f b c\n\u22a2 \u2211' (p : \u03b3 \u00d7 \u03b2), uncurry f (Prod.swap p) = \u2211' (c : \u03b3 \u00d7 \u03b2), uncurry f (\u2191(Equiv.prodComm \u03b3 \u03b2) c)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\n\u22a2 \u2211' (i : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b) = \u2211' (b : \u03b3), m (s b)\n[PROOFSTEP]\nhave H : \u2200 n, m (\u2a06 b \u2208 decode\u2082 \u03b3 n, s b) \u2260 0 \u2192 (decode\u2082 \u03b3 n).isSome :=\n by\n intro n h\n generalize decode\u2082 \u03b3 n = foo at *\n cases' foo with b\n \u00b7 refine' (h <| by simp [m0]).elim\n \u00b7 exact rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\n\u22a2 \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\n[PROOFSTEP]\nintro n h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nn : \u2115\nh : m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0\n\u22a2 Option.isSome (decode\u2082 \u03b3 n) = true\n[PROOFSTEP]\ngeneralize decode\u2082 \u03b3 n = foo at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nn : \u2115\nfoo : Option \u03b3\nh : m (\u2a06 (b : \u03b3) (_ : b \u2208 foo), s b) \u2260 0\n\u22a2 Option.isSome foo = true\n[PROOFSTEP]\ncases' foo with b\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nn : \u2115\nh : m (\u2a06 (b : \u03b3) (_ : b \u2208 none), s b) \u2260 0\n\u22a2 Option.isSome none = true\n[PROOFSTEP]\nrefine' (h <| by simp [m0]).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nn : \u2115\nh : m (\u2a06 (b : \u03b3) (_ : b \u2208 none), s b) \u2260 0\n\u22a2 m (\u2a06 (b : \u03b3) (_ : b \u2208 none), s b) = 0\n[PROOFSTEP]\nsimp [m0]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nn : \u2115\nb : \u03b3\nh : m (\u2a06 (b_1 : \u03b3) (_ : b_1 \u2208 some b), s b_1) \u2260 0\n\u22a2 Option.isSome (some b) = true\n[PROOFSTEP]\nexact rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\n\u22a2 \u2211' (i : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b) = \u2211' (b : \u03b3), m (s b)\n[PROOFSTEP]\nsymm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\n\u22a2 \u2211' (b : \u03b3), m (s b) = \u2211' (i : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)\n[PROOFSTEP]\nrefine' tsum_eq_tsum_of_ne_zero_bij (fun a => Option.get _ (H a.1 a.2)) _ _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\n\u22a2 \u2200 \u2983x y : \u2191(support fun i => m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b))\u2984,\n (fun a => Option.get (decode\u2082 \u03b3 \u2191a) (_ : Option.isSome (decode\u2082 \u03b3 \u2191a) = true)) x =\n (fun a => Option.get (decode\u2082 \u03b3 \u2191a) (_ : Option.isSome (decode\u2082 \u03b3 \u2191a) = true)) y \u2192\n \u2191x = \u2191y\n[PROOFSTEP]\ndsimp only []\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\n\u22a2 \u2200 \u2983x y : \u2191(support fun i => m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b))\u2984,\n Option.get (decode\u2082 \u03b3 \u2191x) (_ : Option.isSome (decode\u2082 \u03b3 \u2191x) = true) =\n Option.get (decode\u2082 \u03b3 \u2191y) (_ : Option.isSome (decode\u2082 \u03b3 \u2191y) = true) \u2192\n \u2191x = \u2191y\n[PROOFSTEP]\nrintro \u27e8m, hm\u27e9 \u27e8n, hn\u27e9 e\n[GOAL]\ncase refine'_1.mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm\u271d : \u03b2 \u2192 \u03b1\nm0 : m\u271d \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m\u271d (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\nm : \u2115\nhm : m \u2208 support fun i => m\u271d (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)\nn : \u2115\nhn : n \u2208 support fun i => m\u271d (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)\ne :\n Option.get (decode\u2082 \u03b3 \u2191{ val := m, property := hm })\n (_ : Option.isSome (decode\u2082 \u03b3 \u2191{ val := m, property := hm }) = true) =\n Option.get (decode\u2082 \u03b3 \u2191{ val := n, property := hn })\n (_ : Option.isSome (decode\u2082 \u03b3 \u2191{ val := n, property := hn }) = true)\n\u22a2 \u2191{ val := m, property := hm } = \u2191{ val := n, property := hn }\n[PROOFSTEP]\nhave := mem_decode\u2082.1 (Option.get_mem (H n hn))\n[GOAL]\ncase refine'_1.mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm\u271d : \u03b2 \u2192 \u03b1\nm0 : m\u271d \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m\u271d (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\nm : \u2115\nhm : m \u2208 support fun i => m\u271d (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)\nn : \u2115\nhn : n \u2208 support fun i => m\u271d (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)\ne :\n Option.get (decode\u2082 \u03b3 \u2191{ val := m, property := hm })\n (_ : Option.isSome (decode\u2082 \u03b3 \u2191{ val := m, property := hm }) = true) =\n Option.get (decode\u2082 \u03b3 \u2191{ val := n, property := hn })\n (_ : Option.isSome (decode\u2082 \u03b3 \u2191{ val := n, property := hn }) = true)\nthis : encode (Option.get (decode\u2082 \u03b3 n) (_ : Option.isSome (decode\u2082 \u03b3 n) = true)) = n\n\u22a2 \u2191{ val := m, property := hm } = \u2191{ val := n, property := hn }\n[PROOFSTEP]\nrwa [\u2190 e, mem_decode\u2082.1 (Option.get_mem (H m hm))] at this \n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\n\u22a2 (support fun b => m (s b)) \u2286 Set.range fun a => Option.get (decode\u2082 \u03b3 \u2191a) (_ : Option.isSome (decode\u2082 \u03b3 \u2191a) = true)\n[PROOFSTEP]\nintro b h\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\nb : \u03b3\nh : b \u2208 support fun b => m (s b)\n\u22a2 b \u2208 Set.range fun a => Option.get (decode\u2082 \u03b3 \u2191a) (_ : Option.isSome (decode\u2082 \u03b3 \u2191a) = true)\n[PROOFSTEP]\nrefine' \u27e8\u27e8encode b, _\u27e9, _\u27e9\n[GOAL]\ncase refine'_2.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\nb : \u03b3\nh : b \u2208 support fun b => m (s b)\n\u22a2 encode b \u2208 support fun i => m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)\n[PROOFSTEP]\nsimp only [mem_support, encodek\u2082] at h \u22a2\n[GOAL]\ncase refine'_2.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\nb : \u03b3\nh : m (s b) \u2260 0\n\u22a2 m (\u2a06 (b_1 : \u03b3) (_ : b_1 \u2208 some b), s b_1) \u2260 0\n[PROOFSTEP]\nconvert h\n[GOAL]\ncase h.e'_2.h.e'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\nb : \u03b3\nh : m (s b) \u2260 0\n\u22a2 \u2a06 (b_1 : \u03b3) (_ : b_1 \u2208 some b), s b_1 = s b\n[PROOFSTEP]\nsimp [Set.ext_iff, encodek\u2082]\n[GOAL]\ncase refine'_2.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\nb : \u03b3\nh : b \u2208 support fun b => m (s b)\n\u22a2 (fun a => Option.get (decode\u2082 \u03b3 \u2191a) (_ : Option.isSome (decode\u2082 \u03b3 \u2191a) = true))\n { val := encode b, property := (_ : encode b \u2208 support fun i => m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)) } =\n b\n[PROOFSTEP]\nexact Option.get_of_mem _ (encodek\u2082 _)\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\n\u22a2 \u2200 (x : \u2191(support fun i => m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b))),\n m (s ((fun a => Option.get (decode\u2082 \u03b3 \u2191a) (_ : Option.isSome (decode\u2082 \u03b3 \u2191a) = true)) x)) =\n m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 \u2191x), s b)\n[PROOFSTEP]\nrintro \u27e8n, h\u27e9\n[GOAL]\ncase refine'_3.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\nn : \u2115\nh : n \u2208 support fun i => m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)\n\u22a2 m (s ((fun a => Option.get (decode\u2082 \u03b3 \u2191a) (_ : Option.isSome (decode\u2082 \u03b3 \u2191a) = true)) { val := n, property := h })) =\n m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 \u2191{ val := n, property := h }), s b)\n[PROOFSTEP]\ndsimp only [Subtype.coe_mk]\n[GOAL]\ncase refine'_3.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\nn : \u2115\nh : n \u2208 support fun i => m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)\n\u22a2 m (s (Option.get (decode\u2082 \u03b3 n) (_ : Option.isSome (decode\u2082 \u03b3 n) = true))) = m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b)\n[PROOFSTEP]\ntrans\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\nn : \u2115\nh : n \u2208 support fun i => m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)\n\u22a2 m (s (Option.get (decode\u2082 \u03b3 n) (_ : Option.isSome (decode\u2082 \u03b3 n) = true))) = ?m.392413\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\nn : \u2115\nh : n \u2208 support fun i => m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)\n\u22a2 ?m.392413 = m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b)\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\nn : \u2115\nh : n \u2208 support fun i => m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)\n\u22a2 \u03b1\n[PROOFSTEP]\nswap\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\nn : \u2115\nh : n \u2208 support fun i => m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)\n\u22a2 ?m.392413 = m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b)\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\nn : \u2115\nh : n \u2208 support fun i => m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)\n\u22a2 m (s (Option.get (decode\u2082 \u03b3 n) (_ : Option.isSome (decode\u2082 \u03b3 n) = true))) = ?m.392413\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\nn : \u2115\nh : n \u2208 support fun i => m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)\n\u22a2 \u03b1\n[PROOFSTEP]\nrw [show decode\u2082 \u03b3 n = _ from Option.get_mem (H n h)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\nn : \u2115\nh : n \u2208 support fun i => m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)\n\u22a2 m (s (Option.get (decode\u2082 \u03b3 n) (_ : Option.isSome (decode\u2082 \u03b3 n) = true))) =\n m (\u2a06 (b : \u03b3) (_ : b \u2208 some (Option.get (decode\u2082 \u03b3 n) (_ : Option.isSome (decode\u2082 \u03b3 n) = true))), s b)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Encodable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\ns : \u03b3 \u2192 \u03b2\nH : \u2200 (n : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 n), s b) \u2260 0 \u2192 Option.isSome (decode\u2082 \u03b3 n) = true\nn : \u2115\nh : n \u2208 support fun i => m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)\n\u22a2 s (Option.get (decode\u2082 \u03b3 n) (_ : Option.isSome (decode\u2082 \u03b3 n) = true)) =\n \u2a06 (b : \u03b3) (_ : b \u2208 some (Option.get (decode\u2082 \u03b3 n) (_ : Option.isSome (decode\u2082 \u03b3 n) = true))), s b\n[PROOFSTEP]\nsimp [ext_iff, -Option.some_get]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Countable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nm_iSup : \u2200 (s : \u2115 \u2192 \u03b2), R (m (\u2a06 (i : \u2115), s i)) (\u2211' (i : \u2115), m (s i))\ns : \u03b3 \u2192 \u03b2\n\u22a2 R (m (\u2a06 (b : \u03b3), s b)) (\u2211' (b : \u03b3), m (s b))\n[PROOFSTEP]\ncases nonempty_encodable \u03b3\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Countable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nm_iSup : \u2200 (s : \u2115 \u2192 \u03b2), R (m (\u2a06 (i : \u2115), s i)) (\u2211' (i : \u2115), m (s i))\ns : \u03b3 \u2192 \u03b2\nval\u271d : Encodable \u03b3\n\u22a2 R (m (\u2a06 (b : \u03b3), s b)) (\u2211' (b : \u03b3), m (s b))\n[PROOFSTEP]\nrw [\u2190 iSup_decode\u2082, \u2190 tsum_iSup_decode\u2082 _ m0 s]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Countable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nm_iSup : \u2200 (s : \u2115 \u2192 \u03b2), R (m (\u2a06 (i : \u2115), s i)) (\u2211' (i : \u2115), m (s i))\ns : \u03b3 \u2192 \u03b2\nval\u271d : Encodable \u03b3\n\u22a2 R (m (\u2a06 (i : \u2115) (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b)) (\u2211' (i : \u2115), m (\u2a06 (b : \u03b3) (_ : b \u2208 decode\u2082 \u03b3 i), s b))\n[PROOFSTEP]\nexact m_iSup _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Countable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nm_iSup : \u2200 (s : \u2115 \u2192 \u03b2), R (m (\u2a06 (i : \u2115), s i)) (\u2211' (i : \u2115), m (s i))\ns : \u03b4 \u2192 \u03b2\nt : Finset \u03b4\n\u22a2 R (m (\u2a06 (d : \u03b4) (_ : d \u2208 t), s d)) (\u2211 d in t, m (s d))\n[PROOFSTEP]\nrw [iSup_subtype', \u2190 Finset.tsum_subtype]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Countable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nm_iSup : \u2200 (s : \u2115 \u2192 \u03b2), R (m (\u2a06 (i : \u2115), s i)) (\u2211' (i : \u2115), m (s i))\ns : \u03b4 \u2192 \u03b2\nt : Finset \u03b4\n\u22a2 R (m (\u2a06 (x : { d // d \u2208 t }), s \u2191x)) (\u2211' (x : { x // x \u2208 t }), m (s \u2191x))\n[PROOFSTEP]\nexact rel_iSup_tsum m m0 R m_iSup _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Countable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nm_iSup : \u2200 (s : \u2115 \u2192 \u03b2), R (m (\u2a06 (i : \u2115), s i)) (\u2211' (i : \u2115), m (s i))\ns\u2081 s\u2082 : \u03b2\n\u22a2 R (m (s\u2081 \u2294 s\u2082)) (m s\u2081 + m s\u2082)\n[PROOFSTEP]\nconvert rel_iSup_tsum m m0 R m_iSup fun b => cond b s\u2081 s\u2082\n[GOAL]\ncase h.e'_1.h.e'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Countable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nm_iSup : \u2200 (s : \u2115 \u2192 \u03b2), R (m (\u2a06 (i : \u2115), s i)) (\u2211' (i : \u2115), m (s i))\ns\u2081 s\u2082 : \u03b2\n\u22a2 s\u2081 \u2294 s\u2082 = \u2a06 (b : Bool), bif b then s\u2081 else s\u2082\n[PROOFSTEP]\nsimp only [iSup_bool_eq, cond]\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : Countable \u03b3\ninst\u271d : CompleteLattice \u03b2\nm : \u03b2 \u2192 \u03b1\nm0 : m \u22a5 = 0\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nm_iSup : \u2200 (s : \u2115 \u2192 \u03b2), R (m (\u2a06 (i : \u2115), s i)) (\u2211' (i : \u2115), m (s i))\ns\u2081 s\u2082 : \u03b2\n\u22a2 m s\u2081 + m s\u2082 = \u2211' (b : Bool), m (bif b then s\u2081 else s\u2082)\n[PROOFSTEP]\nrw [tsum_fintype, Fintype.sum_bool, cond, cond]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : HasSum f a\n\u22a2 HasSum (fun b => -f b) (-a)\n[PROOFSTEP]\nsimpa only using h.map (-AddMonoidHom.id \u03b1) continuous_neg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nhf : Summable fun b => -f b\n\u22a2 Summable f\n[PROOFSTEP]\nsimpa only [neg_neg] using hf.neg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nhf : HasSum f a\u2081\nhg : HasSum g a\u2082\n\u22a2 HasSum (fun b => f b - g b) (a\u2081 - a\u2082)\n[PROOFSTEP]\nsimp only [sub_eq_add_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nhf : HasSum f a\u2081\nhg : HasSum g a\u2082\n\u22a2 HasSum (fun b => f b + -g b) (a\u2081 + -a\u2082)\n[PROOFSTEP]\nexact hf.add hg.neg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nhg : Summable g\nhfg : Summable fun b => f b - g b\n\u22a2 Summable f\n[PROOFSTEP]\nsimpa only [sub_add_cancel] using hfg.add hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nhfg : Summable fun b => f b - g b\nhf : Summable f\n\u22a2 Summable fun b => g b - f b\n[PROOFSTEP]\nsimpa only [neg_sub] using hfg.neg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nhf : HasSum f a\u2081\nb : \u03b2\ninst\u271d : DecidableEq \u03b2\na : \u03b1\n\u22a2 HasSum (Function.update f b a) (a - f b + a\u2081)\n[PROOFSTEP]\nconvert (hasSum_ite_eq b (a - f b)).add hf\n[GOAL]\ncase h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nhf : HasSum f a\u2081\nb : \u03b2\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nx\u271d : \u03b2\n\u22a2 Function.update f b a x\u271d = (if x\u271d = b then a - f b else 0) + f x\u271d\n[PROOFSTEP]\nrename_i b'\n[GOAL]\ncase h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nhf : HasSum f a\u2081\nb : \u03b2\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nb' : \u03b2\n\u22a2 Function.update f b a b' = (if b' = b then a - f b else 0) + f b'\n[PROOFSTEP]\nby_cases h : b' = b\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nhf : HasSum f a\u2081\nb : \u03b2\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nb' : \u03b2\nh : b' = b\n\u22a2 Function.update f b a b' = (if b' = b then a - f b else 0) + f b'\n[PROOFSTEP]\nrw [h, update_same]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nhf : HasSum f a\u2081\nb : \u03b2\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nb' : \u03b2\nh : b' = b\n\u22a2 a = (if b = b then a - f b else 0) + f b\n[PROOFSTEP]\nsimp [eq_self_iff_true, if_true, sub_add_cancel]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nhf : HasSum f a\u2081\nb : \u03b2\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nb' : \u03b2\nh : \u00acb' = b\n\u22a2 Function.update f b a b' = (if b' = b then a - f b else 0) + f b'\n[PROOFSTEP]\nsimp only [h, update_noteq, if_false, Ne.def, zero_add, not_false_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ns : Set \u03b2\nhf : HasSum (f \u2218 Subtype.val) a\u2081\n\u22a2 HasSum (f \u2218 Subtype.val) a\u2082 \u2194 HasSum f (a\u2081 + a\u2082)\n[PROOFSTEP]\nrefine' \u27e8fun h => hf.add_compl h, fun h => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ns : Set \u03b2\nhf : HasSum (f \u2218 Subtype.val) a\u2081\nh : HasSum f (a\u2081 + a\u2082)\n\u22a2 HasSum (f \u2218 Subtype.val) a\u2082\n[PROOFSTEP]\nrw [hasSum_subtype_iff_indicator] at hf \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ns : Set \u03b2\nhf : HasSum (Set.indicator s f) a\u2081\nh : HasSum f (a\u2081 + a\u2082)\n\u22a2 HasSum (Set.indicator s\u1d9c f) a\u2082\n[PROOFSTEP]\nrw [Set.indicator_compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ns : Set \u03b2\nhf : HasSum (Set.indicator s f) a\u2081\nh : HasSum f (a\u2081 + a\u2082)\n\u22a2 HasSum (f - Set.indicator s f) a\u2082\n[PROOFSTEP]\nsimpa only [add_sub_cancel'] using h.sub hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ns : Set \u03b2\nhf : HasSum (f \u2218 Subtype.val) a\u2081\n\u22a2 HasSum f (a\u2081 + (a\u2082 - a\u2081)) \u2194 HasSum f a\u2082\n[PROOFSTEP]\nrw [add_sub_cancel'_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ns : Finset \u03b2\n\u22a2 HasSum f (\u2211 b in s, f b + a) \u2194 HasSum f (a + \u2211 i in s, f i)\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : DecidableEq \u03b2\nhf : HasSum f a\nb : \u03b2\n\u22a2 HasSum (fun n => if n = b then 0 else f n) (a - f b)\n[PROOFSTEP]\nconvert hf.update b 0 using 1\n[GOAL]\ncase h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : DecidableEq \u03b2\nhf : HasSum f a\nb : \u03b2\n\u22a2 (fun n => if n = b then 0 else f n) = update f b 0\n[PROOFSTEP]\next n\n[GOAL]\ncase h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : DecidableEq \u03b2\nhf : HasSum f a\nb n : \u03b2\n\u22a2 (if n = b then 0 else f n) = update f b 0 n\n[PROOFSTEP]\nrw [Function.update_apply]\n[GOAL]\ncase h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : DecidableEq \u03b2\nhf : HasSum f a\nb : \u03b2\n\u22a2 a - f b = 0 - f b + a\n[PROOFSTEP]\nrw [sub_add_eq_add_sub, zero_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\n\u22a2 \u2211' (b : \u03b2), -f b = -\u2211' (b : \u03b2), f b\n[PROOFSTEP]\nby_cases hf : Summable f\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\nhf : Summable f\n\u22a2 \u2211' (b : \u03b2), -f b = -\u2211' (b : \u03b2), f b\n[PROOFSTEP]\nexact hf.hasSum.neg.tsum_eq\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\nhf : \u00acSummable f\n\u22a2 \u2211' (b : \u03b2), -f b = -\u2211' (b : \u03b2), f b\n[PROOFSTEP]\nsimp [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable (mt Summable.of_neg hf)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommGroup \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : DecidableEq \u03b2\nhf : Summable f\nb : \u03b2\n\u22a2 \u2211' (n : \u03b2), f n = f b + \u2211' (n : \u03b2), if n = b then 0 else f n\n[PROOFSTEP]\nrw [(hasSum_ite_sub_hasSum hf.hasSum b).tsum_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommGroup \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : DecidableEq \u03b2\nhf : Summable f\nb : \u03b2\n\u22a2 \u2211' (n : \u03b2), f n = f b + (\u2211' (b : \u03b2), f b - f b)\n[PROOFSTEP]\nexact (add_sub_cancel'_right _ _).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nf : \u2115 \u2192 \u03b1\nk : \u2115\na : \u03b1\n\u22a2 HasSum (fun n => f (n + k)) a \u2194 HasSum f (a + \u2211 i in range k, f i)\n[PROOFSTEP]\nrefine' Iff.trans _ (range k).hasSum_compl_iff\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nf : \u2115 \u2192 \u03b1\nk : \u2115\na : \u03b1\n\u22a2 HasSum (fun n => f (n + k)) a \u2194 HasSum (fun x => f \u2191x) a\n[PROOFSTEP]\nrw [\u2190 (notMemRangeEquiv k).symm.hasSum_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nf : \u2115 \u2192 \u03b1\nk : \u2115\na : \u03b1\n\u22a2 HasSum (fun n => f (n + k)) a \u2194 HasSum ((fun x => f \u2191x) \u2218 \u2191(notMemRangeEquiv k).symm) a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nf : \u2115 \u2192 \u03b1\nk : \u2115\na : \u03b1\n\u22a2 HasSum (fun n => f (n + k)) (a - \u2211 i in range k, f i) \u2194 HasSum f a\n[PROOFSTEP]\nsimp [hasSum_nat_add_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nM : Type u_5\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ContinuousAdd M\nf : \u2115 \u2192 M\nk : \u2115\na : M\nh : HasSum (fun n => f (n + k)) a\n\u22a2 HasSum f (\u2211 i in range k, f i + a)\n[PROOFSTEP]\nrefine ((range k).hasSum f).add_compl ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nM : Type u_5\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ContinuousAdd M\nf : \u2115 \u2192 M\nk : \u2115\na : M\nh : HasSum (fun n => f (n + k)) a\n\u22a2 HasSum (f \u2218 Subtype.val) a\n[PROOFSTEP]\nrwa [\u2190 (notMemRangeEquiv k).symm.hasSum_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2076 : AddCommGroup \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nM : Type u_5\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ContinuousAdd M\ninst\u271d : T2Space M\nf : \u2115 \u2192 M\nhf : Summable fun n => f (n + 1)\n\u22a2 \u2211' (b : \u2115), f b = f 0 + \u2211' (b : \u2115), f (b + 1)\n[PROOFSTEP]\nsimpa only [sum_range_one] using (sum_add_tsum_nat_add' hf).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\nf : \u2115 \u2192 \u03b1\n\u22a2 Tendsto (fun i => \u2211' (k : \u2115), f (k + i)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nby_cases hf : Summable f\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\nf : \u2115 \u2192 \u03b1\nhf : Summable f\n\u22a2 Tendsto (fun i => \u2211' (k : \u2115), f (k + i)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave h\u2080 : (fun i => \u2211' i, f i - \u2211 j in range i, f j) = fun i => \u2211' k : \u2115, f (k + i) :=\n by\n ext1 i\n rw [sub_eq_iff_eq_add, add_comm, sum_add_tsum_nat_add i hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\nf : \u2115 \u2192 \u03b1\nhf : Summable f\n\u22a2 (fun i => \u2211' (i : \u2115), f i - \u2211 j in range i, f j) = fun i => \u2211' (k : \u2115), f (k + i)\n[PROOFSTEP]\next1 i\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\nf : \u2115 \u2192 \u03b1\nhf : Summable f\ni : \u2115\n\u22a2 \u2211' (i : \u2115), f i - \u2211 j in range i, f j = \u2211' (k : \u2115), f (k + i)\n[PROOFSTEP]\nrw [sub_eq_iff_eq_add, add_comm, sum_add_tsum_nat_add i hf]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\nf : \u2115 \u2192 \u03b1\nhf : Summable f\nh\u2080 : (fun i => \u2211' (i : \u2115), f i - \u2211 j in range i, f j) = fun i => \u2211' (k : \u2115), f (k + i)\n\u22a2 Tendsto (fun i => \u2211' (k : \u2115), f (k + i)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave h\u2081 : Tendsto (fun _ : \u2115 => \u2211' i, f i) atTop (\ud835\udcdd (\u2211' i, f i)) := tendsto_const_nhds\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\nf : \u2115 \u2192 \u03b1\nhf : Summable f\nh\u2080 : (fun i => \u2211' (i : \u2115), f i - \u2211 j in range i, f j) = fun i => \u2211' (k : \u2115), f (k + i)\nh\u2081 : Tendsto (fun x => \u2211' (i : \u2115), f i) atTop (\ud835\udcdd (\u2211' (i : \u2115), f i))\n\u22a2 Tendsto (fun i => \u2211' (k : \u2115), f (k + i)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa only [h\u2080, sub_self] using Tendsto.sub h\u2081 hf.hasSum.tendsto_sum_nat\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\nf : \u2115 \u2192 \u03b1\nhf : \u00acSummable f\n\u22a2 Tendsto (fun i => \u2211' (k : \u2115), f (k + i)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert tendsto_const_nhds (\u03b1 := \u03b1) (\u03b2 := \u2115) (a := 0) (f := atTop)\n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\nf : \u2115 \u2192 \u03b1\nhf : \u00acSummable f\nx\u271d : \u2115\n\u22a2 \u2211' (k : \u2115), f (k + x\u271d) = 0\n[PROOFSTEP]\nrename_i i\n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\nf : \u2115 \u2192 \u03b1\nhf : \u00acSummable f\ni : \u2115\n\u22a2 \u2211' (k : \u2115), f (k + i) = 0\n[PROOFSTEP]\nrw [\u2190 summable_nat_add_iff i] at hf \n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\nf : \u2115 \u2192 \u03b1\ni : \u2115\nhf : \u00acSummable fun n => f (n + i)\n\u22a2 \u2211' (k : \u2115), f (k + i) = 0\n[PROOFSTEP]\nexact tsum_eq_zero_of_not_summable hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf g : \u2115 \u2192 \u03b1\nhf : HasSum f a\nhg : HasSum g b\n\u22a2 HasSum (Int.rec f g) (a + b)\n[PROOFSTEP]\nhave h\u2081 : Injective ((\u2191) : \u2115 \u2192 \u2124) := @Int.ofNat.inj\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf g : \u2115 \u2192 \u03b1\nhf : HasSum f a\nhg : HasSum g b\nh\u2081 : Injective Nat.cast\n\u22a2 HasSum (Int.rec f g) (a + b)\n[PROOFSTEP]\nhave h\u2082 : Injective Int.negSucc := @Int.negSucc.inj\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf g : \u2115 \u2192 \u03b1\nhf : HasSum f a\nhg : HasSum g b\nh\u2081 : Injective Nat.cast\nh\u2082 : Injective Int.negSucc\n\u22a2 HasSum (Int.rec f g) (a + b)\n[PROOFSTEP]\nhave : IsCompl (Set.range ((\u2191) : \u2115 \u2192 \u2124)) (Set.range Int.negSucc) :=\n by\n constructor\n \u00b7 rw [disjoint_iff_inf_le]\n rintro _ \u27e8\u27e8i, rfl\u27e9, \u27e8j, \u27e8\u27e9\u27e9\u27e9\n \u00b7 rw [codisjoint_iff_le_sup]\n rintro (i | j) _\n exacts [Or.inl \u27e8_, rfl\u27e9, Or.inr \u27e8_, rfl\u27e9]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf g : \u2115 \u2192 \u03b1\nhf : HasSum f a\nhg : HasSum g b\nh\u2081 : Injective Nat.cast\nh\u2082 : Injective Int.negSucc\n\u22a2 IsCompl (Set.range Nat.cast) (Set.range Int.negSucc)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase disjoint\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf g : \u2115 \u2192 \u03b1\nhf : HasSum f a\nhg : HasSum g b\nh\u2081 : Injective Nat.cast\nh\u2082 : Injective Int.negSucc\n\u22a2 Disjoint (Set.range Nat.cast) (Set.range Int.negSucc)\n[PROOFSTEP]\nrw [disjoint_iff_inf_le]\n[GOAL]\ncase disjoint\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf g : \u2115 \u2192 \u03b1\nhf : HasSum f a\nhg : HasSum g b\nh\u2081 : Injective Nat.cast\nh\u2082 : Injective Int.negSucc\n\u22a2 Set.range Nat.cast \u2293 Set.range Int.negSucc \u2264 \u22a5\n[PROOFSTEP]\nrintro _ \u27e8\u27e8i, rfl\u27e9, \u27e8j, \u27e8\u27e9\u27e9\u27e9\n[GOAL]\ncase codisjoint\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf g : \u2115 \u2192 \u03b1\nhf : HasSum f a\nhg : HasSum g b\nh\u2081 : Injective Nat.cast\nh\u2082 : Injective Int.negSucc\n\u22a2 Codisjoint (Set.range Nat.cast) (Set.range Int.negSucc)\n[PROOFSTEP]\nrw [codisjoint_iff_le_sup]\n[GOAL]\ncase codisjoint\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf g : \u2115 \u2192 \u03b1\nhf : HasSum f a\nhg : HasSum g b\nh\u2081 : Injective Nat.cast\nh\u2082 : Injective Int.negSucc\n\u22a2 \u22a4 \u2264 Set.range Nat.cast \u2294 Set.range Int.negSucc\n[PROOFSTEP]\nrintro (i | j) _\n[GOAL]\ncase codisjoint.ofNat\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf g : \u2115 \u2192 \u03b1\nhf : HasSum f a\nhg : HasSum g b\nh\u2081 : Injective Nat.cast\nh\u2082 : Injective Int.negSucc\ni : \u2115\na\u271d : Int.ofNat i \u2208 \u22a4\n\u22a2 Int.ofNat i \u2208 Set.range Nat.cast \u2294 Set.range Int.negSucc\ncase codisjoint.negSucc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf g : \u2115 \u2192 \u03b1\nhf : HasSum f a\nhg : HasSum g b\nh\u2081 : Injective Nat.cast\nh\u2082 : Injective Int.negSucc\nj : \u2115\na\u271d : Int.negSucc j \u2208 \u22a4\n\u22a2 Int.negSucc j \u2208 Set.range Nat.cast \u2294 Set.range Int.negSucc\n[PROOFSTEP]\nexacts [Or.inl \u27e8_, rfl\u27e9, Or.inr \u27e8_, rfl\u27e9]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf g : \u2115 \u2192 \u03b1\nhf : HasSum f a\nhg : HasSum g b\nh\u2081 : Injective Nat.cast\nh\u2082 : Injective Int.negSucc\nthis : IsCompl (Set.range Nat.cast) (Set.range Int.negSucc)\n\u22a2 HasSum (Int.rec f g) (a + b)\n[PROOFSTEP]\nexact HasSum.add_isCompl this (h\u2081.hasSum_range_iff.mpr hf) (h\u2082.hasSum_range_iff.mpr hg)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf : \u2124 \u2192 \u03b1\nhnonneg : HasSum (fun n => f \u2191n) a\nhneg : HasSum (fun n => f (-\u2191(Nat.succ n))) b\n\u22a2 HasSum f (a + b)\n[PROOFSTEP]\nsimp_rw [\u2190 Int.negSucc_coe] at hneg \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf : \u2124 \u2192 \u03b1\nhnonneg : HasSum (fun n => f \u2191n) a\nhneg : HasSum (fun n => f (Int.negSucc n)) b\n\u22a2 HasSum f (a + b)\n[PROOFSTEP]\nconvert hnonneg.int_rec hneg using 1\n[GOAL]\ncase h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf : \u2124 \u2192 \u03b1\nhnonneg : HasSum (fun n => f \u2191n) a\nhneg : HasSum (fun n => f (Int.negSucc n)) b\n\u22a2 f = Int.rec (fun n => f \u2191n) fun n => f (Int.negSucc n)\n[PROOFSTEP]\next (i | j)\n[GOAL]\ncase h.e'_5.h.ofNat\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf : \u2124 \u2192 \u03b1\nhnonneg : HasSum (fun n => f \u2191n) a\nhneg : HasSum (fun n => f (Int.negSucc n)) b\ni : \u2115\n\u22a2 f (Int.ofNat i) = Int.rec (fun n => f \u2191n) (fun n => f (Int.negSucc n)) (Int.ofNat i)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_5.h.negSucc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf : \u2124 \u2192 \u03b1\nhnonneg : HasSum (fun n => f \u2191n) a\nhneg : HasSum (fun n => f (Int.negSucc n)) b\nj : \u2115\n\u22a2 f (Int.negSucc j) = Int.rec (fun n => f \u2191n) (fun n => f (Int.negSucc n)) (Int.negSucc j)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf : \u2124 \u2192 \u03b1\nhpos : HasSum (fun n => f (\u2191n + 1)) a\nhneg : HasSum (fun n => f (-\u2191(Nat.succ n))) b\n\u22a2 \u2200 (g : \u2115 \u2192 \u03b1), HasSum (fun k => g (k + 1)) a \u2192 HasSum g (a + g 0)\n[PROOFSTEP]\nintro g hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nf\u271d g\u271d : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 b : \u03b1\nf : \u2124 \u2192 \u03b1\nhpos : HasSum (fun n => f (\u2191n + 1)) a\nhneg : HasSum (fun n => f (-\u2191(Nat.succ n))) b\ng : \u2115 \u2192 \u03b1\nhg : HasSum (fun k => g (k + 1)) a\n\u22a2 HasSum g (a + g 0)\n[PROOFSTEP]\nsimpa using (hasSum_nat_add_iff _).mp hg\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\n\u22a2 HasSum (fun n => f \u2191n + f (-\u2191n)) (a + f 0)\n[PROOFSTEP]\napply (hf.add (hasSum_ite_eq (0 : \u2124) (f 0))).hasSum_of_sum_eq fun u => ?_\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\n\u22a2 \u2203 v,\n \u2200 (v' : Finset \u2115), v \u2286 v' \u2192 \u2203 u', u \u2286 u' \u2227 \u2211 x in u', (f x + if x = 0 then f 0 else 0) = \u2211 b in v', (f \u2191b + f (-\u2191b))\n[PROOFSTEP]\nrefine' \u27e8u.image Int.natAbs, fun v' hv' => _\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\n\u22a2 \u2203 u', u \u2286 u' \u2227 \u2211 x in u', (f x + if x = 0 then f 0 else 0) = \u2211 b in v', (f \u2191b + f (-\u2191b))\n[PROOFSTEP]\nlet u1 := v'.image fun x : \u2115 => (x : \u2124)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\n\u22a2 \u2203 u', u \u2286 u' \u2227 \u2211 x in u', (f x + if x = 0 then f 0 else 0) = \u2211 b in v', (f \u2191b + f (-\u2191b))\n[PROOFSTEP]\nlet u2 := v'.image fun x : \u2115 => -(x : \u2124)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\n\u22a2 \u2203 u', u \u2286 u' \u2227 \u2211 x in u', (f x + if x = 0 then f 0 else 0) = \u2211 b in v', (f \u2191b + f (-\u2191b))\n[PROOFSTEP]\nhave A : u \u2286 u1 \u222a u2 := by\n intro x hx\n simp only [mem_union, mem_image, exists_prop]\n rcases le_total 0 x with (h'x | h'x)\n \u00b7 left\n refine' \u27e8Int.natAbs x, hv' _, _\u27e9\n \u00b7 simp only [mem_image, exists_prop]\n exact \u27e8x, hx, rfl\u27e9\n \u00b7 simp only [h'x, Int.coe_natAbs, abs_eq_self]\n \u00b7 right\n refine' \u27e8Int.natAbs x, hv' _, _\u27e9\n \u00b7 simp only [mem_image, exists_prop]\n exact \u27e8x, hx, rfl\u27e9\n \u00b7 simp only [abs_of_nonpos h'x, Int.coe_natAbs, neg_neg]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\n\u22a2 u \u2286 u1 \u222a u2\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nx : \u2124\nhx : x \u2208 u\n\u22a2 x \u2208 u1 \u222a u2\n[PROOFSTEP]\nsimp only [mem_union, mem_image, exists_prop]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nx : \u2124\nhx : x \u2208 u\n\u22a2 (\u2203 a, a \u2208 v' \u2227 \u2191a = x) \u2228 \u2203 a, a \u2208 v' \u2227 -\u2191a = x\n[PROOFSTEP]\nrcases le_total 0 x with (h'x | h'x)\n[GOAL]\ncase inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nx : \u2124\nhx : x \u2208 u\nh'x : 0 \u2264 x\n\u22a2 (\u2203 a, a \u2208 v' \u2227 \u2191a = x) \u2228 \u2203 a, a \u2208 v' \u2227 -\u2191a = x\n[PROOFSTEP]\nleft\n[GOAL]\ncase inl.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nx : \u2124\nhx : x \u2208 u\nh'x : 0 \u2264 x\n\u22a2 \u2203 a, a \u2208 v' \u2227 \u2191a = x\n[PROOFSTEP]\nrefine' \u27e8Int.natAbs x, hv' _, _\u27e9\n[GOAL]\ncase inl.h.refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nx : \u2124\nhx : x \u2208 u\nh'x : 0 \u2264 x\n\u22a2 Int.natAbs x \u2208 image Int.natAbs u\n[PROOFSTEP]\nsimp only [mem_image, exists_prop]\n[GOAL]\ncase inl.h.refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nx : \u2124\nhx : x \u2208 u\nh'x : 0 \u2264 x\n\u22a2 \u2203 a, a \u2208 u \u2227 Int.natAbs a = Int.natAbs x\n[PROOFSTEP]\nexact \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase inl.h.refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nx : \u2124\nhx : x \u2208 u\nh'x : 0 \u2264 x\n\u22a2 \u2191(Int.natAbs x) = x\n[PROOFSTEP]\nsimp only [h'x, Int.coe_natAbs, abs_eq_self]\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nx : \u2124\nhx : x \u2208 u\nh'x : x \u2264 0\n\u22a2 (\u2203 a, a \u2208 v' \u2227 \u2191a = x) \u2228 \u2203 a, a \u2208 v' \u2227 -\u2191a = x\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nx : \u2124\nhx : x \u2208 u\nh'x : x \u2264 0\n\u22a2 \u2203 a, a \u2208 v' \u2227 -\u2191a = x\n[PROOFSTEP]\nrefine' \u27e8Int.natAbs x, hv' _, _\u27e9\n[GOAL]\ncase inr.h.refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nx : \u2124\nhx : x \u2208 u\nh'x : x \u2264 0\n\u22a2 Int.natAbs x \u2208 image Int.natAbs u\n[PROOFSTEP]\nsimp only [mem_image, exists_prop]\n[GOAL]\ncase inr.h.refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nx : \u2124\nhx : x \u2208 u\nh'x : x \u2264 0\n\u22a2 \u2203 a, a \u2208 u \u2227 Int.natAbs a = Int.natAbs x\n[PROOFSTEP]\nexact \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase inr.h.refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nx : \u2124\nhx : x \u2208 u\nh'x : x \u2264 0\n\u22a2 -\u2191(Int.natAbs x) = x\n[PROOFSTEP]\nsimp only [abs_of_nonpos h'x, Int.coe_natAbs, neg_neg]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\n\u22a2 \u2203 u', u \u2286 u' \u2227 \u2211 x in u', (f x + if x = 0 then f 0 else 0) = \u2211 b in v', (f \u2191b + f (-\u2191b))\n[PROOFSTEP]\nrefine' \u27e8u1 \u222a u2, A, _\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\n\u22a2 \u2211 x in u1 \u222a u2, (f x + if x = 0 then f 0 else 0) = \u2211 b in v', (f \u2191b + f (-\u2191b))\n[PROOFSTEP]\ncalc\n (\u2211 x in u1 \u222a u2, (f x + ite (x = 0) (f 0) 0)) = (\u2211 x in u1 \u222a u2, f x) + \u2211 x in u1 \u2229 u2, f x :=\n by\n rw [sum_add_distrib]\n congr 1\n refine' (sum_subset_zero_on_sdiff inter_subset_union _ _).symm\n \u00b7 intro x hx\n suffices x \u2260 0 by simp only [this, if_false]\n rintro rfl\n simp at hx \n \u00b7 intro x hx\n simp only [mem_inter, mem_image, exists_prop] at hx \n have : x = 0 := by\n apply le_antisymm\n \u00b7 rcases hx.2 with \u27e8a, _, rfl\u27e9\n simp only [Right.neg_nonpos_iff, Nat.cast_nonneg]\n \u00b7 rcases hx.1 with \u27e8a, _, rfl\u27e9\n simp only [Nat.cast_nonneg]\n simp only [this, eq_self_iff_true, if_true]\n _ = (\u2211 x in u1, f x) + \u2211 x in u2, f x := sum_union_inter\n _ = (\u2211 b in v', f b) + \u2211 b in v', f (-b) := by simp\n _ = \u2211 b in v', (f b + f (-b)) := sum_add_distrib.symm\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\n\u22a2 \u2211 x in u1 \u222a u2, (f x + if x = 0 then f 0 else 0) = \u2211 x in u1 \u222a u2, f x + \u2211 x in u1 \u2229 u2, f x\n[PROOFSTEP]\nrw [sum_add_distrib]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\n\u22a2 (\u2211 x in u1 \u222a u2, f x + \u2211 x in u1 \u222a u2, if x = 0 then f 0 else 0) = \u2211 x in u1 \u222a u2, f x + \u2211 x in u1 \u2229 u2, f x\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\n\u22a2 (\u2211 x in u1 \u222a u2, if x = 0 then f 0 else 0) = \u2211 x in u1 \u2229 u2, f x\n[PROOFSTEP]\nrefine' (sum_subset_zero_on_sdiff inter_subset_union _ _).symm\n[GOAL]\ncase e_a.refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\n\u22a2 \u2200 (x : \u2124), x \u2208 (u1 \u222a u2) \\ (u1 \u2229 u2) \u2192 (if x = 0 then f 0 else 0) = 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase e_a.refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\nx : \u2124\nhx : x \u2208 (u1 \u222a u2) \\ (u1 \u2229 u2)\n\u22a2 (if x = 0 then f 0 else 0) = 0\n[PROOFSTEP]\nsuffices x \u2260 0 by simp only [this, if_false]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\nx : \u2124\nhx : x \u2208 (u1 \u222a u2) \\ (u1 \u2229 u2)\nthis : x \u2260 0\n\u22a2 (if x = 0 then f 0 else 0) = 0\n[PROOFSTEP]\nsimp only [this, if_false]\n[GOAL]\ncase e_a.refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\nx : \u2124\nhx : x \u2208 (u1 \u222a u2) \\ (u1 \u2229 u2)\n\u22a2 x \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase e_a.refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\nhx : 0 \u2208 (u1 \u222a u2) \\ (u1 \u2229 u2)\n\u22a2 False\n[PROOFSTEP]\nsimp at hx \n[GOAL]\ncase e_a.refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\n\u22a2 \u2200 (x : \u2124), x \u2208 u1 \u2229 u2 \u2192 f x = if x = 0 then f 0 else 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase e_a.refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\nx : \u2124\nhx : x \u2208 u1 \u2229 u2\n\u22a2 f x = if x = 0 then f 0 else 0\n[PROOFSTEP]\nsimp only [mem_inter, mem_image, exists_prop] at hx \n[GOAL]\ncase e_a.refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\nx : \u2124\nhx : (\u2203 a, a \u2208 v' \u2227 \u2191a = x) \u2227 \u2203 a, a \u2208 v' \u2227 -\u2191a = x\n\u22a2 f x = if x = 0 then f 0 else 0\n[PROOFSTEP]\nhave : x = 0 := by\n apply le_antisymm\n \u00b7 rcases hx.2 with \u27e8a, _, rfl\u27e9\n simp only [Right.neg_nonpos_iff, Nat.cast_nonneg]\n \u00b7 rcases hx.1 with \u27e8a, _, rfl\u27e9\n simp only [Nat.cast_nonneg]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\nx : \u2124\nhx : (\u2203 a, a \u2208 v' \u2227 \u2191a = x) \u2227 \u2203 a, a \u2208 v' \u2227 -\u2191a = x\n\u22a2 x = 0\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\nx : \u2124\nhx : (\u2203 a, a \u2208 v' \u2227 \u2191a = x) \u2227 \u2203 a, a \u2208 v' \u2227 -\u2191a = x\n\u22a2 x \u2264 0\n[PROOFSTEP]\nrcases hx.2 with \u27e8a, _, rfl\u27e9\n[GOAL]\ncase a.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d\u00b9 a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na\u271d : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\u271d\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\na : \u2115\nleft\u271d : a \u2208 v'\nhx : (\u2203 a_1, a_1 \u2208 v' \u2227 \u2191a_1 = -\u2191a) \u2227 \u2203 a_1, a_1 \u2208 v' \u2227 -\u2191a_1 = -\u2191a\n\u22a2 -\u2191a \u2264 0\n[PROOFSTEP]\nsimp only [Right.neg_nonpos_iff, Nat.cast_nonneg]\n[GOAL]\ncase a\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\nx : \u2124\nhx : (\u2203 a, a \u2208 v' \u2227 \u2191a = x) \u2227 \u2203 a, a \u2208 v' \u2227 -\u2191a = x\n\u22a2 0 \u2264 x\n[PROOFSTEP]\nrcases hx.1 with \u27e8a, _, rfl\u27e9\n[GOAL]\ncase a.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d\u00b9 a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na\u271d : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\u271d\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\na : \u2115\nleft\u271d : a \u2208 v'\nhx : (\u2203 a_1, a_1 \u2208 v' \u2227 \u2191a_1 = \u2191a) \u2227 \u2203 a_1, a_1 \u2208 v' \u2227 -\u2191a_1 = \u2191a\n\u22a2 0 \u2264 \u2191a\n[PROOFSTEP]\nsimp only [Nat.cast_nonneg]\n[GOAL]\ncase e_a.refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\nx : \u2124\nhx : (\u2203 a, a \u2208 v' \u2227 \u2191a = x) \u2227 \u2203 a, a \u2208 v' \u2227 -\u2191a = x\nthis : x = 0\n\u22a2 f x = if x = 0 then f 0 else 0\n[PROOFSTEP]\nsimp only [this, eq_self_iff_true, if_true]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : AddCommGroup \u03b1\u271d\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalAddGroup \u03b1\u271d\nf\u271d g : \u03b2 \u2192 \u03b1\u271d\na\u271d a\u2081 a\u2082 : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : ContinuousAdd \u03b1\na : \u03b1\nf : \u2124 \u2192 \u03b1\nhf : HasSum f a\nu : Finset \u2124\nv' : Finset \u2115\nhv' : image Int.natAbs u \u2286 v'\nu1 : Finset \u2124 := image (fun x => \u2191x) v'\nu2 : Finset \u2124 := image (fun x => -\u2191x) v'\nA : u \u2286 u1 \u222a u2\n\u22a2 \u2211 x in u1, f x + \u2211 x in u2, f x = \u2211 b in v', f \u2191b + \u2211 b in v', f (-\u2191b)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\n\u22a2 (CauchySeq fun s => \u2211 b in s, f b) \u2194 \u2200 (e : Set \u03b1), e \u2208 \ud835\udcdd 0 \u2192 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\n[PROOFSTEP]\nsimp only [CauchySeq, cauchy_map_iff, and_iff_right atTop_neBot, prod_atTop_atTop_eq, uniformity_eq_comap_nhds_zero \u03b1,\n tendsto_comap_iff, (\u00b7 \u2218 \u00b7), atTop_neBot, true_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\n\u22a2 Tendsto (fun x => \u2211 b in x.snd, f b - \u2211 b in x.fst, f b) atTop (\ud835\udcdd 0) \u2194\n \u2200 (e : Set \u03b1), e \u2208 \ud835\udcdd 0 \u2192 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\n[PROOFSTEP]\nrw [tendsto_atTop']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\n\u22a2 (\u2200 (s : Set \u03b1), s \u2208 \ud835\udcdd 0 \u2192 \u2203 a, \u2200 (b : Finset \u03b2 \u00d7 Finset \u03b2), b \u2265 a \u2192 \u2211 b in b.snd, f b - \u2211 b in b.fst, f b \u2208 s) \u2194\n \u2200 (e : Set \u03b1), e \u2208 \ud835\udcdd 0 \u2192 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\n\u22a2 (\u2200 (s : Set \u03b1), s \u2208 \ud835\udcdd 0 \u2192 \u2203 a, \u2200 (b : Finset \u03b2 \u00d7 Finset \u03b2), b \u2265 a \u2192 \u2211 b in b.snd, f b - \u2211 b in b.fst, f b \u2208 s) \u2192\n \u2200 (e : Set \u03b1), e \u2208 \ud835\udcdd 0 \u2192 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\n[PROOFSTEP]\nintro h e he\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 (s : Set \u03b1), s \u2208 \ud835\udcdd 0 \u2192 \u2203 a, \u2200 (b : Finset \u03b2 \u00d7 Finset \u03b2), b \u2265 a \u2192 \u2211 b in b.snd, f b - \u2211 b in b.fst, f b \u2208 s\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\n\u22a2 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\n[PROOFSTEP]\nobtain \u27e8\u27e8s\u2081, s\u2082\u27e9, h\u27e9 := h e he\n[GOAL]\ncase mp.intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 (s : Set \u03b1), s \u2208 \ud835\udcdd 0 \u2192 \u2203 a, \u2200 (b : Finset \u03b2 \u00d7 Finset \u03b2), b \u2265 a \u2192 \u2211 b in b.snd, f b - \u2211 b in b.fst, f b \u2208 s\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\ns\u2081 s\u2082 : Finset \u03b2\nh : \u2200 (b : Finset \u03b2 \u00d7 Finset \u03b2), b \u2265 (s\u2081, s\u2082) \u2192 \u2211 b in b.snd, f b - \u2211 b in b.fst, f b \u2208 e\n\u22a2 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\n[PROOFSTEP]\nuse s\u2081 \u222a s\u2082\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 (s : Set \u03b1), s \u2208 \ud835\udcdd 0 \u2192 \u2203 a, \u2200 (b : Finset \u03b2 \u00d7 Finset \u03b2), b \u2265 a \u2192 \u2211 b in b.snd, f b - \u2211 b in b.fst, f b \u2208 s\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\ns\u2081 s\u2082 : Finset \u03b2\nh : \u2200 (b : Finset \u03b2 \u00d7 Finset \u03b2), b \u2265 (s\u2081, s\u2082) \u2192 \u2211 b in b.snd, f b - \u2211 b in b.fst, f b \u2208 e\n\u22a2 \u2200 (t : Finset \u03b2), Disjoint t (s\u2081 \u222a s\u2082) \u2192 \u2211 b in t, f b \u2208 e\n[PROOFSTEP]\nintro t ht\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 (s : Set \u03b1), s \u2208 \ud835\udcdd 0 \u2192 \u2203 a, \u2200 (b : Finset \u03b2 \u00d7 Finset \u03b2), b \u2265 a \u2192 \u2211 b in b.snd, f b - \u2211 b in b.fst, f b \u2208 s\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\ns\u2081 s\u2082 : Finset \u03b2\nh : \u2200 (b : Finset \u03b2 \u00d7 Finset \u03b2), b \u2265 (s\u2081, s\u2082) \u2192 \u2211 b in b.snd, f b - \u2211 b in b.fst, f b \u2208 e\nt : Finset \u03b2\nht : Disjoint t (s\u2081 \u222a s\u2082)\n\u22a2 \u2211 b in t, f b \u2208 e\n[PROOFSTEP]\nspecialize h (s\u2081 \u222a s\u2082, s\u2081 \u222a s\u2082 \u222a t) \u27e8le_sup_left, le_sup_of_le_left le_sup_right\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 (s : Set \u03b1), s \u2208 \ud835\udcdd 0 \u2192 \u2203 a, \u2200 (b : Finset \u03b2 \u00d7 Finset \u03b2), b \u2265 a \u2192 \u2211 b in b.snd, f b - \u2211 b in b.fst, f b \u2208 s\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\ns\u2081 s\u2082 t : Finset \u03b2\nht : Disjoint t (s\u2081 \u222a s\u2082)\nh : \u2211 b in (s\u2081 \u222a s\u2082, s\u2081 \u222a s\u2082 \u222a t).snd, f b - \u2211 b in (s\u2081 \u222a s\u2082, s\u2081 \u222a s\u2082 \u222a t).fst, f b \u2208 e\n\u22a2 \u2211 b in t, f b \u2208 e\n[PROOFSTEP]\nsimpa only [Finset.sum_union ht.symm, add_sub_cancel'] using h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\n\u22a2 (\u2200 (e : Set \u03b1), e \u2208 \ud835\udcdd 0 \u2192 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e) \u2192\n \u2200 (s : Set \u03b1), s \u2208 \ud835\udcdd 0 \u2192 \u2203 a, \u2200 (b : Finset \u03b2 \u00d7 Finset \u03b2), b \u2265 a \u2192 \u2211 b in b.snd, f b - \u2211 b in b.fst, f b \u2208 s\n[PROOFSTEP]\nrintro h e he\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 (e : Set \u03b1), e \u2208 \ud835\udcdd 0 \u2192 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\n\u22a2 \u2203 a, \u2200 (b : Finset \u03b2 \u00d7 Finset \u03b2), b \u2265 a \u2192 \u2211 b in b.snd, f b - \u2211 b in b.fst, f b \u2208 e\n[PROOFSTEP]\nrcases exists_nhds_half_neg he with \u27e8d, hd, hde\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 (e : Set \u03b1), e \u2208 \ud835\udcdd 0 \u2192 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 0\nhde : \u2200 (v : \u03b1), v \u2208 d \u2192 \u2200 (w : \u03b1), w \u2208 d \u2192 v - w \u2208 e\n\u22a2 \u2203 a, \u2200 (b : Finset \u03b2 \u00d7 Finset \u03b2), b \u2265 a \u2192 \u2211 b in b.snd, f b - \u2211 b in b.fst, f b \u2208 e\n[PROOFSTEP]\nrcases h d hd with \u27e8s, h\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 (e : Set \u03b1), e \u2208 \ud835\udcdd 0 \u2192 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 0\nhde : \u2200 (v : \u03b1), v \u2208 d \u2192 \u2200 (w : \u03b1), w \u2208 d \u2192 v - w \u2208 e\ns : Finset \u03b2\nh : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 d\n\u22a2 \u2203 a, \u2200 (b : Finset \u03b2 \u00d7 Finset \u03b2), b \u2265 a \u2192 \u2211 b in b.snd, f b - \u2211 b in b.fst, f b \u2208 e\n[PROOFSTEP]\nuse(s, s)\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 (e : Set \u03b1), e \u2208 \ud835\udcdd 0 \u2192 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 0\nhde : \u2200 (v : \u03b1), v \u2208 d \u2192 \u2200 (w : \u03b1), w \u2208 d \u2192 v - w \u2208 e\ns : Finset \u03b2\nh : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 d\n\u22a2 \u2200 (b : Finset \u03b2 \u00d7 Finset \u03b2), b \u2265 (s, s) \u2192 \u2211 b in b.snd, f b - \u2211 b in b.fst, f b \u2208 e\n[PROOFSTEP]\nrintro \u27e8t\u2081, t\u2082\u27e9 \u27e8ht\u2081, ht\u2082\u27e9\n[GOAL]\ncase h.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 (e : Set \u03b1), e \u2208 \ud835\udcdd 0 \u2192 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 0\nhde : \u2200 (v : \u03b1), v \u2208 d \u2192 \u2200 (w : \u03b1), w \u2208 d \u2192 v - w \u2208 e\ns : Finset \u03b2\nh : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 d\nt\u2081 t\u2082 : Finset \u03b2\nht\u2081 : (s, s).fst \u2264 (t\u2081, t\u2082).fst\nht\u2082 : (s, s).snd \u2264 (t\u2081, t\u2082).snd\n\u22a2 \u2211 b in (t\u2081, t\u2082).snd, f b - \u2211 b in (t\u2081, t\u2082).fst, f b \u2208 e\n[PROOFSTEP]\nhave : ((\u2211 b in t\u2082, f b) - \u2211 b in t\u2081, f b) = (\u2211 b in t\u2082 \\ s, f b) - \u2211 b in t\u2081 \\ s, f b := by\n rw [\u2190 Finset.sum_sdiff ht\u2081, \u2190 Finset.sum_sdiff ht\u2082, add_sub_add_right_eq_sub]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 (e : Set \u03b1), e \u2208 \ud835\udcdd 0 \u2192 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 0\nhde : \u2200 (v : \u03b1), v \u2208 d \u2192 \u2200 (w : \u03b1), w \u2208 d \u2192 v - w \u2208 e\ns : Finset \u03b2\nh : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 d\nt\u2081 t\u2082 : Finset \u03b2\nht\u2081 : (s, s).fst \u2264 (t\u2081, t\u2082).fst\nht\u2082 : (s, s).snd \u2264 (t\u2081, t\u2082).snd\n\u22a2 \u2211 b in t\u2082, f b - \u2211 b in t\u2081, f b = \u2211 b in t\u2082 \\ s, f b - \u2211 b in t\u2081 \\ s, f b\n[PROOFSTEP]\nrw [\u2190 Finset.sum_sdiff ht\u2081, \u2190 Finset.sum_sdiff ht\u2082, add_sub_add_right_eq_sub]\n[GOAL]\ncase h.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 (e : Set \u03b1), e \u2208 \ud835\udcdd 0 \u2192 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 0\nhde : \u2200 (v : \u03b1), v \u2208 d \u2192 \u2200 (w : \u03b1), w \u2208 d \u2192 v - w \u2208 e\ns : Finset \u03b2\nh : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 d\nt\u2081 t\u2082 : Finset \u03b2\nht\u2081 : (s, s).fst \u2264 (t\u2081, t\u2082).fst\nht\u2082 : (s, s).snd \u2264 (t\u2081, t\u2082).snd\nthis : \u2211 b in t\u2082, f b - \u2211 b in t\u2081, f b = \u2211 b in t\u2082 \\ s, f b - \u2211 b in t\u2081 \\ s, f b\n\u22a2 \u2211 b in (t\u2081, t\u2082).snd, f b - \u2211 b in (t\u2081, t\u2082).fst, f b \u2208 e\n[PROOFSTEP]\nsimp only [this]\n[GOAL]\ncase h.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 (e : Set \u03b1), e \u2208 \ud835\udcdd 0 \u2192 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 0\nhde : \u2200 (v : \u03b1), v \u2208 d \u2192 \u2200 (w : \u03b1), w \u2208 d \u2192 v - w \u2208 e\ns : Finset \u03b2\nh : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 d\nt\u2081 t\u2082 : Finset \u03b2\nht\u2081 : (s, s).fst \u2264 (t\u2081, t\u2082).fst\nht\u2082 : (s, s).snd \u2264 (t\u2081, t\u2082).snd\nthis : \u2211 b in t\u2082, f b - \u2211 b in t\u2081, f b = \u2211 b in t\u2082 \\ s, f b - \u2211 b in t\u2081 \\ s, f b\n\u22a2 \u2211 b in t\u2082 \\ s, f b - \u2211 b in t\u2081 \\ s, f b \u2208 e\n[PROOFSTEP]\nexact hde _ (h _ Finset.sdiff_disjoint) _ (h _ Finset.sdiff_disjoint)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\n\u22a2 Tendsto (fun s => \u2211' (b : { x // \u00acx \u2208 s }), f \u2191b) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nby_cases H : Summable f\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : Summable f\n\u22a2 Tendsto (fun s => \u2211' (b : { x // \u00acx \u2208 s }), f \u2191b) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nintro e he\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : Summable f\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\n\u22a2 e \u2208 Filter.map (fun s => \u2211' (b : { x // \u00acx \u2208 s }), f \u2191b) atTop\n[PROOFSTEP]\nrcases exists_mem_nhds_isClosed_subset he with \u27e8o, ho, o_closed, oe\u27e9\n[GOAL]\ncase pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : Summable f\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\no : Set \u03b1\nho : o \u2208 \ud835\udcdd 0\no_closed : IsClosed o\noe : o \u2286 e\n\u22a2 e \u2208 Filter.map (fun s => \u2211' (b : { x // \u00acx \u2208 s }), f \u2191b) atTop\n[PROOFSTEP]\nsimp only [le_eq_subset, Set.mem_preimage, mem_atTop_sets, Filter.mem_map, ge_iff_le]\n[GOAL]\ncase pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : Summable f\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\no : Set \u03b1\nho : o \u2208 \ud835\udcdd 0\no_closed : IsClosed o\noe : o \u2286 e\n\u22a2 \u2203 a, \u2200 (b : Finset \u03b2), a \u2286 b \u2192 \u2211' (b_1 : { x // \u00acx \u2208 b }), f \u2191b_1 \u2208 e\n[PROOFSTEP]\nobtain \u27e8s, hs\u27e9 : \u2203 s : Finset \u03b2, \u2200 t : Finset \u03b2, Disjoint t s \u2192 (\u2211 b : \u03b2 in t, f b) \u2208 o :=\n cauchySeq_finset_iff_vanishing.1 (Tendsto.cauchySeq H.hasSum) o ho\n[GOAL]\ncase pos.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : Summable f\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\no : Set \u03b1\nho : o \u2208 \ud835\udcdd 0\no_closed : IsClosed o\noe : o \u2286 e\ns : Finset \u03b2\nhs : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 o\n\u22a2 \u2203 a, \u2200 (b : Finset \u03b2), a \u2286 b \u2192 \u2211' (b_1 : { x // \u00acx \u2208 b }), f \u2191b_1 \u2208 e\n[PROOFSTEP]\nrefine' \u27e8s, fun a sa => oe _\u27e9\n[GOAL]\ncase pos.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : Summable f\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\no : Set \u03b1\nho : o \u2208 \ud835\udcdd 0\no_closed : IsClosed o\noe : o \u2286 e\ns : Finset \u03b2\nhs : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 o\na : Finset \u03b2\nsa : s \u2286 a\n\u22a2 \u2211' (b : { x // \u00acx \u2208 a }), f \u2191b \u2208 o\n[PROOFSTEP]\nhave A : Summable fun b : { x // x \u2209 a } => f b := a.summable_compl_iff.2 H\n[GOAL]\ncase pos.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : Summable f\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\no : Set \u03b1\nho : o \u2208 \ud835\udcdd 0\no_closed : IsClosed o\noe : o \u2286 e\ns : Finset \u03b2\nhs : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 o\na : Finset \u03b2\nsa : s \u2286 a\nA : Summable fun b => f \u2191b\n\u22a2 \u2211' (b : { x // \u00acx \u2208 a }), f \u2191b \u2208 o\n[PROOFSTEP]\nrefine' IsClosed.mem_of_tendsto o_closed A.hasSum (eventually_of_forall fun b => _)\n[GOAL]\ncase pos.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : Summable f\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\no : Set \u03b1\nho : o \u2208 \ud835\udcdd 0\no_closed : IsClosed o\noe : o \u2286 e\ns : Finset \u03b2\nhs : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 o\na : Finset \u03b2\nsa : s \u2286 a\nA : Summable fun b => f \u2191b\nb : Finset { x // \u00acx \u2208 a }\n\u22a2 \u2211 b in b, (fun b => f \u2191b) b \u2208 o\n[PROOFSTEP]\nhave : Disjoint (Finset.image (fun i : { x // x \u2209 a } => (i : \u03b2)) b) s :=\n by\n refine' disjoint_left.2 fun i hi his => _\n rcases mem_image.1 hi with \u27e8i', _, rfl\u27e9\n exact i'.2 (sa his)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : Summable f\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\no : Set \u03b1\nho : o \u2208 \ud835\udcdd 0\no_closed : IsClosed o\noe : o \u2286 e\ns : Finset \u03b2\nhs : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 o\na : Finset \u03b2\nsa : s \u2286 a\nA : Summable fun b => f \u2191b\nb : Finset { x // \u00acx \u2208 a }\n\u22a2 Disjoint (image (fun i => \u2191i) b) s\n[PROOFSTEP]\nrefine' disjoint_left.2 fun i hi his => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : Summable f\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\no : Set \u03b1\nho : o \u2208 \ud835\udcdd 0\no_closed : IsClosed o\noe : o \u2286 e\ns : Finset \u03b2\nhs : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 o\na : Finset \u03b2\nsa : s \u2286 a\nA : Summable fun b => f \u2191b\nb : Finset { x // \u00acx \u2208 a }\ni : \u03b2\nhi : i \u2208 image (fun i => \u2191i) b\nhis : i \u2208 s\n\u22a2 False\n[PROOFSTEP]\nrcases mem_image.1 hi with \u27e8i', _, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : Summable f\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\no : Set \u03b1\nho : o \u2208 \ud835\udcdd 0\no_closed : IsClosed o\noe : o \u2286 e\ns : Finset \u03b2\nhs : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 o\na : Finset \u03b2\nsa : s \u2286 a\nA : Summable fun b => f \u2191b\nb : Finset { x // \u00acx \u2208 a }\ni' : { x // \u00acx \u2208 a }\nleft\u271d : i' \u2208 b\nhi : \u2191i' \u2208 image (fun i => \u2191i) b\nhis : \u2191i' \u2208 s\n\u22a2 False\n[PROOFSTEP]\nexact i'.2 (sa his)\n[GOAL]\ncase pos.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : Summable f\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\no : Set \u03b1\nho : o \u2208 \ud835\udcdd 0\no_closed : IsClosed o\noe : o \u2286 e\ns : Finset \u03b2\nhs : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 o\na : Finset \u03b2\nsa : s \u2286 a\nA : Summable fun b => f \u2191b\nb : Finset { x // \u00acx \u2208 a }\nthis : Disjoint (image (fun i => \u2191i) b) s\n\u22a2 \u2211 b in b, (fun b => f \u2191b) b \u2208 o\n[PROOFSTEP]\nconvert hs _ this using 1\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : Summable f\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\no : Set \u03b1\nho : o \u2208 \ud835\udcdd 0\no_closed : IsClosed o\noe : o \u2286 e\ns : Finset \u03b2\nhs : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 o\na : Finset \u03b2\nsa : s \u2286 a\nA : Summable fun b => f \u2191b\nb : Finset { x // \u00acx \u2208 a }\nthis : Disjoint (image (fun i => \u2191i) b) s\n\u22a2 \u2211 b in b, (fun b => f \u2191b) b = \u2211 b in image (fun i => \u2191i) b, f b\n[PROOFSTEP]\nrw [sum_image]\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : Summable f\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\no : Set \u03b1\nho : o \u2208 \ud835\udcdd 0\no_closed : IsClosed o\noe : o \u2286 e\ns : Finset \u03b2\nhs : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 o\na : Finset \u03b2\nsa : s \u2286 a\nA : Summable fun b => f \u2191b\nb : Finset { x // \u00acx \u2208 a }\nthis : Disjoint (image (fun i => \u2191i) b) s\n\u22a2 \u2200 (x : { x // \u00acx \u2208 a }), x \u2208 b \u2192 \u2200 (y : { x // \u00acx \u2208 a }), y \u2208 b \u2192 \u2191x = \u2191y \u2192 x = y\n[PROOFSTEP]\nintro i _ j _ hij\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na\u271d\u00b2 a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : Summable f\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\no : Set \u03b1\nho : o \u2208 \ud835\udcdd 0\no_closed : IsClosed o\noe : o \u2286 e\ns : Finset \u03b2\nhs : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 o\na : Finset \u03b2\nsa : s \u2286 a\nA : Summable fun b => f \u2191b\nb : Finset { x // \u00acx \u2208 a }\nthis : Disjoint (image (fun i => \u2191i) b) s\ni : { x // \u00acx \u2208 a }\na\u271d\u00b9 : i \u2208 b\nj : { x // \u00acx \u2208 a }\na\u271d : j \u2208 b\nhij : \u2191i = \u2191j\n\u22a2 i = j\n[PROOFSTEP]\nexact Subtype.ext hij\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : \u00acSummable f\n\u22a2 Tendsto (fun s => \u2211' (b : { x // \u00acx \u2208 s }), f \u2191b) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert tendsto_const_nhds (\u03b1 := \u03b1) (\u03b2 := Finset \u03b2) (f := atTop) (a := 0)\n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : \u00acSummable f\nx\u271d : Finset \u03b2\n\u22a2 \u2211' (b : { x // \u00acx \u2208 x\u271d }), f \u2191b = 0\n[PROOFSTEP]\napply tsum_eq_zero_of_not_summable\n[GOAL]\ncase h.e'_3.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nf : \u03b2 \u2192 \u03b1\nH : \u00acSummable f\nx\u271d : Finset \u03b2\n\u22a2 \u00acSummable fun b => f \u2191b\n[PROOFSTEP]\nrwa [Finset.summable_compl_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : CompleteSpace \u03b1\n\u22a2 Summable f \u2194 \u2200 (e : Set \u03b1), e \u2208 \ud835\udcdd 0 \u2192 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\n[PROOFSTEP]\nrw [summable_iff_cauchySeq_finset, cauchySeq_finset_iff_vanishing]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : CompleteSpace \u03b1\nhf : Summable f\nh : \u2200 (b : \u03b2), g b = 0 \u2228 g b = f b\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\ns : Finset \u03b2\nhs : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\nt : Finset \u03b2\nht : Disjoint t s\n\u22a2 \u2211 b in filter (fun b => g b = f b) t, g b = \u2211 b in t, g b\n[PROOFSTEP]\n{ refine' Finset.sum_subset (Finset.filter_subset _ _) _\n intro b hbt hb\n simp only [Finset.mem_filter, and_iff_right hbt] at hb \n exact (h b).resolve_right hb\n}\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : CompleteSpace \u03b1\nhf : Summable f\nh : \u2200 (b : \u03b2), g b = 0 \u2228 g b = f b\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\ns : Finset \u03b2\nhs : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\nt : Finset \u03b2\nht : Disjoint t s\n\u22a2 \u2211 b in filter (fun b => g b = f b) t, g b = \u2211 b in t, g b\n[PROOFSTEP]\nrefine' Finset.sum_subset (Finset.filter_subset _ _) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : CompleteSpace \u03b1\nhf : Summable f\nh : \u2200 (b : \u03b2), g b = 0 \u2228 g b = f b\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\ns : Finset \u03b2\nhs : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\nt : Finset \u03b2\nht : Disjoint t s\n\u22a2 \u2200 (x : \u03b2), x \u2208 t \u2192 \u00acx \u2208 filter (fun b => g b = f b) t \u2192 g x = 0\n[PROOFSTEP]\nintro b hbt hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : CompleteSpace \u03b1\nhf : Summable f\nh : \u2200 (b : \u03b2), g b = 0 \u2228 g b = f b\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\ns : Finset \u03b2\nhs : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\nt : Finset \u03b2\nht : Disjoint t s\nb : \u03b2\nhbt : b \u2208 t\nhb : \u00acb \u2208 filter (fun b => g b = f b) t\n\u22a2 g b = 0\n[PROOFSTEP]\nsimp only [Finset.mem_filter, and_iff_right hbt] at hb \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : CompleteSpace \u03b1\nhf : Summable f\nh : \u2200 (b : \u03b2), g b = 0 \u2228 g b = f b\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 0\ns : Finset \u03b2\nhs : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u2211 b in t, f b \u2208 e\nt : Finset \u03b2\nht : Disjoint t s\nb : \u03b2\nhbt : b \u2208 t\nhb : \u00acg b = f b\n\u22a2 g b = 0\n[PROOFSTEP]\nexact (h b).resolve_right hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommGroup \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformAddGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : CompleteSpace \u03b1\ni : \u03b3 \u2192 \u03b2\nhf : Summable f\nhi : Injective i\n\u22a2 Summable (f \u2218 i)\n[PROOFSTEP]\nsimpa only [Set.indicator_range_comp] using\n (hi.summable_iff (fun x hx => Set.indicator_of_not_mem hx _)).2 (hf.indicator (Set.range i))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommGroup \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : CompleteSpace \u03b1\ninst\u271d : T2Space \u03b1\nf : \u03b2 \u2192 \u03b1\nhf : Summable f\ns : Finset \u03b2\n\u22a2 \u2211 x in s, f x + \u2211' (x : { x // \u00acx \u2208 s }), f \u2191x = \u2211' (x : \u03b2), f x\n[PROOFSTEP]\nrw [\u2190 tsum_subtype_add_tsum_subtype_compl hf s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommGroup \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : CompleteSpace \u03b1\ninst\u271d : T2Space \u03b1\nf : \u03b2 \u2192 \u03b1\nhf : Summable f\ns : Finset \u03b2\n\u22a2 \u2211 x in s, f x + \u2211' (x : { x // \u00acx \u2208 s }), f \u2191x = \u2211' (x : \u2191\u2191s), f \u2191x + \u2211' (x : \u2191(\u2191s)\u1d9c), f \u2191x\n[PROOFSTEP]\nsimp only [Finset.tsum_subtype', add_right_inj]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommGroup \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : UniformAddGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : CompleteSpace \u03b1\ninst\u271d : T2Space \u03b1\nf : \u03b2 \u2192 \u03b1\nhf : Summable f\ns : Finset \u03b2\n\u22a2 \u2211' (x : { x // \u00acx \u2208 s }), f \u2191x = \u2211' (x : \u2191(\u2191s)\u1d9c), f \u2191x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : TopologicalAddGroup G\nf : \u03b1 \u2192 G\nhf : Summable f\ne : Set G\nhe : e \u2208 \ud835\udcdd 0\n\u22a2 \u2203 s, \u2200 (t : Finset \u03b1), Disjoint t s \u2192 \u2211 k in t, f k \u2208 e\n[PROOFSTEP]\nletI : UniformSpace G := TopologicalAddGroup.toUniformSpace G\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : TopologicalAddGroup G\nf : \u03b1 \u2192 G\nhf : Summable f\ne : Set G\nhe : e \u2208 \ud835\udcdd 0\nthis : UniformSpace G := TopologicalAddGroup.toUniformSpace G\n\u22a2 \u2203 s, \u2200 (t : Finset \u03b1), Disjoint t s \u2192 \u2211 k in t, f k \u2208 e\n[PROOFSTEP]\nletI : UniformAddGroup G := comm_topologicalAddGroup_is_uniform\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : TopologicalAddGroup G\nf : \u03b1 \u2192 G\nhf : Summable f\ne : Set G\nhe : e \u2208 \ud835\udcdd 0\nthis\u271d : UniformSpace G := TopologicalAddGroup.toUniformSpace G\nthis : UniformAddGroup G := comm_topologicalAddGroup_is_uniform\n\u22a2 \u2203 s, \u2200 (t : Finset \u03b1), Disjoint t s \u2192 \u2211 k in t, f k \u2208 e\n[PROOFSTEP]\nrcases hf with \u27e8y, hy\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : TopologicalAddGroup G\nf : \u03b1 \u2192 G\ne : Set G\nhe : e \u2208 \ud835\udcdd 0\nthis\u271d : UniformSpace G := TopologicalAddGroup.toUniformSpace G\nthis : UniformAddGroup G := comm_topologicalAddGroup_is_uniform\ny : G\nhy : HasSum f y\n\u22a2 \u2203 s, \u2200 (t : Finset \u03b1), Disjoint t s \u2192 \u2211 k in t, f k \u2208 e\n[PROOFSTEP]\nexact cauchySeq_finset_iff_vanishing.1 hy.cauchySeq e he\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : TopologicalAddGroup G\nf : \u03b1 \u2192 G\nhf : Summable f\n\u22a2 Tendsto f cofinite (\ud835\udcdd 0)\n[PROOFSTEP]\nintro e he\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : TopologicalAddGroup G\nf : \u03b1 \u2192 G\nhf : Summable f\ne : Set G\nhe : e \u2208 \ud835\udcdd 0\n\u22a2 e \u2208 Filter.map f cofinite\n[PROOFSTEP]\nrw [Filter.mem_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : TopologicalAddGroup G\nf : \u03b1 \u2192 G\nhf : Summable f\ne : Set G\nhe : e \u2208 \ud835\udcdd 0\n\u22a2 f \u207b\u00b9' e \u2208 cofinite\n[PROOFSTEP]\nrcases hf.vanishing he with \u27e8s, hs\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : TopologicalAddGroup G\nf : \u03b1 \u2192 G\nhf : Summable f\ne : Set G\nhe : e \u2208 \ud835\udcdd 0\ns : Finset \u03b1\nhs : \u2200 (t : Finset \u03b1), Disjoint t s \u2192 \u2211 k in t, f k \u2208 e\n\u22a2 f \u207b\u00b9' e \u2208 cofinite\n[PROOFSTEP]\nrefine' s.eventually_cofinite_nmem.mono fun x hx => _\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : TopologicalAddGroup G\nf : \u03b1 \u2192 G\nhf : Summable f\ne : Set G\nhe : e \u2208 \ud835\udcdd 0\ns : Finset \u03b1\nhs : \u2200 (t : Finset \u03b1), Disjoint t s \u2192 \u2211 k in t, f k \u2208 e\nx : \u03b1\nhx : \u00acx \u2208 s\n\u22a2 f x \u2208 e\n[PROOFSTEP]\nsimpa using hs { x } (disjoint_singleton_left.2 hx)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : TopologicalAddGroup G\nf\u271d : \u03b1 \u2192 G\nf : \u2115 \u2192 G\nhf : Summable f\n\u22a2 Tendsto f atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 Nat.cofinite_eq_atTop]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : TopologicalAddGroup G\nf\u271d : \u03b1 \u2192 G\nf : \u2115 \u2192 G\nhf : Summable f\n\u22a2 Tendsto f cofinite (\ud835\udcdd 0)\n[PROOFSTEP]\nexact hf.tendsto_cofinite_zero\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : TopologicalAddGroup G\nf : \u03b1 \u2192 G\ninst\u271d\u00b9 : TopologicalSpace.FirstCountableTopology G\ninst\u271d : T1Space G\nhf : Summable f\n\u22a2 Set.Countable (support f)\n[PROOFSTEP]\nsimpa only [sInter_sets_nhds] using hf.tendsto_cofinite_zero.countable_compl_preimage_sInter_sets\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : AddCommMonoid \u03b3\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1\ng : \u03b2 \u2192 \u03b3\na : \u03b1\nb : \u03b3\nhf : HasSum f a\nhg : HasSum g b\n\u22a2 HasSum (fun x => (f x, g x)) (a, b)\n[PROOFSTEP]\nsimp [HasSum, \u2190 prod_mk_sum, Filter.Tendsto.prod_mk_nhds hf hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b1 \u2192 Type u_6\ninst\u271d\u00b9 : (x : \u03b1) \u2192 AddCommMonoid (\u03c0 x)\ninst\u271d : (x : \u03b1) \u2192 TopologicalSpace (\u03c0 x)\nf : \u03b9 \u2192 (x : \u03b1) \u2192 \u03c0 x\ng : (x : \u03b1) \u2192 \u03c0 x\n\u22a2 HasSum f g \u2194 \u2200 (x : \u03b1), HasSum (fun i => f i x) (g x)\n[PROOFSTEP]\nsimp only [HasSum, tendsto_pi_nhds, sum_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b1 \u2192 Type u_6\ninst\u271d\u00b9 : (x : \u03b1) \u2192 AddCommMonoid (\u03c0 x)\ninst\u271d : (x : \u03b1) \u2192 TopologicalSpace (\u03c0 x)\nf : \u03b9 \u2192 (x : \u03b1) \u2192 \u03c0 x\n\u22a2 Summable f \u2194 \u2200 (x : \u03b1), Summable fun i => f i x\n[PROOFSTEP]\nsimp only [Summable, Pi.hasSum, skolem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d : T2Space \u03b1\n\u22a2 \u2211' (x : \u03b2), op (f x) = op (\u2211' (x : \u03b2), f x)\n[PROOFSTEP]\nby_cases h : Summable f\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d : T2Space \u03b1\nh : Summable f\n\u22a2 \u2211' (x : \u03b2), op (f x) = op (\u2211' (x : \u03b2), f x)\n[PROOFSTEP]\nexact h.hasSum.op.tsum_eq\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d : T2Space \u03b1\nh : \u00acSummable f\n\u22a2 \u2211' (x : \u03b2), op (f x) = op (\u2211' (x : \u03b2), f x)\n[PROOFSTEP]\nhave ho := summable_op.not.mpr h\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d : T2Space \u03b1\nh : \u00acSummable f\nho : \u00acSummable fun a => op (f a)\n\u22a2 \u2211' (x : \u03b2), op (f x) = op (\u2211' (x : \u03b2), f x)\n[PROOFSTEP]\nrw [tsum_eq_zero_of_not_summable h, tsum_eq_zero_of_not_summable ho, MulOpposite.op_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : StarAddMonoid \u03b1\ninst\u271d : ContinuousStar \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nh : HasSum f a\n\u22a2 HasSum (fun b => Star.star (f b)) (Star.star a)\n[PROOFSTEP]\nsimpa only using h.map (starAddEquiv : \u03b1 \u2243+ \u03b1) continuous_star\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : AddCommMonoid \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : StarAddMonoid \u03b1\ninst\u271d : ContinuousStar \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nhf : Summable fun b => Star.star (f b)\n\u22a2 Summable f\n[PROOFSTEP]\nsimpa only [star_star] using hf.star\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : StarAddMonoid \u03b1\ninst\u271d\u00b9 : ContinuousStar \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d : T2Space \u03b1\n\u22a2 star (\u2211' (b : \u03b2), f b) = \u2211' (b : \u03b2), star (f b)\n[PROOFSTEP]\nby_cases hf : Summable f\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : StarAddMonoid \u03b1\ninst\u271d\u00b9 : ContinuousStar \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d : T2Space \u03b1\nhf : Summable f\n\u22a2 star (\u2211' (b : \u03b2), f b) = \u2211' (b : \u03b2), star (f b)\n[PROOFSTEP]\nexact hf.hasSum.star.tsum_eq.symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : AddCommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : StarAddMonoid \u03b1\ninst\u271d\u00b9 : ContinuousStar \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d : T2Space \u03b1\nhf : \u00acSummable f\n\u22a2 star (\u2211' (b : \u03b2), f b) = \u2211' (b : \u03b2), star (f b)\n[PROOFSTEP]\nrw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable (mt Summable.ofStar hf), star_zero]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.InfiniteSum.Basic", "llama_tokens": 75911, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.531209388216861, "lm_q2_score": 0.4649015713733885, "lm_q1q2_score": 0.24696007931031505}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Cover J X\n\u22a2 \u2200 {X_1 Y : WalkingMulticospan (index S (P \u22d9 F)).fstTo (index S (P \u22d9 F)).sndTo} (f : X_1 \u27f6 Y),\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).map f \u226b\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n Y).hom =\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n X_1).hom \u226b\n (MulticospanIndex.multicospan (index S P) \u22d9 F).map f\n[PROOFSTEP]\nrintro (a | b) (a | b) (f | f | f)\n[GOAL]\ncase left.left.id\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Cover J X\na : (index S (P \u22d9 F)).L\n\u22a2 (MulticospanIndex.multicospan (index S (P \u22d9 F))).map (WalkingMulticospan.Hom.id (WalkingMulticospan.left a)) \u226b\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n (WalkingMulticospan.left a)).hom =\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n (WalkingMulticospan.left a)).hom \u226b\n (MulticospanIndex.multicospan (index S P) \u22d9 F).map (WalkingMulticospan.Hom.id (WalkingMulticospan.left a))\ncase left.right.fst\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Cover J X\nb : (index S (P \u22d9 F)).R\n\u22a2 (MulticospanIndex.multicospan (index S (P \u22d9 F))).map (WalkingMulticospan.Hom.fst b) \u226b\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n (WalkingMulticospan.right b)).hom =\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n (WalkingMulticospan.left (MulticospanIndex.fstTo (index S (P \u22d9 F)) b))).hom \u226b\n (MulticospanIndex.multicospan (index S P) \u22d9 F).map (WalkingMulticospan.Hom.fst b)\ncase left.right.snd\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Cover J X\nb : (index S (P \u22d9 F)).R\n\u22a2 (MulticospanIndex.multicospan (index S (P \u22d9 F))).map (WalkingMulticospan.Hom.snd b) \u226b\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n (WalkingMulticospan.right b)).hom =\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n (WalkingMulticospan.left (MulticospanIndex.sndTo (index S (P \u22d9 F)) b))).hom \u226b\n (MulticospanIndex.multicospan (index S P) \u22d9 F).map (WalkingMulticospan.Hom.snd b)\ncase right.right.id\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Cover J X\nb : (index S (P \u22d9 F)).R\n\u22a2 (MulticospanIndex.multicospan (index S (P \u22d9 F))).map (WalkingMulticospan.Hom.id (WalkingMulticospan.right b)) \u226b\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n (WalkingMulticospan.right b)).hom =\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n (WalkingMulticospan.right b)).hom \u226b\n (MulticospanIndex.multicospan (index S P) \u22d9 F).map (WalkingMulticospan.Hom.id (WalkingMulticospan.right b))\n[PROOFSTEP]\nall_goals aesop_cat\n[GOAL]\ncase left.left.id\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Cover J X\na : (index S (P \u22d9 F)).L\n\u22a2 (MulticospanIndex.multicospan (index S (P \u22d9 F))).map (WalkingMulticospan.Hom.id (WalkingMulticospan.left a)) \u226b\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n (WalkingMulticospan.left a)).hom =\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n (WalkingMulticospan.left a)).hom \u226b\n (MulticospanIndex.multicospan (index S P) \u22d9 F).map (WalkingMulticospan.Hom.id (WalkingMulticospan.left a))\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase left.right.fst\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Cover J X\nb : (index S (P \u22d9 F)).R\n\u22a2 (MulticospanIndex.multicospan (index S (P \u22d9 F))).map (WalkingMulticospan.Hom.fst b) \u226b\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n (WalkingMulticospan.right b)).hom =\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n (WalkingMulticospan.left (MulticospanIndex.fstTo (index S (P \u22d9 F)) b))).hom \u226b\n (MulticospanIndex.multicospan (index S P) \u22d9 F).map (WalkingMulticospan.Hom.fst b)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase left.right.snd\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Cover J X\nb : (index S (P \u22d9 F)).R\n\u22a2 (MulticospanIndex.multicospan (index S (P \u22d9 F))).map (WalkingMulticospan.Hom.snd b) \u226b\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n (WalkingMulticospan.right b)).hom =\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n (WalkingMulticospan.left (MulticospanIndex.sndTo (index S (P \u22d9 F)) b))).hom \u226b\n (MulticospanIndex.multicospan (index S P) \u22d9 F).map (WalkingMulticospan.Hom.snd b)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase right.right.id\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Cover J X\nb : (index S (P \u22d9 F)).R\n\u22a2 (MulticospanIndex.multicospan (index S (P \u22d9 F))).map (WalkingMulticospan.Hom.id (WalkingMulticospan.right b)) \u226b\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n (WalkingMulticospan.right b)).hom =\n ((fun t =>\n match t with\n | WalkingMulticospan.left a =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.left a))\n | WalkingMulticospan.right b =>\n eqToIso\n (_ :\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b) =\n (MulticospanIndex.multicospan (index S (P \u22d9 F))).obj (WalkingMulticospan.right b)))\n (WalkingMulticospan.right b)).hom \u226b\n (MulticospanIndex.multicospan (index S P) \u22d9 F).map (WalkingMulticospan.Hom.id (WalkingMulticospan.right b))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Cover J X\n\u22a2 \u2200 (j : WalkingMulticospan (index S P).fstTo (index S P).sndTo),\n NatTrans.app (F.mapCone (multifork S P)).\u03c0 j =\n (eqToIso (_ : (F.mapCone (multifork S P)).pt = (F.mapCone (multifork S P)).pt)).hom \u226b\n NatTrans.app ((Cones.postcompose (multicospanComp F P S).hom).obj (multifork S (P \u22d9 F))).\u03c0 j\n[PROOFSTEP]\nrintro (a | b)\n[GOAL]\ncase left\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Cover J X\na : (index S P).L\n\u22a2 NatTrans.app (F.mapCone (multifork S P)).\u03c0 (WalkingMulticospan.left a) =\n (eqToIso (_ : (F.mapCone (multifork S P)).pt = (F.mapCone (multifork S P)).pt)).hom \u226b\n NatTrans.app ((Cones.postcompose (multicospanComp F P S).hom).obj (multifork S (P \u22d9 F))).\u03c0\n (WalkingMulticospan.left a)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase left\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Cover J X\na : (index S P).L\n\u22a2 F.map (P.map a.f.op) =\n \ud835\udfd9 (F.obj (P.obj (Opposite.op X))) \u226b\n F.map (P.map a.f.op) \u226b NatTrans.app (multicospanComp F P S).hom (WalkingMulticospan.left a)\n[PROOFSTEP]\nerw [Category.id_comp, multicospanComp_hom_app_left, eqToHom_refl, Category.comp_id]\n[GOAL]\ncase right\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Cover J X\nb : (index S P).R\n\u22a2 NatTrans.app (F.mapCone (multifork S P)).\u03c0 (WalkingMulticospan.right b) =\n (eqToIso (_ : (F.mapCone (multifork S P)).pt = (F.mapCone (multifork S P)).pt)).hom \u226b\n NatTrans.app ((Cones.postcompose (multicospanComp F P S).hom).obj (multifork S (P \u22d9 F))).\u03c0\n (WalkingMulticospan.right b)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase right\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Cover J X\nb : (index S P).R\n\u22a2 F.map (P.map (MulticospanIndex.fstTo (index S P) b).f.op \u226b MulticospanIndex.fst (index S P) b) =\n \ud835\udfd9 (F.obj (P.obj (Opposite.op X))) \u226b\n (F.map (P.map (MulticospanIndex.fstTo (index S (P \u22d9 F)) b).f.op) \u226b MulticospanIndex.fst (index S (P \u22d9 F)) b) \u226b\n NatTrans.app (multicospanComp F P S).hom (WalkingMulticospan.right b)\n[PROOFSTEP]\nerw [Functor.map_comp, Category.assoc, Category.id_comp, multicospanComp_hom_app_right, eqToHom_refl, Category.comp_id]\n[GOAL]\ncase right\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Cover J X\nb : (index S P).R\n\u22a2 F.map (P.map (MulticospanIndex.fstTo (index S P) b).f.op) \u226b F.map (MulticospanIndex.fst (index S P) b) =\n F.map (P.map (MulticospanIndex.fstTo (index S (P \u22d9 F)) b).f.op) \u226b MulticospanIndex.fst (index S (P \u22d9 F)) b\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\ninst\u271d\u00b2 :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) F\ninst\u271d\u00b9 :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) G\ninst\u271d :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) H\nP : C\u1d52\u1d56 \u2964 A\nhP : IsSheaf J P\n\u22a2 IsSheaf J (P \u22d9 F)\n[PROOFSTEP]\nrw [Presheaf.isSheaf_iff_multifork] at hP \u22a2\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\ninst\u271d\u00b2 :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) F\ninst\u271d\u00b9 :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) G\ninst\u271d :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) H\nP : C\u1d52\u1d56 \u2964 A\nhP : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\n\u22a2 \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S (P \u22d9 F)))\n[PROOFSTEP]\nintro X S\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\ninst\u271d\u00b2 :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) F\ninst\u271d\u00b9 :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) G\ninst\u271d :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) H\nP : C\u1d52\u1d56 \u2964 A\nhP : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nX : C\nS : GrothendieckTopology.Cover J X\n\u22a2 Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S (P \u22d9 F)))\n[PROOFSTEP]\nobtain \u27e8h\u27e9 := hP X S\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\ninst\u271d\u00b2 :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) F\ninst\u271d\u00b9 :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) G\ninst\u271d :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) H\nP : C\u1d52\u1d56 \u2964 A\nhP : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nX : C\nS : GrothendieckTopology.Cover J X\nh : IsLimit (GrothendieckTopology.Cover.multifork S P)\n\u22a2 Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S (P \u22d9 F)))\n[PROOFSTEP]\nreplace h := isLimitOfPreserves F h\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\ninst\u271d\u00b2 :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) F\ninst\u271d\u00b9 :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) G\ninst\u271d :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) H\nP : C\u1d52\u1d56 \u2964 A\nhP : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nX : C\nS : GrothendieckTopology.Cover J X\nh : IsLimit (F.mapCone (GrothendieckTopology.Cover.multifork S P))\n\u22a2 Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S (P \u22d9 F)))\n[PROOFSTEP]\nreplace h := Limits.IsLimit.ofIsoLimit h (S.mapMultifork F P)\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nB : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF G H : A \u2964 B\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\ninst\u271d\u00b2 :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) F\ninst\u271d\u00b9 :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) G\ninst\u271d :\n (X : C) \u2192\n (S : GrothendieckTopology.Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 A) \u2192 PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) H\nP : C\u1d52\u1d56 \u2964 A\nhP : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nX : C\nS : GrothendieckTopology.Cover J X\nh :\n IsLimit\n ((Cones.postcompose (GrothendieckTopology.Cover.multicospanComp F P S).hom).obj\n (GrothendieckTopology.Cover.multifork S (P \u22d9 F)))\n\u22a2 Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S (P \u22d9 F)))\n[PROOFSTEP]\nexact \u27e8Limits.IsLimit.postcomposeHomEquiv (S.multicospanComp F P) _ h\u27e9\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.Whiskering", "llama_tokens": 11402, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.3849121444839335, "lm_q1q2_score": 0.2465885207197231}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I \u2264 Ideal.jacobson \u22a5\n\u22a2 IsLocalRingHom (Ideal.Quotient.mk I)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase map_nonunit\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I \u2264 Ideal.jacobson \u22a5\n\u22a2 \u2200 (a : R), IsUnit (\u2191(Ideal.Quotient.mk I) a) \u2192 IsUnit a\n[PROOFSTEP]\nintro a h\n[GOAL]\ncase map_nonunit\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 Ideal.jacobson \u22a5\na : R\nh : IsUnit (\u2191(Ideal.Quotient.mk I) a)\n\u22a2 IsUnit a\n[PROOFSTEP]\nhave : IsUnit (Ideal.Quotient.mk (Ideal.jacobson \u22a5) a) :=\n by\n rw [isUnit_iff_exists_inv] at *\n obtain \u27e8b, hb\u27e9 := h\n obtain \u27e8b, rfl\u27e9 := Ideal.Quotient.mk_surjective b\n use Ideal.Quotient.mk _ b\n rw [\u2190 (Ideal.Quotient.mk _).map_one, \u2190 (Ideal.Quotient.mk _).map_mul, Ideal.Quotient.eq] at hb \u22a2\n exact h hb\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 Ideal.jacobson \u22a5\na : R\nh : IsUnit (\u2191(Ideal.Quotient.mk I) a)\n\u22a2 IsUnit (\u2191(Ideal.Quotient.mk (Ideal.jacobson \u22a5)) a)\n[PROOFSTEP]\nrw [isUnit_iff_exists_inv] at *\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 Ideal.jacobson \u22a5\na : R\nh : \u2203 b, \u2191(Ideal.Quotient.mk I) a * b = 1\n\u22a2 \u2203 b, \u2191(Ideal.Quotient.mk (Ideal.jacobson \u22a5)) a * b = 1\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 := h\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I \u2264 Ideal.jacobson \u22a5\na : R\nb : R \u29f8 I\nhb : \u2191(Ideal.Quotient.mk I) a * b = 1\n\u22a2 \u2203 b, \u2191(Ideal.Quotient.mk (Ideal.jacobson \u22a5)) a * b = 1\n[PROOFSTEP]\nobtain \u27e8b, rfl\u27e9 := Ideal.Quotient.mk_surjective b\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I \u2264 Ideal.jacobson \u22a5\na b : R\nhb : \u2191(Ideal.Quotient.mk I) a * \u2191(Ideal.Quotient.mk I) b = 1\n\u22a2 \u2203 b, \u2191(Ideal.Quotient.mk (Ideal.jacobson \u22a5)) a * b = 1\n[PROOFSTEP]\nuse Ideal.Quotient.mk _ b\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I \u2264 Ideal.jacobson \u22a5\na b : R\nhb : \u2191(Ideal.Quotient.mk I) a * \u2191(Ideal.Quotient.mk I) b = 1\n\u22a2 \u2191(Ideal.Quotient.mk (Ideal.jacobson \u22a5)) a * \u2191(Ideal.Quotient.mk (Ideal.jacobson \u22a5)) b = 1\n[PROOFSTEP]\nrw [\u2190 (Ideal.Quotient.mk _).map_one, \u2190 (Ideal.Quotient.mk _).map_mul, Ideal.Quotient.eq] at hb \u22a2\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I \u2264 Ideal.jacobson \u22a5\na b : R\nhb : a * b - 1 \u2208 I\n\u22a2 a * b - 1 \u2208 Ideal.jacobson \u22a5\n[PROOFSTEP]\nexact h hb\n[GOAL]\ncase map_nonunit\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 Ideal.jacobson \u22a5\na : R\nh : IsUnit (\u2191(Ideal.Quotient.mk I) a)\nthis : IsUnit (\u2191(Ideal.Quotient.mk (Ideal.jacobson \u22a5)) a)\n\u22a2 IsUnit a\n[PROOFSTEP]\nobtain \u27e8\u27e8x, y, h1, h2\u27e9, rfl : x = _\u27e9 := this\n[GOAL]\ncase map_nonunit.intro.mk\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 Ideal.jacobson \u22a5\na : R\nh : IsUnit (\u2191(Ideal.Quotient.mk I) a)\ny : R \u29f8 Ideal.jacobson \u22a5\nh1 : \u2191(Ideal.Quotient.mk (Ideal.jacobson \u22a5)) a * y = 1\nh2 : y * \u2191(Ideal.Quotient.mk (Ideal.jacobson \u22a5)) a = 1\n\u22a2 IsUnit a\n[PROOFSTEP]\nobtain \u27e8y, rfl\u27e9 := Ideal.Quotient.mk_surjective y\n[GOAL]\ncase map_nonunit.intro.mk.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 Ideal.jacobson \u22a5\na : R\nh : IsUnit (\u2191(Ideal.Quotient.mk I) a)\ny : R\nh1 : \u2191(Ideal.Quotient.mk (Ideal.jacobson \u22a5)) a * \u2191(Ideal.Quotient.mk (Ideal.jacobson \u22a5)) y = 1\nh2 : \u2191(Ideal.Quotient.mk (Ideal.jacobson \u22a5)) y * \u2191(Ideal.Quotient.mk (Ideal.jacobson \u22a5)) a = 1\n\u22a2 IsUnit a\n[PROOFSTEP]\nrw [\u2190 (Ideal.Quotient.mk _).map_mul, \u2190 (Ideal.Quotient.mk _).map_one, Ideal.Quotient.eq, Ideal.mem_jacobson_bot] at h1\n h2 \n[GOAL]\ncase map_nonunit.intro.mk.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 Ideal.jacobson \u22a5\na : R\nh : IsUnit (\u2191(Ideal.Quotient.mk I) a)\ny : R\nh1 : \u2200 (y_1 : R), IsUnit ((a * y - 1) * y_1 + 1)\nh2 : \u2200 (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\n\u22a2 IsUnit a\n[PROOFSTEP]\nspecialize h1 1\n[GOAL]\ncase map_nonunit.intro.mk.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 Ideal.jacobson \u22a5\na : R\nh : IsUnit (\u2191(Ideal.Quotient.mk I) a)\ny : R\nh2 : \u2200 (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\nh1 : IsUnit ((a * y - 1) * 1 + 1)\n\u22a2 IsUnit a\n[PROOFSTEP]\nsimp at h1 \n[GOAL]\ncase map_nonunit.intro.mk.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 Ideal.jacobson \u22a5\na : R\nh : IsUnit (\u2191(Ideal.Quotient.mk I) a)\ny : R\nh2 : \u2200 (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\nh1 : IsUnit a \u2227 IsUnit y\n\u22a2 IsUnit a\n[PROOFSTEP]\nexact h1.1\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\nx\u271d\u00b9 : Monic f\na\u2080 : K\nh\u2081 : Polynomial.eval a\u2080 f \u2208 maximalIdeal K\nx\u271d : IsUnit (Polynomial.eval a\u2080 (\u2191derivative f))\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 maximalIdeal K\n[PROOFSTEP]\nsimp only [(maximalIdeal K).eq_bot_of_prime, Ideal.mem_bot] at h\u2081 \u22a2\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\nx\u271d\u00b9 : Monic f\na\u2080 : K\nx\u271d : IsUnit (Polynomial.eval a\u2080 (\u2191derivative f))\nh\u2081 : Polynomial.eval a\u2080 f = 0\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 = 0\n[PROOFSTEP]\nexact \u27e8a\u2080, h\u2081, sub_self _\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n\u22a2 List.TFAE\n [HenselianLocalRing R,\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080,\n \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080]\n[PROOFSTEP]\ntfae_have _3_2 : 3 \u2192 2\n[GOAL]\ncase _3_2\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n\u22a2 (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\n[PROOFSTEP]\nintro H\n[GOAL]\ncase _3_2\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nH :\n \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080\n\u22a2 \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\n[PROOFSTEP]\nexact H (residue R) Ideal.Quotient.mk_surjective\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\n\u22a2 List.TFAE\n [HenselianLocalRing R,\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080,\n \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080]\n[PROOFSTEP]\ntfae_have _2_1 : 2 \u2192 1\n[GOAL]\ncase _2_1\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\n\u22a2 (\u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080) \u2192\n HenselianLocalRing R\n[PROOFSTEP]\nintro H\n[GOAL]\ncase _2_1\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\nH :\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\n\u22a2 HenselianLocalRing R\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase _2_1.is_henselian\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\nH :\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\n\u22a2 \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : R),\n Polynomial.eval a\u2080 f \u2208 maximalIdeal R \u2192\n IsUnit (Polynomial.eval a\u2080 (\u2191derivative f)) \u2192 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 maximalIdeal R\n[PROOFSTEP]\nintro f hf a\u2080 h\u2081 h\u2082\n[GOAL]\ncase _2_1.is_henselian\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\nH :\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\nf : R[X]\nhf : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 maximalIdeal R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (\u2191derivative f))\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 maximalIdeal R\n[PROOFSTEP]\nspecialize H f hf (residue R a\u2080)\n[GOAL]\ncase _2_1.is_henselian\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\nf : R[X]\nhf : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 maximalIdeal R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (\u2191derivative f))\nH :\n \u2191(aeval (\u2191(residue R) a\u2080)) f = 0 \u2192\n \u2191(aeval (\u2191(residue R) a\u2080)) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = \u2191(residue R) a\u2080\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 maximalIdeal R\n[PROOFSTEP]\nhave aux := flip mem_nonunits_iff.mp h\u2082\n[GOAL]\ncase _2_1.is_henselian\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\nf : R[X]\nhf : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 maximalIdeal R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (\u2191derivative f))\nH :\n \u2191(aeval (\u2191(residue R) a\u2080)) f = 0 \u2192\n \u2191(aeval (\u2191(residue R) a\u2080)) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = \u2191(residue R) a\u2080\naux : Polynomial.eval a\u2080 (\u2191derivative f) \u2208 nonunits R \u2192 False\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 maximalIdeal R\n[PROOFSTEP]\nsimp only [aeval_def, ResidueField.algebraMap_eq, eval\u2082_at_apply, \u2190 Ideal.Quotient.eq_zero_iff_mem, \u2190\n LocalRing.mem_maximalIdeal] at H h\u2081 aux \n[GOAL]\ncase _2_1.is_henselian\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\nf : R[X]\nhf : Monic f\na\u2080 : R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (\u2191derivative f))\nH :\n \u2191(residue R) (Polynomial.eval a\u2080 f) = 0 \u2192\n \u2191(residue R) (Polynomial.eval a\u2080 (\u2191derivative f)) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = \u2191(residue R) a\u2080\nh\u2081 : \u2191(Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 f) = 0\naux : \u2191(Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (\u2191derivative f)) = 0 \u2192 False\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 maximalIdeal R\n[PROOFSTEP]\nobtain \u27e8a, ha\u2081, ha\u2082\u27e9 := H h\u2081 aux\n[GOAL]\ncase _2_1.is_henselian.intro.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\nf : R[X]\nhf : Monic f\na\u2080 : R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (\u2191derivative f))\nH :\n \u2191(residue R) (Polynomial.eval a\u2080 f) = 0 \u2192\n \u2191(residue R) (Polynomial.eval a\u2080 (\u2191derivative f)) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = \u2191(residue R) a\u2080\nh\u2081 : \u2191(Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 f) = 0\naux : \u2191(Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (\u2191derivative f)) = 0 \u2192 False\na : R\nha\u2081 : IsRoot f a\nha\u2082 : \u2191(residue R) a = \u2191(residue R) a\u2080\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 maximalIdeal R\n[PROOFSTEP]\nrefine' \u27e8a, ha\u2081, _\u27e9\n[GOAL]\ncase _2_1.is_henselian.intro.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\nf : R[X]\nhf : Monic f\na\u2080 : R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (\u2191derivative f))\nH :\n \u2191(residue R) (Polynomial.eval a\u2080 f) = 0 \u2192\n \u2191(residue R) (Polynomial.eval a\u2080 (\u2191derivative f)) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = \u2191(residue R) a\u2080\nh\u2081 : \u2191(Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 f) = 0\naux : \u2191(Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (\u2191derivative f)) = 0 \u2192 False\na : R\nha\u2081 : IsRoot f a\nha\u2082 : \u2191(residue R) a = \u2191(residue R) a\u2080\n\u22a2 a - a\u2080 \u2208 maximalIdeal R\n[PROOFSTEP]\nrw [\u2190 Ideal.Quotient.eq_zero_iff_mem]\n[GOAL]\ncase _2_1.is_henselian.intro.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\nf : R[X]\nhf : Monic f\na\u2080 : R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (\u2191derivative f))\nH :\n \u2191(residue R) (Polynomial.eval a\u2080 f) = 0 \u2192\n \u2191(residue R) (Polynomial.eval a\u2080 (\u2191derivative f)) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = \u2191(residue R) a\u2080\nh\u2081 : \u2191(Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 f) = 0\naux : \u2191(Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (\u2191derivative f)) = 0 \u2192 False\na : R\nha\u2081 : IsRoot f a\nha\u2082 : \u2191(residue R) a = \u2191(residue R) a\u2080\n\u22a2 \u2191(Ideal.Quotient.mk (maximalIdeal R)) (a - a\u2080) = 0\n[PROOFSTEP]\nrwa [\u2190 sub_eq_zero, \u2190 RingHom.map_sub] at ha\u2082 \n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\n_2_1 :\n (\u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080) \u2192\n HenselianLocalRing R\n\u22a2 List.TFAE\n [HenselianLocalRing R,\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080,\n \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080]\n[PROOFSTEP]\ntfae_have _1_3 : 1 \u2192 3\n[GOAL]\ncase _1_3\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\n_2_1 :\n (\u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080) \u2192\n HenselianLocalRing R\n\u22a2 HenselianLocalRing R \u2192\n \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080\n[PROOFSTEP]\nintro hR K _K \u03c6 h\u03c6 f hf a\u2080 h\u2081 h\u2082\n[GOAL]\ncase _1_3\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\n_2_1 :\n (\u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080) \u2192\n HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u2191\u03c6\nf : R[X]\nhf : Monic f\na\u2080 : K\nh\u2081 : eval\u2082 \u03c6 a\u2080 f = 0\nh\u2082 : eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0\n\u22a2 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080\n[PROOFSTEP]\nobtain \u27e8a\u2080, rfl\u27e9 := h\u03c6 a\u2080\n[GOAL]\ncase _1_3.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\n_2_1 :\n (\u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080) \u2192\n HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u2191\u03c6\nf : R[X]\nhf : Monic f\na\u2080 : R\nh\u2081 : eval\u2082 \u03c6 (\u2191\u03c6 a\u2080) f = 0\nh\u2082 : eval\u2082 \u03c6 (\u2191\u03c6 a\u2080) (\u2191derivative f) \u2260 0\n\u22a2 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = \u2191\u03c6 a\u2080\n[PROOFSTEP]\nhave H := HenselianLocalRing.is_henselian f hf a\u2080\n[GOAL]\ncase _1_3.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\n_2_1 :\n (\u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080) \u2192\n HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u2191\u03c6\nf : R[X]\nhf : Monic f\na\u2080 : R\nh\u2081 : eval\u2082 \u03c6 (\u2191\u03c6 a\u2080) f = 0\nh\u2082 : eval\u2082 \u03c6 (\u2191\u03c6 a\u2080) (\u2191derivative f) \u2260 0\nH :\n Polynomial.eval a\u2080 f \u2208 maximalIdeal R \u2192\n IsUnit (Polynomial.eval a\u2080 (\u2191derivative f)) \u2192 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 maximalIdeal R\n\u22a2 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = \u2191\u03c6 a\u2080\n[PROOFSTEP]\nsimp only [\u2190 ker_eq_maximalIdeal \u03c6 h\u03c6, eval\u2082_at_apply, RingHom.mem_ker \u03c6] at H h\u2081 h\u2082 \n[GOAL]\ncase _1_3.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\n_2_1 :\n (\u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080) \u2192\n HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u2191\u03c6\nf : R[X]\nhf : Monic f\na\u2080 : R\nH : \u2191\u03c6 (Polynomial.eval a\u2080 f) = 0 \u2192 IsUnit (Polynomial.eval a\u2080 (\u2191derivative f)) \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 (a - a\u2080) = 0\nh\u2081 : \u2191\u03c6 (Polynomial.eval a\u2080 f) = 0\nh\u2082 : \u2191\u03c6 (Polynomial.eval a\u2080 (\u2191derivative f)) \u2260 0\n\u22a2 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = \u2191\u03c6 a\u2080\n[PROOFSTEP]\nobtain \u27e8a, ha\u2081, ha\u2082\u27e9 :=\n H h\u2081\n (by\n contrapose! h\u2082\n rwa [\u2190 mem_nonunits_iff, \u2190 LocalRing.mem_maximalIdeal, \u2190 LocalRing.ker_eq_maximalIdeal \u03c6 h\u03c6, RingHom.mem_ker] at\n h\u2082 )\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\n_2_1 :\n (\u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080) \u2192\n HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u2191\u03c6\nf : R[X]\nhf : Monic f\na\u2080 : R\nH : \u2191\u03c6 (Polynomial.eval a\u2080 f) = 0 \u2192 IsUnit (Polynomial.eval a\u2080 (\u2191derivative f)) \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 (a - a\u2080) = 0\nh\u2081 : \u2191\u03c6 (Polynomial.eval a\u2080 f) = 0\nh\u2082 : \u2191\u03c6 (Polynomial.eval a\u2080 (\u2191derivative f)) \u2260 0\n\u22a2 IsUnit (Polynomial.eval a\u2080 (\u2191derivative f))\n[PROOFSTEP]\ncontrapose! h\u2082\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\n_2_1 :\n (\u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080) \u2192\n HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u2191\u03c6\nf : R[X]\nhf : Monic f\na\u2080 : R\nH : \u2191\u03c6 (Polynomial.eval a\u2080 f) = 0 \u2192 IsUnit (Polynomial.eval a\u2080 (\u2191derivative f)) \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 (a - a\u2080) = 0\nh\u2081 : \u2191\u03c6 (Polynomial.eval a\u2080 f) = 0\nh\u2082 : \u00acIsUnit (Polynomial.eval a\u2080 (\u2191derivative f))\n\u22a2 \u2191\u03c6 (Polynomial.eval a\u2080 (\u2191derivative f)) = 0\n[PROOFSTEP]\nrwa [\u2190 mem_nonunits_iff, \u2190 LocalRing.mem_maximalIdeal, \u2190 LocalRing.ker_eq_maximalIdeal \u03c6 h\u03c6, RingHom.mem_ker] at h\u2082 \n[GOAL]\ncase _1_3.intro.intro.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\n_2_1 :\n (\u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080) \u2192\n HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u2191\u03c6\nf : R[X]\nhf : Monic f\na\u2080 : R\nH : \u2191\u03c6 (Polynomial.eval a\u2080 f) = 0 \u2192 IsUnit (Polynomial.eval a\u2080 (\u2191derivative f)) \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 (a - a\u2080) = 0\nh\u2081 : \u2191\u03c6 (Polynomial.eval a\u2080 f) = 0\nh\u2082 : \u2191\u03c6 (Polynomial.eval a\u2080 (\u2191derivative f)) \u2260 0\na : R\nha\u2081 : IsRoot f a\nha\u2082 : \u2191\u03c6 (a - a\u2080) = 0\n\u22a2 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = \u2191\u03c6 a\u2080\n[PROOFSTEP]\nrefine' \u27e8a, ha\u2081, _\u27e9\n[GOAL]\ncase _1_3.intro.intro.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\n_2_1 :\n (\u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080) \u2192\n HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u2191\u03c6\nf : R[X]\nhf : Monic f\na\u2080 : R\nH : \u2191\u03c6 (Polynomial.eval a\u2080 f) = 0 \u2192 IsUnit (Polynomial.eval a\u2080 (\u2191derivative f)) \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 (a - a\u2080) = 0\nh\u2081 : \u2191\u03c6 (Polynomial.eval a\u2080 f) = 0\nh\u2082 : \u2191\u03c6 (Polynomial.eval a\u2080 (\u2191derivative f)) \u2260 0\na : R\nha\u2081 : IsRoot f a\nha\u2082 : \u2191\u03c6 (a - a\u2080) = 0\n\u22a2 \u2191\u03c6 a = \u2191\u03c6 a\u2080\n[PROOFSTEP]\nrwa [\u03c6.map_sub, sub_eq_zero] at ha\u2082 \n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n_3_2 :\n (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080) \u2192\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080\n_2_1 :\n (\u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080) \u2192\n HenselianLocalRing R\n_1_3 :\n HenselianLocalRing R \u2192\n \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080\n\u22a2 List.TFAE\n [HenselianLocalRing R,\n \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : ResidueField R),\n \u2191(aeval a\u2080) f = 0 \u2192 \u2191(aeval a\u2080) (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191(residue R) a = a\u2080,\n \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n Surjective \u2191\u03c6 \u2192\n \u2200 (f : R[X]),\n Monic f \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (\u2191derivative f) \u2260 0 \u2192 \u2203 a, IsRoot f a \u2227 \u2191\u03c6 a = a\u2080]\n[PROOFSTEP]\ntfae_finish\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nhR : HenselianLocalRing R\n\u22a2 maximalIdeal R \u2264 Ideal.jacobson \u22a5\n[PROOFSTEP]\nrw [Ideal.jacobson, le_sInf_iff]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nhR : HenselianLocalRing R\n\u22a2 \u2200 (b : Ideal R), b \u2208 {J | \u22a5 \u2264 J \u2227 Ideal.IsMaximal J} \u2192 maximalIdeal R \u2264 b\n[PROOFSTEP]\nrintro I \u27e8-, hI\u27e9\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d : CommRing R\nhR : HenselianLocalRing R\nI : Ideal R\nhI : Ideal.IsMaximal I\n\u22a2 maximalIdeal R \u2264 I\n[PROOFSTEP]\nexact (eq_maximalIdeal hI).ge\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nhR : HenselianLocalRing R\n\u22a2 \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : R),\n Polynomial.eval a\u2080 f \u2208 maximalIdeal R \u2192\n IsUnit (\u2191(Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (\u2191derivative f))) \u2192\n \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 maximalIdeal R\n[PROOFSTEP]\nintro f hf a\u2080 h\u2081 h\u2082\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nhR : HenselianLocalRing R\nf : R[X]\nhf : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 maximalIdeal R\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (\u2191derivative f)))\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 maximalIdeal R\n[PROOFSTEP]\nrefine' HenselianLocalRing.is_henselian f hf a\u2080 h\u2081 _\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nhR : HenselianLocalRing R\nf : R[X]\nhf : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 maximalIdeal R\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (\u2191derivative f)))\n\u22a2 IsUnit (Polynomial.eval a\u2080 (\u2191derivative f))\n[PROOFSTEP]\ncontrapose! h\u2082\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nhR : HenselianLocalRing R\nf : R[X]\nhf : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 maximalIdeal R\nh\u2082 : \u00acIsUnit (Polynomial.eval a\u2080 (\u2191derivative f))\n\u22a2 \u00acIsUnit (\u2191(Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (\u2191derivative f)))\n[PROOFSTEP]\nrw [\u2190 mem_nonunits_iff, \u2190 LocalRing.mem_maximalIdeal, \u2190 Ideal.Quotient.eq_zero_iff_mem] at h\u2082 \n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nhR : HenselianLocalRing R\nf : R[X]\nhf : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 maximalIdeal R\nh\u2082\u271d : Polynomial.eval a\u2080 (\u2191derivative f) \u2208 nonunits R\nh\u2082 : \u2191(Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (\u2191derivative f)) = 0\n\u22a2 \u00acIsUnit (\u2191(Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (\u2191derivative f)))\n[PROOFSTEP]\nrw [h\u2082]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nhR : HenselianLocalRing R\nf : R[X]\nhf : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 maximalIdeal R\nh\u2082\u271d : Polynomial.eval a\u2080 (\u2191derivative f) \u2208 nonunits R\nh\u2082 : \u2191(Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (\u2191derivative f)) = 0\n\u22a2 \u00acIsUnit 0\n[PROOFSTEP]\nexact not_isUnit_zero\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\n\u22a2 \u2200 (f : R[X]),\n Monic f \u2192\n \u2200 (a\u2080 : R),\n Polynomial.eval a\u2080 f \u2208 I \u2192\n IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f))) \u2192 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 I\n[PROOFSTEP]\nintro f _ a\u2080 h\u2081 h\u2082\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 I\n[PROOFSTEP]\nclassical\nlet f' := derivative f\nlet c : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun _ b => b - f.eval b * Ring.inverse (f'.eval b)\nhave hc : \u2200 n, c (n + 1) = c n - f.eval (c n) * Ring.inverse (f'.eval (c n)) :=\n by\n intro n\n simp only [Nat.rec_add_one]\n -- we now spend some time determining properties of the sequence `c : \u2115 \u2192 R`\n -- `hc_mod`: for every `n`, we have `c n \u2261 a\u2080 [SMOD I]`\n -- `hf'c` : for every `n`, `f'.eval (c n)` is a unit\n -- `hfcI` : for every `n`, `f.eval (c n)` is contained in `I ^ (n+1)`\nhave hc_mod : \u2200 n, c n \u2261 a\u2080 [SMOD I] := by\n intro n\n induction' n with n ih\n \u00b7 rfl\n rw [Nat.succ_eq_add_one, hc, sub_eq_add_neg, \u2190 add_zero a\u2080]\n refine' ih.add _\n rw [SModEq.zero, Ideal.neg_mem_iff]\n refine' I.mul_mem_right _ _\n rw [\u2190 SModEq.zero] at h\u2081 \u22a2\n exact (ih.eval f).trans h\u2081\nhave hf'c : \u2200 n, IsUnit (f'.eval (c n)) := by\n intro n\n haveI := isLocalRingHom_of_le_jacobson_bot I (IsAdicComplete.le_jacobson_bot I)\n apply isUnit_of_map_unit (Ideal.Quotient.mk I)\n convert h\u2082 using 1\n exact SModEq.def.mp ((hc_mod n).eval _)\nhave hfcI : \u2200 n, f.eval (c n) \u2208 I ^ (n + 1) := by\n intro n\n induction' n with n ih\n \u00b7 simpa only [Nat.zero_eq, Nat.rec_zero, zero_add, pow_one] using h\u2081\n rw [Nat.succ_eq_add_one, \u2190 taylor_eval_sub (c n), hc, sub_eq_add_neg, sub_eq_add_neg, add_neg_cancel_comm]\n rw [eval_eq_sum, sum_over_range' _ _ _ (lt_add_of_pos_right _ zero_lt_two), \u2190\n Finset.sum_range_add_sum_Ico _ (Nat.le_add_left _ _)]\n swap\n \u00b7 intro i\n rw [zero_mul]\n refine' Ideal.add_mem _ _ _\n \u00b7 erw [Finset.sum_range_succ]\n rw [Finset.range_one, Finset.sum_singleton, taylor_coeff_zero, taylor_coeff_one, pow_zero, pow_one, mul_one,\n mul_neg, mul_left_comm, Ring.mul_inverse_cancel _ (hf'c n), mul_one, add_neg_self]\n exact Ideal.zero_mem _\n \u00b7 refine' Submodule.sum_mem _ _\n simp only [Finset.mem_Ico]\n rintro i \u27e8h2i, _\u27e9\n have aux : n + 2 \u2264 i * (n + 1) := by trans 2 * (n + 1) <;> nlinarith only [h2i]\n refine' Ideal.mul_mem_left _ _ (Ideal.pow_le_pow aux _)\n rw [pow_mul']\n refine'\n Ideal.pow_mem_pow ((Ideal.neg_mem_iff _).2 <| Ideal.mul_mem_right _ _ ih)\n _\n -- we are now in the position to show that `c : \u2115 \u2192 R` is a Cauchy sequence\nhave aux : \u2200 m n, m \u2264 n \u2192 c m \u2261 c n [SMOD (I ^ m \u2022 \u22a4 : Ideal R)] :=\n by\n intro m n hmn\n rw [\u2190 Ideal.one_eq_top, Ideal.smul_eq_mul, mul_one]\n obtain \u27e8k, rfl\u27e9 := Nat.exists_eq_add_of_le hmn\n clear hmn\n induction' k with k ih\n \u00b7 rw [Nat.zero_eq, add_zero]\n rw [Nat.succ_eq_add_one, \u2190 add_assoc, hc, \u2190 add_zero (c m), sub_eq_add_neg]\n refine' ih.add _\n symm\n rw [SModEq.zero, Ideal.neg_mem_iff]\n refine' Ideal.mul_mem_right _ _ (Ideal.pow_le_pow _ (hfcI _))\n rw [add_assoc]\n exact le_self_add\nobtain \u27e8a, ha\u27e9 := IsPrecomplete.prec' c (aux _ _)\nrefine' \u27e8a, _, _\u27e9\n\u00b7 show f.IsRoot a\n suffices \u2200 n, f.eval a \u2261 0 [SMOD (I ^ n \u2022 \u22a4 : Ideal R)] by exact IsHausdorff.haus' _ this\n intro n\n specialize ha n\n rw [\u2190 Ideal.one_eq_top, Ideal.smul_eq_mul, mul_one] at ha \u22a2\n refine' (ha.symm.eval f).trans _\n rw [SModEq.zero]\n exact Ideal.pow_le_pow le_self_add (hfcI _)\n\u00b7 show a - a\u2080 \u2208 I\n specialize ha 1\n rw [hc, pow_one, \u2190 Ideal.one_eq_top, Ideal.smul_eq_mul, mul_one, sub_eq_add_neg] at ha \n rw [\u2190 SModEq.sub_mem, \u2190 add_zero a\u2080]\n refine' ha.symm.trans (SModEq.rfl.add _)\n rw [SModEq.zero, Ideal.neg_mem_iff]\n exact Ideal.mul_mem_right _ _ h\u2081\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 I\n[PROOFSTEP]\nlet f' := derivative f\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 I\n[PROOFSTEP]\nlet c : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun _ b => b - f.eval b * Ring.inverse (f'.eval b)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 I\n[PROOFSTEP]\nhave hc : \u2200 n, c (n + 1) = c n - f.eval (c n) * Ring.inverse (f'.eval (c n)) :=\n by\n intro n\n simp only [Nat.rec_add_one]\n -- we now spend some time determining properties of the sequence `c : \u2115 \u2192 R`\n -- `hc_mod`: for every `n`, we have `c n \u2261 a\u2080 [SMOD I]`\n -- `hf'c` : for every `n`, `f'.eval (c n)` is a unit\n -- `hfcI` : for every `n`, `f.eval (c n)` is contained in `I ^ (n+1)`\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\n\u22a2 \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\n[PROOFSTEP]\nintro n\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nn : \u2115\n\u22a2 c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\n[PROOFSTEP]\nsimp only [Nat.rec_add_one]\n -- we now spend some time determining properties of the sequence `c : \u2115 \u2192 R`\n -- `hc_mod`: for every `n`, we have `c n \u2261 a\u2080 [SMOD I]`\n -- `hf'c` : for every `n`, `f'.eval (c n)` is a unit\n -- `hfcI` : for every `n`, `f.eval (c n)` is contained in `I ^ (n+1)`\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 I\n[PROOFSTEP]\nhave hc_mod : \u2200 n, c n \u2261 a\u2080 [SMOD I] := by\n intro n\n induction' n with n ih\n \u00b7 rfl\n rw [Nat.succ_eq_add_one, hc, sub_eq_add_neg, \u2190 add_zero a\u2080]\n refine' ih.add _\n rw [SModEq.zero, Ideal.neg_mem_iff]\n refine' I.mul_mem_right _ _\n rw [\u2190 SModEq.zero] at h\u2081 \u22a2\n exact (ih.eval f).trans h\u2081\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\n\u22a2 \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\n[PROOFSTEP]\nintro n\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nn : \u2115\n\u22a2 c n \u2261 a\u2080 [SMOD I]\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\n\u22a2 c Nat.zero \u2261 a\u2080 [SMOD I]\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nn : \u2115\nih : c n \u2261 a\u2080 [SMOD I]\n\u22a2 c (Nat.succ n) \u2261 a\u2080 [SMOD I]\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one, hc, sub_eq_add_neg, \u2190 add_zero a\u2080]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nn : \u2115\nih : c n \u2261 a\u2080 [SMOD I]\n\u22a2 c n + -(Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')) \u2261 a\u2080 + 0 [SMOD I]\n[PROOFSTEP]\nrefine' ih.add _\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nn : \u2115\nih : c n \u2261 a\u2080 [SMOD I]\n\u22a2 -(Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')) \u2261 0 [SMOD I]\n[PROOFSTEP]\nrw [SModEq.zero, Ideal.neg_mem_iff]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nn : \u2115\nih : c n \u2261 a\u2080 [SMOD I]\n\u22a2 Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f') \u2208 I\n[PROOFSTEP]\nrefine' I.mul_mem_right _ _\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nn : \u2115\nih : c n \u2261 a\u2080 [SMOD I]\n\u22a2 Polynomial.eval (c n) f \u2208 I\n[PROOFSTEP]\nrw [\u2190 SModEq.zero] at h\u2081 \u22a2\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2261 0 [SMOD I]\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : R[X] := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nn : \u2115\nih : c n \u2261 a\u2080 [SMOD I]\n\u22a2 Polynomial.eval (c n) f \u2261 0 [SMOD I]\n[PROOFSTEP]\nexact (ih.eval f).trans h\u2081\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 I\n[PROOFSTEP]\nhave hf'c : \u2200 n, IsUnit (f'.eval (c n)) := by\n intro n\n haveI := isLocalRingHom_of_le_jacobson_bot I (IsAdicComplete.le_jacobson_bot I)\n apply isUnit_of_map_unit (Ideal.Quotient.mk I)\n convert h\u2082 using 1\n exact SModEq.def.mp ((hc_mod n).eval _)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\n\u22a2 \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\n[PROOFSTEP]\nintro n\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nn : \u2115\n\u22a2 IsUnit (Polynomial.eval (c n) f')\n[PROOFSTEP]\nhaveI := isLocalRingHom_of_le_jacobson_bot I (IsAdicComplete.le_jacobson_bot I)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nn : \u2115\nthis : IsLocalRingHom (Ideal.Quotient.mk I)\n\u22a2 IsUnit (Polynomial.eval (c n) f')\n[PROOFSTEP]\napply isUnit_of_map_unit (Ideal.Quotient.mk I)\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nn : \u2115\nthis : IsLocalRingHom (Ideal.Quotient.mk I)\n\u22a2 IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval (c n) f'))\n[PROOFSTEP]\nconvert h\u2082 using 1\n[GOAL]\ncase h.e'_3\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nn : \u2115\nthis : IsLocalRingHom (Ideal.Quotient.mk I)\n\u22a2 \u2191(Ideal.Quotient.mk I) (Polynomial.eval (c n) f') = \u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f))\n[PROOFSTEP]\nexact SModEq.def.mp ((hc_mod n).eval _)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 I\n[PROOFSTEP]\nhave hfcI : \u2200 n, f.eval (c n) \u2208 I ^ (n + 1) := by\n intro n\n induction' n with n ih\n \u00b7 simpa only [Nat.zero_eq, Nat.rec_zero, zero_add, pow_one] using h\u2081\n rw [Nat.succ_eq_add_one, \u2190 taylor_eval_sub (c n), hc, sub_eq_add_neg, sub_eq_add_neg, add_neg_cancel_comm]\n rw [eval_eq_sum, sum_over_range' _ _ _ (lt_add_of_pos_right _ zero_lt_two), \u2190\n Finset.sum_range_add_sum_Ico _ (Nat.le_add_left _ _)]\n swap\n \u00b7 intro i\n rw [zero_mul]\n refine' Ideal.add_mem _ _ _\n \u00b7 erw [Finset.sum_range_succ]\n rw [Finset.range_one, Finset.sum_singleton, taylor_coeff_zero, taylor_coeff_one, pow_zero, pow_one, mul_one,\n mul_neg, mul_left_comm, Ring.mul_inverse_cancel _ (hf'c n), mul_one, add_neg_self]\n exact Ideal.zero_mem _\n \u00b7 refine' Submodule.sum_mem _ _\n simp only [Finset.mem_Ico]\n rintro i \u27e8h2i, _\u27e9\n have aux : n + 2 \u2264 i * (n + 1) := by trans 2 * (n + 1) <;> nlinarith only [h2i]\n refine' Ideal.mul_mem_left _ _ (Ideal.pow_le_pow aux _)\n rw [pow_mul']\n refine'\n Ideal.pow_mem_pow ((Ideal.neg_mem_iff _).2 <| Ideal.mul_mem_right _ _ ih)\n _\n -- we are now in the position to show that `c : \u2115 \u2192 R` is a Cauchy sequence\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\n\u22a2 \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\n[PROOFSTEP]\nintro n\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\n\u22a2 Polynomial.eval (c n) f \u2208 I ^ (n + 1)\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\n\u22a2 Polynomial.eval (c Nat.zero) f \u2208 I ^ (Nat.zero + 1)\n[PROOFSTEP]\nsimpa only [Nat.zero_eq, Nat.rec_zero, zero_add, pow_one] using h\u2081\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\n\u22a2 Polynomial.eval (c (Nat.succ n)) f \u2208 I ^ (Nat.succ n + 1)\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one, \u2190 taylor_eval_sub (c n), hc, sub_eq_add_neg, sub_eq_add_neg, add_neg_cancel_comm]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\n\u22a2 Polynomial.eval (-(Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f'))) (\u2191(taylor (c n)) f) \u2208\n I ^ (n + 1 + 1)\n[PROOFSTEP]\nrw [eval_eq_sum, sum_over_range' _ _ _ (lt_add_of_pos_right _ zero_lt_two), \u2190\n Finset.sum_range_add_sum_Ico _ (Nat.le_add_left _ _)]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\n\u22a2 \u2211 k in Finset.range 2,\n coeff (\u2191(taylor (c n)) f) k * (-(Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f'))) ^ k +\n \u2211 k in Finset.Ico 2 (natDegree (\u2191(taylor (c n)) f) + 2),\n coeff (\u2191(taylor (c n)) f) k * (-(Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f'))) ^ k \u2208\n I ^ (n + 1 + 1)\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\n\u22a2 \u2200 (n_1 : \u2115), 0 * (-(Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f'))) ^ n_1 = 0\n[PROOFSTEP]\nswap\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\n\u22a2 \u2200 (n_1 : \u2115), 0 * (-(Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f'))) ^ n_1 = 0\n[PROOFSTEP]\nintro i\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\ni : \u2115\n\u22a2 0 * (-(Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f'))) ^ i = 0\n[PROOFSTEP]\nrw [zero_mul]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\n\u22a2 \u2211 k in Finset.range 2,\n coeff (\u2191(taylor (c n)) f) k * (-(Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f'))) ^ k +\n \u2211 k in Finset.Ico 2 (natDegree (\u2191(taylor (c n)) f) + 2),\n coeff (\u2191(taylor (c n)) f) k * (-(Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f'))) ^ k \u2208\n I ^ (n + 1 + 1)\n[PROOFSTEP]\nrefine' Ideal.add_mem _ _ _\n[GOAL]\ncase succ.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\n\u22a2 \u2211 k in Finset.range 2,\n coeff (\u2191(taylor (c n)) f) k * (-(Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f'))) ^ k \u2208\n I ^ (n + 1 + 1)\n[PROOFSTEP]\nerw [Finset.sum_range_succ]\n[GOAL]\ncase succ.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\n\u22a2 \u2211 x in Finset.range 1,\n coeff (\u2191(taylor (c n)) f) x * (-(Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f'))) ^ x +\n coeff (\u2191(taylor (c n)) f) 1 * (-(Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f'))) ^ 1 \u2208\n I ^ (n + 1 + 1)\n[PROOFSTEP]\nrw [Finset.range_one, Finset.sum_singleton, taylor_coeff_zero, taylor_coeff_one, pow_zero, pow_one, mul_one, mul_neg,\n mul_left_comm, Ring.mul_inverse_cancel _ (hf'c n), mul_one, add_neg_self]\n[GOAL]\ncase succ.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\n\u22a2 0 \u2208 I ^ (n + 1 + 1)\n[PROOFSTEP]\nexact Ideal.zero_mem _\n[GOAL]\ncase succ.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\n\u22a2 \u2211 k in Finset.Ico 2 (natDegree (\u2191(taylor (c n)) f) + 2),\n coeff (\u2191(taylor (c n)) f) k * (-(Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f'))) ^ k \u2208\n I ^ (n + 1 + 1)\n[PROOFSTEP]\nrefine' Submodule.sum_mem _ _\n[GOAL]\ncase succ.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\n\u22a2 \u2200 (c_1 : \u2115),\n c_1 \u2208 Finset.Ico 2 (natDegree (\u2191(taylor (c n)) f) + 2) \u2192\n coeff (\u2191(taylor (c n)) f) c_1 * (-(Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f'))) ^ c_1 \u2208\n I ^ (n + 1 + 1)\n[PROOFSTEP]\nsimp only [Finset.mem_Ico]\n[GOAL]\ncase succ.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\n\u22a2 \u2200 (c : \u2115),\n 2 \u2264 c \u2227\n c <\n natDegree\n (\u2191(taylor\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f)))\n n))\n f) +\n 2 \u2192\n coeff\n (\u2191(taylor\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f)))\n n))\n f)\n c *\n (-(Polynomial.eval\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n)\n f *\n Ring.inverse\n (Polynomial.eval\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f)))\n n)\n (\u2191derivative f)))) ^\n c \u2208\n I ^ (n + 1 + 1)\n[PROOFSTEP]\nrintro i \u27e8h2i, _\u27e9\n[GOAL]\ncase succ.refine'_2.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\ni : \u2115\nh2i : 2 \u2264 i\nright\u271d :\n i <\n natDegree\n (\u2191(taylor\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n))\n f) +\n 2\n\u22a2 coeff\n (\u2191(taylor\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n))\n f)\n i *\n (-(Polynomial.eval\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n) f *\n Ring.inverse\n (Polynomial.eval\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n)\n (\u2191derivative f)))) ^\n i \u2208\n I ^ (n + 1 + 1)\n[PROOFSTEP]\nhave aux : n + 2 \u2264 i * (n + 1) := by trans 2 * (n + 1) <;> nlinarith only [h2i]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\ni : \u2115\nh2i : 2 \u2264 i\nright\u271d :\n i <\n natDegree\n (\u2191(taylor\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n))\n f) +\n 2\n\u22a2 n + 2 \u2264 i * (n + 1)\n[PROOFSTEP]\ntrans 2 * (n + 1)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\ni : \u2115\nh2i : 2 \u2264 i\nright\u271d :\n i <\n natDegree\n (\u2191(taylor\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n))\n f) +\n 2\n\u22a2 n + 2 \u2264 2 * (n + 1)\n[PROOFSTEP]\nnlinarith only [h2i]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\ni : \u2115\nh2i : 2 \u2264 i\nright\u271d :\n i <\n natDegree\n (\u2191(taylor\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n))\n f) +\n 2\n\u22a2 2 * (n + 1) \u2264 i * (n + 1)\n[PROOFSTEP]\nnlinarith only [h2i]\n[GOAL]\ncase succ.refine'_2.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\ni : \u2115\nh2i : 2 \u2264 i\nright\u271d :\n i <\n natDegree\n (\u2191(taylor\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n))\n f) +\n 2\naux : n + 2 \u2264 i * (n + 1)\n\u22a2 coeff\n (\u2191(taylor\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n))\n f)\n i *\n (-(Polynomial.eval\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n) f *\n Ring.inverse\n (Polynomial.eval\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n)\n (\u2191derivative f)))) ^\n i \u2208\n I ^ (n + 1 + 1)\n[PROOFSTEP]\nrefine' Ideal.mul_mem_left _ _ (Ideal.pow_le_pow aux _)\n[GOAL]\ncase succ.refine'_2.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\ni : \u2115\nh2i : 2 \u2264 i\nright\u271d :\n i <\n natDegree\n (\u2191(taylor\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n))\n f) +\n 2\naux : n + 2 \u2264 i * (n + 1)\n\u22a2 (-(Polynomial.eval\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n) f *\n Ring.inverse\n (Polynomial.eval\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n)\n (\u2191derivative f)))) ^\n i \u2208\n I ^ (i * (n + 1))\n[PROOFSTEP]\nrw [pow_mul']\n[GOAL]\ncase succ.refine'_2.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nn : \u2115\nih : Polynomial.eval (c n) f \u2208 I ^ (n + 1)\ni : \u2115\nh2i : 2 \u2264 i\nright\u271d :\n i <\n natDegree\n (\u2191(taylor\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n))\n f) +\n 2\naux : n + 2 \u2264 i * (n + 1)\n\u22a2 (-(Polynomial.eval\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n) f *\n Ring.inverse\n (Polynomial.eval\n (Nat.rec a\u2080 (fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b (\u2191derivative f))) n)\n (\u2191derivative f)))) ^\n i \u2208\n (I ^ (n + 1)) ^ i\n[PROOFSTEP]\nrefine'\n Ideal.pow_mem_pow ((Ideal.neg_mem_iff _).2 <| Ideal.mul_mem_right _ _ ih)\n _\n -- we are now in the position to show that `c : \u2115 \u2192 R` is a Cauchy sequence\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 I\n[PROOFSTEP]\nhave aux : \u2200 m n, m \u2264 n \u2192 c m \u2261 c n [SMOD (I ^ m \u2022 \u22a4 : Ideal R)] :=\n by\n intro m n hmn\n rw [\u2190 Ideal.one_eq_top, Ideal.smul_eq_mul, mul_one]\n obtain \u27e8k, rfl\u27e9 := Nat.exists_eq_add_of_le hmn\n clear hmn\n induction' k with k ih\n \u00b7 rw [Nat.zero_eq, add_zero]\n rw [Nat.succ_eq_add_one, \u2190 add_assoc, hc, \u2190 add_zero (c m), sub_eq_add_neg]\n refine' ih.add _\n symm\n rw [SModEq.zero, Ideal.neg_mem_iff]\n refine' Ideal.mul_mem_right _ _ (Ideal.pow_le_pow _ (hfcI _))\n rw [add_assoc]\n exact le_self_add\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\n\u22a2 \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\n[PROOFSTEP]\nintro m n hmn\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\nm n : \u2115\nhmn : m \u2264 n\n\u22a2 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\n[PROOFSTEP]\nrw [\u2190 Ideal.one_eq_top, Ideal.smul_eq_mul, mul_one]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\nm n : \u2115\nhmn : m \u2264 n\n\u22a2 c m \u2261 c n [SMOD I ^ m]\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := Nat.exists_eq_add_of_le hmn\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\nm k : \u2115\nhmn : m \u2264 m + k\n\u22a2 c m \u2261 c (m + k) [SMOD I ^ m]\n[PROOFSTEP]\nclear hmn\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\nm k : \u2115\n\u22a2 c m \u2261 c (m + k) [SMOD I ^ m]\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase intro.zero\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\nm : \u2115\n\u22a2 c m \u2261 c (m + Nat.zero) [SMOD I ^ m]\n[PROOFSTEP]\nrw [Nat.zero_eq, add_zero]\n[GOAL]\ncase intro.succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\nm k : \u2115\nih : c m \u2261 c (m + k) [SMOD I ^ m]\n\u22a2 c m \u2261 c (m + Nat.succ k) [SMOD I ^ m]\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one, \u2190 add_assoc, hc, \u2190 add_zero (c m), sub_eq_add_neg]\n[GOAL]\ncase intro.succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\nm k : \u2115\nih : c m \u2261 c (m + k) [SMOD I ^ m]\n\u22a2 c m + 0 \u2261 c (m + k) + -(Polynomial.eval (c (m + k)) f * Ring.inverse (Polynomial.eval (c (m + k)) f')) [SMOD I ^ m]\n[PROOFSTEP]\nrefine' ih.add _\n[GOAL]\ncase intro.succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\nm k : \u2115\nih : c m \u2261 c (m + k) [SMOD I ^ m]\n\u22a2 0 \u2261 -(Polynomial.eval (c (m + k)) f * Ring.inverse (Polynomial.eval (c (m + k)) f')) [SMOD I ^ m]\n[PROOFSTEP]\nsymm\n[GOAL]\ncase intro.succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\nm k : \u2115\nih : c m \u2261 c (m + k) [SMOD I ^ m]\n\u22a2 -(Polynomial.eval (c (m + k)) f * Ring.inverse (Polynomial.eval (c (m + k)) f')) \u2261 0 [SMOD I ^ m]\n[PROOFSTEP]\nrw [SModEq.zero, Ideal.neg_mem_iff]\n[GOAL]\ncase intro.succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\nm k : \u2115\nih : c m \u2261 c (m + k) [SMOD I ^ m]\n\u22a2 Polynomial.eval (c (m + k)) f * Ring.inverse (Polynomial.eval (c (m + k)) f') \u2208 I ^ m\n[PROOFSTEP]\nrefine' Ideal.mul_mem_right _ _ (Ideal.pow_le_pow _ (hfcI _))\n[GOAL]\ncase intro.succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\nm k : \u2115\nih : c m \u2261 c (m + k) [SMOD I ^ m]\n\u22a2 m \u2264 m + k + 1\n[PROOFSTEP]\nrw [add_assoc]\n[GOAL]\ncase intro.succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\nm k : \u2115\nih : c m \u2261 c (m + k) [SMOD I ^ m]\n\u22a2 m \u2264 m + (k + 1)\n[PROOFSTEP]\nexact le_self_add\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 I\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := IsPrecomplete.prec' c (aux _ _)\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\na : R\nha : \u2200 (n : \u2115), c n \u2261 a [SMOD I ^ n \u2022 \u22a4]\n\u22a2 \u2203 a, IsRoot f a \u2227 a - a\u2080 \u2208 I\n[PROOFSTEP]\nrefine' \u27e8a, _, _\u27e9\n[GOAL]\ncase intro.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\na : R\nha : \u2200 (n : \u2115), c n \u2261 a [SMOD I ^ n \u2022 \u22a4]\n\u22a2 IsRoot f a\n[PROOFSTEP]\nshow f.IsRoot a\n[GOAL]\ncase intro.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\na : R\nha : \u2200 (n : \u2115), c n \u2261 a [SMOD I ^ n \u2022 \u22a4]\n\u22a2 IsRoot f a\n[PROOFSTEP]\nsuffices \u2200 n, f.eval a \u2261 0 [SMOD (I ^ n \u2022 \u22a4 : Ideal R)] by exact IsHausdorff.haus' _ this\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\na : R\nha : \u2200 (n : \u2115), c n \u2261 a [SMOD I ^ n \u2022 \u22a4]\nthis : \u2200 (n : \u2115), Polynomial.eval a f \u2261 0 [SMOD I ^ n \u2022 \u22a4]\n\u22a2 IsRoot f a\n[PROOFSTEP]\nexact IsHausdorff.haus' _ this\n[GOAL]\ncase intro.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\na : R\nha : \u2200 (n : \u2115), c n \u2261 a [SMOD I ^ n \u2022 \u22a4]\n\u22a2 \u2200 (n : \u2115), Polynomial.eval a f \u2261 0 [SMOD I ^ n \u2022 \u22a4]\n[PROOFSTEP]\nintro n\n[GOAL]\ncase intro.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\na : R\nha : \u2200 (n : \u2115), c n \u2261 a [SMOD I ^ n \u2022 \u22a4]\nn : \u2115\n\u22a2 Polynomial.eval a f \u2261 0 [SMOD I ^ n \u2022 \u22a4]\n[PROOFSTEP]\nspecialize ha n\n[GOAL]\ncase intro.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\na : R\nn : \u2115\nha : c n \u2261 a [SMOD I ^ n \u2022 \u22a4]\n\u22a2 Polynomial.eval a f \u2261 0 [SMOD I ^ n \u2022 \u22a4]\n[PROOFSTEP]\nrw [\u2190 Ideal.one_eq_top, Ideal.smul_eq_mul, mul_one] at ha \u22a2\n[GOAL]\ncase intro.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\na : R\nn : \u2115\nha : c n \u2261 a [SMOD I ^ n]\n\u22a2 Polynomial.eval a f \u2261 0 [SMOD I ^ n]\n[PROOFSTEP]\nrefine' (ha.symm.eval f).trans _\n[GOAL]\ncase intro.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\na : R\nn : \u2115\nha : c n \u2261 a [SMOD I ^ n]\n\u22a2 Polynomial.eval (c n) f \u2261 0 [SMOD I ^ n]\n[PROOFSTEP]\nrw [SModEq.zero]\n[GOAL]\ncase intro.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\na : R\nn : \u2115\nha : c n \u2261 a [SMOD I ^ n]\n\u22a2 Polynomial.eval (c n) f \u2208 I ^ n\n[PROOFSTEP]\nexact Ideal.pow_le_pow le_self_add (hfcI _)\n[GOAL]\ncase intro.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\na : R\nha : \u2200 (n : \u2115), c n \u2261 a [SMOD I ^ n \u2022 \u22a4]\n\u22a2 a - a\u2080 \u2208 I\n[PROOFSTEP]\nshow a - a\u2080 \u2208 I\n[GOAL]\ncase intro.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\na : R\nha : \u2200 (n : \u2115), c n \u2261 a [SMOD I ^ n \u2022 \u22a4]\n\u22a2 a - a\u2080 \u2208 I\n[PROOFSTEP]\nspecialize ha 1\n[GOAL]\ncase intro.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\na : R\nha : c 1 \u2261 a [SMOD I ^ 1 \u2022 \u22a4]\n\u22a2 a - a\u2080 \u2208 I\n[PROOFSTEP]\nrw [hc, pow_one, \u2190 Ideal.one_eq_top, Ideal.smul_eq_mul, mul_one, sub_eq_add_neg] at ha \n[GOAL]\ncase intro.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\na : R\nha : c 0 + -(Polynomial.eval (c 0) f * Ring.inverse (Polynomial.eval (c 0) f')) \u2261 a [SMOD I]\n\u22a2 a - a\u2080 \u2208 I\n[PROOFSTEP]\nrw [\u2190 SModEq.sub_mem, \u2190 add_zero a\u2080]\n[GOAL]\ncase intro.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\na : R\nha : c 0 + -(Polynomial.eval (c 0) f * Ring.inverse (Polynomial.eval (c 0) f')) \u2261 a [SMOD I]\n\u22a2 a \u2261 a\u2080 + 0 [SMOD I]\n[PROOFSTEP]\nrefine' ha.symm.trans (SModEq.rfl.add _)\n[GOAL]\ncase intro.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\na : R\nha : c 0 + -(Polynomial.eval (c 0) f * Ring.inverse (Polynomial.eval (c 0) f')) \u2261 a [SMOD I]\n\u22a2 -(Polynomial.eval (c 0) f * Ring.inverse (Polynomial.eval (c 0) f')) \u2261 0 [SMOD I]\n[PROOFSTEP]\nrw [SModEq.zero, Ideal.neg_mem_iff]\n[GOAL]\ncase intro.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nI : Ideal R\ninst\u271d : IsAdicComplete I R\nf : R[X]\nx\u271d : Monic f\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 I\nh\u2082 : IsUnit (\u2191(Ideal.Quotient.mk I) (Polynomial.eval a\u2080 (\u2191derivative f)))\nf' : (fun x => R[X]) f := \u2191derivative f\nc : \u2115 \u2192 R := fun n => Nat.recOn n a\u2080 fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')\nhc : \u2200 (n : \u2115), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')\nhc_mod : \u2200 (n : \u2115), c n \u2261 a\u2080 [SMOD I]\nhf'c : \u2200 (n : \u2115), IsUnit (Polynomial.eval (c n) f')\nhfcI : \u2200 (n : \u2115), Polynomial.eval (c n) f \u2208 I ^ (n + 1)\naux : \u2200 (m n : \u2115), m \u2264 n \u2192 c m \u2261 c n [SMOD I ^ m \u2022 \u22a4]\na : R\nha : c 0 + -(Polynomial.eval (c 0) f * Ring.inverse (Polynomial.eval (c 0) f')) \u2261 a [SMOD I]\n\u22a2 Polynomial.eval (c 0) f * Ring.inverse (Polynomial.eval (c 0) f') \u2208 I\n[PROOFSTEP]\nexact Ideal.mul_mem_right _ _ h\u2081\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Henselian", "llama_tokens": 48036, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.46879062662624377, "lm_q1q2_score": 0.2453745533667865}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : f.domain = g.domain\nh' : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 g.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\n\u22a2 f = g\n[PROOFSTEP]\nrcases f with \u27e8f_dom, f\u27e9\n[GOAL]\ncase mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : E \u2192\u2097.[R] F\nf_dom : Submodule R E\nf : { x // x \u2208 f_dom } \u2192\u2097[R] F\nh : { domain := f_dom, toFun := f }.domain = g.domain\nh' :\n \u2200 \u2983x : { x // x \u2208 { domain := f_dom, toFun := f }.domain }\u2984 \u2983y : { x // x \u2208 g.domain }\u2984,\n \u2191x = \u2191y \u2192 \u2191{ domain := f_dom, toFun := f } x = \u2191g y\n\u22a2 { domain := f_dom, toFun := f } = g\n[PROOFSTEP]\nrcases g with \u27e8g_dom, g\u27e9\n[GOAL]\ncase mk.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf_dom : Submodule R E\nf : { x // x \u2208 f_dom } \u2192\u2097[R] F\ng_dom : Submodule R E\ng : { x // x \u2208 g_dom } \u2192\u2097[R] F\nh : { domain := f_dom, toFun := f }.domain = { domain := g_dom, toFun := g }.domain\nh' :\n \u2200 \u2983x : { x // x \u2208 { domain := f_dom, toFun := f }.domain }\u2984 \u2983y : { x // x \u2208 { domain := g_dom, toFun := g }.domain }\u2984,\n \u2191x = \u2191y \u2192 \u2191{ domain := f_dom, toFun := f } x = \u2191{ domain := g_dom, toFun := g } y\n\u22a2 { domain := f_dom, toFun := f } = { domain := g_dom, toFun := g }\n[PROOFSTEP]\nobtain rfl : f_dom = g_dom := h\n[GOAL]\ncase mk.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf_dom : Submodule R E\nf g : { x // x \u2208 f_dom } \u2192\u2097[R] F\nh' :\n \u2200 \u2983x : { x // x \u2208 { domain := f_dom, toFun := f }.domain }\u2984 \u2983y : { x // x \u2208 { domain := f_dom, toFun := g }.domain }\u2984,\n \u2191x = \u2191y \u2192 \u2191{ domain := f_dom, toFun := f } x = \u2191{ domain := f_dom, toFun := g } y\n\u22a2 { domain := f_dom, toFun := f } = { domain := f_dom, toFun := g }\n[PROOFSTEP]\nobtain rfl : f = g := LinearMap.ext fun x => h' rfl\n[GOAL]\ncase mk.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf_dom : Submodule R E\nf : { x // x \u2208 f_dom } \u2192\u2097[R] F\nh' :\n \u2200 \u2983x y : { x // x \u2208 { domain := f_dom, toFun := f }.domain }\u2984,\n \u2191x = \u2191y \u2192 \u2191{ domain := f_dom, toFun := f } x = \u2191{ domain := f_dom, toFun := f } y\n\u22a2 { domain := f_dom, toFun := f } = { domain := f_dom, toFun := f }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nEQ : f = g\nx y : { x // x \u2208 f.domain }\nh : \u2191x = \u2191y\n\u22a2 \u2191f x = \u2191f y\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nEQ : f = g\nx y : { x // x \u2208 f.domain }\nh : \u2191x = \u2191y\n\u22a2 x = y\n[PROOFSTEP]\nexact_mod_cast h\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\n\u22a2 \u2200 (c\u2081 c\u2082 : R), c\u2081 \u2022 x = c\u2082 \u2022 x \u2192 c\u2081 \u2022 y = c\u2082 \u2022 y\n[PROOFSTEP]\nintro c\u2081 c\u2082 h\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nc\u2081 c\u2082 : R\nh : c\u2081 \u2022 x = c\u2082 \u2022 x\n\u22a2 c\u2081 \u2022 y = c\u2082 \u2022 y\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 sub_smul] at h \u22a2\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nc\u2081 c\u2082 : R\nh\u271d : c\u2081 \u2022 x = c\u2082 \u2022 x\nh : (c\u2081 - c\u2082) \u2022 x = 0\n\u22a2 (c\u2081 - c\u2082) \u2022 y = 0\n[PROOFSTEP]\nexact H _ h\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny\u271d : F\nH\u271d : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y\u271d = 0\nH : \u2200 (c\u2081 c\u2082 : R), c\u2081 \u2022 x = c\u2082 \u2022 x \u2192 c\u2081 \u2022 y\u271d = c\u2082 \u2022 y\u271d\ny z : { x_1 // x_1 \u2208 span R {x} }\n\u22a2 (fun z => Classical.choose (_ : \u2203 a, a \u2022 x = \u2191z) \u2022 y\u271d) (y + z) =\n (fun z => Classical.choose (_ : \u2203 a, a \u2022 x = \u2191z) \u2022 y\u271d) y + (fun z => Classical.choose (_ : \u2203 a, a \u2022 x = \u2191z) \u2022 y\u271d) z\n[PROOFSTEP]\ndsimp only []\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny\u271d : F\nH\u271d : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y\u271d = 0\nH : \u2200 (c\u2081 c\u2082 : R), c\u2081 \u2022 x = c\u2082 \u2022 x \u2192 c\u2081 \u2022 y\u271d = c\u2082 \u2022 y\u271d\ny z : { x_1 // x_1 \u2208 span R {x} }\n\u22a2 Classical.choose (_ : \u2203 a, a \u2022 x = \u2191(y + z)) \u2022 y\u271d =\n Classical.choose (_ : \u2203 a, a \u2022 x = \u2191y) \u2022 y\u271d + Classical.choose (_ : \u2203 a, a \u2022 x = \u2191z) \u2022 y\u271d\n[PROOFSTEP]\nrw [\u2190 add_smul]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny\u271d : F\nH\u271d : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y\u271d = 0\nH : \u2200 (c\u2081 c\u2082 : R), c\u2081 \u2022 x = c\u2082 \u2022 x \u2192 c\u2081 \u2022 y\u271d = c\u2082 \u2022 y\u271d\ny z : { x_1 // x_1 \u2208 span R {x} }\n\u22a2 Classical.choose (_ : \u2203 a, a \u2022 x = \u2191(y + z)) \u2022 y\u271d =\n (Classical.choose (_ : \u2203 a, a \u2022 x = \u2191y) + Classical.choose (_ : \u2203 a, a \u2022 x = \u2191z)) \u2022 y\u271d\n[PROOFSTEP]\napply H\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny\u271d : F\nH\u271d : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y\u271d = 0\nH : \u2200 (c\u2081 c\u2082 : R), c\u2081 \u2022 x = c\u2082 \u2022 x \u2192 c\u2081 \u2022 y\u271d = c\u2082 \u2022 y\u271d\ny z : { x_1 // x_1 \u2208 span R {x} }\n\u22a2 Classical.choose (_ : \u2203 a, a \u2022 x = \u2191(y + z)) \u2022 x =\n (Classical.choose (_ : \u2203 a, a \u2022 x = \u2191y) + Classical.choose (_ : \u2203 a, a \u2022 x = \u2191z)) \u2022 x\n[PROOFSTEP]\nsimp only [add_smul, sub_smul, fun w : R \u2219 x => Classical.choose_spec (mem_span_singleton.1 w.prop)]\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny\u271d : F\nH\u271d : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y\u271d = 0\nH : \u2200 (c\u2081 c\u2082 : R), c\u2081 \u2022 x = c\u2082 \u2022 x \u2192 c\u2081 \u2022 y\u271d = c\u2082 \u2022 y\u271d\ny z : { x_1 // x_1 \u2208 span R {x} }\n\u22a2 \u2191(y + z) = \u2191y + \u2191z\n[PROOFSTEP]\napply coe_add\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH\u271d : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nH : \u2200 (c\u2081 c\u2082 : R), c\u2081 \u2022 x = c\u2082 \u2022 x \u2192 c\u2081 \u2022 y = c\u2082 \u2022 y\nc : R\nz : { x_1 // x_1 \u2208 span R {x} }\n\u22a2 AddHom.toFun\n { toFun := fun z => Classical.choose (_ : \u2203 a, a \u2022 x = \u2191z) \u2022 y,\n map_add' :=\n (_ :\n \u2200 (y_1 z : { x_1 // x_1 \u2208 span R {x} }),\n (fun z => Classical.choose (_ : \u2203 a, a \u2022 x = \u2191z) \u2022 y) (y_1 + z) =\n (fun z => Classical.choose (_ : \u2203 a, a \u2022 x = \u2191z) \u2022 y) y_1 +\n (fun z => Classical.choose (_ : \u2203 a, a \u2022 x = \u2191z) \u2022 y) z) }\n (c \u2022 z) =\n \u2191(RingHom.id R) c \u2022\n AddHom.toFun\n { toFun := fun z => Classical.choose (_ : \u2203 a, a \u2022 x = \u2191z) \u2022 y,\n map_add' :=\n (_ :\n \u2200 (y_1 z : { x_1 // x_1 \u2208 span R {x} }),\n (fun z => Classical.choose (_ : \u2203 a, a \u2022 x = \u2191z) \u2022 y) (y_1 + z) =\n (fun z => Classical.choose (_ : \u2203 a, a \u2022 x = \u2191z) \u2022 y) y_1 +\n (fun z => Classical.choose (_ : \u2203 a, a \u2022 x = \u2191z) \u2022 y) z) }\n z\n[PROOFSTEP]\ndsimp only []\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH\u271d : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nH : \u2200 (c\u2081 c\u2082 : R), c\u2081 \u2022 x = c\u2082 \u2022 x \u2192 c\u2081 \u2022 y = c\u2082 \u2022 y\nc : R\nz : { x_1 // x_1 \u2208 span R {x} }\n\u22a2 Classical.choose (_ : \u2203 a, a \u2022 x = \u2191(c \u2022 z)) \u2022 y = \u2191(RingHom.id R) c \u2022 Classical.choose (_ : \u2203 a, a \u2022 x = \u2191z) \u2022 y\n[PROOFSTEP]\nrw [smul_smul]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH\u271d : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nH : \u2200 (c\u2081 c\u2082 : R), c\u2081 \u2022 x = c\u2082 \u2022 x \u2192 c\u2081 \u2022 y = c\u2082 \u2022 y\nc : R\nz : { x_1 // x_1 \u2208 span R {x} }\n\u22a2 Classical.choose (_ : \u2203 a, a \u2022 x = \u2191(c \u2022 z)) \u2022 y = (\u2191(RingHom.id R) c * Classical.choose (_ : \u2203 a, a \u2022 x = \u2191z)) \u2022 y\n[PROOFSTEP]\napply H\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH\u271d : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nH : \u2200 (c\u2081 c\u2082 : R), c\u2081 \u2022 x = c\u2082 \u2022 x \u2192 c\u2081 \u2022 y = c\u2082 \u2022 y\nc : R\nz : { x_1 // x_1 \u2208 span R {x} }\n\u22a2 Classical.choose (_ : \u2203 a, a \u2022 x = \u2191(c \u2022 z)) \u2022 x = (\u2191(RingHom.id R) c * Classical.choose (_ : \u2203 a, a \u2022 x = \u2191z)) \u2022 x\n[PROOFSTEP]\nsimp only [mul_smul, fun w : R \u2219 x => Classical.choose_spec (mem_span_singleton.1 w.prop)]\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH\u271d : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nH : \u2200 (c\u2081 c\u2082 : R), c\u2081 \u2022 x = c\u2082 \u2022 x \u2192 c\u2081 \u2022 y = c\u2082 \u2022 y\nc : R\nz : { x_1 // x_1 \u2208 span R {x} }\n\u22a2 \u2191(c \u2022 z) = \u2191(RingHom.id R) c \u2022 \u2191z\n[PROOFSTEP]\napply coe_smul\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nc : R\nh : c \u2022 x \u2208 (mkSpanSingleton' x y H).domain\n\u22a2 \u2191(mkSpanSingleton' x y H) { val := c \u2022 x, property := h } = c \u2022 y\n[PROOFSTEP]\ndsimp [mkSpanSingleton']\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nc : R\nh : c \u2022 x \u2208 (mkSpanSingleton' x y H).domain\n\u22a2 Classical.choose (_ : \u2203 a, a \u2022 x = c \u2022 x) \u2022 y = c \u2022 y\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 sub_smul]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nc : R\nh : c \u2022 x \u2208 (mkSpanSingleton' x y H).domain\n\u22a2 (Classical.choose (_ : \u2203 a, a \u2022 x = c \u2022 x) - c) \u2022 y = 0\n[PROOFSTEP]\napply H\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nc : R\nh : c \u2022 x \u2208 (mkSpanSingleton' x y H).domain\n\u22a2 (Classical.choose (_ : \u2203 a, a \u2022 x = c \u2022 x) - c) \u2022 x = 0\n[PROOFSTEP]\nsimp only [sub_smul, one_smul, sub_eq_zero]\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nc : R\nh : c \u2022 x \u2208 (mkSpanSingleton' x y H).domain\n\u22a2 Classical.choose (_ : \u2203 a, a \u2022 x = c \u2022 x) \u2022 x = c \u2022 x\n[PROOFSTEP]\napply Classical.choose_spec (mem_span_singleton.1 h)\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nh : x \u2208 (mkSpanSingleton' x y H).domain\n\u22a2 \u2191(mkSpanSingleton' x y H) { val := x, property := h } = y\n[PROOFSTEP]\nhave := by refine mkSpanSingleton'_apply x y H 1 ?_; rwa [one_smul]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nh : x \u2208 (mkSpanSingleton' x y H).domain\n\u22a2 ?m.88594\n[PROOFSTEP]\nrefine mkSpanSingleton'_apply x y H 1 ?_\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nh : x \u2208 (mkSpanSingleton' x y H).domain\n\u22a2 1 \u2022 x \u2208 (mkSpanSingleton' x y H).domain\n[PROOFSTEP]\nrwa [one_smul]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nh : x \u2208 (mkSpanSingleton' x y H).domain\nthis : \u2191(mkSpanSingleton' x y H) { val := 1 \u2022 x, property := (_ : 1 \u2022 x \u2208 (mkSpanSingleton' x y H).domain) } = 1 \u2022 y\n\u22a2 \u2191(mkSpanSingleton' x y H) { val := x, property := h } = y\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_2.h.e'_10.h.e'_3\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nh : x \u2208 (mkSpanSingleton' x y H).domain\nthis : \u2191(mkSpanSingleton' x y H) { val := 1 \u2022 x, property := (_ : 1 \u2022 x \u2208 (mkSpanSingleton' x y H).domain) } = 1 \u2022 y\n\u22a2 x = 1 \u2022 x\n[PROOFSTEP]\nrw [one_smul]\n[GOAL]\ncase h.e'_3\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nx : E\ny : F\nH : \u2200 (c : R), c \u2022 x = 0 \u2192 c \u2022 y = 0\nh : x \u2208 (mkSpanSingleton' x y H).domain\nthis : \u2191(mkSpanSingleton' x y H) { val := 1 \u2022 x, property := (_ : 1 \u2022 x \u2208 (mkSpanSingleton' x y H).domain) } = 1 \u2022 y\n\u22a2 y = 1 \u2022 y\n[PROOFSTEP]\nrw [one_smul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : Ring R\nE\u271d : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup E\u271d\ninst\u271d\u2079 : Module R E\u271d\nF\u271d : Type u_3\ninst\u271d\u2078 : AddCommGroup F\u271d\ninst\u271d\u2077 : Module R F\u271d\nG : Type u_4\ninst\u271d\u2076 : AddCommGroup G\ninst\u271d\u2075 : Module R G\nK : Type u_5\nE : Type u_6\nF : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module K E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module K F\nx : E\ny : F\nhx : x \u2260 0\nc : K\nhc\u271d : c \u2022 x = 0\nhc : c = 0\n\u22a2 c \u2022 y = 0\n[PROOFSTEP]\nrw [hc, zero_smul]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nx y : E\nx\u271d\u00b9 : x \u2208 {x | \u2203 hf hg, \u2191f { val := x, property := hf } = \u2191g { val := x, property := hg }}\nx\u271d : y \u2208 {x | \u2203 hf hg, \u2191f { val := x, property := hf } = \u2191g { val := x, property := hg }}\nhfx : x \u2208 f.domain\nhgx : x \u2208 g.domain\nhx : \u2191f { val := x, property := hfx } = \u2191g { val := x, property := hgx }\nhfy : y \u2208 f.domain\nhgy : y \u2208 g.domain\nhy : \u2191f { val := y, property := hfy } = \u2191g { val := y, property := hgy }\n\u22a2 \u2191f { val := x + y, property := (_ : x + y \u2208 f.domain) } = \u2191g { val := x + y, property := (_ : x + y \u2208 g.domain) }\n[PROOFSTEP]\nerw [f.map_add \u27e8x, hfx\u27e9 \u27e8y, hfy\u27e9, g.map_add \u27e8x, hgx\u27e9 \u27e8y, hgy\u27e9, hx, hy]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\n\u22a2 \u2200 (c : R) {x : E},\n x \u2208\n {\n toAddSubsemigroup :=\n { carrier := {x | \u2203 hf hg, \u2191f { val := x, property := hf } = \u2191g { val := x, property := hg }},\n add_mem' :=\n (_ :\n \u2200 {x y : E},\n x \u2208 {x | \u2203 hf hg, \u2191f { val := x, property := hf } = \u2191g { val := x, property := hg }} \u2192\n y \u2208 {x | \u2203 hf hg, \u2191f { val := x, property := hf } = \u2191g { val := x, property := hg }} \u2192\n x + y \u2208 {x | \u2203 hf hg, \u2191f { val := x, property := hf } = \u2191g { val := x, property := hg }}) },\n zero_mem' :=\n (_ :\n \u2203 hf hg,\n \u2191f { val := 0, property := hf } = \u2191g { val := 0, property := hg }) }.toAddSubsemigroup.carrier \u2192\n c \u2022 x \u2208\n {\n toAddSubsemigroup :=\n { carrier := {x | \u2203 hf hg, \u2191f { val := x, property := hf } = \u2191g { val := x, property := hg }},\n add_mem' :=\n (_ :\n \u2200 {x y : E},\n x \u2208 {x | \u2203 hf hg, \u2191f { val := x, property := hf } = \u2191g { val := x, property := hg }} \u2192\n y \u2208 {x | \u2203 hf hg, \u2191f { val := x, property := hf } = \u2191g { val := x, property := hg }} \u2192\n x + y \u2208 {x | \u2203 hf hg, \u2191f { val := x, property := hf } = \u2191g { val := x, property := hg }}) },\n zero_mem' :=\n (_ :\n \u2203 hf hg,\n \u2191f { val := 0, property := hf } = \u2191g { val := 0, property := hg }) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrintro c x \u27e8hfx, hgx, hx\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nc : R\nx : E\nhfx : x \u2208 f.domain\nhgx : x \u2208 g.domain\nhx : \u2191f { val := x, property := hfx } = \u2191g { val := x, property := hgx }\n\u22a2 c \u2022 x \u2208\n {\n toAddSubsemigroup :=\n { carrier := {x | \u2203 hf hg, \u2191f { val := x, property := hf } = \u2191g { val := x, property := hg }},\n add_mem' :=\n (_ :\n \u2200 {x y : E},\n x \u2208 {x | \u2203 hf hg, \u2191f { val := x, property := hf } = \u2191g { val := x, property := hg }} \u2192\n y \u2208 {x | \u2203 hf hg, \u2191f { val := x, property := hf } = \u2191g { val := x, property := hg }} \u2192\n x + y \u2208 {x | \u2203 hf hg, \u2191f { val := x, property := hf } = \u2191g { val := x, property := hg }}) },\n zero_mem' :=\n (_ : \u2203 hf hg, \u2191f { val := 0, property := hf } = \u2191g { val := 0, property := hg }) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nexact \u27e8smul_mem _ c hfx, smul_mem _ c hgx, by erw [f.map_smul c \u27e8x, hfx\u27e9, g.map_smul c \u27e8x, hgx\u27e9, hx]\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nc : R\nx : E\nhfx : x \u2208 f.domain\nhgx : x \u2208 g.domain\nhx : \u2191f { val := x, property := hfx } = \u2191g { val := x, property := hgx }\n\u22a2 \u2191f { val := c \u2022 x, property := (_ : c \u2022 x \u2208 f.domain) } = \u2191g { val := c \u2022 x, property := (_ : c \u2022 x \u2208 g.domain) }\n[PROOFSTEP]\nerw [f.map_smul c \u27e8x, hfx\u27e9, g.map_smul c \u27e8x, hgx\u27e9, hx]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\n\u22a2 \u2200 (a b c : E \u2192\u2097.[R] F), a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\n[PROOFSTEP]\nrintro f g h \u27e8fg_le, fg_eq\u27e9 \u27e8fh_le, fh_eq\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g h : E \u2192\u2097.[R] F\nfg_le : f.domain \u2264 g.domain\nfg_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 g.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nfh_le : f.domain \u2264 h.domain\nfh_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 h.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191h y\n\u22a2 f \u2264 g \u2293 h\n[PROOFSTEP]\nexact\n \u27e8fun x hx =>\n \u27e8fg_le hx, fh_le hx, by\n -- Porting note: `[exact \u27e8x, hx\u27e9, rfl, rfl]` \u2192 `[skip, exact \u27e8x, hx\u27e9, skip] <;> rfl`refine'\n (fg_eq _).symm.trans (fh_eq _) <;>\n [skip; exact \u27e8x, hx\u27e9; skip] <;>\n rfl\u27e9,\n fun x \u27e8y, yg, hy\u27e9 h => by\n apply fg_eq\n exact h\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g h : E \u2192\u2097.[R] F\nfg_le : f.domain \u2264 g.domain\nfg_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 g.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nfh_le : f.domain \u2264 h.domain\nfh_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 h.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191h y\nx : E\nhx : x \u2208 f.domain\n\u22a2 \u2191g { val := x, property := (_ : x \u2208 g.domain) } = \u2191h { val := x, property := (_ : x \u2208 h.domain) }\n[PROOFSTEP]\nrefine' (fg_eq _).symm.trans (fh_eq _) <;> [skip; exact \u27e8x, hx\u27e9; skip]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g h : E \u2192\u2097.[R] F\nfg_le : f.domain \u2264 g.domain\nfg_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 g.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nfh_le : f.domain \u2264 h.domain\nfh_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 h.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191h y\nx : E\nhx : x \u2208 f.domain\n\u22a2 \u2191g { val := x, property := (_ : x \u2208 g.domain) } = \u2191h { val := x, property := (_ : x \u2208 h.domain) }\n[PROOFSTEP]\nrefine' (fg_eq _).symm.trans (fh_eq _)\n[GOAL]\ncase refine'_1\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g h : E \u2192\u2097.[R] F\nfg_le : f.domain \u2264 g.domain\nfg_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 g.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nfh_le : f.domain \u2264 h.domain\nfh_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 h.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191h y\nx : E\nhx : x \u2208 f.domain\n\u22a2 \u2191?refine'_2 = \u2191{ val := x, property := (_ : x \u2208 g.domain) }\n[PROOFSTEP]\nskip\n[GOAL]\ncase refine'_2\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g h : E \u2192\u2097.[R] F\nfg_le : f.domain \u2264 g.domain\nfg_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 g.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nfh_le : f.domain \u2264 h.domain\nfh_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 h.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191h y\nx : E\nhx : x \u2208 f.domain\n\u22a2 { x // x \u2208 f.domain }\n[PROOFSTEP]\nexact \u27e8x, hx\u27e9\n[GOAL]\ncase refine'_3\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g h : E \u2192\u2097.[R] F\nfg_le : f.domain \u2264 g.domain\nfg_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 g.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nfh_le : f.domain \u2264 h.domain\nfh_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 h.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191h y\nx : E\nhx : x \u2208 f.domain\n\u22a2 \u2191{ val := x, property := hx } = \u2191{ val := x, property := (_ : x \u2208 h.domain) }\n[PROOFSTEP]\nskip\n[GOAL]\ncase refine'_1\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g h : E \u2192\u2097.[R] F\nfg_le : f.domain \u2264 g.domain\nfg_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 g.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nfh_le : f.domain \u2264 h.domain\nfh_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 h.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191h y\nx : E\nhx : x \u2208 f.domain\n\u22a2 \u2191{ val := x, property := hx } = \u2191{ val := x, property := (_ : x \u2208 g.domain) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_3\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g h : E \u2192\u2097.[R] F\nfg_le : f.domain \u2264 g.domain\nfg_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 g.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nfh_le : f.domain \u2264 h.domain\nfh_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 h.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191h y\nx : E\nhx : x \u2208 f.domain\n\u22a2 \u2191{ val := x, property := hx } = \u2191{ val := x, property := (_ : x \u2208 h.domain) }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g h\u271d : E \u2192\u2097.[R] F\nfg_le : f.domain \u2264 g.domain\nfg_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 g.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nfh_le : f.domain \u2264 h\u271d.domain\nfh_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 h\u271d.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191h\u271d y\nx : { x // x \u2208 f.domain }\nx\u271d : { x // x \u2208 (g \u2293 h\u271d).domain }\ny : E\nyg : y \u2208 g.domain\nhy : \u2203 hg, \u2191g { val := y, property := yg } = \u2191h\u271d { val := y, property := hg }\nh : \u2191x = \u2191{ val := y, property := (_ : \u2203 hf hg, \u2191g { val := y, property := hf } = \u2191h\u271d { val := y, property := hg }) }\n\u22a2 \u2191f x =\n \u2191(g \u2293 h\u271d)\n { val := y, property := (_ : \u2203 hf hg, \u2191g { val := y, property := hf } = \u2191h\u271d { val := y, property := hg }) }\n[PROOFSTEP]\napply fg_eq\n[GOAL]\ncase _h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g h\u271d : E \u2192\u2097.[R] F\nfg_le : f.domain \u2264 g.domain\nfg_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 g.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nfh_le : f.domain \u2264 h\u271d.domain\nfh_eq : \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 h\u271d.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191h\u271d y\nx : { x // x \u2208 f.domain }\nx\u271d : { x // x \u2208 (g \u2293 h\u271d).domain }\ny : E\nyg : y \u2208 g.domain\nhy : \u2203 hg, \u2191g { val := y, property := yg } = \u2191h\u271d { val := y, property := hg }\nh : \u2191x = \u2191{ val := y, property := (_ : \u2203 hf hg, \u2191g { val := y, property := hf } = \u2191h\u271d { val := y, property := hg }) }\n\u22a2 \u2191x =\n \u2191(\u2191(ofLe (_ : \u2200 (_x : E), _x \u2208 eqLocus g h\u271d \u2192 _x \u2208 g.domain))\n { val := y, property := (_ : \u2203 hf hg, \u2191g { val := y, property := hf } = \u2191h\u271d { val := y, property := hg }) })\n[PROOFSTEP]\nexact h\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 \u22a5.domain }\ny : { x // x \u2208 f.domain }\nh : \u2191x = \u2191y\n\u22a2 \u2191\u22a5 x = \u2191f y\n[PROOFSTEP]\nhave hx : x = 0 := Subtype.eq ((mem_bot R).1 x.2)\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 \u22a5.domain }\ny : { x // x \u2208 f.domain }\nh : \u2191x = \u2191y\nhx : x = 0\n\u22a2 \u2191\u22a5 x = \u2191f y\n[PROOFSTEP]\nhave hy : y = 0 := Subtype.eq (h.symm.trans (congr_arg _ hx))\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 \u22a5.domain }\ny : { x // x \u2208 f.domain }\nh : \u2191x = \u2191y\nhx : x = 0\nhy : y = 0\n\u22a2 \u2191\u22a5 x = \u2191f y\n[PROOFSTEP]\nrw [hx, hy, map_zero, map_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\n\u22a2 \u2203 fg,\n \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }) (z : { x // x \u2208 f.domain \u2294 g.domain }),\n \u2191x + \u2191y = \u2191z \u2192 \u2191fg z = \u2191f x + \u2191g y\n[PROOFSTEP]\nchoose x hx y hy hxy using fun z : \u21a5(f.domain \u2294 g.domain) => mem_sup.1 z.prop\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z + y z = \u2191z\n\u22a2 \u2203 fg,\n \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }) (z : { x // x \u2208 f.domain \u2294 g.domain }),\n \u2191x + \u2191y = \u2191z \u2192 \u2191fg z = \u2191f x + \u2191g y\n[PROOFSTEP]\nset fg := fun z => f \u27e8x z, hx z\u27e9 + g \u27e8y z, hy z\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z + y z = \u2191z\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\n\u22a2 \u2203 fg,\n \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }) (z : { x // x \u2208 f.domain \u2294 g.domain }),\n \u2191x + \u2191y = \u2191z \u2192 \u2191fg z = \u2191f x + \u2191g y\n[PROOFSTEP]\nhave fg_eq :\n \u2200 (x' : f.domain) (y' : g.domain) (z' : \u21a5(f.domain \u2294 g.domain)) (_H : (x' : E) + y' = z'), fg z' = f x' + g y' :=\n by\n intro x' y' z' H\n dsimp\n rw [add_comm, \u2190 sub_eq_sub_iff_add_eq_add, eq_comm, \u2190 map_sub, \u2190 map_sub]\n apply h\n simp only [\u2190 eq_sub_iff_add_eq] at hxy \n simp only [AddSubgroupClass.coe_sub, coe_mk, coe_mk, hxy, \u2190 sub_add, \u2190 sub_sub, sub_self, zero_sub, \u2190 H]\n apply neg_add_eq_sub\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z + y z = \u2191z\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\n\u22a2 \u2200 (x' : { x // x \u2208 f.domain }) (y' : { x // x \u2208 g.domain }) (z' : { x // x \u2208 f.domain \u2294 g.domain }),\n \u2191x' + \u2191y' = \u2191z' \u2192 fg z' = \u2191f x' + \u2191g y'\n[PROOFSTEP]\nintro x' y' z' H\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z + y z = \u2191z\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\nx' : { x // x \u2208 f.domain }\ny' : { x // x \u2208 g.domain }\nz' : { x // x \u2208 f.domain \u2294 g.domain }\nH : \u2191x' + \u2191y' = \u2191z'\n\u22a2 fg z' = \u2191f x' + \u2191g y'\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z + y z = \u2191z\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\nx' : { x // x \u2208 f.domain }\ny' : { x // x \u2208 g.domain }\nz' : { x // x \u2208 f.domain \u2294 g.domain }\nH : \u2191x' + \u2191y' = \u2191z'\n\u22a2 \u2191f { val := x z', property := (_ : x z' \u2208 f.domain) } + \u2191g { val := y z', property := (_ : y z' \u2208 g.domain) } =\n \u2191f x' + \u2191g y'\n[PROOFSTEP]\nrw [add_comm, \u2190 sub_eq_sub_iff_add_eq_add, eq_comm, \u2190 map_sub, \u2190 map_sub]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z + y z = \u2191z\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\nx' : { x // x \u2208 f.domain }\ny' : { x // x \u2208 g.domain }\nz' : { x // x \u2208 f.domain \u2294 g.domain }\nH : \u2191x' + \u2191y' = \u2191z'\n\u22a2 \u2191f (x' - { val := x z', property := (_ : x z' \u2208 f.domain) }) =\n \u2191g ({ val := y z', property := (_ : y z' \u2208 g.domain) } - y')\n[PROOFSTEP]\napply h\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z + y z = \u2191z\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\nx' : { x // x \u2208 f.domain }\ny' : { x // x \u2208 g.domain }\nz' : { x // x \u2208 f.domain \u2294 g.domain }\nH : \u2191x' + \u2191y' = \u2191z'\n\u22a2 \u2191(x' - { val := x z', property := (_ : x z' \u2208 f.domain) }) =\n \u2191({ val := y z', property := (_ : y z' \u2208 g.domain) } - y')\n[PROOFSTEP]\nsimp only [\u2190 eq_sub_iff_add_eq] at hxy \n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\nx' : { x // x \u2208 f.domain }\ny' : { x // x \u2208 g.domain }\nz' : { x // x \u2208 f.domain \u2294 g.domain }\nH : \u2191x' + \u2191y' = \u2191z'\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z = \u2191z - y z\n\u22a2 \u2191(x' - { val := x z', property := (_ : x z' \u2208 f.domain) }) =\n \u2191({ val := y z', property := (_ : y z' \u2208 g.domain) } - y')\n[PROOFSTEP]\nsimp only [AddSubgroupClass.coe_sub, coe_mk, coe_mk, hxy, \u2190 sub_add, \u2190 sub_sub, sub_self, zero_sub, \u2190 H]\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\nx' : { x // x \u2208 f.domain }\ny' : { x // x \u2208 g.domain }\nz' : { x // x \u2208 f.domain \u2294 g.domain }\nH : \u2191x' + \u2191y' = \u2191z'\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z = \u2191z - y z\n\u22a2 -\u2191y' + y z' = y z' - \u2191y'\n[PROOFSTEP]\napply neg_add_eq_sub\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z + y z = \u2191z\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\nfg_eq :\n \u2200 (x' : { x // x \u2208 f.domain }) (y' : { x // x \u2208 g.domain }) (z' : { x // x \u2208 f.domain \u2294 g.domain }),\n \u2191x' + \u2191y' = \u2191z' \u2192 fg z' = \u2191f x' + \u2191g y'\n\u22a2 \u2203 fg,\n \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }) (z : { x // x \u2208 f.domain \u2294 g.domain }),\n \u2191x + \u2191y = \u2191z \u2192 \u2191fg z = \u2191f x + \u2191g y\n[PROOFSTEP]\nrefine' \u27e8{ toFun := fg .. }, fg_eq\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z + y z = \u2191z\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\nfg_eq :\n \u2200 (x' : { x // x \u2208 f.domain }) (y' : { x // x \u2208 g.domain }) (z' : { x // x \u2208 f.domain \u2294 g.domain }),\n \u2191x' + \u2191y' = \u2191z' \u2192 fg z' = \u2191f x' + \u2191g y'\n\u22a2 \u2200 (x y : { x // x \u2208 f.domain \u2294 g.domain }), fg (x + y) = fg x + fg y\n[PROOFSTEP]\nrintro \u27e8z\u2081, hz\u2081\u27e9 \u27e8z\u2082, hz\u2082\u27e9\n[GOAL]\ncase refine'_1.mk.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z + y z = \u2191z\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\nfg_eq :\n \u2200 (x' : { x // x \u2208 f.domain }) (y' : { x // x \u2208 g.domain }) (z' : { x // x \u2208 f.domain \u2294 g.domain }),\n \u2191x' + \u2191y' = \u2191z' \u2192 fg z' = \u2191f x' + \u2191g y'\nz\u2081 : E\nhz\u2081 : z\u2081 \u2208 f.domain \u2294 g.domain\nz\u2082 : E\nhz\u2082 : z\u2082 \u2208 f.domain \u2294 g.domain\n\u22a2 fg ({ val := z\u2081, property := hz\u2081 } + { val := z\u2082, property := hz\u2082 }) =\n fg { val := z\u2081, property := hz\u2081 } + fg { val := z\u2082, property := hz\u2082 }\n[PROOFSTEP]\nrw [\u2190 add_assoc, add_right_comm (f _), \u2190 map_add, add_assoc, \u2190 map_add]\n[GOAL]\ncase refine'_1.mk.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z + y z = \u2191z\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\nfg_eq :\n \u2200 (x' : { x // x \u2208 f.domain }) (y' : { x // x \u2208 g.domain }) (z' : { x // x \u2208 f.domain \u2294 g.domain }),\n \u2191x' + \u2191y' = \u2191z' \u2192 fg z' = \u2191f x' + \u2191g y'\nz\u2081 : E\nhz\u2081 : z\u2081 \u2208 f.domain \u2294 g.domain\nz\u2082 : E\nhz\u2082 : z\u2082 \u2208 f.domain \u2294 g.domain\n\u22a2 fg ({ val := z\u2081, property := hz\u2081 } + { val := z\u2082, property := hz\u2082 }) =\n \u2191f\n ({ val := x { val := z\u2081, property := hz\u2081 }, property := (_ : x { val := z\u2081, property := hz\u2081 } \u2208 f.domain) } +\n { val := x { val := z\u2082, property := hz\u2082 }, property := (_ : x { val := z\u2082, property := hz\u2082 } \u2208 f.domain) }) +\n \u2191g\n ({ val := y { val := z\u2081, property := hz\u2081 }, property := (_ : y { val := z\u2081, property := hz\u2081 } \u2208 g.domain) } +\n { val := y { val := z\u2082, property := hz\u2082 }, property := (_ : y { val := z\u2082, property := hz\u2082 } \u2208 g.domain) })\n[PROOFSTEP]\napply fg_eq\n[GOAL]\ncase refine'_1.mk.mk._H\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z + y z = \u2191z\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\nfg_eq :\n \u2200 (x' : { x // x \u2208 f.domain }) (y' : { x // x \u2208 g.domain }) (z' : { x // x \u2208 f.domain \u2294 g.domain }),\n \u2191x' + \u2191y' = \u2191z' \u2192 fg z' = \u2191f x' + \u2191g y'\nz\u2081 : E\nhz\u2081 : z\u2081 \u2208 f.domain \u2294 g.domain\nz\u2082 : E\nhz\u2082 : z\u2082 \u2208 f.domain \u2294 g.domain\n\u22a2 \u2191({ val := x { val := z\u2081, property := hz\u2081 }, property := (_ : x { val := z\u2081, property := hz\u2081 } \u2208 f.domain) } +\n { val := x { val := z\u2082, property := hz\u2082 }, property := (_ : x { val := z\u2082, property := hz\u2082 } \u2208 f.domain) }) +\n \u2191({ val := y { val := z\u2081, property := hz\u2081 }, property := (_ : y { val := z\u2081, property := hz\u2081 } \u2208 g.domain) } +\n { val := y { val := z\u2082, property := hz\u2082 }, property := (_ : y { val := z\u2082, property := hz\u2082 } \u2208 g.domain) }) =\n \u2191({ val := z\u2081, property := hz\u2081 } + { val := z\u2082, property := hz\u2082 })\n[PROOFSTEP]\nsimp only [coe_add, coe_mk, \u2190 add_assoc]\n[GOAL]\ncase refine'_1.mk.mk._H\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z + y z = \u2191z\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\nfg_eq :\n \u2200 (x' : { x // x \u2208 f.domain }) (y' : { x // x \u2208 g.domain }) (z' : { x // x \u2208 f.domain \u2294 g.domain }),\n \u2191x' + \u2191y' = \u2191z' \u2192 fg z' = \u2191f x' + \u2191g y'\nz\u2081 : E\nhz\u2081 : z\u2081 \u2208 f.domain \u2294 g.domain\nz\u2082 : E\nhz\u2082 : z\u2082 \u2208 f.domain \u2294 g.domain\n\u22a2 x { val := z\u2081, property := hz\u2081 } + x { val := z\u2082, property := hz\u2082 } + y { val := z\u2081, property := hz\u2081 } +\n y { val := z\u2082, property := hz\u2082 } =\n z\u2081 + z\u2082\n[PROOFSTEP]\nrw [add_right_comm (x _), hxy, add_assoc, hxy, coe_mk, coe_mk]\n[GOAL]\ncase refine'_2\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z + y z = \u2191z\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\nfg_eq :\n \u2200 (x' : { x // x \u2208 f.domain }) (y' : { x // x \u2208 g.domain }) (z' : { x // x \u2208 f.domain \u2294 g.domain }),\n \u2191x' + \u2191y' = \u2191z' \u2192 fg z' = \u2191f x' + \u2191g y'\n\u22a2 \u2200 (r : R) (x_1 : { x // x \u2208 f.domain \u2294 g.domain }),\n AddHom.toFun { toFun := fg, map_add' := (_ : \u2200 (x y : { x // x \u2208 f.domain \u2294 g.domain }), fg (x + y) = fg x + fg y) }\n (r \u2022 x_1) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fg, map_add' := (_ : \u2200 (x y : { x // x \u2208 f.domain \u2294 g.domain }), fg (x + y) = fg x + fg y) } x_1\n[PROOFSTEP]\nintro c z\n[GOAL]\ncase refine'_2\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z + y z = \u2191z\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\nfg_eq :\n \u2200 (x' : { x // x \u2208 f.domain }) (y' : { x // x \u2208 g.domain }) (z' : { x // x \u2208 f.domain \u2294 g.domain }),\n \u2191x' + \u2191y' = \u2191z' \u2192 fg z' = \u2191f x' + \u2191g y'\nc : R\nz : { x // x \u2208 f.domain \u2294 g.domain }\n\u22a2 AddHom.toFun { toFun := fg, map_add' := (_ : \u2200 (x y : { x // x \u2208 f.domain \u2294 g.domain }), fg (x + y) = fg x + fg y) }\n (c \u2022 z) =\n \u2191(RingHom.id R) c \u2022\n AddHom.toFun\n { toFun := fg, map_add' := (_ : \u2200 (x y : { x // x \u2208 f.domain \u2294 g.domain }), fg (x + y) = fg x + fg y) } z\n[PROOFSTEP]\nrw [smul_add, \u2190 map_smul, \u2190 map_smul]\n[GOAL]\ncase refine'_2\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z + y z = \u2191z\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\nfg_eq :\n \u2200 (x' : { x // x \u2208 f.domain }) (y' : { x // x \u2208 g.domain }) (z' : { x // x \u2208 f.domain \u2294 g.domain }),\n \u2191x' + \u2191y' = \u2191z' \u2192 fg z' = \u2191f x' + \u2191g y'\nc : R\nz : { x // x \u2208 f.domain \u2294 g.domain }\n\u22a2 AddHom.toFun { toFun := fg, map_add' := (_ : \u2200 (x y : { x // x \u2208 f.domain \u2294 g.domain }), fg (x + y) = fg x + fg y) }\n (c \u2022 z) =\n \u2191f (\u2191(RingHom.id R) c \u2022 { val := x z, property := (_ : x z \u2208 f.domain) }) +\n \u2191g (\u2191(RingHom.id R) c \u2022 { val := y z, property := (_ : y z \u2208 g.domain) })\n[PROOFSTEP]\napply fg_eq\n[GOAL]\ncase refine'_2._H\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nx : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhx : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z \u2208 f.domain\ny : { x // x \u2208 f.domain \u2294 g.domain } \u2192 E\nhy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), y z \u2208 g.domain\nhxy : \u2200 (z : { x // x \u2208 f.domain \u2294 g.domain }), x z + y z = \u2191z\nfg : { x // x \u2208 f.domain \u2294 g.domain } \u2192 F :=\n fun z => \u2191f { val := x z, property := (_ : x z \u2208 f.domain) } + \u2191g { val := y z, property := (_ : y z \u2208 g.domain) }\nfg_eq :\n \u2200 (x' : { x // x \u2208 f.domain }) (y' : { x // x \u2208 g.domain }) (z' : { x // x \u2208 f.domain \u2294 g.domain }),\n \u2191x' + \u2191y' = \u2191z' \u2192 fg z' = \u2191f x' + \u2191g y'\nc : R\nz : { x // x \u2208 f.domain \u2294 g.domain }\n\u22a2 \u2191(\u2191(RingHom.id R) c \u2022 { val := x z, property := (_ : x z \u2208 f.domain) }) +\n \u2191(\u2191(RingHom.id R) c \u2022 { val := y z, property := (_ : y z \u2208 g.domain) }) =\n \u2191(c \u2022 z)\n[PROOFSTEP]\nsimp only [coe_smul, coe_mk, \u2190 smul_add, hxy, RingHom.id_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\n\u22a2 f \u2264 LinearPMap.sup f g h\n[PROOFSTEP]\nrefine' \u27e8le_sup_left, fun z\u2081 z\u2082 hz => _\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nz\u2081 : { x // x \u2208 f.domain }\nz\u2082 : { x // x \u2208 (LinearPMap.sup f g h).domain }\nhz : \u2191z\u2081 = \u2191z\u2082\n\u22a2 \u2191f z\u2081 = \u2191(LinearPMap.sup f g h) z\u2082\n[PROOFSTEP]\nrw [\u2190 add_zero (f _), \u2190 g.map_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nz\u2081 : { x // x \u2208 f.domain }\nz\u2082 : { x // x \u2208 (LinearPMap.sup f g h).domain }\nhz : \u2191z\u2081 = \u2191z\u2082\n\u22a2 \u2191f z\u2081 + \u2191g 0 = \u2191(LinearPMap.sup f g h) z\u2082\n[PROOFSTEP]\nrefine' (sup_apply h _ _ _ _).symm\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nz\u2081 : { x // x \u2208 f.domain }\nz\u2082 : { x // x \u2208 (LinearPMap.sup f g h).domain }\nhz : \u2191z\u2081 = \u2191z\u2082\n\u22a2 \u2191z\u2081 + \u21910 = \u2191z\u2082\n[PROOFSTEP]\nsimpa\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\n\u22a2 g \u2264 LinearPMap.sup f g h\n[PROOFSTEP]\nrefine' \u27e8le_sup_right, fun z\u2081 z\u2082 hz => _\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nz\u2081 : { x // x \u2208 g.domain }\nz\u2082 : { x // x \u2208 (LinearPMap.sup f g h).domain }\nhz : \u2191z\u2081 = \u2191z\u2082\n\u22a2 \u2191g z\u2081 = \u2191(LinearPMap.sup f g h) z\u2082\n[PROOFSTEP]\nrw [\u2190 zero_add (g _), \u2190 f.map_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nz\u2081 : { x // x \u2208 g.domain }\nz\u2082 : { x // x \u2208 (LinearPMap.sup f g h).domain }\nhz : \u2191z\u2081 = \u2191z\u2082\n\u22a2 \u2191f 0 + \u2191g z\u2081 = \u2191(LinearPMap.sup f g h) z\u2082\n[PROOFSTEP]\nrefine' (sup_apply h _ _ _ _).symm\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : { x // x \u2208 f.domain }) (y : { x // x \u2208 g.domain }), \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\nz\u2081 : { x // x \u2208 g.domain }\nz\u2082 : { x // x \u2208 (LinearPMap.sup f g h).domain }\nhz : \u2191z\u2081 = \u2191z\u2082\n\u22a2 \u21910 + \u2191z\u2081 = \u2191z\u2082\n[PROOFSTEP]\nsimpa\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : Disjoint f.domain g.domain\nx : { x // x \u2208 f.domain }\ny : { x // x \u2208 g.domain }\nhxy : \u2191x = \u2191y\n\u22a2 \u2191f x = \u2191g y\n[PROOFSTEP]\nrw [disjoint_def] at h \n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : E), x \u2208 f.domain \u2192 x \u2208 g.domain \u2192 x = 0\nx : { x // x \u2208 f.domain }\ny : { x // x \u2208 g.domain }\nhxy : \u2191x = \u2191y\n\u22a2 \u2191f x = \u2191g y\n[PROOFSTEP]\nhave hy : y = 0 := Subtype.eq (h y (hxy \u25b8 x.2) y.2)\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : E), x \u2208 f.domain \u2192 x \u2208 g.domain \u2192 x = 0\nx : { x // x \u2208 f.domain }\ny : { x // x \u2208 g.domain }\nhxy : \u2191x = \u2191y\nhy : y = 0\n\u22a2 \u2191f x = \u2191g y\n[PROOFSTEP]\nhave hx : x = 0 := Subtype.eq (hxy.trans <| congr_arg _ hy)\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh : \u2200 (x : E), x \u2208 f.domain \u2192 x \u2208 g.domain \u2192 x = 0\nx : { x // x \u2208 f.domain }\ny : { x // x \u2208 g.domain }\nhxy : \u2191x = \u2191y\nhy : y = 0\nhx : x = 0\n\u22a2 \u2191f x = \u2191g y\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\n\u22a2 - -f = f\n[PROOFSTEP]\next x y hxy\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : E\n\u22a2 x \u2208 (- -f).domain \u2194 x \u2208 f.domain\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h'\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 (- -f).domain }\ny : { x // x \u2208 f.domain }\nhxy : \u2191x = \u2191y\n\u22a2 \u2191(- -f) x = \u2191f y\n[PROOFSTEP]\nsimp only [neg_apply, neg_neg]\n[GOAL]\ncase h'\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 (- -f).domain }\ny : { x // x \u2208 f.domain }\nhxy : \u2191x = \u2191y\n\u22a2 \u2191f x = \u2191f y\n[PROOFSTEP]\ncases x\n[GOAL]\ncase h'.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\ny : { x // x \u2208 f.domain }\nval\u271d : E\nproperty\u271d : val\u271d \u2208 (- -f).domain\nhxy : \u2191{ val := val\u271d, property := property\u271d } = \u2191y\n\u22a2 \u2191f { val := val\u271d, property := property\u271d } = \u2191f y\n[PROOFSTEP]\ncongr\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g h : E \u2192\u2097.[R] F\n\u22a2 f + g + h = f + (g + h)\n[PROOFSTEP]\next x y hxy\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g h : E \u2192\u2097.[R] F\nx : E\n\u22a2 x \u2208 (f + g + h).domain \u2194 x \u2208 (f + (g + h)).domain\n[PROOFSTEP]\nsimp only [add_domain, inf_assoc]\n[GOAL]\ncase h'\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g h : E \u2192\u2097.[R] F\nx : { x // x \u2208 (f + g + h).domain }\ny : { x // x \u2208 (f + (g + h)).domain }\nhxy : \u2191x = \u2191y\n\u22a2 \u2191(f + g + h) x = \u2191(f + (g + h)) y\n[PROOFSTEP]\nsimp only [add_apply, hxy, add_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\n\u22a2 0 + f = f\n[PROOFSTEP]\next x y hxy\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : E\n\u22a2 x \u2208 (0 + f).domain \u2194 x \u2208 f.domain\n[PROOFSTEP]\nsimp [add_domain]\n[GOAL]\ncase h'\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 (0 + f).domain }\ny : { x // x \u2208 f.domain }\nhxy : \u2191x = \u2191y\n\u22a2 \u2191(0 + f) x = \u2191f y\n[PROOFSTEP]\nsimp only [add_apply, hxy, zero_apply, zero_add]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\n\u22a2 f + 0 = f\n[PROOFSTEP]\next x y hxy\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : E\n\u22a2 x \u2208 (f + 0).domain \u2194 x \u2208 f.domain\n[PROOFSTEP]\nsimp [add_domain]\n[GOAL]\ncase h'\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 (f + 0).domain }\ny : { x // x \u2208 f.domain }\nhxy : \u2191x = \u2191y\n\u22a2 \u2191(f + 0) x = \u2191f y\n[PROOFSTEP]\nsimp only [add_apply, hxy, zero_apply, add_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\n\u22a2 0 + f = f\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\n\u22a2 \u2200 (a : E \u2192\u2097.[R] F), a + 0 = a\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\n\u22a2 f + g = g + f\n[PROOFSTEP]\next x y hxy\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nx : E\n\u22a2 x \u2208 (f + g).domain \u2194 x \u2208 (g + f).domain\n[PROOFSTEP]\nsimp only [add_domain, inf_comm]\n[GOAL]\ncase h'\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nx : { x // x \u2208 (f + g).domain }\ny : { x // x \u2208 (g + f).domain }\nhxy : \u2191x = \u2191y\n\u22a2 \u2191(f + g) x = \u2191(g + f) y\n[PROOFSTEP]\nsimp only [add_apply, hxy, add_comm]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\n\u22a2 f - g = f + -g\n[PROOFSTEP]\next x y h\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nx : E\n\u22a2 x \u2208 (f - g).domain \u2194 x \u2208 (f + -g).domain\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h'\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nx : { x // x \u2208 (f - g).domain }\ny : { x // x \u2208 (f + -g).domain }\nh : \u2191x = \u2191y\n\u22a2 \u2191(f - g) x = \u2191(f + -g) y\n[PROOFSTEP]\nsimp [sub_apply, add_apply, neg_apply, \u2190 sub_eq_add_neg, h]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\n\u22a2 -(f + g) = -g + -f\n[PROOFSTEP]\next x y h\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nx : E\n\u22a2 x \u2208 (-(f + g)).domain \u2194 x \u2208 (-g + -f).domain\n[PROOFSTEP]\nsimp [add_domain, sub_domain, neg_domain, And.comm]\n[GOAL]\ncase h'\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nx : { x // x \u2208 (-(f + g)).domain }\ny : { x // x \u2208 (-g + -f).domain }\nh : \u2191x = \u2191y\n\u22a2 \u2191(-(f + g)) x = \u2191(-g + -f) y\n[PROOFSTEP]\nsimp [sub_apply, add_apply, neg_apply, \u2190 sub_eq_add_neg, h]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh' : f + g = 0\n\u22a2 -f = g\n[PROOFSTEP]\next x y h\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh' : f + g = 0\nx : E\n\u22a2 x \u2208 (-f).domain \u2194 x \u2208 g.domain\n[PROOFSTEP]\nhave : (0 : E \u2192\u2097.[R] F).domain = \u22a4 := zero_domain\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh' : f + g = 0\nx : E\nthis : 0.domain = \u22a4\n\u22a2 x \u2208 (-f).domain \u2194 x \u2208 g.domain\n[PROOFSTEP]\nsimp only [\u2190 h', add_domain, ge_iff_le, inf_eq_top_iff] at this \n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh' : f + g = 0\nx : E\nthis : f.domain = \u22a4 \u2227 g.domain = \u22a4\n\u22a2 x \u2208 (-f).domain \u2194 x \u2208 g.domain\n[PROOFSTEP]\nrw [neg_domain, this.1, this.2]\n[GOAL]\ncase h'\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh' : f + g = 0\nx : { x // x \u2208 (-f).domain }\ny : { x // x \u2208 g.domain }\nh : \u2191x = \u2191y\n\u22a2 \u2191(-f) x = \u2191g y\n[PROOFSTEP]\nsimp only [inf_coe, neg_domain, Eq.ndrec, Int.ofNat_eq_coe, neg_apply]\n[GOAL]\ncase h'\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh' : f + g = 0\nx : { x // x \u2208 (-f).domain }\ny : { x // x \u2208 g.domain }\nh : \u2191x = \u2191y\n\u22a2 -\u2191f x = \u2191g y\n[PROOFSTEP]\nrw [ext_iff] at h' \n[GOAL]\ncase h'\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nh' : \u2203 _domain_eq, \u2200 \u2983x : { x // x \u2208 (f + g).domain }\u2984 \u2983y : { x // x \u2208 0.domain }\u2984, \u2191x = \u2191y \u2192 \u2191(f + g) x = \u21910 y\nx : { x // x \u2208 (-f).domain }\ny : { x // x \u2208 g.domain }\nh : \u2191x = \u2191y\n\u22a2 -\u2191f x = \u2191g y\n[PROOFSTEP]\nrcases h' with \u27e8hdom, h'\u27e9\n[GOAL]\ncase h'.intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nx : { x // x \u2208 (-f).domain }\ny : { x // x \u2208 g.domain }\nh : \u2191x = \u2191y\nhdom : (f + g).domain = 0.domain\nh' : \u2200 \u2983x : { x // x \u2208 (f + g).domain }\u2984 \u2983y : { x // x \u2208 0.domain }\u2984, \u2191x = \u2191y \u2192 \u2191(f + g) x = \u21910 y\n\u22a2 -\u2191f x = \u2191g y\n[PROOFSTEP]\nrw [zero_domain] at hdom \n[GOAL]\ncase h'.intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nx : { x // x \u2208 (-f).domain }\ny : { x // x \u2208 g.domain }\nh : \u2191x = \u2191y\nhdom : (f + g).domain = \u22a4\nh' : \u2200 \u2983x : { x // x \u2208 (f + g).domain }\u2984 \u2983y : { x // x \u2208 0.domain }\u2984, \u2191x = \u2191y \u2192 \u2191(f + g) x = \u21910 y\n\u22a2 -\u2191f x = \u2191g y\n[PROOFSTEP]\nsimp only [inf_coe, neg_domain, Eq.ndrec, Int.ofNat_eq_coe, zero_domain, top_coe, zero_apply, Subtype.forall, mem_top,\n forall_true_left, forall_eq'] at h' \n[GOAL]\ncase h'.intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nx : { x // x \u2208 (-f).domain }\ny : { x // x \u2208 g.domain }\nh : \u2191x = \u2191y\nhdom : (f + g).domain = \u22a4\nh' : \u2200 (a : E) (b : a \u2208 (f + g).domain), \u2191(f + g) { val := a, property := b } = 0\n\u22a2 -\u2191f x = \u2191g y\n[PROOFSTEP]\nspecialize h' x.1 (by simp [hdom])\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nx : { x // x \u2208 (-f).domain }\ny : { x // x \u2208 g.domain }\nh : \u2191x = \u2191y\nhdom : (f + g).domain = \u22a4\nh' : \u2200 (a : E) (b : a \u2208 (f + g).domain), \u2191(f + g) { val := a, property := b } = 0\n\u22a2 \u2191x \u2208 (f + g).domain\n[PROOFSTEP]\nsimp [hdom]\n[GOAL]\ncase h'.intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nx : { x // x \u2208 (-f).domain }\ny : { x // x \u2208 g.domain }\nh : \u2191x = \u2191y\nhdom : (f + g).domain = \u22a4\nh' : \u2191(f + g) { val := \u2191x, property := (_ : \u2191x \u2208 (f + g).domain) } = 0\n\u22a2 -\u2191f x = \u2191g y\n[PROOFSTEP]\nsimp only [inf_coe, neg_domain, Eq.ndrec, Int.ofNat_eq_coe, add_apply, Subtype.coe_eta, \u2190 neg_eq_iff_add_eq_zero] at h' \n[GOAL]\ncase h'.intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf g : E \u2192\u2097.[R] F\nx : { x // x \u2208 (-f).domain }\ny : { x // x \u2208 g.domain }\nh : \u2191x = \u2191y\nhdom : (f + g).domain = \u22a4\nh' : -\u2191f x = \u2191g { val := \u2191x, property := (_ : \u2191{ val := \u2191x, property := (_ : \u2191x \u2208 (f + g).domain) } \u2208 \u2191g.domain) }\n\u22a2 -\u2191f x = \u2191g y\n[PROOFSTEP]\nrw [h', h]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nK : Type u_5\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : Module K E\ninst\u271d : Module K F\nf : E \u2192\u2097.[K] F\nx : E\ny : F\nhx : \u00acx \u2208 f.domain\n\u22a2 Disjoint f.domain (mkSpanSingleton x y (_ : x = 0 \u2192 False)).domain\n[PROOFSTEP]\nsimp [disjoint_span_singleton]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nK : Type u_5\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : Module K E\ninst\u271d : Module K F\nf : E \u2192\u2097.[K] F\nx : E\ny : F\nhx : \u00acx \u2208 f.domain\n\u22a2 x \u2208 f.domain \u2192 x = 0\n[PROOFSTEP]\nexact fun h => False.elim <| hx h\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nK : Type u_5\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : Module K E\ninst\u271d : Module K F\nf : E \u2192\u2097.[K] F\nx : E\ny : F\nhx : \u00acx \u2208 f.domain\nx' : E\nhx' : x' \u2208 f.domain\nc : K\n\u22a2 \u2191(supSpanSingleton f x y hx)\n { val := x' + c \u2022 x,\n property := (_ : x' + c \u2022 x \u2208 f.domain \u2294 (mkSpanSingleton x y (_ : x = 0 \u2192 False)).domain) } =\n \u2191f { val := x', property := hx' } + c \u2022 y\n[PROOFSTEP]\nerw [sup_apply _ \u27e8x', hx'\u27e9 \u27e8c \u2022 x, _\u27e9, mkSpanSingleton'_apply]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nK : Type u_5\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : Module K E\ninst\u271d : Module K F\nf : E \u2192\u2097.[K] F\nx : E\ny : F\nhx : \u00acx \u2208 f.domain\nx' : E\nhx' : x' \u2208 f.domain\nc : K\n\u22a2 c \u2022 x \u2208 (mkSpanSingleton' x y (_ : \u2200 (c : K), c \u2022 x = 0 \u2192 c \u2022 y = 0)).domain\n[PROOFSTEP]\nexact mem_span_singleton.2 \u27e8c, rfl\u27e9\n[GOAL]\ncase hz\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nK : Type u_5\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : Module K E\ninst\u271d : Module K F\nf : E \u2192\u2097.[K] F\nx : E\ny : F\nhx : \u00acx \u2208 f.domain\nx' : E\nhx' : x' \u2208 f.domain\nc : K\n\u22a2 \u2191{ val := x', property := hx' } + \u2191{ val := c \u2022 x, property := (_ : c \u2022 x \u2208 span K {x}) } =\n \u2191{ val := x' + c \u2022 x, property := (_ : x' + c \u2022 x \u2208 f.domain \u2294 (mkSpanSingleton x y (_ : x = 0 \u2192 False)).domain) }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\n\u22a2 \u2203 f, { domain := sSup (domain '' c), toFun := f } \u2208 upperBounds c\n[PROOFSTEP]\ncases' c.eq_empty_or_nonempty with ceq cne\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\nceq : c = \u2205\n\u22a2 \u2203 f, { domain := sSup (domain '' c), toFun := f } \u2208 upperBounds c\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) \u2205\n\u22a2 \u2203 f, { domain := sSup (domain '' \u2205), toFun := f } \u2208 upperBounds \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\n\u22a2 \u2203 f, { domain := sSup (domain '' c), toFun := f } \u2208 upperBounds c\n[PROOFSTEP]\nhave hdir : DirectedOn (\u00b7 \u2264 \u00b7) (domain '' c) := directedOn_image.2 (hc.mono @(domain_mono.monotone))\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\n\u22a2 \u2203 f, { domain := sSup (domain '' c), toFun := f } \u2208 upperBounds c\n[PROOFSTEP]\nhave P : \u2200 x : \u21a5(sSup (domain '' c)), { p : c // (x : E) \u2208 p.val.domain } :=\n by\n rintro x\n apply Classical.indefiniteDescription\n have :=\n (mem_sSup_of_directed (cne.image _) hdir).1\n x.2\n -- Porting note: + `\u2190 bex_def`\n rwa [bex_image_iff, \u2190 bex_def, SetCoe.exists'] at this \n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\n\u22a2 (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\n[PROOFSTEP]\nrintro x\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nx : { x // x \u2208 sSup (domain '' c) }\n\u22a2 { p // \u2191x \u2208 (\u2191p).domain }\n[PROOFSTEP]\napply Classical.indefiniteDescription\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nx : { x // x \u2208 sSup (domain '' c) }\n\u22a2 \u2203 x_1, \u2191x \u2208 (\u2191x_1).domain\n[PROOFSTEP]\nhave :=\n (mem_sSup_of_directed (cne.image _) hdir).1\n x.2\n -- Porting note: + `\u2190 bex_def`\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nx : { x // x \u2208 sSup (domain '' c) }\nthis : \u2203 y, y \u2208 domain '' c \u2227 \u2191x \u2208 y\n\u22a2 \u2203 x_1, \u2191x \u2208 (\u2191x_1).domain\n[PROOFSTEP]\nrwa [bex_image_iff, \u2190 bex_def, SetCoe.exists'] at this \n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\n\u22a2 \u2203 f, { domain := sSup (domain '' c), toFun := f } \u2208 upperBounds c\n[PROOFSTEP]\nset f : \u21a5(sSup (domain '' c)) \u2192 F := fun x => (P x).val.val \u27e8x, (P x).property\u27e9\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\n\u22a2 \u2203 f, { domain := sSup (domain '' c), toFun := f } \u2208 upperBounds c\n[PROOFSTEP]\nhave f_eq : \u2200 (p : c) (x : \u21a5(sSup (domain '' c))) (y : p.1.1) (_hxy : (x : E) = y), f x = p.1 y :=\n by\n intro p x y hxy\n rcases hc (P x).1.1 (P x).1.2 p.1 p.2 with\n \u27e8q, _hqc, hxq, hpq\u27e9\n -- Porting note: `refine' ..; exacts [ofLe hpq.1 y, hxy, rfl]`\n -- \u2192 `refine' .. <;> [skip; exact ofLe hpq.1 y; rfl]; exact hxy`\n refine' (hxq.2 _).trans (hpq.2 _).symm <;> [skip; exact ofLe hpq.1 y; rfl]; exact hxy\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\n\u22a2 \u2200 (p : \u2191c) (x : { x // x \u2208 sSup (domain '' c) }) (y : { x // x \u2208 (\u2191p).domain }), \u2191x = \u2191y \u2192 f x = \u2191\u2191p y\n[PROOFSTEP]\nintro p x y hxy\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\np : \u2191c\nx : { x // x \u2208 sSup (domain '' c) }\ny : { x // x \u2208 (\u2191p).domain }\nhxy : \u2191x = \u2191y\n\u22a2 f x = \u2191\u2191p y\n[PROOFSTEP]\nrcases hc (P x).1.1 (P x).1.2 p.1 p.2 with\n \u27e8q, _hqc, hxq, hpq\u27e9\n -- Porting note: `refine' ..; exacts [ofLe hpq.1 y, hxy, rfl]`\n -- \u2192 `refine' .. <;> [skip; exact ofLe hpq.1 y; rfl]; exact hxy`\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\np : \u2191c\nx : { x // x \u2208 sSup (domain '' c) }\ny : { x // x \u2208 (\u2191p).domain }\nhxy : \u2191x = \u2191y\nq : E \u2192\u2097.[R] F\n_hqc : q \u2208 c\nhxq : \u2191\u2191(P x) \u2264 q\nhpq : \u2191p \u2264 q\n\u22a2 f x = \u2191\u2191p y\n[PROOFSTEP]\nrefine' (hxq.2 _).trans (hpq.2 _).symm <;> [skip; exact ofLe hpq.1 y; rfl]\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\np : \u2191c\nx : { x // x \u2208 sSup (domain '' c) }\ny : { x // x \u2208 (\u2191p).domain }\nhxy : \u2191x = \u2191y\nq : E \u2192\u2097.[R] F\n_hqc : q \u2208 c\nhxq : \u2191\u2191(P x) \u2264 q\nhpq : \u2191p \u2264 q\n\u22a2 f x = \u2191\u2191p y\n[PROOFSTEP]\nrefine' (hxq.2 _).trans (hpq.2 _).symm\n[GOAL]\ncase intro.intro.intro.refine'_1\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\np : \u2191c\nx : { x // x \u2208 sSup (domain '' c) }\ny : { x // x \u2208 (\u2191p).domain }\nhxy : \u2191x = \u2191y\nq : E \u2192\u2097.[R] F\n_hqc : q \u2208 c\nhxq : \u2191\u2191(P x) \u2264 q\nhpq : \u2191p \u2264 q\n\u22a2 \u2191{ val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) } = \u2191?intro.intro.intro.refine'_2\n[PROOFSTEP]\nskip\n[GOAL]\ncase intro.intro.intro.refine'_2\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\np : \u2191c\nx : { x // x \u2208 sSup (domain '' c) }\ny : { x // x \u2208 (\u2191p).domain }\nhxy : \u2191x = \u2191y\nq : E \u2192\u2097.[R] F\n_hqc : q \u2208 c\nhxq : \u2191\u2191(P x) \u2264 q\nhpq : \u2191p \u2264 q\n\u22a2 { x // x \u2208 q.domain }\n[PROOFSTEP]\nexact ofLe hpq.1 y\n[GOAL]\ncase intro.intro.intro.refine'_3\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\np : \u2191c\nx : { x // x \u2208 sSup (domain '' c) }\ny : { x // x \u2208 (\u2191p).domain }\nhxy : \u2191x = \u2191y\nq : E \u2192\u2097.[R] F\n_hqc : q \u2208 c\nhxq : \u2191\u2191(P x) \u2264 q\nhpq : \u2191p \u2264 q\n\u22a2 \u2191y = \u2191(\u2191(ofLe (_ : (\u2191p).domain \u2264 q.domain)) y)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro.intro.refine'_1\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\np : \u2191c\nx : { x // x \u2208 sSup (domain '' c) }\ny : { x // x \u2208 (\u2191p).domain }\nhxy : \u2191x = \u2191y\nq : E \u2192\u2097.[R] F\n_hqc : q \u2208 c\nhxq : \u2191\u2191(P x) \u2264 q\nhpq : \u2191p \u2264 q\n\u22a2 \u2191{ val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) } = \u2191(\u2191(ofLe (_ : (\u2191p).domain \u2264 q.domain)) y)\n[PROOFSTEP]\nexact hxy\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\nf_eq : \u2200 (p : \u2191c) (x : { x // x \u2208 sSup (domain '' c) }) (y : { x // x \u2208 (\u2191p).domain }), \u2191x = \u2191y \u2192 f x = \u2191\u2191p y\n\u22a2 \u2203 f, { domain := sSup (domain '' c), toFun := f } \u2208 upperBounds c\n[PROOFSTEP]\nrefine' \u27e8{ toFun := f .. }, _\u27e9\n[GOAL]\ncase inr.refine'_1\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\nf_eq : \u2200 (p : \u2191c) (x : { x // x \u2208 sSup (domain '' c) }) (y : { x // x \u2208 (\u2191p).domain }), \u2191x = \u2191y \u2192 f x = \u2191\u2191p y\n\u22a2 \u2200 (x y : { x // x \u2208 sSup (domain '' c) }), f (x + y) = f x + f y\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase inr.refine'_1\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\nf_eq : \u2200 (p : \u2191c) (x : { x // x \u2208 sSup (domain '' c) }) (y : { x // x \u2208 (\u2191p).domain }), \u2191x = \u2191y \u2192 f x = \u2191\u2191p y\nx y : { x // x \u2208 sSup (domain '' c) }\n\u22a2 f (x + y) = f x + f y\n[PROOFSTEP]\nrcases hc (P x).1.1 (P x).1.2 (P y).1.1 (P y).1.2 with \u27e8p, hpc, hpx, hpy\u27e9\n[GOAL]\ncase inr.refine'_1.intro.intro.intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\nf_eq : \u2200 (p : \u2191c) (x : { x // x \u2208 sSup (domain '' c) }) (y : { x // x \u2208 (\u2191p).domain }), \u2191x = \u2191y \u2192 f x = \u2191\u2191p y\nx y : { x // x \u2208 sSup (domain '' c) }\np : E \u2192\u2097.[R] F\nhpc : p \u2208 c\nhpx : \u2191\u2191(P x) \u2264 p\nhpy : \u2191\u2191(P y) \u2264 p\n\u22a2 f (x + y) = f x + f y\n[PROOFSTEP]\nset x' := ofLe hpx.1 \u27e8x, (P x).2\u27e9\n[GOAL]\ncase inr.refine'_1.intro.intro.intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\nf_eq : \u2200 (p : \u2191c) (x : { x // x \u2208 sSup (domain '' c) }) (y : { x // x \u2208 (\u2191p).domain }), \u2191x = \u2191y \u2192 f x = \u2191\u2191p y\nx y : { x // x \u2208 sSup (domain '' c) }\np : E \u2192\u2097.[R] F\nhpc : p \u2208 c\nhpx : \u2191\u2191(P x) \u2264 p\nhpy : \u2191\u2191(P y) \u2264 p\nx' : { x // x \u2208 p.domain } :=\n \u2191(ofLe (_ : (\u2191\u2191(P x)).domain \u2264 p.domain)) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\n\u22a2 f (x + y) = f x + f y\n[PROOFSTEP]\nset y' := ofLe hpy.1 \u27e8y, (P y).2\u27e9\n[GOAL]\ncase inr.refine'_1.intro.intro.intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\nf_eq : \u2200 (p : \u2191c) (x : { x // x \u2208 sSup (domain '' c) }) (y : { x // x \u2208 (\u2191p).domain }), \u2191x = \u2191y \u2192 f x = \u2191\u2191p y\nx y : { x // x \u2208 sSup (domain '' c) }\np : E \u2192\u2097.[R] F\nhpc : p \u2208 c\nhpx : \u2191\u2191(P x) \u2264 p\nhpy : \u2191\u2191(P y) \u2264 p\nx' : { x // x \u2208 p.domain } :=\n \u2191(ofLe (_ : (\u2191\u2191(P x)).domain \u2264 p.domain)) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\ny' : { x // x \u2208 p.domain } :=\n \u2191(ofLe (_ : (\u2191\u2191(P y)).domain \u2264 p.domain)) { val := \u2191y, property := (_ : \u2191y \u2208 (\u2191\u2191(P y)).domain) }\n\u22a2 f (x + y) = f x + f y\n[PROOFSTEP]\nrw [f_eq \u27e8p, hpc\u27e9 x x' rfl, f_eq \u27e8p, hpc\u27e9 y y' rfl, f_eq \u27e8p, hpc\u27e9 (x + y) (x' + y') rfl, map_add]\n[GOAL]\ncase inr.refine'_2\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\nf_eq : \u2200 (p : \u2191c) (x : { x // x \u2208 sSup (domain '' c) }) (y : { x // x \u2208 (\u2191p).domain }), \u2191x = \u2191y \u2192 f x = \u2191\u2191p y\n\u22a2 \u2200 (r : R) (x : { x // x \u2208 sSup (domain '' c) }),\n AddHom.toFun { toFun := f, map_add' := (_ : \u2200 (x y : { x // x \u2208 sSup (domain '' c) }), f (x + y) = f x + f y) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun { toFun := f, map_add' := (_ : \u2200 (x y : { x // x \u2208 sSup (domain '' c) }), f (x + y) = f x + f y) }\n x\n[PROOFSTEP]\nintro c x\n[GOAL]\ncase inr.refine'_2\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc\u271d : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\u271d\ncne : Set.Nonempty c\u271d\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c\u271d)\nP : (x : { x // x \u2208 sSup (domain '' c\u271d) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c\u271d) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\nf_eq : \u2200 (p : \u2191c\u271d) (x : { x // x \u2208 sSup (domain '' c\u271d) }) (y : { x // x \u2208 (\u2191p).domain }), \u2191x = \u2191y \u2192 f x = \u2191\u2191p y\nc : R\nx : { x // x \u2208 sSup (domain '' c\u271d) }\n\u22a2 AddHom.toFun { toFun := f, map_add' := (_ : \u2200 (x y : { x // x \u2208 sSup (domain '' c\u271d) }), f (x + y) = f x + f y) }\n (c \u2022 x) =\n \u2191(RingHom.id R) c \u2022\n AddHom.toFun { toFun := f, map_add' := (_ : \u2200 (x y : { x // x \u2208 sSup (domain '' c\u271d) }), f (x + y) = f x + f y) } x\n[PROOFSTEP]\nsimp only [f_eq (P x).1 (c \u2022 x) (c \u2022 \u27e8x, (P x).2\u27e9) rfl, \u2190 map_smul, RingHom.id_apply]\n[GOAL]\ncase inr.refine'_3\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\nf_eq : \u2200 (p : \u2191c) (x : { x // x \u2208 sSup (domain '' c) }) (y : { x // x \u2208 (\u2191p).domain }), \u2191x = \u2191y \u2192 f x = \u2191\u2191p y\n\u22a2 { domain := sSup (domain '' c),\n toFun :=\n {\n toAddHom :=\n { toFun := f, map_add' := (_ : \u2200 (x y : { x // x \u2208 sSup (domain '' c) }), f (x + y) = f x + f y) },\n map_smul' :=\n (_ :\n \u2200 (c_1 : R) (x : { x // x \u2208 sSup (domain '' c) }),\n f (c_1 \u2022 x) = c_1 \u2022 \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }) } } \u2208\n upperBounds c\n[PROOFSTEP]\nintro p hpc\n[GOAL]\ncase inr.refine'_3\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\nf_eq : \u2200 (p : \u2191c) (x : { x // x \u2208 sSup (domain '' c) }) (y : { x // x \u2208 (\u2191p).domain }), \u2191x = \u2191y \u2192 f x = \u2191\u2191p y\np : E \u2192\u2097.[R] F\nhpc : p \u2208 c\n\u22a2 p \u2264\n { domain := sSup (domain '' c),\n toFun :=\n {\n toAddHom :=\n { toFun := f, map_add' := (_ : \u2200 (x y : { x // x \u2208 sSup (domain '' c) }), f (x + y) = f x + f y) },\n map_smul' :=\n (_ :\n \u2200 (c_1 : R) (x : { x // x \u2208 sSup (domain '' c) }),\n f (c_1 \u2022 x) = c_1 \u2022 \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }) } }\n[PROOFSTEP]\nrefine' \u27e8le_sSup <| mem_image_of_mem domain hpc, fun x y hxy => Eq.symm _\u27e9\n[GOAL]\ncase inr.refine'_3\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\ncne : Set.Nonempty c\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) (domain '' c)\nP : (x : { x // x \u2208 sSup (domain '' c) }) \u2192 { p // \u2191x \u2208 (\u2191p).domain }\nf : { x // x \u2208 sSup (domain '' c) } \u2192 F := fun x => \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }\nf_eq : \u2200 (p : \u2191c) (x : { x // x \u2208 sSup (domain '' c) }) (y : { x // x \u2208 (\u2191p).domain }), \u2191x = \u2191y \u2192 f x = \u2191\u2191p y\np : E \u2192\u2097.[R] F\nhpc : p \u2208 c\nx : { x // x \u2208 p.domain }\ny :\n { x //\n x \u2208\n { domain := sSup (domain '' c),\n toFun :=\n {\n toAddHom :=\n { toFun := f, map_add' := (_ : \u2200 (x y : { x // x \u2208 sSup (domain '' c) }), f (x + y) = f x + f y) },\n map_smul' :=\n (_ :\n \u2200 (c_1 : R) (x : { x // x \u2208 sSup (domain '' c) }),\n f (c_1 \u2022 x) = c_1 \u2022 \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }) } }.domain }\nhxy : \u2191x = \u2191y\n\u22a2 \u2191{ domain := sSup (domain '' c),\n toFun :=\n {\n toAddHom :=\n { toFun := f, map_add' := (_ : \u2200 (x y : { x // x \u2208 sSup (domain '' c) }), f (x + y) = f x + f y) },\n map_smul' :=\n (_ :\n \u2200 (c_1 : R) (x : { x // x \u2208 sSup (domain '' c) }),\n f (c_1 \u2022 x) = c_1 \u2022 \u2191\u2191\u2191(P x) { val := \u2191x, property := (_ : \u2191x \u2208 (\u2191\u2191(P x)).domain) }) } }\n y =\n \u2191p x\n[PROOFSTEP]\nexact f_eq \u27e8p, hpc\u27e9 _ _ hxy.symm\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\nl : E \u2192\u2097.[R] F\nhl : l \u2208 c\nx : { x // x \u2208 l.domain }\n\u22a2 \u2191(LinearPMap.sSup c hc) { val := \u2191x, property := (_ : \u2191x \u2208 (LinearPMap.sSup c hc).domain) } = \u2191l x\n[PROOFSTEP]\nsymm\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\nl : E \u2192\u2097.[R] F\nhl : l \u2208 c\nx : { x // x \u2208 l.domain }\n\u22a2 \u2191l x = \u2191(LinearPMap.sSup c hc) { val := \u2191x, property := (_ : \u2191x \u2208 (LinearPMap.sSup c hc).domain) }\n[PROOFSTEP]\napply (Classical.choose_spec (sSup_aux c hc) hl).2\n[GOAL]\ncase _h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\nl : E \u2192\u2097.[R] F\nhl : l \u2208 c\nx : { x // x \u2208 l.domain }\n\u22a2 \u2191x = \u2191{ val := \u2191x, property := (_ : \u2191x \u2208 (LinearPMap.sSup c hc).domain) }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nS : Submodule R E\n\u22a2 S \u2293 f.domain \u2264 f.domain\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nS : Submodule R E\nx : { x // x \u2208 S \u2293 f.domain }\ny : { x // x \u2208 f.domain }\nh : \u2191x = \u2191y\n\u22a2 \u2191(domRestrict f S) x = \u2191f y\n[PROOFSTEP]\nhave : Submodule.ofLe (by simp) x = y := by\n ext\n simp [h]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nS : Submodule R E\nx : { x // x \u2208 S \u2293 f.domain }\ny : { x // x \u2208 f.domain }\nh : \u2191x = \u2191y\n\u22a2 S \u2293 f.domain \u2264 f.domain\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nS : Submodule R E\nx : { x // x \u2208 S \u2293 f.domain }\ny : { x // x \u2208 f.domain }\nh : \u2191x = \u2191y\n\u22a2 \u2191(Submodule.ofLe (_ : S \u2293 f.domain \u2264 f.domain)) x = y\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nS : Submodule R E\nx : { x // x \u2208 S \u2293 f.domain }\ny : { x // x \u2208 f.domain }\nh : \u2191x = \u2191y\n\u22a2 \u2191(\u2191(Submodule.ofLe (_ : S \u2293 f.domain \u2264 f.domain)) x) = \u2191y\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nS : Submodule R E\nx : { x // x \u2208 S \u2293 f.domain }\ny : { x // x \u2208 f.domain }\nh : \u2191x = \u2191y\nthis : \u2191(Submodule.ofLe (_ : S \u2293 f.domain \u2264 f.domain)) x = y\n\u22a2 \u2191(domRestrict f S) x = \u2191f y\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nS : Submodule R E\nx : { x // x \u2208 S \u2293 f.domain }\ny : { x // x \u2208 f.domain }\nh : \u2191x = \u2191y\nthis : \u2191(Submodule.ofLe (_ : S \u2293 f.domain \u2264 f.domain)) x = y\n\u22a2 \u2191(domRestrict f S) x = \u2191f (\u2191(Submodule.ofLe (_ : S \u2293 f.domain \u2264 f.domain)) x)\n[PROOFSTEP]\nexact LinearPMap.mk_apply _ _ _\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nS : Submodule R E\n\u22a2 (domRestrict f S).domain \u2264 f.domain\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : E \u00d7 F\n\u22a2 x \u2208 graph f \u2194 \u2203 y, (\u2191y, \u2191f y) = x\n[PROOFSTEP]\nsimp [graph]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : E \u00d7 F\n\u22a2 x \u2208 graph f \u2194 \u2203 y, \u2191y = x.fst \u2227 \u2191f y = x.snd\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nfst\u271d : E\nsnd\u271d : F\n\u22a2 (fst\u271d, snd\u271d) \u2208 graph f \u2194 \u2203 y, \u2191y = (fst\u271d, snd\u271d).fst \u2227 \u2191f y = (fst\u271d, snd\u271d).snd\n[PROOFSTEP]\nsimp_rw [mem_graph_iff', Prod.mk.inj_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 f.domain }\n\u22a2 (\u2191x, \u2191f x) \u2208 graph f\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\n\u22a2 Submodule.map (LinearMap.fst R E F) (graph f) = f.domain\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : E\n\u22a2 x \u2208 Submodule.map (LinearMap.fst R E F) (graph f) \u2194 x \u2208 f.domain\n[PROOFSTEP]\nsimp only [Submodule.mem_map, mem_graph_iff, Subtype.exists, exists_and_left, exists_eq_left, LinearMap.fst_apply,\n Prod.exists, exists_and_right, exists_eq_right]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : E\n\u22a2 (\u2203 x_1 h, \u2191f { val := x, property := (_ : (x, x_1).fst \u2208 f.domain) } = x_1) \u2194 x \u2208 f.domain\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : E\n\u22a2 (\u2203 x_1 h, \u2191f { val := x, property := (_ : (x, x_1).fst \u2208 f.domain) } = x_1) \u2192 x \u2208 f.domain\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mpr\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : E\n\u22a2 x \u2208 f.domain \u2192 \u2203 x_1 h, \u2191f { val := x, property := (_ : (x, x_1).fst \u2208 f.domain) } = x_1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mp\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : E\nh : \u2203 x_1 h, \u2191f { val := x, property := (_ : (x, x_1).fst \u2208 f.domain) } = x_1\n\u22a2 x \u2208 f.domain\n[PROOFSTEP]\nrcases h with \u27e8x, hx, _\u27e9\n[GOAL]\ncase h.mp.intro.intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx\u271d : E\nx : F\nhx : x\u271d \u2208 f.domain\nh\u271d : \u2191f { val := x\u271d, property := (_ : (x\u271d, x).fst \u2208 f.domain) } = x\n\u22a2 x\u271d \u2208 f.domain\n[PROOFSTEP]\nexact hx\n[GOAL]\ncase h.mpr\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : E\nh : x \u2208 f.domain\n\u22a2 \u2203 x_1 h, \u2191f { val := x, property := (_ : (x, x_1).fst \u2208 f.domain) } = x_1\n[PROOFSTEP]\nuse f \u27e8x, h\u27e9\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx : E\nh : x \u2208 f.domain\n\u22a2 \u2203 h_1,\n \u2191f { val := x, property := (_ : (x, \u2191f { val := x, property := h }).fst \u2208 f.domain) } =\n \u2191f { val := x, property := h }\n[PROOFSTEP]\nsimp only [h, exists_prop]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\n\u22a2 Submodule.map (LinearMap.snd R E F) (graph f) = LinearMap.range f.toFun\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx\u271d : F\n\u22a2 x\u271d \u2208 Submodule.map (LinearMap.snd R E F) (graph f) \u2194 x\u271d \u2208 LinearMap.range f.toFun\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nz : M\n\u22a2 graph (z \u2022 f) = Submodule.map (LinearMap.prodMap LinearMap.id (z \u2022 LinearMap.id)) (graph f)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nz : M\nx : E \u00d7 F\n\u22a2 x \u2208 graph (z \u2022 f) \u2194 x \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (z \u2022 LinearMap.id)) (graph f)\n[PROOFSTEP]\ncases' x with x_fst x_snd\n[GOAL]\ncase h.mk\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\n\u22a2 (x_fst, x_snd) \u2208 graph (z \u2022 f) \u2194\n (x_fst, x_snd) \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (z \u2022 LinearMap.id)) (graph f)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mk.mp\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\n\u22a2 (x_fst, x_snd) \u2208 graph (z \u2022 f) \u2192\n (x_fst, x_snd) \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (z \u2022 LinearMap.id)) (graph f)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mk.mpr\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\n\u22a2 (x_fst, x_snd) \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (z \u2022 LinearMap.id)) (graph f) \u2192\n (x_fst, x_snd) \u2208 graph (z \u2022 f)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mk.mp\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\nh : (x_fst, x_snd) \u2208 graph (z \u2022 f)\n\u22a2 (x_fst, x_snd) \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (z \u2022 LinearMap.id)) (graph f)\n[PROOFSTEP]\nrw [mem_graph_iff] at h \n[GOAL]\ncase h.mk.mp\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\nh : \u2203 y, \u2191y = (x_fst, x_snd).fst \u2227 \u2191(z \u2022 f) y = (x_fst, x_snd).snd\n\u22a2 (x_fst, x_snd) \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (z \u2022 LinearMap.id)) (graph f)\n[PROOFSTEP]\nrcases h with \u27e8y, hy, h\u27e9\n[GOAL]\ncase h.mk.mp.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\ny : { x // x \u2208 (z \u2022 f).domain }\nhy : \u2191y = (x_fst, x_snd).fst\nh : \u2191(z \u2022 f) y = (x_fst, x_snd).snd\n\u22a2 (x_fst, x_snd) \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (z \u2022 LinearMap.id)) (graph f)\n[PROOFSTEP]\nrw [LinearPMap.smul_apply] at h \n[GOAL]\ncase h.mk.mp.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\ny : { x // x \u2208 (z \u2022 f).domain }\nhy : \u2191y = (x_fst, x_snd).fst\nh : z \u2022 \u2191f y = (x_fst, x_snd).snd\n\u22a2 (x_fst, x_snd) \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (z \u2022 LinearMap.id)) (graph f)\n[PROOFSTEP]\nrw [Submodule.mem_map]\n[GOAL]\ncase h.mk.mp.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\ny : { x // x \u2208 (z \u2022 f).domain }\nhy : \u2191y = (x_fst, x_snd).fst\nh : z \u2022 \u2191f y = (x_fst, x_snd).snd\n\u22a2 \u2203 y, y \u2208 graph f \u2227 \u2191(LinearMap.prodMap LinearMap.id (z \u2022 LinearMap.id)) y = (x_fst, x_snd)\n[PROOFSTEP]\nsimp only [mem_graph_iff, LinearMap.prodMap_apply, LinearMap.id_coe, id.def, LinearMap.smul_apply, Prod.mk.inj_iff,\n Prod.exists, exists_exists_and_eq_and]\n[GOAL]\ncase h.mk.mp.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\ny : { x // x \u2208 (z \u2022 f).domain }\nhy : \u2191y = (x_fst, x_snd).fst\nh : z \u2022 \u2191f y = (x_fst, x_snd).snd\n\u22a2 \u2203 a a_1, \u2191a_1 = a \u2227 a = x_fst \u2227 z \u2022 \u2191f a_1 = x_snd\n[PROOFSTEP]\nuse x_fst, y, hy\n[GOAL]\ncase h.mk.mpr\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\nh : (x_fst, x_snd) \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (z \u2022 LinearMap.id)) (graph f)\n\u22a2 (x_fst, x_snd) \u2208 graph (z \u2022 f)\n[PROOFSTEP]\nrw [Submodule.mem_map] at h \n[GOAL]\ncase h.mk.mpr\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\nh : \u2203 y, y \u2208 graph f \u2227 \u2191(LinearMap.prodMap LinearMap.id (z \u2022 LinearMap.id)) y = (x_fst, x_snd)\n\u22a2 (x_fst, x_snd) \u2208 graph (z \u2022 f)\n[PROOFSTEP]\nrcases h with \u27e8x', hx', h\u27e9\n[GOAL]\ncase h.mk.mpr.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\nx' : E \u00d7 F\nhx' : x' \u2208 graph f\nh : \u2191(LinearMap.prodMap LinearMap.id (z \u2022 LinearMap.id)) x' = (x_fst, x_snd)\n\u22a2 (x_fst, x_snd) \u2208 graph (z \u2022 f)\n[PROOFSTEP]\ncases x'\n[GOAL]\ncase h.mk.mpr.intro.intro.mk\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\nfst\u271d : E\nsnd\u271d : F\nhx' : (fst\u271d, snd\u271d) \u2208 graph f\nh : \u2191(LinearMap.prodMap LinearMap.id (z \u2022 LinearMap.id)) (fst\u271d, snd\u271d) = (x_fst, x_snd)\n\u22a2 (x_fst, x_snd) \u2208 graph (z \u2022 f)\n[PROOFSTEP]\nsimp only [LinearMap.prodMap_apply, LinearMap.id_coe, id.def, LinearMap.smul_apply, Prod.mk.inj_iff] at h \n[GOAL]\ncase h.mk.mpr.intro.intro.mk\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\nfst\u271d : E\nsnd\u271d : F\nhx' : (fst\u271d, snd\u271d) \u2208 graph f\nh : fst\u271d = x_fst \u2227 z \u2022 snd\u271d = x_snd\n\u22a2 (x_fst, x_snd) \u2208 graph (z \u2022 f)\n[PROOFSTEP]\nrw [mem_graph_iff] at hx' \u22a2\n[GOAL]\ncase h.mk.mpr.intro.intro.mk\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\nfst\u271d : E\nsnd\u271d : F\nhx' : \u2203 y, \u2191y = (fst\u271d, snd\u271d).fst \u2227 \u2191f y = (fst\u271d, snd\u271d).snd\nh : fst\u271d = x_fst \u2227 z \u2022 snd\u271d = x_snd\n\u22a2 \u2203 y, \u2191y = (x_fst, x_snd).fst \u2227 \u2191(z \u2022 f) y = (x_fst, x_snd).snd\n[PROOFSTEP]\nrcases hx' with \u27e8y, hy, hx'\u27e9\n[GOAL]\ncase h.mk.mpr.intro.intro.mk.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\nfst\u271d : E\nsnd\u271d : F\nh : fst\u271d = x_fst \u2227 z \u2022 snd\u271d = x_snd\ny : { x // x \u2208 f.domain }\nhy : \u2191y = (fst\u271d, snd\u271d).fst\nhx' : \u2191f y = (fst\u271d, snd\u271d).snd\n\u22a2 \u2203 y, \u2191y = (x_fst, x_snd).fst \u2227 \u2191(z \u2022 f) y = (x_fst, x_snd).snd\n[PROOFSTEP]\nuse y\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\nfst\u271d : E\nsnd\u271d : F\nh : fst\u271d = x_fst \u2227 z \u2022 snd\u271d = x_snd\ny : { x // x \u2208 f.domain }\nhy : \u2191y = (fst\u271d, snd\u271d).fst\nhx' : \u2191f y = (fst\u271d, snd\u271d).snd\n\u22a2 \u2191y = (x_fst, x_snd).fst \u2227 \u2191(z \u2022 f) y = (x_fst, x_snd).snd\n[PROOFSTEP]\nrw [\u2190 h.1, \u2190 h.2]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nz : M\nx_fst : E\nx_snd : F\nfst\u271d : E\nsnd\u271d : F\nh : fst\u271d = x_fst \u2227 z \u2022 snd\u271d = x_snd\ny : { x // x \u2208 f.domain }\nhy : \u2191y = (fst\u271d, snd\u271d).fst\nhx' : \u2191f y = (fst\u271d, snd\u271d).snd\n\u22a2 \u2191y = (fst\u271d, z \u2022 snd\u271d).fst \u2227 \u2191(z \u2022 f) y = (fst\u271d, z \u2022 snd\u271d).snd\n[PROOFSTEP]\nsimp [hy, hx']\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\n\u22a2 graph (-f) = Submodule.map (LinearMap.prodMap LinearMap.id (-LinearMap.id)) (graph f)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx : E \u00d7 F\n\u22a2 x \u2208 graph (-f) \u2194 x \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (-LinearMap.id)) (graph f)\n[PROOFSTEP]\ncases' x with x_fst x_snd\n[GOAL]\ncase h.mk\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\n\u22a2 (x_fst, x_snd) \u2208 graph (-f) \u2194\n (x_fst, x_snd) \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (-LinearMap.id)) (graph f)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mk.mp\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\n\u22a2 (x_fst, x_snd) \u2208 graph (-f) \u2192\n (x_fst, x_snd) \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (-LinearMap.id)) (graph f)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mk.mpr\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\n\u22a2 (x_fst, x_snd) \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (-LinearMap.id)) (graph f) \u2192\n (x_fst, x_snd) \u2208 graph (-f)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mk.mp\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\nh : (x_fst, x_snd) \u2208 graph (-f)\n\u22a2 (x_fst, x_snd) \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (-LinearMap.id)) (graph f)\n[PROOFSTEP]\nrw [mem_graph_iff] at h \n[GOAL]\ncase h.mk.mp\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\nh : \u2203 y, \u2191y = (x_fst, x_snd).fst \u2227 \u2191(-f) y = (x_fst, x_snd).snd\n\u22a2 (x_fst, x_snd) \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (-LinearMap.id)) (graph f)\n[PROOFSTEP]\nrcases h with \u27e8y, hy, h\u27e9\n[GOAL]\ncase h.mk.mp.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\ny : { x // x \u2208 (-f).domain }\nhy : \u2191y = (x_fst, x_snd).fst\nh : \u2191(-f) y = (x_fst, x_snd).snd\n\u22a2 (x_fst, x_snd) \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (-LinearMap.id)) (graph f)\n[PROOFSTEP]\nrw [LinearPMap.neg_apply] at h \n[GOAL]\ncase h.mk.mp.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\ny : { x // x \u2208 (-f).domain }\nhy : \u2191y = (x_fst, x_snd).fst\nh : -\u2191f y = (x_fst, x_snd).snd\n\u22a2 (x_fst, x_snd) \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (-LinearMap.id)) (graph f)\n[PROOFSTEP]\nrw [Submodule.mem_map]\n[GOAL]\ncase h.mk.mp.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\ny : { x // x \u2208 (-f).domain }\nhy : \u2191y = (x_fst, x_snd).fst\nh : -\u2191f y = (x_fst, x_snd).snd\n\u22a2 \u2203 y, y \u2208 graph f \u2227 \u2191(LinearMap.prodMap LinearMap.id (-LinearMap.id)) y = (x_fst, x_snd)\n[PROOFSTEP]\nsimp only [mem_graph_iff, LinearMap.prodMap_apply, LinearMap.id_coe, id.def, LinearMap.neg_apply, Prod.mk.inj_iff,\n Prod.exists, exists_exists_and_eq_and]\n[GOAL]\ncase h.mk.mp.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\ny : { x // x \u2208 (-f).domain }\nhy : \u2191y = (x_fst, x_snd).fst\nh : -\u2191f y = (x_fst, x_snd).snd\n\u22a2 \u2203 a a_1, \u2191a_1 = a \u2227 a = x_fst \u2227 -\u2191f a_1 = x_snd\n[PROOFSTEP]\nuse x_fst, y, hy\n[GOAL]\ncase h.mk.mpr\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\nh : (x_fst, x_snd) \u2208 Submodule.map (LinearMap.prodMap LinearMap.id (-LinearMap.id)) (graph f)\n\u22a2 (x_fst, x_snd) \u2208 graph (-f)\n[PROOFSTEP]\nrw [Submodule.mem_map] at h \n[GOAL]\ncase h.mk.mpr\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\nh : \u2203 y, y \u2208 graph f \u2227 \u2191(LinearMap.prodMap LinearMap.id (-LinearMap.id)) y = (x_fst, x_snd)\n\u22a2 (x_fst, x_snd) \u2208 graph (-f)\n[PROOFSTEP]\nrcases h with \u27e8x', hx', h\u27e9\n[GOAL]\ncase h.mk.mpr.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\nx' : E \u00d7 F\nhx' : x' \u2208 graph f\nh : \u2191(LinearMap.prodMap LinearMap.id (-LinearMap.id)) x' = (x_fst, x_snd)\n\u22a2 (x_fst, x_snd) \u2208 graph (-f)\n[PROOFSTEP]\ncases x'\n[GOAL]\ncase h.mk.mpr.intro.intro.mk\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\nfst\u271d : E\nsnd\u271d : F\nhx' : (fst\u271d, snd\u271d) \u2208 graph f\nh : \u2191(LinearMap.prodMap LinearMap.id (-LinearMap.id)) (fst\u271d, snd\u271d) = (x_fst, x_snd)\n\u22a2 (x_fst, x_snd) \u2208 graph (-f)\n[PROOFSTEP]\nsimp only [LinearMap.prodMap_apply, LinearMap.id_coe, id.def, LinearMap.neg_apply, Prod.mk.inj_iff] at h \n[GOAL]\ncase h.mk.mpr.intro.intro.mk\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\nfst\u271d : E\nsnd\u271d : F\nhx' : (fst\u271d, snd\u271d) \u2208 graph f\nh : fst\u271d = x_fst \u2227 -snd\u271d = x_snd\n\u22a2 (x_fst, x_snd) \u2208 graph (-f)\n[PROOFSTEP]\nrw [mem_graph_iff] at hx' \u22a2\n[GOAL]\ncase h.mk.mpr.intro.intro.mk\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\nfst\u271d : E\nsnd\u271d : F\nhx' : \u2203 y, \u2191y = (fst\u271d, snd\u271d).fst \u2227 \u2191f y = (fst\u271d, snd\u271d).snd\nh : fst\u271d = x_fst \u2227 -snd\u271d = x_snd\n\u22a2 \u2203 y, \u2191y = (x_fst, x_snd).fst \u2227 \u2191(-f) y = (x_fst, x_snd).snd\n[PROOFSTEP]\nrcases hx' with \u27e8y, hy, hx'\u27e9\n[GOAL]\ncase h.mk.mpr.intro.intro.mk.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\nfst\u271d : E\nsnd\u271d : F\nh : fst\u271d = x_fst \u2227 -snd\u271d = x_snd\ny : { x // x \u2208 f.domain }\nhy : \u2191y = (fst\u271d, snd\u271d).fst\nhx' : \u2191f y = (fst\u271d, snd\u271d).snd\n\u22a2 \u2203 y, \u2191y = (x_fst, x_snd).fst \u2227 \u2191(-f) y = (x_fst, x_snd).snd\n[PROOFSTEP]\nuse y\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\nfst\u271d : E\nsnd\u271d : F\nh : fst\u271d = x_fst \u2227 -snd\u271d = x_snd\ny : { x // x \u2208 f.domain }\nhy : \u2191y = (fst\u271d, snd\u271d).fst\nhx' : \u2191f y = (fst\u271d, snd\u271d).snd\n\u22a2 \u2191y = (x_fst, x_snd).fst \u2227 \u2191(-f) y = (x_fst, x_snd).snd\n[PROOFSTEP]\nrw [\u2190 h.1, \u2190 h.2]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx_fst : E\nx_snd : F\nfst\u271d : E\nsnd\u271d : F\nh : fst\u271d = x_fst \u2227 -snd\u271d = x_snd\ny : { x // x \u2208 f.domain }\nhy : \u2191y = (fst\u271d, snd\u271d).fst\nhx' : \u2191f y = (fst\u271d, snd\u271d).snd\n\u22a2 \u2191y = (fst\u271d, -snd\u271d).fst \u2227 \u2191(-f) y = (fst\u271d, -snd\u271d).snd\n[PROOFSTEP]\nsimp [hy, hx']\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx y : E\nx' y' : F\nhx : (x, x') \u2208 graph f\nhy : (y, y') \u2208 graph f\nhxy : x = y\n\u22a2 x' = y'\n[PROOFSTEP]\nrw [mem_graph_iff] at hx hy \n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx y : E\nx' y' : F\nhx : \u2203 y, \u2191y = (x, x').fst \u2227 \u2191f y = (x, x').snd\nhy : \u2203 y_1, \u2191y_1 = (y, y').fst \u2227 \u2191f y_1 = (y, y').snd\nhxy : x = y\n\u22a2 x' = y'\n[PROOFSTEP]\nrcases hx with \u27e8x'', hx1, hx2\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx y : E\nx' y' : F\nhy : \u2203 y_1, \u2191y_1 = (y, y').fst \u2227 \u2191f y_1 = (y, y').snd\nhxy : x = y\nx'' : { x // x \u2208 f.domain }\nhx1 : \u2191x'' = (x, x').fst\nhx2 : \u2191f x'' = (x, x').snd\n\u22a2 x' = y'\n[PROOFSTEP]\nrcases hy with \u27e8y'', hy1, hy2\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx y : E\nx' y' : F\nhxy : x = y\nx'' : { x // x \u2208 f.domain }\nhx1 : \u2191x'' = (x, x').fst\nhx2 : \u2191f x'' = (x, x').snd\ny'' : { x // x \u2208 f.domain }\nhy1 : \u2191y'' = (y, y').fst\nhy2 : \u2191f y'' = (y, y').snd\n\u22a2 x' = y'\n[PROOFSTEP]\nsimp only at hx1 hx2 hy1 hy2 \n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx y : E\nx' y' : F\nhxy : x = y\nx'' : { x // x \u2208 f.domain }\nhx1 : \u2191x'' = x\nhx2 : \u2191f x'' = x'\ny'' : { x // x \u2208 f.domain }\nhy1 : \u2191y'' = y\nhy2 : \u2191f y'' = y'\n\u22a2 x' = y'\n[PROOFSTEP]\nrw [\u2190 hx1, \u2190 hy1, SetLike.coe_eq_coe] at hxy \n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx y : E\nx' y' : F\nx'' : { x // x \u2208 f.domain }\nhx1 : \u2191x'' = x\nhx2 : \u2191f x'' = x'\ny'' : { x // x \u2208 f.domain }\nhxy : x'' = y''\nhy1 : \u2191y'' = y\nhy2 : \u2191f y'' = y'\n\u22a2 x' = y'\n[PROOFSTEP]\nrw [\u2190 hx2, \u2190 hy2, hxy]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx y : E \u00d7 F\nhx : x \u2208 graph f\nhy : y \u2208 graph f\nhxy : x.fst = y.fst\n\u22a2 x.snd = y.snd\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\ny : E \u00d7 F\nhy : y \u2208 graph f\nfst\u271d : E\nsnd\u271d : F\nhx : (fst\u271d, snd\u271d) \u2208 graph f\nhxy : (fst\u271d, snd\u271d).fst = y.fst\n\u22a2 (fst\u271d, snd\u271d).snd = y.snd\n[PROOFSTEP]\ncases y\n[GOAL]\ncase mk.mk\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nfst\u271d\u00b9 : E\nsnd\u271d\u00b9 : F\nhx : (fst\u271d\u00b9, snd\u271d\u00b9) \u2208 graph f\nfst\u271d : E\nsnd\u271d : F\nhy : (fst\u271d, snd\u271d) \u2208 graph f\nhxy : (fst\u271d\u00b9, snd\u271d\u00b9).fst = (fst\u271d, snd\u271d).fst\n\u22a2 (fst\u271d\u00b9, snd\u271d\u00b9).snd = (fst\u271d, snd\u271d).snd\n[PROOFSTEP]\nexact f.mem_graph_snd_inj hx hy hxy\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx : E\n\u22a2 x \u2208 f.domain \u2194 \u2203 y, (x, y) \u2208 graph f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx : E\n\u22a2 x \u2208 f.domain \u2192 \u2203 y, (x, y) \u2208 graph f\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx : E\n\u22a2 (\u2203 y, (x, y) \u2208 graph f) \u2192 x \u2208 f.domain\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx : E\nh : x \u2208 f.domain\n\u22a2 \u2203 y, (x, y) \u2208 graph f\n[PROOFSTEP]\nuse f \u27e8x, h\u27e9\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx : E\nh : x \u2208 f.domain\n\u22a2 (x, \u2191f { val := x, property := h }) \u2208 graph f\n[PROOFSTEP]\nexact f.mem_graph \u27e8x, h\u27e9\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf : E \u2192\u2097.[R] F\nx : E\nh : \u2203 y, (x, y) \u2208 graph f\n\u22a2 x \u2208 f.domain\n[PROOFSTEP]\ncases' h with y h\n[GOAL]\ncase mpr.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx : E\ny : F\nh : (x, y) \u2208 graph f\n\u22a2 x \u2208 f.domain\n[PROOFSTEP]\nrw [mem_graph_iff] at h \n[GOAL]\ncase mpr.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx : E\ny : F\nh : \u2203 y_1, \u2191y_1 = (x, y).fst \u2227 \u2191f y_1 = (x, y).snd\n\u22a2 x \u2208 f.domain\n[PROOFSTEP]\ncases' h with x' h\n[GOAL]\ncase mpr.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx : E\ny : F\nx' : { x // x \u2208 f.domain }\nh : \u2191x' = (x, y).fst \u2227 \u2191f x' = (x, y).snd\n\u22a2 x \u2208 f.domain\n[PROOFSTEP]\nsimp only at h \n[GOAL]\ncase mpr.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx : E\ny : F\nx' : { x // x \u2208 f.domain }\nh : \u2191x' = x \u2227 \u2191f x' = y\n\u22a2 x \u2208 f.domain\n[PROOFSTEP]\nrw [\u2190 h.1]\n[GOAL]\ncase mpr.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx : E\ny : F\nx' : { x // x \u2208 f.domain }\nh : \u2191x' = x \u2227 \u2191f x' = y\n\u22a2 \u2191x' \u2208 f.domain\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx : E\ny : F\nh : (x, y) \u2208 graph f\n\u22a2 x \u2208 f.domain\n[PROOFSTEP]\nrw [mem_domain_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx : E\ny : F\nh : (x, y) \u2208 graph f\n\u22a2 \u2203 y, (x, y) \u2208 graph f\n[PROOFSTEP]\nexact \u27e8y, h\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx : E\ny : F\nhx : x \u2208 f.domain\n\u22a2 y = \u2191f { val := x, property := hx } \u2194 (x, y) \u2208 graph f\n[PROOFSTEP]\nrw [mem_graph_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx : E\ny : F\nhx : x \u2208 f.domain\n\u22a2 y = \u2191f { val := x, property := hx } \u2194 \u2203 y_1, \u2191y_1 = (x, y).fst \u2227 \u2191f y_1 = (x, y).snd\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx : E\ny : F\nhx : x \u2208 f.domain\n\u22a2 y = \u2191f { val := x, property := hx } \u2192 \u2203 y_1, \u2191y_1 = (x, y).fst \u2227 \u2191f y_1 = (x, y).snd\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx : E\ny : F\nhx : x \u2208 f.domain\n\u22a2 (\u2203 y_1, \u2191y_1 = (x, y).fst \u2227 \u2191f y_1 = (x, y).snd) \u2192 y = \u2191f { val := x, property := hx }\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx : E\ny : F\nhx : x \u2208 f.domain\nh : y = \u2191f { val := x, property := hx }\n\u22a2 \u2203 y_1, \u2191y_1 = (x, y).fst \u2227 \u2191f y_1 = (x, y).snd\n[PROOFSTEP]\nuse\u27e8x, hx\u27e9\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx : E\ny : F\nhx : x \u2208 f.domain\nh : y = \u2191f { val := x, property := hx }\n\u22a2 \u2191{ val := x, property := hx } = (x, y).fst \u2227 \u2191f { val := x, property := hx } = (x, y).snd\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx : E\ny : F\nhx : x \u2208 f.domain\nh : \u2203 y_1, \u2191y_1 = (x, y).fst \u2227 \u2191f y_1 = (x, y).snd\n\u22a2 y = \u2191f { val := x, property := hx }\n[PROOFSTEP]\nrcases h with \u27e8\u27e8x', hx'\u27e9, \u27e8h1, h2\u27e9\u27e9\n[GOAL]\ncase mpr.intro.mk.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx : E\ny : F\nhx : x \u2208 f.domain\nx' : E\nhx' : x' \u2208 f.domain\nh1 : \u2191{ val := x', property := hx' } = (x, y).fst\nh2 : \u2191f { val := x', property := hx' } = (x, y).snd\n\u22a2 y = \u2191f { val := x, property := hx }\n[PROOFSTEP]\nsimp only [Submodule.coe_mk] at h1 h2 \n[GOAL]\ncase mpr.intro.mk.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\nx : E\ny : F\nhx : x \u2208 f.domain\nx' : E\nhx' : x' \u2208 f.domain\nh1 : x' = x\nh2 : \u2191f { val := x', property := hx' } = y\n\u22a2 y = \u2191f { val := x, property := hx }\n[PROOFSTEP]\nsimp only [\u2190 h2, h1]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\ny : F\n\u22a2 y \u2208 range \u2191f \u2194 \u2203 x, (x, y) \u2208 graph f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\ny : F\n\u22a2 y \u2208 range \u2191f \u2192 \u2203 x, (x, y) \u2208 graph f\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\ny : F\n\u22a2 (\u2203 x, (x, y) \u2208 graph f) \u2192 y \u2208 range \u2191f\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\ny : F\nh : y \u2208 range \u2191f\n\u22a2 \u2203 x, (x, y) \u2208 graph f\n[PROOFSTEP]\nrw [Set.mem_range] at h \n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\ny : F\nh : \u2203 y_1, \u2191f y_1 = y\n\u22a2 \u2203 x, (x, y) \u2208 graph f\n[PROOFSTEP]\nrcases h with \u27e8\u27e8x, hx\u27e9, h\u27e9\n[GOAL]\ncase mp.intro.mk\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\ny : F\nx : E\nhx : x \u2208 f.domain\nh : \u2191f { val := x, property := hx } = y\n\u22a2 \u2203 x, (x, y) \u2208 graph f\n[PROOFSTEP]\nuse x\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\ny : F\nx : E\nhx : x \u2208 f.domain\nh : \u2191f { val := x, property := hx } = y\n\u22a2 (x, y) \u2208 graph f\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\ny : F\nx : E\nhx : x \u2208 f.domain\nh : \u2191f { val := x, property := hx } = y\n\u22a2 (x, \u2191f { val := x, property := hx }) \u2208 graph f\n[PROOFSTEP]\nexact f.mem_graph \u27e8x, hx\u27e9\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\ny : F\nh : \u2203 x, (x, y) \u2208 graph f\n\u22a2 y \u2208 range \u2191f\n[PROOFSTEP]\ncases' h with x h\n[GOAL]\ncase mpr.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\ny : F\nx : E\nh : (x, y) \u2208 graph f\n\u22a2 y \u2208 range \u2191f\n[PROOFSTEP]\nrw [mem_graph_iff] at h \n[GOAL]\ncase mpr.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\ny : F\nx : E\nh : \u2203 y_1, \u2191y_1 = (x, y).fst \u2227 \u2191f y_1 = (x, y).snd\n\u22a2 y \u2208 range \u2191f\n[PROOFSTEP]\ncases' h with x h\n[GOAL]\ncase mpr.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\ny : F\nx\u271d : E\nx : { x // x \u2208 f.domain }\nh : \u2191x = (x\u271d, y).fst \u2227 \u2191f x = (x\u271d, y).snd\n\u22a2 y \u2208 range \u2191f\n[PROOFSTEP]\nrw [Set.mem_range]\n[GOAL]\ncase mpr.intro.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\ny : F\nx\u271d : E\nx : { x // x \u2208 f.domain }\nh : \u2191x = (x\u271d, y).fst \u2227 \u2191f x = (x\u271d, y).snd\n\u22a2 \u2203 y_1, \u2191f y_1 = y\n[PROOFSTEP]\nuse x\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\ny : F\nx\u271d : E\nx : { x // x \u2208 f.domain }\nh : \u2191x = (x\u271d, y).fst \u2227 \u2191f x = (x\u271d, y).snd\n\u22a2 \u2191f x = y\n[PROOFSTEP]\nsimp only at h \n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf : E \u2192\u2097.[R] F\ny : F\nx\u271d : E\nx : { x // x \u2208 f.domain }\nh : \u2191x = x\u271d \u2227 \u2191f x = y\n\u22a2 \u2191f x = y\n[PROOFSTEP]\nrw [h.2]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf g : E \u2192\u2097.[R] F\nh : graph f = graph g\nx : E\n\u22a2 x \u2208 f.domain \u2194 x \u2208 g.domain\n[PROOFSTEP]\nsimp_rw [mem_domain_iff, h]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf g : E \u2192\u2097.[R] F\nh : graph f \u2264 graph g\n\u22a2 f \u2264 g\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf g : E \u2192\u2097.[R] F\nh : graph f \u2264 graph g\n\u22a2 f.domain \u2264 g.domain\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase left\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf g : E \u2192\u2097.[R] F\nh : graph f \u2264 graph g\nx : E\nhx : x \u2208 f.domain\n\u22a2 x \u2208 g.domain\n[PROOFSTEP]\nrw [mem_domain_iff] at hx \u22a2\n[GOAL]\ncase left\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf g : E \u2192\u2097.[R] F\nh : graph f \u2264 graph g\nx : E\nhx : \u2203 y, (x, y) \u2208 graph f\n\u22a2 \u2203 y, (x, y) \u2208 graph g\n[PROOFSTEP]\ncases' hx with y hx\n[GOAL]\ncase left.intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf g : E \u2192\u2097.[R] F\nh : graph f \u2264 graph g\nx : E\ny : F\nhx : (x, y) \u2208 graph f\n\u22a2 \u2203 y, (x, y) \u2208 graph g\n[PROOFSTEP]\nuse y\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf g : E \u2192\u2097.[R] F\nh : graph f \u2264 graph g\nx : E\ny : F\nhx : (x, y) \u2208 graph f\n\u22a2 (x, y) \u2208 graph g\n[PROOFSTEP]\nexact h hx\n[GOAL]\ncase right\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf g : E \u2192\u2097.[R] F\nh : graph f \u2264 graph g\n\u22a2 \u2200 \u2983x : { x // x \u2208 f.domain }\u2984 \u2983y : { x // x \u2208 g.domain }\u2984, \u2191x = \u2191y \u2192 \u2191f x = \u2191g y\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9 \u27e8y, hy\u27e9 hxy\n[GOAL]\ncase right.mk.mk\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf g : E \u2192\u2097.[R] F\nh : graph f \u2264 graph g\nx : E\nhx : x \u2208 f.domain\ny : E\nhy : y \u2208 g.domain\nhxy : \u2191{ val := x, property := hx } = \u2191{ val := y, property := hy }\n\u22a2 \u2191f { val := x, property := hx } = \u2191g { val := y, property := hy }\n[PROOFSTEP]\nrw [image_iff]\n[GOAL]\ncase right.mk.mk\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf g : E \u2192\u2097.[R] F\nh : graph f \u2264 graph g\nx : E\nhx : x \u2208 f.domain\ny : E\nhy : y \u2208 g.domain\nhxy : \u2191{ val := x, property := hx } = \u2191{ val := y, property := hy }\n\u22a2 (y, \u2191f { val := x, property := hx }) \u2208 graph g\n[PROOFSTEP]\nrefine' h _\n[GOAL]\ncase right.mk.mk\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf g : E \u2192\u2097.[R] F\nh : graph f \u2264 graph g\nx : E\nhx : x \u2208 f.domain\ny : E\nhy : y \u2208 g.domain\nhxy : \u2191{ val := x, property := hx } = \u2191{ val := y, property := hy }\n\u22a2 (y, \u2191f { val := x, property := hx }) \u2208 graph f\n[PROOFSTEP]\nsimp only [Submodule.coe_mk] at hxy \n[GOAL]\ncase right.mk.mk\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf g : E \u2192\u2097.[R] F\nh : graph f \u2264 graph g\nx : E\nhx : x \u2208 f.domain\ny : E\nhy : y \u2208 g.domain\nhxy : x = y\n\u22a2 (y, \u2191f { val := x, property := hx }) \u2208 graph f\n[PROOFSTEP]\nrw [hxy] at hx \n[GOAL]\ncase right.mk.mk\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf g : E \u2192\u2097.[R] F\nh : graph f \u2264 graph g\nx : E\nhx\u271d : x \u2208 f.domain\ny : E\nhx : y \u2208 f.domain\nhy : y \u2208 g.domain\nhxy : x = y\n\u22a2 (y, \u2191f { val := x, property := hx\u271d }) \u2208 graph f\n[PROOFSTEP]\nrw [\u2190 image_iff hx]\n[GOAL]\ncase right.mk.mk\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf g : E \u2192\u2097.[R] F\nh : graph f \u2264 graph g\nx : E\nhx\u271d : x \u2208 f.domain\ny : E\nhx : y \u2208 f.domain\nhy : y \u2208 g.domain\nhxy : x = y\n\u22a2 \u2191f { val := x, property := hx\u271d } = \u2191f { val := y, property := hx }\n[PROOFSTEP]\nsimp [hxy]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf g : E \u2192\u2097.[R] F\nh : f \u2264 g\n\u22a2 graph f \u2264 graph g\n[PROOFSTEP]\nintro x hx\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf g : E \u2192\u2097.[R] F\nh : f \u2264 g\nx : E \u00d7 F\nhx : x \u2208 graph f\n\u22a2 x \u2208 graph g\n[PROOFSTEP]\nrw [mem_graph_iff] at hx \u22a2\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf g : E \u2192\u2097.[R] F\nh : f \u2264 g\nx : E \u00d7 F\nhx : \u2203 y, \u2191y = x.fst \u2227 \u2191f y = x.snd\n\u22a2 \u2203 y, \u2191y = x.fst \u2227 \u2191g y = x.snd\n[PROOFSTEP]\ncases' hx with y hx\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf g : E \u2192\u2097.[R] F\nh : f \u2264 g\nx : E \u00d7 F\ny : { x // x \u2208 f.domain }\nhx : \u2191y = x.fst \u2227 \u2191f y = x.snd\n\u22a2 \u2203 y, \u2191y = x.fst \u2227 \u2191g y = x.snd\n[PROOFSTEP]\nuse\u27e8y, h.1 y.2\u27e9\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf g : E \u2192\u2097.[R] F\nh : f \u2264 g\nx : E \u00d7 F\ny : { x // x \u2208 f.domain }\nhx : \u2191y = x.fst \u2227 \u2191f y = x.snd\n\u22a2 \u2191{ val := \u2191y, property := (_ : \u2191y \u2208 g.domain) } = x.fst \u2227 \u2191g { val := \u2191y, property := (_ : \u2191y \u2208 g.domain) } = x.snd\n[PROOFSTEP]\nsimp only [hx, Submodule.coe_mk, eq_self_iff_true, true_and_iff]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf g : E \u2192\u2097.[R] F\nh : f \u2264 g\nx : E \u00d7 F\ny : { x // x \u2208 f.domain }\nhx : \u2191y = x.fst \u2227 \u2191f y = x.snd\n\u22a2 \u2191g { val := x.fst, property := (_ : (fun x => x \u2208 g.domain) x.fst) } = x.snd\n[PROOFSTEP]\nconvert hx.2 using 1\n[GOAL]\ncase h.e'_2\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf g : E \u2192\u2097.[R] F\nh : f \u2264 g\nx : E \u00d7 F\ny : { x // x \u2208 f.domain }\nhx : \u2191y = x.fst \u2227 \u2191f y = x.snd\n\u22a2 \u2191g { val := x.fst, property := (_ : (fun x => x \u2208 g.domain) x.fst) } = \u2191f y\n[PROOFSTEP]\nrefine' (h.2 _).symm\n[GOAL]\ncase h.e'_2\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf g : E \u2192\u2097.[R] F\nh : f \u2264 g\nx : E \u00d7 F\ny : { x // x \u2208 f.domain }\nhx : \u2191y = x.fst \u2227 \u2191f y = x.snd\n\u22a2 \u2191y = \u2191{ val := x.fst, property := (_ : (fun x => x \u2208 g.domain) x.fst) }\n[PROOFSTEP]\nsimp only [hx.1, Submodule.coe_mk]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf g : E \u2192\u2097.[R] F\nh : graph f = graph g\n\u22a2 f = g\n[PROOFSTEP]\nrefine ext (Submodule.ext fun x => ?_) (fun x y h' => ?_)\n[GOAL]\ncase refine_1\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny : M\nf g : E \u2192\u2097.[R] F\nh : graph f = graph g\nx : E\n\u22a2 x \u2208 f.domain \u2194 x \u2208 g.domain\ncase refine_2\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf g : E \u2192\u2097.[R] F\nh : graph f = graph g\nx : { x // x \u2208 f.domain }\ny : { x // x \u2208 g.domain }\nh' : \u2191x = \u2191y\n\u22a2 \u2191f x = \u2191g y\n[PROOFSTEP]\nexact mem_domain_iff_of_eq_graph h\n[GOAL]\ncase refine_2\nR : Type u_1\ninst\u271d\u2079 : Ring R\nE : Type u_2\ninst\u271d\u2078 : AddCommGroup E\ninst\u271d\u2077 : Module R E\nF : Type u_3\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : Module R F\nG : Type u_4\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module R G\nM : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : DistribMulAction M F\ninst\u271d : SMulCommClass R M F\ny\u271d : M\nf g : E \u2192\u2097.[R] F\nh : graph f = graph g\nx : { x // x \u2208 f.domain }\ny : { x // x \u2208 g.domain }\nh' : \u2191x = \u2191y\n\u22a2 \u2191f x = \u2191g y\n[PROOFSTEP]\nexact (le_of_le_graph h.le).2 h'\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 {x : E \u00d7 F}, x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\na : E\nha : a \u2208 map (LinearMap.fst R E F) g\n\u22a2 \u2203! b, (a, b) \u2208 g\n[PROOFSTEP]\nrefine' exists_unique_of_exists_of_unique _ _\n[GOAL]\ncase refine'_1\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 {x : E \u00d7 F}, x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\na : E\nha : a \u2208 map (LinearMap.fst R E F) g\n\u22a2 \u2203 x, (a, x) \u2208 g\n[PROOFSTEP]\nconvert ha\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 {x : E \u00d7 F}, x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\na : E\nha : a \u2208 map (LinearMap.fst R E F) g\n\u22a2 (\u2203 x, (a, x) \u2208 g) \u2194 a \u2208 map (LinearMap.fst R E F) g\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 {x : E \u00d7 F}, x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\na : E\nha : a \u2208 map (LinearMap.fst R E F) g\n\u22a2 \u2200 (y\u2081 y\u2082 : F), (a, y\u2081) \u2208 g \u2192 (a, y\u2082) \u2208 g \u2192 y\u2081 = y\u2082\n[PROOFSTEP]\nintro y\u2081 y\u2082 hy\u2081 hy\u2082\n[GOAL]\ncase refine'_2\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 {x : E \u00d7 F}, x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\na : E\nha : a \u2208 map (LinearMap.fst R E F) g\ny\u2081 y\u2082 : F\nhy\u2081 : (a, y\u2081) \u2208 g\nhy\u2082 : (a, y\u2082) \u2208 g\n\u22a2 y\u2081 = y\u2082\n[PROOFSTEP]\nhave hy : ((0 : E), y\u2081 - y\u2082) \u2208 g := by\n convert g.sub_mem hy\u2081 hy\u2082\n exact (sub_self _).symm\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 {x : E \u00d7 F}, x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\na : E\nha : a \u2208 map (LinearMap.fst R E F) g\ny\u2081 y\u2082 : F\nhy\u2081 : (a, y\u2081) \u2208 g\nhy\u2082 : (a, y\u2082) \u2208 g\n\u22a2 (0, y\u2081 - y\u2082) \u2208 g\n[PROOFSTEP]\nconvert g.sub_mem hy\u2081 hy\u2082\n[GOAL]\ncase h.e'_4.h.e'_3\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 {x : E \u00d7 F}, x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\na : E\nha : a \u2208 map (LinearMap.fst R E F) g\ny\u2081 y\u2082 : F\nhy\u2081 : (a, y\u2081) \u2208 g\nhy\u2082 : (a, y\u2082) \u2208 g\n\u22a2 0 = ((a, y\u2081) - (a, y\u2082)).fst\n[PROOFSTEP]\nexact (sub_self _).symm\n[GOAL]\ncase refine'_2\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 {x : E \u00d7 F}, x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\na : E\nha : a \u2208 map (LinearMap.fst R E F) g\ny\u2081 y\u2082 : F\nhy\u2081 : (a, y\u2081) \u2208 g\nhy\u2082 : (a, y\u2082) \u2208 g\nhy : (0, y\u2081 - y\u2082) \u2208 g\n\u22a2 y\u2081 = y\u2082\n[PROOFSTEP]\nexact sub_eq_zero.mp (hg hy (by simp))\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 {x : E \u00d7 F}, x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\na : E\nha : a \u2208 map (LinearMap.fst R E F) g\ny\u2081 y\u2082 : F\nhy\u2081 : (a, y\u2081) \u2208 g\nhy\u2082 : (a, y\u2082) \u2208 g\nhy : (0, y\u2081 - y\u2082) \u2208 g\n\u22a2 (0, y\u2081 - y\u2082).fst = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nv w : { x // x \u2208 map (LinearMap.fst R E F) g }\n\u22a2 (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) (v + w) =\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) v +\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) w\n[PROOFSTEP]\nhave hadd := (g.map (LinearMap.fst R E F)).add_mem v.2 w.2\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nv w : { x // x \u2208 map (LinearMap.fst R E F) g }\nhadd : \u2191v + \u2191w \u2208 map (LinearMap.fst R E F) g\n\u22a2 (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) (v + w) =\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) v +\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) w\n[PROOFSTEP]\nhave hvw := valFromGraph_mem hg hadd\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nv w : { x // x \u2208 map (LinearMap.fst R E F) g }\nhadd : \u2191v + \u2191w \u2208 map (LinearMap.fst R E F) g\nhvw : (\u2191v + \u2191w, valFromGraph hg hadd) \u2208 g\n\u22a2 (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) (v + w) =\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) v +\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) w\n[PROOFSTEP]\nhave hvw' := g.add_mem (valFromGraph_mem hg v.2) (valFromGraph_mem hg w.2)\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nv w : { x // x \u2208 map (LinearMap.fst R E F) g }\nhadd : \u2191v + \u2191w \u2208 map (LinearMap.fst R E F) g\nhvw : (\u2191v + \u2191w, valFromGraph hg hadd) \u2208 g\nhvw' :\n (\u2191v, valFromGraph hg (_ : \u2191v \u2208 map (LinearMap.fst R E F) g)) +\n (\u2191w, valFromGraph hg (_ : \u2191w \u2208 map (LinearMap.fst R E F) g)) \u2208\n g\n\u22a2 (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) (v + w) =\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) v +\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) w\n[PROOFSTEP]\nrw [Prod.mk_add_mk] at hvw' \n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nv w : { x // x \u2208 map (LinearMap.fst R E F) g }\nhadd : \u2191v + \u2191w \u2208 map (LinearMap.fst R E F) g\nhvw : (\u2191v + \u2191w, valFromGraph hg hadd) \u2208 g\nhvw' :\n (\u2191v + \u2191w,\n valFromGraph hg (_ : \u2191v \u2208 map (LinearMap.fst R E F) g) + valFromGraph hg (_ : \u2191w \u2208 map (LinearMap.fst R E F) g)) \u2208\n g\n\u22a2 (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) (v + w) =\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) v +\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) w\n[PROOFSTEP]\nexact (existsUnique_from_graph (@hg) hadd).unique hvw hvw'\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\na : R\nv : { x // x \u2208 map (LinearMap.fst R E F) g }\n\u22a2 AddHom.toFun\n { toFun := fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g),\n map_add' :=\n (_ :\n \u2200 (v w : { x // x \u2208 map (LinearMap.fst R E F) g }),\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) (v + w) =\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) v +\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) w) }\n (a \u2022 v) =\n \u2191(RingHom.id R) a \u2022\n AddHom.toFun\n { toFun := fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g),\n map_add' :=\n (_ :\n \u2200 (v w : { x // x \u2208 map (LinearMap.fst R E F) g }),\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) (v + w) =\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) v +\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) w) }\n v\n[PROOFSTEP]\nhave hsmul := (g.map (LinearMap.fst R E F)).smul_mem a v.2\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\na : R\nv : { x // x \u2208 map (LinearMap.fst R E F) g }\nhsmul : a \u2022 \u2191v \u2208 map (LinearMap.fst R E F) g\n\u22a2 AddHom.toFun\n { toFun := fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g),\n map_add' :=\n (_ :\n \u2200 (v w : { x // x \u2208 map (LinearMap.fst R E F) g }),\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) (v + w) =\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) v +\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) w) }\n (a \u2022 v) =\n \u2191(RingHom.id R) a \u2022\n AddHom.toFun\n { toFun := fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g),\n map_add' :=\n (_ :\n \u2200 (v w : { x // x \u2208 map (LinearMap.fst R E F) g }),\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) (v + w) =\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) v +\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) w) }\n v\n[PROOFSTEP]\nhave hav := valFromGraph_mem hg hsmul\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\na : R\nv : { x // x \u2208 map (LinearMap.fst R E F) g }\nhsmul : a \u2022 \u2191v \u2208 map (LinearMap.fst R E F) g\nhav : (a \u2022 \u2191v, valFromGraph hg hsmul) \u2208 g\n\u22a2 AddHom.toFun\n { toFun := fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g),\n map_add' :=\n (_ :\n \u2200 (v w : { x // x \u2208 map (LinearMap.fst R E F) g }),\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) (v + w) =\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) v +\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) w) }\n (a \u2022 v) =\n \u2191(RingHom.id R) a \u2022\n AddHom.toFun\n { toFun := fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g),\n map_add' :=\n (_ :\n \u2200 (v w : { x // x \u2208 map (LinearMap.fst R E F) g }),\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) (v + w) =\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) v +\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) w) }\n v\n[PROOFSTEP]\nhave hav' := g.smul_mem a (valFromGraph_mem hg v.2)\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\na : R\nv : { x // x \u2208 map (LinearMap.fst R E F) g }\nhsmul : a \u2022 \u2191v \u2208 map (LinearMap.fst R E F) g\nhav : (a \u2022 \u2191v, valFromGraph hg hsmul) \u2208 g\nhav' : a \u2022 (\u2191v, valFromGraph hg (_ : \u2191v \u2208 map (LinearMap.fst R E F) g)) \u2208 g\n\u22a2 AddHom.toFun\n { toFun := fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g),\n map_add' :=\n (_ :\n \u2200 (v w : { x // x \u2208 map (LinearMap.fst R E F) g }),\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) (v + w) =\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) v +\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) w) }\n (a \u2022 v) =\n \u2191(RingHom.id R) a \u2022\n AddHom.toFun\n { toFun := fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g),\n map_add' :=\n (_ :\n \u2200 (v w : { x // x \u2208 map (LinearMap.fst R E F) g }),\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) (v + w) =\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) v +\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) w) }\n v\n[PROOFSTEP]\nrw [Prod.smul_mk] at hav' \n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\na : R\nv : { x // x \u2208 map (LinearMap.fst R E F) g }\nhsmul : a \u2022 \u2191v \u2208 map (LinearMap.fst R E F) g\nhav : (a \u2022 \u2191v, valFromGraph hg hsmul) \u2208 g\nhav' : (a \u2022 \u2191v, a \u2022 valFromGraph hg (_ : \u2191v \u2208 map (LinearMap.fst R E F) g)) \u2208 g\n\u22a2 AddHom.toFun\n { toFun := fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g),\n map_add' :=\n (_ :\n \u2200 (v w : { x // x \u2208 map (LinearMap.fst R E F) g }),\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) (v + w) =\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) v +\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) w) }\n (a \u2022 v) =\n \u2191(RingHom.id R) a \u2022\n AddHom.toFun\n { toFun := fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g),\n map_add' :=\n (_ :\n \u2200 (v w : { x // x \u2208 map (LinearMap.fst R E F) g }),\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) (v + w) =\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) v +\n (fun x => valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) w) }\n v\n[PROOFSTEP]\nexact (existsUnique_from_graph (@hg) hsmul).unique hav hav'\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx : { x // x \u2208 map (LinearMap.fst R E F) g }\n\u22a2 \u2191(toLinearPMap g) x = valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)\n[PROOFSTEP]\nclassical\nchange (if hg : _ then g.toLinearPMapAux hg else 0) x = _\nrw [dif_pos]\n\u00b7 rfl\n\u00b7 exact hg\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx : { x // x \u2208 map (LinearMap.fst R E F) g }\n\u22a2 \u2191(toLinearPMap g) x = valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)\n[PROOFSTEP]\nchange (if hg : _ then g.toLinearPMapAux hg else 0) x = _\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx : { x // x \u2208 map (LinearMap.fst R E F) g }\n\u22a2 \u2191(if hg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0 then toLinearPMapAux g hg else 0) x =\n valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)\n[PROOFSTEP]\nrw [dif_pos]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx : { x // x \u2208 map (LinearMap.fst R E F) g }\n\u22a2 \u2191(toLinearPMapAux g ?hc) x = valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase hc\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx : { x // x \u2208 map (LinearMap.fst R E F) g }\n\u22a2 \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\n[PROOFSTEP]\nexact hg\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx : { x // x \u2208 map (LinearMap.fst R E F) g }\n\u22a2 (\u2191x, \u2191(toLinearPMap g) x) \u2208 g\n[PROOFSTEP]\nrw [toLinearPMap_apply_aux hg]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx : { x // x \u2208 map (LinearMap.fst R E F) g }\n\u22a2 (\u2191x, valFromGraph hg (_ : \u2191x \u2208 map (LinearMap.fst R E F) g)) \u2208 g\n[PROOFSTEP]\nexact valFromGraph_mem hg x.2\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\n\u22a2 LinearPMap.graph (toLinearPMap g) = g\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx : E \u00d7 F\n\u22a2 x \u2208 LinearPMap.graph (toLinearPMap g) \u2194 x \u2208 g\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx : E \u00d7 F\n\u22a2 x \u2208 LinearPMap.graph (toLinearPMap g) \u2192 x \u2208 g\n[PROOFSTEP]\nintro hx\n[GOAL]\ncase h.mpr\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx : E \u00d7 F\n\u22a2 x \u2208 g \u2192 x \u2208 LinearPMap.graph (toLinearPMap g)\n[PROOFSTEP]\nintro hx\n[GOAL]\ncase h.mp\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx : E \u00d7 F\nhx : x \u2208 LinearPMap.graph (toLinearPMap g)\n\u22a2 x \u2208 g\n[PROOFSTEP]\nrw [LinearPMap.mem_graph_iff] at hx \n[GOAL]\ncase h.mp\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx : E \u00d7 F\nhx : \u2203 y, \u2191y = x.fst \u2227 \u2191(toLinearPMap g) y = x.snd\n\u22a2 x \u2208 g\n[PROOFSTEP]\nrcases hx with \u27e8y, hx1, hx2\u27e9\n[GOAL]\ncase h.mp.intro.intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx : E \u00d7 F\ny : { x // x \u2208 (toLinearPMap g).domain }\nhx1 : \u2191y = x.fst\nhx2 : \u2191(toLinearPMap g) y = x.snd\n\u22a2 x \u2208 g\n[PROOFSTEP]\nconvert g.mem_graph_toLinearPMap hg y using 1\n[GOAL]\ncase h.e'_4\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx : E \u00d7 F\ny : { x // x \u2208 (toLinearPMap g).domain }\nhx1 : \u2191y = x.fst\nhx2 : \u2191(toLinearPMap g) y = x.snd\n\u22a2 x = (\u2191y, \u2191(toLinearPMap g) y)\n[PROOFSTEP]\nexact Prod.ext hx1.symm hx2.symm\n[GOAL]\ncase h.mpr\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx : E \u00d7 F\nhx : x \u2208 g\n\u22a2 x \u2208 LinearPMap.graph (toLinearPMap g)\n[PROOFSTEP]\nrw [LinearPMap.mem_graph_iff]\n[GOAL]\ncase h.mpr\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx : E \u00d7 F\nhx : x \u2208 g\n\u22a2 \u2203 y, \u2191y = x.fst \u2227 \u2191(toLinearPMap g) y = x.snd\n[PROOFSTEP]\ncases' x with x_fst x_snd\n[GOAL]\ncase h.mpr.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx_fst : E\nx_snd : F\nhx : (x_fst, x_snd) \u2208 g\n\u22a2 \u2203 y, \u2191y = (x_fst, x_snd).fst \u2227 \u2191(toLinearPMap g) y = (x_fst, x_snd).snd\n[PROOFSTEP]\nhave hx_fst : x_fst \u2208 g.map (LinearMap.fst R E F) :=\n by\n simp only [mem_map, LinearMap.fst_apply, Prod.exists, exists_and_right, exists_eq_right]\n exact \u27e8x_snd, hx\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx_fst : E\nx_snd : F\nhx : (x_fst, x_snd) \u2208 g\n\u22a2 x_fst \u2208 map (LinearMap.fst R E F) g\n[PROOFSTEP]\nsimp only [mem_map, LinearMap.fst_apply, Prod.exists, exists_and_right, exists_eq_right]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx_fst : E\nx_snd : F\nhx : (x_fst, x_snd) \u2208 g\n\u22a2 \u2203 x, (x_fst, x) \u2208 g\n[PROOFSTEP]\nexact \u27e8x_snd, hx\u27e9\n[GOAL]\ncase h.mpr.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx_fst : E\nx_snd : F\nhx : (x_fst, x_snd) \u2208 g\nhx_fst : x_fst \u2208 map (LinearMap.fst R E F) g\n\u22a2 \u2203 y, \u2191y = (x_fst, x_snd).fst \u2227 \u2191(toLinearPMap g) y = (x_fst, x_snd).snd\n[PROOFSTEP]\nrefine' \u27e8\u27e8x_fst, hx_fst\u27e9, Subtype.coe_mk x_fst hx_fst, _\u27e9\n[GOAL]\ncase h.mpr.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx_fst : E\nx_snd : F\nhx : (x_fst, x_snd) \u2208 g\nhx_fst : x_fst \u2208 map (LinearMap.fst R E F) g\n\u22a2 \u2191(toLinearPMap g) { val := x_fst, property := hx_fst } = (x_fst, x_snd).snd\n[PROOFSTEP]\nrw [toLinearPMap_apply_aux hg]\n[GOAL]\ncase h.mpr.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\nx_fst : E\nx_snd : F\nhx : (x_fst, x_snd) \u2208 g\nhx_fst : x_fst \u2208 map (LinearMap.fst R E F) g\n\u22a2 valFromGraph hg (_ : \u2191{ val := x_fst, property := hx_fst } \u2208 map (LinearMap.fst R E F) g) = (x_fst, x_snd).snd\n[PROOFSTEP]\nexact (existsUnique_from_graph (@hg) hx_fst).unique (valFromGraph_mem hg hx_fst) hx\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.fst = 0 \u2192 x.snd = 0\n\u22a2 LinearMap.range (toLinearPMap g).toFun = map (LinearMap.snd R E F) g\n[PROOFSTEP]\nrwa [\u2190 LinearPMap.graph_map_snd_eq_range, toLinearPMap_graph_eq]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\n\u22a2 (inverse f).domain = LinearMap.range f.toFun\n[PROOFSTEP]\nrw [inverse, Submodule.toLinearPMap_domain, \u2190 graph_map_snd_eq_range, \u2190 LinearEquiv.fst_comp_prodComm,\n Submodule.map_comp]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\n\u22a2 Submodule.map (LinearMap.fst R F E) (Submodule.map (LinearEquiv.prodComm R E F) (graph f)) =\n Submodule.map (LinearMap.fst R F E) (Submodule.map (\u2191(LinearEquiv.prodComm R E F)) (graph f))\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\nx : F \u00d7 E\nhv : x \u2208 Submodule.map (LinearEquiv.prodComm R E F) (graph f)\nhv' : x.fst = 0\n\u22a2 x.snd = 0\n[PROOFSTEP]\nsimp only [Submodule.mem_map, mem_graph_iff, Subtype.exists, exists_and_left, exists_eq_left,\n LinearEquiv.prodComm_apply, Prod.exists, Prod.swap_prod_mk] at hv \n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\nx : F \u00d7 E\nhv' : x.fst = 0\nhv : \u2203 a b, (\u2203 h, \u2191f { val := a, property := (_ : (a, b).fst \u2208 f.domain) } = b) \u2227 (b, a) = x\n\u22a2 x.snd = 0\n[PROOFSTEP]\nrcases hv with \u27e8a, b, \u27e8ha, h1\u27e9, \u27e8h2, h3\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refl\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\na : E\nb : F\nha : a \u2208 f.domain\nh1 : \u2191f { val := a, property := (_ : (a, b).fst \u2208 f.domain) } = b\nhv' : (b, a).fst = 0\n\u22a2 (b, a).snd = 0\n[PROOFSTEP]\nsimp only at hv' \u22a2\n[GOAL]\ncase intro.intro.intro.intro.refl\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\na : E\nb : F\nha : a \u2208 f.domain\nh1 : \u2191f { val := a, property := (_ : (a, b).fst \u2208 f.domain) } = b\nhv' : b = 0\n\u22a2 a = 0\n[PROOFSTEP]\nrw [hv'] at h1 \n[GOAL]\ncase intro.intro.intro.intro.refl\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\na : E\nb : F\nha : a \u2208 f.domain\nh1 : \u2191f { val := a, property := (_ : (a, 0).fst \u2208 f.domain) } = 0\nhv' : b = 0\n\u22a2 a = 0\n[PROOFSTEP]\nrw [LinearMap.ker_eq_bot'] at hf \n[GOAL]\ncase intro.intro.intro.intro.refl\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : \u2200 (m : { x // x \u2208 f.domain }), \u2191f.toFun m = 0 \u2192 m = 0\na : E\nb : F\nha : a \u2208 f.domain\nh1 : \u2191f { val := a, property := (_ : (a, 0).fst \u2208 f.domain) } = 0\nhv' : b = 0\n\u22a2 a = 0\n[PROOFSTEP]\nspecialize hf \u27e8a, ha\u27e9 h1\n[GOAL]\ncase intro.intro.intro.intro.refl\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\na : E\nb : F\nha : a \u2208 f.domain\nh1 : \u2191f { val := a, property := (_ : (a, 0).fst \u2208 f.domain) } = 0\nhv' : b = 0\nhf : { val := a, property := ha } = 0\n\u22a2 a = 0\n[PROOFSTEP]\nsimp only [Submodule.mk_eq_zero] at hf \n[GOAL]\ncase intro.intro.intro.intro.refl\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\na : E\nb : F\nha : a \u2208 f.domain\nh1 : \u2191f { val := a, property := (_ : (a, 0).fst \u2208 f.domain) } = 0\nhv' : b = 0\nhf : a = 0\n\u22a2 a = 0\n[PROOFSTEP]\nexact hf\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\n\u22a2 graph (inverse f) = Submodule.map (LinearEquiv.prodComm R E F) (graph f)\n[PROOFSTEP]\nrw [inverse, Submodule.toLinearPMap_graph_eq _ (mem_inverse_graph_snd_eq_zero hf)]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\n\u22a2 LinearMap.range (inverse f).toFun = f.domain\n[PROOFSTEP]\nrw [inverse, Submodule.toLinearPMap_range _ (mem_inverse_graph_snd_eq_zero hf), \u2190 graph_map_fst_eq_domain, \u2190\n LinearEquiv.snd_comp_prodComm, Submodule.map_comp]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\n\u22a2 Submodule.map (LinearMap.snd R F E) (Submodule.map (LinearEquiv.prodComm R E F) (graph f)) =\n Submodule.map (LinearMap.snd R F E) (Submodule.map (\u2191(LinearEquiv.prodComm R E F)) (graph f))\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\nx : { x // x \u2208 f.domain }\n\u22a2 (\u2191f x, \u2191x) \u2208 graph (inverse f)\n[PROOFSTEP]\nsimp only [inverse_graph hf, Submodule.mem_map, mem_graph_iff, Subtype.exists, exists_and_left, exists_eq_left,\n LinearEquiv.prodComm_apply, Prod.exists, Prod.swap_prod_mk, Prod.mk.injEq]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\nx : { x // x \u2208 f.domain }\n\u22a2 \u2203 a b, (\u2203 h, \u2191f { val := a, property := (_ : (a, b).fst \u2208 f.domain) } = b) \u2227 b = \u2191f x \u2227 a = \u2191x\n[PROOFSTEP]\nexact \u27e8(x : E), f x, \u27e8x.2, Eq.refl _\u27e9, Eq.refl _, Eq.refl _\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\ny : { x // x \u2208 (inverse f).domain }\nx : { x // x \u2208 f.domain }\nhxy : \u2191f x = \u2191y\n\u22a2 \u2191(inverse f) y = \u2191x\n[PROOFSTEP]\nhave := mem_inverse_graph hf x\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\ny : { x // x \u2208 (inverse f).domain }\nx : { x // x \u2208 f.domain }\nhxy : \u2191f x = \u2191y\nthis : (\u2191f x, \u2191x) \u2208 graph (inverse f)\n\u22a2 \u2191(inverse f) y = \u2191x\n[PROOFSTEP]\nsimp only [mem_graph_iff, Subtype.exists, exists_and_left, exists_eq_left] at this \n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\ny : { x // x \u2208 (inverse f).domain }\nx : { x // x \u2208 f.domain }\nhxy : \u2191f x = \u2191y\nthis : \u2203 x_1, \u2191(inverse f) { val := \u2191f x, property := (_ : \u2191f x \u2208 (inverse f).domain) } = \u2191x\n\u22a2 \u2191(inverse f) y = \u2191x\n[PROOFSTEP]\nrcases this with \u27e8hx, h\u27e9\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\ny : { x // x \u2208 (inverse f).domain }\nx : { x // x \u2208 f.domain }\nhxy : \u2191f x = \u2191y\nhx : \u2191f x \u2208 (inverse f).domain\nh : \u2191(inverse f) { val := \u2191f x, property := (_ : \u2191f x \u2208 (inverse f).domain) } = \u2191x\n\u22a2 \u2191(inverse f) y = \u2191x\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\ny : { x // x \u2208 (inverse f).domain }\nx : { x // x \u2208 f.domain }\nhxy : \u2191f x = \u2191y\nhx : \u2191f x \u2208 (inverse f).domain\nh : \u2191(inverse f) { val := \u2191f x, property := (_ : \u2191f x \u2208 (inverse f).domain) } = \u2191x\n\u22a2 \u2191(inverse f) y = \u2191(inverse f) { val := \u2191f x, property := (_ : \u2191f x \u2208 (inverse f).domain) }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase intro.e_a\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\ny : { x // x \u2208 (inverse f).domain }\nx : { x // x \u2208 f.domain }\nhxy : \u2191f x = \u2191y\nhx : \u2191f x \u2208 (inverse f).domain\nh : \u2191(inverse f) { val := \u2191f x, property := (_ : \u2191f x \u2208 (inverse f).domain) } = \u2191x\n\u22a2 y = { val := \u2191f x, property := (_ : \u2191f x \u2208 (inverse f).domain) }\n[PROOFSTEP]\nsimp only [hxy, Subtype.coe_eta]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.LinearPMap", "llama_tokens": 101722, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.4263215925474903, "lm_q1q2_score": 0.24457149227526195}} {"text": "[GOAL]\nt : Type u \u2192 Type u \u2192 Type u\ninst\u271d\u2075 : Bitraversable t\n\u03b2\u271d : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulBitraversable t\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1\u2080 \u03b1\u2081 \u03b1\u2082 \u03b2 : Type u\nf : \u03b1\u2080 \u2192 F \u03b1\u2081\nf' : \u03b1\u2081 \u2192 G \u03b1\u2082\nx : t \u03b1\u2080 \u03b2\n\u22a2 Comp.mk (tfst f' <$> tfst f x) = tfst (Comp.mk \u2218 map f' \u2218 f) x\n[PROOFSTEP]\nrw [\u2190 comp_bitraverse]\n[GOAL]\nt : Type u \u2192 Type u \u2192 Type u\ninst\u271d\u2075 : Bitraversable t\n\u03b2\u271d : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulBitraversable t\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1\u2080 \u03b1\u2081 \u03b1\u2082 \u03b2 : Type u\nf : \u03b1\u2080 \u2192 F \u03b1\u2081\nf' : \u03b1\u2081 \u2192 G \u03b1\u2082\nx : t \u03b1\u2080 \u03b2\n\u22a2 bitraverse (Comp.mk \u2218 map f' \u2218 f) (Comp.mk \u2218 map pure \u2218 pure) x = tfst (Comp.mk \u2218 map f' \u2218 f) x\n[PROOFSTEP]\nsimp only [Function.comp, tfst, map_pure, Pure.pure]\n[GOAL]\nt : Type u \u2192 Type u \u2192 Type u\ninst\u271d\u2075 : Bitraversable t\n\u03b2 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulBitraversable t\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1\u2080 \u03b1\u2081 \u03b2\u2080 \u03b2\u2081 : Type u\nf : \u03b1\u2080 \u2192 F \u03b1\u2081\nf' : \u03b2\u2080 \u2192 G \u03b2\u2081\nx : t \u03b1\u2080 \u03b2\u2080\n\u22a2 Comp.mk (tfst f <$> tsnd f' x) = bitraverse (Comp.mk \u2218 pure \u2218 f) (Comp.mk \u2218 map pure \u2218 f') x\n[PROOFSTEP]\nrw [\u2190 comp_bitraverse]\n[GOAL]\nt : Type u \u2192 Type u \u2192 Type u\ninst\u271d\u2075 : Bitraversable t\n\u03b2 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulBitraversable t\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1\u2080 \u03b1\u2081 \u03b2\u2080 \u03b2\u2081 : Type u\nf : \u03b1\u2080 \u2192 F \u03b1\u2081\nf' : \u03b2\u2080 \u2192 G \u03b2\u2081\nx : t \u03b1\u2080 \u03b2\u2080\n\u22a2 bitraverse (Comp.mk \u2218 map f \u2218 pure) (Comp.mk \u2218 map pure \u2218 f') x =\n bitraverse (Comp.mk \u2218 pure \u2218 f) (Comp.mk \u2218 map pure \u2218 f') x\n[PROOFSTEP]\nsimp only [Function.comp, map_pure]\n[GOAL]\nt : Type u \u2192 Type u \u2192 Type u\ninst\u271d\u2075 : Bitraversable t\n\u03b2 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulBitraversable t\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1\u2080 \u03b1\u2081 \u03b2\u2080 \u03b2\u2081 : Type u\nf : \u03b1\u2080 \u2192 F \u03b1\u2081\nf' : \u03b2\u2080 \u2192 G \u03b2\u2081\nx : t \u03b1\u2080 \u03b2\u2080\n\u22a2 Comp.mk (tsnd f' <$> tfst f x) = bitraverse (Comp.mk \u2218 map pure \u2218 f) (Comp.mk \u2218 pure \u2218 f') x\n[PROOFSTEP]\nrw [\u2190 comp_bitraverse]\n[GOAL]\nt : Type u \u2192 Type u \u2192 Type u\ninst\u271d\u2075 : Bitraversable t\n\u03b2 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulBitraversable t\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1\u2080 \u03b1\u2081 \u03b2\u2080 \u03b2\u2081 : Type u\nf : \u03b1\u2080 \u2192 F \u03b1\u2081\nf' : \u03b2\u2080 \u2192 G \u03b2\u2081\nx : t \u03b1\u2080 \u03b2\u2080\n\u22a2 bitraverse (Comp.mk \u2218 map pure \u2218 f) (Comp.mk \u2218 map f' \u2218 pure) x =\n bitraverse (Comp.mk \u2218 map pure \u2218 f) (Comp.mk \u2218 pure \u2218 f') x\n[PROOFSTEP]\nsimp only [Function.comp, map_pure]\n[GOAL]\nt : Type u \u2192 Type u \u2192 Type u\ninst\u271d\u2075 : Bitraversable t\n\u03b2 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulBitraversable t\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2\u2080 \u03b2\u2081 \u03b2\u2082 : Type u\ng : \u03b2\u2080 \u2192 F \u03b2\u2081\ng' : \u03b2\u2081 \u2192 G \u03b2\u2082\nx : t \u03b1 \u03b2\u2080\n\u22a2 Comp.mk (tsnd g' <$> tsnd g x) = tsnd (Comp.mk \u2218 map g' \u2218 g) x\n[PROOFSTEP]\nrw [\u2190 comp_bitraverse]\n[GOAL]\nt : Type u \u2192 Type u \u2192 Type u\ninst\u271d\u2075 : Bitraversable t\n\u03b2 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulBitraversable t\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2\u2080 \u03b2\u2081 \u03b2\u2082 : Type u\ng : \u03b2\u2080 \u2192 F \u03b2\u2081\ng' : \u03b2\u2081 \u2192 G \u03b2\u2082\nx : t \u03b1 \u03b2\u2080\n\u22a2 bitraverse (Comp.mk \u2218 map pure \u2218 pure) (Comp.mk \u2218 map g' \u2218 g) x = tsnd (Comp.mk \u2218 map g' \u2218 g) x\n[PROOFSTEP]\nsimp only [Function.comp, map_pure]\n[GOAL]\nt : Type u \u2192 Type u \u2192 Type u\ninst\u271d\u2075 : Bitraversable t\n\u03b2 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulBitraversable t\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2\u2080 \u03b2\u2081 \u03b2\u2082 : Type u\ng : \u03b2\u2080 \u2192 F \u03b2\u2081\ng' : \u03b2\u2081 \u2192 G \u03b2\u2082\nx : t \u03b1 \u03b2\u2080\n\u22a2 bitraverse (fun x => Comp.mk (pure (pure x))) (fun x => Comp.mk (g' <$> g x)) x =\n tsnd (fun x => Comp.mk (g' <$> g x)) x\n[PROOFSTEP]\nrfl\n[GOAL]\nt : Type u \u2192 Type u \u2192 Type u\ninst\u271d\u2075 : Bitraversable t\n\u03b2\u271d : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulBitraversable t\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b1' \u03b2 : Type u\nf : \u03b1 \u2192 \u03b1'\nx : t \u03b1 \u03b2\n\u22a2 tfst (pure \u2218 f) x = pure (fst f x)\n[PROOFSTEP]\napply bitraverse_eq_bimap_id\n[GOAL]\nt : Type u \u2192 Type u \u2192 Type u\ninst\u271d\u2075 : Bitraversable t\n\u03b2\u271d : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulBitraversable t\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b2' : Type u\nf : \u03b2 \u2192 \u03b2'\nx : t \u03b1 \u03b2\n\u22a2 tsnd (pure \u2218 f) x = pure (snd f x)\n[PROOFSTEP]\napply bitraverse_eq_bimap_id\n", "meta": {"mathlib_filename": "Mathlib.Control.Bitraversable.Lemmas", "llama_tokens": 2268, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.546738151984614, "lm_q2_score": 0.4416730056646256, "lm_q1q2_score": 0.24147948289856733}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nas : List \u03b1\nx : \u03b1\n\u22a2 x \u2208 List.insert a as \u2194 x \u2208 a :: as\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nas bs : List \u03b1\nx : \u03b1\n\u22a2 x \u2208 as \u222a bs \u2194 x \u2208 as ++ bs\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b : \u03b1\nas : List \u03b1\n\u22a2 b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\n[PROOFSTEP]\ninduction as with\n| nil => simp [remove]\n| cons a' as ih =>\n simp [remove]\n cases Decidable.em (a = a') with\n | inl h =>\n simp only [if_pos h, ih]\n exact \u27e8fun \u27e8h1, h2\u27e9 \u21a6 \u27e8Or.inr h1, h2\u27e9, fun \u27e8h1, h2\u27e9 \u21a6 \u27e8Or.resolve_left h1 (h \u25b8 h2), h2\u27e9\u27e9\n | inr h =>\n simp [if_neg h, ih]\n constructor\n {focus\n intro h'\n cases h' with\n | inl h\u2081 => exact \u27e8Or.inl h\u2081, h\u2081.symm \u25b8 (Ne.symm h)\u27e9\n | inr h\u2081 => exact \u27e8Or.inr h\u2081.1, h\u2081.2\u27e9\n }\n intro \u27e8h1, h2\u27e9\n cases h1 with\n | inl h1' => exact Or.inl h1'\n | inr h1' => exact Or.inr \u27e8h1', h2\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b : \u03b1\nas : List \u03b1\n\u22a2 b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\n[PROOFSTEP]\ninduction as with\n| nil => simp [remove]\n| cons a' as ih =>\n simp [remove]\n cases Decidable.em (a = a') with\n | inl h =>\n simp only [if_pos h, ih]\n exact \u27e8fun \u27e8h1, h2\u27e9 \u21a6 \u27e8Or.inr h1, h2\u27e9, fun \u27e8h1, h2\u27e9 \u21a6 \u27e8Or.resolve_left h1 (h \u25b8 h2), h2\u27e9\u27e9\n | inr h =>\n simp [if_neg h, ih]\n constructor\n {focus\n intro h'\n cases h' with\n | inl h\u2081 => exact \u27e8Or.inl h\u2081, h\u2081.symm \u25b8 (Ne.symm h)\u27e9\n | inr h\u2081 => exact \u27e8Or.inr h\u2081.1, h\u2081.2\u27e9\n }\n intro \u27e8h1, h2\u27e9\n cases h1 with\n | inl h1' => exact Or.inl h1'\n | inr h1' => exact Or.inr \u27e8h1', h2\u27e9\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b : \u03b1\n\u22a2 b \u2208 remove a [] \u2194 b \u2208 [] \u2227 b \u2260 a\n[PROOFSTEP]\n\n| nil => simp [remove]\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b : \u03b1\n\u22a2 b \u2208 remove a [] \u2194 b \u2208 [] \u2227 b \u2260 a\n[PROOFSTEP]\nsimp [remove]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\n\u22a2 b \u2208 remove a (a' :: as) \u2194 b \u2208 a' :: as \u2227 b \u2260 a\n[PROOFSTEP]\n\n| cons a' as ih =>\n simp [remove]\n cases Decidable.em (a = a') with\n | inl h =>\n simp only [if_pos h, ih]\n exact \u27e8fun \u27e8h1, h2\u27e9 \u21a6 \u27e8Or.inr h1, h2\u27e9, fun \u27e8h1, h2\u27e9 \u21a6 \u27e8Or.resolve_left h1 (h \u25b8 h2), h2\u27e9\u27e9\n | inr h =>\n simp [if_neg h, ih]\n constructor\n {focus\n intro h'\n cases h' with\n | inl h\u2081 => exact \u27e8Or.inl h\u2081, h\u2081.symm \u25b8 (Ne.symm h)\u27e9\n | inr h\u2081 => exact \u27e8Or.inr h\u2081.1, h\u2081.2\u27e9\n }\n intro \u27e8h1, h2\u27e9\n cases h1 with\n | inl h1' => exact Or.inl h1'\n | inr h1' => exact Or.inr \u27e8h1', h2\u27e9\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\n\u22a2 b \u2208 remove a (a' :: as) \u2194 b \u2208 a' :: as \u2227 b \u2260 a\n[PROOFSTEP]\nsimp [remove]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\n\u22a2 (b \u2208 if a = a' then remove a as else a' :: remove a as) \u2194 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a\n[PROOFSTEP]\ncases Decidable.em (a = a') with\n| inl h =>\n simp only [if_pos h, ih]\n exact \u27e8fun \u27e8h1, h2\u27e9 \u21a6 \u27e8Or.inr h1, h2\u27e9, fun \u27e8h1, h2\u27e9 \u21a6 \u27e8Or.resolve_left h1 (h \u25b8 h2), h2\u27e9\u27e9\n| inr h =>\n simp [if_neg h, ih]\n constructor\n {focus\n intro h'\n cases h' with\n | inl h\u2081 => exact \u27e8Or.inl h\u2081, h\u2081.symm \u25b8 (Ne.symm h)\u27e9\n | inr h\u2081 => exact \u27e8Or.inr h\u2081.1, h\u2081.2\u27e9\n }\n intro \u27e8h1, h2\u27e9\n cases h1 with\n | inl h1' => exact Or.inl h1'\n | inr h1' => exact Or.inr \u27e8h1', h2\u27e9\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nx\u271d : a = a' \u2228 \u00aca = a'\n\u22a2 (b \u2208 if a = a' then remove a as else a' :: remove a as) \u2194 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a\n[PROOFSTEP]\ncases Decidable.em (a = a') with\n| inl h =>\n simp only [if_pos h, ih]\n exact \u27e8fun \u27e8h1, h2\u27e9 \u21a6 \u27e8Or.inr h1, h2\u27e9, fun \u27e8h1, h2\u27e9 \u21a6 \u27e8Or.resolve_left h1 (h \u25b8 h2), h2\u27e9\u27e9\n| inr h =>\n simp [if_neg h, ih]\n constructor\n {focus\n intro h'\n cases h' with\n | inl h\u2081 => exact \u27e8Or.inl h\u2081, h\u2081.symm \u25b8 (Ne.symm h)\u27e9\n | inr h\u2081 => exact \u27e8Or.inr h\u2081.1, h\u2081.2\u27e9\n }\n intro \u27e8h1, h2\u27e9\n cases h1 with\n | inl h1' => exact Or.inl h1'\n | inr h1' => exact Or.inr \u27e8h1', h2\u27e9\n[GOAL]\ncase cons.inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : a = a'\n\u22a2 (b \u2208 if a = a' then remove a as else a' :: remove a as) \u2194 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a\n[PROOFSTEP]\n\n| inl h =>\n simp only [if_pos h, ih]\n exact \u27e8fun \u27e8h1, h2\u27e9 \u21a6 \u27e8Or.inr h1, h2\u27e9, fun \u27e8h1, h2\u27e9 \u21a6 \u27e8Or.resolve_left h1 (h \u25b8 h2), h2\u27e9\u27e9\n[GOAL]\ncase cons.inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : a = a'\n\u22a2 (b \u2208 if a = a' then remove a as else a' :: remove a as) \u2194 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a\n[PROOFSTEP]\nsimp only [if_pos h, ih]\n[GOAL]\ncase cons.inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : a = a'\n\u22a2 b \u2208 as \u2227 b \u2260 a \u2194 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a\n[PROOFSTEP]\nexact \u27e8fun \u27e8h1, h2\u27e9 \u21a6 \u27e8Or.inr h1, h2\u27e9, fun \u27e8h1, h2\u27e9 \u21a6 \u27e8Or.resolve_left h1 (h \u25b8 h2), h2\u27e9\u27e9\n[GOAL]\ncase cons.inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\n\u22a2 (b \u2208 if a = a' then remove a as else a' :: remove a as) \u2194 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a\n[PROOFSTEP]\n\n| inr h =>\n simp [if_neg h, ih]\n constructor\n {focus\n intro h'\n cases h' with\n | inl h\u2081 => exact \u27e8Or.inl h\u2081, h\u2081.symm \u25b8 (Ne.symm h)\u27e9\n | inr h\u2081 => exact \u27e8Or.inr h\u2081.1, h\u2081.2\u27e9\n }\n intro \u27e8h1, h2\u27e9\n cases h1 with\n | inl h1' => exact Or.inl h1'\n | inr h1' => exact Or.inr \u27e8h1', h2\u27e9\n[GOAL]\ncase cons.inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\n\u22a2 (b \u2208 if a = a' then remove a as else a' :: remove a as) \u2194 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a\n[PROOFSTEP]\nsimp [if_neg h, ih]\n[GOAL]\ncase cons.inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\n\u22a2 b = a' \u2228 b \u2208 as \u2227 \u00acb = a \u2194 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons.inr.mp\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\n\u22a2 b = a' \u2228 b \u2208 as \u2227 \u00acb = a \u2192 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a\ncase cons.inr.mpr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\n\u22a2 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a \u2192 b = a' \u2228 b \u2208 as \u2227 \u00acb = a\n[PROOFSTEP]\n{focus\n intro h'\n cases h' with\n | inl h\u2081 => exact \u27e8Or.inl h\u2081, h\u2081.symm \u25b8 (Ne.symm h)\u27e9\n | inr h\u2081 => exact \u27e8Or.inr h\u2081.1, h\u2081.2\u27e9\n}\n[GOAL]\ncase cons.inr.mp\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\n\u22a2 b = a' \u2228 b \u2208 as \u2227 \u00acb = a \u2192 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a\n[PROOFSTEP]\nfocus\n intro h'\n cases h' with\n | inl h\u2081 => exact \u27e8Or.inl h\u2081, h\u2081.symm \u25b8 (Ne.symm h)\u27e9\n | inr h\u2081 => exact \u27e8Or.inr h\u2081.1, h\u2081.2\u27e9\n[GOAL]\ncase cons.inr.mp\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\n\u22a2 b = a' \u2228 b \u2208 as \u2227 \u00acb = a \u2192 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a\n[PROOFSTEP]\nintro h'\n[GOAL]\ncase cons.inr.mp\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\nh' : b = a' \u2228 b \u2208 as \u2227 \u00acb = a\n\u22a2 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a\n[PROOFSTEP]\ncases h' with\n| inl h\u2081 => exact \u27e8Or.inl h\u2081, h\u2081.symm \u25b8 (Ne.symm h)\u27e9\n| inr h\u2081 => exact \u27e8Or.inr h\u2081.1, h\u2081.2\u27e9\n[GOAL]\ncase cons.inr.mp\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\nh' : b = a' \u2228 b \u2208 as \u2227 \u00acb = a\n\u22a2 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a\n[PROOFSTEP]\ncases h' with\n| inl h\u2081 => exact \u27e8Or.inl h\u2081, h\u2081.symm \u25b8 (Ne.symm h)\u27e9\n| inr h\u2081 => exact \u27e8Or.inr h\u2081.1, h\u2081.2\u27e9\n[GOAL]\ncase cons.inr.mp.inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\nh\u2081 : b = a'\n\u22a2 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a\n[PROOFSTEP]\n\n| inl h\u2081 => exact \u27e8Or.inl h\u2081, h\u2081.symm \u25b8 (Ne.symm h)\u27e9\n[GOAL]\ncase cons.inr.mp.inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\nh\u2081 : b = a'\n\u22a2 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a\n[PROOFSTEP]\nexact \u27e8Or.inl h\u2081, h\u2081.symm \u25b8 (Ne.symm h)\u27e9\n[GOAL]\ncase cons.inr.mp.inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\nh\u2081 : b \u2208 as \u2227 \u00acb = a\n\u22a2 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a\n[PROOFSTEP]\n\n| inr h\u2081 => exact \u27e8Or.inr h\u2081.1, h\u2081.2\u27e9\n[GOAL]\ncase cons.inr.mp.inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\nh\u2081 : b \u2208 as \u2227 \u00acb = a\n\u22a2 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a\n[PROOFSTEP]\nexact \u27e8Or.inr h\u2081.1, h\u2081.2\u27e9\n[GOAL]\ncase cons.inr.mpr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\n\u22a2 (b = a' \u2228 b \u2208 as) \u2227 \u00acb = a \u2192 b = a' \u2228 b \u2208 as \u2227 \u00acb = a\n[PROOFSTEP]\nintro \u27e8h1, h2\u27e9\n[GOAL]\ncase cons.inr.mpr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\nh1 : b = a' \u2228 b \u2208 as\nh2 : \u00acb = a\n\u22a2 b = a' \u2228 b \u2208 as \u2227 \u00acb = a\n[PROOFSTEP]\ncases h1 with\n| inl h1' => exact Or.inl h1'\n| inr h1' => exact Or.inr \u27e8h1', h2\u27e9\n[GOAL]\ncase cons.inr.mpr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\nh1 : b = a' \u2228 b \u2208 as\nh2 : \u00acb = a\n\u22a2 b = a' \u2228 b \u2208 as \u2227 \u00acb = a\n[PROOFSTEP]\ncases h1 with\n| inl h1' => exact Or.inl h1'\n| inr h1' => exact Or.inr \u27e8h1', h2\u27e9\n[GOAL]\ncase cons.inr.mpr.inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\nh2 : \u00acb = a\nh1' : b = a'\n\u22a2 b = a' \u2228 b \u2208 as \u2227 \u00acb = a\n[PROOFSTEP]\n\n| inl h1' => exact Or.inl h1'\n[GOAL]\ncase cons.inr.mpr.inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\nh2 : \u00acb = a\nh1' : b = a'\n\u22a2 b = a' \u2228 b \u2208 as \u2227 \u00acb = a\n[PROOFSTEP]\nexact Or.inl h1'\n[GOAL]\ncase cons.inr.mpr.inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\nh2 : \u00acb = a\nh1' : b \u2208 as\n\u22a2 b = a' \u2228 b \u2208 as \u2227 \u00acb = a\n[PROOFSTEP]\n\n| inr h1' => exact Or.inr \u27e8h1', h2\u27e9\n[GOAL]\ncase cons.inr.mpr.inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.8926\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b a' : \u03b1\nas : List \u03b1\nih : b \u2208 remove a as \u2194 b \u2208 as \u2227 b \u2260 a\nh : \u00aca = a'\nh2 : \u00acb = a\nh1' : b \u2208 as\n\u22a2 b = a' \u2228 b \u2208 as \u2227 \u00acb = a\n[PROOFSTEP]\nexact Or.inr \u27e8h1', h2\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.19025\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nx\u271d : \u00aca \u2208 []\n\u22a2 remove a [] = []\n[PROOFSTEP]\nsimp [remove]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.19025\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : \u00aca \u2208 a' :: as\n\u22a2 remove a (a' :: as) = a' :: as\n[PROOFSTEP]\nhave h1 : a \u2260 a' := fun h' \u21a6 h (by rw [h']; apply mem_cons_self)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.19025\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : \u00aca \u2208 a' :: as\nh' : a = a'\n\u22a2 a \u2208 a' :: as\n[PROOFSTEP]\nrw [h']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.19025\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : \u00aca \u2208 a' :: as\nh' : a = a'\n\u22a2 a' \u2208 a' :: as\n[PROOFSTEP]\napply mem_cons_self\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.19025\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : \u00aca \u2208 a' :: as\nh1 : a \u2260 a'\n\u22a2 remove a (a' :: as) = a' :: as\n[PROOFSTEP]\nhave h2 : a \u2209 as := fun h' \u21a6 h (mem_cons_of_mem _ h')\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.19025\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : \u00aca \u2208 a' :: as\nh1 : a \u2260 a'\nh2 : \u00aca \u2208 as\n\u22a2 remove a (a' :: as) = a' :: as\n[PROOFSTEP]\nsimp [remove, h1, remove_eq_of_not_mem h2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.19685\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b : \u03b1\nas : List \u03b1\nh : b \u2208 remove a as\n\u22a2 b \u2208 as\n[PROOFSTEP]\nrw [mem_remove_iff] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.19685\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na b : \u03b1\nas : List \u03b1\nh : b \u2208 as \u2227 b \u2260 a\n\u22a2 b \u2208 as\n[PROOFSTEP]\nexact\n h.1\n /- card -/\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.20767\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas : List \u03b1\nh : a \u2208 as\n\u22a2 card (a :: as) = card as\n[PROOFSTEP]\nsimp [card, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.21944\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas : List \u03b1\nh : \u00aca \u2208 as\n\u22a2 card (a :: as) = card as + 1\n[PROOFSTEP]\nsimp [card, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.22769\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas : List \u03b1\n\u22a2 card as \u2264 card (a :: as)\n[PROOFSTEP]\ncases Decidable.em (a \u2208 as) with\n| inl h => simp [h, Nat.le_refl]\n| inr h => simp [h, Nat.le_succ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.22769\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas : List \u03b1\nx\u271d : a \u2208 as \u2228 \u00aca \u2208 as\n\u22a2 card as \u2264 card (a :: as)\n[PROOFSTEP]\ncases Decidable.em (a \u2208 as) with\n| inl h => simp [h, Nat.le_refl]\n| inr h => simp [h, Nat.le_succ]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.22769\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas : List \u03b1\nh : a \u2208 as\n\u22a2 card as \u2264 card (a :: as)\n[PROOFSTEP]\n\n| inl h => simp [h, Nat.le_refl]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.22769\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas : List \u03b1\nh : a \u2208 as\n\u22a2 card as \u2264 card (a :: as)\n[PROOFSTEP]\nsimp [h, Nat.le_refl]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.22769\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas : List \u03b1\nh : \u00aca \u2208 as\n\u22a2 card as \u2264 card (a :: as)\n[PROOFSTEP]\n\n| inr h => simp [h, Nat.le_succ]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.22769\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas : List \u03b1\nh : \u00aca \u2208 as\n\u22a2 card as \u2264 card (a :: as)\n[PROOFSTEP]\nsimp [h, Nat.le_succ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.25043\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas : List \u03b1\nh : a \u2208 as\n\u22a2 card (List.insert a as) = card as\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.25854\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas : List \u03b1\nh : \u00aca \u2208 as\n\u22a2 card (List.insert a as) = card as + 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\n\u22a2 card (a' :: as) = card (remove a (a' :: as)) + 1\n[PROOFSTEP]\ncases Decidable.em (a = a') with\n| inl h' =>\n simp [remove, if_pos h']\n cases Decidable.em (a \u2208 as) with\n | inl h'' =>\n have h\u2083 : a' \u2208 as := h' \u25b8 h''\n simp [card_remove_of_mem h'', h\u2083]\n | inr h'' =>\n have h\u2083 : a' \u2209 as := h' \u25b8 h''\n simp [card_cons_of_not_mem h\u2083, remove_eq_of_not_mem h'']\n| inr h' =>\n have h\u2083 : a \u2208 as := (mem_cons.1 h).resolve_left h'\n simp [remove, h']\n cases Decidable.em (a' \u2208 as) with\n | inl h'' =>\n have : a' \u2208 remove a as := by rw [mem_remove_iff]; exact \u27e8h'', Ne.symm h'\u27e9\n simp [h'', this, card_remove_of_mem h\u2083]\n | inr h'' =>\n have : a' \u2209 remove a as := fun h \u21a6 h'' (mem_of_mem_remove h)\n simp [h'', this, card_remove_of_mem h\u2083]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nx\u271d : a = a' \u2228 \u00aca = a'\n\u22a2 card (a' :: as) = card (remove a (a' :: as)) + 1\n[PROOFSTEP]\ncases Decidable.em (a = a') with\n| inl h' =>\n simp [remove, if_pos h']\n cases Decidable.em (a \u2208 as) with\n | inl h'' =>\n have h\u2083 : a' \u2208 as := h' \u25b8 h''\n simp [card_remove_of_mem h'', h\u2083]\n | inr h'' =>\n have h\u2083 : a' \u2209 as := h' \u25b8 h''\n simp [card_cons_of_not_mem h\u2083, remove_eq_of_not_mem h'']\n| inr h' =>\n have h\u2083 : a \u2208 as := (mem_cons.1 h).resolve_left h'\n simp [remove, h']\n cases Decidable.em (a' \u2208 as) with\n | inl h'' =>\n have : a' \u2208 remove a as := by rw [mem_remove_iff]; exact \u27e8h'', Ne.symm h'\u27e9\n simp [h'', this, card_remove_of_mem h\u2083]\n | inr h'' =>\n have : a' \u2209 remove a as := fun h \u21a6 h'' (mem_of_mem_remove h)\n simp [h'', this, card_remove_of_mem h\u2083]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : a = a'\n\u22a2 card (a' :: as) = card (remove a (a' :: as)) + 1\n[PROOFSTEP]\n\n| inl h' =>\n simp [remove, if_pos h']\n cases Decidable.em (a \u2208 as) with\n | inl h'' =>\n have h\u2083 : a' \u2208 as := h' \u25b8 h''\n simp [card_remove_of_mem h'', h\u2083]\n | inr h'' =>\n have h\u2083 : a' \u2209 as := h' \u25b8 h''\n simp [card_cons_of_not_mem h\u2083, remove_eq_of_not_mem h'']\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : a = a'\n\u22a2 card (a' :: as) = card (remove a (a' :: as)) + 1\n[PROOFSTEP]\nsimp [remove, if_pos h']\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : a = a'\n\u22a2 card (a' :: as) = card (remove a as) + 1\n[PROOFSTEP]\ncases Decidable.em (a \u2208 as) with\n| inl h'' =>\n have h\u2083 : a' \u2208 as := h' \u25b8 h''\n simp [card_remove_of_mem h'', h\u2083]\n| inr h'' =>\n have h\u2083 : a' \u2209 as := h' \u25b8 h''\n simp [card_cons_of_not_mem h\u2083, remove_eq_of_not_mem h'']\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : a = a'\nx\u271d : a \u2208 as \u2228 \u00aca \u2208 as\n\u22a2 card (a' :: as) = card (remove a as) + 1\n[PROOFSTEP]\ncases Decidable.em (a \u2208 as) with\n| inl h'' =>\n have h\u2083 : a' \u2208 as := h' \u25b8 h''\n simp [card_remove_of_mem h'', h\u2083]\n| inr h'' =>\n have h\u2083 : a' \u2209 as := h' \u25b8 h''\n simp [card_cons_of_not_mem h\u2083, remove_eq_of_not_mem h'']\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : a = a'\nh'' : a \u2208 as\n\u22a2 card (a' :: as) = card (remove a as) + 1\n[PROOFSTEP]\n\n| inl h'' =>\n have h\u2083 : a' \u2208 as := h' \u25b8 h''\n simp [card_remove_of_mem h'', h\u2083]\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : a = a'\nh'' : a \u2208 as\n\u22a2 card (a' :: as) = card (remove a as) + 1\n[PROOFSTEP]\nhave h\u2083 : a' \u2208 as := h' \u25b8 h''\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : a = a'\nh'' : a \u2208 as\nh\u2083 : a' \u2208 as\n\u22a2 card (a' :: as) = card (remove a as) + 1\n[PROOFSTEP]\nsimp [card_remove_of_mem h'', h\u2083]\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : a = a'\nh'' : \u00aca \u2208 as\n\u22a2 card (a' :: as) = card (remove a as) + 1\n[PROOFSTEP]\n\n| inr h'' =>\n have h\u2083 : a' \u2209 as := h' \u25b8 h''\n simp [card_cons_of_not_mem h\u2083, remove_eq_of_not_mem h'']\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : a = a'\nh'' : \u00aca \u2208 as\n\u22a2 card (a' :: as) = card (remove a as) + 1\n[PROOFSTEP]\nhave h\u2083 : a' \u2209 as := h' \u25b8 h''\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : a = a'\nh'' : \u00aca \u2208 as\nh\u2083 : \u00aca' \u2208 as\n\u22a2 card (a' :: as) = card (remove a as) + 1\n[PROOFSTEP]\nsimp [card_cons_of_not_mem h\u2083, remove_eq_of_not_mem h'']\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : \u00aca = a'\n\u22a2 card (a' :: as) = card (remove a (a' :: as)) + 1\n[PROOFSTEP]\n\n| inr h' =>\n have h\u2083 : a \u2208 as := (mem_cons.1 h).resolve_left h'\n simp [remove, h']\n cases Decidable.em (a' \u2208 as) with\n | inl h'' =>\n have : a' \u2208 remove a as := by rw [mem_remove_iff]; exact \u27e8h'', Ne.symm h'\u27e9\n simp [h'', this, card_remove_of_mem h\u2083]\n | inr h'' =>\n have : a' \u2209 remove a as := fun h \u21a6 h'' (mem_of_mem_remove h)\n simp [h'', this, card_remove_of_mem h\u2083]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : \u00aca = a'\n\u22a2 card (a' :: as) = card (remove a (a' :: as)) + 1\n[PROOFSTEP]\nhave h\u2083 : a \u2208 as := (mem_cons.1 h).resolve_left h'\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : \u00aca = a'\nh\u2083 : a \u2208 as\n\u22a2 card (a' :: as) = card (remove a (a' :: as)) + 1\n[PROOFSTEP]\nsimp [remove, h']\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : \u00aca = a'\nh\u2083 : a \u2208 as\n\u22a2 card (a' :: as) = card (a' :: remove a as) + 1\n[PROOFSTEP]\ncases Decidable.em (a' \u2208 as) with\n| inl h'' =>\n have : a' \u2208 remove a as := by rw [mem_remove_iff]; exact \u27e8h'', Ne.symm h'\u27e9\n simp [h'', this, card_remove_of_mem h\u2083]\n| inr h'' =>\n have : a' \u2209 remove a as := fun h \u21a6 h'' (mem_of_mem_remove h)\n simp [h'', this, card_remove_of_mem h\u2083]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : \u00aca = a'\nh\u2083 : a \u2208 as\nx\u271d : a' \u2208 as \u2228 \u00aca' \u2208 as\n\u22a2 card (a' :: as) = card (a' :: remove a as) + 1\n[PROOFSTEP]\ncases Decidable.em (a' \u2208 as) with\n| inl h'' =>\n have : a' \u2208 remove a as := by rw [mem_remove_iff]; exact \u27e8h'', Ne.symm h'\u27e9\n simp [h'', this, card_remove_of_mem h\u2083]\n| inr h'' =>\n have : a' \u2209 remove a as := fun h \u21a6 h'' (mem_of_mem_remove h)\n simp [h'', this, card_remove_of_mem h\u2083]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : \u00aca = a'\nh\u2083 : a \u2208 as\nh'' : a' \u2208 as\n\u22a2 card (a' :: as) = card (a' :: remove a as) + 1\n[PROOFSTEP]\n\n| inl h'' =>\n have : a' \u2208 remove a as := by rw [mem_remove_iff]; exact \u27e8h'', Ne.symm h'\u27e9\n simp [h'', this, card_remove_of_mem h\u2083]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : \u00aca = a'\nh\u2083 : a \u2208 as\nh'' : a' \u2208 as\n\u22a2 card (a' :: as) = card (a' :: remove a as) + 1\n[PROOFSTEP]\nhave : a' \u2208 remove a as := by rw [mem_remove_iff]; exact \u27e8h'', Ne.symm h'\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : \u00aca = a'\nh\u2083 : a \u2208 as\nh'' : a' \u2208 as\n\u22a2 a' \u2208 remove a as\n[PROOFSTEP]\nrw [mem_remove_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : \u00aca = a'\nh\u2083 : a \u2208 as\nh'' : a' \u2208 as\n\u22a2 a' \u2208 as \u2227 a' \u2260 a\n[PROOFSTEP]\nexact \u27e8h'', Ne.symm h'\u27e9\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : \u00aca = a'\nh\u2083 : a \u2208 as\nh'' : a' \u2208 as\nthis : a' \u2208 remove a as\n\u22a2 card (a' :: as) = card (a' :: remove a as) + 1\n[PROOFSTEP]\nsimp [h'', this, card_remove_of_mem h\u2083]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : \u00aca = a'\nh\u2083 : a \u2208 as\nh'' : \u00aca' \u2208 as\n\u22a2 card (a' :: as) = card (a' :: remove a as) + 1\n[PROOFSTEP]\n\n| inr h'' =>\n have : a' \u2209 remove a as := fun h \u21a6 h'' (mem_of_mem_remove h)\n simp [h'', this, card_remove_of_mem h\u2083]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : \u00aca = a'\nh\u2083 : a \u2208 as\nh'' : \u00aca' \u2208 as\n\u22a2 card (a' :: as) = card (a' :: remove a as) + 1\n[PROOFSTEP]\nhave : a' \u2209 remove a as := fun h \u21a6 h'' (mem_of_mem_remove h)\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.27827\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na a' : \u03b1\nas : List \u03b1\nh : a \u2208 a' :: as\nh' : \u00aca = a'\nh\u2083 : a \u2208 as\nh'' : \u00aca' \u2208 as\nthis : \u00aca' \u2208 remove a as\n\u22a2 card (a' :: as) = card (a' :: remove a as) + 1\n[PROOFSTEP]\nsimp [h'', this, card_remove_of_mem h\u2083]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.43663\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nbs : List \u03b1\nx\u271d : [] \u2286 bs\n\u22a2 card [] \u2264 card bs\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.43663\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\nhsub : a :: as \u2286 bs\n\u22a2 card (a :: as) \u2264 card bs\n[PROOFSTEP]\ncases Decidable.em (a \u2208 as) with\n| inl h' =>\n have hsub' : as \u2286 bs := fun _ xmem \u21a6 hsub (mem_cons_of_mem a xmem)\n simp [h', card_subset_le hsub']\n| inr h' =>\n have : a \u2208 bs := hsub (Mem.head ..)\n rw [card_cons_of_not_mem h', card_remove_of_mem this]\n apply Nat.add_le_add_right\n apply card_subset_le\n intro x xmem\n rw [mem_remove_iff]\n exact \u27e8hsub (mem_cons_of_mem _ xmem), fun h \u21a6 h' (h \u25b8 xmem)\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.43663\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\nhsub : a :: as \u2286 bs\nx\u271d : a \u2208 as \u2228 \u00aca \u2208 as\n\u22a2 card (a :: as) \u2264 card bs\n[PROOFSTEP]\ncases Decidable.em (a \u2208 as) with\n| inl h' =>\n have hsub' : as \u2286 bs := fun _ xmem \u21a6 hsub (mem_cons_of_mem a xmem)\n simp [h', card_subset_le hsub']\n| inr h' =>\n have : a \u2208 bs := hsub (Mem.head ..)\n rw [card_cons_of_not_mem h', card_remove_of_mem this]\n apply Nat.add_le_add_right\n apply card_subset_le\n intro x xmem\n rw [mem_remove_iff]\n exact \u27e8hsub (mem_cons_of_mem _ xmem), fun h \u21a6 h' (h \u25b8 xmem)\u27e9\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.43663\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\nhsub : a :: as \u2286 bs\nh' : a \u2208 as\n\u22a2 card (a :: as) \u2264 card bs\n[PROOFSTEP]\n\n| inl h' =>\n have hsub' : as \u2286 bs := fun _ xmem \u21a6 hsub (mem_cons_of_mem a xmem)\n simp [h', card_subset_le hsub']\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.43663\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\nhsub : a :: as \u2286 bs\nh' : a \u2208 as\n\u22a2 card (a :: as) \u2264 card bs\n[PROOFSTEP]\nhave hsub' : as \u2286 bs := fun _ xmem \u21a6 hsub (mem_cons_of_mem a xmem)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.43663\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\nhsub : a :: as \u2286 bs\nh' : a \u2208 as\nhsub' : as \u2286 bs\n\u22a2 card (a :: as) \u2264 card bs\n[PROOFSTEP]\nsimp [h', card_subset_le hsub']\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.43663\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\nhsub : a :: as \u2286 bs\nh' : \u00aca \u2208 as\n\u22a2 card (a :: as) \u2264 card bs\n[PROOFSTEP]\n\n| inr h' =>\n have : a \u2208 bs := hsub (Mem.head ..)\n rw [card_cons_of_not_mem h', card_remove_of_mem this]\n apply Nat.add_le_add_right\n apply card_subset_le\n intro x xmem\n rw [mem_remove_iff]\n exact \u27e8hsub (mem_cons_of_mem _ xmem), fun h \u21a6 h' (h \u25b8 xmem)\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.43663\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\nhsub : a :: as \u2286 bs\nh' : \u00aca \u2208 as\n\u22a2 card (a :: as) \u2264 card bs\n[PROOFSTEP]\nhave : a \u2208 bs := hsub (Mem.head ..)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.43663\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\nhsub : a :: as \u2286 bs\nh' : \u00aca \u2208 as\nthis : a \u2208 bs\n\u22a2 card (a :: as) \u2264 card bs\n[PROOFSTEP]\nrw [card_cons_of_not_mem h', card_remove_of_mem this]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.43663\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\nhsub : a :: as \u2286 bs\nh' : \u00aca \u2208 as\nthis : a \u2208 bs\n\u22a2 card as + 1 \u2264 card (remove a bs) + 1\n[PROOFSTEP]\napply Nat.add_le_add_right\n[GOAL]\ncase inr.h\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.43663\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\nhsub : a :: as \u2286 bs\nh' : \u00aca \u2208 as\nthis : a \u2208 bs\n\u22a2 card as \u2264 card (remove a bs)\n[PROOFSTEP]\napply card_subset_le\n[GOAL]\ncase inr.h.a\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.43663\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\nhsub : a :: as \u2286 bs\nh' : \u00aca \u2208 as\nthis : a \u2208 bs\n\u22a2 as \u2286 remove a bs\n[PROOFSTEP]\nintro x xmem\n[GOAL]\ncase inr.h.a\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.43663\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\nhsub : a :: as \u2286 bs\nh' : \u00aca \u2208 as\nthis : a \u2208 bs\nx : \u03b1\nxmem : x \u2208 as\n\u22a2 x \u2208 remove a bs\n[PROOFSTEP]\nrw [mem_remove_iff]\n[GOAL]\ncase inr.h.a\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.43663\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\nhsub : a :: as \u2286 bs\nh' : \u00aca \u2208 as\nthis : a \u2208 bs\nx : \u03b1\nxmem : x \u2208 as\n\u22a2 x \u2208 bs \u2227 x \u2260 a\n[PROOFSTEP]\nexact \u27e8hsub (mem_cons_of_mem _ xmem), fun h \u21a6 h' (h \u25b8 xmem)\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\nas : List \u03b1\n\u22a2 card (map f as) \u2264 card as\n[PROOFSTEP]\ninduction as with\n| nil => simp\n| cons a as ih =>\n cases Decidable.em (f a \u2208 map f as) with\n | inl h =>\n rw [map, card_cons_of_mem h]\n apply Nat.le_trans ih (card_le_card_cons ..)\n | inr h =>\n have : a \u2209 as := fun h'' \u21a6 h (mem_map_of_mem _ h'')\n rw [map, card_cons_of_not_mem h, card_cons_of_not_mem this]\n exact Nat.add_le_add_right ih _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\nas : List \u03b1\n\u22a2 card (map f as) \u2264 card as\n[PROOFSTEP]\ninduction as with\n| nil => simp\n| cons a as ih =>\n cases Decidable.em (f a \u2208 map f as) with\n | inl h =>\n rw [map, card_cons_of_mem h]\n apply Nat.le_trans ih (card_le_card_cons ..)\n | inr h =>\n have : a \u2209 as := fun h'' \u21a6 h (mem_map_of_mem _ h'')\n rw [map, card_cons_of_not_mem h, card_cons_of_not_mem this]\n exact Nat.add_le_add_right ih _\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 card (map f []) \u2264 card []\n[PROOFSTEP]\n\n| nil => simp\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 card (map f []) \u2264 card []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : card (map f as) \u2264 card as\n\u22a2 card (map f (a :: as)) \u2264 card (a :: as)\n[PROOFSTEP]\n\n| cons a as ih =>\n cases Decidable.em (f a \u2208 map f as) with\n | inl h =>\n rw [map, card_cons_of_mem h]\n apply Nat.le_trans ih (card_le_card_cons ..)\n | inr h =>\n have : a \u2209 as := fun h'' \u21a6 h (mem_map_of_mem _ h'')\n rw [map, card_cons_of_not_mem h, card_cons_of_not_mem this]\n exact Nat.add_le_add_right ih _\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : card (map f as) \u2264 card as\n\u22a2 card (map f (a :: as)) \u2264 card (a :: as)\n[PROOFSTEP]\ncases Decidable.em (f a \u2208 map f as) with\n| inl h =>\n rw [map, card_cons_of_mem h]\n apply Nat.le_trans ih (card_le_card_cons ..)\n| inr h =>\n have : a \u2209 as := fun h'' \u21a6 h (mem_map_of_mem _ h'')\n rw [map, card_cons_of_not_mem h, card_cons_of_not_mem this]\n exact Nat.add_le_add_right ih _\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : card (map f as) \u2264 card as\nx\u271d : f a \u2208 map f as \u2228 \u00acf a \u2208 map f as\n\u22a2 card (map f (a :: as)) \u2264 card (a :: as)\n[PROOFSTEP]\ncases Decidable.em (f a \u2208 map f as) with\n| inl h =>\n rw [map, card_cons_of_mem h]\n apply Nat.le_trans ih (card_le_card_cons ..)\n| inr h =>\n have : a \u2209 as := fun h'' \u21a6 h (mem_map_of_mem _ h'')\n rw [map, card_cons_of_not_mem h, card_cons_of_not_mem this]\n exact Nat.add_le_add_right ih _\n[GOAL]\ncase cons.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : card (map f as) \u2264 card as\nh : f a \u2208 map f as\n\u22a2 card (map f (a :: as)) \u2264 card (a :: as)\n[PROOFSTEP]\n\n| inl h =>\n rw [map, card_cons_of_mem h]\n apply Nat.le_trans ih (card_le_card_cons ..)\n[GOAL]\ncase cons.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : card (map f as) \u2264 card as\nh : f a \u2208 map f as\n\u22a2 card (map f (a :: as)) \u2264 card (a :: as)\n[PROOFSTEP]\nrw [map, card_cons_of_mem h]\n[GOAL]\ncase cons.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : card (map f as) \u2264 card as\nh : f a \u2208 map f as\n\u22a2 card (map f as) \u2264 card (a :: as)\n[PROOFSTEP]\napply Nat.le_trans ih (card_le_card_cons ..)\n[GOAL]\ncase cons.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : card (map f as) \u2264 card as\nh : \u00acf a \u2208 map f as\n\u22a2 card (map f (a :: as)) \u2264 card (a :: as)\n[PROOFSTEP]\n\n| inr h =>\n have : a \u2209 as := fun h'' \u21a6 h (mem_map_of_mem _ h'')\n rw [map, card_cons_of_not_mem h, card_cons_of_not_mem this]\n exact Nat.add_le_add_right ih _\n[GOAL]\ncase cons.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : card (map f as) \u2264 card as\nh : \u00acf a \u2208 map f as\n\u22a2 card (map f (a :: as)) \u2264 card (a :: as)\n[PROOFSTEP]\nhave : a \u2209 as := fun h'' \u21a6 h (mem_map_of_mem _ h'')\n[GOAL]\ncase cons.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : card (map f as) \u2264 card as\nh : \u00acf a \u2208 map f as\nthis : \u00aca \u2208 as\n\u22a2 card (map f (a :: as)) \u2264 card (a :: as)\n[PROOFSTEP]\nrw [map, card_cons_of_not_mem h, card_cons_of_not_mem this]\n[GOAL]\ncase cons.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : card (map f as) \u2264 card as\nh : \u00acf a \u2208 map f as\nthis : \u00aca \u2208 as\n\u22a2 card (map f as) + 1 \u2264 card as + 1\n[PROOFSTEP]\nexact Nat.add_le_add_right ih _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\nas : List \u03b1\n\u22a2 inj_on f as \u2192 card (map f as) = card as\n[PROOFSTEP]\ninduction as with\n| nil => simp\n| cons a as ih =>\n cases Decidable.em (f a \u2208 map f as) with\n | inl h =>\n intro inj_on'\n cases (exists_of_mem_map h) with\n | intro x hx =>\n have : x = a := inj_on' (mem_cons_of_mem _ hx.1) (mem_cons_self ..) hx.2\n have h1 : a \u2208 as := this \u25b8 hx.1\n have h2 : inj_on f as := inj_on_of_subset inj_on' (subset_cons _ _)\n rw [map, card_cons_of_mem h, ih h2, card_cons_of_mem h1]\n | inr h =>\n intro inj_on'\n have h1 : a \u2209 as := fun h'' \u21a6 h (mem_map_of_mem _ h'')\n have h2 : inj_on f as := inj_on_of_subset inj_on' (subset_cons _ _)\n rw [map, card_cons_of_not_mem h, card_cons_of_not_mem h1, ih h2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\nas : List \u03b1\n\u22a2 inj_on f as \u2192 card (map f as) = card as\n[PROOFSTEP]\ninduction as with\n| nil => simp\n| cons a as ih =>\n cases Decidable.em (f a \u2208 map f as) with\n | inl h =>\n intro inj_on'\n cases (exists_of_mem_map h) with\n | intro x hx =>\n have : x = a := inj_on' (mem_cons_of_mem _ hx.1) (mem_cons_self ..) hx.2\n have h1 : a \u2208 as := this \u25b8 hx.1\n have h2 : inj_on f as := inj_on_of_subset inj_on' (subset_cons _ _)\n rw [map, card_cons_of_mem h, ih h2, card_cons_of_mem h1]\n | inr h =>\n intro inj_on'\n have h1 : a \u2209 as := fun h'' \u21a6 h (mem_map_of_mem _ h'')\n have h2 : inj_on f as := inj_on_of_subset inj_on' (subset_cons _ _)\n rw [map, card_cons_of_not_mem h, card_cons_of_not_mem h1, ih h2]\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 inj_on f [] \u2192 card (map f []) = card []\n[PROOFSTEP]\n\n| nil => simp\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 inj_on f [] \u2192 card (map f []) = card []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : inj_on f as \u2192 card (map f as) = card as\n\u22a2 inj_on f (a :: as) \u2192 card (map f (a :: as)) = card (a :: as)\n[PROOFSTEP]\n\n| cons a as ih =>\n cases Decidable.em (f a \u2208 map f as) with\n | inl h =>\n intro inj_on'\n cases (exists_of_mem_map h) with\n | intro x hx =>\n have : x = a := inj_on' (mem_cons_of_mem _ hx.1) (mem_cons_self ..) hx.2\n have h1 : a \u2208 as := this \u25b8 hx.1\n have h2 : inj_on f as := inj_on_of_subset inj_on' (subset_cons _ _)\n rw [map, card_cons_of_mem h, ih h2, card_cons_of_mem h1]\n | inr h =>\n intro inj_on'\n have h1 : a \u2209 as := fun h'' \u21a6 h (mem_map_of_mem _ h'')\n have h2 : inj_on f as := inj_on_of_subset inj_on' (subset_cons _ _)\n rw [map, card_cons_of_not_mem h, card_cons_of_not_mem h1, ih h2]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : inj_on f as \u2192 card (map f as) = card as\n\u22a2 inj_on f (a :: as) \u2192 card (map f (a :: as)) = card (a :: as)\n[PROOFSTEP]\ncases Decidable.em (f a \u2208 map f as) with\n| inl h =>\n intro inj_on'\n cases (exists_of_mem_map h) with\n | intro x hx =>\n have : x = a := inj_on' (mem_cons_of_mem _ hx.1) (mem_cons_self ..) hx.2\n have h1 : a \u2208 as := this \u25b8 hx.1\n have h2 : inj_on f as := inj_on_of_subset inj_on' (subset_cons _ _)\n rw [map, card_cons_of_mem h, ih h2, card_cons_of_mem h1]\n| inr h =>\n intro inj_on'\n have h1 : a \u2209 as := fun h'' \u21a6 h (mem_map_of_mem _ h'')\n have h2 : inj_on f as := inj_on_of_subset inj_on' (subset_cons _ _)\n rw [map, card_cons_of_not_mem h, card_cons_of_not_mem h1, ih h2]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : inj_on f as \u2192 card (map f as) = card as\nx\u271d : f a \u2208 map f as \u2228 \u00acf a \u2208 map f as\n\u22a2 inj_on f (a :: as) \u2192 card (map f (a :: as)) = card (a :: as)\n[PROOFSTEP]\ncases Decidable.em (f a \u2208 map f as) with\n| inl h =>\n intro inj_on'\n cases (exists_of_mem_map h) with\n | intro x hx =>\n have : x = a := inj_on' (mem_cons_of_mem _ hx.1) (mem_cons_self ..) hx.2\n have h1 : a \u2208 as := this \u25b8 hx.1\n have h2 : inj_on f as := inj_on_of_subset inj_on' (subset_cons _ _)\n rw [map, card_cons_of_mem h, ih h2, card_cons_of_mem h1]\n| inr h =>\n intro inj_on'\n have h1 : a \u2209 as := fun h'' \u21a6 h (mem_map_of_mem _ h'')\n have h2 : inj_on f as := inj_on_of_subset inj_on' (subset_cons _ _)\n rw [map, card_cons_of_not_mem h, card_cons_of_not_mem h1, ih h2]\n[GOAL]\ncase cons.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : inj_on f as \u2192 card (map f as) = card as\nh : f a \u2208 map f as\n\u22a2 inj_on f (a :: as) \u2192 card (map f (a :: as)) = card (a :: as)\n[PROOFSTEP]\n\n| inl h =>\n intro inj_on'\n cases (exists_of_mem_map h) with\n | intro x hx =>\n have : x = a := inj_on' (mem_cons_of_mem _ hx.1) (mem_cons_self ..) hx.2\n have h1 : a \u2208 as := this \u25b8 hx.1\n have h2 : inj_on f as := inj_on_of_subset inj_on' (subset_cons _ _)\n rw [map, card_cons_of_mem h, ih h2, card_cons_of_mem h1]\n[GOAL]\ncase cons.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : inj_on f as \u2192 card (map f as) = card as\nh : f a \u2208 map f as\n\u22a2 inj_on f (a :: as) \u2192 card (map f (a :: as)) = card (a :: as)\n[PROOFSTEP]\nintro inj_on'\n[GOAL]\ncase cons.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : inj_on f as \u2192 card (map f as) = card as\nh : f a \u2208 map f as\ninj_on' : inj_on f (a :: as)\n\u22a2 card (map f (a :: as)) = card (a :: as)\n[PROOFSTEP]\ncases (exists_of_mem_map h) with\n| intro x hx =>\n have : x = a := inj_on' (mem_cons_of_mem _ hx.1) (mem_cons_self ..) hx.2\n have h1 : a \u2208 as := this \u25b8 hx.1\n have h2 : inj_on f as := inj_on_of_subset inj_on' (subset_cons _ _)\n rw [map, card_cons_of_mem h, ih h2, card_cons_of_mem h1]\n[GOAL]\ncase cons.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : inj_on f as \u2192 card (map f as) = card as\nh : f a \u2208 map f as\ninj_on' : inj_on f (a :: as)\nx\u271d : \u2203 a_1, a_1 \u2208 as \u2227 f a_1 = f a\n\u22a2 card (map f (a :: as)) = card (a :: as)\n[PROOFSTEP]\ncases (exists_of_mem_map h) with\n| intro x hx =>\n have : x = a := inj_on' (mem_cons_of_mem _ hx.1) (mem_cons_self ..) hx.2\n have h1 : a \u2208 as := this \u25b8 hx.1\n have h2 : inj_on f as := inj_on_of_subset inj_on' (subset_cons _ _)\n rw [map, card_cons_of_mem h, ih h2, card_cons_of_mem h1]\n[GOAL]\ncase cons.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : inj_on f as \u2192 card (map f as) = card as\nh : f a \u2208 map f as\ninj_on' : inj_on f (a :: as)\nx : \u03b1\nhx : x \u2208 as \u2227 f x = f a\n\u22a2 card (map f (a :: as)) = card (a :: as)\n[PROOFSTEP]\n\n| intro x hx =>\n have : x = a := inj_on' (mem_cons_of_mem _ hx.1) (mem_cons_self ..) hx.2\n have h1 : a \u2208 as := this \u25b8 hx.1\n have h2 : inj_on f as := inj_on_of_subset inj_on' (subset_cons _ _)\n rw [map, card_cons_of_mem h, ih h2, card_cons_of_mem h1]\n[GOAL]\ncase cons.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : inj_on f as \u2192 card (map f as) = card as\nh : f a \u2208 map f as\ninj_on' : inj_on f (a :: as)\nx : \u03b1\nhx : x \u2208 as \u2227 f x = f a\n\u22a2 card (map f (a :: as)) = card (a :: as)\n[PROOFSTEP]\nhave : x = a := inj_on' (mem_cons_of_mem _ hx.1) (mem_cons_self ..) hx.2\n[GOAL]\ncase cons.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : inj_on f as \u2192 card (map f as) = card as\nh : f a \u2208 map f as\ninj_on' : inj_on f (a :: as)\nx : \u03b1\nhx : x \u2208 as \u2227 f x = f a\nthis : x = a\n\u22a2 card (map f (a :: as)) = card (a :: as)\n[PROOFSTEP]\nhave h1 : a \u2208 as := this \u25b8 hx.1\n[GOAL]\ncase cons.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : inj_on f as \u2192 card (map f as) = card as\nh : f a \u2208 map f as\ninj_on' : inj_on f (a :: as)\nx : \u03b1\nhx : x \u2208 as \u2227 f x = f a\nthis : x = a\nh1 : a \u2208 as\n\u22a2 card (map f (a :: as)) = card (a :: as)\n[PROOFSTEP]\nhave h2 : inj_on f as := inj_on_of_subset inj_on' (subset_cons _ _)\n[GOAL]\ncase cons.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : inj_on f as \u2192 card (map f as) = card as\nh : f a \u2208 map f as\ninj_on' : inj_on f (a :: as)\nx : \u03b1\nhx : x \u2208 as \u2227 f x = f a\nthis : x = a\nh1 : a \u2208 as\nh2 : inj_on f as\n\u22a2 card (map f (a :: as)) = card (a :: as)\n[PROOFSTEP]\nrw [map, card_cons_of_mem h, ih h2, card_cons_of_mem h1]\n[GOAL]\ncase cons.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : inj_on f as \u2192 card (map f as) = card as\nh : \u00acf a \u2208 map f as\n\u22a2 inj_on f (a :: as) \u2192 card (map f (a :: as)) = card (a :: as)\n[PROOFSTEP]\n\n| inr h =>\n intro inj_on'\n have h1 : a \u2209 as := fun h'' \u21a6 h (mem_map_of_mem _ h'')\n have h2 : inj_on f as := inj_on_of_subset inj_on' (subset_cons _ _)\n rw [map, card_cons_of_not_mem h, card_cons_of_not_mem h1, ih h2]\n[GOAL]\ncase cons.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : inj_on f as \u2192 card (map f as) = card as\nh : \u00acf a \u2208 map f as\n\u22a2 inj_on f (a :: as) \u2192 card (map f (a :: as)) = card (a :: as)\n[PROOFSTEP]\nintro inj_on'\n[GOAL]\ncase cons.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : inj_on f as \u2192 card (map f as) = card as\nh : \u00acf a \u2208 map f as\ninj_on' : inj_on f (a :: as)\n\u22a2 card (map f (a :: as)) = card (a :: as)\n[PROOFSTEP]\nhave h1 : a \u2209 as := fun h'' \u21a6 h (mem_map_of_mem _ h'')\n[GOAL]\ncase cons.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : inj_on f as \u2192 card (map f as) = card as\nh : \u00acf a \u2208 map f as\ninj_on' : inj_on f (a :: as)\nh1 : \u00aca \u2208 as\n\u22a2 card (map f (a :: as)) = card (a :: as)\n[PROOFSTEP]\nhave h2 : inj_on f as := inj_on_of_subset inj_on' (subset_cons _ _)\n[GOAL]\ncase cons.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nas : List \u03b1\nih : inj_on f as \u2192 card (map f as) = card as\nh : \u00acf a \u2208 map f as\ninj_on' : inj_on f (a :: as)\nh1 : \u00aca \u2208 as\nh2 : inj_on f as\n\u22a2 card (map f (a :: as)) = card (a :: as)\n[PROOFSTEP]\nrw [map, card_cons_of_not_mem h, card_cons_of_not_mem h1, ih h2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.48627\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nx\u271d\u00b9 : List \u03b1\nx\u271d : Disjoint [] x\u271d\u00b9\n\u22a2 card ([] ++ x\u271d\u00b9) = card [] + card x\u271d\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.48627\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\ndisj : Disjoint (a :: as) bs\n\u22a2 card (a :: as ++ bs) = card (a :: as) + card bs\n[PROOFSTEP]\nhave disj' : Disjoint as bs := fun _ h1 h2 \u21a6 disj (mem_cons_of_mem a h1) h2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.48627\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\ndisj : Disjoint (a :: as) bs\ndisj' : Disjoint as bs\n\u22a2 card (a :: as ++ bs) = card (a :: as) + card bs\n[PROOFSTEP]\ncases Decidable.em (a \u2208 as) with\n| inl h => simp [h, card_append_disjoint disj']\n| inr h =>\n have h1 : a \u2209 bs := fun h' \u21a6 disj (mem_cons_self a as) h'\n simp [h, h1, card_append_disjoint disj']\n rw [Nat.add_right_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.48627\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\ndisj : Disjoint (a :: as) bs\ndisj' : Disjoint as bs\nx\u271d : a \u2208 as \u2228 \u00aca \u2208 as\n\u22a2 card (a :: as ++ bs) = card (a :: as) + card bs\n[PROOFSTEP]\ncases Decidable.em (a \u2208 as) with\n| inl h => simp [h, card_append_disjoint disj']\n| inr h =>\n have h1 : a \u2209 bs := fun h' \u21a6 disj (mem_cons_self a as) h'\n simp [h, h1, card_append_disjoint disj']\n rw [Nat.add_right_comm]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.48627\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\ndisj : Disjoint (a :: as) bs\ndisj' : Disjoint as bs\nh : a \u2208 as\n\u22a2 card (a :: as ++ bs) = card (a :: as) + card bs\n[PROOFSTEP]\n\n| inl h => simp [h, card_append_disjoint disj']\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.48627\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\ndisj : Disjoint (a :: as) bs\ndisj' : Disjoint as bs\nh : a \u2208 as\n\u22a2 card (a :: as ++ bs) = card (a :: as) + card bs\n[PROOFSTEP]\nsimp [h, card_append_disjoint disj']\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.48627\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\ndisj : Disjoint (a :: as) bs\ndisj' : Disjoint as bs\nh : \u00aca \u2208 as\n\u22a2 card (a :: as ++ bs) = card (a :: as) + card bs\n[PROOFSTEP]\n\n| inr h =>\n have h1 : a \u2209 bs := fun h' \u21a6 disj (mem_cons_self a as) h'\n simp [h, h1, card_append_disjoint disj']\n rw [Nat.add_right_comm]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.48627\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\ndisj : Disjoint (a :: as) bs\ndisj' : Disjoint as bs\nh : \u00aca \u2208 as\n\u22a2 card (a :: as ++ bs) = card (a :: as) + card bs\n[PROOFSTEP]\nhave h1 : a \u2209 bs := fun h' \u21a6 disj (mem_cons_self a as) h'\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.48627\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\ndisj : Disjoint (a :: as) bs\ndisj' : Disjoint as bs\nh : \u00aca \u2208 as\nh1 : \u00aca \u2208 bs\n\u22a2 card (a :: as ++ bs) = card (a :: as) + card bs\n[PROOFSTEP]\nsimp [h, h1, card_append_disjoint disj']\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.48627\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nas bs : List \u03b1\ndisj : Disjoint (a :: as) bs\ndisj' : Disjoint as bs\nh : \u00aca \u2208 as\nh1 : \u00aca \u2208 bs\n\u22a2 card as + card bs + 1 = card as + 1 + card bs\n[PROOFSTEP]\nrw [Nat.add_right_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.54019\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nas bs : List \u03b1\nh : Disjoint as bs\n\u22a2 card (as \u222a bs) = card as + card bs\n[PROOFSTEP]\nrw [card_eq_of_equiv (union_equiv_append as bs), card_append_disjoint h]\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Card", "llama_tokens": 24582, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5312093733737563, "lm_q2_score": 0.45326184801538616, "lm_q1q2_score": 0.24077694225848403}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 g : Set \u03b1 \u2192 Filter \u03b2\n\u22a2 Filter.lift \u22a4 g = g univ\n[PROOFSTEP]\nsimp [Filter.lift]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\u271d\n\u03b9 : Sort u_6\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nf : Filter \u03b1\nhf : HasBasis f p s\u271d\n\u03b2 : \u03b9 \u2192 Type u_5\npg : (i : \u03b9) \u2192 \u03b2 i \u2192 Prop\nsg : (i : \u03b9) \u2192 \u03b2 i \u2192 Set \u03b3\ng : Set \u03b1 \u2192 Filter \u03b3\nhg : \u2200 (i : \u03b9), HasBasis (g (s\u271d i)) (pg i) (sg i)\ngm : Monotone g\ns : Set \u03b3\n\u22a2 s \u2208 Filter.lift f g \u2194 \u2203 i, p i \u2227 \u2203 x, pg i x \u2227 sg i x \u2286 s\n[PROOFSTEP]\nrefine' (mem_biInf_of_directed _ \u27e8univ, univ_sets _\u27e9).trans _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\u271d\n\u03b9 : Sort u_6\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nf : Filter \u03b1\nhf : HasBasis f p s\u271d\n\u03b2 : \u03b9 \u2192 Type u_5\npg : (i : \u03b9) \u2192 \u03b2 i \u2192 Prop\nsg : (i : \u03b9) \u2192 \u03b2 i \u2192 Set \u03b3\ng : Set \u03b1 \u2192 Filter \u03b3\nhg : \u2200 (i : \u03b9), HasBasis (g (s\u271d i)) (pg i) (sg i)\ngm : Monotone g\ns : Set \u03b3\n\u22a2 DirectedOn ((fun s => g s) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) f.sets\n[PROOFSTEP]\nintro t\u2081 ht\u2081 t\u2082 ht\u2082\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\u271d\n\u03b9 : Sort u_6\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nf : Filter \u03b1\nhf : HasBasis f p s\u271d\n\u03b2 : \u03b9 \u2192 Type u_5\npg : (i : \u03b9) \u2192 \u03b2 i \u2192 Prop\nsg : (i : \u03b9) \u2192 \u03b2 i \u2192 Set \u03b3\ng : Set \u03b1 \u2192 Filter \u03b3\nhg : \u2200 (i : \u03b9), HasBasis (g (s\u271d i)) (pg i) (sg i)\ngm : Monotone g\ns : Set \u03b3\nt\u2081 : Set \u03b1\nht\u2081 : t\u2081 \u2208 f.sets\nt\u2082 : Set \u03b1\nht\u2082 : t\u2082 \u2208 f.sets\n\u22a2 \u2203 z, z \u2208 f.sets \u2227 ((fun s => g s) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) t\u2081 z \u2227 ((fun s => g s) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) t\u2082 z\n[PROOFSTEP]\nexact \u27e8t\u2081 \u2229 t\u2082, inter_mem ht\u2081 ht\u2082, gm <| inter_subset_left _ _, gm <| inter_subset_right _ _\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\u271d\n\u03b9 : Sort u_6\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nf : Filter \u03b1\nhf : HasBasis f p s\u271d\n\u03b2 : \u03b9 \u2192 Type u_5\npg : (i : \u03b9) \u2192 \u03b2 i \u2192 Prop\nsg : (i : \u03b9) \u2192 \u03b2 i \u2192 Set \u03b3\ng : Set \u03b1 \u2192 Filter \u03b3\nhg : \u2200 (i : \u03b9), HasBasis (g (s\u271d i)) (pg i) (sg i)\ngm : Monotone g\ns : Set \u03b3\n\u22a2 (\u2203 i, i \u2208 f.sets \u2227 s \u2208 g i) \u2194 \u2203 i, p i \u2227 \u2203 x, pg i x \u2227 sg i x \u2286 s\n[PROOFSTEP]\nsimp only [\u2190 (hg _).mem_iff]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\u271d\n\u03b9 : Sort u_6\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nf : Filter \u03b1\nhf : HasBasis f p s\u271d\n\u03b2 : \u03b9 \u2192 Type u_5\npg : (i : \u03b9) \u2192 \u03b2 i \u2192 Prop\nsg : (i : \u03b9) \u2192 \u03b2 i \u2192 Set \u03b3\ng : Set \u03b1 \u2192 Filter \u03b3\nhg : \u2200 (i : \u03b9), HasBasis (g (s\u271d i)) (pg i) (sg i)\ngm : Monotone g\ns : Set \u03b3\n\u22a2 (\u2203 i, i \u2208 f.sets \u2227 s \u2208 g i) \u2194 \u2203 i, p i \u2227 s \u2208 g (s\u271d i)\n[PROOFSTEP]\nexact hf.exists_iff fun t\u2081 t\u2082 ht H => gm ht H\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\u271d\n\u03b9 : Type u_6\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nf : Filter \u03b1\nhf : HasBasis f p s\n\u03b2 : \u03b9 \u2192 Type u_5\npg : (i : \u03b9) \u2192 \u03b2 i \u2192 Prop\nsg : (i : \u03b9) \u2192 \u03b2 i \u2192 Set \u03b3\ng : Set \u03b1 \u2192 Filter \u03b3\nhg : \u2200 (i : \u03b9), HasBasis (g (s i)) (pg i) (sg i)\ngm : Monotone g\n\u22a2 HasBasis (Filter.lift f g) (fun i => p i.fst \u2227 pg i.fst i.snd) fun i => sg i.fst i.snd\n[PROOFSTEP]\nrefine' \u27e8fun t => (hf.mem_lift_iff hg gm).trans _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\u271d\n\u03b9 : Type u_6\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nf : Filter \u03b1\nhf : HasBasis f p s\n\u03b2 : \u03b9 \u2192 Type u_5\npg : (i : \u03b9) \u2192 \u03b2 i \u2192 Prop\nsg : (i : \u03b9) \u2192 \u03b2 i \u2192 Set \u03b3\ng : Set \u03b1 \u2192 Filter \u03b3\nhg : \u2200 (i : \u03b9), HasBasis (g (s i)) (pg i) (sg i)\ngm : Monotone g\nt : Set \u03b3\n\u22a2 (\u2203 i, p i \u2227 \u2203 x, pg i x \u2227 sg i x \u2286 t) \u2194 \u2203 i, (p i.fst \u2227 pg i.fst i.snd) \u2227 sg i.fst i.snd \u2286 t\n[PROOFSTEP]\nsimp [Sigma.exists, and_assoc, exists_and_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\nhg : Monotone g\ns : Set \u03b2\n\u22a2 (\u2203 i, i \u2208 f \u2227 \u2203 x, x \u2208 g i \u2227 id x \u2286 s) \u2194 \u2203 t, t \u2208 f \u2227 s \u2208 g t\n[PROOFSTEP]\nsimp only [id, exists_mem_subset_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\nhg : Monotone g\n\u22a2 \u22c2\u2080 {s | s \u2208 Filter.lift f g} = \u22c2 (s : Set \u03b1) (_ : s \u2208 f), \u22c2\u2080 {t | t \u2208 g s}\n[PROOFSTEP]\nsimp only [sInter_eq_biInter, mem_setOf_eq, Filter.mem_sets, mem_lift_sets hg, iInter_exists, iInter_and,\n @iInter_comm _ (Set \u03b2)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\nm : \u03b3 \u2192 \u03b2\nl : Filter \u03b3\n\u22a2 Tendsto m l (Filter.lift f g) \u2194 \u2200 (s : Set \u03b1), s \u2208 f \u2192 Tendsto m l (g s)\n[PROOFSTEP]\nsimp only [Filter.lift, tendsto_iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\nm : \u03b2 \u2192 \u03b3\nhg : Monotone g\nthis : Monotone (map m \u2218 g)\ns : Set \u03b3\n\u22a2 s \u2208 map m (Filter.lift f g) \u2194 s \u2208 Filter.lift f (map m \u2218 g)\n[PROOFSTEP]\nsimp only [mem_lift_sets hg, mem_lift_sets this, exists_prop, mem_map, Function.comp_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\nm : \u03b3 \u2192 \u03b2\n\u22a2 comap m (Filter.lift f g) = Filter.lift f (comap m \u2218 g)\n[PROOFSTEP]\nsimp only [Filter.lift, comap_iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\nm : \u03b3 \u2192 \u03b2\n\u22a2 \u2a05 (i : Set \u03b1) (_ : i \u2208 f), comap m (g i) = \u2a05 (s : Set \u03b1) (_ : s \u2208 f), (comap m \u2218 g) s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\nhm : Monotone g\n\u22a2 NeBot (Filter.lift f g) \u2194 \u2200 (s : Set \u03b1), s \u2208 f \u2192 NeBot (g s)\n[PROOFSTEP]\nsimp only [neBot_iff, Ne.def, \u2190 empty_mem_iff_bot, mem_lift_sets hm, not_exists, not_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\nf : Filter \u03b1\ng h : Set \u03b1 \u2192 Filter \u03b2\n\u22a2 (Filter.lift f fun x => g x \u2293 h x) = Filter.lift f g \u2293 Filter.lift f h\n[PROOFSTEP]\nsimp only [Filter.lift, iInf_inf_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\nf : Filter \u03b1\ns : Set \u03b1\nhs : s \u2208 f\n\u22a2 f \u2264 \ud835\udcdf s\n[PROOFSTEP]\nsimp only [hs, le_principal_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Filter \u03b2\nhg : \u2200 (s t : Set \u03b1), g (s \u2229 t) = g s \u2293 g t\n\u22a2 Filter.lift (iInf f) g = \u2a05 (i : \u03b9), Filter.lift (f i) g\n[PROOFSTEP]\nrefine' lift_iInf_le.antisymm fun s => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Filter \u03b2\nhg : \u2200 (s t : Set \u03b1), g (s \u2229 t) = g s \u2293 g t\ns : Set \u03b2\n\u22a2 s \u2208 Filter.lift (iInf f) g \u2192 s \u2208 \u2a05 (i : \u03b9), Filter.lift (f i) g\n[PROOFSTEP]\nhave H : \u2200 t \u2208 iInf f, \u2a05 i, (f i).lift g \u2264 g t := by\n intro t ht\n refine' iInf_sets_induct ht _ fun hs ht => _\n \u00b7 inhabit \u03b9\n exact iInf\u2082_le_of_le default univ (iInf_le _ univ_mem)\n \u00b7 rw [hg]\n exact le_inf (iInf\u2082_le_of_le _ _ <| iInf_le _ hs) ht\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Filter \u03b2\nhg : \u2200 (s t : Set \u03b1), g (s \u2229 t) = g s \u2293 g t\ns : Set \u03b2\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 iInf f \u2192 \u2a05 (i : \u03b9), Filter.lift (f i) g \u2264 g t\n[PROOFSTEP]\nintro t ht\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Filter \u03b2\nhg : \u2200 (s t : Set \u03b1), g (s \u2229 t) = g s \u2293 g t\ns : Set \u03b2\nt : Set \u03b1\nht : t \u2208 iInf f\n\u22a2 \u2a05 (i : \u03b9), Filter.lift (f i) g \u2264 g t\n[PROOFSTEP]\nrefine' iInf_sets_induct ht _ fun hs ht => _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Filter \u03b2\nhg : \u2200 (s t : Set \u03b1), g (s \u2229 t) = g s \u2293 g t\ns : Set \u03b2\nt : Set \u03b1\nht : t \u2208 iInf f\n\u22a2 \u2a05 (i : \u03b9), Filter.lift (f i) g \u2264 g univ\n[PROOFSTEP]\ninhabit \u03b9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Filter \u03b2\nhg : \u2200 (s t : Set \u03b1), g (s \u2229 t) = g s \u2293 g t\ns : Set \u03b2\nt : Set \u03b1\nht : t \u2208 iInf f\ninhabited_h : Inhabited \u03b9\n\u22a2 \u2a05 (i : \u03b9), Filter.lift (f i) g \u2264 g univ\n[PROOFSTEP]\nexact iInf\u2082_le_of_le default univ (iInf_le _ univ_mem)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Filter \u03b2\nhg : \u2200 (s t : Set \u03b1), g (s \u2229 t) = g s \u2293 g t\ns : Set \u03b2\nt : Set \u03b1\nht\u271d : t \u2208 iInf f\ni\u271d : \u03b9\ns\u2081\u271d s\u2082\u271d : Set \u03b1\nhs : s\u2081\u271d \u2208 f i\u271d\nht : \u2a05 (i : \u03b9), Filter.lift (f i) g \u2264 g s\u2082\u271d\n\u22a2 \u2a05 (i : \u03b9), Filter.lift (f i) g \u2264 g (s\u2081\u271d \u2229 s\u2082\u271d)\n[PROOFSTEP]\nrw [hg]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Filter \u03b2\nhg : \u2200 (s t : Set \u03b1), g (s \u2229 t) = g s \u2293 g t\ns : Set \u03b2\nt : Set \u03b1\nht\u271d : t \u2208 iInf f\ni\u271d : \u03b9\ns\u2081\u271d s\u2082\u271d : Set \u03b1\nhs : s\u2081\u271d \u2208 f i\u271d\nht : \u2a05 (i : \u03b9), Filter.lift (f i) g \u2264 g s\u2082\u271d\n\u22a2 \u2a05 (i : \u03b9), Filter.lift (f i) g \u2264 g s\u2081\u271d \u2293 g s\u2082\u271d\n[PROOFSTEP]\nexact le_inf (iInf\u2082_le_of_le _ _ <| iInf_le _ hs) ht\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Filter \u03b2\nhg : \u2200 (s t : Set \u03b1), g (s \u2229 t) = g s \u2293 g t\ns : Set \u03b2\nH : \u2200 (t : Set \u03b1), t \u2208 iInf f \u2192 \u2a05 (i : \u03b9), Filter.lift (f i) g \u2264 g t\n\u22a2 s \u2208 Filter.lift (iInf f) g \u2192 s \u2208 \u2a05 (i : \u03b9), Filter.lift (f i) g\n[PROOFSTEP]\nsimp only [mem_lift_sets (Monotone.of_map_inf hg), exists_imp, and_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Filter \u03b2\nhg : \u2200 (s t : Set \u03b1), g (s \u2229 t) = g s \u2293 g t\ns : Set \u03b2\nH : \u2200 (t : Set \u03b1), t \u2208 iInf f \u2192 \u2a05 (i : \u03b9), Filter.lift (f i) g \u2264 g t\n\u22a2 \u2200 (x : Set \u03b1), x \u2208 iInf f \u2192 s \u2208 g x \u2192 s \u2208 \u2a05 (i : \u03b9), Filter.lift (f i) g\n[PROOFSTEP]\nexact fun t ht hs => H t ht hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Filter \u03b2\nhf : Directed (fun x x_1 => x \u2265 x_1) f\nhg : Monotone g\ns : Set \u03b2\n\u22a2 s \u2208 Filter.lift (iInf f) g \u2192 s \u2208 \u2a05 (i : \u03b9), Filter.lift (f i) g\n[PROOFSTEP]\nsimp only [mem_lift_sets hg, exists_imp, and_imp, mem_iInf_of_directed hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Filter \u03b2\nhf : Directed (fun x x_1 => x \u2265 x_1) f\nhg : Monotone g\ns : Set \u03b2\n\u22a2 \u2200 (x : Set \u03b1) (x_1 : \u03b9), x \u2208 f x_1 \u2192 s \u2208 g x \u2192 s \u2208 \u2a05 (i : \u03b9), Filter.lift (f i) g\n[PROOFSTEP]\nexact fun t i ht hs => mem_iInf_of_mem i <| mem_lift ht hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Filter \u03b2\nhg : \u2200 (s t : Set \u03b1), g (s \u2229 t) = g s \u2293 g t\nhg' : g univ = \u22a4\n\u22a2 Filter.lift (iInf f) g = \u2a05 (i : \u03b9), Filter.lift (f i) g\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b9\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Filter \u03b2\nhg : \u2200 (s t : Set \u03b1), g (s \u2229 t) = g s \u2293 g t\nhg' : g univ = \u22a4\nh\u271d : IsEmpty \u03b9\n\u22a2 Filter.lift (iInf f) g = \u2a05 (i : \u03b9), Filter.lift (f i) g\n[PROOFSTEP]\nsimp [iInf_of_empty, hg']\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Filter \u03b2\nhg : \u2200 (s t : Set \u03b1), g (s \u2229 t) = g s \u2293 g t\nhg' : g univ = \u22a4\nh\u271d : Nonempty \u03b9\n\u22a2 Filter.lift (iInf f) g = \u2a05 (i : \u03b9), Filter.lift (f i) g\n[PROOFSTEP]\nexact lift_iInf hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\nh h\u2081 h\u2082 : Set \u03b1 \u2192 Set \u03b2\nm : \u03b3 \u2192 \u03b2\nl : Filter \u03b3\n\u22a2 Tendsto m l (Filter.lift' f h) \u2194 \u2200 (s : Set \u03b1), s \u2208 f \u2192 \u2200\u1da0 (a : \u03b3) in l, m a \u2208 h s\n[PROOFSTEP]\nsimp only [Filter.lift', tendsto_lift, tendsto_principal, comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\nh h\u2081 h\u2082 : Set \u03b1 \u2192 Set \u03b2\n\u03b9 : Sort u_5\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nhf : HasBasis f p s\nhh : Monotone h\nt : Set \u03b2\n\u22a2 (\u2203 i, p i \u2227 \u2203 x, True \u2227 h (s i) \u2286 t) \u2194 \u2203 i, p i \u2227 (h \u2218 s) i \u2286 t\n[PROOFSTEP]\nsimp only [exists_const, true_and, comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\nh h\u2081 h\u2082 : Set \u03b1 \u2192 Set \u03b2\nm : \u03b2 \u2192 \u03b3\nhh : Monotone h\n\u22a2 Filter.lift f (map m \u2218 \ud835\udcdf \u2218 h) = Filter.lift' f (image m \u2218 h)\n[PROOFSTEP]\nsimp only [comp, Filter.lift', map_principal]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\nh h\u2081 h\u2082 : Set \u03b1 \u2192 Set \u03b2\nm : \u03b3 \u2192 \u03b2\n\u22a2 comap m (Filter.lift' f h) = Filter.lift' f (preimage m \u2218 h)\n[PROOFSTEP]\nsimp only [Filter.lift', comap_lift_eq, (\u00b7 \u2218 \u00b7), comap_principal]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\nh h\u2081 h\u2082 : Set \u03b1 \u2192 Set \u03b2\na : \u03b1\nhh : Monotone h\n\u22a2 Filter.lift' (pure a) h = \ud835\udcdf (h {a})\n[PROOFSTEP]\nrw [\u2190 principal_singleton, lift'_principal hh]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\nh h\u2081 h\u2082 : Set \u03b1 \u2192 Set \u03b2\nhh : Monotone h\n\u22a2 Filter.lift' \u22a5 h = \ud835\udcdf (h \u2205)\n[PROOFSTEP]\nrw [\u2190 principal_empty, lift'_principal hh]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\nh\u271d h\u2081 h\u2082 g : Set \u03b1 \u2192 Set \u03b2\nh : Set \u03b2 \u2192 Filter \u03b3\nhg : Monotone g\nhh : Monotone h\n\u22a2 (Filter.lift f fun s => Filter.lift (\ud835\udcdf (g s)) h) = Filter.lift f fun s => h (g s)\n[PROOFSTEP]\nsimp only [lift_principal, hh, eq_self_iff_true]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\nh\u271d h\u2081 h\u2082 h : Set \u03b1 \u2192 Set \u03b2\ns : Set \u03b2\n\u22a2 Filter.lift' f h \u2293 \ud835\udcdf s = Filter.lift' f fun t => h t \u2229 s\n[PROOFSTEP]\nsimp only [Filter.lift', Filter.lift, (\u00b7 \u2218 \u00b7), \u2190 inf_principal, iInf_subtype', \u2190 iInf_inf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\nh h\u2081 h\u2082 : Set \u03b1 \u2192 Set \u03b2\nhh : Monotone h\n\u22a2 (\u2200 (s : Set \u03b1), s \u2208 f \u2192 NeBot (\ud835\udcdf (h s))) \u2194 \u2200 (s : Set \u03b1), s \u2208 f \u2192 Set.Nonempty (h s)\n[PROOFSTEP]\nsimp only [principal_neBot_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\nh h\u2081 h\u2082 : Set \u03b1 \u2192 Set \u03b2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Set \u03b2\nhg : \u2200 (s t : Set \u03b1), g (s \u2229 t) = g s \u2229 g t\ns t : Set \u03b1\n\u22a2 (\ud835\udcdf \u2218 g) (s \u2229 t) = (\ud835\udcdf \u2218 g) s \u2293 (\ud835\udcdf \u2218 g) t\n[PROOFSTEP]\nsimp only [inf_principal, comp, hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\nh h\u2081 h\u2082 : Set \u03b1 \u2192 Set \u03b2\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Set \u03b2\nhg : \u2200 {s t : Set \u03b1}, g (s \u2229 t) = g s \u2229 g t\nhg' : g univ = univ\ns t : Set \u03b1\n\u22a2 (\ud835\udcdf \u2218 g) (s \u2229 t) = (\ud835\udcdf \u2218 g) s \u2293 (\ud835\udcdf \u2218 g) t\n[PROOFSTEP]\nsimp only [inf_principal, comp, hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\nh h\u2081 h\u2082 : Set \u03b1 \u2192 Set \u03b2\nf : \u03b9 \u2192 Filter \u03b1\ng : Set \u03b1 \u2192 Set \u03b2\nhg : \u2200 {s t : Set \u03b1}, g (s \u2229 t) = g s \u2229 g t\nhg' : g univ = univ\n\u22a2 (\ud835\udcdf \u2218 g) univ = \u22a4\n[PROOFSTEP]\nrw [Function.comp_apply, hg', principal_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\nh h\u2081 h\u2082 : Set \u03b1 \u2192 Set \u03b2\nf g : Filter \u03b1\ns : Set \u03b1 \u2192 Set \u03b2\nhs : \u2200 (t\u2081 t\u2082 : Set \u03b1), s (t\u2081 \u2229 t\u2082) = s t\u2081 \u2229 s t\u2082\n\u22a2 Filter.lift' (f \u2293 g) s = Filter.lift' f s \u2293 Filter.lift' g s\n[PROOFSTEP]\nrw [inf_eq_iInf, inf_eq_iInf, lift'_iInf hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\nh h\u2081 h\u2082 : Set \u03b1 \u2192 Set \u03b2\nf g : Filter \u03b1\ns : Set \u03b1 \u2192 Set \u03b2\nhs : \u2200 (t\u2081 t\u2082 : Set \u03b1), s (t\u2081 \u2229 t\u2082) = s t\u2081 \u2229 s t\u2082\n\u22a2 \u2a05 (i : Bool), Filter.lift' (bif i then f else g) s = \u2a05 (b : Bool), bif b then Filter.lift' f s else Filter.lift' g s\n[PROOFSTEP]\nrefine iInf_congr ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\nh h\u2081 h\u2082 : Set \u03b1 \u2192 Set \u03b2\nf g : Filter \u03b1\ns : Set \u03b1 \u2192 Set \u03b2\nhs : \u2200 (t\u2081 t\u2082 : Set \u03b1), s (t\u2081 \u2229 t\u2082) = s t\u2081 \u2229 s t\u2082\n\u22a2 \u2200 (i : Bool), Filter.lift' (bif i then f else g) s = bif i then Filter.lift' f s else Filter.lift' g s\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase false\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\nh h\u2081 h\u2082 : Set \u03b1 \u2192 Set \u03b2\nf g : Filter \u03b1\ns : Set \u03b1 \u2192 Set \u03b2\nhs : \u2200 (t\u2081 t\u2082 : Set \u03b1), s (t\u2081 \u2229 t\u2082) = s t\u2081 \u2229 s t\u2082\n\u22a2 Filter.lift' (bif false then f else g) s = bif false then Filter.lift' f s else Filter.lift' g s\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\nh h\u2081 h\u2082 : Set \u03b1 \u2192 Set \u03b2\nf g : Filter \u03b1\ns : Set \u03b1 \u2192 Set \u03b2\nhs : \u2200 (t\u2081 t\u2082 : Set \u03b1), s (t\u2081 \u2229 t\u2082) = s t\u2081 \u2229 s t\u2082\n\u22a2 Filter.lift' (bif true then f else g) s = bif true then Filter.lift' f s else Filter.lift' g s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f : Filter \u03b1\ng : Filter \u03b2\n\u22a2 f \u00d7\u02e2 g = Filter.lift f fun s => Filter.lift' g fun t => s \u00d7\u02e2 t\n[PROOFSTEP]\nsimpa only [Filter.lift', Filter.lift, (f.basis_sets.prod g.basis_sets).eq_biInf, iInf_prod, iInf_and] using\n iInf_congr fun i => iInf_comm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d : Filter \u03b1\nf : \u03b1 \u00d7 \u03b1 \u2192 \u03b2\nx : Filter \u03b1\ny : Filter \u03b2\n\u22a2 Tendsto f (x \u00d7\u02e2 x) y \u2194 \u2200 (W : Set \u03b2), W \u2208 y \u2192 \u2203 U, U \u2208 x \u2227 \u2200 (x x' : \u03b1), x \u2208 U \u2192 x' \u2208 U \u2192 f (x, x') \u2208 W\n[PROOFSTEP]\nsimp only [tendsto_def, mem_prod_same_iff, prod_sub_preimage_iff, exists_prop, iff_self_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf : Filter \u03b1\n\u03b1\u2081 : Type u_5\n\u03b1\u2082 : Type u_6\n\u03b2\u2081 : Type u_7\n\u03b2\u2082 : Type u_8\nf\u2081 : Filter \u03b1\u2081\nf\u2082 : Filter \u03b1\u2082\ng\u2081 : Set \u03b1\u2081 \u2192 Filter \u03b2\u2081\ng\u2082 : Set \u03b1\u2082 \u2192 Filter \u03b2\u2082\nhg\u2081 : Monotone g\u2081\nhg\u2082 : Monotone g\u2082\n\u22a2 Filter.lift f\u2081 g\u2081 \u00d7\u02e2 Filter.lift f\u2082 g\u2082 = Filter.lift f\u2081 fun s => Filter.lift f\u2082 fun t => g\u2081 s \u00d7\u02e2 g\u2082 t\n[PROOFSTEP]\nsimp only [prod_def, lift_assoc hg\u2081]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf : Filter \u03b1\n\u03b1\u2081 : Type u_5\n\u03b1\u2082 : Type u_6\n\u03b2\u2081 : Type u_7\n\u03b2\u2082 : Type u_8\nf\u2081 : Filter \u03b1\u2081\nf\u2082 : Filter \u03b1\u2082\ng\u2081 : Set \u03b1\u2081 \u2192 Filter \u03b2\u2081\ng\u2082 : Set \u03b1\u2082 \u2192 Filter \u03b2\u2082\nhg\u2081 : Monotone g\u2081\nhg\u2082 : Monotone g\u2082\n\u22a2 (Filter.lift f\u2081 fun s => Filter.lift (g\u2081 s) fun s => Filter.lift' (Filter.lift f\u2082 g\u2082) fun t => s \u00d7\u02e2 t) =\n Filter.lift f\u2081 fun s => Filter.lift f\u2082 fun t => Filter.lift (g\u2081 s) fun s => Filter.lift' (g\u2082 t) fun t => s \u00d7\u02e2 t\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf : Filter \u03b1\n\u03b1\u2081 : Type u_5\n\u03b1\u2082 : Type u_6\n\u03b2\u2081 : Type u_7\n\u03b2\u2082 : Type u_8\nf\u2081 : Filter \u03b1\u2081\nf\u2082 : Filter \u03b1\u2082\ng\u2081 : Set \u03b1\u2081 \u2192 Filter \u03b2\u2081\ng\u2082 : Set \u03b1\u2082 \u2192 Filter \u03b2\u2082\nhg\u2081 : Monotone g\u2081\nhg\u2082 : Monotone g\u2082\n\u22a2 (fun s => Filter.lift (g\u2081 s) fun s => Filter.lift' (Filter.lift f\u2082 g\u2082) fun t => s \u00d7\u02e2 t) = fun s =>\n Filter.lift f\u2082 fun t => Filter.lift (g\u2081 s) fun s => Filter.lift' (g\u2082 t) fun t => s \u00d7\u02e2 t\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf : Filter \u03b1\n\u03b1\u2081 : Type u_5\n\u03b1\u2082 : Type u_6\n\u03b2\u2081 : Type u_7\n\u03b2\u2082 : Type u_8\nf\u2081 : Filter \u03b1\u2081\nf\u2082 : Filter \u03b1\u2082\ng\u2081 : Set \u03b1\u2081 \u2192 Filter \u03b2\u2081\ng\u2082 : Set \u03b1\u2082 \u2192 Filter \u03b2\u2082\nhg\u2081 : Monotone g\u2081\nhg\u2082 : Monotone g\u2082\nx : Set \u03b1\u2081\n\u22a2 (Filter.lift (g\u2081 x) fun s => Filter.lift' (Filter.lift f\u2082 g\u2082) fun t => s \u00d7\u02e2 t) =\n Filter.lift f\u2082 fun t => Filter.lift (g\u2081 x) fun s => Filter.lift' (g\u2082 t) fun t => s \u00d7\u02e2 t\n[PROOFSTEP]\nrw [lift_comm]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf : Filter \u03b1\n\u03b1\u2081 : Type u_5\n\u03b1\u2082 : Type u_6\n\u03b2\u2081 : Type u_7\n\u03b2\u2082 : Type u_8\nf\u2081 : Filter \u03b1\u2081\nf\u2082 : Filter \u03b1\u2082\ng\u2081 : Set \u03b1\u2081 \u2192 Filter \u03b2\u2081\ng\u2082 : Set \u03b1\u2082 \u2192 Filter \u03b2\u2082\nhg\u2081 : Monotone g\u2081\nhg\u2082 : Monotone g\u2082\nx : Set \u03b1\u2081\n\u22a2 (Filter.lift (g\u2081 x) fun s => Filter.lift' (Filter.lift f\u2082 g\u2082) fun t => s \u00d7\u02e2 t) =\n Filter.lift (g\u2081 x) fun t => Filter.lift f\u2082 fun s => Filter.lift' (g\u2082 s) fun t_1 => t \u00d7\u02e2 t_1\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf : Filter \u03b1\n\u03b1\u2081 : Type u_5\n\u03b1\u2082 : Type u_6\n\u03b2\u2081 : Type u_7\n\u03b2\u2082 : Type u_8\nf\u2081 : Filter \u03b1\u2081\nf\u2082 : Filter \u03b1\u2082\ng\u2081 : Set \u03b1\u2081 \u2192 Filter \u03b2\u2081\ng\u2082 : Set \u03b1\u2082 \u2192 Filter \u03b2\u2082\nhg\u2081 : Monotone g\u2081\nhg\u2082 : Monotone g\u2082\nx : Set \u03b1\u2081\n\u22a2 (fun s => Filter.lift' (Filter.lift f\u2082 g\u2082) fun t => s \u00d7\u02e2 t) = fun t =>\n Filter.lift f\u2082 fun s => Filter.lift' (g\u2082 s) fun t_1 => t \u00d7\u02e2 t_1\n[PROOFSTEP]\nfunext y\n[GOAL]\ncase h.h.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf : Filter \u03b1\n\u03b1\u2081 : Type u_5\n\u03b1\u2082 : Type u_6\n\u03b2\u2081 : Type u_7\n\u03b2\u2082 : Type u_8\nf\u2081 : Filter \u03b1\u2081\nf\u2082 : Filter \u03b1\u2082\ng\u2081 : Set \u03b1\u2081 \u2192 Filter \u03b2\u2081\ng\u2082 : Set \u03b1\u2082 \u2192 Filter \u03b2\u2082\nhg\u2081 : Monotone g\u2081\nhg\u2082 : Monotone g\u2082\nx : Set \u03b1\u2081\ny : Set \u03b2\u2081\n\u22a2 (Filter.lift' (Filter.lift f\u2082 g\u2082) fun t => y \u00d7\u02e2 t) = Filter.lift f\u2082 fun s => Filter.lift' (g\u2082 s) fun t => y \u00d7\u02e2 t\n[PROOFSTEP]\napply lift'_lift_assoc hg\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf : Filter \u03b1\n\u03b1\u2081 : Type u_5\n\u03b1\u2082 : Type u_6\n\u03b2\u2081 : Type u_7\n\u03b2\u2082 : Type u_8\nf\u2081 : Filter \u03b1\u2081\nf\u2082 : Filter \u03b1\u2082\ng\u2081 : Set \u03b1\u2081 \u2192 Set \u03b2\u2081\ng\u2082 : Set \u03b1\u2082 \u2192 Set \u03b2\u2082\nhg\u2081 : Monotone g\u2081\nhg\u2082 : Monotone g\u2082\n\u22a2 (Filter.lift f\u2081 fun s => Filter.lift f\u2082 fun t => \ud835\udcdf (g\u2081 s) \u00d7\u02e2 \ud835\udcdf (g\u2082 t)) =\n Filter.lift f\u2081 fun s => Filter.lift f\u2082 fun t => \ud835\udcdf (g\u2081 s \u00d7\u02e2 g\u2082 t)\n[PROOFSTEP]\n{simp only [prod_principal_principal]\n}\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf : Filter \u03b1\n\u03b1\u2081 : Type u_5\n\u03b1\u2082 : Type u_6\n\u03b2\u2081 : Type u_7\n\u03b2\u2082 : Type u_8\nf\u2081 : Filter \u03b1\u2081\nf\u2082 : Filter \u03b1\u2082\ng\u2081 : Set \u03b1\u2081 \u2192 Set \u03b2\u2081\ng\u2082 : Set \u03b1\u2082 \u2192 Set \u03b2\u2082\nhg\u2081 : Monotone g\u2081\nhg\u2082 : Monotone g\u2082\n\u22a2 (Filter.lift f\u2081 fun s => Filter.lift f\u2082 fun t => \ud835\udcdf (g\u2081 s) \u00d7\u02e2 \ud835\udcdf (g\u2082 t)) =\n Filter.lift f\u2081 fun s => Filter.lift f\u2082 fun t => \ud835\udcdf (g\u2081 s \u00d7\u02e2 g\u2082 t)\n[PROOFSTEP]\nsimp only [prod_principal_principal]\n", "meta": {"mathlib_filename": "Mathlib.Order.Filter.Lift", "llama_tokens": 11513, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.4035668537353746, "lm_q1q2_score": 0.23918059505750586}} {"text": "[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191(toTopCat X)\n\u22a2 \u00acr \u2208 (to\u0393SpecFun X x).asIdeal \u2194 IsUnit (\u2191(\u0393ToStalk X x) r)\n[PROOFSTEP]\nerw [LocalRing.mem_maximalIdeal, Classical.not_not]\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\n\u22a2 to\u0393SpecFun X \u207b\u00b9' (basicOpen r).carrier = (RingedSpace.basicOpen (toRingedSpace X) r).carrier\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx\u271d : \u2191(toTopCat X)\n\u22a2 x\u271d \u2208 to\u0393SpecFun X \u207b\u00b9' (basicOpen r).carrier \u2194 x\u271d \u2208 (RingedSpace.basicOpen (toRingedSpace X) r).carrier\n[PROOFSTEP]\nerw [X.toRingedSpace.mem_top_basicOpen]\n[GOAL]\ncase h\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx\u271d : \u2191(toTopCat X)\n\u22a2 x\u271d \u2208 to\u0393SpecFun X \u207b\u00b9' (basicOpen r).carrier \u2194\n IsUnit (\u2191(germ (toRingedSpace X).toPresheafedSpace.presheaf { val := x\u271d, property := (_ : x\u271d \u2208 \u22a4) }) r)\n[PROOFSTEP]\napply not_mem_prime_iff_unit_in_stalk\n[GOAL]\nX : LocallyRingedSpace\n\u22a2 Continuous (to\u0393SpecFun X)\n[PROOFSTEP]\napply isTopologicalBasis_basic_opens.continuous\n[GOAL]\ncase hf\nX : LocallyRingedSpace\n\u22a2 \u2200 (s : Set (PrimeSpectrum \u2191(\u0393.obj (op X)))), (s \u2208 Set.range fun r => \u2191(basicOpen r)) \u2192 IsOpen (to\u0393SpecFun X \u207b\u00b9' s)\n[PROOFSTEP]\nrintro _ \u27e8r, rfl\u27e9\n[GOAL]\ncase hf.intro\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\n\u22a2 IsOpen (to\u0393SpecFun X \u207b\u00b9' (fun r => \u2191(basicOpen r)) r)\n[PROOFSTEP]\nerw [X.to\u0393Spec_preim_basicOpen_eq r]\n[GOAL]\ncase hf.intro\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\n\u22a2 IsOpen (RingedSpace.basicOpen (toRingedSpace X) r).carrier\n[PROOFSTEP]\nexact (X.toRingedSpace.basicOpen r).2\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\n\u22a2 IsUnit (\u2191(toTo\u0393SpecMapBasicOpen X r) r)\n[PROOFSTEP]\nconvert\n (X.presheaf.map <| (eqToHom <| X.to\u0393SpecMapBasicOpen_eq r).op).isUnit_map\n (X.toRingedSpace.isUnit_res_basicOpen r)\n -- Porting note : `rw [comp_apply]` to `erw [comp_apply]`\n[GOAL]\ncase h.e'_3\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\n\u22a2 \u2191(toTo\u0393SpecMapBasicOpen X r) r =\n \u2191(X.presheaf.map (eqToHom (_ : to\u0393SpecMapBasicOpen X r = RingedSpace.basicOpen (toRingedSpace X) r)).op)\n (\u2191((toRingedSpace X).toPresheafedSpace.presheaf.map\n (homOfLE (_ : RingedSpace.basicOpen (toRingedSpace X) r \u2264 \u22a4)).op)\n r)\n[PROOFSTEP]\nerw [\u2190 comp_apply, \u2190 Functor.map_comp]\n[GOAL]\ncase h.e'_3\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\n\u22a2 \u2191(toTo\u0393SpecMapBasicOpen X r) r =\n \u2191((toRingedSpace X).toPresheafedSpace.presheaf.map\n ((homOfLE (_ : RingedSpace.basicOpen (toRingedSpace X) r \u2264 \u22a4)).op \u226b\n (eqToHom (_ : to\u0393SpecMapBasicOpen X r = RingedSpace.basicOpen (toRingedSpace X) r)).op))\n r\n[PROOFSTEP]\ncongr\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nf : (structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r)) \u27f6 X.presheaf.obj (op (to\u0393SpecMapBasicOpen X r))\n\u22a2 toOpen (\u2191(\u0393.obj (op X))) (basicOpen r) \u226b f = toTo\u0393SpecMapBasicOpen X r \u2194 f = to\u0393SpecCApp X r\n[PROOFSTEP]\nhave loc_inst := IsLocalization.to_basicOpen (\u0393.obj (op X)) r\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nf : (structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r)) \u27f6 X.presheaf.obj (op (to\u0393SpecMapBasicOpen X r))\nloc_inst : IsLocalization.Away r \u2191((structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r)))\n\u22a2 toOpen (\u2191(\u0393.obj (op X))) (basicOpen r) \u226b f = toTo\u0393SpecMapBasicOpen X r \u2194 f = to\u0393SpecCApp X r\n[PROOFSTEP]\nrw [\u2190 @IsLocalization.Away.AwayMap.lift_comp _ _ _ _ _ _ _ r loc_inst _ (X.isUnit_res_to\u0393SpecMapBasicOpen r)]\n --pick_goal 5; exact is_localization.to_basic_open _ r\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nf : (structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r)) \u27f6 X.presheaf.obj (op (to\u0393SpecMapBasicOpen X r))\nloc_inst : IsLocalization.Away r \u2191((structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r)))\n\u22a2 toOpen (\u2191(\u0393.obj (op X))) (basicOpen r) \u226b f =\n RingHom.comp (IsLocalization.Away.lift r (_ : IsUnit (\u2191(toTo\u0393SpecMapBasicOpen X r) r)))\n (algebraMap \u2191(\u0393.obj (op X)) \u2191((structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r)))) \u2194\n f = to\u0393SpecCApp X r\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nf : (structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r)) \u27f6 X.presheaf.obj (op (to\u0393SpecMapBasicOpen X r))\nloc_inst : IsLocalization.Away r \u2191((structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r)))\n\u22a2 toOpen (\u2191(\u0393.obj (op X))) (basicOpen r) \u226b f =\n RingHom.comp (IsLocalization.Away.lift r (_ : IsUnit (\u2191(toTo\u0393SpecMapBasicOpen X r) r)))\n (algebraMap \u2191(\u0393.obj (op X)) \u2191((structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r)))) \u2192\n f = to\u0393SpecCApp X r\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nf : (structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r)) \u27f6 X.presheaf.obj (op (to\u0393SpecMapBasicOpen X r))\nloc_inst : IsLocalization.Away r \u2191((structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r)))\nh :\n toOpen (\u2191(\u0393.obj (op X))) (basicOpen r) \u226b f =\n RingHom.comp (IsLocalization.Away.lift r (_ : IsUnit (\u2191(toTo\u0393SpecMapBasicOpen X r) r)))\n (algebraMap \u2191(\u0393.obj (op X)) \u2191((structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r))))\n\u22a2 f = to\u0393SpecCApp X r\n[PROOFSTEP]\nrefine' @IsLocalization.ringHom_ext _ _ _ _ _ _ _ _ loc_inst _ _ _\n[GOAL]\ncase mp\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nf : (structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r)) \u27f6 X.presheaf.obj (op (to\u0393SpecMapBasicOpen X r))\nloc_inst : IsLocalization.Away r \u2191((structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r)))\nh :\n toOpen (\u2191(\u0393.obj (op X))) (basicOpen r) \u226b f =\n RingHom.comp (IsLocalization.Away.lift r (_ : IsUnit (\u2191(toTo\u0393SpecMapBasicOpen X r) r)))\n (algebraMap \u2191(\u0393.obj (op X)) \u2191((structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r))))\n\u22a2 RingHom.comp f (algebraMap \u2191(\u0393.obj (op X)) \u2191((structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r)))) =\n RingHom.comp (to\u0393SpecCApp X r)\n (algebraMap \u2191(\u0393.obj (op X)) \u2191((structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r))))\n[PROOFSTEP]\nexact h\n[GOAL]\ncase mpr\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nf : (structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r)) \u27f6 X.presheaf.obj (op (to\u0393SpecMapBasicOpen X r))\nloc_inst : IsLocalization.Away r \u2191((structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r)))\n\u22a2 f = to\u0393SpecCApp X r \u2192\n toOpen (\u2191(\u0393.obj (op X))) (basicOpen r) \u226b f =\n RingHom.comp (IsLocalization.Away.lift r (_ : IsUnit (\u2191(toTo\u0393SpecMapBasicOpen X r) r)))\n (algebraMap \u2191(\u0393.obj (op X)) \u2191((structureSheaf \u2191(\u0393.obj (op X))).val.obj (op (basicOpen r))))\n[PROOFSTEP]\napply congr_arg\n[GOAL]\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nr s : (InducedCategory (Opens (PrimeSpectrum \u2191(\u0393.obj (op X)))) basicOpen)\u1d52\u1d56\nf : r \u27f6 s\n\u22a2 ((inducedFunctor basicOpen).op \u22d9 (structureSheaf \u2191(\u0393.obj (op X))).val).map f \u226b (fun r => to\u0393SpecCApp X r.unop) s =\n (fun r => to\u0393SpecCApp X r.unop) r \u226b\n ((inducedFunctor basicOpen).op \u22d9 ((TopCat.Sheaf.pushforward (to\u0393SpecBase X)).obj (\ud835\udcaa X)).val).map f\n[PROOFSTEP]\napply (StructureSheaf.to_basicOpen_epi (\u0393.obj (op X)) r.unop).1\n[GOAL]\ncase a\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nr s : (InducedCategory (Opens (PrimeSpectrum \u2191(\u0393.obj (op X)))) basicOpen)\u1d52\u1d56\nf : r \u27f6 s\n\u22a2 toOpen (\u2191(\u0393.obj (op X))) (basicOpen r.unop) \u226b\n ((inducedFunctor basicOpen).op \u22d9 (structureSheaf \u2191(\u0393.obj (op X))).val).map f \u226b (fun r => to\u0393SpecCApp X r.unop) s =\n toOpen (\u2191(\u0393.obj (op X))) (basicOpen r.unop) \u226b\n (fun r => to\u0393SpecCApp X r.unop) r \u226b\n ((inducedFunctor basicOpen).op \u22d9 ((TopCat.Sheaf.pushforward (to\u0393SpecBase X)).obj (\ud835\udcaa X)).val).map f\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc]\n[GOAL]\ncase a\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nr s : (InducedCategory (Opens (PrimeSpectrum \u2191(\u0393.obj (op X)))) basicOpen)\u1d52\u1d56\nf : r \u27f6 s\n\u22a2 (toOpen (\u2191(\u0393.obj (op X))) (basicOpen r.unop) \u226b\n ((inducedFunctor basicOpen).op \u22d9 (structureSheaf \u2191(\u0393.obj (op X))).val).map f) \u226b\n to\u0393SpecCApp X s.unop =\n (toOpen (\u2191(\u0393.obj (op X))) (basicOpen r.unop) \u226b to\u0393SpecCApp X r.unop) \u226b\n ((inducedFunctor basicOpen).op \u22d9 ((TopCat.Sheaf.pushforward (to\u0393SpecBase X)).obj (\ud835\udcaa X)).val).map f\n[PROOFSTEP]\nerw [X.to\u0393SpecCApp_spec r.unop]\n[GOAL]\ncase a\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nr s : (InducedCategory (Opens (PrimeSpectrum \u2191(\u0393.obj (op X)))) basicOpen)\u1d52\u1d56\nf : r \u27f6 s\n\u22a2 (toOpen (\u2191(\u0393.obj (op X))) (basicOpen r.unop) \u226b\n ((inducedFunctor basicOpen).op \u22d9 (structureSheaf \u2191(\u0393.obj (op X))).val).map f) \u226b\n to\u0393SpecCApp X s.unop =\n toTo\u0393SpecMapBasicOpen X r.unop \u226b\n ((inducedFunctor basicOpen).op \u22d9 ((TopCat.Sheaf.pushforward (to\u0393SpecBase X)).obj (\ud835\udcaa X)).val).map f\n[PROOFSTEP]\nconvert X.to\u0393SpecCApp_spec s.unop\n[GOAL]\ncase h.e'_3.h\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nr s : (InducedCategory (Opens (PrimeSpectrum \u2191(\u0393.obj (op X)))) basicOpen)\u1d52\u1d56\nf : r \u27f6 s\ne_1\u271d :\n (CommRingCat.of \u2191(\u0393.obj (op X)) \u27f6\n ((inducedFunctor basicOpen).op \u22d9 ((TopCat.Sheaf.pushforward (to\u0393SpecBase X)).obj (\ud835\udcaa X)).val).obj s) =\n (CommRingCat.of \u2191(\u0393.obj (op X)) \u27f6 X.presheaf.obj (op (to\u0393SpecMapBasicOpen X s.unop)))\n\u22a2 toTo\u0393SpecMapBasicOpen X r.unop \u226b\n ((inducedFunctor basicOpen).op \u22d9 ((TopCat.Sheaf.pushforward (to\u0393SpecBase X)).obj (\ud835\udcaa X)).val).map f =\n toTo\u0393SpecMapBasicOpen X s.unop\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.e'_3.h\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nr s : (InducedCategory (Opens (PrimeSpectrum \u2191(\u0393.obj (op X)))) basicOpen)\u1d52\u1d56\nf : r \u27f6 s\ne_1\u271d :\n (CommRingCat.of \u2191(\u0393.obj (op X)) \u27f6\n ((inducedFunctor basicOpen).op \u22d9 ((TopCat.Sheaf.pushforward (to\u0393SpecBase X)).obj (\ud835\udcaa X)).val).obj s) =\n (CommRingCat.of \u2191(\u0393.obj (op X)) \u27f6 X.presheaf.obj (op (to\u0393SpecMapBasicOpen X s.unop)))\n\u22a2 toTo\u0393SpecMapBasicOpen X s.unop =\n toTo\u0393SpecMapBasicOpen X r.unop \u226b\n ((inducedFunctor basicOpen).op \u22d9 ((TopCat.Sheaf.pushforward (to\u0393SpecBase X)).obj (\ud835\udcaa X)).val).map f\n[PROOFSTEP]\napply X.presheaf.map_comp\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\n\u22a2 NatTrans.app (to\u0393SpecSheafedSpace X).c (op (basicOpen r)) = to\u0393SpecCApp X r\n[PROOFSTEP]\nhave :=\n TopCat.Sheaf.extend_hom_app (Spec.toSheafedSpace.obj (op (\u0393.obj (op X)))).presheaf\n ((TopCat.Sheaf.pushforward X.to\u0393SpecBase).obj X.\ud835\udcaa) isBasis_basic_opens X.to\u0393SpecCBasicOpens r\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nthis :\n NatTrans.app\n (\u2191(TopCat.Sheaf.restrictHomEquivHom (Spec.toSheafedSpace.obj (op (\u0393.obj (op X)))).presheaf\n ((TopCat.Sheaf.pushforward (to\u0393SpecBase X)).obj (\ud835\udcaa X)) (_ : Opens.IsBasis (Set.range basicOpen)))\n (to\u0393SpecCBasicOpens X))\n (op (basicOpen r)) =\n NatTrans.app (to\u0393SpecCBasicOpens X) (op r)\n\u22a2 NatTrans.app (to\u0393SpecSheafedSpace X).c (op (basicOpen r)) = to\u0393SpecCApp X r\n[PROOFSTEP]\ndsimp at this \n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nthis :\n NatTrans.app\n (\u2191(TopCat.Sheaf.restrictHomEquivHom (structureSheaf \u2191(X.presheaf.obj (op \u22a4))).val\n ((TopCat.Sheaf.pushforward (to\u0393SpecBase X)).obj (\ud835\udcaa X)) (_ : Opens.IsBasis (Set.range basicOpen)))\n (to\u0393SpecCBasicOpens X))\n (op (basicOpen r)) =\n to\u0393SpecCApp X r\n\u22a2 NatTrans.app (to\u0393SpecSheafedSpace X).c (op (basicOpen r)) = to\u0393SpecCApp X r\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nthis :\n NatTrans.app\n (\u2191(TopCat.Sheaf.restrictHomEquivHom (structureSheaf \u2191(X.presheaf.obj (op \u22a4))).val\n ((TopCat.Sheaf.pushforward (to\u0393SpecBase X)).obj (\ud835\udcaa X)) (_ : Opens.IsBasis (Set.range basicOpen)))\n (to\u0393SpecCBasicOpens X))\n (op (basicOpen r)) =\n to\u0393SpecCApp X r\n\u22a2 NatTrans.app (to\u0393SpecSheafedSpace X).c (op (basicOpen r)) =\n NatTrans.app\n (\u2191(TopCat.Sheaf.restrictHomEquivHom (structureSheaf \u2191(X.presheaf.obj (op \u22a4))).val\n ((TopCat.Sheaf.pushforward (to\u0393SpecBase X)).obj (\ud835\udcaa X)) (_ : Opens.IsBasis (Set.range basicOpen)))\n (to\u0393SpecCBasicOpens X))\n (op (basicOpen r))\n[PROOFSTEP]\ndsimp\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nthis :\n NatTrans.app\n (\u2191(TopCat.Sheaf.restrictHomEquivHom (structureSheaf \u2191(X.presheaf.obj (op \u22a4))).val\n ((TopCat.Sheaf.pushforward (to\u0393SpecBase X)).obj (\ud835\udcaa X)) (_ : Opens.IsBasis (Set.range basicOpen)))\n (to\u0393SpecCBasicOpens X))\n (op (basicOpen r)) =\n to\u0393SpecCApp X r\n\u22a2 NatTrans.app\n (\u2191(TopCat.Sheaf.restrictHomEquivHom (structureSheaf \u2191(X.presheaf.obj (op \u22a4))).val\n ((TopCat.Sheaf.pushforward (to\u0393SpecBase X)).obj (\ud835\udcaa X)) (_ : Opens.IsBasis (Set.range basicOpen)))\n (to\u0393SpecCBasicOpens X))\n (op (basicOpen r)) =\n NatTrans.app\n (\u2191(TopCat.Sheaf.restrictHomEquivHom (structureSheaf \u2191(X.presheaf.obj (op \u22a4))).val\n ((TopCat.Sheaf.pushforward (to\u0393SpecBase X)).obj (\ud835\udcaa X)) (_ : Opens.IsBasis (Set.range basicOpen)))\n (to\u0393SpecCBasicOpens X))\n (op (basicOpen r))\n[PROOFSTEP]\ncongr\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191(toTopCat X)\n\u22a2 toStalk (\u2191(op (\u0393.obj (op X))).unop) (\u2191(to\u0393SpecSheafedSpace X).base x) \u226b\n PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x =\n \u0393ToStalk X x\n[PROOFSTEP]\nrw [PresheafedSpace.stalkMap]\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191(toTopCat X)\n\u22a2 toStalk (\u2191(op (\u0393.obj (op X))).unop) (\u2191(to\u0393SpecSheafedSpace X).base x) \u226b\n (stalkFunctor CommRingCat (\u2191(to\u0393SpecSheafedSpace X).base x)).map (to\u0393SpecSheafedSpace X).c \u226b\n stalkPushforward CommRingCat (to\u0393SpecSheafedSpace X).base X.presheaf x =\n \u0393ToStalk X x\n[PROOFSTEP]\nerw [\u2190 toOpen_germ _ (basicOpen (1 : \u0393.obj (op X))) \u27e8X.to\u0393SpecFun x, by rw [basicOpen_one]; trivial\u27e9]\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191(toTopCat X)\n\u22a2 to\u0393SpecFun X x \u2208 basicOpen 1\n[PROOFSTEP]\nrw [basicOpen_one]\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191(toTopCat X)\n\u22a2 to\u0393SpecFun X x \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191(toTopCat X)\n\u22a2 (toOpen (\u2191(\u0393.obj (op X))) (basicOpen 1) \u226b\n germ (TopCat.Sheaf.presheaf (structureSheaf \u2191(\u0393.obj (op X))))\n { val := to\u0393SpecFun X x, property := (_ : to\u0393SpecFun X x \u2208 basicOpen 1) }) \u226b\n (stalkFunctor CommRingCat (\u2191(to\u0393SpecSheafedSpace X).base x)).map (to\u0393SpecSheafedSpace X).c \u226b\n stalkPushforward CommRingCat (to\u0393SpecSheafedSpace X).base X.presheaf x =\n \u0393ToStalk X x\n[PROOFSTEP]\nrw [\u2190 Category.assoc, Category.assoc (toOpen _ _)]\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191(toTopCat X)\n\u22a2 (toOpen (\u2191(\u0393.obj (op X))) (basicOpen 1) \u226b\n germ (TopCat.Sheaf.presheaf (structureSheaf \u2191(\u0393.obj (op X))))\n { val := to\u0393SpecFun X x, property := (_ : to\u0393SpecFun X x \u2208 basicOpen 1) } \u226b\n (stalkFunctor CommRingCat (\u2191(to\u0393SpecSheafedSpace X).base x)).map (to\u0393SpecSheafedSpace X).c) \u226b\n stalkPushforward CommRingCat (to\u0393SpecSheafedSpace X).base X.presheaf x =\n \u0393ToStalk X x\n[PROOFSTEP]\nerw [stalkFunctor_map_germ]\n -- Porting note : was `rw [\u2190assoc, to\u0393SpecSheafedSpace_app_spec]`, but Lean did not like it.\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191(toTopCat X)\n\u22a2 (toOpen (\u2191(\u0393.obj (op X))) (basicOpen 1) \u226b\n NatTrans.app (to\u0393SpecSheafedSpace X).c (op (basicOpen 1)) \u226b\n germ ((to\u0393SpecSheafedSpace X).base _* X.presheaf)\n { val := to\u0393SpecFun X x, property := (_ : to\u0393SpecFun X x \u2208 basicOpen 1) }) \u226b\n stalkPushforward CommRingCat (to\u0393SpecSheafedSpace X).base X.presheaf x =\n \u0393ToStalk X x\n[PROOFSTEP]\nrw [to\u0393SpecSheafedSpace_app_spec_assoc]\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191(toTopCat X)\n\u22a2 (toTo\u0393SpecMapBasicOpen X 1 \u226b\n germ ((to\u0393SpecSheafedSpace X).base _* X.presheaf)\n { val := to\u0393SpecFun X x, property := (_ : to\u0393SpecFun X x \u2208 basicOpen 1) }) \u226b\n stalkPushforward CommRingCat (to\u0393SpecSheafedSpace X).base X.presheaf x =\n \u0393ToStalk X x\n[PROOFSTEP]\nunfold \u0393ToStalk\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191(toTopCat X)\n\u22a2 (toTo\u0393SpecMapBasicOpen X 1 \u226b\n germ ((to\u0393SpecSheafedSpace X).base _* X.presheaf)\n { val := to\u0393SpecFun X x, property := (_ : to\u0393SpecFun X x \u2208 basicOpen 1) }) \u226b\n stalkPushforward CommRingCat (to\u0393SpecSheafedSpace X).base X.presheaf x =\n germ X.presheaf { val := x, property := trivial }\n[PROOFSTEP]\nrw [\u2190 stalkPushforward_germ _ X.to\u0393SpecBase X.presheaf \u22a4]\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191(toTopCat X)\n\u22a2 (toTo\u0393SpecMapBasicOpen X 1 \u226b\n germ ((to\u0393SpecSheafedSpace X).base _* X.presheaf)\n { val := to\u0393SpecFun X x, property := (_ : to\u0393SpecFun X x \u2208 basicOpen 1) }) \u226b\n stalkPushforward CommRingCat (to\u0393SpecSheafedSpace X).base X.presheaf x =\n germ (to\u0393SpecBase X _* X.presheaf)\n { val := \u2191(to\u0393SpecBase X) \u2191{ val := x, property := trivial },\n property := (_ : \u2191{ val := x, property := trivial } \u2208 (Opens.map (to\u0393SpecBase X)).obj \u22a4) } \u226b\n stalkPushforward CommRingCat (to\u0393SpecBase X) X.presheaf \u2191{ val := x, property := trivial }\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191(toTopCat X)\n\u22a2 toTo\u0393SpecMapBasicOpen X 1 \u226b\n germ ((to\u0393SpecSheafedSpace X).base _* X.presheaf)\n { val := to\u0393SpecFun X x, property := (_ : to\u0393SpecFun X x \u2208 basicOpen 1) } =\n germ (to\u0393SpecBase X _* X.presheaf)\n { val := \u2191(to\u0393SpecBase X) \u2191{ val := x, property := trivial },\n property := (_ : \u2191{ val := x, property := trivial } \u2208 (Opens.map (to\u0393SpecBase X)).obj \u22a4) }\n[PROOFSTEP]\nchange (X.to\u0393SpecBase _* X.presheaf).map le_top.hom.op \u226b _ = _\n[GOAL]\ncase e_a\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191(toTopCat X)\n\u22a2 (to\u0393SpecBase X _* X.presheaf).map\n (LE.le.hom\n (_ :\n { carrier := {x | \u00ac1 \u2208 x.asIdeal}, is_open' := (_ : \u2203 y, zeroLocus y = {x | \u00ac1 \u2208 x.asIdeal}\u1d9c) } \u2264 \u22a4)).op \u226b\n germ ((to\u0393SpecSheafedSpace X).base _* X.presheaf)\n { val := to\u0393SpecFun X x, property := (_ : to\u0393SpecFun X x \u2208 basicOpen 1) } =\n germ (to\u0393SpecBase X _* X.presheaf)\n { val := \u2191(to\u0393SpecBase X) \u2191{ val := x, property := trivial },\n property := (_ : \u2191{ val := x, property := trivial } \u2208 (Opens.map (to\u0393SpecBase X)).obj \u22a4) }\n[PROOFSTEP]\napply germ_res\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\n\u22a2 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsLocalRingHom (PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x)\n[PROOFSTEP]\nintro x\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 IsLocalRingHom (PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x)\n[PROOFSTEP]\nlet p : PrimeSpectrum (\u0393.obj (op X)) := X.to\u0393SpecFun x\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\np : PrimeSpectrum \u2191(\u0393.obj (op X)) := to\u0393SpecFun X x\n\u22a2 IsLocalRingHom (PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x)\n[PROOFSTEP]\nconstructor\n -- show stalk map is local hom \u2193\n[GOAL]\ncase map_nonunit\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\np : PrimeSpectrum \u2191(\u0393.obj (op X)) := to\u0393SpecFun X x\n\u22a2 \u2200\n (a :\n \u2191(PresheafedSpace.stalk (Spec.locallyRingedSpaceObj (\u0393.obj (op X))).toSheafedSpace.toPresheafedSpace\n (\u2191(to\u0393SpecSheafedSpace X).base x))),\n IsUnit (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) a) \u2192 IsUnit a\n[PROOFSTEP]\nlet S := (structureSheaf _).presheaf.stalk p\n[GOAL]\ncase map_nonunit\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\np : PrimeSpectrum \u2191(\u0393.obj (op X)) := to\u0393SpecFun X x\nS : CommRingCat := TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191(\u0393.obj (op X)))) p\n\u22a2 \u2200\n (a :\n \u2191(PresheafedSpace.stalk (Spec.locallyRingedSpaceObj (\u0393.obj (op X))).toSheafedSpace.toPresheafedSpace\n (\u2191(to\u0393SpecSheafedSpace X).base x))),\n IsUnit (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) a) \u2192 IsUnit a\n[PROOFSTEP]\nrintro (t : S) ht\n[GOAL]\ncase map_nonunit\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\np : PrimeSpectrum \u2191(\u0393.obj (op X)) := to\u0393SpecFun X x\nS : CommRingCat := TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191(\u0393.obj (op X)))) p\nt : \u2191S\nht : IsUnit (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t)\n\u22a2 IsUnit t\n[PROOFSTEP]\nobtain \u27e8\u27e8r, s\u27e9, he\u27e9 := IsLocalization.surj p.asIdeal.primeCompl t\n[GOAL]\ncase map_nonunit.intro.mk\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\np : PrimeSpectrum \u2191(\u0393.obj (op X)) := to\u0393SpecFun X x\nS : CommRingCat := TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191(\u0393.obj (op X)))) p\nt : \u2191S\nht : IsUnit (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t)\nr : \u2191(\u0393.obj (op X))\ns : { x // x \u2208 Ideal.primeCompl p.asIdeal }\nhe : t * \u2191(algebraMap \u2191(\u0393.obj (op X)) \u2191S) \u2191(r, s).snd = \u2191(algebraMap \u2191(\u0393.obj (op X)) \u2191S) (r, s).fst\n\u22a2 IsUnit t\n[PROOFSTEP]\ndsimp at he \n[GOAL]\ncase map_nonunit.intro.mk\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\np : PrimeSpectrum \u2191(\u0393.obj (op X)) := to\u0393SpecFun X x\nS : CommRingCat := TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191(\u0393.obj (op X)))) p\nt : \u2191S\nht : IsUnit (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t)\nr : \u2191(\u0393.obj (op X))\ns : { x // x \u2208 Ideal.primeCompl p.asIdeal }\nhe :\n t * \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) \u2191s =\n \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) r\n\u22a2 IsUnit t\n[PROOFSTEP]\nset t' := _\n[GOAL]\ncase map_nonunit.intro.mk\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\np : PrimeSpectrum \u2191(\u0393.obj (op X)) := to\u0393SpecFun X x\nS : CommRingCat := TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191(\u0393.obj (op X)))) p\nt : \u2191S\nht : IsUnit (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t)\nr : \u2191(\u0393.obj (op X))\ns : { x // x \u2208 Ideal.primeCompl p.asIdeal }\nhe :\n t * \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) \u2191s =\n \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) r\nt' : ?m.142259 := ?m.142260\n\u22a2 IsUnit t\n[PROOFSTEP]\nchange t * t' = _ at he \n[GOAL]\ncase map_nonunit.intro.mk\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\np : PrimeSpectrum \u2191(\u0393.obj (op X)) := to\u0393SpecFun X x\nS : CommRingCat := TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191(\u0393.obj (op X)))) p\nt : \u2191S\nht : IsUnit (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t)\nr : \u2191(\u0393.obj (op X))\ns : { x // x \u2208 Ideal.primeCompl p.asIdeal }\nt' : \u2191S := \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) \u2191s\nhe : t * t' = \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) r\n\u22a2 IsUnit t\n[PROOFSTEP]\napply isUnit_of_mul_isUnit_left (y := t')\n[GOAL]\ncase map_nonunit.intro.mk\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\np : PrimeSpectrum \u2191(\u0393.obj (op X)) := to\u0393SpecFun X x\nS : CommRingCat := TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191(\u0393.obj (op X)))) p\nt : \u2191S\nht : IsUnit (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t)\nr : \u2191(\u0393.obj (op X))\ns : { x // x \u2208 Ideal.primeCompl p.asIdeal }\nt' : \u2191S := \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) \u2191s\nhe : t * t' = \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) r\n\u22a2 IsUnit (t * t')\n[PROOFSTEP]\nrw [he]\n[GOAL]\ncase map_nonunit.intro.mk\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\np : PrimeSpectrum \u2191(\u0393.obj (op X)) := to\u0393SpecFun X x\nS : CommRingCat := TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191(\u0393.obj (op X)))) p\nt : \u2191S\nht : IsUnit (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t)\nr : \u2191(\u0393.obj (op X))\ns : { x // x \u2208 Ideal.primeCompl p.asIdeal }\nt' : \u2191S := \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) \u2191s\nhe : t * t' = \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) r\n\u22a2 IsUnit (\u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) r)\n[PROOFSTEP]\nrefine' IsLocalization.map_units S (\u27e8r, _\u27e9 : p.asIdeal.primeCompl)\n[GOAL]\ncase map_nonunit.intro.mk\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\np : PrimeSpectrum \u2191(\u0393.obj (op X)) := to\u0393SpecFun X x\nS : CommRingCat := TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191(\u0393.obj (op X)))) p\nt : \u2191S\nht : IsUnit (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t)\nr : \u2191(\u0393.obj (op X))\ns : { x // x \u2208 Ideal.primeCompl p.asIdeal }\nt' : \u2191S := \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) \u2191s\nhe : t * t' = \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) r\n\u22a2 r \u2208 Ideal.primeCompl p.asIdeal\n[PROOFSTEP]\napply (not_mem_prime_iff_unit_in_stalk _ _ _).mpr\n[GOAL]\ncase map_nonunit.intro.mk\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\np : PrimeSpectrum \u2191(\u0393.obj (op X)) := to\u0393SpecFun X x\nS : CommRingCat := TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191(\u0393.obj (op X)))) p\nt : \u2191S\nht : IsUnit (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t)\nr : \u2191(\u0393.obj (op X))\ns : { x // x \u2208 Ideal.primeCompl p.asIdeal }\nt' : \u2191S := \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) \u2191s\nhe : t * t' = \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) r\n\u22a2 IsUnit (\u2191(\u0393ToStalk X x) r)\n[PROOFSTEP]\nrw [\u2190 toStalk_stalkMap_to\u0393Spec]\n[GOAL]\ncase map_nonunit.intro.mk\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\np : PrimeSpectrum \u2191(\u0393.obj (op X)) := to\u0393SpecFun X x\nS : CommRingCat := TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191(\u0393.obj (op X)))) p\nt : \u2191S\nht : IsUnit (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t)\nr : \u2191(\u0393.obj (op X))\ns : { x // x \u2208 Ideal.primeCompl p.asIdeal }\nt' : \u2191S := \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) \u2191s\nhe : t * t' = \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) r\n\u22a2 IsUnit\n (\u2191(toStalk (\u2191(op (\u0393.obj (op X))).unop) (\u2191(to\u0393SpecSheafedSpace X).base x) \u226b\n PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x)\n r)\n[PROOFSTEP]\nerw [comp_apply, \u2190 he]\n[GOAL]\ncase map_nonunit.intro.mk\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\np : PrimeSpectrum \u2191(\u0393.obj (op X)) := to\u0393SpecFun X x\nS : CommRingCat := TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191(\u0393.obj (op X)))) p\nt : \u2191S\nht : IsUnit (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t)\nr : \u2191(\u0393.obj (op X))\ns : { x // x \u2208 Ideal.primeCompl p.asIdeal }\nt' : \u2191S := \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) \u2191s\nhe : t * t' = \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) r\n\u22a2 IsUnit (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) (t * t'))\n[PROOFSTEP]\nrw [RingHom.map_mul]\n -- Porting note : `IsLocalization.map_units` and the goal needs to be simplified before Lean\n -- realize it is useful\n[GOAL]\ncase map_nonunit.intro.mk\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\np : PrimeSpectrum \u2191(\u0393.obj (op X)) := to\u0393SpecFun X x\nS : CommRingCat := TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191(\u0393.obj (op X)))) p\nt : \u2191S\nht : IsUnit (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t)\nr : \u2191(\u0393.obj (op X))\ns : { x // x \u2208 Ideal.primeCompl p.asIdeal }\nt' : \u2191S := \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) \u2191s\nhe : t * t' = \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) r\n\u22a2 IsUnit\n (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t * \u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t')\n[PROOFSTEP]\nhave := IsLocalization.map_units (R := \u0393.obj (op X)) S s\n[GOAL]\ncase map_nonunit.intro.mk\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\np : PrimeSpectrum \u2191(\u0393.obj (op X)) := to\u0393SpecFun X x\nS : CommRingCat := TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191(\u0393.obj (op X)))) p\nt : \u2191S\nht : IsUnit (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t)\nr : \u2191(\u0393.obj (op X))\ns : { x // x \u2208 Ideal.primeCompl p.asIdeal }\nt' : \u2191S := \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) \u2191s\nhe : t * t' = \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) r\nthis : IsUnit (\u2191(algebraMap \u2191(\u0393.obj (op X)) \u2191S) \u2191s)\n\u22a2 IsUnit\n (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t * \u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t')\n[PROOFSTEP]\ndsimp at this \u22a2\n[GOAL]\ncase map_nonunit.intro.mk\nX : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X))\nx : \u2191\u2191X.toPresheafedSpace\np : PrimeSpectrum \u2191(\u0393.obj (op X)) := to\u0393SpecFun X x\nS : CommRingCat := TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191(\u0393.obj (op X)))) p\nt : \u2191S\nht : IsUnit (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t)\nr : \u2191(\u0393.obj (op X))\ns : { x // x \u2208 Ideal.primeCompl p.asIdeal }\nt' : \u2191S := \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) \u2191s\nhe : t * t' = \u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) r\nthis : IsUnit (\u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) \u2191s)\n\u22a2 IsUnit\n (\u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) t *\n \u2191(PresheafedSpace.stalkMap (to\u0393SpecSheafedSpace X) x) (\u2191(toStalk (\u2191(X.presheaf.obj (op \u22a4))) (to\u0393SpecFun X x)) \u2191s))\n[PROOFSTEP]\nrefine ht.mul <| this.map _\n[GOAL]\nX\u271d : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X\u271d))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R \u27f6 \u0393.obj (op X)\n\u03b2 : X \u27f6 Spec.locallyRingedSpaceObj R\nw : (to\u0393Spec X).val.base \u226b (Spec.locallyRingedSpaceMap f).val.base = \u03b2.val.base\nh :\n \u2200 (r : \u2191R),\n f \u226b X.presheaf.map (homOfLE (_ : (Opens.map \u03b2.val.base).obj (basicOpen r) \u2264 \u22a4)).op =\n toOpen (\u2191R) (basicOpen r) \u226b NatTrans.app \u03b2.val.c (op (basicOpen r))\n\u22a2 to\u0393Spec X \u226b Spec.locallyRingedSpaceMap f = \u03b2\n[PROOFSTEP]\next1\n -- Porting note : need more hand holding here\n[GOAL]\ncase h\nX\u271d : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X\u271d))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R \u27f6 \u0393.obj (op X)\n\u03b2 : X \u27f6 Spec.locallyRingedSpaceObj R\nw : (to\u0393Spec X).val.base \u226b (Spec.locallyRingedSpaceMap f).val.base = \u03b2.val.base\nh :\n \u2200 (r : \u2191R),\n f \u226b X.presheaf.map (homOfLE (_ : (Opens.map \u03b2.val.base).obj (basicOpen r) \u2264 \u22a4)).op =\n toOpen (\u2191R) (basicOpen r) \u226b NatTrans.app \u03b2.val.c (op (basicOpen r))\n\u22a2 (to\u0393Spec X \u226b Spec.locallyRingedSpaceMap f).val = \u03b2.val\n[PROOFSTEP]\nchange (X.to\u0393Spec.1 \u226b _).base = _ at w \n[GOAL]\ncase h\nX\u271d : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X\u271d))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R \u27f6 \u0393.obj (op X)\n\u03b2 : X \u27f6 Spec.locallyRingedSpaceObj R\nh :\n \u2200 (r : \u2191R),\n f \u226b X.presheaf.map (homOfLE (_ : (Opens.map \u03b2.val.base).obj (basicOpen r) \u2264 \u22a4)).op =\n toOpen (\u2191R) (basicOpen r) \u226b NatTrans.app \u03b2.val.c (op (basicOpen r))\nw : ((to\u0393Spec X).val \u226b (Spec.locallyRingedSpaceMap f).val).base = \u03b2.val.base\n\u22a2 (to\u0393Spec X \u226b Spec.locallyRingedSpaceMap f).val = \u03b2.val\n[PROOFSTEP]\napply Spec.basicOpen_hom_ext w\n[GOAL]\ncase h\nX\u271d : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X\u271d))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R \u27f6 \u0393.obj (op X)\n\u03b2 : X \u27f6 Spec.locallyRingedSpaceObj R\nh :\n \u2200 (r : \u2191R),\n f \u226b X.presheaf.map (homOfLE (_ : (Opens.map \u03b2.val.base).obj (basicOpen r) \u2264 \u22a4)).op =\n toOpen (\u2191R) (basicOpen r) \u226b NatTrans.app \u03b2.val.c (op (basicOpen r))\nw : ((to\u0393Spec X).val \u226b (Spec.locallyRingedSpaceMap f).val).base = \u03b2.val.base\n\u22a2 \u2200 (r : \u2191R),\n let U := basicOpen r;\n (toOpen (\u2191R) U \u226b NatTrans.app ((to\u0393Spec X).val \u226b (Spec.locallyRingedSpaceMap f).val).c (op U)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (Opens.map ((to\u0393Spec X).val \u226b (Spec.locallyRingedSpaceMap f).val).base).op.obj (op U) =\n (Opens.map \u03b2.val.base).op.obj (op U))) =\n toOpen (\u2191R) U \u226b NatTrans.app \u03b2.val.c (op U)\n[PROOFSTEP]\nintro r U\n[GOAL]\ncase h\nX\u271d : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X\u271d))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R \u27f6 \u0393.obj (op X)\n\u03b2 : X \u27f6 Spec.locallyRingedSpaceObj R\nh :\n \u2200 (r : \u2191R),\n f \u226b X.presheaf.map (homOfLE (_ : (Opens.map \u03b2.val.base).obj (basicOpen r) \u2264 \u22a4)).op =\n toOpen (\u2191R) (basicOpen r) \u226b NatTrans.app \u03b2.val.c (op (basicOpen r))\nw : ((to\u0393Spec X).val \u226b (Spec.locallyRingedSpaceMap f).val).base = \u03b2.val.base\nr : \u2191R\nU : Opens (PrimeSpectrum \u2191R) := basicOpen r\n\u22a2 (toOpen (\u2191R) U \u226b NatTrans.app ((to\u0393Spec X).val \u226b (Spec.locallyRingedSpaceMap f).val).c (op U)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (Opens.map ((to\u0393Spec X).val \u226b (Spec.locallyRingedSpaceMap f).val).base).op.obj (op U) =\n (Opens.map \u03b2.val.base).op.obj (op U))) =\n toOpen (\u2191R) U \u226b NatTrans.app \u03b2.val.c (op U)\n[PROOFSTEP]\nerw [LocallyRingedSpace.comp_val_c_app]\n[GOAL]\ncase h\nX\u271d : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X\u271d))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R \u27f6 \u0393.obj (op X)\n\u03b2 : X \u27f6 Spec.locallyRingedSpaceObj R\nh :\n \u2200 (r : \u2191R),\n f \u226b X.presheaf.map (homOfLE (_ : (Opens.map \u03b2.val.base).obj (basicOpen r) \u2264 \u22a4)).op =\n toOpen (\u2191R) (basicOpen r) \u226b NatTrans.app \u03b2.val.c (op (basicOpen r))\nw : ((to\u0393Spec X).val \u226b (Spec.locallyRingedSpaceMap f).val).base = \u03b2.val.base\nr : \u2191R\nU : Opens (PrimeSpectrum \u2191R) := basicOpen r\n\u22a2 (toOpen (\u2191R) U \u226b\n NatTrans.app (Spec.locallyRingedSpaceMap f).val.c (op U) \u226b\n NatTrans.app (to\u0393Spec X).val.c (op ((Opens.map (Spec.locallyRingedSpaceMap f).val.base).obj (op U).unop))) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (Opens.map ((to\u0393Spec X).val \u226b (Spec.locallyRingedSpaceMap f).val).base).op.obj (op U) =\n (Opens.map \u03b2.val.base).op.obj (op U))) =\n toOpen (\u2191R) U \u226b NatTrans.app \u03b2.val.c (op U)\n[PROOFSTEP]\nerw [toOpen_comp_comap_assoc]\n[GOAL]\ncase h\nX\u271d : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X\u271d))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R \u27f6 \u0393.obj (op X)\n\u03b2 : X \u27f6 Spec.locallyRingedSpaceObj R\nh :\n \u2200 (r : \u2191R),\n f \u226b X.presheaf.map (homOfLE (_ : (Opens.map \u03b2.val.base).obj (basicOpen r) \u2264 \u22a4)).op =\n toOpen (\u2191R) (basicOpen r) \u226b NatTrans.app \u03b2.val.c (op (basicOpen r))\nw : ((to\u0393Spec X).val \u226b (Spec.locallyRingedSpaceMap f).val).base = \u03b2.val.base\nr : \u2191R\nU : Opens (PrimeSpectrum \u2191R) := basicOpen r\n\u22a2 (CommRingCat.ofHom f \u226b\n toOpen (\u2191(\u0393.obj (op X))) (\u2191(Opens.comap (PrimeSpectrum.comap f)) U) \u226b\n NatTrans.app (to\u0393Spec X).val.c (op ((Opens.map (Spec.locallyRingedSpaceMap f).val.base).obj (op U).unop))) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (Opens.map ((to\u0393Spec X).val \u226b (Spec.locallyRingedSpaceMap f).val).base).op.obj (op U) =\n (Opens.map \u03b2.val.base).op.obj (op U))) =\n toOpen (\u2191R) U \u226b NatTrans.app \u03b2.val.c (op U)\n[PROOFSTEP]\nrw [Category.assoc]\n[GOAL]\ncase h\nX\u271d : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X\u271d))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R \u27f6 \u0393.obj (op X)\n\u03b2 : X \u27f6 Spec.locallyRingedSpaceObj R\nh :\n \u2200 (r : \u2191R),\n f \u226b X.presheaf.map (homOfLE (_ : (Opens.map \u03b2.val.base).obj (basicOpen r) \u2264 \u22a4)).op =\n toOpen (\u2191R) (basicOpen r) \u226b NatTrans.app \u03b2.val.c (op (basicOpen r))\nw : ((to\u0393Spec X).val \u226b (Spec.locallyRingedSpaceMap f).val).base = \u03b2.val.base\nr : \u2191R\nU : Opens (PrimeSpectrum \u2191R) := basicOpen r\n\u22a2 CommRingCat.ofHom f \u226b\n (toOpen (\u2191(\u0393.obj (op X))) (\u2191(Opens.comap (PrimeSpectrum.comap f)) U) \u226b\n NatTrans.app (to\u0393Spec X).val.c (op ((Opens.map (Spec.locallyRingedSpaceMap f).val.base).obj (op U).unop))) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (Opens.map ((to\u0393Spec X).val \u226b (Spec.locallyRingedSpaceMap f).val).base).op.obj (op U) =\n (Opens.map \u03b2.val.base).op.obj (op U))) =\n toOpen (\u2191R) U \u226b NatTrans.app \u03b2.val.c (op U)\n[PROOFSTEP]\nerw [to\u0393SpecSheafedSpace_app_spec, \u2190 X.presheaf.map_comp]\n[GOAL]\ncase h\nX\u271d : LocallyRingedSpace\nr\u271d : \u2191(\u0393.obj (op X\u271d))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R \u27f6 \u0393.obj (op X)\n\u03b2 : X \u27f6 Spec.locallyRingedSpaceObj R\nh :\n \u2200 (r : \u2191R),\n f \u226b X.presheaf.map (homOfLE (_ : (Opens.map \u03b2.val.base).obj (basicOpen r) \u2264 \u22a4)).op =\n toOpen (\u2191R) (basicOpen r) \u226b NatTrans.app \u03b2.val.c (op (basicOpen r))\nw : ((to\u0393Spec X).val \u226b (Spec.locallyRingedSpaceMap f).val).base = \u03b2.val.base\nr : \u2191R\nU : Opens (PrimeSpectrum \u2191R) := basicOpen r\n\u22a2 CommRingCat.ofHom f \u226b\n X.presheaf.map\n ((Opens.leTop (to\u0393SpecMapBasicOpen X (\u2191f r))).op \u226b\n eqToHom\n (_ :\n (Opens.map ((to\u0393Spec X).val \u226b (Spec.locallyRingedSpaceMap f).val).base).op.obj (op U) =\n (Opens.map \u03b2.val.base).op.obj (op U))) =\n toOpen (\u2191R) U \u226b NatTrans.app \u03b2.val.c (op U)\n[PROOFSTEP]\nconvert h r using 1\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\n\u22a2 toSpec\u0393 (\u0393.obj (op X)) \u226b NatTrans.app (to\u0393Spec X).val.c (op \u22a4) = \ud835\udfd9 (\u0393.obj (op X))\n[PROOFSTEP]\nunfold toSpec\u0393\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\n\u22a2 toOpen \u2191(\u0393.obj (op X)) \u22a4 \u226b NatTrans.app (to\u0393Spec X).val.c (op \u22a4) = \ud835\udfd9 (\u0393.obj (op X))\n[PROOFSTEP]\nrw [\u2190 toOpen_res _ (basicOpen (1 : \u0393.obj (op X))) \u22a4 (eqToHom basicOpen_one.symm)]\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\n\u22a2 (toOpen (\u2191(\u0393.obj (op X))) (basicOpen 1) \u226b\n (structureSheaf \u2191(\u0393.obj (op X))).val.map (eqToHom (_ : \u22a4 = basicOpen 1)).op) \u226b\n NatTrans.app (to\u0393Spec X).val.c (op \u22a4) =\n \ud835\udfd9 (\u0393.obj (op X))\n[PROOFSTEP]\nerw [Category.assoc]\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\n\u22a2 toOpen (\u2191(\u0393.obj (op X))) (basicOpen 1) \u226b\n (structureSheaf \u2191(\u0393.obj (op X))).val.map (eqToHom (_ : \u22a4 = basicOpen 1)).op \u226b\n NatTrans.app (to\u0393Spec X).val.c (op \u22a4) =\n \ud835\udfd9 (\u0393.obj (op X))\n[PROOFSTEP]\nrw [NatTrans.naturality, \u2190 Category.assoc]\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\n\u22a2 (toOpen (\u2191(\u0393.obj (op X))) (basicOpen 1) \u226b NatTrans.app (to\u0393Spec X).val.c (op (basicOpen 1))) \u226b\n ((to\u0393Spec X).val.base _* X.presheaf).map (eqToHom (_ : \u22a4 = basicOpen 1)).op =\n \ud835\udfd9 (\u0393.obj (op X))\n[PROOFSTEP]\nerw [X.to\u0393SpecSheafedSpace_app_spec 1, \u2190 Functor.map_comp]\n[GOAL]\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\n\u22a2 X.presheaf.map\n ((Opens.leTop (to\u0393SpecMapBasicOpen X 1)).op \u226b\n (Opens.map (to\u0393Spec X).val.base).op.map (eqToHom (_ : \u22a4 = basicOpen 1)).op) =\n \ud835\udfd9 (\u0393.obj (op X))\n[PROOFSTEP]\nconvert eqToHom_map X.presheaf _\n[GOAL]\ncase convert_3\nX : LocallyRingedSpace\nr : \u2191(\u0393.obj (op X))\n\u22a2 op \u22a4 = (Opens.map (to\u0393Spec X).val.base).op.obj (op \u22a4)\n[PROOFSTEP]\nrfl\n[GOAL]\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\n\u22a2 (\ud835\udfed LocallyRingedSpace).map f \u226b to\u0393Spec Y = to\u0393Spec X \u226b (\u0393.rightOp \u22d9 Spec.toLocallyRingedSpace).map f\n[PROOFSTEP]\nsymm\n[GOAL]\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\n\u22a2 to\u0393Spec X \u226b (\u0393.rightOp \u22d9 Spec.toLocallyRingedSpace).map f = (\ud835\udfed LocallyRingedSpace).map f \u226b to\u0393Spec Y\n[PROOFSTEP]\napply LocallyRingedSpace.comp_ring_hom_ext\n[GOAL]\ncase w\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\n\u22a2 (to\u0393Spec ((\ud835\udfed LocallyRingedSpace).obj X)).val.base \u226b (Spec.locallyRingedSpaceMap (\u0393.rightOp.map f).unop).val.base =\n ((\ud835\udfed LocallyRingedSpace).map f \u226b to\u0393Spec Y).val.base\n[PROOFSTEP]\next1 x\n[GOAL]\ncase w.w\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nx : (forget TopCat).obj \u2191((\ud835\udfed LocallyRingedSpace).obj X).toPresheafedSpace\n\u22a2 \u2191((to\u0393Spec ((\ud835\udfed LocallyRingedSpace).obj X)).val.base \u226b (Spec.locallyRingedSpaceMap (\u0393.rightOp.map f).unop).val.base)\n x =\n \u2191((\ud835\udfed LocallyRingedSpace).map f \u226b to\u0393Spec Y).val.base x\n[PROOFSTEP]\ndsimp [Spec.topMap, LocallyRingedSpace.to\u0393SpecFun]\n --Porting Note: Had to add the next four lines\n[GOAL]\ncase w.w\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nx : (forget TopCat).obj \u2191((\ud835\udfed LocallyRingedSpace).obj X).toPresheafedSpace\n\u22a2 \u2191(to\u0393SpecBase X \u226b PrimeSpectrum.comap (NatTrans.app f.val.c (op \u22a4))) x = \u2191(f.val.base \u226b to\u0393SpecBase Y) x\n[PROOFSTEP]\nrw [comp_apply, comp_apply]\n[GOAL]\ncase w.w\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nx : (forget TopCat).obj \u2191((\ud835\udfed LocallyRingedSpace).obj X).toPresheafedSpace\n\u22a2 \u2191(PrimeSpectrum.comap (NatTrans.app f.val.c (op \u22a4))) (\u2191(to\u0393SpecBase X) x) = \u2191(to\u0393SpecBase Y) (\u2191f.val.base x)\n[PROOFSTEP]\ndsimp [to\u0393SpecBase]\n[GOAL]\ncase w.w\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nx : (forget TopCat).obj \u2191((\ud835\udfed LocallyRingedSpace).obj X).toPresheafedSpace\n\u22a2 \u2191(PrimeSpectrum.comap (NatTrans.app f.val.c (op \u22a4))) (\u2191(ContinuousMap.mk (to\u0393SpecFun X)) x) =\n \u2191(ContinuousMap.mk (to\u0393SpecFun Y)) (\u2191f.val.base x)\n[PROOFSTEP]\nrw [ContinuousMap.coe_mk, ContinuousMap.coe_mk]\n[GOAL]\ncase w.w\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nx : (forget TopCat).obj \u2191((\ud835\udfed LocallyRingedSpace).obj X).toPresheafedSpace\n\u22a2 \u2191(PrimeSpectrum.comap (NatTrans.app f.val.c (op \u22a4))) (to\u0393SpecFun X x) = to\u0393SpecFun Y (\u2191f.val.base x)\n[PROOFSTEP]\ndsimp [to\u0393SpecFun]\n[GOAL]\ncase w.w\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nx : (forget TopCat).obj \u2191((\ud835\udfed LocallyRingedSpace).obj X).toPresheafedSpace\n\u22a2 \u2191(PrimeSpectrum.comap (NatTrans.app f.val.c (op \u22a4)))\n (\u2191(PrimeSpectrum.comap (\u0393ToStalk X x)) (LocalRing.closedPoint \u2191(TopCat.Presheaf.stalk X.presheaf x))) =\n \u2191(PrimeSpectrum.comap (\u0393ToStalk Y (\u2191f.val.base x)))\n (LocalRing.closedPoint \u2191(TopCat.Presheaf.stalk Y.presheaf (\u2191f.val.base x)))\n[PROOFSTEP]\nrw [\u2190 LocalRing.comap_closedPoint (PresheafedSpace.stalkMap f.val x), \u2190 PrimeSpectrum.comap_comp_apply, \u2190\n PrimeSpectrum.comap_comp_apply]\n[GOAL]\ncase w.w\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nx : (forget TopCat).obj \u2191((\ud835\udfed LocallyRingedSpace).obj X).toPresheafedSpace\n\u22a2 \u2191(PrimeSpectrum.comap (RingHom.comp (\u0393ToStalk X x) (NatTrans.app f.val.c (op \u22a4))))\n (LocalRing.closedPoint \u2191(TopCat.Presheaf.stalk X.presheaf x)) =\n \u2191(PrimeSpectrum.comap (RingHom.comp (PresheafedSpace.stalkMap f.val x) (\u0393ToStalk Y (\u2191f.val.base x))))\n (LocalRing.closedPoint \u2191(PresheafedSpace.stalk X.toPresheafedSpace x))\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase w.w.e_a.e_f\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nx : (forget TopCat).obj \u2191((\ud835\udfed LocallyRingedSpace).obj X).toPresheafedSpace\n\u22a2 RingHom.comp (\u0393ToStalk X x) (NatTrans.app f.val.c (op \u22a4)) =\n RingHom.comp (PresheafedSpace.stalkMap f.val x) (\u0393ToStalk Y (\u2191f.val.base x))\n[PROOFSTEP]\nexact (PresheafedSpace.stalkMap_germ f.1 \u22a4 \u27e8x, trivial\u27e9).symm\n[GOAL]\ncase h\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\n\u22a2 \u2200 (r : \u2191(\u0393.rightOp.obj Y).unop),\n (\u0393.rightOp.map f).unop \u226b\n ((\ud835\udfed LocallyRingedSpace).obj X).presheaf.map\n (homOfLE (_ : (Opens.map ((\ud835\udfed LocallyRingedSpace).map f \u226b to\u0393Spec Y).val.base).obj (basicOpen r) \u2264 \u22a4)).op =\n toOpen (\u2191(\u0393.rightOp.obj Y).unop) (basicOpen r) \u226b\n NatTrans.app ((\ud835\udfed LocallyRingedSpace).map f \u226b to\u0393Spec Y).val.c (op (basicOpen r))\n[PROOFSTEP]\nintro r\n[GOAL]\ncase h\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nr : \u2191(\u0393.rightOp.obj Y).unop\n\u22a2 (\u0393.rightOp.map f).unop \u226b\n ((\ud835\udfed LocallyRingedSpace).obj X).presheaf.map\n (homOfLE (_ : (Opens.map ((\ud835\udfed LocallyRingedSpace).map f \u226b to\u0393Spec Y).val.base).obj (basicOpen r) \u2264 \u22a4)).op =\n toOpen (\u2191(\u0393.rightOp.obj Y).unop) (basicOpen r) \u226b\n NatTrans.app ((\ud835\udfed LocallyRingedSpace).map f \u226b to\u0393Spec Y).val.c (op (basicOpen r))\n[PROOFSTEP]\nrw [LocallyRingedSpace.comp_val_c_app, \u2190 Category.assoc]\n[GOAL]\ncase h\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nr : \u2191(\u0393.rightOp.obj Y).unop\n\u22a2 (\u0393.rightOp.map f).unop \u226b\n ((\ud835\udfed LocallyRingedSpace).obj X).presheaf.map\n (homOfLE (_ : (Opens.map ((\ud835\udfed LocallyRingedSpace).map f \u226b to\u0393Spec Y).val.base).obj (basicOpen r) \u2264 \u22a4)).op =\n (toOpen (\u2191(\u0393.rightOp.obj Y).unop) (basicOpen r) \u226b NatTrans.app (to\u0393Spec Y).val.c (op (basicOpen r))) \u226b\n NatTrans.app ((\ud835\udfed LocallyRingedSpace).map f).val.c\n (op ((Opens.map (to\u0393Spec Y).val.base).obj (op (basicOpen r)).unop))\n[PROOFSTEP]\nerw [Y.to\u0393SpecSheafedSpace_app_spec, f.1.c.naturality]\n[GOAL]\ncase h\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nr : \u2191(\u0393.rightOp.obj Y).unop\n\u22a2 (\u0393.rightOp.map f).unop \u226b\n ((\ud835\udfed LocallyRingedSpace).obj X).presheaf.map\n (homOfLE (_ : (Opens.map ((\ud835\udfed LocallyRingedSpace).map f \u226b to\u0393Spec Y).val.base).obj (basicOpen r) \u2264 \u22a4)).op =\n NatTrans.app f.val.c (op \u22a4) \u226b (f.val.base _* X.presheaf).map (Opens.leTop (to\u0393SpecMapBasicOpen Y r)).op\n[PROOFSTEP]\nrfl\n[GOAL]\nR : CommRingCat\n\u22a2 NatTrans.app identityTo\u0393Spec (Spec.toLocallyRingedSpace.obj (op R)) \u226b\n Spec.toLocallyRingedSpace.map (NatTrans.app Spec\u0393Identity.inv R).op =\n \ud835\udfd9 ((\ud835\udfed LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R)))\n[PROOFSTEP]\napply LocallyRingedSpace.comp_ring_hom_ext\n[GOAL]\ncase w\nR : CommRingCat\n\u22a2 (to\u0393Spec ((\ud835\udfed LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R)))).val.base \u226b\n (Spec.locallyRingedSpaceMap (NatTrans.app Spec\u0393Identity.inv R).op.unop).val.base =\n (\ud835\udfd9 ((\ud835\udfed LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R)))).val.base\n[PROOFSTEP]\next (p : PrimeSpectrum R)\n[GOAL]\ncase w.w\nR : CommRingCat\np : PrimeSpectrum \u2191R\n\u22a2 \u2191((to\u0393Spec ((\ud835\udfed LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R)))).val.base \u226b\n (Spec.locallyRingedSpaceMap (NatTrans.app Spec\u0393Identity.inv R).op.unop).val.base)\n p =\n \u2191(\ud835\udfd9 ((\ud835\udfed LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R)))).val.base p\n[PROOFSTEP]\nchange _ = p\n[GOAL]\ncase w.w\nR : CommRingCat\np : PrimeSpectrum \u2191R\n\u22a2 \u2191((to\u0393Spec ((\ud835\udfed LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R)))).val.base \u226b\n (Spec.locallyRingedSpaceMap (NatTrans.app Spec\u0393Identity.inv R).op.unop).val.base)\n p =\n p\n[PROOFSTEP]\next x\n[GOAL]\ncase w.w.asIdeal.h\nR : CommRingCat\np : PrimeSpectrum \u2191R\nx : \u2191(op ((\ud835\udfed CommRingCat).obj R)).unop\n\u22a2 x \u2208\n (\u2191((to\u0393Spec ((\ud835\udfed LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R)))).val.base \u226b\n (Spec.locallyRingedSpaceMap (NatTrans.app Spec\u0393Identity.inv R).op.unop).val.base)\n p).asIdeal \u2194\n x \u2208 p.asIdeal\n[PROOFSTEP]\nerw [\u2190 IsLocalization.AtPrime.to_map_mem_maximal_iff ((structureSheaf R).presheaf.stalk p) p.asIdeal x]\n[GOAL]\ncase w.w.asIdeal.h\nR : CommRingCat\np : PrimeSpectrum \u2191R\nx : \u2191(op ((\ud835\udfed CommRingCat).obj R)).unop\n\u22a2 x \u2208\n (\u2191((to\u0393Spec ((\ud835\udfed LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R)))).val.base \u226b\n (Spec.locallyRingedSpaceMap (NatTrans.app Spec\u0393Identity.inv R).op.unop).val.base)\n p).asIdeal \u2194\n \u2191(algebraMap \u2191R \u2191(TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191R)) p)) x \u2208\n LocalRing.maximalIdeal \u2191(TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191R)) p)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\nR : CommRingCat\n\u22a2 \u2200 (r : \u2191(op ((\ud835\udfed CommRingCat).obj R)).unop),\n (NatTrans.app Spec\u0393Identity.inv R).op.unop \u226b\n ((\ud835\udfed LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R))).presheaf.map\n (homOfLE\n (_ :\n (Opens.map (\ud835\udfd9 ((\ud835\udfed LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R)))).val.base).obj\n (basicOpen r) \u2264\n \u22a4)).op =\n toOpen (\u2191(op ((\ud835\udfed CommRingCat).obj R)).unop) (basicOpen r) \u226b\n NatTrans.app (\ud835\udfd9 ((\ud835\udfed LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R)))).val.c (op (basicOpen r))\n[PROOFSTEP]\nintro r\n[GOAL]\ncase h\nR : CommRingCat\nr : \u2191(op ((\ud835\udfed CommRingCat).obj R)).unop\n\u22a2 (NatTrans.app Spec\u0393Identity.inv R).op.unop \u226b\n ((\ud835\udfed LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R))).presheaf.map\n (homOfLE\n (_ :\n (Opens.map (\ud835\udfd9 ((\ud835\udfed LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R)))).val.base).obj\n (basicOpen r) \u2264\n \u22a4)).op =\n toOpen (\u2191(op ((\ud835\udfed CommRingCat).obj R)).unop) (basicOpen r) \u226b\n NatTrans.app (\ud835\udfd9 ((\ud835\udfed LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R)))).val.c (op (basicOpen r))\n[PROOFSTEP]\napply toOpen_res\n[GOAL]\n\u22a2 whiskerRight identityTo\u0393Spec \u0393.rightOp \u226b\n (Functor.associator \u0393.rightOp Spec.toLocallyRingedSpace \u0393.rightOp).hom \u226b\n whiskerLeft \u0393.rightOp (NatIso.op Spec\u0393Identity).inv =\n NatTrans.id (\ud835\udfed LocallyRingedSpace \u22d9 \u0393.rightOp)\n[PROOFSTEP]\next X\n[GOAL]\ncase w.h\nX : LocallyRingedSpace\n\u22a2 NatTrans.app\n (whiskerRight identityTo\u0393Spec \u0393.rightOp \u226b\n (Functor.associator \u0393.rightOp Spec.toLocallyRingedSpace \u0393.rightOp).hom \u226b\n whiskerLeft \u0393.rightOp (NatIso.op Spec\u0393Identity).inv)\n X =\n NatTrans.app (NatTrans.id (\ud835\udfed LocallyRingedSpace \u22d9 \u0393.rightOp)) X\n[PROOFSTEP]\nerw [Category.id_comp]\n[GOAL]\ncase w.h\nX : LocallyRingedSpace\n\u22a2 NatTrans.app (whiskerRight identityTo\u0393Spec \u0393.rightOp \u226b whiskerLeft \u0393.rightOp (NatIso.op Spec\u0393Identity).inv) X =\n NatTrans.app (NatTrans.id (\ud835\udfed LocallyRingedSpace \u22d9 \u0393.rightOp)) X\n[PROOFSTEP]\nexact congr_arg Quiver.Hom.op (left_triangle X)\n[GOAL]\n\u22a2 whiskerLeft Spec.toLocallyRingedSpace identityTo\u0393Spec \u226b\n (Functor.associator Spec.toLocallyRingedSpace \u0393.rightOp Spec.toLocallyRingedSpace).inv \u226b\n whiskerRight (NatIso.op Spec\u0393Identity).inv Spec.toLocallyRingedSpace =\n NatTrans.id (Spec.toLocallyRingedSpace \u22d9 \ud835\udfed LocallyRingedSpace)\n[PROOFSTEP]\next R : 2\n -- Porting note : a little bit hand holding\n[GOAL]\ncase w.h\nR : CommRingCat\u1d52\u1d56\n\u22a2 NatTrans.app\n (whiskerLeft Spec.toLocallyRingedSpace identityTo\u0393Spec \u226b\n (Functor.associator Spec.toLocallyRingedSpace \u0393.rightOp Spec.toLocallyRingedSpace).inv \u226b\n whiskerRight (NatIso.op Spec\u0393Identity).inv Spec.toLocallyRingedSpace)\n R =\n NatTrans.app (NatTrans.id (Spec.toLocallyRingedSpace \u22d9 \ud835\udfed LocallyRingedSpace)) R\n[PROOFSTEP]\nchange identityTo\u0393Spec.app _ \u226b \ud835\udfd9 _ \u226b Spec.toLocallyRingedSpace.map _ = \ud835\udfd9 _\n[GOAL]\ncase w.h\nR : CommRingCat\u1d52\u1d56\n\u22a2 NatTrans.app identityTo\u0393Spec (Spec.locallyRingedSpaceObj R.unop) \u226b\n \ud835\udfd9 ((\u0393.rightOp \u22d9 Spec.toLocallyRingedSpace).obj (Spec.locallyRingedSpaceObj R.unop)) \u226b\n Spec.toLocallyRingedSpace.map (NatTrans.app (NatIso.op Spec\u0393Identity).inv R) =\n \ud835\udfd9 ((\ud835\udfed LocallyRingedSpace).obj (Spec.locallyRingedSpaceObj R.unop))\n[PROOFSTEP]\nsimp_rw [Category.id_comp, show (NatIso.op Spec\u0393Identity).inv.app R = (Spec\u0393Identity.inv.app R.unop).op from rfl]\n[GOAL]\ncase w.h\nR : CommRingCat\u1d52\u1d56\n\u22a2 NatTrans.app identityTo\u0393Spec (Spec.locallyRingedSpaceObj R.unop) \u226b\n Spec.toLocallyRingedSpace.map (NatTrans.app Spec\u0393Identity.inv R.unop).op =\n \ud835\udfd9 ((\ud835\udfed LocallyRingedSpace).obj (Spec.locallyRingedSpaceObj R.unop))\n[PROOFSTEP]\nexact right_triangle R.unop\n[GOAL]\nX : Scheme\nR : CommRingCat\u1d52\u1d56\nf : op (Scheme.\u0393.obj (op X)) \u27f6 R\n\u22a2 \u2191(Adjunction.homEquiv adjunction X R) f =\n \u2191(Adjunction.homEquiv locallyRingedSpaceAdjunction X.toLocallyRingedSpace R) f\n[PROOFSTEP]\ndsimp [adjunction, Adjunction.restrictFullyFaithful]\n[GOAL]\nX : Scheme\nR : CommRingCat\u1d52\u1d56\nf : op (Scheme.\u0393.obj (op X)) \u27f6 R\n\u22a2 \u2191(equivOfFullyFaithful Scheme.forgetToLocallyRingedSpace).symm\n (\ud835\udfd9 X.toLocallyRingedSpace \u226b\n \u2191(Adjunction.homEquiv locallyRingedSpaceAdjunction X.toLocallyRingedSpace R)\n (\ud835\udfd9 (op (X.presheaf.obj (op \u22a4))) \u226b \u2191(equivOfFullyFaithful (\ud835\udfed CommRingCat\u1d52\u1d56)) f \u226b \ud835\udfd9 R) \u226b\n \ud835\udfd9 (Spec.locallyRingedSpaceObj R.unop)) =\n \u2191(Adjunction.homEquiv locallyRingedSpaceAdjunction X.toLocallyRingedSpace R) f\n[PROOFSTEP]\nsimp only [Category.comp_id, Category.id_comp]\n[GOAL]\nX : Scheme\nR : CommRingCat\u1d52\u1d56\nf : op (Scheme.\u0393.obj (op X)) \u27f6 R\n\u22a2 \u2191(equivOfFullyFaithful Scheme.forgetToLocallyRingedSpace).symm\n (\u2191(Adjunction.homEquiv locallyRingedSpaceAdjunction X.toLocallyRingedSpace R)\n (\u2191(equivOfFullyFaithful (\ud835\udfed CommRingCat\u1d52\u1d56)) f)) =\n \u2191(Adjunction.homEquiv locallyRingedSpaceAdjunction X.toLocallyRingedSpace R) f\n[PROOFSTEP]\nrfl\n --Porting Note: Added\n[GOAL]\nX : Scheme\nR : CommRingCat\u1d52\u1d56\nf : X \u27f6 Scheme.Spec.obj R\n\u22a2 \u2191(Adjunction.homEquiv adjunction X R).symm f =\n \u2191(Adjunction.homEquiv locallyRingedSpaceAdjunction X.toLocallyRingedSpace R).symm f\n[PROOFSTEP]\nrw [adjunction_homEquiv]\n[GOAL]\nX : Scheme\nR : CommRingCat\u1d52\u1d56\nf : X \u27f6 Scheme.Spec.obj R\n\u22a2 \u2191(Adjunction.homEquiv locallyRingedSpaceAdjunction X.toLocallyRingedSpace R).symm f =\n \u2191(Adjunction.homEquiv locallyRingedSpaceAdjunction X.toLocallyRingedSpace R).symm f\n[PROOFSTEP]\nrfl\n[GOAL]\nR : CommRingCat\u1d52\u1d56\n\u22a2 NatTrans.app adjunction.counit R = NatTrans.app locallyRingedSpaceAdjunction.counit R\n[PROOFSTEP]\nrw [\u2190 Adjunction.homEquiv_symm_id, \u2190 Adjunction.homEquiv_symm_id, adjunction_homEquiv_symm_apply]\n[GOAL]\nR : CommRingCat\u1d52\u1d56\n\u22a2 \u2191(Adjunction.homEquiv locallyRingedSpaceAdjunction (Scheme.Spec.obj R).toLocallyRingedSpace R).symm\n (\ud835\udfd9 (Scheme.Spec.obj R)) =\n \u2191(Adjunction.homEquiv locallyRingedSpaceAdjunction (Spec.toLocallyRingedSpace.obj R) R).symm\n (\ud835\udfd9 (Spec.toLocallyRingedSpace.obj R))\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Scheme\n\u22a2 NatTrans.app adjunction.unit X = NatTrans.app locallyRingedSpaceAdjunction.unit X.toLocallyRingedSpace\n[PROOFSTEP]\nrw [\u2190 Adjunction.homEquiv_id, \u2190 Adjunction.homEquiv_id, adjunction_homEquiv_apply]\n[GOAL]\nX : Scheme\n\u22a2 \u2191(Adjunction.homEquiv locallyRingedSpaceAdjunction X.toLocallyRingedSpace (Scheme.\u0393.rightOp.obj X))\n (\ud835\udfd9 (Scheme.\u0393.rightOp.obj X)) =\n \u2191(Adjunction.homEquiv locallyRingedSpaceAdjunction X.toLocallyRingedSpace (\u0393.rightOp.obj X.toLocallyRingedSpace))\n (\ud835\udfd9 (\u0393.rightOp.obj X.toLocallyRingedSpace))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 IsIso locallyRingedSpaceAdjunction.counit\n[PROOFSTEP]\ndsimp only [locallyRingedSpaceAdjunction, Adjunction.mkOfUnitCounit_counit]\n -- Porting Note: `dsimp` was unnecessary and had to make this explicit\n[GOAL]\n\u22a2 IsIso (NatIso.op Spec\u0393Identity).inv\n[PROOFSTEP]\nconvert IsIso.of_iso_inv (NatIso.op Spec\u0393Identity) using 1\n[GOAL]\n\u22a2 IsIso adjunction.counit\n[PROOFSTEP]\napply (config := { allowSynthFailures := true }) NatIso.isIso_of_isIso_app\n[GOAL]\ncase inst\n\u22a2 \u2200 (X : CommRingCat\u1d52\u1d56), IsIso (NatTrans.app adjunction.counit X)\n[PROOFSTEP]\nintro R\n[GOAL]\ncase inst\nR : CommRingCat\u1d52\u1d56\n\u22a2 IsIso (NatTrans.app adjunction.counit R)\n[PROOFSTEP]\nrw [adjunction_counit_app]\n[GOAL]\ncase inst\nR : CommRingCat\u1d52\u1d56\n\u22a2 IsIso (NatTrans.app locallyRingedSpaceAdjunction.counit R)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX : Scheme\n\u22a2 NatTrans.app (NatTrans.app adjunction.unit X).val.c (op \u22a4) = NatTrans.app Spec\u0393Identity.hom (X.presheaf.obj (op \u22a4))\n[PROOFSTEP]\nhave := congr_app \u0393Spec.adjunction.left_triangle X\n[GOAL]\nX : Scheme\nthis :\n NatTrans.app (whiskerRight adjunction.unit Scheme.\u0393.rightOp \u226b whiskerLeft Scheme.\u0393.rightOp adjunction.counit) X =\n NatTrans.app (\ud835\udfd9 (\ud835\udfed Scheme \u22d9 Scheme.\u0393.rightOp)) X\n\u22a2 NatTrans.app (NatTrans.app adjunction.unit X).val.c (op \u22a4) = NatTrans.app Spec\u0393Identity.hom (X.presheaf.obj (op \u22a4))\n[PROOFSTEP]\ndsimp at this \n[GOAL]\nX : Scheme\nthis :\n (NatTrans.app (NatTrans.app adjunction.unit X).val.c (op \u22a4)).op \u226b\n NatTrans.app adjunction.counit (op (X.presheaf.obj (op \u22a4))) =\n \ud835\udfd9 (op (X.presheaf.obj (op \u22a4)))\n\u22a2 NatTrans.app (NatTrans.app adjunction.unit X).val.c (op \u22a4) = NatTrans.app Spec\u0393Identity.hom (X.presheaf.obj (op \u22a4))\n[PROOFSTEP]\nrw [\u2190 IsIso.eq_comp_inv] at this \n[GOAL]\nX : Scheme\nthis :\n (NatTrans.app (NatTrans.app adjunction.unit X).val.c (op \u22a4)).op =\n \ud835\udfd9 (op (X.presheaf.obj (op \u22a4))) \u226b inv (NatTrans.app adjunction.counit (op (X.presheaf.obj (op \u22a4))))\n\u22a2 NatTrans.app (NatTrans.app adjunction.unit X).val.c (op \u22a4) = NatTrans.app Spec\u0393Identity.hom (X.presheaf.obj (op \u22a4))\n[PROOFSTEP]\nsimp only [adjunction_counit_app, locallyRingedSpaceAdjunction_counit, NatIso.op_inv, NatTrans.op_app, unop_op,\n Functor.id_obj, Functor.comp_obj, Functor.rightOp_obj, Spec.toLocallyRingedSpace_obj, \u0393_obj,\n Spec.locallyRingedSpaceObj_toSheafedSpace, Spec.sheafedSpaceObj_carrier, Spec.sheafedSpaceObj_presheaf,\n Spec\u0393Identity_inv_app, Category.id_comp] at this \n[GOAL]\nX : Scheme\nthis : (NatTrans.app (NatTrans.app adjunction.unit X).val.c (op \u22a4)).op = inv (toSpec\u0393 (X.presheaf.obj (op \u22a4))).op\n\u22a2 NatTrans.app (NatTrans.app adjunction.unit X).val.c (op \u22a4) = NatTrans.app Spec\u0393Identity.hom (X.presheaf.obj (op \u22a4))\n[PROOFSTEP]\nrw [\u2190 op_inv, Quiver.Hom.op_inj.eq_iff] at this \n[GOAL]\nX : Scheme\nthis : NatTrans.app (NatTrans.app adjunction.unit X).val.c (op \u22a4) = inv (toSpec\u0393 (X.presheaf.obj (op \u22a4)))\n\u22a2 NatTrans.app (NatTrans.app adjunction.unit X).val.c (op \u22a4) = NatTrans.app Spec\u0393Identity.hom (X.presheaf.obj (op \u22a4))\n[PROOFSTEP]\nrw [Spec\u0393Identity_hom_app]\n[GOAL]\nX : Scheme\nthis : NatTrans.app (NatTrans.app adjunction.unit X).val.c (op \u22a4) = inv (toSpec\u0393 (X.presheaf.obj (op \u22a4)))\n\u22a2 NatTrans.app (NatTrans.app adjunction.unit X).val.c (op \u22a4) = inv (toSpec\u0393 (X.presheaf.obj (op \u22a4)))\n[PROOFSTEP]\nconvert this using 1\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.GammaSpecAdjunction", "llama_tokens": 25179, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.3886180267058489, "lm_q1q2_score": 0.23903422866138604}} {"text": "[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nM : Monad C\n\u22a2 MonoidalCategory.tensorHom (\u03bc M) (\ud835\udfd9 M.toFunctor) \u226b \u03bc M =\n (MonoidalCategory.associator M.toFunctor M.toFunctor M.toFunctor).hom \u226b\n MonoidalCategory.tensorHom (\ud835\udfd9 M.toFunctor) (\u03bc M) \u226b \u03bc M\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d : Category.{v, u} C\nM : Monad C\nx\u271d : C\n\u22a2 NatTrans.app (MonoidalCategory.tensorHom (\u03bc M) (\ud835\udfd9 M.toFunctor) \u226b \u03bc M) x\u271d =\n NatTrans.app\n ((MonoidalCategory.associator M.toFunctor M.toFunctor M.toFunctor).hom \u226b\n MonoidalCategory.tensorHom (\ud835\udfd9 M.toFunctor) (\u03bc M) \u226b \u03bc M)\n x\u271d\n[PROOFSTEP]\nsimp [M.assoc]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nM : Mon_ (C \u2964 C)\nX : C\n\u22a2 M.X.map (NatTrans.app M.mul X) \u226b NatTrans.app M.mul X = NatTrans.app M.mul (M.X.obj X) \u226b NatTrans.app M.mul X\n[PROOFSTEP]\nrw [\u2190 NatTrans.hcomp_id_app, \u2190 NatTrans.comp_app]\n -- Porting note: had to add this step:\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nM : Mon_ (C \u2964 C)\nX : C\n\u22a2 NatTrans.app ((M.mul \u25eb \ud835\udfd9 M.X) \u226b M.mul) X = NatTrans.app M.mul (M.X.obj X) \u226b NatTrans.app M.mul X\n[PROOFSTEP]\nerw [M.mul_assoc]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nM : Mon_ (C \u2964 C)\nX : C\n\u22a2 NatTrans.app ((MonoidalCategory.associator M.X M.X M.X).hom \u226b MonoidalCategory.tensorHom (\ud835\udfd9 M.X) M.mul \u226b M.mul) X =\n NatTrans.app M.mul (M.X.obj X) \u226b NatTrans.app M.mul X\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nM : Mon_ (C \u2964 C)\nX : C\n\u22a2 NatTrans.app M.one (M.X.obj X) \u226b NatTrans.app M.mul X = \ud835\udfd9 ((\ud835\udfed C).obj (M.X.obj X))\n[PROOFSTEP]\nerw [\u2190 NatTrans.id_hcomp_app M.one, \u2190 NatTrans.comp_app, M.mul_one]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nM : Mon_ (C \u2964 C)\nX : C\n\u22a2 NatTrans.app (MonoidalCategory.rightUnitor M.X).hom X = \ud835\udfd9 ((\ud835\udfed C).obj (M.X.obj X))\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nM : Mon_ (C \u2964 C)\nX : C\n\u22a2 M.X.map (NatTrans.app M.one X) \u226b NatTrans.app M.mul X = \ud835\udfd9 (M.X.obj ((\ud835\udfed C).obj X))\n[PROOFSTEP]\nerw [\u2190 NatTrans.hcomp_id_app M.one, \u2190 NatTrans.comp_app, M.one_mul]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nM : Mon_ (C \u2964 C)\nX : C\n\u22a2 NatTrans.app (MonoidalCategory.leftUnitor M.X).hom X = \ud835\udfd9 (M.X.obj ((\ud835\udfed C).obj X))\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Mon_ (C \u2964 C)\nf : X \u27f6 Y\nsrc\u271d : X.X \u27f6 Y.X := f.hom\n\u22a2 \u2200 (X_1 : C), NatTrans.app (\u03b7 (ofMon X)) X_1 \u226b NatTrans.app (NatTrans.mk src\u271d.app) X_1 = NatTrans.app (\u03b7 (ofMon Y)) X_1\n[PROOFSTEP]\nintro X\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y : Mon_ (C \u2964 C)\nf : X\u271d \u27f6 Y\nsrc\u271d : X\u271d.X \u27f6 Y.X := f.hom\nX : C\n\u22a2 NatTrans.app (\u03b7 (ofMon X\u271d)) X \u226b NatTrans.app (NatTrans.mk src\u271d.app) X = NatTrans.app (\u03b7 (ofMon Y)) X\n[PROOFSTEP]\nerw [\u2190 NatTrans.comp_app, f.one_hom]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y : Mon_ (C \u2964 C)\nf : X\u271d \u27f6 Y\nsrc\u271d : X\u271d.X \u27f6 Y.X := f.hom\nX : C\n\u22a2 NatTrans.app Y.one X = NatTrans.app (\u03b7 (ofMon Y)) X\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Mon_ (C \u2964 C)\nf : X \u27f6 Y\nsrc\u271d : X.X \u27f6 Y.X := f.hom\n\u22a2 \u2200 (X_1 : C),\n NatTrans.app (\u03bc (ofMon X)) X_1 \u226b NatTrans.app (NatTrans.mk src\u271d.app) X_1 =\n ((ofMon X).toFunctor.map (NatTrans.app (NatTrans.mk src\u271d.app) X_1) \u226b\n NatTrans.app (NatTrans.mk src\u271d.app) ((ofMon Y).toFunctor.obj X_1)) \u226b\n NatTrans.app (\u03bc (ofMon Y)) X_1\n[PROOFSTEP]\nintro Z\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Mon_ (C \u2964 C)\nf : X \u27f6 Y\nsrc\u271d : X.X \u27f6 Y.X := f.hom\nZ : C\n\u22a2 NatTrans.app (\u03bc (ofMon X)) Z \u226b NatTrans.app (NatTrans.mk src\u271d.app) Z =\n ((ofMon X).toFunctor.map (NatTrans.app (NatTrans.mk src\u271d.app) Z) \u226b\n NatTrans.app (NatTrans.mk src\u271d.app) ((ofMon Y).toFunctor.obj Z)) \u226b\n NatTrans.app (\u03bc (ofMon Y)) Z\n[PROOFSTEP]\nerw [\u2190 NatTrans.comp_app, f.mul_hom]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Mon_ (C \u2964 C)\nf : X \u27f6 Y\nsrc\u271d : X.X \u27f6 Y.X := f.hom\nZ : C\n\u22a2 NatTrans.app (MonoidalCategory.tensorHom f.hom f.hom \u226b Y.mul) Z =\n ((ofMon X).toFunctor.map (NatTrans.app (NatTrans.mk src\u271d.app) Z) \u226b\n NatTrans.app (NatTrans.mk src\u271d.app) ((ofMon Y).toFunctor.obj Z)) \u226b\n NatTrans.app (\u03bc (ofMon Y)) Z\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Mon_ (C \u2964 C)\nf : X \u27f6 Y\nsrc\u271d : X.X \u27f6 Y.X := f.hom\nZ : C\n\u22a2 (NatTrans.app f.hom (X.X.obj Z) \u226b Y.X.map (NatTrans.app f.hom Z)) \u226b NatTrans.app Y.mul Z =\n ((ofMon X).toFunctor.map (NatTrans.app f.hom Z) \u226b NatTrans.app f.hom (Y.X.obj Z)) \u226b NatTrans.app Y.mul Z\n[PROOFSTEP]\nsimp only [NatTrans.naturality, NatTrans.hcomp_app, assoc, NatTrans.comp_app, ofMon_\u03bc]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monad.EquivMon", "llama_tokens": 2307, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.4532618480153861, "lm_q1q2_score": 0.2372464746477365}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nT : Monad C\nA B : Algebra T\nh : A.A \u2245 B.A\nw : autoParam (T.map h.hom \u226b B.a = A.a \u226b h.hom) _auto\u271d\n\u22a2 T.map h.inv \u226b A.a = B.a \u226b h.inv\n[PROOFSTEP]\nrw [h.eq_comp_inv, Category.assoc, \u2190 w, \u2190 Functor.map_comp_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nT : Monad C\nA B : Algebra T\nh : A.A \u2245 B.A\nw : autoParam (T.map h.hom \u226b B.a = A.a \u226b h.hom) _auto\u271d\n\u22a2 T.map (h.inv \u226b h.hom) \u226b B.a = B.a\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nT : Monad C\nX : C\nY : Algebra T\nf : X \u27f6 (forget T).obj Y\n\u22a2 T.map (T.map f \u226b Y.a) \u226b Y.a = ((free T).obj X).a \u226b T.map f \u226b Y.a\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nT : Monad C\nX : C\nY : Algebra T\nf : X \u27f6 (forget T).obj Y\n\u22a2 T.map (T.map f \u226b Y.a) \u226b Y.a = NatTrans.app (\u03bc T) X \u226b T.map f \u226b Y.a\n[PROOFSTEP]\nsimp [\u2190 Y.assoc, \u2190 T.\u03bc.naturality_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nT : Monad C\nX : C\nY : Algebra T\nf : (free T).obj X \u27f6 Y\n\u22a2 (fun f => Algebra.Hom.mk (T.map f \u226b Y.a)) ((fun f => NatTrans.app (\u03b7 T) X \u226b f.f) f) = f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nT : Monad C\nX : C\nY : Algebra T\nf : (free T).obj X \u27f6 Y\n\u22a2 ((fun f => Algebra.Hom.mk (T.map f \u226b Y.a)) ((fun f => NatTrans.app (\u03b7 T) X \u226b f.f) f)).f = f.f\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nT : Monad C\nX : C\nY : Algebra T\nf : (free T).obj X \u27f6 Y\n\u22a2 T.map (NatTrans.app (\u03b7 T) X \u226b f.f) \u226b Y.a = f.f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nT : Monad C\nX : C\nY : Algebra T\nf : X \u27f6 (forget T).obj Y\n\u22a2 (fun f => NatTrans.app (\u03b7 T) X \u226b f.f) ((fun f => Algebra.Hom.mk (T.map f \u226b Y.a)) f) = f\n[PROOFSTEP]\ndsimp only [forget_obj]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nT : Monad C\nX : C\nY : Algebra T\nf : X \u27f6 (forget T).obj Y\n\u22a2 NatTrans.app (\u03b7 T) X \u226b T.map f \u226b Y.a = f\n[PROOFSTEP]\nrw [\u2190 T.\u03b7.naturality_assoc, Y.unit]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nT : Monad C\nX : C\nY : Algebra T\nf : X \u27f6 (forget T).obj Y\n\u22a2 (\ud835\udfed C).map f \u226b \ud835\udfd9 Y.A = f\n[PROOFSTEP]\napply Category.comp_id\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nT : Monad C\nA B : Algebra T\nf : A \u27f6 B\ninst\u271d : IsIso f.f\n\u22a2 T.map (inv f.f) \u226b A.a = B.a \u226b inv f.f\n[PROOFSTEP]\nrw [IsIso.eq_comp_inv f.f, Category.assoc, \u2190 f.h]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nT : Monad C\nA B : Algebra T\nf : A \u27f6 B\ninst\u271d : IsIso f.f\n\u22a2 T.map (inv f.f) \u226b T.map f.f \u226b B.a = B.a\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nT : Monad C\nA B : Algebra T\nf : A \u27f6 B\ninst\u271d : IsIso f.f\n\u22a2 f \u226b Algebra.Hom.mk (inv f.f) = \ud835\udfd9 A \u2227 Algebra.Hom.mk (inv f.f) \u226b f = \ud835\udfd9 B\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nT T\u2081 T\u2082 : Monad C\nh : T\u2082 \u27f6 T\u2081\nA : Algebra T\u2081\n\u22a2 NatTrans.app (\u03b7 T\u2082) A.A \u226b NatTrans.app h.toNatTrans A.A \u226b A.a = \ud835\udfd9 A.A\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nT T\u2081 T\u2082 : Monad C\nh : T\u2082 \u27f6 T\u2081\nA : Algebra T\u2081\n\u22a2 NatTrans.app (\u03b7 T\u2082) A.A \u226b NatTrans.app h.toNatTrans A.A \u226b A.a = \ud835\udfd9 A.A\n[PROOFSTEP]\nsimp [A.unit]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nT T\u2081 T\u2082 : Monad C\nh : T\u2082 \u27f6 T\u2081\nA : Algebra T\u2081\n\u22a2 NatTrans.app (\u03bc T\u2082) A.A \u226b NatTrans.app h.toNatTrans A.A \u226b A.a =\n T\u2082.map (NatTrans.app h.toNatTrans A.A \u226b A.a) \u226b NatTrans.app h.toNatTrans A.A \u226b A.a\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nT T\u2081 T\u2082 : Monad C\nh : T\u2082 \u27f6 T\u2081\nA : Algebra T\u2081\n\u22a2 NatTrans.app (\u03bc T\u2082) A.A \u226b NatTrans.app h.toNatTrans A.A \u226b A.a =\n T\u2082.map (NatTrans.app h.toNatTrans A.A \u226b A.a) \u226b NatTrans.app h.toNatTrans A.A \u226b A.a\n[PROOFSTEP]\nsimp [A.assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nT T\u2081 T\u2082 : Monad C\nh : T\u2081 \u2245 T\u2082\n\u22a2 \ud835\udfd9 T\u2081 = h.hom \u226b h.inv\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nT T\u2081 T\u2082 : Monad C\nh : T\u2081 \u2245 T\u2082\n\u22a2 h.inv \u226b h.hom = \ud835\udfd9 T\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nG : Comonad C\nA B : Coalgebra G\nh : A.A \u2245 B.A\nw : autoParam (A.a \u226b G.map h.hom = h.hom \u226b B.a) _auto\u271d\n\u22a2 B.a \u226b G.map h.inv = h.inv \u226b A.a\n[PROOFSTEP]\nrw [h.eq_inv_comp, \u2190 reassoc_of% w, \u2190 Functor.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nG : Comonad C\nA B : Coalgebra G\nh : A.A \u2245 B.A\nw : autoParam (A.a \u226b G.map h.hom = h.hom \u226b B.a) _auto\u271d\n\u22a2 A.a \u226b G.map (h.hom \u226b h.inv) = A.a\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nG : Comonad C\nX : Coalgebra G\nY : C\nf : (forget G).obj X \u27f6 Y\n\u22a2 X.a \u226b G.map (X.a \u226b G.map f) = (X.a \u226b G.map f) \u226b ((cofree G).obj Y).a\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nG : Comonad C\nX : Coalgebra G\nY : C\nf : (forget G).obj X \u27f6 Y\n\u22a2 X.a \u226b G.map (X.a \u226b G.map f) = (X.a \u226b G.map f) \u226b NatTrans.app (\u03b4 G) Y\n[PROOFSTEP]\nsimp [\u2190 Coalgebra.coassoc_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nG : Comonad C\nX : Coalgebra G\nY : C\nf : (forget G).obj X \u27f6 Y\n\u22a2 (fun g => g.f \u226b NatTrans.app (\u03b5 G) Y) ((fun f => Coalgebra.Hom.mk (X.a \u226b G.map f)) f) = f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nG : Comonad C\nX : Coalgebra G\nY : C\nf : (forget G).obj X \u27f6 Y\n\u22a2 (X.a \u226b G.map f) \u226b NatTrans.app (\u03b5 G) Y = f\n[PROOFSTEP]\nrw [Category.assoc, G.\u03b5.naturality, Functor.id_map, X.counit_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nG : Comonad C\nX : Coalgebra G\nY : C\ng : X \u27f6 (cofree G).obj Y\n\u22a2 (fun f => Coalgebra.Hom.mk (X.a \u226b G.map f)) ((fun g => g.f \u226b NatTrans.app (\u03b5 G) Y) g) = g\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nG : Comonad C\nX : Coalgebra G\nY : C\ng : X \u27f6 (cofree G).obj Y\n\u22a2 ((fun f => Coalgebra.Hom.mk (X.a \u226b G.map f)) ((fun g => g.f \u226b NatTrans.app (\u03b5 G) Y) g)).f = g.f\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nG : Comonad C\nX : Coalgebra G\nY : C\ng : X \u27f6 (cofree G).obj Y\n\u22a2 X.a \u226b G.map (g.f \u226b NatTrans.app (\u03b5 G) Y) = g.f\n[PROOFSTEP]\nrw [Functor.map_comp, g.h_assoc, cofree_obj_a, Comonad.right_counit]\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nG : Comonad C\nX : Coalgebra G\nY : C\ng : X \u27f6 (cofree G).obj Y\n\u22a2 g.f \u226b \ud835\udfd9 (G.obj Y) = g.f\n[PROOFSTEP]\napply comp_id\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nG : Comonad C\nA B : Coalgebra G\nf : A \u27f6 B\ninst\u271d : IsIso f.f\n\u22a2 B.a \u226b G.map (inv f.f) = inv f.f \u226b A.a\n[PROOFSTEP]\nrw [IsIso.eq_inv_comp f.f, \u2190 f.h_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nG : Comonad C\nA B : Coalgebra G\nf : A \u27f6 B\ninst\u271d : IsIso f.f\n\u22a2 A.a \u226b G.map f.f \u226b G.map (inv f.f) = A.a\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nG : Comonad C\nA B : Coalgebra G\nf : A \u27f6 B\ninst\u271d : IsIso f.f\n\u22a2 f \u226b Coalgebra.Hom.mk (inv f.f) = \ud835\udfd9 A \u2227 Coalgebra.Hom.mk (inv f.f) \u226b f = \ud835\udfd9 B\n[PROOFSTEP]\naesop_cat\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monad.Algebra", "llama_tokens": 3658, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.4301473485858429, "lm_q1q2_score": 0.23684229839681967}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ns : KernelFork f\n\u22a2 Fork.\u03b9 s \u226b f = 0\n[PROOFSTEP]\nerw [Fork.condition, HasZeroMorphisms.comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ns : KernelFork f\n\u22a2 NatTrans.app s.\u03c0 one = 0\n[PROOFSTEP]\nsimp [Fork.app_one_eq_\u03b9_comp_right]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\n\u03b9 : Z \u27f6 X\nw : \u03b9 \u226b f = 0\n\u22a2 \u03b9 \u226b f = \u03b9 \u226b 0\n[PROOFSTEP]\nrw [w, HasZeroMorphisms.comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ns : Fork f 0\n\u22a2 \u2200 (j : WalkingParallelPair),\n NatTrans.app s.\u03c0 j = (Iso.refl s.pt).hom \u226b NatTrans.app (Fork.of\u03b9 (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b f = Fork.\u03b9 s \u226b 0)).\u03c0 j\n[PROOFSTEP]\nrintro \u27e8j\u27e9\n[GOAL]\ncase zero\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ns : Fork f 0\n\u22a2 NatTrans.app s.\u03c0 zero =\n (Iso.refl s.pt).hom \u226b NatTrans.app (Fork.of\u03b9 (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b f = Fork.\u03b9 s \u226b 0)).\u03c0 zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase one\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ns : Fork f 0\n\u22a2 NatTrans.app s.\u03c0 one =\n (Iso.refl s.pt).hom \u226b NatTrans.app (Fork.of\u03b9 (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b f = Fork.\u03b9 s \u226b 0)).\u03c0 one\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nP : C\n\u03b9 \u03b9' : P \u27f6 X\nw : \u03b9 \u226b f = 0\nh : \u03b9 = \u03b9'\n\u22a2 \u03b9' \u226b f = 0\n[PROOFSTEP]\nrw [\u2190 h, w]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u'\ninst\u271d\u00b2 : Category.{v, u'} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : IsEquivalence F\napp : (j : WalkingParallelPair) \u2192 (parallelPair f 0 \u22d9 F).obj j \u2245 (parallelPair (F.map f) 0).obj j :=\n fun j =>\n match j with\n | zero => Iso.refl ((parallelPair f 0 \u22d9 F).obj zero)\n | one => Iso.refl ((parallelPair f 0 \u22d9 F).obj one)\n\u22a2 \u2200 {X_1 Y_1 : WalkingParallelPair} (f_1 : X_1 \u27f6 Y_1),\n (parallelPair f 0 \u22d9 F).map f_1 \u226b (app Y_1).hom = (app X_1).hom \u226b (parallelPair (F.map f) 0).map f_1\n[PROOFSTEP]\nrintro \u27e8i\u27e9 \u27e8j\u27e9\n[GOAL]\ncase zero.zero\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u'\ninst\u271d\u00b2 : Category.{v, u'} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : IsEquivalence F\napp : (j : WalkingParallelPair) \u2192 (parallelPair f 0 \u22d9 F).obj j \u2245 (parallelPair (F.map f) 0).obj j :=\n fun j =>\n match j with\n | zero => Iso.refl ((parallelPair f 0 \u22d9 F).obj zero)\n | one => Iso.refl ((parallelPair f 0 \u22d9 F).obj one)\n\u22a2 \u2200 (f_1 : zero \u27f6 zero),\n (parallelPair f 0 \u22d9 F).map f_1 \u226b (app zero).hom = (app zero).hom \u226b (parallelPair (F.map f) 0).map f_1\n[PROOFSTEP]\nintro g\n[GOAL]\ncase zero.one\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u'\ninst\u271d\u00b2 : Category.{v, u'} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : IsEquivalence F\napp : (j : WalkingParallelPair) \u2192 (parallelPair f 0 \u22d9 F).obj j \u2245 (parallelPair (F.map f) 0).obj j :=\n fun j =>\n match j with\n | zero => Iso.refl ((parallelPair f 0 \u22d9 F).obj zero)\n | one => Iso.refl ((parallelPair f 0 \u22d9 F).obj one)\n\u22a2 \u2200 (f_1 : zero \u27f6 one),\n (parallelPair f 0 \u22d9 F).map f_1 \u226b (app one).hom = (app zero).hom \u226b (parallelPair (F.map f) 0).map f_1\n[PROOFSTEP]\nintro g\n[GOAL]\ncase one.zero\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u'\ninst\u271d\u00b2 : Category.{v, u'} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : IsEquivalence F\napp : (j : WalkingParallelPair) \u2192 (parallelPair f 0 \u22d9 F).obj j \u2245 (parallelPair (F.map f) 0).obj j :=\n fun j =>\n match j with\n | zero => Iso.refl ((parallelPair f 0 \u22d9 F).obj zero)\n | one => Iso.refl ((parallelPair f 0 \u22d9 F).obj one)\n\u22a2 \u2200 (f_1 : one \u27f6 zero),\n (parallelPair f 0 \u22d9 F).map f_1 \u226b (app zero).hom = (app one).hom \u226b (parallelPair (F.map f) 0).map f_1\n[PROOFSTEP]\nintro g\n[GOAL]\ncase one.one\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u'\ninst\u271d\u00b2 : Category.{v, u'} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : IsEquivalence F\napp : (j : WalkingParallelPair) \u2192 (parallelPair f 0 \u22d9 F).obj j \u2245 (parallelPair (F.map f) 0).obj j :=\n fun j =>\n match j with\n | zero => Iso.refl ((parallelPair f 0 \u22d9 F).obj zero)\n | one => Iso.refl ((parallelPair f 0 \u22d9 F).obj one)\n\u22a2 \u2200 (f_1 : one \u27f6 one),\n (parallelPair f 0 \u22d9 F).map f_1 \u226b (app one).hom = (app one).hom \u226b (parallelPair (F.map f) 0).map f_1\n[PROOFSTEP]\nintro g\n[GOAL]\ncase zero.zero\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u'\ninst\u271d\u00b2 : Category.{v, u'} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : IsEquivalence F\napp : (j : WalkingParallelPair) \u2192 (parallelPair f 0 \u22d9 F).obj j \u2245 (parallelPair (F.map f) 0).obj j :=\n fun j =>\n match j with\n | zero => Iso.refl ((parallelPair f 0 \u22d9 F).obj zero)\n | one => Iso.refl ((parallelPair f 0 \u22d9 F).obj one)\ng : zero \u27f6 zero\n\u22a2 (parallelPair f 0 \u22d9 F).map g \u226b (app zero).hom = (app zero).hom \u226b (parallelPair (F.map f) 0).map g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase zero.one\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u'\ninst\u271d\u00b2 : Category.{v, u'} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : IsEquivalence F\napp : (j : WalkingParallelPair) \u2192 (parallelPair f 0 \u22d9 F).obj j \u2245 (parallelPair (F.map f) 0).obj j :=\n fun j =>\n match j with\n | zero => Iso.refl ((parallelPair f 0 \u22d9 F).obj zero)\n | one => Iso.refl ((parallelPair f 0 \u22d9 F).obj one)\ng : zero \u27f6 one\n\u22a2 (parallelPair f 0 \u22d9 F).map g \u226b (app one).hom = (app zero).hom \u226b (parallelPair (F.map f) 0).map g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase one.zero\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u'\ninst\u271d\u00b2 : Category.{v, u'} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : IsEquivalence F\napp : (j : WalkingParallelPair) \u2192 (parallelPair f 0 \u22d9 F).obj j \u2245 (parallelPair (F.map f) 0).obj j :=\n fun j =>\n match j with\n | zero => Iso.refl ((parallelPair f 0 \u22d9 F).obj zero)\n | one => Iso.refl ((parallelPair f 0 \u22d9 F).obj one)\ng : one \u27f6 zero\n\u22a2 (parallelPair f 0 \u22d9 F).map g \u226b (app zero).hom = (app one).hom \u226b (parallelPair (F.map f) 0).map g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase one.one\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u'\ninst\u271d\u00b2 : Category.{v, u'} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : IsEquivalence F\napp : (j : WalkingParallelPair) \u2192 (parallelPair f 0 \u22d9 F).obj j \u2245 (parallelPair (F.map f) 0).obj j :=\n fun j =>\n match j with\n | zero => Iso.refl ((parallelPair f 0 \u22d9 F).obj zero)\n | one => Iso.refl ((parallelPair f 0 \u22d9 F).obj one)\ng : one \u27f6 one\n\u22a2 (parallelPair f 0 \u22d9 F).map g \u226b (app one).hom = (app one).hom \u226b (parallelPair (F.map f) 0).map g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase zero.zero.id\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u'\ninst\u271d\u00b2 : Category.{v, u'} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : IsEquivalence F\napp : (j : WalkingParallelPair) \u2192 (parallelPair f 0 \u22d9 F).obj j \u2245 (parallelPair (F.map f) 0).obj j :=\n fun j =>\n match j with\n | zero => Iso.refl ((parallelPair f 0 \u22d9 F).obj zero)\n | one => Iso.refl ((parallelPair f 0 \u22d9 F).obj one)\n\u22a2 (parallelPair f 0 \u22d9 F).map (WalkingParallelPairHom.id zero) \u226b (app zero).hom =\n (app zero).hom \u226b (parallelPair (F.map f) 0).map (WalkingParallelPairHom.id zero)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase zero.one.left\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u'\ninst\u271d\u00b2 : Category.{v, u'} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : IsEquivalence F\napp : (j : WalkingParallelPair) \u2192 (parallelPair f 0 \u22d9 F).obj j \u2245 (parallelPair (F.map f) 0).obj j :=\n fun j =>\n match j with\n | zero => Iso.refl ((parallelPair f 0 \u22d9 F).obj zero)\n | one => Iso.refl ((parallelPair f 0 \u22d9 F).obj one)\n\u22a2 (parallelPair f 0 \u22d9 F).map WalkingParallelPairHom.left \u226b (app one).hom =\n (app zero).hom \u226b (parallelPair (F.map f) 0).map WalkingParallelPairHom.left\n[PROOFSTEP]\nsimp\n[GOAL]\ncase zero.one.right\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u'\ninst\u271d\u00b2 : Category.{v, u'} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : IsEquivalence F\napp : (j : WalkingParallelPair) \u2192 (parallelPair f 0 \u22d9 F).obj j \u2245 (parallelPair (F.map f) 0).obj j :=\n fun j =>\n match j with\n | zero => Iso.refl ((parallelPair f 0 \u22d9 F).obj zero)\n | one => Iso.refl ((parallelPair f 0 \u22d9 F).obj one)\n\u22a2 (parallelPair f 0 \u22d9 F).map WalkingParallelPairHom.right \u226b (app one).hom =\n (app zero).hom \u226b (parallelPair (F.map f) 0).map WalkingParallelPairHom.right\n[PROOFSTEP]\nsimp\n[GOAL]\ncase one.one.id\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u'\ninst\u271d\u00b2 : Category.{v, u'} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : IsEquivalence F\napp : (j : WalkingParallelPair) \u2192 (parallelPair f 0 \u22d9 F).obj j \u2245 (parallelPair (F.map f) 0).obj j :=\n fun j =>\n match j with\n | zero => Iso.refl ((parallelPair f 0 \u22d9 F).obj zero)\n | one => Iso.refl ((parallelPair f 0 \u22d9 F).obj one)\n\u22a2 (parallelPair f 0 \u22d9 F).map (WalkingParallelPairHom.id one) \u226b (app one).hom =\n (app one).hom \u226b (parallelPair (F.map f) 0).map (WalkingParallelPairHom.id one)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nt : KernelFork f\nlift : (s : KernelFork f) \u2192 s.pt \u27f6 t.pt\nfac : \u2200 (s : KernelFork f), lift s \u226b Fork.\u03b9 t = Fork.\u03b9 s\nuniq : \u2200 (s : KernelFork f) (m : s.pt \u27f6 t.pt), m \u226b Fork.\u03b9 t = Fork.\u03b9 s \u2192 m = lift s\ns : Cone (parallelPair f 0)\nj : WalkingParallelPair\n\u22a2 lift s \u226b NatTrans.app t.\u03c0 j = NatTrans.app s.\u03c0 j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase zero\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nt : KernelFork f\nlift : (s : KernelFork f) \u2192 s.pt \u27f6 t.pt\nfac : \u2200 (s : KernelFork f), lift s \u226b Fork.\u03b9 t = Fork.\u03b9 s\nuniq : \u2200 (s : KernelFork f) (m : s.pt \u27f6 t.pt), m \u226b Fork.\u03b9 t = Fork.\u03b9 s \u2192 m = lift s\ns : Cone (parallelPair f 0)\n\u22a2 lift s \u226b NatTrans.app t.\u03c0 zero = NatTrans.app s.\u03c0 zero\n[PROOFSTEP]\nexact fac s\n[GOAL]\ncase one\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nt : KernelFork f\nlift : (s : KernelFork f) \u2192 s.pt \u27f6 t.pt\nfac : \u2200 (s : KernelFork f), lift s \u226b Fork.\u03b9 t = Fork.\u03b9 s\nuniq : \u2200 (s : KernelFork f) (m : s.pt \u27f6 t.pt), m \u226b Fork.\u03b9 t = Fork.\u03b9 s \u2192 m = lift s\ns : Cone (parallelPair f 0)\n\u22a2 lift s \u226b NatTrans.app t.\u03c0 one = NatTrans.app s.\u03c0 one\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\ni : IsLimit c\nZ : C\ng : Y \u27f6 Z\nhg : Mono g\nh : X \u27f6 Z\nhh : h = f \u226b g\n\u22a2 Fork.\u03b9 c \u226b h = 0\n[PROOFSTEP]\nsimp [hh]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\ni : IsLimit c\nZ : C\ng : Y \u27f6 Z\nhg : Mono g\nh : X \u27f6 Z\nhh : h = f \u226b g\ns : Fork h 0\n\u22a2 Fork.\u03b9 s \u226b f = Fork.\u03b9 s \u226b 0\n[PROOFSTEP]\nrw [\u2190 cancel_mono g]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\ni : IsLimit c\nZ : C\ng : Y \u27f6 Z\nhg : Mono g\nh : X \u27f6 Z\nhh : h = f \u226b g\ns : Fork h 0\n\u22a2 (Fork.\u03b9 s \u226b f) \u226b g = (Fork.\u03b9 s \u226b 0) \u226b g\n[PROOFSTEP]\nsimp [\u2190 hh, s.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\ni : IsLimit c\nZ : C\ng : Y \u27f6 Z\nhg : Mono g\nh : X \u27f6 Z\nhh : h = f \u226b g\ns : Fork h 0\ns' : KernelFork f := Fork.of\u03b9 (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b f = Fork.\u03b9 s \u226b 0)\nl : { l // l \u226b Fork.\u03b9 c = Fork.\u03b9 s' } := KernelFork.IsLimit.lift' i (Fork.\u03b9 s') (_ : Fork.\u03b9 s' \u226b f = 0)\nm\u271d :\n ((Functor.const WalkingParallelPair).obj s.pt).obj zero \u27f6\n ((Functor.const WalkingParallelPair).obj (KernelFork.of\u03b9 (Fork.\u03b9 c) (_ : Fork.\u03b9 c \u226b h = 0)).pt).obj zero\nhm : m\u271d \u226b Fork.\u03b9 (KernelFork.of\u03b9 (Fork.\u03b9 c) (_ : Fork.\u03b9 c \u226b h = 0)) = Fork.\u03b9 s\n\u22a2 m\u271d = \u2191l\n[PROOFSTEP]\napply Fork.IsLimit.hom_ext i\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\ni : IsLimit c\nZ : C\ng : Y \u27f6 Z\nhg : Mono g\nh : X \u27f6 Z\nhh : h = f \u226b g\ns : Fork h 0\ns' : KernelFork f := Fork.of\u03b9 (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b f = Fork.\u03b9 s \u226b 0)\nl : { l // l \u226b Fork.\u03b9 c = Fork.\u03b9 s' } := KernelFork.IsLimit.lift' i (Fork.\u03b9 s') (_ : Fork.\u03b9 s' \u226b f = 0)\nm\u271d :\n ((Functor.const WalkingParallelPair).obj s.pt).obj zero \u27f6\n ((Functor.const WalkingParallelPair).obj (KernelFork.of\u03b9 (Fork.\u03b9 c) (_ : Fork.\u03b9 c \u226b h = 0)).pt).obj zero\nhm : m\u271d \u226b Fork.\u03b9 (KernelFork.of\u03b9 (Fork.\u03b9 c) (_ : Fork.\u03b9 c \u226b h = 0)) = Fork.\u03b9 s\n\u22a2 m\u271d \u226b Fork.\u03b9 c = \u2191l \u226b Fork.\u03b9 c\n[PROOFSTEP]\nrw [Fork.\u03b9_of\u03b9] at hm \n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\ni : IsLimit c\nZ : C\ng : Y \u27f6 Z\nhg : Mono g\nh : X \u27f6 Z\nhh : h = f \u226b g\ns : Fork h 0\ns' : KernelFork f := Fork.of\u03b9 (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b f = Fork.\u03b9 s \u226b 0)\nl : { l // l \u226b Fork.\u03b9 c = Fork.\u03b9 s' } := KernelFork.IsLimit.lift' i (Fork.\u03b9 s') (_ : Fork.\u03b9 s' \u226b f = 0)\nm\u271d :\n ((Functor.const WalkingParallelPair).obj s.pt).obj zero \u27f6\n ((Functor.const WalkingParallelPair).obj (KernelFork.of\u03b9 (Fork.\u03b9 c) (_ : Fork.\u03b9 c \u226b h = 0)).pt).obj zero\nhm : m\u271d \u226b Fork.\u03b9 c = Fork.\u03b9 s\n\u22a2 m\u271d \u226b Fork.\u03b9 c = \u2191l \u226b Fork.\u03b9 c\n[PROOFSTEP]\nrw [hm]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\ni : IsLimit c\nZ : C\ng : Y \u27f6 Z\nhg : Mono g\nh : X \u27f6 Z\nhh : h = f \u226b g\ns : Fork h 0\ns' : KernelFork f := Fork.of\u03b9 (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b f = Fork.\u03b9 s \u226b 0)\nl : { l // l \u226b Fork.\u03b9 c = Fork.\u03b9 s' } := KernelFork.IsLimit.lift' i (Fork.\u03b9 s') (_ : Fork.\u03b9 s' \u226b f = 0)\nm\u271d :\n ((Functor.const WalkingParallelPair).obj s.pt).obj zero \u27f6\n ((Functor.const WalkingParallelPair).obj (KernelFork.of\u03b9 (Fork.\u03b9 c) (_ : Fork.\u03b9 c \u226b h = 0)).pt).obj zero\nhm : m\u271d \u226b Fork.\u03b9 c = Fork.\u03b9 s\n\u22a2 Fork.\u03b9 s = \u2191l \u226b Fork.\u03b9 c\n[PROOFSTEP]\nexact l.2.symm\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\ni : IsLimit c\nZ : C\ng : Y \u27f6 Z\nhg : Mono g\nh : X \u27f6 Z\nhh : h = f \u226b g\ns : KernelFork h\n\u22a2 Fork.\u03b9 s \u226b f = Fork.\u03b9 s \u226b 0\n[PROOFSTEP]\nrw [\u2190 cancel_mono g, Category.assoc, \u2190 hh]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\ni : IsLimit c\nZ : C\ng : Y \u27f6 Z\nhg : Mono g\nh : X \u27f6 Z\nhh : h = f \u226b g\ns : KernelFork h\n\u22a2 Fork.\u03b9 s \u226b h = (Fork.\u03b9 s \u226b 0) \u226b g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nW : C\ng : Y \u27f6 W\nh : X \u27f6 W\nc : KernelFork h\ni : IsLimit c\nhf : Fork.\u03b9 c \u226b f = 0\nhfg : f \u226b g = h\ns : Fork f 0\n\u22a2 Fork.\u03b9 s \u226b h = 0\n[PROOFSTEP]\nsimp [\u2190 hfg]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nW : C\ng : Y \u27f6 W\nh : X \u27f6 W\nc : KernelFork h\ni : IsLimit c\nhf : Fork.\u03b9 c \u226b f = 0\nhfg : f \u226b g = h\ns : Fork f 0\n\u22a2 (fun s => IsLimit.lift i (KernelFork.of\u03b9 (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b h = 0))) s \u226b\n Fork.\u03b9 (KernelFork.of\u03b9 (Fork.\u03b9 c) hf) =\n Fork.\u03b9 s\n[PROOFSTEP]\nsimp only [KernelFork.\u03b9_of\u03b9, Fork.IsLimit.lift_\u03b9]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nW : C\ng : Y \u27f6 W\nh\u271d : X \u27f6 W\nc : KernelFork h\u271d\ni : IsLimit c\nhf : Fork.\u03b9 c \u226b f = 0\nhfg : f \u226b g = h\u271d\ns : Fork f 0\nm : s.pt \u27f6 (KernelFork.of\u03b9 (Fork.\u03b9 c) hf).pt\nh : m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (Fork.\u03b9 c) hf) = Fork.\u03b9 s\n\u22a2 m = (fun s => IsLimit.lift i (KernelFork.of\u03b9 (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b h\u271d = 0))) s\n[PROOFSTEP]\napply Fork.IsLimit.hom_ext i\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nW : C\ng : Y \u27f6 W\nh\u271d : X \u27f6 W\nc : KernelFork h\u271d\ni : IsLimit c\nhf : Fork.\u03b9 c \u226b f = 0\nhfg : f \u226b g = h\u271d\ns : Fork f 0\nm : s.pt \u27f6 (KernelFork.of\u03b9 (Fork.\u03b9 c) hf).pt\nh : m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (Fork.\u03b9 c) hf) = Fork.\u03b9 s\n\u22a2 m \u226b Fork.\u03b9 c = (fun s => IsLimit.lift i (KernelFork.of\u03b9 (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b h\u271d = 0))) s \u226b Fork.\u03b9 c\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nhf : f = 0\n\u22a2 \ud835\udfd9 X \u226b f = 0\n[PROOFSTEP]\nrw [hf, comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nhf : f = 0\nW'\u271d : C\nx\u271d\u00b2 : W'\u271d \u27f6 X\nx\u271d\u00b9 : x\u271d\u00b2 \u226b f = 0\nx\u271d : W'\u271d \u27f6 X\nhb : x\u271d \u226b \ud835\udfd9 X = x\u271d\u00b2\n\u22a2 x\u271d = (fun {W'} x x_1 => x) x\u271d\u00b2 x\u271d\u00b9\n[PROOFSTEP]\nsimp only [\u2190 hb, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\nhf : Mono f\nh : IsZero c.pt\ns : KernelFork f\n\u22a2 (fun s => 0) s \u226b Fork.\u03b9 c = Fork.\u03b9 s\n[PROOFSTEP]\nrw [zero_comp, \u2190 cancel_mono f, zero_comp, s.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\nhc : IsLimit c\nhf : f = 0\n\u22a2 IsIso (Fork.\u03b9 c)\n[PROOFSTEP]\nlet e : c.pt \u2245 X := IsLimit.conePointUniqueUpToIso hc (KernelFork.IsLimit.ofId (f : X \u27f6 Y) hf)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\nhc : IsLimit c\nhf : f = 0\ne : c.pt \u2245 X := IsLimit.conePointUniqueUpToIso hc (ofId f hf)\n\u22a2 IsIso (Fork.\u03b9 c)\n[PROOFSTEP]\nhave eq : e.inv \u226b c.\u03b9 = \ud835\udfd9 X := Fork.IsLimit.lift_\u03b9 hc\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\nhc : IsLimit c\nhf : f = 0\ne : c.pt \u2245 X := IsLimit.conePointUniqueUpToIso hc (ofId f hf)\neq : e.inv \u226b Fork.\u03b9 c = \ud835\udfd9 X\n\u22a2 IsIso (Fork.\u03b9 c)\n[PROOFSTEP]\nhaveI : IsIso (e.inv \u226b c.\u03b9) := by\n rw [eq]\n infer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\nhc : IsLimit c\nhf : f = 0\ne : c.pt \u2245 X := IsLimit.conePointUniqueUpToIso hc (ofId f hf)\neq : e.inv \u226b Fork.\u03b9 c = \ud835\udfd9 X\n\u22a2 IsIso (e.inv \u226b Fork.\u03b9 c)\n[PROOFSTEP]\nrw [eq]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\nhc : IsLimit c\nhf : f = 0\ne : c.pt \u2245 X := IsLimit.conePointUniqueUpToIso hc (ofId f hf)\neq : e.inv \u226b Fork.\u03b9 c = \ud835\udfd9 X\n\u22a2 IsIso (\ud835\udfd9 X)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\nhc : IsLimit c\nhf : f = 0\ne : c.pt \u2245 X := IsLimit.conePointUniqueUpToIso hc (ofId f hf)\neq : e.inv \u226b Fork.\u03b9 c = \ud835\udfd9 X\nthis : IsIso (e.inv \u226b Fork.\u03b9 c)\n\u22a2 IsIso (Fork.\u03b9 c)\n[PROOFSTEP]\nexact IsIso.of_isIso_comp_left e.inv c.\u03b9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\n\u22a2 (Iso.refl (limit.cone (parallelPair f 0)).pt).hom \u226b\n Fork.\u03b9 (Fork.of\u03b9 (kernel.\u03b9 f) (_ : kernel.\u03b9 f \u226b f = kernel.\u03b9 f \u226b 0)) =\n Fork.\u03b9 (limit.cone (parallelPair f 0))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nW : C\nh : 0 \u226b f = 0\n\u22a2 lift f 0 h = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nW : C\nh : 0 \u226b f = 0\n\u22a2 lift f 0 h \u226b equalizer.\u03b9 f 0 = 0 \u226b equalizer.\u03b9 f 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\nW : C\nk : W \u27f6 X\nh : k \u226b f = 0\ninst\u271d : Mono k\nZ : C\ng g' : Z \u27f6 W\nw : g \u226b lift f k h = g' \u226b lift f k h\n\u22a2 g = g'\n[PROOFSTEP]\nreplace w := w =\u226b kernel.\u03b9 f\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\nW : C\nk : W \u27f6 X\nh : k \u226b f = 0\ninst\u271d : Mono k\nZ : C\ng g' : Z \u27f6 W\nw : (g \u226b lift f k h) \u226b \u03b9 f = (g' \u226b lift f k h) \u226b \u03b9 f\n\u22a2 g = g'\n[PROOFSTEP]\nsimp only [Category.assoc, kernel.lift_\u03b9] at w \n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\nW : C\nk : W \u27f6 X\nh : k \u226b f = 0\ninst\u271d : Mono k\nZ : C\ng g' : Z \u27f6 W\nw : g \u226b k = g' \u226b k\n\u22a2 g = g'\n[PROOFSTEP]\nexact (cancel_mono k).1 w\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\nX' Y' : C\nf' : X' \u27f6 Y'\ninst\u271d : HasKernel f'\np : X \u27f6 X'\nq : Y \u27f6 Y'\nw : f \u226b q = p \u226b f'\n\u22a2 (\u03b9 f \u226b p) \u226b f' = 0\n[PROOFSTEP]\nsimp [\u2190 w]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasKernel f\u271d\nX Y Z X' Y' Z' : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasKernel g\nw : f \u226b g = 0\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\ninst\u271d : HasKernel g'\nw' : f' \u226b g' = 0\np : X \u27f6 X'\nq : Y \u27f6 Y'\nr : Z \u27f6 Z'\nh\u2081 : f \u226b q = p \u226b f'\nh\u2082 : g \u226b r = q \u226b g'\n\u22a2 lift g f w \u226b map g g' q r h\u2082 = p \u226b lift g' f' w'\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasKernel f\u271d\nX Y Z X' Y' Z' : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasKernel g\nw : f \u226b g = 0\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\ninst\u271d : HasKernel g'\nw' : f' \u226b g' = 0\np : X \u27f6 X'\nq : Y \u27f6 Y'\nr : Z \u27f6 Z'\nh\u2081 : f \u226b q = p \u226b f'\nh\u2082 : g \u226b r = q \u226b g'\n\u22a2 (lift g f w \u226b map g g' q r h\u2082) \u226b equalizer.\u03b9 g' 0 = (p \u226b lift g' f' w') \u226b equalizer.\u03b9 g' 0\n[PROOFSTEP]\nsimp [h\u2081]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\nX' Y' : C\nf' : X' \u27f6 Y'\ninst\u271d : HasKernel f'\np : X \u2245 X'\nq : Y \u2245 Y'\nw : f \u226b q.hom = p.hom \u226b f'\n\u22a2 f' \u226b q.inv = p.inv \u226b f\n[PROOFSTEP]\nrefine' (cancel_mono q.hom).1 _\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\nX' Y' : C\nf' : X' \u27f6 Y'\ninst\u271d : HasKernel f'\np : X \u2245 X'\nq : Y \u2245 Y'\nw : f \u226b q.hom = p.hom \u226b f'\n\u22a2 (f' \u226b q.inv) \u226b q.hom = (p.inv \u226b f) \u226b q.hom\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : Epi (kernel.\u03b9 f)\n\u22a2 kernel.\u03b9 f \u226b f = kernel.\u03b9 f \u226b 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\n\u22a2 \ud835\udfd9 X \u226b 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\n\u22a2 kernelZeroIsoSource.inv = kernel.lift 0 (\ud835\udfd9 X) (_ : \ud835\udfd9 X \u226b 0 = 0)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\n\u22a2 kernelZeroIsoSource.inv \u226b equalizer.\u03b9 0 0 = kernel.lift 0 (\ud835\udfd9 X) (_ : \ud835\udfd9 X \u226b 0 = 0) \u226b equalizer.\u03b9 0 0\n[PROOFSTEP]\nsimp [kernelZeroIsoSource]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nf g : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasKernel g\nh : f = g\n\u22a2 parallelPair f 0 \u2245 parallelPair g 0\n[PROOFSTEP]\nrw [h]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nh : f = f\n\u22a2 kernelIsoOfEq h = Iso.refl (kernel f)\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nh : f = f\n\u22a2 (kernelIsoOfEq h).hom \u226b equalizer.\u03b9 f 0 = (Iso.refl (kernel f)).hom \u226b equalizer.\u03b9 f 0\n[PROOFSTEP]\nsimp [kernelIsoOfEq]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nf g : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasKernel g\nh : f = g\n\u22a2 (kernelIsoOfEq h).hom \u226b kernel.\u03b9 g = kernel.\u03b9 f\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b9 inst\u271d : HasKernel f\n\u22a2 (kernelIsoOfEq (_ : f = f)).hom \u226b kernel.\u03b9 f = kernel.\u03b9 f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nf g : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasKernel g\nh : f = g\n\u22a2 (kernelIsoOfEq h).inv \u226b kernel.\u03b9 f = kernel.\u03b9 g\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b9 inst\u271d : HasKernel f\n\u22a2 (kernelIsoOfEq (_ : f = f)).inv \u226b kernel.\u03b9 f = kernel.\u03b9 f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nZ : C\nf g : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasKernel g\nh : f = g\ne : Z \u27f6 X\nhe : e \u226b f = 0\n\u22a2 e \u226b g = 0\n[PROOFSTEP]\nsimp [\u2190 h, he]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nZ : C\nf g : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasKernel g\nh : f = g\ne : Z \u27f6 X\nhe : e \u226b f = 0\n\u22a2 kernel.lift f e he \u226b (kernelIsoOfEq h).hom = kernel.lift g e (_ : e \u226b g = 0)\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nZ : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ne : Z \u27f6 X\nhe : e \u226b f = 0\ninst\u271d : HasKernel f\n\u22a2 kernel.lift f e he \u226b (kernelIsoOfEq (_ : f = f)).hom = kernel.lift f e (_ : e \u226b f = 0)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nZ : C\nf g : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasKernel g\nh : f = g\ne : Z \u27f6 X\nhe : e \u226b g = 0\n\u22a2 e \u226b f = 0\n[PROOFSTEP]\nsimp [h, he]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nZ : C\nf g : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasKernel g\nh : f = g\ne : Z \u27f6 X\nhe : e \u226b g = 0\n\u22a2 kernel.lift g e he \u226b (kernelIsoOfEq h).inv = kernel.lift f e (_ : e \u226b f = 0)\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nZ : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ne : Z \u27f6 X\ninst\u271d : HasKernel f\nhe : e \u226b f = 0\n\u22a2 kernel.lift f e he \u226b (kernelIsoOfEq (_ : f = f)).inv = kernel.lift f e (_ : e \u226b f = 0)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasKernel f\u271d\nf g h : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\ninst\u271d\u00b9 : HasKernel g\ninst\u271d : HasKernel h\nw\u2081 : f = g\nw\u2082 : g = h\n\u22a2 kernelIsoOfEq w\u2081 \u226a\u226b kernelIsoOfEq w\u2082 = kernelIsoOfEq (_ : f = h)\n[PROOFSTEP]\ncases w\u2081\n[GOAL]\ncase refl\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasKernel f\u271d\nf h : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\ninst\u271d\u00b9 : HasKernel h\ninst\u271d : HasKernel f\nw\u2082 : f = h\n\u22a2 kernelIsoOfEq (_ : f = f) \u226a\u226b kernelIsoOfEq w\u2082 = kernelIsoOfEq (_ : f = h)\n[PROOFSTEP]\ncases w\u2082\n[GOAL]\ncase refl.refl\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasKernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b2 inst\u271d\u00b9 inst\u271d : HasKernel f\n\u22a2 kernelIsoOfEq (_ : f = f) \u226a\u226b kernelIsoOfEq (_ : f = f) = kernelIsoOfEq (_ : f = f)\n[PROOFSTEP]\next\n[GOAL]\ncase refl.refl.w.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasKernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b2 inst\u271d\u00b9 inst\u271d : HasKernel f\n\u22a2 (kernelIsoOfEq (_ : f = f) \u226a\u226b kernelIsoOfEq (_ : f = f)).hom \u226b equalizer.\u03b9 f 0 =\n (kernelIsoOfEq (_ : f = f)).hom \u226b equalizer.\u03b9 f 0\n[PROOFSTEP]\nsimp [kernelIsoOfEq]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nw : f \u2260 0\nI : IsIso (kernel.\u03b9 f)\n\u22a2 Epi (kernel.\u03b9 f)\n[PROOFSTEP]\nskip\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nw : f \u2260 0\nI : IsIso (kernel.\u03b9 f)\n\u22a2 Epi (kernel.\u03b9 f)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasKernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : Mono g\n\u22a2 kernel.\u03b9 (f \u226b g) \u226b f = 0\n[PROOFSTEP]\nrw [\u2190 cancel_mono g]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasKernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : Mono g\n\u22a2 (kernel.\u03b9 (f \u226b g) \u226b f) \u226b g = 0 \u226b g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasKernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : Mono g\n\u22a2 kernel.\u03b9 f \u226b f \u226b g = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasKernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\ninst\u271d : HasKernel g\n\u22a2 (kernel.\u03b9 g \u226b inv f) \u226b f \u226b g = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasKernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\ninst\u271d : HasKernel g\ns : KernelFork (f \u226b g)\n\u22a2 (Fork.\u03b9 s \u226b f) \u226b g = 0\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasKernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\ninst\u271d : HasKernel g\n\u22a2 \u2200 (s : KernelFork (f \u226b g)),\n (fun s => kernel.lift g (Fork.\u03b9 s \u226b f) (_ : (Fork.\u03b9 s \u226b f) \u226b g = 0)) s \u226b\n Fork.\u03b9 (KernelFork.of\u03b9 (kernel.\u03b9 g \u226b inv f) (_ : (kernel.\u03b9 g \u226b inv f) \u226b f \u226b g = 0)) =\n Fork.\u03b9 s\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasKernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\ninst\u271d : HasKernel g\ns : KernelFork (f \u226b g)\nm : s.pt \u27f6 (KernelFork.of\u03b9 (kernel.\u03b9 g \u226b inv f) (_ : (kernel.\u03b9 g \u226b inv f) \u226b f \u226b g = 0)).pt\nw : m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (kernel.\u03b9 g \u226b inv f) (_ : (kernel.\u03b9 g \u226b inv f) \u226b f \u226b g = 0)) = Fork.\u03b9 s\n\u22a2 m = (fun s => kernel.lift g (Fork.\u03b9 s \u226b f) (_ : (Fork.\u03b9 s \u226b f) \u226b g = 0)) s\n[PROOFSTEP]\nsimp_rw [\u2190 w]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasKernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\ninst\u271d : HasKernel g\ns : KernelFork (f \u226b g)\nm : s.pt \u27f6 (KernelFork.of\u03b9 (kernel.\u03b9 g \u226b inv f) (_ : (kernel.\u03b9 g \u226b inv f) \u226b f \u226b g = 0)).pt\nw : m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (kernel.\u03b9 g \u226b inv f) (_ : (kernel.\u03b9 g \u226b inv f) \u226b f \u226b g = 0)) = Fork.\u03b9 s\n\u22a2 m =\n kernel.lift g ((m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (kernel.\u03b9 g \u226b inv f) (_ : (kernel.\u03b9 g \u226b inv f) \u226b f \u226b g = 0))) \u226b f)\n (_ : ((m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (kernel.\u03b9 g \u226b inv f) (_ : (kernel.\u03b9 g \u226b inv f) \u226b f \u226b g = 0))) \u226b f) \u226b g = 0)\n[PROOFSTEP]\napply equalizer.hom_ext\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasKernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\ninst\u271d : HasKernel g\ns : KernelFork (f \u226b g)\nm : s.pt \u27f6 (KernelFork.of\u03b9 (kernel.\u03b9 g \u226b inv f) (_ : (kernel.\u03b9 g \u226b inv f) \u226b f \u226b g = 0)).pt\nw : m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (kernel.\u03b9 g \u226b inv f) (_ : (kernel.\u03b9 g \u226b inv f) \u226b f \u226b g = 0)) = Fork.\u03b9 s\n\u22a2 m \u226b equalizer.\u03b9 g 0 =\n kernel.lift g ((m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (kernel.\u03b9 g \u226b inv f) (_ : (kernel.\u03b9 g \u226b inv f) \u226b f \u226b g = 0))) \u226b f)\n (_ : ((m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (kernel.\u03b9 g \u226b inv f) (_ : (kernel.\u03b9 g \u226b inv f) \u226b f \u226b g = 0))) \u226b f) \u226b g = 0) \u226b\n equalizer.\u03b9 g 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasKernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\ninst\u271d : HasKernel g\n\u22a2 (kernel.\u03b9 (f \u226b g) \u226b f) \u226b g = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasKernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\ninst\u271d : HasKernel g\n\u22a2 (kernel.\u03b9 g \u226b inv f) \u226b f \u226b g = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : Mono f\ns : Fork f 0\n\u22a2 (fun s => 0) s \u226b Fork.\u03b9 (zeroKernelFork f) = Fork.\u03b9 s\n[PROOFSTEP]\nerw [zero_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : Mono f\ns : Fork f 0\n\u22a2 0 = Fork.\u03b9 s\n[PROOFSTEP]\nrefine (zero_of_comp_mono f ?_).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : Mono f\ns : Fork f 0\n\u22a2 Fork.\u03b9 s \u226b f = 0\n[PROOFSTEP]\nexact KernelFork.condition _\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d : HasZeroObject C\nX Y : C\nf : X \u27f6 Y\nhf : \u2200 (Z : C) (g : Z \u27f6 X), g \u226b f = 0 \u2192 g = 0\n\u22a2 0 \u226b f = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d : HasZeroObject C\nX Y : C\nf : X \u27f6 Y\nhf : \u2200 (Z : C) (g : Z \u27f6 X), g \u226b f = 0 \u2192 g = 0\ns : Fork f 0\n\u22a2 (fun s => 0) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0)) = Fork.\u03b9 s\n[PROOFSTEP]\nrw [hf _ _ (KernelFork.condition s), zero_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d : HasZeroObject C\nX Y : C\nf : X \u27f6 Y\nhf : \u2200 (Z : C) (g : Z \u27f6 X), g \u226b f = 0 \u2192 g = 0\ns : Fork f 0\nm : s.pt \u27f6 (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0)).pt\nx\u271d : m \u226b Fork.\u03b9 (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0)) = Fork.\u03b9 s\n\u22a2 m = (fun s => 0) s\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d : HasZeroObject C\nX Y : C\nf : X \u27f6 Y\nhf : \u2200 (Z : C) (g : Z \u27f6 X), g \u226b f = 0 \u2192 g = 0\ns : Fork f 0\nm : s.pt \u27f6 (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0)).pt\nx\u271d : m \u226b Fork.\u03b9 (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0)) = Fork.\u03b9 s\n\u22a2 m = 0\n[PROOFSTEP]\napply HasZeroObject.to_zero_ext\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : X \u27f6 Z\ni : Z \u2245 Y\nh : l \u226b i.hom = f\ns : KernelFork f\nhs : IsLimit s\n\u22a2 Fork.\u03b9 s \u226b l = 0\n[PROOFSTEP]\nsimp [\u2190 i.comp_inv_eq.2 h.symm]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : X \u27f6 Z\ni : Z \u2245 Y\nh : l \u226b i.hom = f\ns\u271d : KernelFork f\nhs : IsLimit s\u271d\ns : Fork l 0\n\u22a2 Fork.\u03b9 s \u226b f = 0\n[PROOFSTEP]\nsimp [\u2190 h]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : X \u27f6 Z\ni : Z \u2245 Y\nh : l \u226b i.hom = f\ns\u271d : KernelFork f\nhs : IsLimit s\u271d\ns : Fork l 0\n\u22a2 (fun s => IsLimit.lift hs (KernelFork.of\u03b9 (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b f = 0))) s \u226b\n Fork.\u03b9 (KernelFork.of\u03b9 (Fork.\u03b9 s\u271d) (_ : Fork.\u03b9 s\u271d \u226b l = 0)) =\n Fork.\u03b9 s\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : X \u27f6 Z\ni : Z \u2245 Y\nh\u271d : l \u226b i.hom = f\ns\u271d : KernelFork f\nhs : IsLimit s\u271d\ns : Fork l 0\nm : s.pt \u27f6 (KernelFork.of\u03b9 (Fork.\u03b9 s\u271d) (_ : Fork.\u03b9 s\u271d \u226b l = 0)).pt\nh : m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (Fork.\u03b9 s\u271d) (_ : Fork.\u03b9 s\u271d \u226b l = 0)) = Fork.\u03b9 s\n\u22a2 m = (fun s => IsLimit.lift hs (KernelFork.of\u03b9 (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b f = 0))) s\n[PROOFSTEP]\napply Fork.IsLimit.hom_ext hs\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : X \u27f6 Z\ni : Z \u2245 Y\nh\u271d : l \u226b i.hom = f\ns\u271d : KernelFork f\nhs : IsLimit s\u271d\ns : Fork l 0\nm : s.pt \u27f6 (KernelFork.of\u03b9 (Fork.\u03b9 s\u271d) (_ : Fork.\u03b9 s\u271d \u226b l = 0)).pt\nh : m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (Fork.\u03b9 s\u271d) (_ : Fork.\u03b9 s\u271d \u226b l = 0)) = Fork.\u03b9 s\n\u22a2 m \u226b Fork.\u03b9 s\u271d = (fun s => IsLimit.lift hs (KernelFork.of\u03b9 (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b f = 0))) s \u226b Fork.\u03b9 s\u271d\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nZ : C\nl : X \u27f6 Z\ni : Z \u2245 Y\nh : l \u226b i.hom = f\n\u22a2 \u03b9 f \u226b l = 0\n[PROOFSTEP]\nsimp [\u2190 i.comp_inv_eq.2 h.symm]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : Z \u27f6 X\ns : KernelFork f\nhs : IsLimit s\ni : Z \u2245 s.pt\nh : i.hom \u226b Fork.\u03b9 s = l\n\u22a2 l \u226b f = 0\n[PROOFSTEP]\nsimp [\u2190 h]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : Z \u27f6 X\ns : KernelFork f\nhs : IsLimit s\ni : Z \u2245 s.pt\nh : i.hom \u226b Fork.\u03b9 s = l\nj : WalkingParallelPair\n\u22a2 NatTrans.app s.\u03c0 j = i.symm.hom \u226b NatTrans.app (KernelFork.of\u03b9 l (_ : l \u226b f = 0)).\u03c0 j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase zero\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : Z \u27f6 X\ns : KernelFork f\nhs : IsLimit s\ni : Z \u2245 s.pt\nh : i.hom \u226b Fork.\u03b9 s = l\n\u22a2 NatTrans.app s.\u03c0 zero = i.symm.hom \u226b NatTrans.app (KernelFork.of\u03b9 l (_ : l \u226b f = 0)).\u03c0 zero\n[PROOFSTEP]\nexact (Iso.eq_inv_comp i).2 h\n[GOAL]\ncase one\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : Z \u27f6 X\ns : KernelFork f\nhs : IsLimit s\ni : Z \u2245 s.pt\nh : i.hom \u226b Fork.\u03b9 s = l\n\u22a2 NatTrans.app s.\u03c0 one = i.symm.hom \u226b NatTrans.app (KernelFork.of\u03b9 l (_ : l \u226b f = 0)).\u03c0 one\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase one\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : Z \u27f6 X\ns : KernelFork f\nhs : IsLimit s\ni : Z \u2245 s.pt\nh : i.hom \u226b Fork.\u03b9 s = l\n\u22a2 NatTrans.app s.\u03c0 one = i.inv \u226b l \u226b f\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase one\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : Z \u27f6 X\ns : KernelFork f\nhs : IsLimit s\ni : Z \u2245 s.pt\nh : i.hom \u226b Fork.\u03b9 s = l\n\u22a2 NatTrans.app s.\u03c0 one = i.inv \u226b (i.hom \u226b Fork.\u03b9 s) \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nZ : C\nl : Z \u27f6 X\ni : Z \u2245 kernel f\nh : i.hom \u226b \u03b9 f = l\n\u22a2 l \u226b f = 0\n[PROOFSTEP]\nsimp [\u2190 h]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ns : CokernelCofork f\n\u22a2 f \u226b Cofork.\u03c0 s = 0\n[PROOFSTEP]\nrw [Cofork.condition, zero_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ns : CokernelCofork f\n\u22a2 NatTrans.app s.\u03b9 zero = 0\n[PROOFSTEP]\nsimp [Cofork.app_zero_eq_comp_\u03c0_right]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\n\u03c0 : Y \u27f6 Z\nw : f \u226b \u03c0 = 0\n\u22a2 f \u226b \u03c0 = 0 \u226b \u03c0\n[PROOFSTEP]\nrw [w, zero_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ns : Cofork f 0\nj : WalkingParallelPair\n\u22a2 NatTrans.app s.\u03b9 j \u226b (Iso.refl s.pt).hom =\n NatTrans.app (Cofork.of\u03c0 (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = 0 \u226b Cofork.\u03c0 s)).\u03b9 j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase zero\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ns : Cofork f 0\n\u22a2 NatTrans.app s.\u03b9 zero \u226b (Iso.refl s.pt).hom =\n NatTrans.app (Cofork.of\u03c0 (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = 0 \u226b Cofork.\u03c0 s)).\u03b9 zero\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase one\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ns : Cofork f 0\n\u22a2 NatTrans.app s.\u03b9 one \u226b (Iso.refl s.pt).hom =\n NatTrans.app (Cofork.of\u03c0 (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = 0 \u226b Cofork.\u03c0 s)).\u03b9 one\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nP : C\n\u03c0 \u03c0' : Y \u27f6 P\nw : f \u226b \u03c0 = 0\nh : \u03c0 = \u03c0'\n\u22a2 f \u226b \u03c0' = 0\n[PROOFSTEP]\nrw [\u2190 h, w]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nP : C\n\u03c0 \u03c0' : Y \u27f6 P\nw : f \u226b \u03c0 = 0\nh : \u03c0 = \u03c0'\nj : WalkingParallelPair\n\u22a2 NatTrans.app (CokernelCofork.of\u03c0 \u03c0 w).\u03b9 j \u226b (Iso.refl (CokernelCofork.of\u03c0 \u03c0 w).pt).hom =\n NatTrans.app (CokernelCofork.of\u03c0 \u03c0' (_ : f \u226b \u03c0' = 0)).\u03b9 j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase zero\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nP : C\n\u03c0 \u03c0' : Y \u27f6 P\nw : f \u226b \u03c0 = 0\nh : \u03c0 = \u03c0'\n\u22a2 NatTrans.app (CokernelCofork.of\u03c0 \u03c0 w).\u03b9 zero \u226b (Iso.refl (CokernelCofork.of\u03c0 \u03c0 w).pt).hom =\n NatTrans.app (CokernelCofork.of\u03c0 \u03c0' (_ : f \u226b \u03c0' = 0)).\u03b9 zero\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase one\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nP : C\n\u03c0 \u03c0' : Y \u27f6 P\nw : f \u226b \u03c0 = 0\nh : \u03c0 = \u03c0'\n\u22a2 NatTrans.app (CokernelCofork.of\u03c0 \u03c0 w).\u03b9 one \u226b (Iso.refl (CokernelCofork.of\u03c0 \u03c0 w).pt).hom =\n NatTrans.app (CokernelCofork.of\u03c0 \u03c0' (_ : f \u226b \u03c0' = 0)).\u03b9 one\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nt : CokernelCofork f\ndesc : (s : CokernelCofork f) \u2192 t.pt \u27f6 s.pt\nfac : \u2200 (s : CokernelCofork f), Cofork.\u03c0 t \u226b desc s = Cofork.\u03c0 s\nuniq : \u2200 (s : CokernelCofork f) (m : t.pt \u27f6 s.pt), Cofork.\u03c0 t \u226b m = Cofork.\u03c0 s \u2192 m = desc s\ns : Cocone (parallelPair f 0)\nj : WalkingParallelPair\n\u22a2 NatTrans.app t.\u03b9 j \u226b desc s = NatTrans.app s.\u03b9 j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase zero\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nt : CokernelCofork f\ndesc : (s : CokernelCofork f) \u2192 t.pt \u27f6 s.pt\nfac : \u2200 (s : CokernelCofork f), Cofork.\u03c0 t \u226b desc s = Cofork.\u03c0 s\nuniq : \u2200 (s : CokernelCofork f) (m : t.pt \u27f6 s.pt), Cofork.\u03c0 t \u226b m = Cofork.\u03c0 s \u2192 m = desc s\ns : Cocone (parallelPair f 0)\n\u22a2 NatTrans.app t.\u03b9 zero \u226b desc s = NatTrans.app s.\u03b9 zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase one\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nt : CokernelCofork f\ndesc : (s : CokernelCofork f) \u2192 t.pt \u27f6 s.pt\nfac : \u2200 (s : CokernelCofork f), Cofork.\u03c0 t \u226b desc s = Cofork.\u03c0 s\nuniq : \u2200 (s : CokernelCofork f) (m : t.pt \u27f6 s.pt), Cofork.\u03c0 t \u226b m = Cofork.\u03c0 s \u2192 m = desc s\ns : Cocone (parallelPair f 0)\n\u22a2 NatTrans.app t.\u03b9 one \u226b desc s = NatTrans.app s.\u03b9 one\n[PROOFSTEP]\nexact fac s\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\ni : IsColimit c\nW : C\ng : W \u27f6 X\nhg : Epi g\nh : W \u27f6 Y\nhh : h = g \u226b f\n\u22a2 h \u226b Cofork.\u03c0 c = 0\n[PROOFSTEP]\nrw [hh]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\ni : IsColimit c\nW : C\ng : W \u27f6 X\nhg : Epi g\nh : W \u27f6 Y\nhh : h = g \u226b f\n\u22a2 (g \u226b f) \u226b Cofork.\u03c0 c = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\ni : IsColimit c\nW : C\ng : W \u27f6 X\nhg : Epi g\nh : W \u27f6 Y\nhh : h = g \u226b f\ns : Cofork h 0\n\u22a2 f \u226b Cofork.\u03c0 s = 0 \u226b Cofork.\u03c0 s\n[PROOFSTEP]\napply hg.left_cancellation\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\ni : IsColimit c\nW : C\ng : W \u27f6 X\nhg : Epi g\nh : W \u27f6 Y\nhh : h = g \u226b f\ns : Cofork h 0\n\u22a2 g \u226b f \u226b Cofork.\u03c0 s = g \u226b 0 \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nrw [\u2190 Category.assoc, \u2190 hh, s.condition]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\ni : IsColimit c\nW : C\ng : W \u27f6 X\nhg : Epi g\nh : W \u27f6 Y\nhh : h = g \u226b f\ns : Cofork h 0\n\u22a2 0 \u226b Cofork.\u03c0 s = g \u226b 0 \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\ni : IsColimit c\nW : C\ng : W \u27f6 X\nhg : Epi g\nh : W \u27f6 Y\nhh : h = g \u226b f\ns : Cofork h 0\ns' : CokernelCofork f := Cofork.of\u03c0 (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = 0 \u226b Cofork.\u03c0 s)\nl : { l // Cofork.\u03c0 c \u226b l = Cofork.\u03c0 s' } := CokernelCofork.IsColimit.desc' i (Cofork.\u03c0 s') (_ : f \u226b Cofork.\u03c0 s' = 0)\nm\u271d :\n ((Functor.const WalkingParallelPair).obj (CokernelCofork.of\u03c0 (Cofork.\u03c0 c) (_ : h \u226b Cofork.\u03c0 c = 0)).pt).obj one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj one\nhm : Cofork.\u03c0 (CokernelCofork.of\u03c0 (Cofork.\u03c0 c) (_ : h \u226b Cofork.\u03c0 c = 0)) \u226b m\u271d = Cofork.\u03c0 s\n\u22a2 m\u271d = \u2191l\n[PROOFSTEP]\napply Cofork.IsColimit.hom_ext i\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\ni : IsColimit c\nW : C\ng : W \u27f6 X\nhg : Epi g\nh : W \u27f6 Y\nhh : h = g \u226b f\ns : Cofork h 0\ns' : CokernelCofork f := Cofork.of\u03c0 (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = 0 \u226b Cofork.\u03c0 s)\nl : { l // Cofork.\u03c0 c \u226b l = Cofork.\u03c0 s' } := CokernelCofork.IsColimit.desc' i (Cofork.\u03c0 s') (_ : f \u226b Cofork.\u03c0 s' = 0)\nm\u271d :\n ((Functor.const WalkingParallelPair).obj (CokernelCofork.of\u03c0 (Cofork.\u03c0 c) (_ : h \u226b Cofork.\u03c0 c = 0)).pt).obj one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj one\nhm : Cofork.\u03c0 (CokernelCofork.of\u03c0 (Cofork.\u03c0 c) (_ : h \u226b Cofork.\u03c0 c = 0)) \u226b m\u271d = Cofork.\u03c0 s\n\u22a2 Cofork.\u03c0 c \u226b m\u271d = Cofork.\u03c0 c \u226b \u2191l\n[PROOFSTEP]\nrw [Cofork.\u03c0_of\u03c0] at hm \n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\ni : IsColimit c\nW : C\ng : W \u27f6 X\nhg : Epi g\nh : W \u27f6 Y\nhh : h = g \u226b f\ns : Cofork h 0\ns' : CokernelCofork f := Cofork.of\u03c0 (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = 0 \u226b Cofork.\u03c0 s)\nl : { l // Cofork.\u03c0 c \u226b l = Cofork.\u03c0 s' } := CokernelCofork.IsColimit.desc' i (Cofork.\u03c0 s') (_ : f \u226b Cofork.\u03c0 s' = 0)\nm\u271d :\n ((Functor.const WalkingParallelPair).obj (CokernelCofork.of\u03c0 (Cofork.\u03c0 c) (_ : h \u226b Cofork.\u03c0 c = 0)).pt).obj one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj one\nhm : Cofork.\u03c0 c \u226b m\u271d = Cofork.\u03c0 s\n\u22a2 Cofork.\u03c0 c \u226b m\u271d = Cofork.\u03c0 c \u226b \u2191l\n[PROOFSTEP]\nrw [hm]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\ni : IsColimit c\nW : C\ng : W \u27f6 X\nhg : Epi g\nh : W \u27f6 Y\nhh : h = g \u226b f\ns : Cofork h 0\ns' : CokernelCofork f := Cofork.of\u03c0 (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = 0 \u226b Cofork.\u03c0 s)\nl : { l // Cofork.\u03c0 c \u226b l = Cofork.\u03c0 s' } := CokernelCofork.IsColimit.desc' i (Cofork.\u03c0 s') (_ : f \u226b Cofork.\u03c0 s' = 0)\nm\u271d :\n ((Functor.const WalkingParallelPair).obj (CokernelCofork.of\u03c0 (Cofork.\u03c0 c) (_ : h \u226b Cofork.\u03c0 c = 0)).pt).obj one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj one\nhm : Cofork.\u03c0 c \u226b m\u271d = Cofork.\u03c0 s\n\u22a2 Cofork.\u03c0 s = Cofork.\u03c0 c \u226b \u2191l\n[PROOFSTEP]\nexact l.2.symm\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\ni : IsColimit c\nW : C\ng : W \u27f6 X\nhg : Epi g\nh : W \u27f6 Y\nhh : h = g \u226b f\ns : CokernelCofork h\n\u22a2 f \u226b Cofork.\u03c0 s = 0 \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nrw [\u2190 cancel_epi g, \u2190 Category.assoc, \u2190 hh]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\ni : IsColimit c\nW : C\ng : W \u27f6 X\nhg : Epi g\nh : W \u27f6 Y\nhh : h = g \u226b f\ns : CokernelCofork h\n\u22a2 h \u226b Cofork.\u03c0 s = g \u226b 0 \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nW : C\ng : W \u27f6 X\nh : W \u27f6 Y\nc : CokernelCofork h\ni : IsColimit c\nhf : f \u226b Cofork.\u03c0 c = 0\nhfg : g \u226b f = h\ns : Cofork f 0\n\u22a2 h \u226b Cofork.\u03c0 s = 0\n[PROOFSTEP]\nsimp [\u2190 hfg]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nW : C\ng : W \u27f6 X\nh : W \u27f6 Y\nc : CokernelCofork h\ni : IsColimit c\nhf : f \u226b Cofork.\u03c0 c = 0\nhfg : g \u226b f = h\ns : Cofork f 0\n\u22a2 Cofork.\u03c0 (CokernelCofork.of\u03c0 (Cofork.\u03c0 c) hf) \u226b\n (fun s => IsColimit.desc i (CokernelCofork.of\u03c0 (Cofork.\u03c0 s) (_ : h \u226b Cofork.\u03c0 s = 0))) s =\n Cofork.\u03c0 s\n[PROOFSTEP]\nsimp only [CokernelCofork.\u03c0_of\u03c0, Cofork.IsColimit.\u03c0_desc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nW : C\ng : W \u27f6 X\nh\u271d : W \u27f6 Y\nc : CokernelCofork h\u271d\ni : IsColimit c\nhf : f \u226b Cofork.\u03c0 c = 0\nhfg : g \u226b f = h\u271d\ns : Cofork f 0\nm : (CokernelCofork.of\u03c0 (Cofork.\u03c0 c) hf).pt \u27f6 s.pt\nh : Cofork.\u03c0 (CokernelCofork.of\u03c0 (Cofork.\u03c0 c) hf) \u226b m = Cofork.\u03c0 s\n\u22a2 m = (fun s => IsColimit.desc i (CokernelCofork.of\u03c0 (Cofork.\u03c0 s) (_ : h\u271d \u226b Cofork.\u03c0 s = 0))) s\n[PROOFSTEP]\napply Cofork.IsColimit.hom_ext i\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nW : C\ng : W \u27f6 X\nh\u271d : W \u27f6 Y\nc : CokernelCofork h\u271d\ni : IsColimit c\nhf : f \u226b Cofork.\u03c0 c = 0\nhfg : g \u226b f = h\u271d\ns : Cofork f 0\nm : (CokernelCofork.of\u03c0 (Cofork.\u03c0 c) hf).pt \u27f6 s.pt\nh : Cofork.\u03c0 (CokernelCofork.of\u03c0 (Cofork.\u03c0 c) hf) \u226b m = Cofork.\u03c0 s\n\u22a2 Cofork.\u03c0 c \u226b m =\n Cofork.\u03c0 c \u226b (fun s => IsColimit.desc i (CokernelCofork.of\u03c0 (Cofork.\u03c0 s) (_ : h\u271d \u226b Cofork.\u03c0 s = 0))) s\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nhf : f = 0\n\u22a2 f \u226b \ud835\udfd9 Y = 0\n[PROOFSTEP]\nrw [hf, zero_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nhf : f = 0\nZ'\u271d : C\nx\u271d\u00b2 : Y \u27f6 Z'\u271d\nx\u271d\u00b9 : f \u226b x\u271d\u00b2 = 0\nx\u271d : Y \u27f6 Z'\u271d\nhb : \ud835\udfd9 Y \u226b x\u271d = x\u271d\u00b2\n\u22a2 x\u271d = (fun {Z'} x x_1 => x) x\u271d\u00b2 x\u271d\u00b9\n[PROOFSTEP]\nsimp only [\u2190 hb, Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\nhf : Epi f\nh : IsZero c.pt\ns : CokernelCofork f\n\u22a2 Cofork.\u03c0 c \u226b (fun s => 0) s = Cofork.\u03c0 s\n[PROOFSTEP]\nrw [comp_zero, \u2190 cancel_epi f, comp_zero, s.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\nhc : IsColimit c\nhf : f = 0\n\u22a2 IsIso (Cofork.\u03c0 c)\n[PROOFSTEP]\nlet e : c.pt \u2245 Y := IsColimit.coconePointUniqueUpToIso hc (CokernelCofork.IsColimit.ofId (f : X \u27f6 Y) hf)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\nhc : IsColimit c\nhf : f = 0\ne : c.pt \u2245 Y := IsColimit.coconePointUniqueUpToIso hc (ofId f hf)\n\u22a2 IsIso (Cofork.\u03c0 c)\n[PROOFSTEP]\nhave eq : c.\u03c0 \u226b e.hom = \ud835\udfd9 Y := Cofork.IsColimit.\u03c0_desc hc\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\nhc : IsColimit c\nhf : f = 0\ne : c.pt \u2245 Y := IsColimit.coconePointUniqueUpToIso hc (ofId f hf)\neq : Cofork.\u03c0 c \u226b e.hom = \ud835\udfd9 Y\n\u22a2 IsIso (Cofork.\u03c0 c)\n[PROOFSTEP]\nhaveI : IsIso (c.\u03c0 \u226b e.hom) := by\n rw [eq]\n dsimp\n infer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\nhc : IsColimit c\nhf : f = 0\ne : c.pt \u2245 Y := IsColimit.coconePointUniqueUpToIso hc (ofId f hf)\neq : Cofork.\u03c0 c \u226b e.hom = \ud835\udfd9 Y\n\u22a2 IsIso (Cofork.\u03c0 c \u226b e.hom)\n[PROOFSTEP]\nrw [eq]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\nhc : IsColimit c\nhf : f = 0\ne : c.pt \u2245 Y := IsColimit.coconePointUniqueUpToIso hc (ofId f hf)\neq : Cofork.\u03c0 c \u226b e.hom = \ud835\udfd9 Y\n\u22a2 IsIso (\ud835\udfd9 Y)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\nhc : IsColimit c\nhf : f = 0\ne : c.pt \u2245 Y := IsColimit.coconePointUniqueUpToIso hc (ofId f hf)\neq : Cofork.\u03c0 c \u226b e.hom = \ud835\udfd9 Y\n\u22a2 IsIso (\ud835\udfd9 Y)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\nc : CokernelCofork f\nhc : IsColimit c\nhf : f = 0\ne : c.pt \u2245 Y := IsColimit.coconePointUniqueUpToIso hc (ofId f hf)\neq : Cofork.\u03c0 c \u226b e.hom = \ud835\udfd9 Y\nthis : IsIso (Cofork.\u03c0 c \u226b e.hom)\n\u22a2 IsIso (Cofork.\u03c0 c)\n[PROOFSTEP]\nexact IsIso.of_isIso_comp_right c.\u03c0 e.hom\n[GOAL]\nC\u271d : Type u\ninst\u271d\u2075 : Category.{v, u} C\u271d\ninst\u271d\u2074 : HasZeroMorphisms C\u271d\nX\u271d Y\u271d : C\u271d\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b3 : HasCokernel f\u271d\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nh : f \u226b g = 0\ninst\u271d : HasCokernel f\n\u22a2 colimit.\u03b9 (parallelPair f 0) zero \u226b cokernel.desc f g h = 0\n[PROOFSTEP]\nrw [(colimit.w (parallelPair f 0) WalkingParallelPairHom.left).symm]\n[GOAL]\nC\u271d : Type u\ninst\u271d\u2075 : Category.{v, u} C\u271d\ninst\u271d\u2074 : HasZeroMorphisms C\u271d\nX\u271d Y\u271d : C\u271d\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b3 : HasCokernel f\u271d\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nh : f \u226b g = 0\ninst\u271d : HasCokernel f\n\u22a2 ((parallelPair f 0).map WalkingParallelPairHom.left \u226b colimit.\u03b9 (parallelPair f 0) one) \u226b cokernel.desc f g h = 0\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasCokernel f\nW : C\nh : f \u226b 0 = 0\n\u22a2 desc f 0 h = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasCokernel f\nW : C\nh : f \u226b 0 = 0\n\u22a2 coequalizer.\u03c0 f 0 \u226b desc f 0 h = coequalizer.\u03c0 f 0 \u226b 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\nW : C\nk : Y \u27f6 W\nh : f \u226b k = 0\ninst\u271d : Epi k\nZ : C\ng g' : W \u27f6 Z\nw : desc f k h \u226b g = desc f k h \u226b g'\n\u22a2 g = g'\n[PROOFSTEP]\nreplace w := cokernel.\u03c0 f \u226b= w\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\nW : C\nk : Y \u27f6 W\nh : f \u226b k = 0\ninst\u271d : Epi k\nZ : C\ng g' : W \u27f6 Z\nw : \u03c0 f \u226b desc f k h \u226b g = \u03c0 f \u226b desc f k h \u226b g'\n\u22a2 g = g'\n[PROOFSTEP]\nsimp only [cokernel.\u03c0_desc_assoc] at w \n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\nW : C\nk : Y \u27f6 W\nh : f \u226b k = 0\ninst\u271d : Epi k\nZ : C\ng g' : W \u27f6 Z\nw : k \u226b g = k \u226b g'\n\u22a2 g = g'\n[PROOFSTEP]\nexact (cancel_epi k).1 w\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\nX' Y' : C\nf' : X' \u27f6 Y'\ninst\u271d : HasCokernel f'\np : X \u27f6 X'\nq : Y \u27f6 Y'\nw : f \u226b q = p \u226b f'\n\u22a2 f \u226b q \u226b \u03c0 f' = 0\n[PROOFSTEP]\nhave : f \u226b q \u226b \u03c0 f' = p \u226b f' \u226b \u03c0 f' := by\n simp only [\u2190 Category.assoc]\n apply congrArg (\u00b7 \u226b \u03c0 f') w\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\nX' Y' : C\nf' : X' \u27f6 Y'\ninst\u271d : HasCokernel f'\np : X \u27f6 X'\nq : Y \u27f6 Y'\nw : f \u226b q = p \u226b f'\n\u22a2 f \u226b q \u226b \u03c0 f' = p \u226b f' \u226b \u03c0 f'\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\nX' Y' : C\nf' : X' \u27f6 Y'\ninst\u271d : HasCokernel f'\np : X \u27f6 X'\nq : Y \u27f6 Y'\nw : f \u226b q = p \u226b f'\n\u22a2 (f \u226b q) \u226b \u03c0 f' = (p \u226b f') \u226b \u03c0 f'\n[PROOFSTEP]\napply congrArg (\u00b7 \u226b \u03c0 f') w\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\nX' Y' : C\nf' : X' \u27f6 Y'\ninst\u271d : HasCokernel f'\np : X \u27f6 X'\nq : Y \u27f6 Y'\nw : f \u226b q = p \u226b f'\nthis : f \u226b q \u226b \u03c0 f' = p \u226b f' \u226b \u03c0 f'\n\u22a2 f \u226b q \u226b \u03c0 f' = 0\n[PROOFSTEP]\nsimp [this]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasCokernel f\u271d\nX Y Z X' Y' Z' : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\ng : Y \u27f6 Z\nw : f \u226b g = 0\nf' : X' \u27f6 Y'\ninst\u271d : HasCokernel f'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\np : X \u27f6 X'\nq : Y \u27f6 Y'\nr : Z \u27f6 Z'\nh\u2081 : f \u226b q = p \u226b f'\nh\u2082 : g \u226b r = q \u226b g'\n\u22a2 map f f' p q h\u2081 \u226b desc f' g' w' = desc f g w \u226b r\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasCokernel f\u271d\nX Y Z X' Y' Z' : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\ng : Y \u27f6 Z\nw : f \u226b g = 0\nf' : X' \u27f6 Y'\ninst\u271d : HasCokernel f'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\np : X \u27f6 X'\nq : Y \u27f6 Y'\nr : Z \u27f6 Z'\nh\u2081 : f \u226b q = p \u226b f'\nh\u2082 : g \u226b r = q \u226b g'\n\u22a2 coequalizer.\u03c0 f 0 \u226b map f f' p q h\u2081 \u226b desc f' g' w' = coequalizer.\u03c0 f 0 \u226b desc f g w \u226b r\n[PROOFSTEP]\nsimp [h\u2082]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\nX' Y' : C\nf' : X' \u27f6 Y'\ninst\u271d : HasCokernel f'\np : X \u2245 X'\nq : Y \u2245 Y'\nw : f \u226b q.hom = p.hom \u226b f'\n\u22a2 f' \u226b q.inv = p.inv \u226b f\n[PROOFSTEP]\nrefine' (cancel_mono q.hom).1 _\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\nX' Y' : C\nf' : X' \u27f6 Y'\ninst\u271d : HasCokernel f'\np : X \u2245 X'\nq : Y \u2245 Y'\nw : f \u226b q.hom = p.hom \u226b f'\n\u22a2 (f' \u226b q.inv) \u226b q.hom = (p.inv \u226b f) \u226b q.hom\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Mono (cokernel.\u03c0 f)\n\u22a2 f \u226b cokernel.\u03c0 f = 0 \u226b cokernel.\u03c0 f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasCokernel f\n\u22a2 0 \u226b \ud835\udfd9 Y = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasCokernel f\n\u22a2 cokernelZeroIsoTarget.hom = cokernel.desc 0 (\ud835\udfd9 Y) (_ : 0 \u226b \ud835\udfd9 Y = 0)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasCokernel f\n\u22a2 coequalizer.\u03c0 0 0 \u226b cokernelZeroIsoTarget.hom = coequalizer.\u03c0 0 0 \u226b cokernel.desc 0 (\ud835\udfd9 Y) (_ : 0 \u226b \ud835\udfd9 Y = 0)\n[PROOFSTEP]\nsimp [cokernelZeroIsoTarget]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasCokernel f\u271d\nf g : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : HasCokernel g\nh : f = g\n\u22a2 parallelPair f 0 \u2245 parallelPair g 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasCokernel f\u271d\nf g : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : HasCokernel g\nh : f = g\n\u22a2 parallelPair g 0 \u2245 parallelPair g 0\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasCokernel f\nh : f = f\n\u22a2 cokernelIsoOfEq h = Iso.refl (cokernel f)\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasCokernel f\nh : f = f\n\u22a2 coequalizer.\u03c0 f 0 \u226b (cokernelIsoOfEq h).hom = coequalizer.\u03c0 f 0 \u226b (Iso.refl (cokernel f)).hom\n[PROOFSTEP]\nsimp [cokernelIsoOfEq]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasCokernel f\u271d\nf g : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : HasCokernel g\nh : f = g\n\u22a2 cokernel.\u03c0 f \u226b (cokernelIsoOfEq h).hom = cokernel.\u03c0 g\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasCokernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b9 inst\u271d : HasCokernel f\n\u22a2 cokernel.\u03c0 f \u226b (cokernelIsoOfEq (_ : f = f)).hom = cokernel.\u03c0 f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasCokernel f\u271d\nf g : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : HasCokernel g\nh : f = g\n\u22a2 cokernel.\u03c0 g \u226b (cokernelIsoOfEq h).inv = cokernel.\u03c0 f\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasCokernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b9 inst\u271d : HasCokernel f\n\u22a2 cokernel.\u03c0 f \u226b (cokernelIsoOfEq (_ : f = f)).inv = cokernel.\u03c0 f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasCokernel f\u271d\nZ : C\nf g : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : HasCokernel g\nh : f = g\ne : Y \u27f6 Z\nhe : g \u226b e = 0\n\u22a2 f \u226b e = 0\n[PROOFSTEP]\nsimp [h, he]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasCokernel f\u271d\nZ : C\nf g : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : HasCokernel g\nh : f = g\ne : Y \u27f6 Z\nhe : g \u226b e = 0\n\u22a2 (cokernelIsoOfEq h).hom \u226b cokernel.desc g e he = cokernel.desc f e (_ : f \u226b e = 0)\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasCokernel f\u271d\nZ : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\ne : Y \u27f6 Z\ninst\u271d : HasCokernel f\nhe : f \u226b e = 0\n\u22a2 (cokernelIsoOfEq (_ : f = f)).hom \u226b cokernel.desc f e he = cokernel.desc f e (_ : f \u226b e = 0)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasCokernel f\u271d\nZ : C\nf g : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : HasCokernel g\nh : f = g\ne : Y \u27f6 Z\nhe : f \u226b e = 0\n\u22a2 g \u226b e = 0\n[PROOFSTEP]\nsimp [\u2190 h, he]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasCokernel f\u271d\nZ : C\nf g : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : HasCokernel g\nh : f = g\ne : Y \u27f6 Z\nhe : f \u226b e = 0\n\u22a2 (cokernelIsoOfEq h).inv \u226b cokernel.desc f e he = cokernel.desc g e (_ : g \u226b e = 0)\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasCokernel f\u271d\nZ : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasCokernel f\ne : Y \u27f6 Z\nhe : f \u226b e = 0\ninst\u271d : HasCokernel f\n\u22a2 (cokernelIsoOfEq (_ : f = f)).inv \u226b cokernel.desc f e he = cokernel.desc f e (_ : f \u226b e = 0)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasCokernel f\u271d\nf g h : X \u27f6 Y\ninst\u271d\u00b2 : HasCokernel f\ninst\u271d\u00b9 : HasCokernel g\ninst\u271d : HasCokernel h\nw\u2081 : f = g\nw\u2082 : g = h\n\u22a2 cokernelIsoOfEq w\u2081 \u226a\u226b cokernelIsoOfEq w\u2082 = cokernelIsoOfEq (_ : f = h)\n[PROOFSTEP]\ncases w\u2081\n[GOAL]\ncase refl\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasCokernel f\u271d\nf h : X \u27f6 Y\ninst\u271d\u00b2 : HasCokernel f\ninst\u271d\u00b9 : HasCokernel h\ninst\u271d : HasCokernel f\nw\u2082 : f = h\n\u22a2 cokernelIsoOfEq (_ : f = f) \u226a\u226b cokernelIsoOfEq w\u2082 = cokernelIsoOfEq (_ : f = h)\n[PROOFSTEP]\ncases w\u2082\n[GOAL]\ncase refl.refl\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasCokernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b2 inst\u271d\u00b9 inst\u271d : HasCokernel f\n\u22a2 cokernelIsoOfEq (_ : f = f) \u226a\u226b cokernelIsoOfEq (_ : f = f) = cokernelIsoOfEq (_ : f = f)\n[PROOFSTEP]\next\n[GOAL]\ncase refl.refl.w.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasCokernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b2 inst\u271d\u00b9 inst\u271d : HasCokernel f\n\u22a2 coequalizer.\u03c0 f 0 \u226b (cokernelIsoOfEq (_ : f = f) \u226a\u226b cokernelIsoOfEq (_ : f = f)).hom =\n coequalizer.\u03c0 f 0 \u226b (cokernelIsoOfEq (_ : f = f)).hom\n[PROOFSTEP]\nsimp [cokernelIsoOfEq]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasCokernel f\nw : f \u2260 0\nI : IsIso (cokernel.\u03c0 f)\n\u22a2 Mono (cokernel.\u03c0 f)\n[PROOFSTEP]\nskip\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasCokernel f\nw : f \u2260 0\nI : IsIso (cokernel.\u03c0 f)\n\u22a2 Mono (cokernel.\u03c0 f)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasCokernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : IsIso g\n\u22a2 (f \u226b g) \u226b inv g \u226b cokernel.\u03c0 f = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasCokernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : IsIso g\ns : CokernelCofork (f \u226b g)\n\u22a2 f \u226b g \u226b Cofork.\u03c0 s = 0\n[PROOFSTEP]\nrw [\u2190 Category.assoc, CokernelCofork.condition]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasCokernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : IsIso g\n\u22a2 \u2200 (s : CokernelCofork (f \u226b g)),\n Cofork.\u03c0 (CokernelCofork.of\u03c0 (inv g \u226b cokernel.\u03c0 f) (_ : (f \u226b g) \u226b inv g \u226b cokernel.\u03c0 f = 0)) \u226b\n (fun s => cokernel.desc f (g \u226b Cofork.\u03c0 s) (_ : f \u226b g \u226b Cofork.\u03c0 s = 0)) s =\n Cofork.\u03c0 s\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasCokernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : IsIso g\ns : CokernelCofork (f \u226b g)\nm : (CokernelCofork.of\u03c0 (inv g \u226b cokernel.\u03c0 f) (_ : (f \u226b g) \u226b inv g \u226b cokernel.\u03c0 f = 0)).pt \u27f6 s.pt\nw : Cofork.\u03c0 (CokernelCofork.of\u03c0 (inv g \u226b cokernel.\u03c0 f) (_ : (f \u226b g) \u226b inv g \u226b cokernel.\u03c0 f = 0)) \u226b m = Cofork.\u03c0 s\n\u22a2 m = (fun s => cokernel.desc f (g \u226b Cofork.\u03c0 s) (_ : f \u226b g \u226b Cofork.\u03c0 s = 0)) s\n[PROOFSTEP]\nsimp_rw [\u2190 w]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasCokernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : IsIso g\ns : CokernelCofork (f \u226b g)\nm : (CokernelCofork.of\u03c0 (inv g \u226b cokernel.\u03c0 f) (_ : (f \u226b g) \u226b inv g \u226b cokernel.\u03c0 f = 0)).pt \u27f6 s.pt\nw : Cofork.\u03c0 (CokernelCofork.of\u03c0 (inv g \u226b cokernel.\u03c0 f) (_ : (f \u226b g) \u226b inv g \u226b cokernel.\u03c0 f = 0)) \u226b m = Cofork.\u03c0 s\n\u22a2 m =\n cokernel.desc f\n (g \u226b Cofork.\u03c0 (CokernelCofork.of\u03c0 (inv g \u226b cokernel.\u03c0 f) (_ : (f \u226b g) \u226b inv g \u226b cokernel.\u03c0 f = 0)) \u226b m)\n (_ :\n f \u226b g \u226b Cofork.\u03c0 (CokernelCofork.of\u03c0 (inv g \u226b cokernel.\u03c0 f) (_ : (f \u226b g) \u226b inv g \u226b cokernel.\u03c0 f = 0)) \u226b m = 0)\n[PROOFSTEP]\napply coequalizer.hom_ext\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasCokernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : IsIso g\ns : CokernelCofork (f \u226b g)\nm : (CokernelCofork.of\u03c0 (inv g \u226b cokernel.\u03c0 f) (_ : (f \u226b g) \u226b inv g \u226b cokernel.\u03c0 f = 0)).pt \u27f6 s.pt\nw : Cofork.\u03c0 (CokernelCofork.of\u03c0 (inv g \u226b cokernel.\u03c0 f) (_ : (f \u226b g) \u226b inv g \u226b cokernel.\u03c0 f = 0)) \u226b m = Cofork.\u03c0 s\n\u22a2 coequalizer.\u03c0 f 0 \u226b m =\n coequalizer.\u03c0 f 0 \u226b\n cokernel.desc f\n (g \u226b Cofork.\u03c0 (CokernelCofork.of\u03c0 (inv g \u226b cokernel.\u03c0 f) (_ : (f \u226b g) \u226b inv g \u226b cokernel.\u03c0 f = 0)) \u226b m)\n (_ :\n f \u226b g \u226b Cofork.\u03c0 (CokernelCofork.of\u03c0 (inv g \u226b cokernel.\u03c0 f) (_ : (f \u226b g) \u226b inv g \u226b cokernel.\u03c0 f = 0)) \u226b m = 0)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasCokernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : IsIso g\n\u22a2 (f \u226b g) \u226b inv g \u226b cokernel.\u03c0 f = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasCokernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : IsIso g\n\u22a2 f \u226b g \u226b cokernel.\u03c0 (f \u226b g) = 0\n[PROOFSTEP]\nrw [\u2190 Category.assoc, cokernel.condition]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasCokernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : Epi f\ninst\u271d : HasCokernel g\n\u22a2 (f \u226b g) \u226b cokernel.\u03c0 g = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasCokernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : Epi f\ninst\u271d : HasCokernel g\n\u22a2 g \u226b cokernel.\u03c0 (f \u226b g) = 0\n[PROOFSTEP]\nrw [\u2190 cancel_epi f, \u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b2 : HasCokernel f\u271d\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : Epi f\ninst\u271d : HasCokernel g\n\u22a2 (f \u226b g) \u226b cokernel.\u03c0 (f \u226b g) = f \u226b 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : Epi f\ns : Cofork f 0\n\u22a2 Cofork.\u03c0 (zeroCokernelCofork f) \u226b (fun s => 0) s = Cofork.\u03c0 s\n[PROOFSTEP]\nerw [zero_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : Epi f\ns : Cofork f 0\n\u22a2 0 = Cofork.\u03c0 s\n[PROOFSTEP]\nrefine (zero_of_epi_comp f ?_).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : Epi f\ns : Cofork f 0\n\u22a2 f \u226b Cofork.\u03c0 s = 0\n[PROOFSTEP]\nexact CokernelCofork.condition _\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nF : MonoFactorisation f\n\u22a2 kernel.\u03b9 f \u226b F.e = 0\n[PROOFSTEP]\nrw [\u2190 cancel_mono F.m, zero_comp, Category.assoc, F.fac, kernel.condition]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\n\u22a2 image.\u03b9 f \u226b cokernel.\u03c0 f = 0\n[PROOFSTEP]\nhave w := cokernel.condition f\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\nw : f \u226b cokernel.\u03c0 f = 0\n\u22a2 image.\u03b9 f \u226b cokernel.\u03c0 f = 0\n[PROOFSTEP]\nconv at w =>\n lhs\n congr\n rw [\u2190 image.fac f]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\nw : f \u226b cokernel.\u03c0 f = 0\n| f \u226b cokernel.\u03c0 f = 0\n[PROOFSTEP]\n lhs\n congr\n rw [\u2190 image.fac f]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\nw : f \u226b cokernel.\u03c0 f = 0\n| f \u226b cokernel.\u03c0 f = 0\n[PROOFSTEP]\n lhs\n congr\n rw [\u2190 image.fac f]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\nw : f \u226b cokernel.\u03c0 f = 0\n| f \u226b cokernel.\u03c0 f = 0\n[PROOFSTEP]\nlhs\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\nw : f \u226b cokernel.\u03c0 f = 0\n| f \u226b cokernel.\u03c0 f\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\nw : f \u226b cokernel.\u03c0 f = 0\n| f\ncase a\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\nw : f \u226b cokernel.\u03c0 f = 0\n| cokernel.\u03c0 f\n[PROOFSTEP]\nrw [\u2190 image.fac f]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\nw : (factorThruImage f \u226b image.\u03b9 f) \u226b cokernel.\u03c0 f = 0\n\u22a2 image.\u03b9 f \u226b cokernel.\u03c0 f = 0\n[PROOFSTEP]\nrw [\u2190 HasZeroMorphisms.comp_zero (Limits.factorThruImage f), Category.assoc, cancel_epi] at w \n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\nw : image.\u03b9 f \u226b cokernel.\u03c0 f = 0\n\u22a2 image.\u03b9 f \u226b cokernel.\u03c0 f = 0\n[PROOFSTEP]\nexact w\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\n\u22a2 f \u226b cokernel.\u03c0 (image.\u03b9 f) = 0\n[PROOFSTEP]\nconv =>\n lhs\n congr\n rw [\u2190 image.fac f]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\n| f \u226b cokernel.\u03c0 (image.\u03b9 f) = 0\n[PROOFSTEP]\n lhs\n congr\n rw [\u2190 image.fac f]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\n| f \u226b cokernel.\u03c0 (image.\u03b9 f) = 0\n[PROOFSTEP]\n lhs\n congr\n rw [\u2190 image.fac f]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\n| f \u226b cokernel.\u03c0 (image.\u03b9 f) = 0\n[PROOFSTEP]\nlhs\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\n| f \u226b cokernel.\u03c0 (image.\u03b9 f)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\n| f\ncase a\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\n| cokernel.\u03c0 (image.\u03b9 f)\n[PROOFSTEP]\nrw [\u2190 image.fac f]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\ninst\u271d\u00b2 : HasCokernel (image.\u03b9 f)\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : Epi (factorThruImage f)\n\u22a2 (factorThruImage f \u226b image.\u03b9 f) \u226b cokernel.\u03c0 (image.\u03b9 f) = 0\n[PROOFSTEP]\nrw [Category.assoc, cokernel.condition, HasZeroMorphisms.comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d : HasZeroObject C\nX Y : C\nf : X \u27f6 Y\nhf : \u2200 (Z : C) (g : Y \u27f6 Z), f \u226b g = 0 \u2192 g = 0\n\u22a2 f \u226b 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d : HasZeroObject C\nX Y : C\nf : X \u27f6 Y\nhf : \u2200 (Z : C) (g : Y \u27f6 Z), f \u226b g = 0 \u2192 g = 0\ns : Cofork f 0\n\u22a2 Cofork.\u03c0 (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0)) \u226b (fun s => 0) s = Cofork.\u03c0 s\n[PROOFSTEP]\nrw [hf _ _ (CokernelCofork.condition s), comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d : HasZeroObject C\nX Y : C\nf : X \u27f6 Y\nhf : \u2200 (Z : C) (g : Y \u27f6 Z), f \u226b g = 0 \u2192 g = 0\ns : Cofork f 0\nm : (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0)).pt \u27f6 s.pt\nx\u271d : Cofork.\u03c0 (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0)) \u226b m = Cofork.\u03c0 s\n\u22a2 m = (fun s => 0) s\n[PROOFSTEP]\napply HasZeroObject.from_zero_ext\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : Z \u27f6 Y\ni : X \u2245 Z\nh : i.hom \u226b l = f\ns : CokernelCofork f\nhs : IsColimit s\n\u22a2 l \u226b Cofork.\u03c0 s = 0\n[PROOFSTEP]\nsimp [i.eq_inv_comp.2 h]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : Z \u27f6 Y\ni : X \u2245 Z\nh : i.hom \u226b l = f\ns\u271d : CokernelCofork f\nhs : IsColimit s\u271d\ns : Cofork l 0\n\u22a2 f \u226b Cofork.\u03c0 s = 0\n[PROOFSTEP]\nsimp [\u2190 h]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : Z \u27f6 Y\ni : X \u2245 Z\nh : i.hom \u226b l = f\ns\u271d : CokernelCofork f\nhs : IsColimit s\u271d\ns : Cofork l 0\n\u22a2 Cofork.\u03c0 (CokernelCofork.of\u03c0 (Cofork.\u03c0 s\u271d) (_ : l \u226b Cofork.\u03c0 s\u271d = 0)) \u226b\n (fun s => IsColimit.desc hs (CokernelCofork.of\u03c0 (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = 0))) s =\n Cofork.\u03c0 s\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : Z \u27f6 Y\ni : X \u2245 Z\nh\u271d : i.hom \u226b l = f\ns\u271d : CokernelCofork f\nhs : IsColimit s\u271d\ns : Cofork l 0\nm : (CokernelCofork.of\u03c0 (Cofork.\u03c0 s\u271d) (_ : l \u226b Cofork.\u03c0 s\u271d = 0)).pt \u27f6 s.pt\nh : Cofork.\u03c0 (CokernelCofork.of\u03c0 (Cofork.\u03c0 s\u271d) (_ : l \u226b Cofork.\u03c0 s\u271d = 0)) \u226b m = Cofork.\u03c0 s\n\u22a2 m = (fun s => IsColimit.desc hs (CokernelCofork.of\u03c0 (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = 0))) s\n[PROOFSTEP]\napply Cofork.IsColimit.hom_ext hs\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : Z \u27f6 Y\ni : X \u2245 Z\nh\u271d : i.hom \u226b l = f\ns\u271d : CokernelCofork f\nhs : IsColimit s\u271d\ns : Cofork l 0\nm : (CokernelCofork.of\u03c0 (Cofork.\u03c0 s\u271d) (_ : l \u226b Cofork.\u03c0 s\u271d = 0)).pt \u27f6 s.pt\nh : Cofork.\u03c0 (CokernelCofork.of\u03c0 (Cofork.\u03c0 s\u271d) (_ : l \u226b Cofork.\u03c0 s\u271d = 0)) \u226b m = Cofork.\u03c0 s\n\u22a2 Cofork.\u03c0 s\u271d \u226b m =\n Cofork.\u03c0 s\u271d \u226b (fun s => IsColimit.desc hs (CokernelCofork.of\u03c0 (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = 0))) s\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasCokernel f\nZ : C\nl : Z \u27f6 Y\ni : X \u2245 Z\nh : i.hom \u226b l = f\n\u22a2 l \u226b \u03c0 f = 0\n[PROOFSTEP]\nsimp [i.eq_inv_comp.2 h]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : Y \u27f6 Z\ns : CokernelCofork f\nhs : IsColimit s\ni : s.pt \u2245 Z\nh : Cofork.\u03c0 s \u226b i.hom = l\n\u22a2 f \u226b l = 0\n[PROOFSTEP]\nsimp [\u2190 h]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : Y \u27f6 Z\ns : CokernelCofork f\nhs : IsColimit s\ni : s.pt \u2245 Z\nh : Cofork.\u03c0 s \u226b i.hom = l\nj : WalkingParallelPair\n\u22a2 NatTrans.app s.\u03b9 j \u226b i.hom = NatTrans.app (CokernelCofork.of\u03c0 l (_ : f \u226b l = 0)).\u03b9 j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase zero\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : Y \u27f6 Z\ns : CokernelCofork f\nhs : IsColimit s\ni : s.pt \u2245 Z\nh : Cofork.\u03c0 s \u226b i.hom = l\n\u22a2 NatTrans.app s.\u03b9 zero \u226b i.hom = NatTrans.app (CokernelCofork.of\u03c0 l (_ : f \u226b l = 0)).\u03b9 zero\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase zero\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : Y \u27f6 Z\ns : CokernelCofork f\nhs : IsColimit s\ni : s.pt \u2245 Z\nh : Cofork.\u03c0 s \u226b i.hom = l\n\u22a2 NatTrans.app s.\u03b9 zero \u226b i.hom = f \u226b l\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase zero\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : Y \u27f6 Z\ns : CokernelCofork f\nhs : IsColimit s\ni : s.pt \u2245 Z\nh : Cofork.\u03c0 s \u226b i.hom = l\n\u22a2 NatTrans.app s.\u03b9 zero \u226b i.hom = f \u226b Cofork.\u03c0 s \u226b i.hom\n[PROOFSTEP]\nsimp\n[GOAL]\ncase one\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : Y \u27f6 Z\ns : CokernelCofork f\nhs : IsColimit s\ni : s.pt \u2245 Z\nh : Cofork.\u03c0 s \u226b i.hom = l\n\u22a2 NatTrans.app s.\u03b9 one \u226b i.hom = NatTrans.app (CokernelCofork.of\u03c0 l (_ : f \u226b l = 0)).\u03b9 one\n[PROOFSTEP]\nexact h\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasCokernel f\nZ : C\nl : Y \u27f6 Z\ni : cokernel f \u2245 Z\nh : \u03c0 f \u226b i.hom = l\n\u22a2 f \u226b l = 0\n[PROOFSTEP]\nsimp [\u2190 h]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b3 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u00b2 : Functor.PreservesZeroMorphisms G\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasKernel (G.map f)\n\u22a2 G.map (kernel.\u03b9 f) \u226b G.map f = 0\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp, kernel.condition, Functor.map_zero]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b3 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u00b2 : Functor.PreservesZeroMorphisms G\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasKernel (G.map f)\nZ : C\nh : Z \u27f6 X\nw : h \u226b f = 0\n\u22a2 G.map h \u226b G.map f = 0\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp, w, Functor.map_zero]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b3 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u00b2 : Functor.PreservesZeroMorphisms G\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasKernel (G.map f)\nZ : C\nh : Z \u27f6 X\nw : h \u226b f = 0\n\u22a2 G.map (kernel.lift f h w) \u226b kernelComparison f G = kernel.lift (G.map f) (G.map h) (_ : G.map h \u226b G.map f = 0)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b3 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u00b2 : Functor.PreservesZeroMorphisms G\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasKernel (G.map f)\nZ : C\nh : Z \u27f6 X\nw : h \u226b f = 0\n\u22a2 (G.map (kernel.lift f h w) \u226b kernelComparison f G) \u226b equalizer.\u03b9 (G.map f) 0 =\n kernel.lift (G.map f) (G.map h) (_ : G.map h \u226b G.map f = 0) \u226b equalizer.\u03b9 (G.map f) 0\n[PROOFSTEP]\nsimp [\u2190 G.map_comp]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\ninst\u271d\u2077 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2075 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\nX' Y' : C\ninst\u271d\u00b3 : HasKernel f\ninst\u271d\u00b2 : HasKernel (G.map f)\ng : X' \u27f6 Y'\ninst\u271d\u00b9 : HasKernel g\ninst\u271d : HasKernel (G.map g)\np : X \u27f6 X'\nq : Y \u27f6 Y'\nhpq : f \u226b q = p \u226b g\n\u22a2 G.map f \u226b G.map q = G.map p \u226b G.map g\n[PROOFSTEP]\nrw [\u2190 G.map_comp, hpq, G.map_comp]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\ninst\u271d\u2077 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2075 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\nX' Y' : C\ninst\u271d\u00b3 : HasKernel f\ninst\u271d\u00b2 : HasKernel (G.map f)\ng : X' \u27f6 Y'\ninst\u271d\u00b9 : HasKernel g\ninst\u271d : HasKernel (G.map g)\np : X \u27f6 X'\nq : Y \u27f6 Y'\nhpq : f \u226b q = p \u226b g\n\u22a2 G.map (kernel.\u03b9 f) \u226b G.map f = 0\n[PROOFSTEP]\nrw [\u2190 G.map_comp, kernel.condition, G.map_zero]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\ninst\u271d\u2077 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2075 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\nX' Y' : C\ninst\u271d\u00b3 : HasKernel f\ninst\u271d\u00b2 : HasKernel (G.map f)\ng : X' \u27f6 Y'\ninst\u271d\u00b9 : HasKernel g\ninst\u271d : HasKernel (G.map g)\np : X \u27f6 X'\nq : Y \u27f6 Y'\nhpq : f \u226b q = p \u226b g\n\u22a2 G.map (kernel.\u03b9 g) \u226b G.map g = 0\n[PROOFSTEP]\nrw [\u2190 G.map_comp, kernel.condition, G.map_zero]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\ninst\u271d\u2077 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2075 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\nX' Y' : C\ninst\u271d\u00b3 : HasKernel f\ninst\u271d\u00b2 : HasKernel (G.map f)\ng : X' \u27f6 Y'\ninst\u271d\u00b9 : HasKernel g\ninst\u271d : HasKernel (G.map g)\np : X \u27f6 X'\nq : Y \u27f6 Y'\nhpq : f \u226b q = p \u226b g\n\u22a2 G.map (kernel.\u03b9 f) \u226b G.map p = G.map (kernel.map f g p q hpq) \u226b G.map (kernel.\u03b9 g)\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\ninst\u271d\u2077 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2075 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\nX' Y' : C\ninst\u271d\u00b3 : HasKernel f\ninst\u271d\u00b2 : HasKernel (G.map f)\ng : X' \u27f6 Y'\ninst\u271d\u00b9 : HasKernel g\ninst\u271d : HasKernel (G.map g)\np : X \u27f6 X'\nq : Y \u27f6 Y'\nhpq : f \u226b q = p \u226b g\n\u22a2 G.map (kernel.\u03b9 f \u226b p) = G.map (kernel.map f g p q hpq \u226b kernel.\u03b9 g)\n[PROOFSTEP]\nexact G.congr_map (kernel.lift_\u03b9 _ _ _).symm\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b3 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u00b2 : Functor.PreservesZeroMorphisms G\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : HasCokernel (G.map f)\n\u22a2 G.map f \u226b G.map (coequalizer.\u03c0 f 0) = 0\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp, cokernel.condition, Functor.map_zero]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b3 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u00b2 : Functor.PreservesZeroMorphisms G\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : HasCokernel (G.map f)\nZ : C\nh : Y \u27f6 Z\nw : f \u226b h = 0\n\u22a2 G.map f \u226b G.map h = 0\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp, w, Functor.map_zero]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b3 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u00b2 : Functor.PreservesZeroMorphisms G\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : HasCokernel (G.map f)\nZ : C\nh : Y \u27f6 Z\nw : f \u226b h = 0\n\u22a2 cokernelComparison f G \u226b G.map (cokernel.desc f h w) = cokernel.desc (G.map f) (G.map h) (_ : G.map f \u226b G.map h = 0)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b3 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u00b2 : Functor.PreservesZeroMorphisms G\ninst\u271d\u00b9 : HasCokernel f\ninst\u271d : HasCokernel (G.map f)\nZ : C\nh : Y \u27f6 Z\nw : f \u226b h = 0\n\u22a2 coequalizer.\u03c0 (G.map f) 0 \u226b cokernelComparison f G \u226b G.map (cokernel.desc f h w) =\n coequalizer.\u03c0 (G.map f) 0 \u226b cokernel.desc (G.map f) (G.map h) (_ : G.map f \u226b G.map h = 0)\n[PROOFSTEP]\nsimp [\u2190 G.map_comp]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\ninst\u271d\u2077 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2075 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\nX' Y' : C\ninst\u271d\u00b3 : HasCokernel f\ninst\u271d\u00b2 : HasCokernel (G.map f)\ng : X' \u27f6 Y'\ninst\u271d\u00b9 : HasCokernel g\ninst\u271d : HasCokernel (G.map g)\np : X \u27f6 X'\nq : Y \u27f6 Y'\nhpq : f \u226b q = p \u226b g\n\u22a2 G.map f \u226b G.map q = G.map p \u226b G.map g\n[PROOFSTEP]\nrw [\u2190 G.map_comp, hpq, G.map_comp]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\ninst\u271d\u2077 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2075 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\nX' Y' : C\ninst\u271d\u00b3 : HasCokernel f\ninst\u271d\u00b2 : HasCokernel (G.map f)\ng : X' \u27f6 Y'\ninst\u271d\u00b9 : HasCokernel g\ninst\u271d : HasCokernel (G.map g)\np : X \u27f6 X'\nq : Y \u27f6 Y'\nhpq : f \u226b q = p \u226b g\n\u22a2 G.map f \u226b G.map (coequalizer.\u03c0 f 0) = 0\n[PROOFSTEP]\nrw [\u2190 G.map_comp, cokernel.condition, G.map_zero]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\ninst\u271d\u2077 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2075 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\nX' Y' : C\ninst\u271d\u00b3 : HasCokernel f\ninst\u271d\u00b2 : HasCokernel (G.map f)\ng : X' \u27f6 Y'\ninst\u271d\u00b9 : HasCokernel g\ninst\u271d : HasCokernel (G.map g)\np : X \u27f6 X'\nq : Y \u27f6 Y'\nhpq : f \u226b q = p \u226b g\n\u22a2 G.map g \u226b G.map (coequalizer.\u03c0 g 0) = 0\n[PROOFSTEP]\nrw [\u2190 G.map_comp, cokernel.condition, G.map_zero]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\ninst\u271d\u2077 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2075 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\nX' Y' : C\ninst\u271d\u00b3 : HasCokernel f\ninst\u271d\u00b2 : HasCokernel (G.map f)\ng : X' \u27f6 Y'\ninst\u271d\u00b9 : HasCokernel g\ninst\u271d : HasCokernel (G.map g)\np : X \u27f6 X'\nq : Y \u27f6 Y'\nhpq : f \u226b q = p \u226b g\n\u22a2 G.map (coequalizer.\u03c0 f 0) \u226b G.map (cokernel.map f g p q hpq) = G.map q \u226b G.map (coequalizer.\u03c0 g 0)\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\ninst\u271d\u2077 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2075 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\nX' Y' : C\ninst\u271d\u00b3 : HasCokernel f\ninst\u271d\u00b2 : HasCokernel (G.map f)\ng : X' \u27f6 Y'\ninst\u271d\u00b9 : HasCokernel g\ninst\u271d : HasCokernel (G.map g)\np : X \u27f6 X'\nq : Y \u27f6 Y'\nhpq : f \u226b q = p \u226b g\n\u22a2 G.map (coequalizer.\u03c0 f 0 \u226b cokernel.map f g p q hpq) = G.map (q \u226b coequalizer.\u03c0 g 0)\n[PROOFSTEP]\nexact G.congr_map (cokernel.\u03c0_desc _ _ _)\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Shapes.Kernels", "llama_tokens": 47217, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6150878696277513, "lm_q2_score": 0.38121956625614994, "lm_q1q2_score": 0.23448353086891066}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\n\u22a2 GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S) \u2194 \u2203 T, Sieve.functorPullback G \u2191T = S\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d\u2074 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\n\u22a2 GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S) \u2192 \u2203 T, Sieve.functorPullback G \u2191T = S\n[PROOFSTEP]\nintro hS\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d\u2074 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\nhS : GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)\n\u22a2 \u2203 T, Sieve.functorPullback G \u2191T = S\n[PROOFSTEP]\nexact \u27e8\u27e8_, hS\u27e9, (Sieve.fullyFaithfulFunctorGaloisCoinsertion G X).u_l_eq S\u27e9\n[GOAL]\ncase mpr\nC : Type u_1\ninst\u271d\u2074 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\n\u22a2 (\u2203 T, Sieve.functorPullback G \u2191T = S) \u2192 GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)\n[PROOFSTEP]\nrintro \u27e8T, rfl\u27e9\n[GOAL]\ncase mpr.intro\nC : Type u_1\ninst\u271d\u2074 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\n\u22a2 GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G (Sieve.functorPullback G \u2191T))\n[PROOFSTEP]\nexact Hld T\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\n\u22a2 \u22a4 \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\n[PROOFSTEP]\nchange K _ _\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\n\u22a2 GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G \u22a4)\n[PROOFSTEP]\nrw [Sieve.functorPushforward_top]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\n\u22a2 GrothendieckTopology.sieves K (G.obj X) \u22a4\n[PROOFSTEP]\nexact K.top_mem _\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX Y : C\nS : Sieve X\nf : Y \u27f6 X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\n\u22a2 Sieve.pullback f S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) Y\n[PROOFSTEP]\nhave : S.pullback f = ((S.functorPushforward G).pullback (G.map f)).functorPullback G :=\n by\n conv_lhs => rw [\u2190 (Sieve.fullyFaithfulFunctorGaloisCoinsertion G X).u_l_eq S]\n ext\n change (S.functorPushforward G) _ \u2194 (S.functorPushforward G) _\n rw [G.map_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX Y : C\nS : Sieve X\nf : Y \u27f6 X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\n\u22a2 Sieve.pullback f S = Sieve.functorPullback G (Sieve.pullback (G.map f) (Sieve.functorPushforward G S))\n[PROOFSTEP]\nconv_lhs => rw [\u2190 (Sieve.fullyFaithfulFunctorGaloisCoinsertion G X).u_l_eq S]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX Y : C\nS : Sieve X\nf : Y \u27f6 X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\n| Sieve.pullback f S\n[PROOFSTEP]\nrw [\u2190 (Sieve.fullyFaithfulFunctorGaloisCoinsertion G X).u_l_eq S]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX Y : C\nS : Sieve X\nf : Y \u27f6 X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\n| Sieve.pullback f S\n[PROOFSTEP]\nrw [\u2190 (Sieve.fullyFaithfulFunctorGaloisCoinsertion G X).u_l_eq S]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX Y : C\nS : Sieve X\nf : Y \u27f6 X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\n| Sieve.pullback f S\n[PROOFSTEP]\nrw [\u2190 (Sieve.fullyFaithfulFunctorGaloisCoinsertion G X).u_l_eq S]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX Y : C\nS : Sieve X\nf : Y \u27f6 X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\n\u22a2 Sieve.pullback f (Sieve.functorPullback G (Sieve.functorPushforward G S)) =\n Sieve.functorPullback G (Sieve.pullback (G.map f) (Sieve.functorPushforward G S))\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX Y : C\nS : Sieve X\nf : Y \u27f6 X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 Y\n\u22a2 (Sieve.pullback f (Sieve.functorPullback G (Sieve.functorPushforward G S))).arrows f\u271d \u2194\n (Sieve.functorPullback G (Sieve.pullback (G.map f) (Sieve.functorPushforward G S))).arrows f\u271d\n[PROOFSTEP]\nchange (S.functorPushforward G) _ \u2194 (S.functorPushforward G) _\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX Y : C\nS : Sieve X\nf : Y \u27f6 X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 Y\n\u22a2 (Sieve.functorPushforward G S).arrows (G.map (f\u271d \u226b f)) \u2194 (Sieve.functorPushforward G S).arrows (G.map f\u271d \u226b G.map f)\n[PROOFSTEP]\nrw [G.map_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX Y : C\nS : Sieve X\nf : Y \u27f6 X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\nthis : Sieve.pullback f S = Sieve.functorPullback G (Sieve.pullback (G.map f) (Sieve.functorPushforward G S))\n\u22a2 Sieve.pullback f S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) Y\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX Y : C\nS : Sieve X\nf : Y \u27f6 X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\nthis : Sieve.pullback f S = Sieve.functorPullback G (Sieve.pullback (G.map f) (Sieve.functorPushforward G S))\n\u22a2 Sieve.functorPullback G (Sieve.pullback (G.map f) (Sieve.functorPushforward G S)) \u2208\n (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) Y\n[PROOFSTEP]\nchange K _ _\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX Y : C\nS : Sieve X\nf : Y \u27f6 X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\nthis : Sieve.pullback f S = Sieve.functorPullback G (Sieve.pullback (G.map f) (Sieve.functorPushforward G S))\n\u22a2 GrothendieckTopology.sieves K (G.obj Y)\n (Sieve.functorPushforward G (Sieve.functorPullback G (Sieve.pullback (G.map f) (Sieve.functorPushforward G S))))\n[PROOFSTEP]\napply Hld \u27e8_, K.pullback_stable (G.map f) hS\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\nS' : Sieve X\nH' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Sieve.pullback f S' \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) Y\n\u22a2 S' \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\n[PROOFSTEP]\napply K.transitive hS\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\nS' : Sieve X\nH' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Sieve.pullback f S' \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) Y\n\u22a2 \u2200 \u2983Y : D\u2984 \u2983f : Y \u27f6 G.obj X\u2984,\n (Sieve.functorPushforward G S).arrows f \u2192\n Sieve.pullback f (Sieve.functorPushforward G S') \u2208 GrothendieckTopology.sieves K Y\n[PROOFSTEP]\nrintro Y _ \u27e8Z, g, i, hg, rfl\u27e9\n[GOAL]\ncase h.intro.intro.intro.intro\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\nS' : Sieve X\nH' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Sieve.pullback f S' \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) Y\nY : D\nZ : C\ng : Z \u27f6 X\ni : Y \u27f6 G.obj Z\nhg : S.arrows g\n\u22a2 Sieve.pullback (i \u226b G.map g) (Sieve.functorPushforward G S') \u2208 GrothendieckTopology.sieves K Y\n[PROOFSTEP]\nrw [Sieve.pullback_comp]\n[GOAL]\ncase h.intro.intro.intro.intro\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\nS' : Sieve X\nH' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Sieve.pullback f S' \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) Y\nY : D\nZ : C\ng : Z \u27f6 X\ni : Y \u27f6 G.obj Z\nhg : S.arrows g\n\u22a2 Sieve.pullback i (Sieve.pullback (G.map g) (Sieve.functorPushforward G S')) \u2208 GrothendieckTopology.sieves K Y\n[PROOFSTEP]\napply K.pullback_stable i\n[GOAL]\ncase h.intro.intro.intro.intro\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\nS' : Sieve X\nH' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Sieve.pullback f S' \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) Y\nY : D\nZ : C\ng : Z \u27f6 X\ni : Y \u27f6 G.obj Z\nhg : S.arrows g\n\u22a2 Sieve.pullback (G.map g) (Sieve.functorPushforward G S') \u2208 GrothendieckTopology.sieves K (G.obj Z)\n[PROOFSTEP]\nrefine' K.superset_covering _ (H' hg)\n[GOAL]\ncase h.intro.intro.intro.intro\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\nS' : Sieve X\nH' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Sieve.pullback f S' \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) Y\nY : D\nZ : C\ng : Z \u27f6 X\ni : Y \u27f6 G.obj Z\nhg : S.arrows g\n\u22a2 Sieve.functorPushforward G (Sieve.pullback g S') \u2264 Sieve.pullback (G.map g) (Sieve.functorPushforward G S')\n[PROOFSTEP]\nrintro W _ \u27e8Z', g', i', hg, rfl\u27e9\n[GOAL]\ncase h.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\nS' : Sieve X\nH' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Sieve.pullback f S' \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) Y\nY : D\nZ : C\ng : Z \u27f6 X\ni : Y \u27f6 G.obj Z\nhg\u271d : S.arrows g\nW : D\nZ' : C\ng' : Z' \u27f6 Z\ni' : W \u27f6 G.obj Z'\nhg : (Sieve.pullback g S').arrows g'\n\u22a2 (Sieve.pullback (G.map g) (Sieve.functorPushforward G S')).arrows (i' \u226b G.map g')\n[PROOFSTEP]\nrefine' \u27e8Z', g' \u226b g, i', hg, _\u27e9\n[GOAL]\ncase h.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.3052, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{?u.3059, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\nhS : S \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) X\nS' : Sieve X\nH' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Sieve.pullback f S' \u2208 (fun X S => GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)) Y\nY : D\nZ : C\ng : Z \u27f6 X\ni : Y \u27f6 G.obj Z\nhg\u271d : S.arrows g\nW : D\nZ' : C\ng' : Z' \u27f6 Z\ni' : W \u27f6 G.obj Z'\nhg : (Sieve.pullback g S').arrows g'\n\u22a2 (i' \u226b G.map g') \u226b G.map g = i' \u226b G.map (g' \u226b g)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b2 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category.{u, v} A\ninst\u271d : Full G\nH : CoverDense K G\n\u22a2 LocallyCoverDense K G\n[PROOFSTEP]\nintro X T\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b2 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category.{u, v} A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\n\u22a2 Sieve.functorPushforward G (Sieve.functorPullback G \u2191T) \u2208 GrothendieckTopology.sieves K (G.obj X)\n[PROOFSTEP]\nrefine' K.superset_covering _ (K.bind_covering T.property fun Y f _ => H.is_cover Y)\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b2 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category.{u, v} A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\n\u22a2 (Sieve.bind (\u2191T).arrows fun Y f x => Sieve.coverByImage G Y) \u2264 Sieve.functorPushforward G (Sieve.functorPullback G \u2191T)\n[PROOFSTEP]\nrintro Y _ \u27e8Z, _, f, hf, \u27e8W, g, f', rfl : _ = _\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b2 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category.{u, v} A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\nY Z : D\nf : Z \u27f6 G.obj X\nhf : (\u2191T).arrows f\nW : C\ng : Y \u27f6 G.obj W\nf' : G.obj W \u27f6 Z\n\u22a2 (Sieve.functorPushforward G (Sieve.functorPullback G \u2191T)).arrows ((g \u226b f') \u226b f)\n[PROOFSTEP]\nuse W\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b2 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category.{u, v} A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\nY Z : D\nf : Z \u27f6 G.obj X\nhf : (\u2191T).arrows f\nW : C\ng : Y \u27f6 G.obj W\nf' : G.obj W \u27f6 Z\n\u22a2 \u2203 g_1 h, (Sieve.functorPullback G \u2191T).arrows g_1 \u2227 (g \u226b f') \u226b f = h \u226b G.map g_1\n[PROOFSTEP]\nuse G.preimage (f' \u226b f)\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b2 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category.{u, v} A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\nY Z : D\nf : Z \u27f6 G.obj X\nhf : (\u2191T).arrows f\nW : C\ng : Y \u27f6 G.obj W\nf' : G.obj W \u27f6 Z\n\u22a2 \u2203 h, (Sieve.functorPullback G \u2191T).arrows (G.preimage (f' \u226b f)) \u2227 (g \u226b f') \u226b f = h \u226b G.map (G.preimage (f' \u226b f))\n[PROOFSTEP]\nuse g\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b2 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category.{u, v} A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\nY Z : D\nf : Z \u27f6 G.obj X\nhf : (\u2191T).arrows f\nW : C\ng : Y \u27f6 G.obj W\nf' : G.obj W \u27f6 Z\n\u22a2 (Sieve.functorPullback G \u2191T).arrows (G.preimage (f' \u226b f)) \u2227 (g \u226b f') \u226b f = g \u226b G.map (G.preimage (f' \u226b f))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b2 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category.{u, v} A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\nY Z : D\nf : Z \u27f6 G.obj X\nhf : (\u2191T).arrows f\nW : C\ng : Y \u27f6 G.obj W\nf' : G.obj W \u27f6 Z\n\u22a2 (Sieve.functorPullback G \u2191T).arrows (G.preimage (f' \u226b f))\ncase h.right\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b2 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category.{u, v} A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\nY Z : D\nf : Z \u27f6 G.obj X\nhf : (\u2191T).arrows f\nW : C\ng : Y \u27f6 G.obj W\nf' : G.obj W \u27f6 Z\n\u22a2 (g \u226b f') \u226b f = g \u226b G.map (G.preimage (f' \u226b f))\n[PROOFSTEP]\nsimpa using T.val.downward_closed hf f'\n[GOAL]\ncase h.right\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b2 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category.{u, v} A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\nY Z : D\nf : Z \u27f6 G.obj X\nhf : (\u2191T).arrows f\nW : C\ng : Y \u27f6 G.obj W\nf' : G.obj W \u27f6 Z\n\u22a2 (g \u226b f') \u226b f = g \u226b G.map (G.preimage (f' \u226b f))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b9 : Category.{?u.13139, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d : Category.{u, v} A\nX : C\n\u22a2 LocallyCoverDense J (Over.forget X)\n[PROOFSTEP]\nintro Y T\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b9 : Category.{?u.13139, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d : Category.{u, v} A\nX : C\nY : Over X\nT : \u2191(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\n\u22a2 Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) \u2191T) \u2208\n GrothendieckTopology.sieves J ((Over.forget X).obj Y)\n[PROOFSTEP]\nconvert T.property\n[GOAL]\ncase h.e'_4\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b9 : Category.{?u.13139, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d : Category.{u, v} A\nX : C\nY : Over X\nT : \u2191(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\n\u22a2 Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) \u2191T) = \u2191T\n[PROOFSTEP]\next Z f\n[GOAL]\ncase h.e'_4.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b9 : Category.{?u.13139, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d : Category.{u, v} A\nX : C\nY : Over X\nT : \u2191(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\nZ : C\nf : Z \u27f6 (Over.forget X).obj Y\n\u22a2 (Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) \u2191T)).arrows f \u2194 (\u2191T).arrows f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.e'_4.h.mp\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b9 : Category.{?u.13139, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d : Category.{u, v} A\nX : C\nY : Over X\nT : \u2191(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\nZ : C\nf : Z \u27f6 (Over.forget X).obj Y\n\u22a2 (Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) \u2191T)).arrows f \u2192 (\u2191T).arrows f\n[PROOFSTEP]\nrintro \u27e8_, _, g', hg, rfl\u27e9\n[GOAL]\ncase h.e'_4.h.mp.intro.intro.intro.intro\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b9 : Category.{?u.13139, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d : Category.{u, v} A\nX : C\nY : Over X\nT : \u2191(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\nZ : C\nw\u271d\u00b9 : Over X\nw\u271d : w\u271d\u00b9 \u27f6 Y\ng' : Z \u27f6 (Over.forget X).obj w\u271d\u00b9\nhg : (Sieve.functorPullback (Over.forget X) \u2191T).arrows w\u271d\n\u22a2 (\u2191T).arrows (g' \u226b (Over.forget X).map w\u271d)\n[PROOFSTEP]\nexact T.val.downward_closed hg g'\n[GOAL]\ncase h.e'_4.h.mpr\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b9 : Category.{?u.13139, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d : Category.{u, v} A\nX : C\nY : Over X\nT : \u2191(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\nZ : C\nf : Z \u27f6 (Over.forget X).obj Y\n\u22a2 (\u2191T).arrows f \u2192 (Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) \u2191T)).arrows f\n[PROOFSTEP]\nintro hf\n[GOAL]\ncase h.e'_4.h.mpr\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b9 : Category.{?u.13139, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d : Category.{u, v} A\nX : C\nY : Over X\nT : \u2191(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\nZ : C\nf : Z \u27f6 (Over.forget X).obj Y\nhf : (\u2191T).arrows f\n\u22a2 (Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) \u2191T)).arrows f\n[PROOFSTEP]\nexact \u27e8Over.mk (f \u226b Y.hom), Over.homMk f, \ud835\udfd9 _, hf, (Category.id_comp _).symm\u27e9\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.InducedTopology", "llama_tokens": 12352, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.43398146480389854, "lm_q1q2_score": 0.23222279986964123}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : DenselyOrdered \u03b1\nf : Filter \u03b2\nu : \u03b2 \u2192 \u03b1\ns : Set \u03b1\nhs : Dense s\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a < b \u2192 \u00ac((\u2203\u1da0 (n : \u03b2) in f, u n < a) \u2227 \u2203\u1da0 (n : \u03b2) in f, b < u n)\nh : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nh' : autoParam (IsBoundedUnder (fun x x_1 => x \u2265 x_1) f u) _auto\u271d\n\u22a2 \u2203 c, Tendsto u f (\ud835\udcdd c)\n[PROOFSTEP]\nrcases f.eq_or_neBot with rfl | hbot\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : DenselyOrdered \u03b1\nu : \u03b2 \u2192 \u03b1\ns : Set \u03b1\nhs : Dense s\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a < b \u2192 \u00ac((\u2203\u1da0 (n : \u03b2) in \u22a5, u n < a) \u2227 \u2203\u1da0 (n : \u03b2) in \u22a5, b < u n)\nh : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) \u22a5 u) _auto\u271d\nh' : autoParam (IsBoundedUnder (fun x x_1 => x \u2265 x_1) \u22a5 u) _auto\u271d\n\u22a2 \u2203 c, Tendsto u \u22a5 (\ud835\udcdd c)\n[PROOFSTEP]\nexact \u27e8sInf \u2205, tendsto_bot\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : DenselyOrdered \u03b1\nf : Filter \u03b2\nu : \u03b2 \u2192 \u03b1\ns : Set \u03b1\nhs : Dense s\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a < b \u2192 \u00ac((\u2203\u1da0 (n : \u03b2) in f, u n < a) \u2227 \u2203\u1da0 (n : \u03b2) in f, b < u n)\nh : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nh' : autoParam (IsBoundedUnder (fun x x_1 => x \u2265 x_1) f u) _auto\u271d\nhbot : NeBot f\n\u22a2 \u2203 c, Tendsto u f (\ud835\udcdd c)\n[PROOFSTEP]\nrefine' \u27e8limsup u f, _\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : DenselyOrdered \u03b1\nf : Filter \u03b2\nu : \u03b2 \u2192 \u03b1\ns : Set \u03b1\nhs : Dense s\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a < b \u2192 \u00ac((\u2203\u1da0 (n : \u03b2) in f, u n < a) \u2227 \u2203\u1da0 (n : \u03b2) in f, b < u n)\nh : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nh' : autoParam (IsBoundedUnder (fun x x_1 => x \u2265 x_1) f u) _auto\u271d\nhbot : NeBot f\n\u22a2 Tendsto u f (\ud835\udcdd (limsup u f))\n[PROOFSTEP]\napply tendsto_of_le_liminf_of_limsup_le _ le_rfl h h'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : DenselyOrdered \u03b1\nf : Filter \u03b2\nu : \u03b2 \u2192 \u03b1\ns : Set \u03b1\nhs : Dense s\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a < b \u2192 \u00ac((\u2203\u1da0 (n : \u03b2) in f, u n < a) \u2227 \u2203\u1da0 (n : \u03b2) in f, b < u n)\nh : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nh' : autoParam (IsBoundedUnder (fun x x_1 => x \u2265 x_1) f u) _auto\u271d\nhbot : NeBot f\n\u22a2 limsup u f \u2264 liminf u f\n[PROOFSTEP]\nby_contra' hlt\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : DenselyOrdered \u03b1\nf : Filter \u03b2\nu : \u03b2 \u2192 \u03b1\ns : Set \u03b1\nhs : Dense s\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a < b \u2192 \u00ac((\u2203\u1da0 (n : \u03b2) in f, u n < a) \u2227 \u2203\u1da0 (n : \u03b2) in f, b < u n)\nh : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nh' : autoParam (IsBoundedUnder (fun x x_1 => x \u2265 x_1) f u) _auto\u271d\nhbot : NeBot f\nhlt : liminf u f < limsup u f\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8a, \u27e8\u27e8la, au\u27e9, as\u27e9\u27e9 : \u2203 a, (f.liminf u < a \u2227 a < f.limsup u) \u2227 a \u2208 s :=\n dense_iff_inter_open.1 hs (Set.Ioo (f.liminf u) (f.limsup u)) isOpen_Ioo (Set.nonempty_Ioo.2 hlt)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : DenselyOrdered \u03b1\nf : Filter \u03b2\nu : \u03b2 \u2192 \u03b1\ns : Set \u03b1\nhs : Dense s\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a < b \u2192 \u00ac((\u2203\u1da0 (n : \u03b2) in f, u n < a) \u2227 \u2203\u1da0 (n : \u03b2) in f, b < u n)\nh : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nh' : autoParam (IsBoundedUnder (fun x x_1 => x \u2265 x_1) f u) _auto\u271d\nhbot : NeBot f\nhlt : liminf u f < limsup u f\na : \u03b1\nas : a \u2208 s\nla : liminf u f < a\nau : a < limsup u f\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8b, \u27e8\u27e8ab, bu\u27e9, bs\u27e9\u27e9 : \u2203 b, (a < b \u2227 b < f.limsup u) \u2227 b \u2208 s :=\n dense_iff_inter_open.1 hs (Set.Ioo a (f.limsup u)) isOpen_Ioo (Set.nonempty_Ioo.2 au)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : DenselyOrdered \u03b1\nf : Filter \u03b2\nu : \u03b2 \u2192 \u03b1\ns : Set \u03b1\nhs : Dense s\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a < b \u2192 \u00ac((\u2203\u1da0 (n : \u03b2) in f, u n < a) \u2227 \u2203\u1da0 (n : \u03b2) in f, b < u n)\nh : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nh' : autoParam (IsBoundedUnder (fun x x_1 => x \u2265 x_1) f u) _auto\u271d\nhbot : NeBot f\nhlt : liminf u f < limsup u f\na : \u03b1\nas : a \u2208 s\nla : liminf u f < a\nau : a < limsup u f\nb : \u03b1\nbs : b \u2208 s\nab : a < b\nbu : b < limsup u f\n\u22a2 False\n[PROOFSTEP]\nhave A : \u2203\u1da0 n in f, u n < a := frequently_lt_of_liminf_lt (IsBounded.isCobounded_ge h) la\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : DenselyOrdered \u03b1\nf : Filter \u03b2\nu : \u03b2 \u2192 \u03b1\ns : Set \u03b1\nhs : Dense s\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a < b \u2192 \u00ac((\u2203\u1da0 (n : \u03b2) in f, u n < a) \u2227 \u2203\u1da0 (n : \u03b2) in f, b < u n)\nh : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nh' : autoParam (IsBoundedUnder (fun x x_1 => x \u2265 x_1) f u) _auto\u271d\nhbot : NeBot f\nhlt : liminf u f < limsup u f\na : \u03b1\nas : a \u2208 s\nla : liminf u f < a\nau : a < limsup u f\nb : \u03b1\nbs : b \u2208 s\nab : a < b\nbu : b < limsup u f\nA : \u2203\u1da0 (n : \u03b2) in f, u n < a\n\u22a2 False\n[PROOFSTEP]\nhave B : \u2203\u1da0 n in f, b < u n := frequently_lt_of_lt_limsup (IsBounded.isCobounded_le h') bu\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : DenselyOrdered \u03b1\nf : Filter \u03b2\nu : \u03b2 \u2192 \u03b1\ns : Set \u03b1\nhs : Dense s\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a < b \u2192 \u00ac((\u2203\u1da0 (n : \u03b2) in f, u n < a) \u2227 \u2203\u1da0 (n : \u03b2) in f, b < u n)\nh : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nh' : autoParam (IsBoundedUnder (fun x x_1 => x \u2265 x_1) f u) _auto\u271d\nhbot : NeBot f\nhlt : liminf u f < limsup u f\na : \u03b1\nas : a \u2208 s\nla : liminf u f < a\nau : a < limsup u f\nb : \u03b1\nbs : b \u2208 s\nab : a < b\nbu : b < limsup u f\nA : \u2203\u1da0 (n : \u03b2) in f, u n < a\nB : \u2203\u1da0 (n : \u03b2) in f, b < u n\n\u22a2 False\n[PROOFSTEP]\nexact H a as b bs ab \u27e8A, B\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : FirstCountableTopology \u03b1\nf : Filter \u03b2\ninst\u271d : CountableInterFilter f\nu : \u03b2 \u2192 \u03b1\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\n\u22a2 \u2200\u1da0 (b : \u03b2) in f, u b \u2264 limsup u f\n[PROOFSTEP]\nobtain ha | ha := isTop_or_exists_gt (f.limsup u)\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : FirstCountableTopology \u03b1\nf : Filter \u03b2\ninst\u271d : CountableInterFilter f\nu : \u03b2 \u2192 \u03b1\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nha : IsTop (limsup u f)\n\u22a2 \u2200\u1da0 (b : \u03b2) in f, u b \u2264 limsup u f\n[PROOFSTEP]\nexact eventually_of_forall fun _ => ha _\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : FirstCountableTopology \u03b1\nf : Filter \u03b2\ninst\u271d : CountableInterFilter f\nu : \u03b2 \u2192 \u03b1\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nha : \u2203 b, limsup u f < b\n\u22a2 \u2200\u1da0 (b : \u03b2) in f, u b \u2264 limsup u f\n[PROOFSTEP]\nby_cases H : IsGLB (Set.Ioi (f.limsup u)) (f.limsup u)\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : FirstCountableTopology \u03b1\nf : Filter \u03b2\ninst\u271d : CountableInterFilter f\nu : \u03b2 \u2192 \u03b1\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nha : \u2203 b, limsup u f < b\nH : IsGLB (Set.Ioi (limsup u f)) (limsup u f)\n\u22a2 \u2200\u1da0 (b : \u03b2) in f, u b \u2264 limsup u f\n[PROOFSTEP]\nobtain \u27e8u, -, -, hua, hu\u27e9 := H.exists_seq_antitone_tendsto ha\n[GOAL]\ncase pos.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : FirstCountableTopology \u03b1\nf : Filter \u03b2\ninst\u271d : CountableInterFilter f\nu\u271d : \u03b2 \u2192 \u03b1\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u\u271d) _auto\u271d\nha : \u2203 b, limsup u\u271d f < b\nH : IsGLB (Set.Ioi (limsup u\u271d f)) (limsup u\u271d f)\nu : \u2115 \u2192 \u03b1\nhua : Tendsto u atTop (\ud835\udcdd (limsup u\u271d f))\nhu : \u2200 (n : \u2115), u n \u2208 Set.Ioi (limsup u\u271d f)\n\u22a2 \u2200\u1da0 (b : \u03b2) in f, u\u271d b \u2264 limsup u\u271d f\n[PROOFSTEP]\nhave := fun n => eventually_lt_of_limsup_lt (hu n) hf\n[GOAL]\ncase pos.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : FirstCountableTopology \u03b1\nf : Filter \u03b2\ninst\u271d : CountableInterFilter f\nu\u271d : \u03b2 \u2192 \u03b1\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u\u271d) _auto\u271d\nha : \u2203 b, limsup u\u271d f < b\nH : IsGLB (Set.Ioi (limsup u\u271d f)) (limsup u\u271d f)\nu : \u2115 \u2192 \u03b1\nhua : Tendsto u atTop (\ud835\udcdd (limsup u\u271d f))\nhu : \u2200 (n : \u2115), u n \u2208 Set.Ioi (limsup u\u271d f)\nthis : \u2200 (n : \u2115), \u2200\u1da0 (a : \u03b2) in f, u\u271d a < u n\n\u22a2 \u2200\u1da0 (b : \u03b2) in f, u\u271d b \u2264 limsup u\u271d f\n[PROOFSTEP]\nexact (eventually_countable_forall.2 this).mono fun b hb => ge_of_tendsto hua <| eventually_of_forall fun n => (hb _).le\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : FirstCountableTopology \u03b1\nf : Filter \u03b2\ninst\u271d : CountableInterFilter f\nu : \u03b2 \u2192 \u03b1\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nha : \u2203 b, limsup u f < b\nH : \u00acIsGLB (Set.Ioi (limsup u f)) (limsup u f)\n\u22a2 \u2200\u1da0 (b : \u03b2) in f, u b \u2264 limsup u f\n[PROOFSTEP]\nobtain \u27e8x, hx, xa\u27e9 : \u2203 x, (\u2200 \u2983b\u2984, f.limsup u < b \u2192 x \u2264 b) \u2227 f.limsup u < x :=\n by\n simp only [IsGLB, IsGreatest, lowerBounds, upperBounds, Set.mem_Ioi, Set.mem_setOf_eq, not_and, not_forall, not_le,\n exists_prop] at H \n exact H fun x => le_of_lt\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : FirstCountableTopology \u03b1\nf : Filter \u03b2\ninst\u271d : CountableInterFilter f\nu : \u03b2 \u2192 \u03b1\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nha : \u2203 b, limsup u f < b\nH : \u00acIsGLB (Set.Ioi (limsup u f)) (limsup u f)\n\u22a2 \u2203 x, (\u2200 \u2983b : \u03b1\u2984, limsup u f < b \u2192 x \u2264 b) \u2227 limsup u f < x\n[PROOFSTEP]\nsimp only [IsGLB, IsGreatest, lowerBounds, upperBounds, Set.mem_Ioi, Set.mem_setOf_eq, not_and, not_forall, not_le,\n exists_prop] at H \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : FirstCountableTopology \u03b1\nf : Filter \u03b2\ninst\u271d : CountableInterFilter f\nu : \u03b2 \u2192 \u03b1\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nha : \u2203 b, limsup u f < b\nH : (\u2200 \u2983a : \u03b1\u2984, limsup u f < a \u2192 limsup u f \u2264 a) \u2192 \u2203 x, (\u2200 \u2983a : \u03b1\u2984, limsup u f < a \u2192 x \u2264 a) \u2227 limsup u f < x\n\u22a2 \u2203 x, (\u2200 \u2983b : \u03b1\u2984, limsup u f < b \u2192 x \u2264 b) \u2227 limsup u f < x\n[PROOFSTEP]\nexact H fun x => le_of_lt\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : FirstCountableTopology \u03b1\nf : Filter \u03b2\ninst\u271d : CountableInterFilter f\nu : \u03b2 \u2192 \u03b1\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nha : \u2203 b, limsup u f < b\nH : \u00acIsGLB (Set.Ioi (limsup u f)) (limsup u f)\nx : \u03b1\nhx : \u2200 \u2983b : \u03b1\u2984, limsup u f < b \u2192 x \u2264 b\nxa : limsup u f < x\n\u22a2 \u2200\u1da0 (b : \u03b2) in f, u b \u2264 limsup u f\n[PROOFSTEP]\nfilter_upwards [eventually_lt_of_limsup_lt xa hf] with y hy\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : FirstCountableTopology \u03b1\nf : Filter \u03b2\ninst\u271d : CountableInterFilter f\nu : \u03b2 \u2192 \u03b1\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nha : \u2203 b, limsup u f < b\nH : \u00acIsGLB (Set.Ioi (limsup u f)) (limsup u f)\nx : \u03b1\nhx : \u2200 \u2983b : \u03b1\u2984, limsup u f < b \u2192 x \u2264 b\nxa : limsup u f < x\ny : \u03b2\nhy : u y < x\n\u22a2 u y \u2264 limsup u f\n[PROOFSTEP]\ncontrapose! hy\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : FirstCountableTopology \u03b1\nf : Filter \u03b2\ninst\u271d : CountableInterFilter f\nu : \u03b2 \u2192 \u03b1\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) f u) _auto\u271d\nha : \u2203 b, limsup u f < b\nH : \u00acIsGLB (Set.Ioi (limsup u f)) (limsup u f)\nx : \u03b1\nhx : \u2200 \u2983b : \u03b1\u2984, limsup u f < b \u2192 x \u2264 b\nxa : limsup u f < x\ny : \u03b2\nhy : limsup u f < u y\n\u22a2 x \u2264 u y\n[PROOFSTEP]\nexact hx hy\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2074 : CompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : FirstCountableTopology \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\nf : Filter \u03b2\ninst\u271d : CountableInterFilter f\nu : \u03b2 \u2192 \u03b1\nh : u =\u1da0[f] \u22a5\n\u22a2 limsup u f = \u22a5\n[PROOFSTEP]\nrw [limsup_congr h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2074 : CompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : FirstCountableTopology \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\nf : Filter \u03b2\ninst\u271d : CountableInterFilter f\nu : \u03b2 \u2192 \u03b1\nh : u =\u1da0[f] \u22a5\n\u22a2 limsup (fun a => \u22a5 a) f = \u22a5\n[PROOFSTEP]\nexact limsup_const_bot\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\n\u22a2 f (limsSup F) = liminf f F\n[PROOFSTEP]\nhave cobdd : F.IsCobounded (\u00b7 \u2264 \u00b7) := bdd_below.isCobounded_flip\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\n\u22a2 f (limsSup F) = liminf f F\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\n\u22a2 f (limsSup F) \u2264 liminf f F\n[PROOFSTEP]\nrw [limsSup, f_decr.map_sInf_of_continuousAt' f_cont bdd_above cobdd]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\n\u22a2 sSup (f '' {a | \u2200\u1da0 (n : R) in F, n \u2264 a}) \u2264 liminf f F\n[PROOFSTEP]\napply le_of_forall_lt\n[GOAL]\ncase a.H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\n\u22a2 \u2200 (c : S), c < sSup (f '' {a | \u2200\u1da0 (n : R) in F, n \u2264 a}) \u2192 c < liminf f F\n[PROOFSTEP]\nintro c hc\n[GOAL]\ncase a.H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nc : S\nhc : c < sSup (f '' {a | \u2200\u1da0 (n : R) in F, n \u2264 a})\n\u22a2 c < liminf f F\n[PROOFSTEP]\nsimp only [liminf, limsInf, eventually_map] at hc \u22a2\n[GOAL]\ncase a.H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nc : S\nhc : c < sSup (f '' {a | \u2200\u1da0 (n : R) in F, n \u2264 a})\n\u22a2 c < sSup {a | \u2200\u1da0 (a_1 : R) in F, a \u2264 f a_1}\n[PROOFSTEP]\nobtain \u27e8d, hd, h'd\u27e9 := exists_lt_of_lt_csSup (bdd_above.recOn fun x hx \u21a6 \u27e8f x, Set.mem_image_of_mem f hx\u27e9) hc\n[GOAL]\ncase a.H.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nc : S\nhc : c < sSup (f '' {a | \u2200\u1da0 (n : R) in F, n \u2264 a})\nd : S\nhd : d \u2208 f '' {a | \u2200\u1da0 (n : R) in F, n \u2264 a}\nh'd : c < d\n\u22a2 c < sSup {a | \u2200\u1da0 (a_1 : R) in F, a \u2264 f a_1}\n[PROOFSTEP]\napply lt_csSup_of_lt ?_ ?_ h'd\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nc : S\nhc : c < sSup (f '' {a | \u2200\u1da0 (n : R) in F, n \u2264 a})\nd : S\nhd : d \u2208 f '' {a | \u2200\u1da0 (n : R) in F, n \u2264 a}\nh'd : c < d\n\u22a2 BddAbove {a | \u2200\u1da0 (a_1 : R) in F, a \u2264 f a_1}\n[PROOFSTEP]\nexact (Antitone.isBoundedUnder_le_comp f_decr bdd_below).isCoboundedUnder_flip\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nc : S\nhc : c < sSup (f '' {a | \u2200\u1da0 (n : R) in F, n \u2264 a})\nd : S\nhd : d \u2208 f '' {a | \u2200\u1da0 (n : R) in F, n \u2264 a}\nh'd : c < d\n\u22a2 d \u2208 {a | \u2200\u1da0 (a_1 : R) in F, a \u2264 f a_1}\n[PROOFSTEP]\nrcases hd with \u27e8e, \u27e8he, fe_eq_d\u27e9\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nc : S\nhc : c < sSup (f '' {a | \u2200\u1da0 (n : R) in F, n \u2264 a})\nd : S\nh'd : c < d\ne : R\nhe : e \u2208 {a | \u2200\u1da0 (n : R) in F, n \u2264 a}\nfe_eq_d : f e = d\n\u22a2 d \u2208 {a | \u2200\u1da0 (a_1 : R) in F, a \u2264 f a_1}\n[PROOFSTEP]\nfilter_upwards [he] with x hx using (fe_eq_d.symm \u25b8 f_decr hx)\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\n\u22a2 liminf f F \u2264 f (limsSup F)\n[PROOFSTEP]\nby_cases h' : \u2203 c, c < F.limsSup \u2227 Set.Ioo c F.limsSup = \u2205\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nh' : \u2203 c, c < limsSup F \u2227 Set.Ioo c (limsSup F) = \u2205\n\u22a2 liminf f F \u2264 f (limsSup F)\n[PROOFSTEP]\nrcases h' with \u27e8c, c_lt, hc\u27e9\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nc : R\nc_lt : c < limsSup F\nhc : Set.Ioo c (limsSup F) = \u2205\n\u22a2 liminf f F \u2264 f (limsSup F)\n[PROOFSTEP]\nhave B : \u2203\u1da0 n in F, F.limsSup \u2264 n :=\n by\n apply (frequently_lt_of_lt_limsSup cobdd c_lt).mono\n intro x hx\n by_contra'\n have : (Set.Ioo c F.limsSup).Nonempty := \u27e8x, \u27e8hx, this\u27e9\u27e9\n simp only [hc, Set.not_nonempty_empty] at this \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nc : R\nc_lt : c < limsSup F\nhc : Set.Ioo c (limsSup F) = \u2205\n\u22a2 \u2203\u1da0 (n : R) in F, limsSup F \u2264 n\n[PROOFSTEP]\napply (frequently_lt_of_lt_limsSup cobdd c_lt).mono\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nc : R\nc_lt : c < limsSup F\nhc : Set.Ioo c (limsSup F) = \u2205\n\u22a2 \u2200 (x : R), c < x \u2192 limsSup F \u2264 x\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nc : R\nc_lt : c < limsSup F\nhc : Set.Ioo c (limsSup F) = \u2205\nx : R\nhx : c < x\n\u22a2 limsSup F \u2264 x\n[PROOFSTEP]\nby_contra'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nc : R\nc_lt : c < limsSup F\nhc : Set.Ioo c (limsSup F) = \u2205\nx : R\nhx : c < x\nthis : x < limsSup F\n\u22a2 False\n[PROOFSTEP]\nhave : (Set.Ioo c F.limsSup).Nonempty := \u27e8x, \u27e8hx, this\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nc : R\nc_lt : c < limsSup F\nhc : Set.Ioo c (limsSup F) = \u2205\nx : R\nhx : c < x\nthis\u271d : x < limsSup F\nthis : Set.Nonempty (Set.Ioo c (limsSup F))\n\u22a2 False\n[PROOFSTEP]\nsimp only [hc, Set.not_nonempty_empty] at this \n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nc : R\nc_lt : c < limsSup F\nhc : Set.Ioo c (limsSup F) = \u2205\nB : \u2203\u1da0 (n : R) in F, limsSup F \u2264 n\n\u22a2 liminf f F \u2264 f (limsSup F)\n[PROOFSTEP]\napply liminf_le_of_frequently_le _ (bdd_above.isBoundedUnder f_decr)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nc : R\nc_lt : c < limsSup F\nhc : Set.Ioo c (limsSup F) = \u2205\nB : \u2203\u1da0 (n : R) in F, limsSup F \u2264 n\n\u22a2 \u2203\u1da0 (x : R) in F, f x \u2264 f (limsSup F)\n[PROOFSTEP]\nexact (B.mono fun x hx \u21a6 f_decr hx)\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nh' : \u00ac\u2203 c, c < limsSup F \u2227 Set.Ioo c (limsSup F) = \u2205\n\u22a2 liminf f F \u2264 f (limsSup F)\n[PROOFSTEP]\npush_neg at h' \n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nh' : \u2200 (c : R), c < limsSup F \u2192 Set.Ioo c (limsSup F) \u2260 \u2205\n\u22a2 liminf f F \u2264 f (limsSup F)\n[PROOFSTEP]\nby_contra' H\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nh' : \u2200 (c : R), c < limsSup F \u2192 Set.Ioo c (limsSup F) \u2260 \u2205\nH : f (limsSup F) < liminf f F\n\u22a2 False\n[PROOFSTEP]\nhave not_bot : \u00acIsBot F.limsSup := fun maybe_bot \u21a6\n lt_irrefl (F.liminf f) <|\n lt_of_le_of_lt\n (liminf_le_of_frequently_le (frequently_of_forall (fun r \u21a6 f_decr (maybe_bot r)))\n (bdd_above.isBoundedUnder f_decr))\n H\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nh' : \u2200 (c : R), c < limsSup F \u2192 Set.Ioo c (limsSup F) \u2260 \u2205\nH : f (limsSup F) < liminf f F\nnot_bot : \u00acIsBot (limsSup F)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8l, l_lt, h'l\u27e9 : \u2203 l < F.limsSup, Set.Ioc l F.limsSup \u2286 {x : R | f x < F.liminf f}\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nh' : \u2200 (c : R), c < limsSup F \u2192 Set.Ioo c (limsSup F) \u2260 \u2205\nH : f (limsSup F) < liminf f F\nnot_bot : \u00acIsBot (limsSup F)\n\u22a2 \u2203 l, l < limsSup F \u2227 Set.Ioc l (limsSup F) \u2286 {x | f x < liminf f F}\n[PROOFSTEP]\napply exists_Ioc_subset_of_mem_nhds ((tendsto_order.1 f_cont.tendsto).2 _ H)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nh' : \u2200 (c : R), c < limsSup F \u2192 Set.Ioo c (limsSup F) \u2260 \u2205\nH : f (limsSup F) < liminf f F\nnot_bot : \u00acIsBot (limsSup F)\n\u22a2 \u2203 l, l < limsSup F\n[PROOFSTEP]\nsimpa [IsBot] using not_bot\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nh' : \u2200 (c : R), c < limsSup F \u2192 Set.Ioo c (limsSup F) \u2260 \u2205\nH : f (limsSup F) < liminf f F\nnot_bot : \u00acIsBot (limsSup F)\nl : R\nl_lt : l < limsSup F\nh'l : Set.Ioc l (limsSup F) \u2286 {x | f x < liminf f F}\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8m, l_m, m_lt\u27e9 : (Set.Ioo l F.limsSup).Nonempty :=\n by\n contrapose! h'\n refine' \u27e8l, l_lt, by rwa [Set.not_nonempty_iff_eq_empty] at h' \u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nh' : \u2200 (c : R), c < limsSup F \u2192 Set.Ioo c (limsSup F) \u2260 \u2205\nH : f (limsSup F) < liminf f F\nnot_bot : \u00acIsBot (limsSup F)\nl : R\nl_lt : l < limsSup F\nh'l : Set.Ioc l (limsSup F) \u2286 {x | f x < liminf f F}\n\u22a2 Set.Nonempty (Set.Ioo l (limsSup F))\n[PROOFSTEP]\ncontrapose! h'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nH : f (limsSup F) < liminf f F\nnot_bot : \u00acIsBot (limsSup F)\nl : R\nl_lt : l < limsSup F\nh'l : Set.Ioc l (limsSup F) \u2286 {x | f x < liminf f F}\nh' : \u00acSet.Nonempty (Set.Ioo l (limsSup F))\n\u22a2 \u2203 c, c < limsSup F \u2227 Set.Ioo c (limsSup F) = \u2205\n[PROOFSTEP]\nrefine' \u27e8l, l_lt, by rwa [Set.not_nonempty_iff_eq_empty] at h' \u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nH : f (limsSup F) < liminf f F\nnot_bot : \u00acIsBot (limsSup F)\nl : R\nl_lt : l < limsSup F\nh'l : Set.Ioc l (limsSup F) \u2286 {x | f x < liminf f F}\nh' : \u00acSet.Nonempty (Set.Ioo l (limsSup F))\n\u22a2 Set.Ioo l (limsSup F) = \u2205\n[PROOFSTEP]\nrwa [Set.not_nonempty_iff_eq_empty] at h' \n[GOAL]\ncase neg.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nh' : \u2200 (c : R), c < limsSup F \u2192 Set.Ioo c (limsSup F) \u2260 \u2205\nH : f (limsSup F) < liminf f F\nnot_bot : \u00acIsBot (limsSup F)\nl : R\nl_lt : l < limsSup F\nh'l : Set.Ioc l (limsSup F) \u2286 {x | f x < liminf f F}\nm : R\nl_m : l < m\nm_lt : m < limsSup F\n\u22a2 False\n[PROOFSTEP]\nhave B : F.liminf f \u2264 f m := by\n apply liminf_le_of_frequently_le _ _\n \u00b7 apply (frequently_lt_of_lt_limsSup cobdd m_lt).mono\n exact fun x hx \u21a6 f_decr hx.le\n \u00b7 exact IsBounded.isBoundedUnder f_decr bdd_above\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nh' : \u2200 (c : R), c < limsSup F \u2192 Set.Ioo c (limsSup F) \u2260 \u2205\nH : f (limsSup F) < liminf f F\nnot_bot : \u00acIsBot (limsSup F)\nl : R\nl_lt : l < limsSup F\nh'l : Set.Ioc l (limsSup F) \u2286 {x | f x < liminf f F}\nm : R\nl_m : l < m\nm_lt : m < limsSup F\n\u22a2 liminf f F \u2264 f m\n[PROOFSTEP]\napply liminf_le_of_frequently_le _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nh' : \u2200 (c : R), c < limsSup F \u2192 Set.Ioo c (limsSup F) \u2260 \u2205\nH : f (limsSup F) < liminf f F\nnot_bot : \u00acIsBot (limsSup F)\nl : R\nl_lt : l < limsSup F\nh'l : Set.Ioc l (limsSup F) \u2286 {x | f x < liminf f F}\nm : R\nl_m : l < m\nm_lt : m < limsSup F\n\u22a2 \u2203\u1da0 (x : R) in F, f x \u2264 f m\n[PROOFSTEP]\napply (frequently_lt_of_lt_limsSup cobdd m_lt).mono\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nh' : \u2200 (c : R), c < limsSup F \u2192 Set.Ioo c (limsSup F) \u2260 \u2205\nH : f (limsSup F) < liminf f F\nnot_bot : \u00acIsBot (limsSup F)\nl : R\nl_lt : l < limsSup F\nh'l : Set.Ioc l (limsSup F) \u2286 {x | f x < liminf f F}\nm : R\nl_m : l < m\nm_lt : m < limsSup F\n\u22a2 \u2200 (x : R), m < x \u2192 f x \u2264 f m\n[PROOFSTEP]\nexact fun x hx \u21a6 f_decr hx.le\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nh' : \u2200 (c : R), c < limsSup F \u2192 Set.Ioo c (limsSup F) \u2260 \u2205\nH : f (limsSup F) < liminf f F\nnot_bot : \u00acIsBot (limsSup F)\nl : R\nl_lt : l < limsSup F\nh'l : Set.Ioc l (limsSup F) \u2286 {x | f x < liminf f F}\nm : R\nl_m : l < m\nm_lt : m < limsSup F\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) F f\n[PROOFSTEP]\nexact IsBounded.isBoundedUnder f_decr bdd_above\n[GOAL]\ncase neg.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nh' : \u2200 (c : R), c < limsSup F \u2192 Set.Ioo c (limsSup F) \u2260 \u2205\nH : f (limsSup F) < liminf f F\nnot_bot : \u00acIsBot (limsSup F)\nl : R\nl_lt : l < limsSup F\nh'l : Set.Ioc l (limsSup F) \u2286 {x | f x < liminf f F}\nm : R\nl_m : l < m\nm_lt : m < limsSup F\nB : liminf f F \u2264 f m\n\u22a2 False\n[PROOFSTEP]\nhave I : f m < F.liminf f := h'l \u27e8l_m, m_lt.le\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\nF\u271d : Filter \u03b9\ninst\u271d\u2077 : NeBot F\u271d\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder R\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : OrderTopology R\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder S\ninst\u271d\u00b2 : TopologicalSpace S\ninst\u271d\u00b9 : OrderTopology S\nF : Filter R\ninst\u271d : NeBot F\nf : R \u2192 S\nf_decr : Antitone f\nf_cont : ContinuousAt f (limsSup F)\nbdd_above : autoParam (IsBounded (fun x x_1 => x \u2264 x_1) F) _auto\u271d\nbdd_below : autoParam (IsBounded (fun x x_1 => x \u2265 x_1) F) _auto\u271d\ncobdd : IsCobounded (fun x x_1 => x \u2264 x_1) F\nh' : \u2200 (c : R), c < limsSup F \u2192 Set.Ioo c (limsSup F) \u2260 \u2205\nH : f (limsSup F) < liminf f F\nnot_bot : \u00acIsBot (limsSup F)\nl : R\nl_lt : l < limsSup F\nh'l : Set.Ioc l (limsSup F) \u2286 {x | f x < liminf f F}\nm : R\nl_m : l < m\nm_lt : m < limsSup F\nB : liminf f F \u2264 f m\nI : f m < liminf f F\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ (B.trans_lt I)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : CompleteLinearOrder R\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : OrderTopology R\nx : R\nas : \u03b9 \u2192 R\nx_le : \u2200 (i : \u03b9), x \u2264 as i\nF : Filter \u03b9\ninst\u271d : NeBot F\nas_lim : Tendsto as F (\ud835\udcdd x)\n\u22a2 \u2a05 (i : \u03b9), as i = x\n[PROOFSTEP]\nrefine' iInf_eq_of_forall_ge_of_forall_gt_exists_lt (fun i \u21a6 x_le i) _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : CompleteLinearOrder R\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : OrderTopology R\nx : R\nas : \u03b9 \u2192 R\nx_le : \u2200 (i : \u03b9), x \u2264 as i\nF : Filter \u03b9\ninst\u271d : NeBot F\nas_lim : Tendsto as F (\ud835\udcdd x)\n\u22a2 \u2200 (w : R), x < w \u2192 \u2203 i, as i < w\n[PROOFSTEP]\napply fun w x_lt_w \u21a6 \u2039Filter.NeBot F\u203a.nonempty_of_mem (eventually_lt_of_tendsto_lt x_lt_w as_lim)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : CompleteLinearOrder R\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : OrderTopology R\n\u03b9 : Type u_3\nx : R\nas : \u03b9 \u2192 R\nx_lt : \u2200 (i : \u03b9), x < as i\nF : Filter \u03b9\ninst\u271d : NeBot F\nas_lim : Tendsto as F (\ud835\udcdd x)\n\u22a2 \u22c3 (i : \u03b9), Ici (as i) = Ioi x\n[PROOFSTEP]\nhave obs : x \u2209 range as := by\n intro maybe_x_is\n rcases mem_range.mp maybe_x_is with \u27e8i, hi\u27e9\n simpa only [hi, lt_self_iff_false] using x_lt i\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : CompleteLinearOrder R\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : OrderTopology R\n\u03b9 : Type u_3\nx : R\nas : \u03b9 \u2192 R\nx_lt : \u2200 (i : \u03b9), x < as i\nF : Filter \u03b9\ninst\u271d : NeBot F\nas_lim : Tendsto as F (\ud835\udcdd x)\n\u22a2 \u00acx \u2208 range as\n[PROOFSTEP]\nintro maybe_x_is\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : CompleteLinearOrder R\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : OrderTopology R\n\u03b9 : Type u_3\nx : R\nas : \u03b9 \u2192 R\nx_lt : \u2200 (i : \u03b9), x < as i\nF : Filter \u03b9\ninst\u271d : NeBot F\nas_lim : Tendsto as F (\ud835\udcdd x)\nmaybe_x_is : x \u2208 range as\n\u22a2 False\n[PROOFSTEP]\nrcases mem_range.mp maybe_x_is with \u27e8i, hi\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : CompleteLinearOrder R\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : OrderTopology R\n\u03b9 : Type u_3\nx : R\nas : \u03b9 \u2192 R\nx_lt : \u2200 (i : \u03b9), x < as i\nF : Filter \u03b9\ninst\u271d : NeBot F\nas_lim : Tendsto as F (\ud835\udcdd x)\nmaybe_x_is : x \u2208 range as\ni : \u03b9\nhi : as i = x\n\u22a2 False\n[PROOFSTEP]\nsimpa only [hi, lt_self_iff_false] using x_lt i\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : CompleteLinearOrder R\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : OrderTopology R\n\u03b9 : Type u_3\nx : R\nas : \u03b9 \u2192 R\nx_lt : \u2200 (i : \u03b9), x < as i\nF : Filter \u03b9\ninst\u271d : NeBot F\nas_lim : Tendsto as F (\ud835\udcdd x)\nobs : \u00acx \u2208 range as\n\u22a2 \u22c3 (i : \u03b9), Ici (as i) = Ioi x\n[PROOFSTEP]\nhave := iInf_eq_of_forall_le_of_tendsto (fun i \u21a6 (x_lt i).le) as_lim\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : CompleteLinearOrder R\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : OrderTopology R\n\u03b9 : Type u_3\nx : R\nas : \u03b9 \u2192 R\nx_lt : \u2200 (i : \u03b9), x < as i\nF : Filter \u03b9\ninst\u271d : NeBot F\nas_lim : Tendsto as F (\ud835\udcdd x)\nobs : \u00acx \u2208 range as\nthis : \u2a05 (i : \u03b9), as i = x\n\u22a2 \u22c3 (i : \u03b9), Ici (as i) = Ioi x\n[PROOFSTEP]\nrw [\u2190 this] at obs \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : CompleteLinearOrder R\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : OrderTopology R\n\u03b9 : Type u_3\nx : R\nas : \u03b9 \u2192 R\nx_lt : \u2200 (i : \u03b9), x < as i\nF : Filter \u03b9\ninst\u271d : NeBot F\nas_lim : Tendsto as F (\ud835\udcdd x)\nobs : \u00ac\u2a05 (i : \u03b9), as i \u2208 range as\nthis : \u2a05 (i : \u03b9), as i = x\n\u22a2 \u22c3 (i : \u03b9), Ici (as i) = Ioi x\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : CompleteLinearOrder R\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : OrderTopology R\n\u03b9 : Type u_3\nx : R\nas : \u03b9 \u2192 R\nx_lt : \u2200 (i : \u03b9), x < as i\nF : Filter \u03b9\ninst\u271d : NeBot F\nas_lim : Tendsto as F (\ud835\udcdd x)\nobs : \u00ac\u2a05 (i : \u03b9), as i \u2208 range as\nthis : \u2a05 (i : \u03b9), as i = x\n\u22a2 \u22c3 (i : \u03b9), Ici (as i) = Ioi (\u2a05 (i : \u03b9), as i)\n[PROOFSTEP]\nexact iUnion_Ici_eq_Ioi_iInf obs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u22a2 limsup s atTop = {\u03c9 | Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop}\n[PROOFSTEP]\next \u03c9\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\n\u22a2 \u03c9 \u2208 limsup s atTop \u2194 \u03c9 \u2208 {\u03c9 | Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop}\n[PROOFSTEP]\nsimp only [limsup_eq_iInf_iSup_of_nat, ge_iff_le, Set.iSup_eq_iUnion, Set.iInf_eq_iInter, Set.mem_iInter,\n Set.mem_iUnion, exists_prop]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\n\u22a2 (\u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1) \u2194\n \u03c9 \u2208 {\u03c9 | Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\n\u22a2 (\u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1) \u2192\n \u03c9 \u2208 {\u03c9 | Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop}\n[PROOFSTEP]\nintro h\u03c9\n[GOAL]\ncase h.mp\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\n\u22a2 \u03c9 \u2208 {\u03c9 | Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop}\n[PROOFSTEP]\nrefine'\n tendsto_atTop_atTop_of_monotone'\n (fun n m hnm \u21a6\n Finset.sum_mono_set_of_nonneg (fun i \u21a6 Set.indicator_nonneg (fun _ _ \u21a6 zero_le_one) _) (Finset.range_mono hnm))\n _\n[GOAL]\ncase h.mp\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\n\u22a2 \u00acBddAbove (Set.range fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9)\n[PROOFSTEP]\nrintro \u27e8i, h\u27e9\n[GOAL]\ncase h.mp.intro\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni : \u2115\nh : i \u2208 upperBounds (Set.range fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9)\n\u22a2 False\n[PROOFSTEP]\nsimp only [mem_upperBounds, Set.mem_range, forall_exists_index, forall_apply_eq_imp_iff'] at h \n[GOAL]\ncase h.mp.intro\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni : \u2115\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\n\u22a2 False\n[PROOFSTEP]\ninduction' i with k hk\n[GOAL]\ncase h.mp.intro.zero\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni : \u2115\nh\u271d : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 Nat.zero\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8j, hj\u2081, hj\u2082\u27e9 := h\u03c9 1\n[GOAL]\ncase h.mp.intro.zero.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni : \u2115\nh\u271d : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 Nat.zero\nj : \u2115\nhj\u2081 : 1 \u2264 j\nhj\u2082 : \u03c9 \u2208 s j\n\u22a2 False\n[PROOFSTEP]\nrefine' not_lt.2 (h <| j + 1) (lt_of_le_of_lt (Finset.sum_const_zero.symm : 0 = \u2211 k in Finset.range (j + 1), 0).le _)\n[GOAL]\ncase h.mp.intro.zero.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni : \u2115\nh\u271d : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 Nat.zero\nj : \u2115\nhj\u2081 : 1 \u2264 j\nhj\u2082 : \u03c9 \u2208 s j\n\u22a2 \u2211 _x in Finset.range (j + 1), 0 < \u2211 k in Finset.range (j + 1), Set.indicator (s (k + 1)) 1 \u03c9\n[PROOFSTEP]\nrefine'\n Finset.sum_lt_sum (fun m _ \u21a6 Set.indicator_nonneg (fun _ _ \u21a6 zero_le_one) _)\n \u27e8j - 1, Finset.mem_range.2 (lt_of_le_of_lt (Nat.sub_le _ _) j.lt_succ_self), _\u27e9\n[GOAL]\ncase h.mp.intro.zero.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni : \u2115\nh\u271d : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 Nat.zero\nj : \u2115\nhj\u2081 : 1 \u2264 j\nhj\u2082 : \u03c9 \u2208 s j\n\u22a2 0 < Set.indicator (s (j - 1 + 1)) 1 \u03c9\n[PROOFSTEP]\nrw [Nat.sub_add_cancel hj\u2081, Set.indicator_of_mem hj\u2082]\n[GOAL]\ncase h.mp.intro.zero.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni : \u2115\nh\u271d : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 Nat.zero\nj : \u2115\nhj\u2081 : 1 \u2264 j\nhj\u2082 : \u03c9 \u2208 s j\n\u22a2 0 < OfNat.ofNat 1 \u03c9\n[PROOFSTEP]\nexact zero_lt_one\n[GOAL]\ncase h.mp.intro.succ\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni : \u2115\nh\u271d : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\nk : \u2115\nhk : (\u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 k) \u2192 False\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 Nat.succ k\n\u22a2 False\n[PROOFSTEP]\nrw [imp_false] at hk \n[GOAL]\ncase h.mp.intro.succ\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni : \u2115\nh\u271d : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\nk : \u2115\nhk : \u00ac\u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 k\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 Nat.succ k\n\u22a2 False\n[PROOFSTEP]\npush_neg at hk \n[GOAL]\ncase h.mp.intro.succ\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni : \u2115\nh\u271d : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\nk : \u2115\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 Nat.succ k\nhk : \u2203 a, k < \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := hk\n[GOAL]\ncase h.mp.intro.succ.intro\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni\u271d : \u2115\nh\u271d : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\u271d\nk : \u2115\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 Nat.succ k\ni : \u2115\nhi : k < \u2211 k in Finset.range i, Set.indicator (s (k + 1)) 1 \u03c9\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8j, hj\u2081, hj\u2082\u27e9 := h\u03c9 (i + 1)\n[GOAL]\ncase h.mp.intro.succ.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni\u271d : \u2115\nh\u271d : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\u271d\nk : \u2115\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 Nat.succ k\ni : \u2115\nhi : k < \u2211 k in Finset.range i, Set.indicator (s (k + 1)) 1 \u03c9\nj : \u2115\nhj\u2081 : i + 1 \u2264 j\nhj\u2082 : \u03c9 \u2208 s j\n\u22a2 False\n[PROOFSTEP]\nreplace hi : (\u2211 k in Finset.range i, (s (k + 1)).indicator 1 \u03c9) = k + 1 := le_antisymm (h i) hi\n[GOAL]\ncase h.mp.intro.succ.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni\u271d : \u2115\nh\u271d : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\u271d\nk : \u2115\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 Nat.succ k\ni j : \u2115\nhj\u2081 : i + 1 \u2264 j\nhj\u2082 : \u03c9 \u2208 s j\nhi : \u2211 k in Finset.range i, Set.indicator (s (k + 1)) 1 \u03c9 = k + 1\n\u22a2 False\n[PROOFSTEP]\nrefine' not_lt.2 (h <| j + 1) _\n[GOAL]\ncase h.mp.intro.succ.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni\u271d : \u2115\nh\u271d : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\u271d\nk : \u2115\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 Nat.succ k\ni j : \u2115\nhj\u2081 : i + 1 \u2264 j\nhj\u2082 : \u03c9 \u2208 s j\nhi : \u2211 k in Finset.range i, Set.indicator (s (k + 1)) 1 \u03c9 = k + 1\n\u22a2 Nat.succ k < \u2211 k in Finset.range (j + 1), Set.indicator (s (k + 1)) 1 \u03c9\n[PROOFSTEP]\nrw [\u2190 Finset.sum_range_add_sum_Ico _ (i.le_succ.trans (hj\u2081.trans j.le_succ)), hi]\n[GOAL]\ncase h.mp.intro.succ.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni\u271d : \u2115\nh\u271d : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\u271d\nk : \u2115\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 Nat.succ k\ni j : \u2115\nhj\u2081 : i + 1 \u2264 j\nhj\u2082 : \u03c9 \u2208 s j\nhi : \u2211 k in Finset.range i, Set.indicator (s (k + 1)) 1 \u03c9 = k + 1\n\u22a2 Nat.succ k < k + 1 + \u2211 k in Finset.Ico i (Nat.succ j), Set.indicator (s (k + 1)) 1 \u03c9\n[PROOFSTEP]\nrefine' lt_add_of_pos_right _ _\n[GOAL]\ncase h.mp.intro.succ.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni\u271d : \u2115\nh\u271d : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\u271d\nk : \u2115\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 Nat.succ k\ni j : \u2115\nhj\u2081 : i + 1 \u2264 j\nhj\u2082 : \u03c9 \u2208 s j\nhi : \u2211 k in Finset.range i, Set.indicator (s (k + 1)) 1 \u03c9 = k + 1\n\u22a2 0 < \u2211 k in Finset.Ico i (Nat.succ j), Set.indicator (s (k + 1)) 1 \u03c9\n[PROOFSTEP]\nrw [(Finset.sum_const_zero.symm : 0 = \u2211 k in Finset.Ico i (j + 1), 0)]\n[GOAL]\ncase h.mp.intro.succ.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni\u271d : \u2115\nh\u271d : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\u271d\nk : \u2115\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 Nat.succ k\ni j : \u2115\nhj\u2081 : i + 1 \u2264 j\nhj\u2082 : \u03c9 \u2208 s j\nhi : \u2211 k in Finset.range i, Set.indicator (s (k + 1)) 1 \u03c9 = k + 1\n\u22a2 \u2211 _x in Finset.Ico i (j + 1), 0 < \u2211 k in Finset.Ico i (Nat.succ j), Set.indicator (s (k + 1)) 1 \u03c9\n[PROOFSTEP]\nrefine'\n Finset.sum_lt_sum (fun m _ \u21a6 Set.indicator_nonneg (fun _ _ \u21a6 zero_le_one) _)\n \u27e8j - 1,\n Finset.mem_Ico.2\n \u27e8(Nat.le_sub_iff_add_le (le_trans ((le_add_iff_nonneg_left _).2 zero_le') hj\u2081)).2 hj\u2081,\n lt_of_le_of_lt (Nat.sub_le _ _) j.lt_succ_self\u27e9,\n _\u27e9\n[GOAL]\ncase h.mp.intro.succ.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni\u271d : \u2115\nh\u271d : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\u271d\nk : \u2115\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 Nat.succ k\ni j : \u2115\nhj\u2081 : i + 1 \u2264 j\nhj\u2082 : \u03c9 \u2208 s j\nhi : \u2211 k in Finset.range i, Set.indicator (s (k + 1)) 1 \u03c9 = k + 1\n\u22a2 0 < Set.indicator (s (j - 1 + 1)) 1 \u03c9\n[PROOFSTEP]\nrw [Nat.sub_add_cancel (le_trans ((le_add_iff_nonneg_left _).2 zero_le') hj\u2081), Set.indicator_of_mem hj\u2082]\n[GOAL]\ncase h.mp.intro.succ.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\ni\u271d : \u2115\nh\u271d : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\u271d\nk : \u2115\nh : \u2200 (a : \u2115), \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 Nat.succ k\ni j : \u2115\nhj\u2081 : i + 1 \u2264 j\nhj\u2082 : \u03c9 \u2208 s j\nhi : \u2211 k in Finset.range i, Set.indicator (s (k + 1)) 1 \u03c9 = k + 1\n\u22a2 0 < OfNat.ofNat 1 \u03c9\n[PROOFSTEP]\nexact zero_lt_one\n[GOAL]\ncase h.mpr\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\n\u22a2 \u03c9 \u2208 {\u03c9 | Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop} \u2192\n \u2200 (i : \u2115), \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\n[PROOFSTEP]\nrintro h\u03c9 i\n[GOAL]\ncase h.mpr\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u03c9 \u2208 {\u03c9 | Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop}\ni : \u2115\n\u22a2 \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\n[PROOFSTEP]\nrw [Set.mem_setOf_eq, tendsto_atTop_atTop] at h\u03c9 \n[GOAL]\ncase h.mpr\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (b : \u2115), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\ni : \u2115\n\u22a2 \u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\n[PROOFSTEP]\nby_contra hcon\n[GOAL]\ncase h.mpr\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (b : \u2115), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\ni : \u2115\nhcon : \u00ac\u2203 i_1, i \u2264 i_1 \u2227 \u03c9 \u2208 s i_1\n\u22a2 False\n[PROOFSTEP]\npush_neg at hcon \n[GOAL]\ncase h.mpr\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (b : \u2115), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\ni : \u2115\nhcon : \u2200 (i_1 : \u2115), i \u2264 i_1 \u2192 \u00ac\u03c9 \u2208 s i_1\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8j, h\u27e9 := h\u03c9 (i + 1)\n[GOAL]\ncase h.mpr.intro\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (b : \u2115), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\ni : \u2115\nhcon : \u2200 (i_1 : \u2115), i \u2264 i_1 \u2192 \u00ac\u03c9 \u2208 s i_1\nj : \u2115\nh : \u2200 (a : \u2115), j \u2264 a \u2192 i + 1 \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\n\u22a2 False\n[PROOFSTEP]\nhave : (\u2211 k in Finset.range j, (s (k + 1)).indicator 1 \u03c9) \u2264 i :=\n by\n have hle : \u2200 j \u2264 i, (\u2211 k in Finset.range j, (s (k + 1)).indicator 1 \u03c9) \u2264 i :=\n by\n refine' fun j hij \u21a6 (Finset.sum_le_card_nsmul _ _ _ _ : _ \u2264 (Finset.range j).card \u2022 1).trans _\n \u00b7 exact fun m _ \u21a6 Set.indicator_apply_le' (fun _ \u21a6 le_rfl) fun _ \u21a6 zero_le_one\n \u00b7 simpa only [Finset.card_range, smul_eq_mul, mul_one]\n by_cases hij : j < i\n \u00b7 exact hle _ hij.le\n \u00b7 rw [\u2190 Finset.sum_range_add_sum_Ico _ (not_lt.1 hij)]\n suffices (\u2211 k in Finset.Ico i j, (s (k + 1)).indicator 1 \u03c9) = 0\n by\n rw [this, add_zero]\n exact hle _ le_rfl\n refine' Finset.sum_eq_zero fun m hm \u21a6 _\n exact Set.indicator_of_not_mem (hcon _ <| (Finset.mem_Ico.1 hm).1.trans m.le_succ) _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (b : \u2115), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\ni : \u2115\nhcon : \u2200 (i_1 : \u2115), i \u2264 i_1 \u2192 \u00ac\u03c9 \u2208 s i_1\nj : \u2115\nh : \u2200 (a : \u2115), j \u2264 a \u2192 i + 1 \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\n\u22a2 \u2211 k in Finset.range j, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\n[PROOFSTEP]\nhave hle : \u2200 j \u2264 i, (\u2211 k in Finset.range j, (s (k + 1)).indicator 1 \u03c9) \u2264 i :=\n by\n refine' fun j hij \u21a6 (Finset.sum_le_card_nsmul _ _ _ _ : _ \u2264 (Finset.range j).card \u2022 1).trans _\n \u00b7 exact fun m _ \u21a6 Set.indicator_apply_le' (fun _ \u21a6 le_rfl) fun _ \u21a6 zero_le_one\n \u00b7 simpa only [Finset.card_range, smul_eq_mul, mul_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (b : \u2115), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\ni : \u2115\nhcon : \u2200 (i_1 : \u2115), i \u2264 i_1 \u2192 \u00ac\u03c9 \u2208 s i_1\nj : \u2115\nh : \u2200 (a : \u2115), j \u2264 a \u2192 i + 1 \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\n\u22a2 \u2200 (j : \u2115), j \u2264 i \u2192 \u2211 k in Finset.range j, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\n[PROOFSTEP]\nrefine' fun j hij \u21a6 (Finset.sum_le_card_nsmul _ _ _ _ : _ \u2264 (Finset.range j).card \u2022 1).trans _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (b : \u2115), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\ni : \u2115\nhcon : \u2200 (i_1 : \u2115), i \u2264 i_1 \u2192 \u00ac\u03c9 \u2208 s i_1\nj\u271d : \u2115\nh : \u2200 (a : \u2115), j\u271d \u2264 a \u2192 i + 1 \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\nj : \u2115\nhij : j \u2264 i\n\u22a2 \u2200 (x : \u2115), x \u2208 Finset.range j \u2192 Set.indicator (s (x + 1)) 1 \u03c9 \u2264 1\n[PROOFSTEP]\nexact fun m _ \u21a6 Set.indicator_apply_le' (fun _ \u21a6 le_rfl) fun _ \u21a6 zero_le_one\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (b : \u2115), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\ni : \u2115\nhcon : \u2200 (i_1 : \u2115), i \u2264 i_1 \u2192 \u00ac\u03c9 \u2208 s i_1\nj\u271d : \u2115\nh : \u2200 (a : \u2115), j\u271d \u2264 a \u2192 i + 1 \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\nj : \u2115\nhij : j \u2264 i\n\u22a2 Finset.card (Finset.range j) \u2022 1 \u2264 i\n[PROOFSTEP]\nsimpa only [Finset.card_range, smul_eq_mul, mul_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (b : \u2115), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\ni : \u2115\nhcon : \u2200 (i_1 : \u2115), i \u2264 i_1 \u2192 \u00ac\u03c9 \u2208 s i_1\nj : \u2115\nh : \u2200 (a : \u2115), j \u2264 a \u2192 i + 1 \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\nhle : \u2200 (j : \u2115), j \u2264 i \u2192 \u2211 k in Finset.range j, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\n\u22a2 \u2211 k in Finset.range j, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\n[PROOFSTEP]\nby_cases hij : j < i\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (b : \u2115), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\ni : \u2115\nhcon : \u2200 (i_1 : \u2115), i \u2264 i_1 \u2192 \u00ac\u03c9 \u2208 s i_1\nj : \u2115\nh : \u2200 (a : \u2115), j \u2264 a \u2192 i + 1 \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\nhle : \u2200 (j : \u2115), j \u2264 i \u2192 \u2211 k in Finset.range j, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\nhij : j < i\n\u22a2 \u2211 k in Finset.range j, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\n[PROOFSTEP]\nexact hle _ hij.le\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (b : \u2115), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\ni : \u2115\nhcon : \u2200 (i_1 : \u2115), i \u2264 i_1 \u2192 \u00ac\u03c9 \u2208 s i_1\nj : \u2115\nh : \u2200 (a : \u2115), j \u2264 a \u2192 i + 1 \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\nhle : \u2200 (j : \u2115), j \u2264 i \u2192 \u2211 k in Finset.range j, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\nhij : \u00acj < i\n\u22a2 \u2211 k in Finset.range j, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\n[PROOFSTEP]\nrw [\u2190 Finset.sum_range_add_sum_Ico _ (not_lt.1 hij)]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (b : \u2115), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\ni : \u2115\nhcon : \u2200 (i_1 : \u2115), i \u2264 i_1 \u2192 \u00ac\u03c9 \u2208 s i_1\nj : \u2115\nh : \u2200 (a : \u2115), j \u2264 a \u2192 i + 1 \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\nhle : \u2200 (j : \u2115), j \u2264 i \u2192 \u2211 k in Finset.range j, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\nhij : \u00acj < i\n\u22a2 \u2211 k in Finset.range i, Set.indicator (s (k + 1)) 1 \u03c9 + \u2211 k in Finset.Ico i j, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\n[PROOFSTEP]\nsuffices (\u2211 k in Finset.Ico i j, (s (k + 1)).indicator 1 \u03c9) = 0\n by\n rw [this, add_zero]\n exact hle _ le_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (b : \u2115), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\ni : \u2115\nhcon : \u2200 (i_1 : \u2115), i \u2264 i_1 \u2192 \u00ac\u03c9 \u2208 s i_1\nj : \u2115\nh : \u2200 (a : \u2115), j \u2264 a \u2192 i + 1 \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\nhle : \u2200 (j : \u2115), j \u2264 i \u2192 \u2211 k in Finset.range j, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\nhij : \u00acj < i\nthis : \u2211 k in Finset.Ico i j, Set.indicator (s (k + 1)) 1 \u03c9 = 0\n\u22a2 \u2211 k in Finset.range i, Set.indicator (s (k + 1)) 1 \u03c9 + \u2211 k in Finset.Ico i j, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\n[PROOFSTEP]\nrw [this, add_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (b : \u2115), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\ni : \u2115\nhcon : \u2200 (i_1 : \u2115), i \u2264 i_1 \u2192 \u00ac\u03c9 \u2208 s i_1\nj : \u2115\nh : \u2200 (a : \u2115), j \u2264 a \u2192 i + 1 \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\nhle : \u2200 (j : \u2115), j \u2264 i \u2192 \u2211 k in Finset.range j, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\nhij : \u00acj < i\nthis : \u2211 k in Finset.Ico i j, Set.indicator (s (k + 1)) 1 \u03c9 = 0\n\u22a2 \u2211 k in Finset.range i, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\n[PROOFSTEP]\nexact hle _ le_rfl\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (b : \u2115), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\ni : \u2115\nhcon : \u2200 (i_1 : \u2115), i \u2264 i_1 \u2192 \u00ac\u03c9 \u2208 s i_1\nj : \u2115\nh : \u2200 (a : \u2115), j \u2264 a \u2192 i + 1 \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\nhle : \u2200 (j : \u2115), j \u2264 i \u2192 \u2211 k in Finset.range j, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\nhij : \u00acj < i\n\u22a2 \u2211 k in Finset.Ico i j, Set.indicator (s (k + 1)) 1 \u03c9 = 0\n[PROOFSTEP]\nrefine' Finset.sum_eq_zero fun m hm \u21a6 _\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (b : \u2115), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\ni : \u2115\nhcon : \u2200 (i_1 : \u2115), i \u2264 i_1 \u2192 \u00ac\u03c9 \u2208 s i_1\nj : \u2115\nh : \u2200 (a : \u2115), j \u2264 a \u2192 i + 1 \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\nhle : \u2200 (j : \u2115), j \u2264 i \u2192 \u2211 k in Finset.range j, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\nhij : \u00acj < i\nm : \u2115\nhm : m \u2208 Finset.Ico i j\n\u22a2 Set.indicator (s (m + 1)) 1 \u03c9 = 0\n[PROOFSTEP]\nexact Set.indicator_of_not_mem (hcon _ <| (Finset.mem_Ico.1 hm).1.trans m.le_succ) _\n[GOAL]\ncase h.mpr.intro\n\u03b1 : Type u\n\u03b2 : Type v\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nh\u03c9 : \u2200 (b : \u2115), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\ni : \u2115\nhcon : \u2200 (i_1 : \u2115), i \u2264 i_1 \u2192 \u00ac\u03c9 \u2208 s i_1\nj : \u2115\nh : \u2200 (a : \u2115), j \u2264 a \u2192 i + 1 \u2264 \u2211 k in Finset.range a, Set.indicator (s (k + 1)) 1 \u03c9\nthis : \u2211 k in Finset.range j, Set.indicator (s (k + 1)) 1 \u03c9 \u2264 i\n\u22a2 False\n[PROOFSTEP]\nexact not_le.2 (lt_of_lt_of_le i.lt_succ_self <| h _ le_rfl) this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring R\ninst\u271d : Archimedean R\ns : \u2115 \u2192 Set \u03b1\n\u22a2 limsup s atTop = {\u03c9 | Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop}\n[PROOFSTEP]\nrw [limsup_eq_tendsto_sum_indicator_nat_atTop s]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring R\ninst\u271d : Archimedean R\ns : \u2115 \u2192 Set \u03b1\n\u22a2 {\u03c9 | Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop} =\n {\u03c9 | Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop}\n[PROOFSTEP]\next \u03c9\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring R\ninst\u271d : Archimedean R\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\n\u22a2 \u03c9 \u2208 {\u03c9 | Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop} \u2194\n \u03c9 \u2208 {\u03c9 | Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop}\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring R\ninst\u271d : Archimedean R\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\n\u22a2 Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop \u2194\n Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop\n[PROOFSTEP]\nrw [(_ :\n (fun n \u21a6 \u2211 k in Finset.range n, (s (k + 1)).indicator (1 : \u03b1 \u2192 R) \u03c9) = fun n \u21a6\n \u2191(\u2211 k in Finset.range n, (s (k + 1)).indicator (1 : \u03b1 \u2192 \u2115) \u03c9))]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring R\ninst\u271d : Archimedean R\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\n\u22a2 Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop \u2194\n Tendsto (fun n => \u2191(\u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9)) atTop atTop\n[PROOFSTEP]\nexact tendsto_nat_cast_atTop_iff.symm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring R\ninst\u271d : Archimedean R\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\n\u22a2 (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) = fun n =>\n \u2191(\u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9)\n[PROOFSTEP]\next n\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring R\ninst\u271d : Archimedean R\ns : \u2115 \u2192 Set \u03b1\n\u03c9 : \u03b1\nn : \u2115\n\u22a2 \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9 = \u2191(\u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9)\n[PROOFSTEP]\nsimp only [Set.indicator, Pi.one_apply, Finset.sum_boole, Nat.cast_id]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Order.LiminfLimsup", "llama_tokens": 35558, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.3702253856469203, "lm_q1q2_score": 0.2318070649772093}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2\nx : \u03b1\n\u22a2 x \u2208 Dom f \u2194 \u2203 y, y \u2208 f x\n[PROOFSTEP]\nsimp [Dom, Part.dom_iff_mem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nx\u271d : (p : \u03b1 \u2192 Prop) \u00d7 (Subtype p \u2192 \u03b2)\np : \u03b1 \u2192 Prop\nf : Subtype p \u2192 \u03b2\n\u22a2 (fun f => { fst := fun a => (f a).Dom, snd := asSubtype f })\n ((fun f x => { Dom := Sigma.fst f x, get := fun h => Sigma.snd f { val := x, property := h } })\n { fst := p, snd := f }) =\n { fst := p, snd := f }\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nx\u271d : (p : \u03b1 \u2192 Prop) \u00d7 (Subtype p \u2192 \u03b2)\np : \u03b1 \u2192 Prop\nf : Subtype p \u2192 \u03b2\n\u22a2 { fst := fun a => p a, snd := asSubtype fun x => { Dom := p x, get := fun h => f { val := x, property := h } } } =\n { fst := p, snd := f }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2\ns : Set \u03b1\nh : s \u2286 Dom f\na : \u03b1\nb : \u03b2\n\u22a2 b \u2208 restrict f h a \u2194 a \u2208 s \u2227 b \u2208 f a\n[PROOFSTEP]\nsimp [restrict]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nb : \u03b2\n\u22a2 b \u2208 res f s a \u2194 a \u2208 s \u2227 f a = b\n[PROOFSTEP]\nsimp [res, @eq_comm _ b]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 (\u2203 x, f a = b) \u2194 f a = b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\n\u03b1\u271d : Type ?u.16924\nf : \u03b1 \u2192. \u03b1\u271d\n\u22a2 id <$> f = f\n[PROOFSTEP]\nfunext a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\n\u03b1\u271d : Type ?u.16924\nf : \u03b1 \u2192. \u03b1\u271d\na : \u03b1\n\u22a2 (id <$> f) a = f a\n[PROOFSTEP]\ndsimp [Functor.map, PFun.map]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\n\u03b1\u271d : Type ?u.16924\nf : \u03b1 \u2192. \u03b1\u271d\na : \u03b1\n\u22a2 Part.map id (f a) = f a\n[PROOFSTEP]\ncases f a\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\n\u03b1\u271d : Type ?u.16924\nf : \u03b1 \u2192. \u03b1\u271d\na : \u03b1\nDom\u271d : Prop\nget\u271d : Dom\u271d \u2192 \u03b1\u271d\n\u22a2 Part.map id { Dom := Dom\u271d, get := get\u271d } = { Dom := Dom\u271d, get := get\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 fix f a\n\u22a2 (f a).Dom\n[PROOFSTEP]\nlet \u27e8h\u2081, h\u2082\u27e9 := Part.mem_assert_iff.1 h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 fix f a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\u2081\n\u22a2 (f a).Dom\n[PROOFSTEP]\nrw [WellFounded.fixFEq] at h\u2082 \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 fix f a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 :\n b \u2208\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' =>\n (fun y p =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n y (_ : Acc (fun y x => Sum.inr y \u2208 f x) y))\n a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a')\n\u22a2 (f a).Dom\n[PROOFSTEP]\nexact h\u2082.fst.fst\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 fix f a\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 fix f a'\n[PROOFSTEP]\nlet \u27e8h\u2081, h\u2082\u27e9 := Part.mem_assert_iff.1 h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 fix f a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\u2081\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 fix f a'\n[PROOFSTEP]\nrw [WellFounded.fixFEq] at h\u2082 \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 fix f a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 :\n b \u2208\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' =>\n (fun y p =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n y (_ : Acc (fun y x => Sum.inr y \u2208 f x) y))\n a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a')\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 fix f a'\n[PROOFSTEP]\nsimp at h\u2082 \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 fix f a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 :\n \u2203 h,\n b \u2208\n match e : Part.get (f a) h with\n | Sum.inl b => Part.some b\n | Sum.inr a' =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' (_ : Acc (fun y x => Sum.inr y \u2208 f x) a')\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 fix f a'\n[PROOFSTEP]\ncases' h\u2082 with h\u2082 h\u2083\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 fix f a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\nh\u2083 :\n b \u2208\n match e : Part.get (f a) h\u2082 with\n | Sum.inl b => Part.some b\n | Sum.inr a' =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' (_ : Acc (fun y x => Sum.inr y \u2208 f x) a')\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 fix f a'\n[PROOFSTEP]\nsplit at h\u2083 \n[GOAL]\ncase intro.h_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 fix f a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\nb\u271d : \u03b2\nheq\u271d : Part.get (f a) h\u2082 = Sum.inl b\u271d\nh\u2083 : b \u2208 Part.some b\u271d\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 fix f a'\ncase intro.h_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 fix f a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\na'\u271d : \u03b1\nheq\u271d : Part.get (f a) h\u2082 = Sum.inr a'\u271d\nh\u2083 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a'\u271d (_ : Acc (fun y x => Sum.inr y \u2208 f x) a'\u271d)\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 fix f a'\n[PROOFSTEP]\nnext e => simp at h\u2083 ; subst b; refine' Or.inl \u27e8h\u2082, e\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 fix f a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\nb\u271d : \u03b2\ne : Part.get (f a) h\u2082 = Sum.inl b\u271d\nh\u2083 : b \u2208 Part.some b\u271d\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 fix f a'\n[PROOFSTEP]\nsimp at h\u2083 \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 fix f a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\nb\u271d : \u03b2\ne : Part.get (f a) h\u2082 = Sum.inl b\u271d\nh\u2083 : b = b\u271d\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 fix f a'\n[PROOFSTEP]\nsubst b\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\nb\u271d : \u03b2\ne : Part.get (f a) h\u2082 = Sum.inl b\u271d\nh : b\u271d \u2208 fix f a\n\u22a2 Sum.inl b\u271d \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b\u271d \u2208 fix f a'\n[PROOFSTEP]\nrefine' Or.inl \u27e8h\u2082, e\u27e9\n[GOAL]\ncase intro.h_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 fix f a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\na'\u271d : \u03b1\nheq\u271d : Part.get (f a) h\u2082 = Sum.inr a'\u271d\nh\u2083 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a'\u271d (_ : Acc (fun y x => Sum.inr y \u2208 f x) a'\u271d)\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 fix f a'\n[PROOFSTEP]\nnext e => exact Or.inr \u27e8_, \u27e8_, e\u27e9, Part.mem_assert _ h\u2083\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 fix f a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\na'\u271d : \u03b1\ne : Part.get (f a) h\u2082 = Sum.inr a'\u271d\nh\u2083 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a'\u271d (_ : Acc (fun y x => Sum.inr y \u2208 f x) a'\u271d)\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 fix f a'\n[PROOFSTEP]\nexact Or.inr \u27e8_, \u27e8_, e\u27e9, Part.mem_assert _ h\u2083\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 fix f a'\n\u22a2 b \u2208 fix f a\n[PROOFSTEP]\nsimp [fix]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 fix f a'\n\u22a2 \u2203 h,\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\n[PROOFSTEP]\nrcases h with (\u27e8h\u2081, h\u2082\u27e9 | \u27e8a', h, h\u2083\u27e9)\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inl b\n\u22a2 \u2203 h,\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\n[PROOFSTEP]\nrefine' \u27e8\u27e8_, fun y h' => _\u27e9, _\u27e9\n[GOAL]\ncase inl.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inl b\ny : \u03b1\nh' : Sum.inr y \u2208 f a\n\u22a2 Acc (fun x y => Sum.inr x \u2208 f y) y\n[PROOFSTEP]\ninjection Part.mem_unique \u27e8h\u2081, h\u2082\u27e9 h'\n[GOAL]\ncase inl.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inl b\n\u22a2 b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a (_ : Acc (fun x y => Sum.inr x \u2208 f y) a)\n[PROOFSTEP]\nrw [WellFounded.fixFEq]\n -- Porting note: used to be simp [h\u2081, h\u2082]\n[GOAL]\ncase inl.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inl b\n\u22a2 b \u2208\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' =>\n (fun y p =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n y (_ : Acc (fun y x => Sum.inr y \u2208 f x) y))\n a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a')\n[PROOFSTEP]\napply Part.mem_assert h\u2081\n[GOAL]\ncase inl.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inl b\n\u22a2 b \u2208\n match e : Part.get (f a) h\u2081 with\n | Sum.inl b => Part.some b\n | Sum.inr a' =>\n (fun y p =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n y (_ : Acc (fun y x => Sum.inr y \u2208 f x) y))\n a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a')\n[PROOFSTEP]\nsplit\n[GOAL]\ncase inl.intro.refine'_2.h_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inl b\nb\u271d : \u03b2\nheq\u271d : Part.get (f a) h\u2081 = Sum.inl b\u271d\n\u22a2 b \u2208 Part.some b\u271d\ncase inl.intro.refine'_2.h_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inl b\na'\u271d : \u03b1\nheq\u271d : Part.get (f a) h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n (fun y p =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n y (_ : Acc (fun y x => Sum.inr y \u2208 f x) y))\n a'\u271d (_ : \u2203 h, Part.get (f a) h = Sum.inr a'\u271d)\n[PROOFSTEP]\nnext e => injection h\u2082.symm.trans e with h; simp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inl b\nb\u271d : \u03b2\ne : Part.get (f a) h\u2081 = Sum.inl b\u271d\n\u22a2 b \u2208 Part.some b\u271d\n[PROOFSTEP]\ninjection h\u2082.symm.trans e with h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inl b\nb\u271d : \u03b2\ne : Part.get (f a) h\u2081 = Sum.inl b\u271d\nh : b = b\u271d\n\u22a2 b \u2208 Part.some b\u271d\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase inl.intro.refine'_2.h_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inl b\na'\u271d : \u03b1\nheq\u271d : Part.get (f a) h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n (fun y p =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n y (_ : Acc (fun y x => Sum.inr y \u2208 f x) y))\n a'\u271d (_ : \u2203 h, Part.get (f a) h = Sum.inr a'\u271d)\n[PROOFSTEP]\nnext e => injection h\u2082.symm.trans e\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inl b\na'\u271d : \u03b1\ne : Part.get (f a) h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n (fun y p =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n y (_ : Acc (fun y x => Sum.inr y \u2208 f x) y))\n a'\u271d (_ : \u2203 h, Part.get (f a) h = Sum.inr a'\u271d)\n[PROOFSTEP]\ninjection h\u2082.symm.trans e\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 : b \u2208 fix f a'\n\u22a2 \u2203 h,\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\n[PROOFSTEP]\nsimp [fix] at h\u2083 \n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 :\n \u2203 h,\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' h\n\u22a2 \u2203 h,\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\n[PROOFSTEP]\ncases' h\u2083 with h\u2083 h\u2084\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' h\u2083\n\u22a2 \u2203 h,\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\n[PROOFSTEP]\nrefine' \u27e8\u27e8_, fun y h' => _\u27e9, _\u27e9\n[GOAL]\ncase inr.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' h\u2083\ny : \u03b1\nh' : Sum.inr y \u2208 f a\n\u22a2 Acc (fun x y => Sum.inr x \u2208 f y) y\n[PROOFSTEP]\ninjection Part.mem_unique h h' with e\n[GOAL]\ncase inr.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' h\u2083\ny : \u03b1\nh' : Sum.inr y \u2208 f a\ne : a' = y\n\u22a2 Acc (fun x y => Sum.inr x \u2208 f y) y\n[PROOFSTEP]\nexact e \u25b8 h\u2083\n[GOAL]\ncase inr.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' h\u2083\n\u22a2 b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a (_ : Acc (fun x y => Sum.inr x \u2208 f y) a)\n[PROOFSTEP]\ncases' h with h\u2081 h\u2082\n[GOAL]\ncase inr.intro.intro.intro.refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inr a'\n\u22a2 b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a (_ : Acc (fun x y => Sum.inr x \u2208 f y) a)\n[PROOFSTEP]\nrw [WellFounded.fixFEq]\n -- Porting note: used to be simp [h\u2081, h\u2082, h\u2084]\n[GOAL]\ncase inr.intro.intro.intro.refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inr a'\n\u22a2 b \u2208\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a'_1 =>\n (fun y p =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n y (_ : Acc (fun y x => Sum.inr y \u2208 f x) y))\n a'_1 (_ : \u2203 h, Part.get (f a) h = Sum.inr a'_1)\n[PROOFSTEP]\napply Part.mem_assert h\u2081\n[GOAL]\ncase inr.intro.intro.intro.refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inr a'\n\u22a2 b \u2208\n match e : Part.get (f a) h\u2081 with\n | Sum.inl b => Part.some b\n | Sum.inr a'_1 =>\n (fun y p =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n y (_ : Acc (fun y x => Sum.inr y \u2208 f x) y))\n a'_1 (_ : \u2203 h, Part.get (f a) h = Sum.inr a'_1)\n[PROOFSTEP]\nsplit\n[GOAL]\ncase inr.intro.intro.intro.refine'_2.intro.h_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inr a'\nb\u271d : \u03b2\nheq\u271d : Part.get (f a) h\u2081 = Sum.inl b\u271d\n\u22a2 b \u2208 Part.some b\u271d\ncase inr.intro.intro.intro.refine'_2.intro.h_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inr a'\na'\u271d : \u03b1\nheq\u271d : Part.get (f a) h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n (fun y p =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n y (_ : Acc (fun y x => Sum.inr y \u2208 f x) y))\n a'\u271d (_ : \u2203 h, Part.get (f a) h = Sum.inr a'\u271d)\n[PROOFSTEP]\nnext e => injection h\u2082.symm.trans e\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inr a'\nb\u271d : \u03b2\ne : Part.get (f a) h\u2081 = Sum.inl b\u271d\n\u22a2 b \u2208 Part.some b\u271d\n[PROOFSTEP]\ninjection h\u2082.symm.trans e\n[GOAL]\ncase inr.intro.intro.intro.refine'_2.intro.h_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inr a'\na'\u271d : \u03b1\nheq\u271d : Part.get (f a) h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n (fun y p =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n y (_ : Acc (fun y x => Sum.inr y \u2208 f x) y))\n a'\u271d (_ : \u2203 h, Part.get (f a) h = Sum.inr a'\u271d)\n[PROOFSTEP]\nnext e => injection h\u2082.symm.trans e; subst a'; exact h\u2084\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inr a'\na'\u271d : \u03b1\ne : Part.get (f a) h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n (fun y p =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n y (_ : Acc (fun y x => Sum.inr y \u2208 f x) y))\n a'\u271d (_ : \u2203 h, Part.get (f a) h = Sum.inr a'\u271d)\n[PROOFSTEP]\ninjection h\u2082.symm.trans e\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : Part.get (f a) h\u2081 = Sum.inr a'\na'\u271d : \u03b1\ne : Part.get (f a) h\u2081 = Sum.inr a'\u271d\nval_eq\u271d : a' = a'\u271d\n\u22a2 b \u2208\n (fun y p =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n y (_ : Acc (fun y x => Sum.inr y \u2208 f x) y))\n a'\u271d (_ : \u2203 h, Part.get (f a) h = Sum.inr a'\u271d)\n[PROOFSTEP]\nsubst a'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\na'\u271d : \u03b1\ne : Part.get (f a) h\u2081 = Sum.inr a'\u271d\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\u271d\nh\u2084 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a'\u271d h\u2083\nh\u2082 : Part.get (f a) h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n (fun y p =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n y (_ : Acc (fun y x => Sum.inr y \u2208 f x) y))\n a'\u271d (_ : \u2203 h, Part.get (f a) h = Sum.inr a'\u271d)\n[PROOFSTEP]\nexact h\u2084\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nhb : Sum.inl b \u2208 f a\n\u22a2 b \u2208 fix f a\n[PROOFSTEP]\nrw [PFun.mem_fix_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nhb : Sum.inl b \u2208 f a\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 fix f a'\n[PROOFSTEP]\nexact Or.inl hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na a' : \u03b1\nha' : Sum.inr a' \u2208 f a\n\u22a2 fix f a = fix f a'\n[PROOFSTEP]\next b\n[GOAL]\ncase H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na a' : \u03b1\nha' : Sum.inr a' \u2208 f a\nb : \u03b2\n\u22a2 b \u2208 fix f a \u2194 b \u2208 fix f a'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase H.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na a' : \u03b1\nha' : Sum.inr a' \u2208 f a\nb : \u03b2\n\u22a2 b \u2208 fix f a \u2192 b \u2208 fix f a'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase H.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na a' : \u03b1\nha' : Sum.inr a' \u2208 f a\nb : \u03b2\nh : b \u2208 fix f a\n\u22a2 b \u2208 fix f a'\n[PROOFSTEP]\nobtain h' | \u27e8a, h', e'\u27e9 := mem_fix_iff.1 h\n[GOAL]\ncase H.mp.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na a' : \u03b1\nha' : Sum.inr a' \u2208 f a\nb : \u03b2\nh : b \u2208 fix f a\nh' : Sum.inl b \u2208 f a\n\u22a2 b \u2208 fix f a'\n[PROOFSTEP]\ncases Part.mem_unique ha' h'\n[GOAL]\ncase H.mp.inr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na\u271d a' : \u03b1\nha' : Sum.inr a' \u2208 f a\u271d\nb : \u03b2\nh : b \u2208 fix f a\u271d\na : \u03b1\nh' : Sum.inr a \u2208 f a\u271d\ne' : b \u2208 fix f a\n\u22a2 b \u2208 fix f a'\n[PROOFSTEP]\ncases Part.mem_unique ha' h'\n[GOAL]\ncase H.mp.inr.intro.intro.refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na a' : \u03b1\nha' : Sum.inr a' \u2208 f a\nb : \u03b2\nh : b \u2208 fix f a\nh' : Sum.inr a' \u2208 f a\ne' : b \u2208 fix f a'\n\u22a2 b \u2208 fix f a'\n[PROOFSTEP]\nexact e'\n[GOAL]\ncase H.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na a' : \u03b1\nha' : Sum.inr a' \u2208 f a\nb : \u03b2\n\u22a2 b \u2208 fix f a' \u2192 b \u2208 fix f a\n[PROOFSTEP]\nintro h\n[GOAL]\ncase H.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na a' : \u03b1\nha' : Sum.inr a' \u2208 f a\nb : \u03b2\nh : b \u2208 fix f a'\n\u22a2 b \u2208 fix f a\n[PROOFSTEP]\nrw [PFun.mem_fix_iff]\n[GOAL]\ncase H.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na a' : \u03b1\nha' : Sum.inr a' \u2208 f a\nb : \u03b2\nh : b \u2208 fix f a'\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 fix f a'\n[PROOFSTEP]\nexact Or.inr \u27e8a', ha', h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na a' : \u03b1\nhb : b \u2208 fix f a\nha' : Sum.inr a' \u2208 f a\n\u22a2 b \u2208 fix f a'\n[PROOFSTEP]\nrwa [\u2190 fix_fwd_eq ha']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\n\u22a2 C a\n[PROOFSTEP]\nhave h\u2082 := (Part.mem_assert_iff.1 h).snd\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\nh\u2082 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a (_ : Acc (fun x y => Sum.inr x \u2208 f y) a)\n\u22a2 C a\n[PROOFSTEP]\nrevert h\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\n\u22a2 b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a (_ : Acc (fun x y => Sum.inr x \u2208 f y) a) \u2192\n C a\n[PROOFSTEP]\ngeneralize_proofs h\u2081\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\n\u22a2 b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\u2081 \u2192\n C a\n[PROOFSTEP]\nintro h\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\u2081\n\u22a2 C a\n[PROOFSTEP]\nclear h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\u2081\n\u22a2 C a\n[PROOFSTEP]\ninduction' h\u2081 with a ha IH\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na\u271d : \u03b1\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\u271d\nh\u2082\u271d :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a\u271d h\u2081\na : \u03b1\nha : \u2200 (y : \u03b1), Sum.inr y \u2208 f a \u2192 Acc (fun x y => Sum.inr x \u2208 f y) y\nIH :\n (y : \u03b1) \u2192\n (a : Sum.inr y \u2208 f a) \u2192\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n y (_ : Acc ?m.29240 y) \u2192\n C y\nh\u2082 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a (_ : Acc (fun x y => Sum.inr x \u2208 f y) a)\n\u22a2 C a\n[PROOFSTEP]\nhave h : b \u2208 f.fix a := Part.mem_assert_iff.2 \u27e8\u27e8a, ha\u27e9, h\u2082\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na\u271d : \u03b1\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\u271d\nh\u2082\u271d :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a\u271d h\u2081\na : \u03b1\nha : \u2200 (y : \u03b1), Sum.inr y \u2208 f a \u2192 Acc (fun x y => Sum.inr x \u2208 f y) y\nIH :\n (y : \u03b1) \u2192\n (a : Sum.inr y \u2208 f a) \u2192\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n y (_ : Acc ?m.29240 y) \u2192\n C y\nh\u2082 :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a (_ : Acc (fun x y => Sum.inr x \u2208 f y) a)\nh : b \u2208 fix f a\n\u22a2 C a\n[PROOFSTEP]\nexact H a h fun a' fa' => IH a' fa' (Part.mem_assert_iff.1 (fix_fwd h fa')).snd\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\n\u22a2 fixInduction h H = H a h fun a' h' => fixInduction (_ : b \u2208 fix f a') H\n[PROOFSTEP]\nunfold fixInduction\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\n\u22a2 (let_fun h\u2082 :=\n (_ :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a (_ : Acc (fun x y => Sum.inr x \u2208 f y) a));\n Acc.rec (motive := fun {a} h\u2081 =>\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\u2081 \u2192\n C a)\n (fun a ha IH h\u2082 =>\n let_fun h :=\n (_ :\n b \u2208\n Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h);\n H a h fun a' fa' =>\n IH a' fa'\n (_ :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n (_ : Acc (fun x y => Sum.inr x \u2208 f y) a) h\u2082) =\n H a h fun a' h' =>\n let_fun h\u2082 :=\n (_ :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a'));\n Acc.rec (motive := fun {a} h\u2081 =>\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\u2081 \u2192\n (fun x => C x) a)\n (fun a ha IH h\u2082 =>\n let_fun h :=\n (_ :\n b \u2208\n Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h);\n H a h fun a' fa' =>\n IH a' fa'\n (_ :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n (_ : Acc (fun x y => Sum.inr x \u2208 f y) a') h\u2082\n[PROOFSTEP]\ngeneralize (Part.mem_assert_iff.1 h).fst = ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\nha : Acc (fun x y => Sum.inr x \u2208 f y) a\n\u22a2 (let_fun h\u2082 :=\n (_ :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a (_ : Acc (fun x y => Sum.inr x \u2208 f y) a));\n Acc.rec (motive := fun {a} h\u2081 =>\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\u2081 \u2192\n C a)\n (fun a ha IH h\u2082 =>\n let_fun h :=\n (_ :\n b \u2208\n Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h);\n H a h fun a' fa' =>\n IH a' fa'\n (_ :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n ha h\u2082) =\n H a h fun a' h' =>\n let_fun h\u2082 :=\n (_ :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a'));\n Acc.rec (motive := fun {a} h\u2081 =>\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\u2081 \u2192\n (fun x => C x) a)\n (fun a ha IH h\u2082 =>\n let_fun h :=\n (_ :\n b \u2208\n Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h);\n H a h fun a' fa' =>\n IH a' fa'\n (_ :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n (_ : Acc (fun x y => Sum.inr x \u2208 f y) a') h\u2082\n[PROOFSTEP]\ninduction ha\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nH : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a'\nx\u271d : \u03b1\nh\u271d : \u2200 (y : \u03b1), Sum.inr y \u2208 f x\u271d \u2192 Acc (fun x y => Sum.inr x \u2208 f y) y\nh_ih\u271d :\n \u2200 (y : \u03b1) (a : Sum.inr y \u2208 f x\u271d) (h : b \u2208 fix f y),\n (let_fun h\u2082 :=\n (_ :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n y (_ : Acc (fun x y => Sum.inr x \u2208 f y) y));\n Acc.rec (motive := fun {a} h\u2081 =>\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\u2081 \u2192\n C a)\n (fun a ha IH h\u2082 =>\n let_fun h :=\n (_ :\n b \u2208\n Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h);\n H a h fun a' fa' =>\n IH a' fa'\n (_ :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n (_ : Acc ?m.30282 y) h\u2082) =\n H y h fun a' h' =>\n let_fun h\u2082 :=\n (_ :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a'));\n Acc.rec (motive := fun {a} h\u2081 =>\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\u2081 \u2192\n (fun x => C x) a)\n (fun a ha IH h\u2082 =>\n let_fun h :=\n (_ :\n b \u2208\n Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h);\n H a h fun a' fa' =>\n IH a' fa'\n (_ :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n (_ : Acc (fun x y => Sum.inr x \u2208 f y) a') h\u2082\nh : b \u2208 fix f x\u271d\n\u22a2 (let_fun h\u2082 :=\n (_ :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n x\u271d (_ : Acc (fun x y => Sum.inr x \u2208 f y) x\u271d));\n Acc.rec (motive := fun {a} h\u2081 =>\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\u2081 \u2192\n C a)\n (fun a ha IH h\u2082 =>\n let_fun h :=\n (_ :\n b \u2208\n Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h);\n H a h fun a' fa' =>\n IH a' fa'\n (_ :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n (_ : Acc (fun x y => Sum.inr x \u2208 f y) x\u271d) h\u2082) =\n H x\u271d h fun a' h' =>\n let_fun h\u2082 :=\n (_ :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a'));\n Acc.rec (motive := fun {a} h\u2081 =>\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h\u2081 \u2192\n (fun x => C x) a)\n (fun a ha IH h\u2082 =>\n let_fun h :=\n (_ :\n b \u2208\n Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a h);\n H a h fun a' fa' =>\n IH a' fa'\n (_ :\n b \u2208\n WellFounded.fixF\n (fun a IH =>\n Part.assert (f a).Dom fun hf =>\n match e : Part.get (f a) hf with\n | Sum.inl b => Part.some b\n | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))\n a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))\n (_ : Acc (fun x y => Sum.inr x \u2208 f y) a') h\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nhbase : (a_final : \u03b1) \u2192 Sum.inl b \u2208 f a_final \u2192 C a_final\nhind : (a\u2080 a\u2081 : \u03b1) \u2192 b \u2208 fix f a\u2081 \u2192 Sum.inr a\u2081 \u2208 f a\u2080 \u2192 C a\u2081 \u2192 C a\u2080\n\u22a2 C a\n[PROOFSTEP]\nrefine' fixInduction h fun a' h ih => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh\u271d : b \u2208 fix f a\nhbase : (a_final : \u03b1) \u2192 Sum.inl b \u2208 f a_final \u2192 C a_final\nhind : (a\u2080 a\u2081 : \u03b1) \u2192 b \u2208 fix f a\u2081 \u2192 Sum.inr a\u2081 \u2208 f a\u2080 \u2192 C a\u2081 \u2192 C a\u2080\na' : \u03b1\nh : b \u2208 fix f a'\nih : (a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a''\n\u22a2 C a'\n[PROOFSTEP]\nrcases e : (f a').get (dom_of_mem_fix h) with b' | a''\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh\u271d : b \u2208 fix f a\nhbase : (a_final : \u03b1) \u2192 Sum.inl b \u2208 f a_final \u2192 C a_final\nhind : (a\u2080 a\u2081 : \u03b1) \u2192 b \u2208 fix f a\u2081 \u2192 Sum.inr a\u2081 \u2208 f a\u2080 \u2192 C a\u2081 \u2192 C a\u2080\na' : \u03b1\nh : b \u2208 fix f a'\nih : (a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a''\nb' : \u03b2\ne : Part.get (f a') (_ : (f a').Dom) = Sum.inl b'\n\u22a2 C a'\n[PROOFSTEP]\nreplace e : _ \u2208 f a' := \u27e8_, e\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh\u271d : b \u2208 fix f a\nhbase : (a_final : \u03b1) \u2192 Sum.inl b \u2208 f a_final \u2192 C a_final\nhind : (a\u2080 a\u2081 : \u03b1) \u2192 b \u2208 fix f a\u2081 \u2192 Sum.inr a\u2081 \u2208 f a\u2080 \u2192 C a\u2081 \u2192 C a\u2080\na' : \u03b1\nh : b \u2208 fix f a'\nih : (a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a''\na'' : \u03b1\ne : Part.get (f a') (_ : (f a').Dom) = Sum.inr a''\n\u22a2 C a'\n[PROOFSTEP]\nreplace e : _ \u2208 f a' := \u27e8_, e\u27e9\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh\u271d : b \u2208 fix f a\nhbase : (a_final : \u03b1) \u2192 Sum.inl b \u2208 f a_final \u2192 C a_final\nhind : (a\u2080 a\u2081 : \u03b1) \u2192 b \u2208 fix f a\u2081 \u2192 Sum.inr a\u2081 \u2208 f a\u2080 \u2192 C a\u2081 \u2192 C a\u2080\na' : \u03b1\nh : b \u2208 fix f a'\nih : (a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a''\nb' : \u03b2\ne : Sum.inl b' \u2208 f a'\n\u22a2 C a'\n[PROOFSTEP]\napply hbase\n[GOAL]\ncase inl.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh\u271d : b \u2208 fix f a\nhbase : (a_final : \u03b1) \u2192 Sum.inl b \u2208 f a_final \u2192 C a_final\nhind : (a\u2080 a\u2081 : \u03b1) \u2192 b \u2208 fix f a\u2081 \u2192 Sum.inr a\u2081 \u2208 f a\u2080 \u2192 C a\u2081 \u2192 C a\u2080\na' : \u03b1\nh : b \u2208 fix f a'\nih : (a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a''\nb' : \u03b2\ne : Sum.inl b' \u2208 f a'\n\u22a2 Sum.inl b \u2208 f a'\n[PROOFSTEP]\nconvert e\n[GOAL]\ncase h.e'_4.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh\u271d : b \u2208 fix f a\nhbase : (a_final : \u03b1) \u2192 Sum.inl b \u2208 f a_final \u2192 C a_final\nhind : (a\u2080 a\u2081 : \u03b1) \u2192 b \u2208 fix f a\u2081 \u2192 Sum.inr a\u2081 \u2208 f a\u2080 \u2192 C a\u2081 \u2192 C a\u2080\na' : \u03b1\nh : b \u2208 fix f a'\nih : (a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a''\nb' : \u03b2\ne : Sum.inl b' \u2208 f a'\n\u22a2 b = b'\n[PROOFSTEP]\nexact Part.mem_unique h (fix_stop e)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh\u271d : b \u2208 fix f a\nhbase : (a_final : \u03b1) \u2192 Sum.inl b \u2208 f a_final \u2192 C a_final\nhind : (a\u2080 a\u2081 : \u03b1) \u2192 b \u2208 fix f a\u2081 \u2192 Sum.inr a\u2081 \u2208 f a\u2080 \u2192 C a\u2081 \u2192 C a\u2080\na' : \u03b1\nh : b \u2208 fix f a'\nih : (a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a''\na'' : \u03b1\ne : Sum.inr a'' \u2208 f a'\n\u22a2 C a'\n[PROOFSTEP]\nexact hind _ _ (fix_fwd h e) e (ih _ e)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nfa : Sum.inl b \u2208 f a\nhbase : (a_final : \u03b1) \u2192 Sum.inl b \u2208 f a_final \u2192 C a_final\nhind : (a\u2080 a\u2081 : \u03b1) \u2192 b \u2208 fix f a\u2081 \u2192 Sum.inr a\u2081 \u2208 f a\u2080 \u2192 C a\u2081 \u2192 C a\u2080\n\u22a2 fixInduction' h hbase hind = hbase a fa\n[PROOFSTEP]\nunfold fixInduction'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nfa : Sum.inl b \u2208 f a\nhbase : (a_final : \u03b1) \u2192 Sum.inl b \u2208 f a_final \u2192 C a_final\nhind : (a\u2080 a\u2081 : \u03b1) \u2192 b \u2208 fix f a\u2081 \u2192 Sum.inr a\u2081 \u2208 f a\u2080 \u2192 C a\u2081 \u2192 C a\u2080\n\u22a2 (fixInduction h fun a' h ih =>\n Sum.casesOn (motive := fun x => Part.get (f a') (_ : (f a').Dom) = x \u2192 C a') (Part.get (f a') (_ : (f a').Dom))\n (fun b' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inl b');\n hbase a' (_ : Sum.inl b \u2208 f a'))\n (fun a'' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inr a'');\n hind a' a'' (_ : b \u2208 fix f a'') e (ih a'' e))\n (_ : Part.get (f a') (_ : (f a').Dom) = Part.get (f a') (_ : (f a').Dom))) =\n hbase a fa\n[PROOFSTEP]\nrw [fixInduction_spec]\n -- Porting note: the explicit motive required because `simp` behaves differently\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nfa : Sum.inl b \u2208 f a\nhbase : (a_final : \u03b1) \u2192 Sum.inl b \u2208 f a_final \u2192 C a_final\nhind : (a\u2080 a\u2081 : \u03b1) \u2192 b \u2208 fix f a\u2081 \u2192 Sum.inr a\u2081 \u2208 f a\u2080 \u2192 C a\u2081 \u2192 C a\u2080\n\u22a2 Sum.casesOn (motive := fun x => Part.get (f a) (_ : (f a).Dom) = x \u2192 C a) (Part.get (f a) (_ : (f a).Dom))\n (fun b' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a) h = Sum.inl b');\n hbase a (_ : Sum.inl b \u2208 f a))\n (fun a'' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a) h = Sum.inr a'');\n hind a a'' (_ : b \u2208 fix f a'') e\n ((fun a' h' =>\n fixInduction (_ : b \u2208 fix f a') fun a' h ih =>\n Sum.casesOn (motive := fun x => Part.get (f a') (_ : (f a').Dom) = x \u2192 C a')\n (Part.get (f a') (_ : (f a').Dom))\n (fun b' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inl b');\n hbase a' (_ : Sum.inl b \u2208 f a'))\n (fun a'' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inr a'');\n hind a' a'' (_ : b \u2208 fix f a'') e (ih a'' e))\n (_ : Part.get (f a') (_ : (f a').Dom) = Part.get (f a') (_ : (f a').Dom)))\n a'' e))\n (_ : Part.get (f a) (_ : (f a).Dom) = Part.get (f a) (_ : (f a).Dom)) =\n hbase a fa\n[PROOFSTEP]\nrefine'\n Eq.rec (motive := fun x e =>\n Sum.casesOn (motive := fun y => (f a).get (dom_of_mem_fix h) = y \u2192 C a) x _ _\n (Eq.trans (Part.get_eq_of_mem fa (dom_of_mem_fix h)) e) =\n hbase a fa)\n _ (Part.get_eq_of_mem fa (dom_of_mem_fix h)).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na : \u03b1\nh : b \u2208 fix f a\nfa : Sum.inl b \u2208 f a\nhbase : (a_final : \u03b1) \u2192 Sum.inl b \u2208 f a_final \u2192 C a_final\nhind : (a\u2080 a\u2081 : \u03b1) \u2192 b \u2208 fix f a\u2081 \u2192 Sum.inr a\u2081 \u2208 f a\u2080 \u2192 C a\u2081 \u2192 C a\u2080\n\u22a2 (fun x e =>\n Sum.casesOn (motive := fun y => Part.get (f a) (_ : (f a).Dom) = y \u2192 C a) x\n (fun val =>\n (fun b' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a) h = Sum.inl b');\n hbase a (_ : Sum.inl b \u2208 f a))\n val)\n (fun val =>\n (fun a'' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a) h = Sum.inr a'');\n hind a a'' (_ : b \u2208 fix f a'') e\n ((fun a' h' =>\n fixInduction (_ : b \u2208 fix f a') fun a' h ih =>\n Sum.casesOn (motive := fun x => Part.get (f a') (_ : (f a').Dom) = x \u2192 C a')\n (Part.get (f a') (_ : (f a').Dom))\n (fun b' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inl b');\n hbase a' (_ : Sum.inl b \u2208 f a'))\n (fun a'' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inr a'');\n hind a' a'' (_ : b \u2208 fix f a'') e (ih a'' e))\n (_ : Part.get (f a') (_ : (f a').Dom) = Part.get (f a') (_ : (f a').Dom)))\n a'' e))\n val)\n (_ : Part.get (f a) (_ : (f a).Dom) = x) =\n hbase a fa)\n (Sum.inl b) (_ : Sum.inl b = Sum.inl b)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na a' : \u03b1\nh : b \u2208 fix f a\nh' : b \u2208 fix f a'\nfa : Sum.inr a' \u2208 f a\nhbase : (a_final : \u03b1) \u2192 Sum.inl b \u2208 f a_final \u2192 C a_final\nhind : (a\u2080 a\u2081 : \u03b1) \u2192 b \u2208 fix f a\u2081 \u2192 Sum.inr a\u2081 \u2208 f a\u2080 \u2192 C a\u2081 \u2192 C a\u2080\n\u22a2 fixInduction' h hbase hind = hind a a' h' fa (fixInduction' h' hbase hind)\n[PROOFSTEP]\nunfold fixInduction'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na a' : \u03b1\nh : b \u2208 fix f a\nh' : b \u2208 fix f a'\nfa : Sum.inr a' \u2208 f a\nhbase : (a_final : \u03b1) \u2192 Sum.inl b \u2208 f a_final \u2192 C a_final\nhind : (a\u2080 a\u2081 : \u03b1) \u2192 b \u2208 fix f a\u2081 \u2192 Sum.inr a\u2081 \u2208 f a\u2080 \u2192 C a\u2081 \u2192 C a\u2080\n\u22a2 (fixInduction h fun a' h ih =>\n Sum.casesOn (motive := fun x => Part.get (f a') (_ : (f a').Dom) = x \u2192 C a') (Part.get (f a') (_ : (f a').Dom))\n (fun b' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inl b');\n hbase a' (_ : Sum.inl b \u2208 f a'))\n (fun a'' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inr a'');\n hind a' a'' (_ : b \u2208 fix f a'') e (ih a'' e))\n (_ : Part.get (f a') (_ : (f a').Dom) = Part.get (f a') (_ : (f a').Dom))) =\n hind a a' h' fa\n (fixInduction h' fun a' h ih =>\n Sum.casesOn (motive := fun x => Part.get (f a') (_ : (f a').Dom) = x \u2192 (fun x => C x) a')\n (Part.get (f a') (_ : (f a').Dom))\n (fun b' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inl b');\n hbase a' (_ : Sum.inl b \u2208 f a'))\n (fun a'' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inr a'');\n hind a' a'' (_ : b \u2208 fix f a'') e (ih a'' e))\n (_ : Part.get (f a') (_ : (f a').Dom) = Part.get (f a') (_ : (f a').Dom)))\n[PROOFSTEP]\nrw [fixInduction_spec]\n -- Porting note: the explicit motive required because `simp` behaves differently\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na a' : \u03b1\nh : b \u2208 fix f a\nh' : b \u2208 fix f a'\nfa : Sum.inr a' \u2208 f a\nhbase : (a_final : \u03b1) \u2192 Sum.inl b \u2208 f a_final \u2192 C a_final\nhind : (a\u2080 a\u2081 : \u03b1) \u2192 b \u2208 fix f a\u2081 \u2192 Sum.inr a\u2081 \u2208 f a\u2080 \u2192 C a\u2081 \u2192 C a\u2080\n\u22a2 Sum.casesOn (motive := fun x => Part.get (f a) (_ : (f a).Dom) = x \u2192 C a) (Part.get (f a) (_ : (f a).Dom))\n (fun b' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a) h = Sum.inl b');\n hbase a (_ : Sum.inl b \u2208 f a))\n (fun a'' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a) h = Sum.inr a'');\n hind a a'' (_ : b \u2208 fix f a'') e\n ((fun a' h' =>\n fixInduction (_ : b \u2208 fix f a') fun a' h ih =>\n Sum.casesOn (motive := fun x => Part.get (f a') (_ : (f a').Dom) = x \u2192 C a')\n (Part.get (f a') (_ : (f a').Dom))\n (fun b' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inl b');\n hbase a' (_ : Sum.inl b \u2208 f a'))\n (fun a'' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inr a'');\n hind a' a'' (_ : b \u2208 fix f a'') e (ih a'' e))\n (_ : Part.get (f a') (_ : (f a').Dom) = Part.get (f a') (_ : (f a').Dom)))\n a'' e))\n (_ : Part.get (f a) (_ : (f a).Dom) = Part.get (f a) (_ : (f a).Dom)) =\n hind a a' h' fa\n (fixInduction h' fun a' h ih =>\n Sum.casesOn (motive := fun x => Part.get (f a') (_ : (f a').Dom) = x \u2192 (fun x => C x) a')\n (Part.get (f a') (_ : (f a').Dom))\n (fun b' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inl b');\n hbase a' (_ : Sum.inl b \u2208 f a'))\n (fun a'' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inr a'');\n hind a' a'' (_ : b \u2208 fix f a'') e (ih a'' e))\n (_ : Part.get (f a') (_ : (f a').Dom) = Part.get (f a') (_ : (f a').Dom)))\n[PROOFSTEP]\nrefine'\n Eq.rec (motive := fun x e =>\n Sum.casesOn (motive := fun y => (f a).get (dom_of_mem_fix h) = y \u2192 C a) x _ _\n (Eq.trans (Part.get_eq_of_mem fa (dom_of_mem_fix h)) e) =\n _)\n _ (Part.get_eq_of_mem fa (dom_of_mem_fix h)).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nC : \u03b1 \u2192 Sort u_7\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\nb : \u03b2\na a' : \u03b1\nh : b \u2208 fix f a\nh' : b \u2208 fix f a'\nfa : Sum.inr a' \u2208 f a\nhbase : (a_final : \u03b1) \u2192 Sum.inl b \u2208 f a_final \u2192 C a_final\nhind : (a\u2080 a\u2081 : \u03b1) \u2192 b \u2208 fix f a\u2081 \u2192 Sum.inr a\u2081 \u2208 f a\u2080 \u2192 C a\u2081 \u2192 C a\u2080\n\u22a2 (fun x e =>\n Sum.casesOn (motive := fun y => Part.get (f a) (_ : (f a).Dom) = y \u2192 C a) x\n (fun val =>\n (fun b' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a) h = Sum.inl b');\n hbase a (_ : Sum.inl b \u2208 f a))\n val)\n (fun val =>\n (fun a'' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a) h = Sum.inr a'');\n hind a a'' (_ : b \u2208 fix f a'') e\n ((fun a' h' =>\n fixInduction (_ : b \u2208 fix f a') fun a' h ih =>\n Sum.casesOn (motive := fun x => Part.get (f a') (_ : (f a').Dom) = x \u2192 C a')\n (Part.get (f a') (_ : (f a').Dom))\n (fun b' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inl b');\n hbase a' (_ : Sum.inl b \u2208 f a'))\n (fun a'' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inr a'');\n hind a' a'' (_ : b \u2208 fix f a'') e (ih a'' e))\n (_ : Part.get (f a') (_ : (f a').Dom) = Part.get (f a') (_ : (f a').Dom)))\n a'' e))\n val)\n (_ : Part.get (f a) (_ : (f a).Dom) = x) =\n hind a a' h' fa\n (fixInduction h' fun a' h ih =>\n Sum.casesOn (motive := fun x => Part.get (f a') (_ : (f a').Dom) = x \u2192 (fun x => C x) a')\n (Part.get (f a') (_ : (f a').Dom))\n (fun b' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inl b');\n hbase a' (_ : Sum.inl b \u2208 f a'))\n (fun a'' e =>\n let_fun e := (_ : \u2203 h, Part.get (f a') h = Sum.inr a'');\n hind a' a'' (_ : b \u2208 fix f a'') e (ih a'' e))\n (_ : Part.get (f a') (_ : (f a').Dom) = Part.get (f a') (_ : (f a').Dom))))\n (Sum.inr a') (_ : Sum.inr a' = Sum.inr a')\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2\n\u22a2 preimage f Set.univ = Dom f\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 preimage f Set.univ \u2194 x\u271d \u2208 Dom f\n[PROOFSTEP]\nsimp [mem_preimage, mem_dom]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u22a2 preimage (\u2191f) s = f \u207b\u00b9' s\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 preimage (\u2191f) s \u2194 x\u271d \u2208 f \u207b\u00b9' s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\n\u22a2 x \u2208 core (res f s) t \u2194 x \u2208 s \u2192 f x \u2208 t\n[PROOFSTEP]\nsimp [mem_core, mem_res]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 core (res f s) t = s\u1d9c \u222a f \u207b\u00b9' t\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\n\u22a2 x \u2208 core (res f s) t \u2194 x \u2208 s\u1d9c \u222a f \u207b\u00b9' t\n[PROOFSTEP]\nrw [mem_core_res]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\n\u22a2 x \u2208 s \u2192 f x \u2208 t \u2194 x \u2208 s\u1d9c \u222a f \u207b\u00b9' t\n[PROOFSTEP]\nby_cases h : x \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\nh : x \u2208 s\n\u22a2 x \u2208 s \u2192 f x \u2208 t \u2194 x \u2208 s\u1d9c \u222a f \u207b\u00b9' t\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\nh : \u00acx \u2208 s\n\u22a2 x \u2208 s \u2192 f x \u2208 t \u2194 x \u2208 s\u1d9c \u222a f \u207b\u00b9' t\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u22a2 core (\u2191f) s = f \u207b\u00b9' s\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nx : \u03b1\n\u22a2 x \u2208 core (\u2191f) s \u2194 x \u2208 f \u207b\u00b9' s\n[PROOFSTEP]\nsimp [core_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d f : \u03b1 \u2192. \u03b2\ns : Set \u03b2\n\u22a2 core f s = preimage f s \u222a (Dom f)\u1d9c\n[PROOFSTEP]\nrw [preimage_eq, Set.union_distrib_right, Set.union_comm (Dom f), Set.compl_union_self, Set.inter_univ,\n Set.union_eq_self_of_subset_right (f.compl_dom_subset_core s)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d f : \u03b1 \u2192. \u03b2\ns : Set \u03b2\n\u22a2 asSubtype f \u207b\u00b9' s = Subtype.val \u207b\u00b9' preimage f s\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d f : \u03b1 \u2192. \u03b2\ns : Set \u03b2\nx : \u2191(Dom f)\n\u22a2 x \u2208 asSubtype f \u207b\u00b9' s \u2194 x \u2208 Subtype.val \u207b\u00b9' preimage f s\n[PROOFSTEP]\nsimp only [Set.mem_preimage, Set.mem_setOf_eq, PFun.asSubtype, PFun.mem_preimage]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d f : \u03b1 \u2192. \u03b2\ns : Set \u03b2\nx : \u2191(Dom f)\n\u22a2 fn f \u2191x (_ : \u2191x \u2208 Dom f) \u2208 s \u2194 \u2203 y, y \u2208 s \u2227 y \u2208 f \u2191x\n[PROOFSTEP]\nshow f.fn x.val _ \u2208 s \u2194 \u2203 y \u2208 s, y \u2208 f x.val\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d f : \u03b1 \u2192. \u03b2\ns : Set \u03b2\nx : \u2191(Dom f)\n\u22a2 fn f \u2191x (_ : \u2191x \u2208 Dom f) \u2208 s \u2194 \u2203 y, y \u2208 s \u2227 y \u2208 f \u2191x\n[PROOFSTEP]\nexact\n Iff.intro (fun h => \u27e8_, h, Part.get_mem _\u27e9) fun \u27e8y, ys, fxy\u27e9 =>\n have : f.fn x.val x.property \u2208 f x.val := Part.get_mem _\n Part.mem_unique fxy this \u25b8 ys\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\np : \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nb : Subtype p\n\u22a2 b \u2208 toSubtype p f a \u2194 \u2191b = f a\n[PROOFSTEP]\nrw [toSubtype_apply, Part.mem_mk_iff, exists_subtype_mk_eq_iff, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d f : \u03b1 \u2192. \u03b2\nx\u271d\u00b9 : \u03b1\nx\u271d : \u03b2\n\u22a2 x\u271d \u2208 comp (PFun.id \u03b2) f x\u271d\u00b9 \u2194 x\u271d \u2208 f x\u271d\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d f : \u03b1 \u2192. \u03b2\nx\u271d\u00b9 : \u03b1\nx\u271d : \u03b2\n\u22a2 x\u271d \u2208 comp f (PFun.id \u03b1) x\u271d\u00b9 \u2194 x\u271d \u2208 f x\u271d\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b2 \u2192. \u03b3\ng : \u03b1 \u2192. \u03b2\n\u22a2 Dom (comp f g) = preimage g (Dom f)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b2 \u2192. \u03b3\ng : \u03b1 \u2192. \u03b2\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 Dom (comp f g) \u2194 x\u271d \u2208 preimage g (Dom f)\n[PROOFSTEP]\nsimp_rw [mem_preimage, mem_dom, comp_apply, Part.mem_bind_iff, \u2190 exists_and_right]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b2 \u2192. \u03b3\ng : \u03b1 \u2192. \u03b2\nx\u271d : \u03b1\n\u22a2 (\u2203 y a, a \u2208 g x\u271d \u2227 y \u2208 f a) \u2194 \u2203 y x, x \u2208 f y \u2227 y \u2208 g x\u271d\n[PROOFSTEP]\nrw [exists_comm]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b2 \u2192. \u03b3\ng : \u03b1 \u2192. \u03b2\nx\u271d : \u03b1\n\u22a2 (\u2203 b a, b \u2208 g x\u271d \u2227 a \u2208 f b) \u2194 \u2203 y x, x \u2208 f y \u2227 y \u2208 g x\u271d\n[PROOFSTEP]\nsimp_rw [and_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b2 \u2192. \u03b3\ng : \u03b1 \u2192. \u03b2\ns : Set \u03b3\n\u22a2 preimage (comp f g) s = preimage g (preimage f s)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b2 \u2192. \u03b3\ng : \u03b1 \u2192. \u03b2\ns : Set \u03b3\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 preimage (comp f g) s \u2194 x\u271d \u2208 preimage g (preimage f s)\n[PROOFSTEP]\nsimp_rw [mem_preimage, comp_apply, Part.mem_bind_iff, \u2190 exists_and_right, \u2190 exists_and_left]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b2 \u2192. \u03b3\ng : \u03b1 \u2192. \u03b2\ns : Set \u03b3\nx\u271d : \u03b1\n\u22a2 (\u2203 y x, y \u2208 s \u2227 x \u2208 g x\u271d \u2227 y \u2208 f x) \u2194 \u2203 y x, (x \u2208 s \u2227 x \u2208 f y) \u2227 y \u2208 g x\u271d\n[PROOFSTEP]\nrw [exists_comm]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b2 \u2192. \u03b3\ng : \u03b1 \u2192. \u03b2\ns : Set \u03b3\nx\u271d : \u03b1\n\u22a2 (\u2203 b a, a \u2208 s \u2227 b \u2208 g x\u271d \u2227 a \u2208 f b) \u2194 \u2203 y x, (x \u2208 s \u2227 x \u2208 f y) \u2227 y \u2208 g x\u271d\n[PROOFSTEP]\nsimp_rw [and_assoc, and_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b2 \u2192. \u03b3\ng : \u03b1 \u2192. \u03b2\na : Part \u03b1\n\u22a2 Part.bind a (comp f g) = Part.bind (Part.bind a g) f\n[PROOFSTEP]\next c\n[GOAL]\ncase H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b2 \u2192. \u03b3\ng : \u03b1 \u2192. \u03b2\na : Part \u03b1\nc : \u03b3\n\u22a2 c \u2208 Part.bind a (comp f g) \u2194 c \u2208 Part.bind (Part.bind a g) f\n[PROOFSTEP]\nsimp_rw [Part.mem_bind_iff, comp_apply, Part.mem_bind_iff, \u2190 exists_and_right, \u2190 exists_and_left]\n[GOAL]\ncase H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b2 \u2192. \u03b3\ng : \u03b1 \u2192. \u03b2\na : Part \u03b1\nc : \u03b3\n\u22a2 (\u2203 a_1 x, a_1 \u2208 a \u2227 x \u2208 g a_1 \u2227 c \u2208 f x) \u2194 \u2203 a_1 x, (x \u2208 a \u2227 a_1 \u2208 g x) \u2227 c \u2208 f a_1\n[PROOFSTEP]\nrw [exists_comm]\n[GOAL]\ncase H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b2 \u2192. \u03b3\ng : \u03b1 \u2192. \u03b2\na : Part \u03b1\nc : \u03b3\n\u22a2 (\u2203 b a_1, a_1 \u2208 a \u2227 b \u2208 g a_1 \u2227 c \u2208 f b) \u2194 \u2203 a_1 x, (x \u2208 a \u2227 a_1 \u2208 g x) \u2227 c \u2208 f a_1\n[PROOFSTEP]\nsimp_rw [and_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b3 \u2192. \u03b4\ng : \u03b2 \u2192. \u03b3\nh : \u03b1 \u2192. \u03b2\nx\u271d\u00b9 : \u03b1\nx\u271d : \u03b4\n\u22a2 x\u271d \u2208 comp (comp f g) h x\u271d\u00b9 \u2194 x\u271d \u2208 comp f (comp g h) x\u271d\u00b9\n[PROOFSTEP]\nsimp only [comp_apply, Part.bind_comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nx\u271d\u00b9 : \u03b1\nx\u271d : \u03b3\n\u22a2 x\u271d \u2208 \u2191(g \u2218 f) x\u271d\u00b9 \u2194 x\u271d \u2208 comp (\u2191g) (\u2191f) x\u271d\u00b9\n[PROOFSTEP]\nsimp only [coe_val, comp_apply, Function.comp, Part.bind_some]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d f : \u03b1 \u2192. \u03b2\ng : \u03b1 \u2192. \u03b3\nx : \u03b1\ny : \u03b2 \u00d7 \u03b3\n\u22a2 y \u2208 prodLift f g x \u2194 y.fst \u2208 f x \u2227 y.snd \u2208 g x\n[PROOFSTEP]\ntrans \u2203 hp hq, (f x).get hp = y.1 \u2227 (g x).get hq = y.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d f : \u03b1 \u2192. \u03b2\ng : \u03b1 \u2192. \u03b3\nx : \u03b1\ny : \u03b2 \u00d7 \u03b3\n\u22a2 y \u2208 prodLift f g x \u2194 \u2203 hp hq, Part.get (f x) hp = y.fst \u2227 Part.get (g x) hq = y.snd\n[PROOFSTEP]\nsimp only [prodLift, Part.mem_mk_iff, And.exists, Prod.ext_iff]\n -- Porting note: was just `[exists_and_left, exists_and_right]`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d f : \u03b1 \u2192. \u03b2\ng : \u03b1 \u2192. \u03b3\nx : \u03b1\ny : \u03b2 \u00d7 \u03b3\n\u22a2 (\u2203 hp hq, Part.get (f x) hp = y.fst \u2227 Part.get (g x) hq = y.snd) \u2194 y.fst \u2208 f x \u2227 y.snd \u2208 g x\n[PROOFSTEP]\nsimp only [exists_and_left, exists_and_right, (\u00b7 \u2208 \u00b7), Part.Mem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192. \u03b3\ng : \u03b2 \u2192. \u03b4\nx : \u03b1 \u00d7 \u03b2\ny : \u03b3 \u00d7 \u03b4\n\u22a2 y \u2208 prodMap f g x \u2194 y.fst \u2208 f x.fst \u2227 y.snd \u2208 g x.snd\n[PROOFSTEP]\ntrans \u2203 hp hq, (f x.1).get hp = y.1 \u2227 (g x.2).get hq = y.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192. \u03b3\ng : \u03b2 \u2192. \u03b4\nx : \u03b1 \u00d7 \u03b2\ny : \u03b3 \u00d7 \u03b4\n\u22a2 y \u2208 prodMap f g x \u2194 \u2203 hp hq, Part.get (f x.fst) hp = y.fst \u2227 Part.get (g x.snd) hq = y.snd\n[PROOFSTEP]\nsimp only [prodMap, Part.mem_mk_iff, And.exists, Prod.ext_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192. \u03b3\ng : \u03b2 \u2192. \u03b4\nx : \u03b1 \u00d7 \u03b2\ny : \u03b3 \u00d7 \u03b4\n\u22a2 (\u2203 hp hq, Part.get (f x.fst) hp = y.fst \u2227 Part.get (g x.snd) hq = y.snd) \u2194 y.fst \u2208 f x.fst \u2227 y.snd \u2208 g x.snd\n[PROOFSTEP]\nsimp only [exists_and_left, exists_and_right, (\u00b7 \u2208 \u00b7), Part.Mem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf\u271d : \u03b1 \u2192. \u03b2\nf : \u03b1 \u2192. \u03b3\ng : \u03b2 \u2192. \u03b4\na : \u03b1 \u00d7 \u03b2\n\u22a2 \u2200 (b : \u03b3 \u00d7 \u03b4), b \u2208 prodLift (comp f \u2191Prod.fst) (comp g \u2191Prod.snd) a \u2194 b \u2208 prodMap f g a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2\nx\u271d\u00b9 x\u271d : \u03b1 \u00d7 \u03b2\n\u22a2 x\u271d \u2208 prodMap (PFun.id \u03b1) (PFun.id \u03b2) x\u271d\u00b9 \u2194 x\u271d \u2208 PFun.id (\u03b1 \u00d7 \u03b2) x\u271d\u00b9\n[PROOFSTEP]\nsimp [eq_comm]\n", "meta": {"mathlib_filename": "Mathlib.Data.PFun", "llama_tokens": 36850, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.44167300566462553, "lm_q1q2_score": 0.23118063874073333}} {"text": "[GOAL]\nR : Type u\ninst\u271d : Ring R\nr : R\nM N : ModuleCat R\nf : M \u27f6 N\n\u22a2 (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat)).obj.map f \u226b\n (fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r)) N =\n (fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r)) M \u226b\n (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat)).obj.map f\n[PROOFSTEP]\next\n[GOAL]\ncase w\nR : Type u\ninst\u271d : Ring R\nr : R\nM N : ModuleCat R\nf : M \u27f6 N\nx\u271d : \u2191((AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat)).obj.obj M)\n\u22a2 \u2191((AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat)).obj.map f \u226b\n (fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r)) N)\n x\u271d =\n \u2191((fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r)) M \u226b\n (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat)).obj.map f)\n x\u271d\n[PROOFSTEP]\nexact (f.map_smul _ _).symm\n[GOAL]\nR : Type u\ninst\u271d : Ring R\n\u22a2 Function.LeftInverse (fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) fun r =>\n NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r)\n[PROOFSTEP]\nintro r\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nr : R\n\u22a2 (fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1)\n ((fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r)) r) =\n r\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d : Ring R\n\u22a2 Function.RightInverse (fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) fun r =>\n NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r)\n[PROOFSTEP]\nintro \u03c6\n[GOAL]\nR : Type u\ninst\u271d : Ring R\n\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))\n\u22a2 (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6\n[PROOFSTEP]\napply NatTrans.ext\n[GOAL]\ncase app\nR : Type u\ninst\u271d : Ring R\n\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))\n\u22a2 ((fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6)).app =\n \u03c6.app\n[PROOFSTEP]\next M (x : M)\n[GOAL]\ncase app.h.w\nR : Type u\ninst\u271d : Ring R\n\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))\nM : ModuleCat R\nx : \u2191M\n\u22a2 \u2191(NatTrans.app\n ((fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6))\n M)\n x =\n \u2191(NatTrans.app \u03c6 M) x\n[PROOFSTEP]\nhave w :=\n congr_fun ((forget _).congr_map (\u03c6.naturality (ModuleCat.asHomRight (LinearMap.toSpanSingleton R M x)))) (1 : R)\n[GOAL]\ncase app.h.w\nR : Type u\ninst\u271d : Ring R\n\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))\nM : ModuleCat R\nx : \u2191M\nw :\n (forget AddCommGroupCat).map\n ((AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat)).obj.map\n (ModuleCat.asHomRight (LinearMap.toSpanSingleton R (\u2191M) x)) \u226b\n NatTrans.app \u03c6 M)\n 1 =\n (forget AddCommGroupCat).map\n (NatTrans.app \u03c6 (ModuleCat.of R R) \u226b\n (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat)).obj.map\n (ModuleCat.asHomRight (LinearMap.toSpanSingleton R (\u2191M) x)))\n 1\n\u22a2 \u2191(NatTrans.app\n ((fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6))\n M)\n x =\n \u2191(NatTrans.app \u03c6 M) x\n[PROOFSTEP]\nexact w.symm.trans (congr_arg (\u03c6.app M) (one_smul R x))\n[GOAL]\nR : Type u\ninst\u271d : Ring R\n\u22a2 \u2200 (x y : R),\n Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n (x * y) =\n Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n x *\n Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n y\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nx\u271d y\u271d : R\n\u22a2 Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n (x\u271d * y\u271d) =\n Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n x\u271d *\n Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n y\u271d\n[PROOFSTEP]\napply NatTrans.ext\n[GOAL]\ncase app\nR : Type u\ninst\u271d : Ring R\nx\u271d y\u271d : R\n\u22a2 (Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n (x\u271d * y\u271d)).app =\n (Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n x\u271d *\n Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n y\u271d).app\n[PROOFSTEP]\next\n[GOAL]\ncase app.h.w\nR : Type u\ninst\u271d : Ring R\nx\u271d\u00b2 y\u271d : R\nx\u271d\u00b9 : ModuleCat R\nx\u271d : \u2191((AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat)).obj.obj x\u271d\u00b9)\n\u22a2 \u2191(NatTrans.app\n (Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n (x\u271d\u00b2 * y\u271d))\n x\u271d\u00b9)\n x\u271d =\n \u2191(NatTrans.app\n (Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n x\u271d\u00b2 *\n Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n y\u271d)\n x\u271d\u00b9)\n x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase app.h.w\nR : Type u\ninst\u271d : Ring R\nx\u271d\u00b2 y\u271d : R\nx\u271d\u00b9 : ModuleCat R\nx\u271d : \u2191((AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat)).obj.obj x\u271d\u00b9)\n\u22a2 (x\u271d\u00b2 * y\u271d) \u2022 x\u271d =\n \u2191(NatTrans.app\n ((NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) x\u271d\u00b2)) *\n NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) y\u271d))\n x\u271d\u00b9)\n x\u271d\n[PROOFSTEP]\nsimp only [AddCommGroupCat.ofHom_apply, DistribMulAction.toAddMonoidHom_apply, mul_smul]\n[GOAL]\ncase app.h.w\nR : Type u\ninst\u271d : Ring R\nx\u271d\u00b2 y\u271d : R\nx\u271d\u00b9 : ModuleCat R\nx\u271d : \u2191((AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat)).obj.obj x\u271d\u00b9)\n\u22a2 x\u271d\u00b2 \u2022 y\u271d \u2022 x\u271d =\n \u2191(NatTrans.app\n ((NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) x\u271d\u00b2)) *\n NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) y\u271d))\n x\u271d\u00b9)\n x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d : Ring R\n\u22a2 \u2200 (x y : R),\n Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n (x + y) =\n Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n x +\n Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n y\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nx\u271d y\u271d : R\n\u22a2 Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n (x\u271d + y\u271d) =\n Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n x\u271d +\n Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n y\u271d\n[PROOFSTEP]\napply NatTrans.ext\n[GOAL]\ncase app\nR : Type u\ninst\u271d : Ring R\nx\u271d y\u271d : R\n\u22a2 (Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n (x\u271d + y\u271d)).app =\n (Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n x\u271d +\n Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n y\u271d).app\n[PROOFSTEP]\next\n[GOAL]\ncase app.h.w\nR : Type u\ninst\u271d : Ring R\nx\u271d\u00b2 y\u271d : R\nx\u271d\u00b9 : ModuleCat R\nx\u271d : \u2191((AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat)).obj.obj x\u271d\u00b9)\n\u22a2 \u2191(NatTrans.app\n (Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n (x\u271d\u00b2 + y\u271d))\n x\u271d\u00b9)\n x\u271d =\n \u2191(NatTrans.app\n (Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n x\u271d\u00b2 +\n Equiv.toFun\n { toFun := fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r),\n invFun := fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1, left_inv := (_ : \u2200 (r : R), r * 1 = r),\n right_inv :=\n (_ :\n \u2200 (\u03c6 : End (AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat))),\n (fun r => NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) r))\n ((fun \u03c6 => \u2191(NatTrans.app \u03c6 (ModuleCat.of R R)) 1) \u03c6) =\n \u03c6) }\n y\u271d)\n x\u271d\u00b9)\n x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase app.h.w\nR : Type u\ninst\u271d : Ring R\nx\u271d\u00b2 y\u271d : R\nx\u271d\u00b9 : ModuleCat R\nx\u271d : \u2191((AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat)).obj.obj x\u271d\u00b9)\n\u22a2 (x\u271d\u00b2 + y\u271d) \u2022 x\u271d =\n \u2191(NatTrans.app\n ((NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) x\u271d\u00b2)) +\n NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) y\u271d))\n x\u271d\u00b9)\n x\u271d\n[PROOFSTEP]\nsimp only [AddCommGroupCat.ofHom_apply, DistribMulAction.toAddMonoidHom_apply, add_smul]\n[GOAL]\ncase app.h.w\nR : Type u\ninst\u271d : Ring R\nx\u271d\u00b2 y\u271d : R\nx\u271d\u00b9 : ModuleCat R\nx\u271d : \u2191((AdditiveFunctor.of (forget\u2082 (ModuleCat R) AddCommGroupCat)).obj.obj x\u271d\u00b9)\n\u22a2 x\u271d\u00b2 \u2022 x\u271d + y\u271d \u2022 x\u271d =\n \u2191(NatTrans.app\n ((NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) x\u271d\u00b2)) +\n NatTrans.mk fun M => AddCommGroupCat.ofHom (DistribMulAction.toAddMonoidHom (\u2191M) y\u271d))\n x\u271d\u00b9)\n x\u271d\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.ModuleCat.Tannaka", "llama_tokens": 8440, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.4186969093556867, "lm_q1q2_score": 0.230537602218082}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasWideEqualizers C\nT : C\nhT : \u2200 (X : C), Nonempty (T \u27f6 X)\n\u22a2 HasInitial C\n[PROOFSTEP]\nlet endos := T \u27f6 T\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasWideEqualizers C\nT : C\nhT : \u2200 (X : C), Nonempty (T \u27f6 X)\nendos : Type v := T \u27f6 T\n\u22a2 HasInitial C\n[PROOFSTEP]\nlet i := wideEqualizer.\u03b9 (id : endos \u2192 endos)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasWideEqualizers C\nT : C\nhT : \u2200 (X : C), Nonempty (T \u27f6 X)\nendos : Type v := T \u27f6 T\ni : wideEqualizer id \u27f6 T := wideEqualizer.\u03b9 id\n\u22a2 HasInitial C\n[PROOFSTEP]\nhaveI : Nonempty endos := \u27e8\ud835\udfd9 _\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasWideEqualizers C\nT : C\nhT : \u2200 (X : C), Nonempty (T \u27f6 X)\nendos : Type v := T \u27f6 T\ni : wideEqualizer id \u27f6 T := wideEqualizer.\u03b9 id\nthis : Nonempty endos\n\u22a2 HasInitial C\n[PROOFSTEP]\nhave : \u2200 X : C, Unique (wideEqualizer (id : endos \u2192 endos) \u27f6 X) :=\n by\n intro X\n refine' \u27e8\u27e8i \u226b Classical.choice (hT X)\u27e9, fun a => _\u27e9\n let E := equalizer a (i \u226b Classical.choice (hT _))\n let e : E \u27f6 wideEqualizer id := equalizer.\u03b9 _ _\n let h : T \u27f6 E := Classical.choice (hT E)\n have : ((i \u226b h) \u226b e) \u226b i = i \u226b \ud835\udfd9 _ := by\n rw [Category.assoc, Category.assoc]\n apply wideEqualizer.condition (id : endos \u2192 endos) (h \u226b e \u226b i)\n rw [Category.comp_id, cancel_mono_id i] at this \n haveI : IsSplitEpi e := IsSplitEpi.mk' \u27e8i \u226b h, this\u27e9\n rw [\u2190 cancel_epi e]\n apply equalizer.condition\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasWideEqualizers C\nT : C\nhT : \u2200 (X : C), Nonempty (T \u27f6 X)\nendos : Type v := T \u27f6 T\ni : wideEqualizer id \u27f6 T := wideEqualizer.\u03b9 id\nthis : Nonempty endos\n\u22a2 (X : C) \u2192 Unique (wideEqualizer id \u27f6 X)\n[PROOFSTEP]\nintro X\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasWideEqualizers C\nT : C\nhT : \u2200 (X : C), Nonempty (T \u27f6 X)\nendos : Type v := T \u27f6 T\ni : wideEqualizer id \u27f6 T := wideEqualizer.\u03b9 id\nthis : Nonempty endos\nX : C\n\u22a2 Unique (wideEqualizer id \u27f6 X)\n[PROOFSTEP]\nrefine' \u27e8\u27e8i \u226b Classical.choice (hT X)\u27e9, fun a => _\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasWideEqualizers C\nT : C\nhT : \u2200 (X : C), Nonempty (T \u27f6 X)\nendos : Type v := T \u27f6 T\ni : wideEqualizer id \u27f6 T := wideEqualizer.\u03b9 id\nthis : Nonempty endos\nX : C\na : wideEqualizer id \u27f6 X\n\u22a2 a = default\n[PROOFSTEP]\nlet E := equalizer a (i \u226b Classical.choice (hT _))\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasWideEqualizers C\nT : C\nhT : \u2200 (X : C), Nonempty (T \u27f6 X)\nendos : Type v := T \u27f6 T\ni : wideEqualizer id \u27f6 T := wideEqualizer.\u03b9 id\nthis : Nonempty endos\nX : C\na : wideEqualizer id \u27f6 X\nE : C := equalizer a (i \u226b Classical.choice (_ : Nonempty (T \u27f6 X)))\n\u22a2 a = default\n[PROOFSTEP]\nlet e : E \u27f6 wideEqualizer id := equalizer.\u03b9 _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasWideEqualizers C\nT : C\nhT : \u2200 (X : C), Nonempty (T \u27f6 X)\nendos : Type v := T \u27f6 T\ni : wideEqualizer id \u27f6 T := wideEqualizer.\u03b9 id\nthis : Nonempty endos\nX : C\na : wideEqualizer id \u27f6 X\nE : C := equalizer a (i \u226b Classical.choice (_ : Nonempty (T \u27f6 X)))\ne : E \u27f6 wideEqualizer id := equalizer.\u03b9 a (i \u226b Classical.choice (_ : Nonempty (T \u27f6 X)))\n\u22a2 a = default\n[PROOFSTEP]\nlet h : T \u27f6 E := Classical.choice (hT E)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasWideEqualizers C\nT : C\nhT : \u2200 (X : C), Nonempty (T \u27f6 X)\nendos : Type v := T \u27f6 T\ni : wideEqualizer id \u27f6 T := wideEqualizer.\u03b9 id\nthis : Nonempty endos\nX : C\na : wideEqualizer id \u27f6 X\nE : C := equalizer a (i \u226b Classical.choice (_ : Nonempty (T \u27f6 X)))\ne : E \u27f6 wideEqualizer id := equalizer.\u03b9 a (i \u226b Classical.choice (_ : Nonempty (T \u27f6 X)))\nh : T \u27f6 E := Classical.choice (_ : Nonempty (T \u27f6 E))\n\u22a2 a = default\n[PROOFSTEP]\nhave : ((i \u226b h) \u226b e) \u226b i = i \u226b \ud835\udfd9 _ := by\n rw [Category.assoc, Category.assoc]\n apply wideEqualizer.condition (id : endos \u2192 endos) (h \u226b e \u226b i)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasWideEqualizers C\nT : C\nhT : \u2200 (X : C), Nonempty (T \u27f6 X)\nendos : Type v := T \u27f6 T\ni : wideEqualizer id \u27f6 T := wideEqualizer.\u03b9 id\nthis : Nonempty endos\nX : C\na : wideEqualizer id \u27f6 X\nE : C := equalizer a (i \u226b Classical.choice (_ : Nonempty (T \u27f6 X)))\ne : E \u27f6 wideEqualizer id := equalizer.\u03b9 a (i \u226b Classical.choice (_ : Nonempty (T \u27f6 X)))\nh : T \u27f6 E := Classical.choice (_ : Nonempty (T \u27f6 E))\n\u22a2 ((i \u226b h) \u226b e) \u226b i = i \u226b \ud835\udfd9 T\n[PROOFSTEP]\nrw [Category.assoc, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasWideEqualizers C\nT : C\nhT : \u2200 (X : C), Nonempty (T \u27f6 X)\nendos : Type v := T \u27f6 T\ni : wideEqualizer id \u27f6 T := wideEqualizer.\u03b9 id\nthis : Nonempty endos\nX : C\na : wideEqualizer id \u27f6 X\nE : C := equalizer a (i \u226b Classical.choice (_ : Nonempty (T \u27f6 X)))\ne : E \u27f6 wideEqualizer id := equalizer.\u03b9 a (i \u226b Classical.choice (_ : Nonempty (T \u27f6 X)))\nh : T \u27f6 E := Classical.choice (_ : Nonempty (T \u27f6 E))\n\u22a2 i \u226b h \u226b e \u226b i = i \u226b \ud835\udfd9 T\n[PROOFSTEP]\napply wideEqualizer.condition (id : endos \u2192 endos) (h \u226b e \u226b i)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasWideEqualizers C\nT : C\nhT : \u2200 (X : C), Nonempty (T \u27f6 X)\nendos : Type v := T \u27f6 T\ni : wideEqualizer id \u27f6 T := wideEqualizer.\u03b9 id\nthis\u271d : Nonempty endos\nX : C\na : wideEqualizer id \u27f6 X\nE : C := equalizer a (i \u226b Classical.choice (_ : Nonempty (T \u27f6 X)))\ne : E \u27f6 wideEqualizer id := equalizer.\u03b9 a (i \u226b Classical.choice (_ : Nonempty (T \u27f6 X)))\nh : T \u27f6 E := Classical.choice (_ : Nonempty (T \u27f6 E))\nthis : ((i \u226b h) \u226b e) \u226b i = i \u226b \ud835\udfd9 T\n\u22a2 a = default\n[PROOFSTEP]\nrw [Category.comp_id, cancel_mono_id i] at this \n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasWideEqualizers C\nT : C\nhT : \u2200 (X : C), Nonempty (T \u27f6 X)\nendos : Type v := T \u27f6 T\ni : wideEqualizer id \u27f6 T := wideEqualizer.\u03b9 id\nthis\u271d : Nonempty endos\nX : C\na : wideEqualizer id \u27f6 X\nE : C := equalizer a (i \u226b Classical.choice (_ : Nonempty (T \u27f6 X)))\ne : E \u27f6 wideEqualizer id := equalizer.\u03b9 a (i \u226b Classical.choice (_ : Nonempty (T \u27f6 X)))\nh : T \u27f6 E := Classical.choice (_ : Nonempty (T \u27f6 E))\nthis : (i \u226b h) \u226b e = \ud835\udfd9 (wideEqualizer id)\n\u22a2 a = default\n[PROOFSTEP]\nhaveI : IsSplitEpi e := IsSplitEpi.mk' \u27e8i \u226b h, this\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasWideEqualizers C\nT : C\nhT : \u2200 (X : C), Nonempty (T \u27f6 X)\nendos : Type v := T \u27f6 T\ni : wideEqualizer id \u27f6 T := wideEqualizer.\u03b9 id\nthis\u271d\u00b9 : Nonempty endos\nX : C\na : wideEqualizer id \u27f6 X\nE : C := equalizer a (i \u226b Classical.choice (_ : Nonempty (T \u27f6 X)))\ne : E \u27f6 wideEqualizer id := equalizer.\u03b9 a (i \u226b Classical.choice (_ : Nonempty (T \u27f6 X)))\nh : T \u27f6 E := Classical.choice (_ : Nonempty (T \u27f6 E))\nthis\u271d : (i \u226b h) \u226b e = \ud835\udfd9 (wideEqualizer id)\nthis : IsSplitEpi e\n\u22a2 a = default\n[PROOFSTEP]\nrw [\u2190 cancel_epi e]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasWideEqualizers C\nT : C\nhT : \u2200 (X : C), Nonempty (T \u27f6 X)\nendos : Type v := T \u27f6 T\ni : wideEqualizer id \u27f6 T := wideEqualizer.\u03b9 id\nthis\u271d\u00b9 : Nonempty endos\nX : C\na : wideEqualizer id \u27f6 X\nE : C := equalizer a (i \u226b Classical.choice (_ : Nonempty (T \u27f6 X)))\ne : E \u27f6 wideEqualizer id := equalizer.\u03b9 a (i \u226b Classical.choice (_ : Nonempty (T \u27f6 X)))\nh : T \u27f6 E := Classical.choice (_ : Nonempty (T \u27f6 E))\nthis\u271d : (i \u226b h) \u226b e = \ud835\udfd9 (wideEqualizer id)\nthis : IsSplitEpi e\n\u22a2 e \u226b a = e \u226b default\n[PROOFSTEP]\napply equalizer.condition\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasWideEqualizers C\nT : C\nhT : \u2200 (X : C), Nonempty (T \u27f6 X)\nendos : Type v := T \u27f6 T\ni : wideEqualizer id \u27f6 T := wideEqualizer.\u03b9 id\nthis\u271d : Nonempty endos\nthis : (X : C) \u2192 Unique (wideEqualizer id \u27f6 X)\n\u22a2 HasInitial C\n[PROOFSTEP]\nexact hasInitial_of_unique (wideEqualizer (id : endos \u2192 endos))\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Constructions.WeaklyInitial", "llama_tokens": 3353, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.41489884579676883, "lm_q1q2_score": 0.23004915594047903}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nx\u271d\u00b9 x\u271d : R\ns : { x // x \u2208 \u22a5 }\nh : \u2191s * x\u271d\u00b9 = \u2191s * x\u271d\n\u22a2 x\u271d\u00b9 * \u2191s = x\u271d * \u2191s\n[PROOFSTEP]\nrcases s with \u27e8s, hs\u27e9\n[GOAL]\ncase mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nx\u271d\u00b9 x\u271d s : R\nhs : s \u2208 \u22a5\nh : \u2191{ val := s, property := hs } * x\u271d\u00b9 = \u2191{ val := s, property := hs } * x\u271d\n\u22a2 x\u271d\u00b9 * \u2191{ val := s, property := hs } = x\u271d * \u2191{ val := s, property := hs }\n[PROOFSTEP]\nrw [Submonoid.mem_bot] at hs \n[GOAL]\ncase mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nx\u271d\u00b9 x\u271d s : R\nhs\u271d : s \u2208 \u22a5\nhs : s = 1\nh : \u2191{ val := s, property := hs\u271d } * x\u271d\u00b9 = \u2191{ val := s, property := hs\u271d } * x\u271d\n\u22a2 x\u271d\u00b9 * \u2191{ val := s, property := hs\u271d } = x\u271d * \u2191{ val := s, property := hs\u271d }\n[PROOFSTEP]\nsubst hs\n[GOAL]\ncase mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nx\u271d\u00b9 x\u271d : R\nhs : 1 \u2208 \u22a5\nh : \u2191{ val := 1, property := hs } * x\u271d\u00b9 = \u2191{ val := 1, property := hs } * x\u271d\n\u22a2 x\u271d\u00b9 * \u2191{ val := 1, property := hs } = x\u271d * \u2191{ val := 1, property := hs }\n[PROOFSTEP]\nrw [one_mul, one_mul] at h \n[GOAL]\ncase mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nx\u271d\u00b9 x\u271d : R\nhs : 1 \u2208 \u22a5\nh : x\u271d\u00b9 = x\u271d\n\u22a2 x\u271d\u00b9 * \u2191{ val := 1, property := hs } = x\u271d * \u2191{ val := 1, property := hs }\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nx\u271d : R\nhs : 1 \u2208 \u22a5\n\u22a2 x\u271d * \u2191{ val := 1, property := hs } = x\u271d * \u2191{ val := 1, property := hs }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nx\u271d : R\ns : { x // x \u2208 \u22a5 }\n\u22a2 x\u271d * \u2191((fun x s => s) x\u271d s) = \u2191s * (fun r x => r) x\u271d s\n[PROOFSTEP]\nrcases s with \u27e8s, hs\u27e9\n[GOAL]\ncase mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nx\u271d s : R\nhs : s \u2208 \u22a5\n\u22a2 x\u271d * \u2191((fun x s => s) x\u271d { val := s, property := hs }) =\n \u2191{ val := s, property := hs } * (fun r x => r) x\u271d { val := s, property := hs }\n[PROOFSTEP]\nrw [Submonoid.mem_bot] at hs \n[GOAL]\ncase mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nx\u271d s : R\nhs\u271d : s \u2208 \u22a5\nhs : s = 1\n\u22a2 x\u271d * \u2191((fun x s => s) x\u271d { val := s, property := hs\u271d }) =\n \u2191{ val := s, property := hs\u271d } * (fun r x => r) x\u271d { val := s, property := hs\u271d }\n[PROOFSTEP]\nsimp [hs]\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b2 : Monoid R\u271d\nS\u271d : Submonoid R\u271d\ninst\u271d\u00b9 : OreSet S\u271d\nR : Type ?u.7831\ninst\u271d : CommMonoid R\nS : Submonoid R\nm n : R\ns : { x // x \u2208 S }\nh : \u2191s * m = \u2191s * n\n\u22a2 m * \u2191s = n * \u2191s\n[PROOFSTEP]\nrw [mul_comm n s, mul_comm m s, h]\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b2 : Monoid R\u271d\nS\u271d : Submonoid R\u271d\ninst\u271d\u00b9 : OreSet S\u271d\nR : Type ?u.7831\ninst\u271d : CommMonoid R\nS : Submonoid R\nr : R\ns : { x // x \u2208 S }\n\u22a2 r * \u2191((fun x s => s) r s) = \u2191s * (fun r x => r) r s\n[PROOFSTEP]\nrw [mul_comm]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.OreLocalization.OreSet", "llama_tokens": 1582, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5078118642792043, "lm_q2_score": 0.44939263446475963, "lm_q1q2_score": 0.2282069115008926}} {"text": "[GOAL]\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\n\u22a2 iSup (coveringOfPresieve U R) = U\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\n\u22a2 iSup (coveringOfPresieve U R) \u2264 U\n[PROOFSTEP]\nrefine' iSup_le _\n[GOAL]\ncase a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\n\u22a2 \u2200 (i : (V : Opens \u2191X) \u00d7 { f // R f }), coveringOfPresieve U R i \u2264 U\n[PROOFSTEP]\nintro f\n[GOAL]\ncase a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\nf : (V : Opens \u2191X) \u00d7 { f // R f }\n\u22a2 coveringOfPresieve U R f \u2264 U\n[PROOFSTEP]\nexact f.2.1.le\n[GOAL]\ncase a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\n\u22a2 U \u2264 iSup (coveringOfPresieve U R)\n[PROOFSTEP]\nintro x hxU\n[GOAL]\ncase a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\nx : \u2191X\nhxU : x \u2208 \u2191U\n\u22a2 x \u2208 \u2191(iSup (coveringOfPresieve U R))\n[PROOFSTEP]\nrw [Opens.coe_iSup, Set.mem_iUnion]\n[GOAL]\ncase a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\nx : \u2191X\nhxU : x \u2208 \u2191U\n\u22a2 \u2203 i, x \u2208 \u2191(coveringOfPresieve U R i)\n[PROOFSTEP]\nobtain \u27e8V, iVU, \u27e8W, iVW, iWU, hiWU, -\u27e9, hxV\u27e9 := hR x hxU\n[GOAL]\ncase a.intro.intro.intro.intro.intro.intro.intro\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\nx : \u2191X\nhxU : x \u2208 \u2191U\nV : Opens \u2191X\niVU : V \u27f6 U\nhxV : x \u2208 V\nW : Opens \u2191X\niVW : V \u27f6 W\niWU : W \u27f6 U\nhiWU : R iWU\n\u22a2 \u2203 i, x \u2208 \u2191(coveringOfPresieve U R i)\n[PROOFSTEP]\nexact \u27e8\u27e8W, \u27e8iWU, hiWU\u27e9\u27e9, iVW.le hxV\u27e9\n[GOAL]\nX : TopCat\nY : Opens \u2191X\nR : Presieve Y\n\u22a2 presieveOfCoveringAux (coveringOfPresieve Y R) Y = R\n[PROOFSTEP]\nfunext Z\n[GOAL]\ncase h\nX : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\n\u22a2 presieveOfCoveringAux (coveringOfPresieve Y R) Y = R\n[PROOFSTEP]\next f\n[GOAL]\ncase h.h\nX : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\nf : Z \u27f6 Y\n\u22a2 f \u2208 presieveOfCoveringAux (coveringOfPresieve Y R) Y \u2194 f \u2208 R\n[PROOFSTEP]\nexact \u27e8fun \u27e8\u27e8_, f', h\u27e9, rfl\u27e9 => by rwa [Subsingleton.elim f f'], fun h => \u27e8\u27e8Z, f, h\u27e9, rfl\u27e9\u27e9\n[GOAL]\nX : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\nf : Z \u27f6 Y\nx\u271d : f \u2208 presieveOfCoveringAux (coveringOfPresieve Y R) Y\nf' : Z \u27f6 Y\nh : R f'\n\u22a2 f \u2208 R\n[PROOFSTEP]\nrwa [Subsingleton.elim f f']\n[GOAL]\nX : TopCat\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 Sieve.generate (presieveOfCovering U) \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) (iSup U)\n[PROOFSTEP]\nintro x hx\n[GOAL]\nX : TopCat\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nx : \u2191X\nhx : x \u2208 iSup U\n\u22a2 \u2203 U_1 f, (Sieve.generate (presieveOfCovering U)).arrows f \u2227 x \u2208 U_1\n[PROOFSTEP]\nobtain \u27e8i, hxi\u27e9 := Opens.mem_iSup.mp hx\n[GOAL]\ncase intro\nX : TopCat\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nx : \u2191X\nhx : x \u2208 iSup U\ni : \u03b9\nhxi : x \u2208 U i\n\u22a2 \u2203 U_1 f, (Sieve.generate (presieveOfCovering U)).arrows f \u2227 x \u2208 U_1\n[PROOFSTEP]\nexact \u27e8U i, Opens.leSupr U i, \u27e8U i, \ud835\udfd9 _, Opens.leSupr U i, \u27e8i, rfl\u27e9, Category.id_comp _\u27e9, hxi\u27e9\n[GOAL]\nX : TopCat\n\u03b9 : Type u_1\ninst\u271d : Category.{u_2, u_1} \u03b9\nB : \u03b9 \u2964 Opens \u2191X\n\u22a2 CoverDense (Opens.grothendieckTopology \u2191X) B \u2194 Opens.IsBasis (Set.range B.obj)\n[PROOFSTEP]\nrw [Opens.isBasis_iff_nbhd]\n[GOAL]\nX : TopCat\n\u03b9 : Type u_1\ninst\u271d : Category.{u_2, u_1} \u03b9\nB : \u03b9 \u2964 Opens \u2191X\n\u22a2 CoverDense (Opens.grothendieckTopology \u2191X) B \u2194\n \u2200 {U : Opens \u2191X} {x : \u2191X}, x \u2208 U \u2192 \u2203 U', U' \u2208 Set.range B.obj \u2227 x \u2208 U' \u2227 U' \u2264 U\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nX : TopCat\n\u03b9 : Type u_1\ninst\u271d : Category.{u_2, u_1} \u03b9\nB : \u03b9 \u2964 Opens \u2191X\n\u22a2 CoverDense (Opens.grothendieckTopology \u2191X) B \u2192\n \u2200 {U : Opens \u2191X} {x : \u2191X}, x \u2208 U \u2192 \u2203 U', U' \u2208 Set.range B.obj \u2227 x \u2208 U' \u2227 U' \u2264 U\ncase mpr\nX : TopCat\n\u03b9 : Type u_1\ninst\u271d : Category.{u_2, u_1} \u03b9\nB : \u03b9 \u2964 Opens \u2191X\n\u22a2 (\u2200 {U : Opens \u2191X} {x : \u2191X}, x \u2208 U \u2192 \u2203 U', U' \u2208 Set.range B.obj \u2227 x \u2208 U' \u2227 U' \u2264 U) \u2192\n CoverDense (Opens.grothendieckTopology \u2191X) B\n[PROOFSTEP]\nintro hd U x hx\n[GOAL]\ncase mp\nX : TopCat\n\u03b9 : Type u_1\ninst\u271d : Category.{u_2, u_1} \u03b9\nB : \u03b9 \u2964 Opens \u2191X\nhd : CoverDense (Opens.grothendieckTopology \u2191X) B\nU : Opens \u2191X\nx : \u2191X\nhx : x \u2208 U\n\u22a2 \u2203 U', U' \u2208 Set.range B.obj \u2227 x \u2208 U' \u2227 U' \u2264 U\ncase mpr\nX : TopCat\n\u03b9 : Type u_1\ninst\u271d : Category.{u_2, u_1} \u03b9\nB : \u03b9 \u2964 Opens \u2191X\n\u22a2 (\u2200 {U : Opens \u2191X} {x : \u2191X}, x \u2208 U \u2192 \u2203 U', U' \u2208 Set.range B.obj \u2227 x \u2208 U' \u2227 U' \u2264 U) \u2192\n CoverDense (Opens.grothendieckTopology \u2191X) B\n[PROOFSTEP]\nrcases hd.1 U x hx with \u27e8V, f, \u27e8i, f\u2081, f\u2082, _\u27e9, hV\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.mk\nX : TopCat\n\u03b9 : Type u_1\ninst\u271d : Category.{u_2, u_1} \u03b9\nB : \u03b9 \u2964 Opens \u2191X\nhd : CoverDense (Opens.grothendieckTopology \u2191X) B\nU : Opens \u2191X\nx : \u2191X\nhx : x \u2208 U\nV : Opens \u2191X\nf : V \u27f6 U\nhV : x \u2208 V\ni : \u03b9\nf\u2081 : V \u27f6 B.obj i\nf\u2082 : B.obj i \u27f6 U\nfac\u271d : f\u2081 \u226b f\u2082 = f\n\u22a2 \u2203 U', U' \u2208 Set.range B.obj \u2227 x \u2208 U' \u2227 U' \u2264 U\ncase mpr\nX : TopCat\n\u03b9 : Type u_1\ninst\u271d : Category.{u_2, u_1} \u03b9\nB : \u03b9 \u2964 Opens \u2191X\n\u22a2 (\u2200 {U : Opens \u2191X} {x : \u2191X}, x \u2208 U \u2192 \u2203 U', U' \u2208 Set.range B.obj \u2227 x \u2208 U' \u2227 U' \u2264 U) \u2192\n CoverDense (Opens.grothendieckTopology \u2191X) B\n[PROOFSTEP]\nexact \u27e8B.obj i, \u27e8i, rfl\u27e9, f\u2081.le hV, f\u2082.le\u27e9\n[GOAL]\ncase mpr\nX : TopCat\n\u03b9 : Type u_1\ninst\u271d : Category.{u_2, u_1} \u03b9\nB : \u03b9 \u2964 Opens \u2191X\n\u22a2 (\u2200 {U : Opens \u2191X} {x : \u2191X}, x \u2208 U \u2192 \u2203 U', U' \u2208 Set.range B.obj \u2227 x \u2208 U' \u2227 U' \u2264 U) \u2192\n CoverDense (Opens.grothendieckTopology \u2191X) B\n[PROOFSTEP]\nintro hb\n[GOAL]\ncase mpr\nX : TopCat\n\u03b9 : Type u_1\ninst\u271d : Category.{u_2, u_1} \u03b9\nB : \u03b9 \u2964 Opens \u2191X\nhb : \u2200 {U : Opens \u2191X} {x : \u2191X}, x \u2208 U \u2192 \u2203 U', U' \u2208 Set.range B.obj \u2227 x \u2208 U' \u2227 U' \u2264 U\n\u22a2 CoverDense (Opens.grothendieckTopology \u2191X) B\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.is_cover\nX : TopCat\n\u03b9 : Type u_1\ninst\u271d : Category.{u_2, u_1} \u03b9\nB : \u03b9 \u2964 Opens \u2191X\nhb : \u2200 {U : Opens \u2191X} {x : \u2191X}, x \u2208 U \u2192 \u2203 U', U' \u2208 Set.range B.obj \u2227 x \u2208 U' \u2227 U' \u2264 U\n\u22a2 \u2200 (U : Opens \u2191X), Sieve.coverByImage B U \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\n[PROOFSTEP]\nintro U x hx\n[GOAL]\ncase mpr.is_cover\nX : TopCat\n\u03b9 : Type u_1\ninst\u271d : Category.{u_2, u_1} \u03b9\nB : \u03b9 \u2964 Opens \u2191X\nhb : \u2200 {U : Opens \u2191X} {x : \u2191X}, x \u2208 U \u2192 \u2203 U', U' \u2208 Set.range B.obj \u2227 x \u2208 U' \u2227 U' \u2264 U\nU : Opens \u2191X\nx : \u2191X\nhx : x \u2208 U\n\u22a2 \u2203 U_1 f, (Sieve.coverByImage B U).arrows f \u2227 x \u2208 U_1\n[PROOFSTEP]\nrcases hb hx with \u27e8_, \u27e8i, rfl\u27e9, hx, hi\u27e9\n[GOAL]\ncase mpr.is_cover.intro.intro.intro.intro\nX : TopCat\n\u03b9 : Type u_1\ninst\u271d : Category.{u_2, u_1} \u03b9\nB : \u03b9 \u2964 Opens \u2191X\nhb : \u2200 {U : Opens \u2191X} {x : \u2191X}, x \u2208 U \u2192 \u2203 U', U' \u2208 Set.range B.obj \u2227 x \u2208 U' \u2227 U' \u2264 U\nU : Opens \u2191X\nx : \u2191X\nhx\u271d : x \u2208 U\ni : \u03b9\nhx : x \u2208 B.obj i\nhi : B.obj i \u2264 U\n\u22a2 \u2203 U_1 f, (Sieve.coverByImage B U).arrows f \u2227 x \u2208 U_1\n[PROOFSTEP]\nexact \u27e8B.obj i, \u27e8\u27e8hi\u27e9\u27e9, \u27e8\u27e8i, \ud835\udfd9 _, \u27e8\u27e8hi\u27e9\u27e9, rfl\u27e9\u27e9, hx\u27e9\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf : X \u27f6 Y\nF : TopCat.Presheaf C Y\nhf : OpenEmbedding \u2191f\n\u22a2 CompatiblePreserving (Opens.grothendieckTopology \u2191Y) (IsOpenMap.functor (_ : IsOpenMap \u2191f))\n[PROOFSTEP]\nhaveI : Mono f := (TopCat.mono_iff_injective f).mpr hf.inj\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf : X \u27f6 Y\nF : TopCat.Presheaf C Y\nhf : OpenEmbedding \u2191f\nthis : Mono f\n\u22a2 CompatiblePreserving (Opens.grothendieckTopology \u2191Y) (IsOpenMap.functor (_ : IsOpenMap \u2191f))\n[PROOFSTEP]\napply compatiblePreservingOfDownwardsClosed\n[GOAL]\ncase hF\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf : X \u27f6 Y\nF : TopCat.Presheaf C Y\nhf : OpenEmbedding \u2191f\nthis : Mono f\n\u22a2 {c : Opens \u2191X} \u2192\n {d : Opens \u2191Y} \u2192\n (d \u27f6 (IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj c) \u2192\n (c' : Opens \u2191X) \u00d7 ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj c' \u2245 d)\n[PROOFSTEP]\nintro U V i\n[GOAL]\ncase hF\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf : X \u27f6 Y\nF : TopCat.Presheaf C Y\nhf : OpenEmbedding \u2191f\nthis : Mono f\nU : Opens \u2191X\nV : Opens \u2191Y\ni : V \u27f6 (IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj U\n\u22a2 (c' : Opens \u2191X) \u00d7 ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj c' \u2245 V)\n[PROOFSTEP]\nrefine' \u27e8(Opens.map f).obj V, eqToIso <| Opens.ext <| Set.image_preimage_eq_of_subset fun x h \u21a6 _\u27e9\n[GOAL]\ncase hF\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf : X \u27f6 Y\nF : TopCat.Presheaf C Y\nhf : OpenEmbedding \u2191f\nthis : Mono f\nU : Opens \u2191X\nV : Opens \u2191Y\ni : V \u27f6 (IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj U\nx : \u2191Y\nh : x \u2208 V.1\n\u22a2 x \u2208 Set.range fun x => \u2191f x\n[PROOFSTEP]\nobtain \u27e8_, _, rfl\u27e9 := i.le h\n[GOAL]\ncase hF.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf : X \u27f6 Y\nF : TopCat.Presheaf C Y\nhf : OpenEmbedding \u2191f\nthis : Mono f\nU : Opens \u2191X\nV : Opens \u2191Y\ni : V \u27f6 (IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj U\nw\u271d : (forget TopCat).obj X\nleft\u271d : w\u271d \u2208 \u2191U\nh : \u2191f w\u271d \u2208 V.1\n\u22a2 \u2191f w\u271d \u2208 Set.range fun x => \u2191f x\n[PROOFSTEP]\nexact \u27e8_, rfl\u27e9\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf : X \u27f6 Y\nF : TopCat.Presheaf C Y\nhf : IsOpenMap \u2191f\n\u22a2 CoverPreserving (Opens.grothendieckTopology \u2191X) (Opens.grothendieckTopology \u2191Y) (functor hf)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cover_preserve\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf : X \u27f6 Y\nF : TopCat.Presheaf C Y\nhf : IsOpenMap \u2191f\n\u22a2 \u2200 {U : Opens \u2191X} {S : Sieve U},\n S \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U \u2192\n Sieve.functorPushforward (functor hf) S \u2208\n GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191Y) ((functor hf).obj U)\n[PROOFSTEP]\nrintro U S hU _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase cover_preserve.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf : X \u27f6 Y\nF : TopCat.Presheaf C Y\nhf : IsOpenMap \u2191f\nU : Opens \u2191X\nS : Sieve U\nhU : S \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\nx : (forget TopCat).obj X\nhx : x \u2208 \u2191U\n\u22a2 \u2203 U_1 f_1, (Sieve.functorPushforward (functor hf) S).arrows f_1 \u2227 \u2191f x \u2208 U_1\n[PROOFSTEP]\nobtain \u27e8V, i, hV, hxV\u27e9 := hU x hx\n[GOAL]\ncase cover_preserve.intro.intro.intro.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf : X \u27f6 Y\nF : TopCat.Presheaf C Y\nhf : IsOpenMap \u2191f\nU : Opens \u2191X\nS : Sieve U\nhU : S \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\nx : (forget TopCat).obj X\nhx : x \u2208 \u2191U\nV : Opens \u2191X\ni : V \u27f6 U\nhV : S.arrows i\nhxV : x \u2208 V\n\u22a2 \u2203 U_1 f_1, (Sieve.functorPushforward (functor hf) S).arrows f_1 \u2227 \u2191f x \u2208 U_1\n[PROOFSTEP]\nexact \u27e8_, hf.functor.map i, \u27e8_, i, \ud835\udfd9 _, hV, rfl\u27e9, Set.mem_image_of_mem f hxV\u27e9\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type u_1\nB : \u03b9 \u2192 Opens \u2191X\nF\u271d : Presheaf C X\nF' : Sheaf C X\nh\u271d : Opens.IsBasis (Set.range B)\nF : Sheaf C X\nU : Opens \u2191X\nh : U = \u22a5\n\u22a2 Limits.IsTerminal (F.val.obj (op U))\n[PROOFSTEP]\nconvert F.isTerminalOfEmpty\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type u_1\nB : \u03b9 \u2192 Opens \u2191X\nF : Presheaf C X\nF' : Sheaf C X\nh : Opens.IsBasis (Set.range B)\n\u03b1 : (inducedFunctor B).op \u22d9 F \u27f6 (inducedFunctor B).op \u22d9 F'.val\ni : \u03b9\n\u22a2 NatTrans.app (\u2191(restrictHomEquivHom F F' h) \u03b1) (op (B i)) = NatTrans.app \u03b1 (op i)\n[PROOFSTEP]\nnth_rw 2 [\u2190 (restrictHomEquivHom F F' h).left_inv \u03b1]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type u_1\nB : \u03b9 \u2192 Opens \u2191X\nF : Presheaf C X\nF' : Sheaf C X\nh : Opens.IsBasis (Set.range B)\n\u03b1 : (inducedFunctor B).op \u22d9 F \u27f6 (inducedFunctor B).op \u22d9 F'.val\ni : \u03b9\n\u22a2 NatTrans.app (\u2191(restrictHomEquivHom F F' h) \u03b1) (op (B i)) =\n NatTrans.app (Equiv.invFun (restrictHomEquivHom F F' h) (Equiv.toFun (restrictHomEquivHom F F' h) \u03b1)) (op i)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type u_1\nB : \u03b9 \u2192 Opens \u2191X\nF : Presheaf C X\nF' : Sheaf C X\nh : Opens.IsBasis (Set.range B)\n\u03b1 \u03b2 : F \u27f6 F'.val\nhe : \u2200 (i : \u03b9), NatTrans.app \u03b1 (op (B i)) = NatTrans.app \u03b2 (op (B i))\n\u22a2 \u03b1 = \u03b2\n[PROOFSTEP]\napply (restrictHomEquivHom F F' h).symm.injective\n[GOAL]\ncase a\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type u_1\nB : \u03b9 \u2192 Opens \u2191X\nF : Presheaf C X\nF' : Sheaf C X\nh : Opens.IsBasis (Set.range B)\n\u03b1 \u03b2 : F \u27f6 F'.val\nhe : \u2200 (i : \u03b9), NatTrans.app \u03b1 (op (B i)) = NatTrans.app \u03b2 (op (B i))\n\u22a2 \u2191(restrictHomEquivHom F F' h).symm \u03b1 = \u2191(restrictHomEquivHom F F' h).symm \u03b2\n[PROOFSTEP]\next i\n[GOAL]\ncase a.w.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type u_1\nB : \u03b9 \u2192 Opens \u2191X\nF : Presheaf C X\nF' : Sheaf C X\nh : Opens.IsBasis (Set.range B)\n\u03b1 \u03b2 : F \u27f6 F'.val\nhe : \u2200 (i : \u03b9), NatTrans.app \u03b1 (op (B i)) = NatTrans.app \u03b2 (op (B i))\ni : (InducedCategory (Opens \u2191X) B)\u1d52\u1d56\n\u22a2 NatTrans.app (\u2191(restrictHomEquivHom F F' h).symm \u03b1) i = NatTrans.app (\u2191(restrictHomEquivHom F F' h).symm \u03b2) i\n[PROOFSTEP]\nexact he i.unop\n", "meta": {"mathlib_filename": "Mathlib.Topology.Sheaves.SheafCondition.Sites", "llama_tokens": 6546, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5156199157230157, "lm_q2_score": 0.4416730056646256, "lm_q1q2_score": 0.22773539795792527}} {"text": "[GOAL]\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\n\u22a2 Function.Injective fun f => f.toFun\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e8f, _\u27e9, _\u27e9, _\u27e9 \u27e8\u27e8\u27e8g, _\u27e9, _\u27e9, _\u27e9 h\n[GOAL]\ncase mk.mk.mk.mk.mk.mk\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf : A \u2192 B\nmap_smul'\u271d\u00b9 : \u2200 (m : R) (x : A), f (m \u2022 x) = m \u2022 f x\nmap_zero'\u271d\u00b9 : MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } 0 = 0\nmap_add'\u271d\u00b9 :\n \u2200 (x y : A),\n MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } (x + y) =\n MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } x +\n MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } y\nmap_mul'\u271d\u00b9 :\n \u2200 (x y : A),\n MulActionHom.toFun\n { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n map_add' := map_add'\u271d\u00b9 }.toMulActionHom\n (x * y) =\n MulActionHom.toFun\n { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n map_add' := map_add'\u271d\u00b9 }.toMulActionHom\n x *\n MulActionHom.toFun\n { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n map_add' := map_add'\u271d\u00b9 }.toMulActionHom\n y\ng : A \u2192 B\nmap_smul'\u271d : \u2200 (m : R) (x : A), g (m \u2022 x) = m \u2022 g x\nmap_zero'\u271d : MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } 0 = 0\nmap_add'\u271d :\n \u2200 (x y : A),\n MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } (x + y) =\n MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } x +\n MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } y\nmap_mul'\u271d :\n \u2200 (x y : A),\n MulActionHom.toFun\n { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d,\n map_add' := map_add'\u271d }.toMulActionHom\n (x * y) =\n MulActionHom.toFun\n { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d,\n map_add' := map_add'\u271d }.toMulActionHom\n x *\n MulActionHom.toFun\n { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d,\n map_add' := map_add'\u271d }.toMulActionHom\n y\nh :\n (fun f => f.toFun)\n {\n toDistribMulActionHom :=\n { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n map_add' := map_add'\u271d\u00b9 },\n map_mul' := map_mul'\u271d\u00b9 } =\n (fun f => f.toFun)\n {\n toDistribMulActionHom :=\n { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d },\n map_mul' := map_mul'\u271d }\n\u22a2 {\n toDistribMulActionHom :=\n { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n map_add' := map_add'\u271d\u00b9 },\n map_mul' := map_mul'\u271d\u00b9 } =\n {\n toDistribMulActionHom :=\n { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d },\n map_mul' := map_mul'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\n\u22a2 Function.Injective FunLike.coe\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e8f, _\u27e9, _\u27e9, _\u27e9 \u27e8\u27e8\u27e8g, _\u27e9, _\u27e9, _\u27e9 h\n[GOAL]\ncase mk.mk.mk.mk.mk.mk\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf : A \u2192 B\nmap_smul'\u271d\u00b9 : \u2200 (m : R) (x : A), f (m \u2022 x) = m \u2022 f x\nmap_zero'\u271d\u00b9 : MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } 0 = 0\nmap_add'\u271d\u00b9 :\n \u2200 (x y : A),\n MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } (x + y) =\n MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } x +\n MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } y\nmap_mul'\u271d\u00b9 :\n \u2200 (x y : A),\n MulActionHom.toFun\n { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n map_add' := map_add'\u271d\u00b9 }.toMulActionHom\n (x * y) =\n MulActionHom.toFun\n { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n map_add' := map_add'\u271d\u00b9 }.toMulActionHom\n x *\n MulActionHom.toFun\n { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n map_add' := map_add'\u271d\u00b9 }.toMulActionHom\n y\ng : A \u2192 B\nmap_smul'\u271d : \u2200 (m : R) (x : A), g (m \u2022 x) = m \u2022 g x\nmap_zero'\u271d : MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } 0 = 0\nmap_add'\u271d :\n \u2200 (x y : A),\n MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } (x + y) =\n MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } x +\n MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } y\nmap_mul'\u271d :\n \u2200 (x y : A),\n MulActionHom.toFun\n { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d,\n map_add' := map_add'\u271d }.toMulActionHom\n (x * y) =\n MulActionHom.toFun\n { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d,\n map_add' := map_add'\u271d }.toMulActionHom\n x *\n MulActionHom.toFun\n { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d,\n map_add' := map_add'\u271d }.toMulActionHom\n y\nh :\n \u2191{\n toDistribMulActionHom :=\n { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n map_add' := map_add'\u271d\u00b9 },\n map_mul' := map_mul'\u271d\u00b9 } =\n \u2191{\n toDistribMulActionHom :=\n { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d },\n map_mul' := map_mul'\u271d }\n\u22a2 {\n toDistribMulActionHom :=\n { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n map_add' := map_add'\u271d\u00b9 },\n map_mul' := map_mul'\u271d\u00b9 } =\n {\n toDistribMulActionHom :=\n { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d },\n map_mul' := map_mul'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf g : A \u2192\u2099\u2090[R] B\n\u22a2 f = g \u2192 \u2200 (x : A), \u2191f x = \u2191g x\n[PROOFSTEP]\nrintro rfl x\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf : A \u2192\u2099\u2090[R] B\nx : A\n\u22a2 \u2191f x = \u2191f x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf : A \u2192\u2099\u2090[R] B\nh\u2081 : \u2200 (m : R) (x : A), \u2191f (m \u2022 x) = m \u2022 \u2191f x\nh\u2082 : MulActionHom.toFun { toFun := \u2191f, map_smul' := h\u2081 } 0 = 0\nh\u2083 :\n \u2200 (x y : A),\n MulActionHom.toFun { toFun := \u2191f, map_smul' := h\u2081 } (x + y) =\n MulActionHom.toFun { toFun := \u2191f, map_smul' := h\u2081 } x + MulActionHom.toFun { toFun := \u2191f, map_smul' := h\u2081 } y\nh\u2084 :\n \u2200 (x y : A),\n MulActionHom.toFun\n { toMulActionHom := { toFun := \u2191f, map_smul' := h\u2081 }, map_zero' := h\u2082, map_add' := h\u2083 }.toMulActionHom (x * y) =\n MulActionHom.toFun\n { toMulActionHom := { toFun := \u2191f, map_smul' := h\u2081 }, map_zero' := h\u2082, map_add' := h\u2083 }.toMulActionHom x *\n MulActionHom.toFun\n { toMulActionHom := { toFun := \u2191f, map_smul' := h\u2081 }, map_zero' := h\u2082, map_add' := h\u2083 }.toMulActionHom y\n\u22a2 { toDistribMulActionHom := { toMulActionHom := { toFun := \u2191f, map_smul' := h\u2081 }, map_zero' := h\u2082, map_add' := h\u2083 },\n map_mul' := h\u2084 } =\n f\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf g : A \u2192\u2099\u2090[R] B\nh : \u2191f = \u2191g\n\u22a2 f = g\n[PROOFSTEP]\next a\n[GOAL]\ncase h\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf g : A \u2192\u2099\u2090[R] B\nh : \u2191f = \u2191g\na : A\n\u22a2 \u2191f a = \u2191g a\n[PROOFSTEP]\nexact DistribMulActionHom.congr_fun h a\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf g : A \u2192\u2099\u2090[R] B\nh : \u2191f = \u2191g\n\u22a2 f = g\n[PROOFSTEP]\next a\n[GOAL]\ncase h\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf g : A \u2192\u2099\u2090[R] B\nh : \u2191f = \u2191g\na : A\n\u22a2 \u2191f a = \u2191g a\n[PROOFSTEP]\nexact FunLike.congr_fun h a\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf : A \u2192\u2099\u2090[R] B\nh\u2081 : \u2200 (m : R) (x : A), \u2191f (m \u2022 x) = m \u2022 \u2191f x\nh\u2082 : MulActionHom.toFun { toFun := \u2191f, map_smul' := h\u2081 } 0 = 0\nh\u2083 :\n \u2200 (x y : A),\n MulActionHom.toFun { toFun := \u2191f, map_smul' := h\u2081 } (x + y) =\n MulActionHom.toFun { toFun := \u2191f, map_smul' := h\u2081 } x + MulActionHom.toFun { toFun := \u2191f, map_smul' := h\u2081 } y\nh\u2084 :\n \u2200 (x y : A),\n MulActionHom.toFun\n { toMulActionHom := { toFun := \u2191f, map_smul' := h\u2081 }, map_zero' := h\u2082, map_add' := h\u2083 }.toMulActionHom (x * y) =\n MulActionHom.toFun\n { toMulActionHom := { toFun := \u2191f, map_smul' := h\u2081 }, map_zero' := h\u2082, map_add' := h\u2083 }.toMulActionHom x *\n MulActionHom.toFun\n { toMulActionHom := { toFun := \u2191f, map_smul' := h\u2081 }, map_zero' := h\u2082, map_add' := h\u2083 }.toMulActionHom y\n\u22a2 \u2191{ toDistribMulActionHom := { toMulActionHom := { toFun := \u2191f, map_smul' := h\u2081 }, map_zero' := h\u2082, map_add' := h\u2083 },\n map_mul' := h\u2084 } =\n { toMulActionHom := { toFun := \u2191f, map_smul' := h\u2081 }, map_zero' := h\u2082, map_add' := h\u2083 }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf : A \u2192\u2099\u2090[R] B\nh\u2081 : \u2200 (m : R) (x : A), \u2191f (m \u2022 x) = m \u2022 \u2191f x\nh\u2082 : MulActionHom.toFun { toFun := \u2191f, map_smul' := h\u2081 } 0 = 0\nh\u2083 :\n \u2200 (x y : A),\n MulActionHom.toFun { toFun := \u2191f, map_smul' := h\u2081 } (x + y) =\n MulActionHom.toFun { toFun := \u2191f, map_smul' := h\u2081 } x + MulActionHom.toFun { toFun := \u2191f, map_smul' := h\u2081 } y\nh\u2084 :\n \u2200 (x y : A),\n MulActionHom.toFun\n { toMulActionHom := { toFun := \u2191f, map_smul' := h\u2081 }, map_zero' := h\u2082, map_add' := h\u2083 }.toMulActionHom (x * y) =\n MulActionHom.toFun\n { toMulActionHom := { toFun := \u2191f, map_smul' := h\u2081 }, map_zero' := h\u2082, map_add' := h\u2083 }.toMulActionHom x *\n MulActionHom.toFun\n { toMulActionHom := { toFun := \u2191f, map_smul' := h\u2081 }, map_zero' := h\u2082, map_add' := h\u2083 }.toMulActionHom y\n\u22a2 \u2191{ toDistribMulActionHom := { toMulActionHom := { toFun := \u2191f, map_smul' := h\u2081 }, map_zero' := h\u2082, map_add' := h\u2083 },\n map_mul' := h\u2084 } =\n { toFun := \u2191f, map_mul' := h\u2084 }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nsrc\u271d : A \u2192+[R] B := 0\n\u22a2 \u2200 (x y : A),\n MulActionHom.toFun\n { toMulActionHom := src\u271d.toMulActionHom, map_zero' := (_ : MulActionHom.toFun src\u271d.toMulActionHom 0 = 0),\n map_add' :=\n (_ :\n \u2200 (x y : A),\n MulActionHom.toFun src\u271d.toMulActionHom (x + y) =\n MulActionHom.toFun src\u271d.toMulActionHom x +\n MulActionHom.toFun src\u271d.toMulActionHom y) }.toMulActionHom\n (x * y) =\n MulActionHom.toFun\n { toMulActionHom := src\u271d.toMulActionHom, map_zero' := (_ : MulActionHom.toFun src\u271d.toMulActionHom 0 = 0),\n map_add' :=\n (_ :\n \u2200 (x y : A),\n MulActionHom.toFun src\u271d.toMulActionHom (x + y) =\n MulActionHom.toFun src\u271d.toMulActionHom x +\n MulActionHom.toFun src\u271d.toMulActionHom y) }.toMulActionHom\n x *\n MulActionHom.toFun\n { toMulActionHom := src\u271d.toMulActionHom, map_zero' := (_ : MulActionHom.toFun src\u271d.toMulActionHom 0 = 0),\n map_add' :=\n (_ :\n \u2200 (x y : A),\n MulActionHom.toFun src\u271d.toMulActionHom (x + y) =\n MulActionHom.toFun src\u271d.toMulActionHom x +\n MulActionHom.toFun src\u271d.toMulActionHom y) }.toMulActionHom\n y\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf : A \u2192\u2099\u2090[R] B\ng : A \u2192\u2099\u2090[R] C\nc : R\nx : A\n\u22a2 Pi.prod (\u2191f) (\u2191g) (c \u2022 x) = c \u2022 Pi.prod (\u2191f) (\u2191g) x\n[PROOFSTEP]\nsimp only [Pi.prod, Prod.smul_mk, map_smul, RingHom.id_apply]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf : A \u2192\u2099\u2090[R] B\ng : A \u2192\u2099\u2090[R] C\n\u22a2 MulActionHom.toFun\n { toFun := Pi.prod \u2191f \u2191g, map_smul' := (_ : \u2200 (c : R) (x : A), (\u2191f (c \u2022 x), \u2191g (c \u2022 x)) = (c \u2022 \u2191f x, c \u2022 \u2191g x)) }\n 0 =\n 0\n[PROOFSTEP]\nsimp only [Pi.prod, Prod.zero_eq_mk, map_zero]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf : A \u2192\u2099\u2090[R] B\ng : A \u2192\u2099\u2090[R] C\nx y : A\n\u22a2 MulActionHom.toFun\n { toFun := Pi.prod \u2191f \u2191g, map_smul' := (_ : \u2200 (c : R) (x : A), (\u2191f (c \u2022 x), \u2191g (c \u2022 x)) = (c \u2022 \u2191f x, c \u2022 \u2191g x)) }\n (x + y) =\n MulActionHom.toFun\n { toFun := Pi.prod \u2191f \u2191g,\n map_smul' := (_ : \u2200 (c : R) (x : A), (\u2191f (c \u2022 x), \u2191g (c \u2022 x)) = (c \u2022 \u2191f x, c \u2022 \u2191g x)) }\n x +\n MulActionHom.toFun\n { toFun := Pi.prod \u2191f \u2191g,\n map_smul' := (_ : \u2200 (c : R) (x : A), (\u2191f (c \u2022 x), \u2191g (c \u2022 x)) = (c \u2022 \u2191f x, c \u2022 \u2191g x)) }\n y\n[PROOFSTEP]\nsimp only [Pi.prod, Prod.mk_add_mk, map_add]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf : A \u2192\u2099\u2090[R] B\ng : A \u2192\u2099\u2090[R] C\nx y : A\n\u22a2 MulActionHom.toFun\n {\n toMulActionHom :=\n { toFun := Pi.prod \u2191f \u2191g,\n map_smul' := (_ : \u2200 (c : R) (x : A), (\u2191f (c \u2022 x), \u2191g (c \u2022 x)) = (c \u2022 \u2191f x, c \u2022 \u2191g x)) },\n map_zero' := (_ : (\u2191f 0, \u2191g 0) = (0, 0)),\n map_add' := (_ : \u2200 (x y : A), (\u2191f (x + y), \u2191g (x + y)) = (\u2191f x + \u2191f y, \u2191g x + \u2191g y)) }.toMulActionHom\n (x * y) =\n MulActionHom.toFun\n {\n toMulActionHom :=\n { toFun := Pi.prod \u2191f \u2191g,\n map_smul' := (_ : \u2200 (c : R) (x : A), (\u2191f (c \u2022 x), \u2191g (c \u2022 x)) = (c \u2022 \u2191f x, c \u2022 \u2191g x)) },\n map_zero' := (_ : (\u2191f 0, \u2191g 0) = (0, 0)),\n map_add' := (_ : \u2200 (x y : A), (\u2191f (x + y), \u2191g (x + y)) = (\u2191f x + \u2191f y, \u2191g x + \u2191g y)) }.toMulActionHom\n x *\n MulActionHom.toFun\n {\n toMulActionHom :=\n { toFun := Pi.prod \u2191f \u2191g,\n map_smul' := (_ : \u2200 (c : R) (x : A), (\u2191f (c \u2022 x), \u2191g (c \u2022 x)) = (c \u2022 \u2191f x, c \u2022 \u2191g x)) },\n map_zero' := (_ : (\u2191f 0, \u2191g 0) = (0, 0)),\n map_add' := (_ : \u2200 (x y : A), (\u2191f (x + y), \u2191g (x + y)) = (\u2191f x + \u2191f y, \u2191g x + \u2191g y)) }.toMulActionHom\n y\n[PROOFSTEP]\nsimp only [Pi.prod, Prod.mk_mul_mk, map_mul]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf : A \u2192\u2099\u2090[R] B\ng : A \u2192\u2099\u2090[R] C\n\u22a2 comp (fst R B C) (prod f g) = f\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf : A \u2192\u2099\u2090[R] B\ng : A \u2192\u2099\u2090[R] C\n\u22a2 comp (snd R B C) (prod f g) = g\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Hom.NonUnitalAlg", "llama_tokens": 8654, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.585101139733739, "lm_q2_score": 0.38861802670584894, "lm_q1q2_score": 0.22738085034666883}} {"text": "[GOAL]\nX : Scheme\n\u22a2 T0Space \u2191\u2191X.toPresheafedSpace\n[PROOFSTEP]\nrefine' T0Space.of_open_cover fun x => _\n[GOAL]\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 \u2203 s, x \u2208 s \u2227 IsOpen s \u2227 T0Space \u2191s\n[PROOFSTEP]\nobtain \u27e8U, R, \u27e8e\u27e9\u27e9 := X.local_affine x\n[GOAL]\ncase intro.intro.intro\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\nU : OpenNhds x\nR : CommRingCat\ne :\n LocallyRingedSpace.restrict X.toLocallyRingedSpace (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)\n\u22a2 \u2203 s, x \u2208 s \u2227 IsOpen s \u2227 T0Space \u2191s\n[PROOFSTEP]\nlet e' : U.1 \u2243\u209c PrimeSpectrum R :=\n homeoOfIso ((LocallyRingedSpace.forgetToSheafedSpace \u22d9 SheafedSpace.forget _).mapIso e)\n[GOAL]\ncase intro.intro.intro\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\nU : OpenNhds x\nR : CommRingCat\ne :\n LocallyRingedSpace.restrict X.toLocallyRingedSpace (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)\ne' : { x_1 // x_1 \u2208 U.obj } \u2243\u209c PrimeSpectrum \u2191R :=\n homeoOfIso ((LocallyRingedSpace.forgetToSheafedSpace \u22d9 SheafedSpace.forget CommRingCat).mapIso e)\n\u22a2 \u2203 s, x \u2208 s \u2227 IsOpen s \u2227 T0Space \u2191s\n[PROOFSTEP]\nexact \u27e8U.1.1, U.2, U.1.2, e'.embedding.t0Space\u27e9\n[GOAL]\nX : Scheme\n\u22a2 QuasiSober \u2191\u2191X.toPresheafedSpace\n[PROOFSTEP]\napply (config := { allowSynthFailures := true })\n quasiSober_of_open_cover (Set.range fun x => Set.range <| (X.affineCover.map x).1.base)\n[GOAL]\ncase hS\nX : Scheme\n\u22a2 \u2200 (s : \u2191(Set.range fun x => Set.range \u2191(Scheme.OpenCover.map (Scheme.affineCover X) x).val.base)), IsOpen \u2191s\n[PROOFSTEP]\nrintro \u27e8_, i, rfl\u27e9\n[GOAL]\ncase hS.mk.intro\nX : Scheme\ni : (Scheme.affineCover X).J\n\u22a2 IsOpen\n \u2191{ val := (fun x => Set.range \u2191(Scheme.OpenCover.map (Scheme.affineCover X) x).val.base) i,\n property :=\n (_ :\n \u2203 y,\n (fun x => Set.range \u2191(Scheme.OpenCover.map (Scheme.affineCover X) x).val.base) y =\n (fun x => Set.range \u2191(Scheme.OpenCover.map (Scheme.affineCover X) x).val.base) i) }\n[PROOFSTEP]\nexact (X.affineCover.IsOpen i).base_open.open_range\n[GOAL]\ncase hS'\nX : Scheme\n\u22a2 \u2200 (s : \u2191(Set.range fun x => Set.range \u2191(Scheme.OpenCover.map (Scheme.affineCover X) x).val.base)), QuasiSober \u2191\u2191s\n[PROOFSTEP]\nrintro \u27e8_, i, rfl\u27e9\n[GOAL]\ncase hS'.mk.intro\nX : Scheme\ni : (Scheme.affineCover X).J\n\u22a2 QuasiSober\n \u2191\u2191{ val := (fun x => Set.range \u2191(Scheme.OpenCover.map (Scheme.affineCover X) x).val.base) i,\n property :=\n (_ :\n \u2203 y,\n (fun x => Set.range \u2191(Scheme.OpenCover.map (Scheme.affineCover X) x).val.base) y =\n (fun x => Set.range \u2191(Scheme.OpenCover.map (Scheme.affineCover X) x).val.base) i) }\n[PROOFSTEP]\nexact\n @OpenEmbedding.quasiSober _ _ _ _ _\n (Homeomorph.ofEmbedding _ (X.affineCover.IsOpen i).base_open.toEmbedding).symm.openEmbedding\n PrimeSpectrum.quasiSober\n[GOAL]\ncase hS''\nX : Scheme\n\u22a2 (\u22c3\u2080 Set.range fun x => Set.range \u2191(Scheme.OpenCover.map (Scheme.affineCover X) x).val.base) = \u22a4\n[PROOFSTEP]\nrw [Set.top_eq_univ, Set.sUnion_range, Set.eq_univ_iff_forall]\n[GOAL]\ncase hS''\nX : Scheme\n\u22a2 \u2200 (x : (forget TopCat).obj \u2191X.toPresheafedSpace),\n x \u2208 \u22c3 (x : (Scheme.affineCover X).J), Set.range \u2191(Scheme.OpenCover.map (Scheme.affineCover X) x).val.base\n[PROOFSTEP]\nintro x\n[GOAL]\ncase hS''\nX : Scheme\nx : (forget TopCat).obj \u2191X.toPresheafedSpace\n\u22a2 x \u2208 \u22c3 (x : (Scheme.affineCover X).J), Set.range \u2191(Scheme.OpenCover.map (Scheme.affineCover X) x).val.base\n[PROOFSTEP]\nexact \u27e8_, \u27e8_, rfl\u27e9, X.affineCover.Covers x\u27e9\n[GOAL]\nX : Scheme\ninst\u271d : \u2200 (x : \u2191\u2191X.toPresheafedSpace), _root_.IsReduced \u2191(Presheaf.stalk X.presheaf x)\n\u22a2 IsReduced X\n[PROOFSTEP]\nrefine' \u27e8fun U => \u27e8fun s hs => _\u27e9\u27e9\n[GOAL]\nX : Scheme\ninst\u271d : \u2200 (x : \u2191\u2191X.toPresheafedSpace), _root_.IsReduced \u2191(Presheaf.stalk X.presheaf x)\nU : Opens \u2191\u2191X.toPresheafedSpace\ns : \u2191(X.presheaf.obj (op U))\nhs : IsNilpotent s\n\u22a2 s = 0\n[PROOFSTEP]\napply Presheaf.section_ext X.sheaf U s 0\n[GOAL]\nX : Scheme\ninst\u271d : \u2200 (x : \u2191\u2191X.toPresheafedSpace), _root_.IsReduced \u2191(Presheaf.stalk X.presheaf x)\nU : Opens \u2191\u2191X.toPresheafedSpace\ns : \u2191(X.presheaf.obj (op U))\nhs : IsNilpotent s\n\u22a2 \u2200 (x : { x // x \u2208 U }),\n \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) 0\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Scheme\ninst\u271d : \u2200 (x : \u2191\u2191X.toPresheafedSpace), _root_.IsReduced \u2191(Presheaf.stalk X.presheaf x)\nU : Opens \u2191\u2191X.toPresheafedSpace\ns : \u2191(X.presheaf.obj (op U))\nhs : IsNilpotent s\nx : { x // x \u2208 U }\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) 0\n[PROOFSTEP]\nrw [RingHom.map_zero]\n[GOAL]\nX : Scheme\ninst\u271d : \u2200 (x : \u2191\u2191X.toPresheafedSpace), _root_.IsReduced \u2191(Presheaf.stalk X.presheaf x)\nU : Opens \u2191\u2191X.toPresheafedSpace\ns : \u2191(X.presheaf.obj (op U))\nhs : IsNilpotent s\nx : { x // x \u2208 U }\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0\n[PROOFSTEP]\nchange X.presheaf.germ x s = 0\n[GOAL]\nX : Scheme\ninst\u271d : \u2200 (x : \u2191\u2191X.toPresheafedSpace), _root_.IsReduced \u2191(Presheaf.stalk X.presheaf x)\nU : Opens \u2191\u2191X.toPresheafedSpace\ns : \u2191(X.presheaf.obj (op U))\nhs : IsNilpotent s\nx : { x // x \u2208 U }\n\u22a2 \u2191(Presheaf.germ X.presheaf x) s = 0\n[PROOFSTEP]\nexact (hs.map _).eq_zero\n[GOAL]\nX : Scheme\ninst\u271d : IsReduced X\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 _root_.IsReduced \u2191(Presheaf.stalk X.presheaf x)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase eq_zero\nX : Scheme\ninst\u271d : IsReduced X\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 \u2200 (x_1 : \u2191(Presheaf.stalk X.presheaf x)), IsNilpotent x_1 \u2192 x_1 = 0\n[PROOFSTEP]\nrintro g \u27e8n, e\u27e9\n[GOAL]\ncase eq_zero.intro\nX : Scheme\ninst\u271d : IsReduced X\nx : \u2191\u2191X.toPresheafedSpace\ng : \u2191(Presheaf.stalk X.presheaf x)\nn : \u2115\ne : g ^ n = 0\n\u22a2 g = 0\n[PROOFSTEP]\nobtain \u27e8U, hxU, s, rfl\u27e9 := X.presheaf.germ_exist x g\n[GOAL]\ncase eq_zero.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsReduced X\nx : \u2191\u2191X.toPresheafedSpace\nn : \u2115\nU : Opens \u2191\u2191X.toPresheafedSpace\nhxU : x \u2208 U\ns : (forget CommRingCat).obj (X.presheaf.obj (op U))\ne : \u2191(Presheaf.germ X.presheaf { val := x, property := hxU }) s ^ n = 0\n\u22a2 \u2191(Presheaf.germ X.presheaf { val := x, property := hxU }) s = 0\n[PROOFSTEP]\nrw [\u2190 map_pow, \u2190 map_zero (X.presheaf.germ \u27e8x, hxU\u27e9)] at e \n[GOAL]\ncase eq_zero.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsReduced X\nx : \u2191\u2191X.toPresheafedSpace\nn : \u2115\nU : Opens \u2191\u2191X.toPresheafedSpace\nhxU : x \u2208 U\ns : (forget CommRingCat).obj (X.presheaf.obj (op U))\ne :\n \u2191(Presheaf.germ X.presheaf { val := x, property := hxU }) (s ^ n) =\n \u2191(Presheaf.germ X.presheaf { val := x, property := hxU }) 0\n\u22a2 \u2191(Presheaf.germ X.presheaf { val := x, property := hxU }) s = 0\n[PROOFSTEP]\nobtain \u27e8V, hxV, iU, iV, e'\u27e9 := X.presheaf.germ_eq x hxU hxU _ 0 e\n[GOAL]\ncase eq_zero.intro.intro.intro.intro.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsReduced X\nx : \u2191\u2191X.toPresheafedSpace\nn : \u2115\nU : Opens \u2191\u2191X.toPresheafedSpace\nhxU : x \u2208 U\ns : (forget CommRingCat).obj (X.presheaf.obj (op U))\ne :\n \u2191(Presheaf.germ X.presheaf { val := x, property := hxU }) (s ^ n) =\n \u2191(Presheaf.germ X.presheaf { val := x, property := hxU }) 0\nV : Opens \u2191\u2191X.toPresheafedSpace\nhxV : x \u2208 V\niU iV : V \u27f6 U\ne' : \u2191(X.presheaf.map iU.op) (s ^ n) = \u2191(X.presheaf.map iV.op) 0\n\u22a2 \u2191(Presheaf.germ X.presheaf { val := x, property := hxU }) s = 0\n[PROOFSTEP]\nrw [map_pow, map_zero] at e' \n[GOAL]\ncase eq_zero.intro.intro.intro.intro.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsReduced X\nx : \u2191\u2191X.toPresheafedSpace\nn : \u2115\nU : Opens \u2191\u2191X.toPresheafedSpace\nhxU : x \u2208 U\ns : (forget CommRingCat).obj (X.presheaf.obj (op U))\ne :\n \u2191(Presheaf.germ X.presheaf { val := x, property := hxU }) (s ^ n) =\n \u2191(Presheaf.germ X.presheaf { val := x, property := hxU }) 0\nV : Opens \u2191\u2191X.toPresheafedSpace\nhxV : x \u2208 V\niU iV : V \u27f6 U\ne' : \u2191(X.presheaf.map iU.op) s ^ n = 0\n\u22a2 \u2191(Presheaf.germ X.presheaf { val := x, property := hxU }) s = 0\n[PROOFSTEP]\nreplace e' := (IsNilpotent.mk _ _ e').eq_zero (R := X.presheaf.obj <| op V)\n[GOAL]\ncase eq_zero.intro.intro.intro.intro.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsReduced X\nx : \u2191\u2191X.toPresheafedSpace\nn : \u2115\nU : Opens \u2191\u2191X.toPresheafedSpace\nhxU : x \u2208 U\ns : (forget CommRingCat).obj (X.presheaf.obj (op U))\ne :\n \u2191(Presheaf.germ X.presheaf { val := x, property := hxU }) (s ^ n) =\n \u2191(Presheaf.germ X.presheaf { val := x, property := hxU }) 0\nV : Opens \u2191\u2191X.toPresheafedSpace\nhxV : x \u2208 V\niU iV : V \u27f6 U\ne' : \u2191(X.presheaf.map iU.op) s = 0\n\u22a2 \u2191(Presheaf.germ X.presheaf { val := x, property := hxU }) s = 0\n[PROOFSTEP]\nerw [\u2190 ConcreteCategory.congr_hom (X.presheaf.germ_res iU \u27e8x, hxV\u27e9) s]\n[GOAL]\ncase eq_zero.intro.intro.intro.intro.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsReduced X\nx : \u2191\u2191X.toPresheafedSpace\nn : \u2115\nU : Opens \u2191\u2191X.toPresheafedSpace\nhxU : x \u2208 U\ns : (forget CommRingCat).obj (X.presheaf.obj (op U))\ne :\n \u2191(Presheaf.germ X.presheaf { val := x, property := hxU }) (s ^ n) =\n \u2191(Presheaf.germ X.presheaf { val := x, property := hxU }) 0\nV : Opens \u2191\u2191X.toPresheafedSpace\nhxV : x \u2208 V\niU iV : V \u27f6 U\ne' : \u2191(X.presheaf.map iU.op) s = 0\n\u22a2 \u2191(X.presheaf.map iU.op \u226b Presheaf.germ X.presheaf { val := x, property := hxV }) s = 0\n[PROOFSTEP]\nrw [comp_apply, e', map_zero]\n[GOAL]\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d : IsReduced Y\n\u22a2 IsReduced X\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase component_reduced\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d : IsReduced Y\n\u22a2 autoParam (\u2200 (U : Opens \u2191\u2191X.toPresheafedSpace), _root_.IsReduced \u2191(X.presheaf.obj (op U))) _auto\u271d\n[PROOFSTEP]\nintro U\n[GOAL]\ncase component_reduced\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d : IsReduced Y\nU : Opens \u2191\u2191X.toPresheafedSpace\n\u22a2 _root_.IsReduced \u2191(X.presheaf.obj (op U))\n[PROOFSTEP]\nhave : U = (Opens.map f.1.base).obj (H.base_open.isOpenMap.functor.obj U) := by ext1;\n exact (Set.preimage_image_eq _ H.base_open.inj).symm\n[GOAL]\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d : IsReduced Y\nU : Opens \u2191\u2191X.toPresheafedSpace\n\u22a2 U = (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f.val.base)).obj U)\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d : IsReduced Y\nU : Opens \u2191\u2191X.toPresheafedSpace\n\u22a2 \u2191U = \u2191((Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f.val.base)).obj U))\n[PROOFSTEP]\nexact (Set.preimage_image_eq _ H.base_open.inj).symm\n[GOAL]\ncase component_reduced\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d : IsReduced Y\nU : Opens \u2191\u2191X.toPresheafedSpace\nthis : U = (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f.val.base)).obj U)\n\u22a2 _root_.IsReduced \u2191(X.presheaf.obj (op U))\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase component_reduced\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d : IsReduced Y\nU : Opens \u2191\u2191X.toPresheafedSpace\nthis : U = (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f.val.base)).obj U)\n\u22a2 _root_.IsReduced\n \u2191(X.presheaf.obj (op ((Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f.val.base)).obj U))))\n[PROOFSTEP]\nexact\n isReduced_of_injective (inv <| f.1.c.app (op <| H.base_open.isOpenMap.functor.obj U))\n (asIso <| f.1.c.app (op <| H.base_open.isOpenMap.functor.obj U) :\n Y.presheaf.obj _ \u2245 _).symm.commRingCatIsoToRingEquiv.injective\n[GOAL]\nX : Scheme\nR : CommRingCat\nH : _root_.IsReduced \u2191R\n\u22a2 IsReduced (Scheme.Spec.obj (op R))\n[PROOFSTEP]\napply (config := { allowSynthFailures := true }) isReducedOfStalkIsReduced\n[GOAL]\ncase inst\nX : Scheme\nR : CommRingCat\nH : _root_.IsReduced \u2191R\n\u22a2 \u2200 (x : \u2191\u2191(Scheme.Spec.obj (op R)).toPresheafedSpace),\n _root_.IsReduced \u2191(Presheaf.stalk (Scheme.Spec.obj (op R)).presheaf x)\n[PROOFSTEP]\nintro x\n[GOAL]\ncase inst\nX : Scheme\nR : CommRingCat\nH : _root_.IsReduced \u2191R\nx : \u2191\u2191(Scheme.Spec.obj (op R)).toPresheafedSpace\n\u22a2 _root_.IsReduced \u2191(Presheaf.stalk (Scheme.Spec.obj (op R)).presheaf x)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase inst\nX : Scheme\nR : CommRingCat\nH : _root_.IsReduced \u2191R\nx : \u2191\u2191(Scheme.Spec.obj (op R)).toPresheafedSpace\n\u22a2 _root_.IsReduced \u2191(Presheaf.stalk (Scheme.Spec.obj (op R)).presheaf x)\n[PROOFSTEP]\nhave : _root_.IsReduced (CommRingCat.of <| Localization.AtPrime (PrimeSpectrum.asIdeal x)) := by dsimp; infer_instance\n[GOAL]\nX : Scheme\nR : CommRingCat\nH : _root_.IsReduced \u2191R\nx : \u2191\u2191(Scheme.Spec.obj (op R)).toPresheafedSpace\n\u22a2 _root_.IsReduced \u2191(CommRingCat.of (Localization.AtPrime x.asIdeal))\n[PROOFSTEP]\ndsimp\n[GOAL]\nX : Scheme\nR : CommRingCat\nH : _root_.IsReduced \u2191R\nx : \u2191\u2191(Scheme.Spec.obj (op R)).toPresheafedSpace\n\u22a2 _root_.IsReduced (Localization.AtPrime x.asIdeal)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase inst\nX : Scheme\nR : CommRingCat\nH : _root_.IsReduced \u2191R\nx : \u2191\u2191(Scheme.Spec.obj (op R)).toPresheafedSpace\nthis : _root_.IsReduced \u2191(CommRingCat.of (Localization.AtPrime x.asIdeal))\n\u22a2 _root_.IsReduced \u2191(Presheaf.stalk (Scheme.Spec.obj (op R)).presheaf x)\n[PROOFSTEP]\nexact\n isReduced_of_injective (StructureSheaf.stalkIso R x).hom\n (StructureSheaf.stalkIso R x).commRingCatIsoToRingEquiv.injective\n[GOAL]\nX : Scheme\nR : CommRingCat\n\u22a2 IsReduced (Scheme.Spec.obj (op R)) \u2194 _root_.IsReduced \u2191R\n[PROOFSTEP]\nrefine' \u27e8_, fun h => inferInstance\u27e9\n[GOAL]\nX : Scheme\nR : CommRingCat\n\u22a2 IsReduced (Scheme.Spec.obj (op R)) \u2192 _root_.IsReduced \u2191R\n[PROOFSTEP]\nintro h\n[GOAL]\nX : Scheme\nR : CommRingCat\nh : IsReduced (Scheme.Spec.obj (op R))\n\u22a2 _root_.IsReduced \u2191R\n[PROOFSTEP]\nhave : _root_.IsReduced (LocallyRingedSpace.\u0393.obj (op <| Spec.toLocallyRingedSpace.obj <| op R)) := by\n change _root_.IsReduced ((Scheme.Spec.obj <| op R).presheaf.obj <| op \u22a4); infer_instance\n[GOAL]\nX : Scheme\nR : CommRingCat\nh : IsReduced (Scheme.Spec.obj (op R))\n\u22a2 _root_.IsReduced \u2191(LocallyRingedSpace.\u0393.obj (op (Spec.toLocallyRingedSpace.obj (op R))))\n[PROOFSTEP]\nchange _root_.IsReduced ((Scheme.Spec.obj <| op R).presheaf.obj <| op \u22a4)\n[GOAL]\nX : Scheme\nR : CommRingCat\nh : IsReduced (Scheme.Spec.obj (op R))\n\u22a2 _root_.IsReduced \u2191((Scheme.Spec.obj (op R)).presheaf.obj (op \u22a4))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX : Scheme\nR : CommRingCat\nh : IsReduced (Scheme.Spec.obj (op R))\nthis : _root_.IsReduced \u2191(LocallyRingedSpace.\u0393.obj (op (Spec.toLocallyRingedSpace.obj (op R))))\n\u22a2 _root_.IsReduced \u2191R\n[PROOFSTEP]\nexact isReduced_of_injective (toSpec\u0393 R) (asIso <| toSpec\u0393 R).commRingCatIsoToRingEquiv.injective\n[GOAL]\nX : Scheme\ninst\u271d : IsAffine X\nh : _root_.IsReduced \u2191(X.presheaf.obj (op \u22a4))\n\u22a2 IsReduced (Scheme.Spec.obj (op (Scheme.\u0393.obj (op X))))\n[PROOFSTEP]\nrw [affine_isReduced_iff]\n[GOAL]\nX : Scheme\ninst\u271d : IsAffine X\nh : _root_.IsReduced \u2191(X.presheaf.obj (op \u22a4))\n\u22a2 _root_.IsReduced \u2191(Scheme.\u0393.obj (op X))\n[PROOFSTEP]\nexact h\n[GOAL]\nX : Scheme\nP : (X : Scheme) \u2192 Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : \u2200 (X : Scheme) (U : Opens \u2191\u2191X.toPresheafedSpace), (\u2200 (x : { x // x \u2208 U }), \u2203 V x x, P X V) \u2192 P X U\nh\u2082 :\n \u2200 {X Y : Scheme} (f : X \u27f6 Y) [hf : IsOpenImmersion f],\n \u2203 U V hU hV, P X { carrier := U, is_open' := (_ : IsOpen U) } \u2192 P Y { carrier := V, is_open' := (_ : IsOpen V) }\nh\u2083 : \u2200 (R : CommRingCat), P (Scheme.Spec.obj (op R)) \u22a4\n\u22a2 \u2200 (X : Scheme) (U : Opens \u2191\u2191X.toPresheafedSpace), P X U\n[PROOFSTEP]\nintro X U\n[GOAL]\nX\u271d : Scheme\nP : (X : Scheme) \u2192 Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : \u2200 (X : Scheme) (U : Opens \u2191\u2191X.toPresheafedSpace), (\u2200 (x : { x // x \u2208 U }), \u2203 V x x, P X V) \u2192 P X U\nh\u2082 :\n \u2200 {X Y : Scheme} (f : X \u27f6 Y) [hf : IsOpenImmersion f],\n \u2203 U V hU hV, P X { carrier := U, is_open' := (_ : IsOpen U) } \u2192 P Y { carrier := V, is_open' := (_ : IsOpen V) }\nh\u2083 : \u2200 (R : CommRingCat), P (Scheme.Spec.obj (op R)) \u22a4\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\n\u22a2 P X U\n[PROOFSTEP]\napply h\u2081\n[GOAL]\ncase a\nX\u271d : Scheme\nP : (X : Scheme) \u2192 Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : \u2200 (X : Scheme) (U : Opens \u2191\u2191X.toPresheafedSpace), (\u2200 (x : { x // x \u2208 U }), \u2203 V x x, P X V) \u2192 P X U\nh\u2082 :\n \u2200 {X Y : Scheme} (f : X \u27f6 Y) [hf : IsOpenImmersion f],\n \u2203 U V hU hV, P X { carrier := U, is_open' := (_ : IsOpen U) } \u2192 P Y { carrier := V, is_open' := (_ : IsOpen V) }\nh\u2083 : \u2200 (R : CommRingCat), P (Scheme.Spec.obj (op R)) \u22a4\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\n\u22a2 \u2200 (x : { x // x \u2208 U }), \u2203 V x x, P X V\n[PROOFSTEP]\nintro x\n[GOAL]\ncase a\nX\u271d : Scheme\nP : (X : Scheme) \u2192 Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : \u2200 (X : Scheme) (U : Opens \u2191\u2191X.toPresheafedSpace), (\u2200 (x : { x // x \u2208 U }), \u2203 V x x, P X V) \u2192 P X U\nh\u2082 :\n \u2200 {X Y : Scheme} (f : X \u27f6 Y) [hf : IsOpenImmersion f],\n \u2203 U V hU hV, P X { carrier := U, is_open' := (_ : IsOpen U) } \u2192 P Y { carrier := V, is_open' := (_ : IsOpen V) }\nh\u2083 : \u2200 (R : CommRingCat), P (Scheme.Spec.obj (op R)) \u22a4\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\n\u22a2 \u2203 V x x, P X V\n[PROOFSTEP]\nobtain \u27e8_, \u27e8j, rfl\u27e9, hx, i\u27e9 := X.affineBasisCover_is_basis.exists_subset_of_mem_open (SetLike.mem_coe.2 x.prop) U.isOpen\n[GOAL]\ncase a.intro.intro.intro.intro\nX\u271d : Scheme\nP : (X : Scheme) \u2192 Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : \u2200 (X : Scheme) (U : Opens \u2191\u2191X.toPresheafedSpace), (\u2200 (x : { x // x \u2208 U }), \u2203 V x x, P X V) \u2192 P X U\nh\u2082 :\n \u2200 {X Y : Scheme} (f : X \u27f6 Y) [hf : IsOpenImmersion f],\n \u2203 U V hU hV, P X { carrier := U, is_open' := (_ : IsOpen U) } \u2192 P Y { carrier := V, is_open' := (_ : IsOpen V) }\nh\u2083 : \u2200 (R : CommRingCat), P (Scheme.Spec.obj (op R)) \u22a4\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\nj : (Scheme.affineBasisCover X).J\nhx : \u2191x \u2208 Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base\ni : Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base \u2286 \u2191U\n\u22a2 \u2203 V x x, P X V\n[PROOFSTEP]\nlet U' : Opens _ := \u27e8_, (X.affineBasisCover.IsOpen j).base_open.open_range\u27e9\n[GOAL]\ncase a.intro.intro.intro.intro\nX\u271d : Scheme\nP : (X : Scheme) \u2192 Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : \u2200 (X : Scheme) (U : Opens \u2191\u2191X.toPresheafedSpace), (\u2200 (x : { x // x \u2208 U }), \u2203 V x x, P X V) \u2192 P X U\nh\u2082 :\n \u2200 {X Y : Scheme} (f : X \u27f6 Y) [hf : IsOpenImmersion f],\n \u2203 U V hU hV, P X { carrier := U, is_open' := (_ : IsOpen U) } \u2192 P Y { carrier := V, is_open' := (_ : IsOpen V) }\nh\u2083 : \u2200 (R : CommRingCat), P (Scheme.Spec.obj (op R)) \u22a4\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\nj : (Scheme.affineBasisCover X).J\nhx : \u2191x \u2208 Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base\ni : Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base \u2286 \u2191U\nU' : Opens ((forget TopCat).obj \u2191X.toPresheafedSpace) :=\n { carrier := Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base,\n is_open' := (_ : IsOpen (Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base)) }\n\u22a2 \u2203 V x x, P X V\n[PROOFSTEP]\nlet i' : U' \u27f6 U := homOfLE i\n[GOAL]\ncase a.intro.intro.intro.intro\nX\u271d : Scheme\nP : (X : Scheme) \u2192 Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : \u2200 (X : Scheme) (U : Opens \u2191\u2191X.toPresheafedSpace), (\u2200 (x : { x // x \u2208 U }), \u2203 V x x, P X V) \u2192 P X U\nh\u2082 :\n \u2200 {X Y : Scheme} (f : X \u27f6 Y) [hf : IsOpenImmersion f],\n \u2203 U V hU hV, P X { carrier := U, is_open' := (_ : IsOpen U) } \u2192 P Y { carrier := V, is_open' := (_ : IsOpen V) }\nh\u2083 : \u2200 (R : CommRingCat), P (Scheme.Spec.obj (op R)) \u22a4\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\nj : (Scheme.affineBasisCover X).J\nhx : \u2191x \u2208 Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base\ni : Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base \u2286 \u2191U\nU' : Opens ((forget TopCat).obj \u2191X.toPresheafedSpace) :=\n { carrier := Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base,\n is_open' := (_ : IsOpen (Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base)) }\ni' : U' \u27f6 U := homOfLE i\n\u22a2 \u2203 V x x, P X V\n[PROOFSTEP]\nrefine' \u27e8U', hx, i', _\u27e9\n[GOAL]\ncase a.intro.intro.intro.intro\nX\u271d : Scheme\nP : (X : Scheme) \u2192 Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : \u2200 (X : Scheme) (U : Opens \u2191\u2191X.toPresheafedSpace), (\u2200 (x : { x // x \u2208 U }), \u2203 V x x, P X V) \u2192 P X U\nh\u2082 :\n \u2200 {X Y : Scheme} (f : X \u27f6 Y) [hf : IsOpenImmersion f],\n \u2203 U V hU hV, P X { carrier := U, is_open' := (_ : IsOpen U) } \u2192 P Y { carrier := V, is_open' := (_ : IsOpen V) }\nh\u2083 : \u2200 (R : CommRingCat), P (Scheme.Spec.obj (op R)) \u22a4\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\nj : (Scheme.affineBasisCover X).J\nhx : \u2191x \u2208 Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base\ni : Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base \u2286 \u2191U\nU' : Opens ((forget TopCat).obj \u2191X.toPresheafedSpace) :=\n { carrier := Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base,\n is_open' := (_ : IsOpen (Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base)) }\ni' : U' \u27f6 U := homOfLE i\n\u22a2 P X U'\n[PROOFSTEP]\nobtain \u27e8_, _, rfl, rfl, h\u2082'\u27e9 := h\u2082 (X.affineBasisCover.map j)\n[GOAL]\ncase a.intro.intro.intro.intro.intro.intro.intro.intro\nX\u271d : Scheme\nP : (X : Scheme) \u2192 Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : \u2200 (X : Scheme) (U : Opens \u2191\u2191X.toPresheafedSpace), (\u2200 (x : { x // x \u2208 U }), \u2203 V x x, P X V) \u2192 P X U\nh\u2082 :\n \u2200 {X Y : Scheme} (f : X \u27f6 Y) [hf : IsOpenImmersion f],\n \u2203 U V hU hV, P X { carrier := U, is_open' := (_ : IsOpen U) } \u2192 P Y { carrier := V, is_open' := (_ : IsOpen V) }\nh\u2083 : \u2200 (R : CommRingCat), P (Scheme.Spec.obj (op R)) \u22a4\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\nj : (Scheme.affineBasisCover X).J\nhx : \u2191x \u2208 Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base\ni : Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base \u2286 \u2191U\nU' : Opens ((forget TopCat).obj \u2191X.toPresheafedSpace) :=\n { carrier := Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base,\n is_open' := (_ : IsOpen (Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base)) }\ni' : U' \u27f6 U := homOfLE i\nh\u2082' :\n P (Scheme.OpenCover.obj (Scheme.affineBasisCover X) j) { carrier := \u22a4, is_open' := (_ : IsOpen \u22a4) } \u2192\n P X\n { carrier := Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base,\n is_open' := (_ : IsOpen (Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base)) }\n\u22a2 P X U'\n[PROOFSTEP]\napply h\u2082'\n[GOAL]\ncase a.intro.intro.intro.intro.intro.intro.intro.intro\nX\u271d : Scheme\nP : (X : Scheme) \u2192 Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : \u2200 (X : Scheme) (U : Opens \u2191\u2191X.toPresheafedSpace), (\u2200 (x : { x // x \u2208 U }), \u2203 V x x, P X V) \u2192 P X U\nh\u2082 :\n \u2200 {X Y : Scheme} (f : X \u27f6 Y) [hf : IsOpenImmersion f],\n \u2203 U V hU hV, P X { carrier := U, is_open' := (_ : IsOpen U) } \u2192 P Y { carrier := V, is_open' := (_ : IsOpen V) }\nh\u2083 : \u2200 (R : CommRingCat), P (Scheme.Spec.obj (op R)) \u22a4\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\nj : (Scheme.affineBasisCover X).J\nhx : \u2191x \u2208 Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base\ni : Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base \u2286 \u2191U\nU' : Opens ((forget TopCat).obj \u2191X.toPresheafedSpace) :=\n { carrier := Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base,\n is_open' := (_ : IsOpen (Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base)) }\ni' : U' \u27f6 U := homOfLE i\nh\u2082' :\n P (Scheme.OpenCover.obj (Scheme.affineBasisCover X) j) { carrier := \u22a4, is_open' := (_ : IsOpen \u22a4) } \u2192\n P X\n { carrier := Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base,\n is_open' := (_ : IsOpen (Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover X) j).val.base)) }\n\u22a2 P (Scheme.OpenCover.obj (Scheme.affineBasisCover X) j) { carrier := \u22a4, is_open' := (_ : IsOpen \u22a4) }\n[PROOFSTEP]\napply h\u2083\n[GOAL]\nX : Scheme\nP : (X : Scheme) \u2192 \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : \u2200 (R : CommRingCat) (x : PrimeSpectrum \u2191R), P (Scheme.Spec.obj (op R)) x\nh\u2082 : \u2200 {X Y : Scheme} (f : X \u27f6 Y) [inst : IsOpenImmersion f] (x : \u2191\u2191X.toPresheafedSpace), P X x \u2192 P Y (\u2191f.val.base x)\n\u22a2 \u2200 (X : Scheme) (x : \u2191\u2191X.toPresheafedSpace), P X x\n[PROOFSTEP]\nintro X x\n[GOAL]\nX\u271d : Scheme\nP : (X : Scheme) \u2192 \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : \u2200 (R : CommRingCat) (x : PrimeSpectrum \u2191R), P (Scheme.Spec.obj (op R)) x\nh\u2082 : \u2200 {X Y : Scheme} (f : X \u27f6 Y) [inst : IsOpenImmersion f] (x : \u2191\u2191X.toPresheafedSpace), P X x \u2192 P Y (\u2191f.val.base x)\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 P X x\n[PROOFSTEP]\nobtain \u27e8y, e\u27e9 := X.affineCover.Covers x\n[GOAL]\ncase intro\nX\u271d : Scheme\nP : (X : Scheme) \u2192 \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : \u2200 (R : CommRingCat) (x : PrimeSpectrum \u2191R), P (Scheme.Spec.obj (op R)) x\nh\u2082 : \u2200 {X Y : Scheme} (f : X \u27f6 Y) [inst : IsOpenImmersion f] (x : \u2191\u2191X.toPresheafedSpace), P X x \u2192 P Y (\u2191f.val.base x)\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\ny :\n (forget TopCat).obj\n \u2191(Scheme.OpenCover.obj (Scheme.affineCover X) (Scheme.OpenCover.f (Scheme.affineCover X) x)).toPresheafedSpace\ne : \u2191(Scheme.OpenCover.map (Scheme.affineCover X) (Scheme.OpenCover.f (Scheme.affineCover X) x)).val.base y = x\n\u22a2 P X x\n[PROOFSTEP]\nconvert h\u2082 (X.affineCover.map (X.affineCover.f x)) y _\n[GOAL]\ncase h.e'_2\nX\u271d : Scheme\nP : (X : Scheme) \u2192 \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : \u2200 (R : CommRingCat) (x : PrimeSpectrum \u2191R), P (Scheme.Spec.obj (op R)) x\nh\u2082 : \u2200 {X Y : Scheme} (f : X \u27f6 Y) [inst : IsOpenImmersion f] (x : \u2191\u2191X.toPresheafedSpace), P X x \u2192 P Y (\u2191f.val.base x)\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\ny :\n (forget TopCat).obj\n \u2191(Scheme.OpenCover.obj (Scheme.affineCover X) (Scheme.OpenCover.f (Scheme.affineCover X) x)).toPresheafedSpace\ne : \u2191(Scheme.OpenCover.map (Scheme.affineCover X) (Scheme.OpenCover.f (Scheme.affineCover X) x)).val.base y = x\n\u22a2 x = \u2191(Scheme.OpenCover.map (Scheme.affineCover X) (Scheme.OpenCover.f (Scheme.affineCover X) x)).val.base y\n[PROOFSTEP]\nrw [e]\n[GOAL]\ncase intro\nX\u271d : Scheme\nP : (X : Scheme) \u2192 \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : \u2200 (R : CommRingCat) (x : PrimeSpectrum \u2191R), P (Scheme.Spec.obj (op R)) x\nh\u2082 : \u2200 {X Y : Scheme} (f : X \u27f6 Y) [inst : IsOpenImmersion f] (x : \u2191\u2191X.toPresheafedSpace), P X x \u2192 P Y (\u2191f.val.base x)\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\ny :\n (forget TopCat).obj\n \u2191(Scheme.OpenCover.obj (Scheme.affineCover X) (Scheme.OpenCover.f (Scheme.affineCover X) x)).toPresheafedSpace\ne : \u2191(Scheme.OpenCover.map (Scheme.affineCover X) (Scheme.OpenCover.f (Scheme.affineCover X) x)).val.base y = x\n\u22a2 P (Scheme.OpenCover.obj (Scheme.affineCover X) (Scheme.OpenCover.f (Scheme.affineCover X) x)) y\n[PROOFSTEP]\napply h\u2081\n[GOAL]\nX\u271d : Scheme\nX : Scheme\nhX : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns : \u2191(X.presheaf.obj (op U))\nhs : Scheme.basicOpen X s = \u22a5\n\u22a2 s = 0\n[PROOFSTEP]\napply TopCat.Presheaf.section_ext X.sheaf U\n[GOAL]\ncase h\nX\u271d : Scheme\nX : Scheme\nhX : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns : \u2191(X.presheaf.obj (op U))\nhs : Scheme.basicOpen X s = \u22a5\n\u22a2 \u2200 (x : { x // x \u2208 U }),\n \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) 0\n[PROOFSTEP]\nconv => intro x; rw [RingHom.map_zero]\n[GOAL]\nX\u271d : Scheme\nX : Scheme\nhX : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns : \u2191(X.presheaf.obj (op U))\nhs : Scheme.basicOpen X s = \u22a5\n| \u2200 (x : { x // x \u2208 U }),\n \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) 0\n[PROOFSTEP]\nintro x; rw [RingHom.map_zero]\n[GOAL]\nX\u271d : Scheme\nX : Scheme\nhX : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns : \u2191(X.presheaf.obj (op U))\nhs : Scheme.basicOpen X s = \u22a5\n| \u2200 (x : { x // x \u2208 U }),\n \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) 0\n[PROOFSTEP]\nintro x; rw [RingHom.map_zero]\n[GOAL]\nX\u271d : Scheme\nX : Scheme\nhX : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns : \u2191(X.presheaf.obj (op U))\nhs : Scheme.basicOpen X s = \u22a5\n| \u2200 (x : { x // x \u2208 U }),\n \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) 0\n[PROOFSTEP]\nintro x\n[GOAL]\nX\u271d : Scheme\nX : Scheme\nhX : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns : \u2191(X.presheaf.obj (op U))\nhs : Scheme.basicOpen X s = \u22a5\nx : { x // x \u2208 U }\n| \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) 0\n[PROOFSTEP]\nrw [RingHom.map_zero]\n[GOAL]\ncase h\nX\u271d : Scheme\nX : Scheme\nhX : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns : \u2191(X.presheaf.obj (op U))\nhs : Scheme.basicOpen X s = \u22a5\n\u22a2 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0\n[PROOFSTEP]\nrefine'\n (@reduce_to_affine_global\n (fun X U => \u2200 [IsReduced X] (s : X.presheaf.obj (op U)), X.basicOpen s = \u22a5 \u2192 \u2200 x, (X.sheaf.presheaf.germ x) s = 0)\n _ _ _)\n X U s hs\n[GOAL]\ncase h.refine'_1\nX\u271d : Scheme\nX : Scheme\nhX : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns : \u2191(X.presheaf.obj (op U))\nhs : Scheme.basicOpen X s = \u22a5\n\u22a2 \u2200 (X : Scheme) (U : Opens \u2191\u2191X.toPresheafedSpace),\n (\u2200 (x : { x // x \u2208 U }),\n \u2203 V x x,\n (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n X V) \u2192\n (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n X U\n[PROOFSTEP]\nintro X U hx hX s hs x\n[GOAL]\ncase h.refine'_1\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU\u271d : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U\u271d))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhx :\n \u2200 (x : { x // x \u2208 U }),\n \u2203 V x x,\n (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n X V\nhX : IsReduced X\ns : \u2191(X.presheaf.obj (op U))\nhs : Scheme.basicOpen X s = \u22a5\nx : { x // x \u2208 U }\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0\n[PROOFSTEP]\nobtain \u27e8V, hx, i, H\u27e9 := hx x\n[GOAL]\ncase h.refine'_1.intro.intro.intro\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU\u271d : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U\u271d))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhx\u271d :\n \u2200 (x : { x // x \u2208 U }),\n \u2203 V x x,\n (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n X V\nhX : IsReduced X\ns : \u2191(X.presheaf.obj (op U))\nhs : Scheme.basicOpen X s = \u22a5\nx : { x // x \u2208 U }\nV : Opens \u2191\u2191X.toPresheafedSpace\nhx : \u2191x \u2208 V\ni : V \u27f6 U\nH :\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op V))),\n Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 V }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0\n[PROOFSTEP]\nspecialize H (X.presheaf.map i.op s)\n[GOAL]\ncase h.refine'_1.intro.intro.intro\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU\u271d : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U\u271d))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhx\u271d :\n \u2200 (x : { x // x \u2208 U }),\n \u2203 V x x,\n (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n X V\nhX : IsReduced X\ns : \u2191(X.presheaf.obj (op U))\nhs : Scheme.basicOpen X s = \u22a5\nx : { x // x \u2208 U }\nV : Opens \u2191\u2191X.toPresheafedSpace\nhx : \u2191x \u2208 V\ni : V \u27f6 U\nH :\n Scheme.basicOpen X (\u2191(X.presheaf.map i.op) s) = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 V }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) (\u2191(X.presheaf.map i.op) s) = 0\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0\n[PROOFSTEP]\nerw [Scheme.basicOpen_res] at H \n[GOAL]\ncase h.refine'_1.intro.intro.intro\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU\u271d : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U\u271d))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhx\u271d :\n \u2200 (x : { x // x \u2208 U }),\n \u2203 V x x,\n (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n X V\nhX : IsReduced X\ns : \u2191(X.presheaf.obj (op U))\nhs : Scheme.basicOpen X s = \u22a5\nx : { x // x \u2208 U }\nV : Opens \u2191\u2191X.toPresheafedSpace\nhx : \u2191x \u2208 V\ni : V \u27f6 U\nH :\n V \u2293 Scheme.basicOpen X s = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 V }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) (\u2191(X.presheaf.map i.op) s) = 0\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0\n[PROOFSTEP]\nrw [hs] at H \n[GOAL]\ncase h.refine'_1.intro.intro.intro\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU\u271d : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U\u271d))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhx\u271d :\n \u2200 (x : { x // x \u2208 U }),\n \u2203 V x x,\n (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n X V\nhX : IsReduced X\ns : \u2191(X.presheaf.obj (op U))\nhs : Scheme.basicOpen X s = \u22a5\nx : { x // x \u2208 U }\nV : Opens \u2191\u2191X.toPresheafedSpace\nhx : \u2191x \u2208 V\ni : V \u27f6 U\nH :\n V \u2293 \u22a5 = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 V }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) (\u2191(X.presheaf.map i.op) s) = 0\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0\n[PROOFSTEP]\nspecialize H inf_bot_eq \u27e8x, hx\u27e9\n[GOAL]\ncase h.refine'_1.intro.intro.intro\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU\u271d : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U\u271d))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhx\u271d :\n \u2200 (x : { x // x \u2208 U }),\n \u2203 V x x,\n (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n X V\nhX : IsReduced X\ns : \u2191(X.presheaf.obj (op U))\nhs : Scheme.basicOpen X s = \u22a5\nx : { x // x \u2208 U }\nV : Opens \u2191\u2191X.toPresheafedSpace\nhx : \u2191x \u2208 V\ni : V \u27f6 U\nH : \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) { val := \u2191x, property := hx }) (\u2191(X.presheaf.map i.op) s) = 0\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0\n[PROOFSTEP]\nerw [TopCat.Presheaf.germ_res_apply] at H \n[GOAL]\ncase h.refine'_1.intro.intro.intro\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU\u271d : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U\u271d))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhx\u271d :\n \u2200 (x : { x // x \u2208 U }),\n \u2203 V x x,\n (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n X V\nhX : IsReduced X\ns : \u2191(X.presheaf.obj (op U))\nhs : Scheme.basicOpen X s = \u22a5\nx : { x // x \u2208 U }\nV : Opens \u2191\u2191X.toPresheafedSpace\nhx : \u2191x \u2208 V\ni : V \u27f6 U\nH :\n \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X))\n ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) { val := \u2191x, property := hx }))\n s =\n 0\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0\n[PROOFSTEP]\nexact H\n[GOAL]\ncase h.refine'_2\nX\u271d : Scheme\nX : Scheme\nhX : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns : \u2191(X.presheaf.obj (op U))\nhs : Scheme.basicOpen X s = \u22a5\n\u22a2 \u2200 {X Y : Scheme} (f : X \u27f6 Y) [hf : IsOpenImmersion f],\n \u2203 U V hU hV,\n (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n X { carrier := U, is_open' := (_ : IsOpen U) } \u2192\n (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n Y { carrier := V, is_open' := (_ : IsOpen V) }\n[PROOFSTEP]\nrintro X Y f hf\n[GOAL]\ncase h.refine'_2\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX : IsReduced X\u271d\nU : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns : \u2191(X\u271d.presheaf.obj (op U))\nhs : Scheme.basicOpen X\u271d s = \u22a5\nX Y : Scheme\nf : X \u27f6 Y\nhf : IsOpenImmersion f\n\u22a2 \u2203 U V hU hV,\n (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n X { carrier := U, is_open' := (_ : IsOpen U) } \u2192\n (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n Y { carrier := V, is_open' := (_ : IsOpen V) }\n[PROOFSTEP]\nhave e : f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ := by\n rw [\u2190 Set.image_univ, Set.preimage_image_eq _ hf.base_open.inj]\n[GOAL]\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX : IsReduced X\u271d\nU : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns : \u2191(X\u271d.presheaf.obj (op U))\nhs : Scheme.basicOpen X\u271d s = \u22a5\nX Y : Scheme\nf : X \u27f6 Y\nhf : IsOpenImmersion f\n\u22a2 \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ\n[PROOFSTEP]\nrw [\u2190 Set.image_univ, Set.preimage_image_eq _ hf.base_open.inj]\n[GOAL]\ncase h.refine'_2\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX : IsReduced X\u271d\nU : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns : \u2191(X\u271d.presheaf.obj (op U))\nhs : Scheme.basicOpen X\u271d s = \u22a5\nX Y : Scheme\nf : X \u27f6 Y\nhf : IsOpenImmersion f\ne : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ\n\u22a2 \u2203 U V hU hV,\n (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n X { carrier := U, is_open' := (_ : IsOpen U) } \u2192\n (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n Y { carrier := V, is_open' := (_ : IsOpen V) }\n[PROOFSTEP]\nrefine' \u27e8_, _, e, rfl, _\u27e9\n[GOAL]\ncase h.refine'_2\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX : IsReduced X\u271d\nU : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns : \u2191(X\u271d.presheaf.obj (op U))\nhs : Scheme.basicOpen X\u271d s = \u22a5\nX Y : Scheme\nf : X \u27f6 Y\nhf : IsOpenImmersion f\ne : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ\n\u22a2 (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n X\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) } \u2192\n (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n Y { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }\n[PROOFSTEP]\nrintro H hX s hs \u27e8_, x, rfl\u27e9\n[GOAL]\ncase h.refine'_2.mk.intro\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX Y : Scheme\nf : X \u27f6 Y\nhf : IsOpenImmersion f\ne : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ\nH :\n \u2200 [inst : IsReduced X]\n (s :\n \u2191(X.presheaf.obj\n (op\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200\n (x :\n { x //\n x \u2208\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) } }),\n \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0\nhX : IsReduced Y\ns : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }))\nhs : Scheme.basicOpen Y s = \u22a5\nx : (forget TopCat).obj \u2191X.toPresheafedSpace\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf Y))\n { val := \u2191f.val.base x, property := (_ : \u2203 y, \u2191f.val.base y = \u2191f.val.base x) })\n s =\n 0\n[PROOFSTEP]\nhaveI := isReducedOfOpenImmersion f\n[GOAL]\ncase h.refine'_2.mk.intro\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX Y : Scheme\nf : X \u27f6 Y\nhf : IsOpenImmersion f\ne : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ\nH :\n \u2200 [inst : IsReduced X]\n (s :\n \u2191(X.presheaf.obj\n (op\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200\n (x :\n { x //\n x \u2208\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) } }),\n \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0\nhX : IsReduced Y\ns : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }))\nhs : Scheme.basicOpen Y s = \u22a5\nx : (forget TopCat).obj \u2191X.toPresheafedSpace\nthis : IsReduced X\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf Y))\n { val := \u2191f.val.base x, property := (_ : \u2203 y, \u2191f.val.base y = \u2191f.val.base x) })\n s =\n 0\n[PROOFSTEP]\nspecialize H (f.1.c.app _ s) _ \u27e8x, by rw [Opens.mem_mk, e]; trivial\u27e9\n[GOAL]\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX Y : Scheme\nf : X \u27f6 Y\nhf : IsOpenImmersion f\ne : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ\nH :\n \u2200 [inst : IsReduced X]\n (s :\n \u2191(X.presheaf.obj\n (op\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200\n (x :\n { x //\n x \u2208\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) } }),\n \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0\nhX : IsReduced Y\ns : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }))\nhs : Scheme.basicOpen Y s = \u22a5\nx : (forget TopCat).obj \u2191X.toPresheafedSpace\nthis : IsReduced X\n\u22a2 x \u2208\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }\n[PROOFSTEP]\nrw [Opens.mem_mk, e]\n[GOAL]\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX Y : Scheme\nf : X \u27f6 Y\nhf : IsOpenImmersion f\ne : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ\nH :\n \u2200 [inst : IsReduced X]\n (s :\n \u2191(X.presheaf.obj\n (op\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200\n (x :\n { x //\n x \u2208\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) } }),\n \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0\nhX : IsReduced Y\ns : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }))\nhs : Scheme.basicOpen Y s = \u22a5\nx : (forget TopCat).obj \u2191X.toPresheafedSpace\nthis : IsReduced X\n\u22a2 x \u2208 Set.univ\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase h.refine'_2.mk.intro\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX Y : Scheme\nf : X \u27f6 Y\nhf : IsOpenImmersion f\ne : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ\nH :\n \u2200 [inst : IsReduced X]\n (s :\n \u2191(X.presheaf.obj\n (op\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200\n (x :\n { x //\n x \u2208\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) } }),\n \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0\nhX : IsReduced Y\ns : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }))\nhs : Scheme.basicOpen Y s = \u22a5\nx : (forget TopCat).obj \u2191X.toPresheafedSpace\nthis : IsReduced X\n\u22a2 Scheme.basicOpen X\n (\u2191(NatTrans.app f.val.c\n (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }))\n s) =\n \u22a5\n[PROOFSTEP]\nrw [\u2190 Scheme.preimage_basicOpen, hs]\n[GOAL]\ncase h.refine'_2.mk.intro\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX Y : Scheme\nf : X \u27f6 Y\nhf : IsOpenImmersion f\ne : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ\nH :\n \u2200 [inst : IsReduced X]\n (s :\n \u2191(X.presheaf.obj\n (op\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200\n (x :\n { x //\n x \u2208\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) } }),\n \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0\nhX : IsReduced Y\ns : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }))\nhs : Scheme.basicOpen Y s = \u22a5\nx : (forget TopCat).obj \u2191X.toPresheafedSpace\nthis : IsReduced X\n\u22a2 (Opens.map f.val.base).obj \u22a5 = \u22a5\n[PROOFSTEP]\next1\n[GOAL]\ncase h.refine'_2.mk.intro.h\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX Y : Scheme\nf : X \u27f6 Y\nhf : IsOpenImmersion f\ne : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ\nH :\n \u2200 [inst : IsReduced X]\n (s :\n \u2191(X.presheaf.obj\n (op\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }))),\n Scheme.basicOpen X s = \u22a5 \u2192\n \u2200\n (x :\n { x //\n x \u2208\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) } }),\n \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0\nhX : IsReduced Y\ns : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }))\nhs : Scheme.basicOpen Y s = \u22a5\nx : (forget TopCat).obj \u2191X.toPresheafedSpace\nthis : IsReduced X\n\u22a2 \u2191((Opens.map f.val.base).obj \u22a5) = \u2191\u22a5\n[PROOFSTEP]\nsimp [Opens.map]\n[GOAL]\ncase h.refine'_2.mk.intro\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX Y : Scheme\nf : X \u27f6 Y\nhf : IsOpenImmersion f\ne : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ\nhX : IsReduced Y\ns : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }))\nhs : Scheme.basicOpen Y s = \u22a5\nx : (forget TopCat).obj \u2191X.toPresheafedSpace\nthis : IsReduced X\nH :\n \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X))\n { val := x,\n property :=\n (_ :\n x \u2208\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) })\n (\u2191(NatTrans.app f.val.c\n (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }))\n s) =\n 0\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf Y))\n { val := \u2191f.val.base x, property := (_ : \u2203 y, \u2191f.val.base y = \u2191f.val.base x) })\n s =\n 0\n[PROOFSTEP]\nerw [\u2190 PresheafedSpace.stalkMap_germ_apply f.1 \u27e8_, _\u27e9 \u27e8x, _\u27e9] at H \n[GOAL]\ncase h.refine'_2.mk.intro\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX Y : Scheme\nf : X \u27f6 Y\nhf : IsOpenImmersion f\ne : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ\nhX : IsReduced Y\ns : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }))\nhs : Scheme.basicOpen Y s = \u22a5\nx : (forget TopCat).obj \u2191X.toPresheafedSpace\nthis : IsReduced X\nH :\n \u2191(PresheafedSpace.stalkMap f.val\n \u2191{ val := x,\n property :=\n (_ :\n x \u2208\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) })\n (\u2191(Presheaf.germ Y.presheaf\n {\n val :=\n \u2191f.val.base\n \u2191{ val := x,\n property :=\n (_ :\n x \u2208\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) },\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ :\n x \u2208\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) } \u2208\n (Opens.map f.val.base).obj\n { carrier := fun x => \u2203 y, \u2191f.val.base y = x,\n is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }) })\n s) =\n 0\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf Y))\n { val := \u2191f.val.base x, property := (_ : \u2203 y, \u2191f.val.base y = \u2191f.val.base x) })\n s =\n 0\n[PROOFSTEP]\napply_fun inv <| PresheafedSpace.stalkMap f.val x at H \n[GOAL]\ncase h.refine'_2.mk.intro\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX Y : Scheme\nf : X \u27f6 Y\nhf : IsOpenImmersion f\ne : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ\nhX : IsReduced Y\ns : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }))\nhs : Scheme.basicOpen Y s = \u22a5\nx : (forget TopCat).obj \u2191X.toPresheafedSpace\nthis : IsReduced X\nH :\n \u2191(inv (PresheafedSpace.stalkMap f.val x))\n (\u2191(PresheafedSpace.stalkMap f.val\n \u2191{ val := x,\n property :=\n (_ :\n x \u2208\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) })\n (\u2191(Presheaf.germ Y.presheaf\n {\n val :=\n \u2191f.val.base\n \u2191{ val := x,\n property :=\n (_ :\n x \u2208\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) },\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ :\n x \u2208\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) } \u2208\n (Opens.map f.val.base).obj\n { carrier := fun x => \u2203 y, \u2191f.val.base y = x,\n is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }) })\n s)) =\n \u2191(inv (PresheafedSpace.stalkMap f.val x)) 0\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf Y))\n { val := \u2191f.val.base x, property := (_ : \u2203 y, \u2191f.val.base y = \u2191f.val.base x) })\n s =\n 0\n[PROOFSTEP]\nerw [CategoryTheory.IsIso.hom_inv_id_apply, map_zero] at H \n[GOAL]\ncase h.refine'_2.mk.intro\nX\u271d\u00b9 : Scheme\nX\u271d : Scheme\nhX\u271d : IsReduced X\u271d\nU : Opens \u2191\u2191X\u271d.toPresheafedSpace\ns\u271d : \u2191(X\u271d.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5\nX Y : Scheme\nf : X \u27f6 Y\nhf : IsOpenImmersion f\ne : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ\nhX : IsReduced Y\ns : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }))\nhs : Scheme.basicOpen Y s = \u22a5\nx : (forget TopCat).obj \u2191X.toPresheafedSpace\nthis : IsReduced X\nH :\n \u2191(Presheaf.germ Y.presheaf\n {\n val :=\n \u2191f.val.base\n \u2191{ val := x,\n property :=\n (_ :\n x \u2208\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) },\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ :\n x \u2208\n { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base,\n is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) } \u2208\n (Opens.map f.val.base).obj\n { carrier := fun x => \u2203 y, \u2191f.val.base y = x, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }) })\n s =\n 0\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf Y))\n { val := \u2191f.val.base x, property := (_ : \u2203 y, \u2191f.val.base y = \u2191f.val.base x) })\n s =\n 0\n[PROOFSTEP]\nexact H\n[GOAL]\ncase h.refine'_3\nX\u271d : Scheme\nX : Scheme\nhX : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns : \u2191(X.presheaf.obj (op U))\nhs : Scheme.basicOpen X s = \u22a5\n\u22a2 \u2200 (R : CommRingCat),\n (fun X U =>\n \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))),\n Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0)\n (Scheme.Spec.obj (op R)) \u22a4\n[PROOFSTEP]\nintro R hX s hs x\n[GOAL]\ncase h.refine'_3\nX\u271d : Scheme\nX : Scheme\nhX\u271d : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns\u271d : \u2191(X.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X s\u271d = \u22a5\nR : CommRingCat\nhX : IsReduced (Scheme.Spec.obj (op R))\ns : \u2191((Scheme.Spec.obj (op R)).presheaf.obj (op \u22a4))\nhs : Scheme.basicOpen (Scheme.Spec.obj (op R)) s = \u22a5\nx : { x // x \u2208 \u22a4 }\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf (Scheme.Spec.obj (op R)))) x) s = 0\n[PROOFSTEP]\nerw [basicOpen_eq_of_affine', PrimeSpectrum.basicOpen_eq_bot_iff] at hs \n[GOAL]\ncase h.refine'_3\nX\u271d : Scheme\nX : Scheme\nhX\u271d : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns\u271d : \u2191(X.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X s\u271d = \u22a5\nR : CommRingCat\nhX : IsReduced (Scheme.Spec.obj (op R))\ns : \u2191((Scheme.Spec.obj (op R)).presheaf.obj (op \u22a4))\nhs : IsNilpotent (\u2191(Spec\u0393Identity.app R).hom s)\nx : { x // x \u2208 \u22a4 }\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf (Scheme.Spec.obj (op R)))) x) s = 0\n[PROOFSTEP]\nreplace hs := hs.map (Spec\u0393Identity.app R).inv\n[GOAL]\ncase h.refine'_3\nX\u271d : Scheme\nX : Scheme\nhX\u271d : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns\u271d : \u2191(X.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X s\u271d = \u22a5\nR : CommRingCat\nhX : IsReduced (Scheme.Spec.obj (op R))\ns : \u2191((Scheme.Spec.obj (op R)).presheaf.obj (op \u22a4))\nx : { x // x \u2208 \u22a4 }\nhs : IsNilpotent (\u2191(Spec\u0393Identity.app R).inv (\u2191(Spec\u0393Identity.app R).hom s))\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf (Scheme.Spec.obj (op R)))) x) s = 0\n[PROOFSTEP]\nreplace hs := @IsNilpotent.eq_zero _ _ _ _ (show _ from ?_) hs\n[GOAL]\ncase h.refine'_3.refine_2\nX\u271d : Scheme\nX : Scheme\nhX\u271d : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns\u271d : \u2191(X.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X s\u271d = \u22a5\nR : CommRingCat\nhX : IsReduced (Scheme.Spec.obj (op R))\ns : \u2191((Scheme.Spec.obj (op R)).presheaf.obj (op \u22a4))\nx : { x // x \u2208 \u22a4 }\nhs : \u2191(Spec\u0393Identity.app R).inv (\u2191(Spec\u0393Identity.app R).hom s) = 0\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf (Scheme.Spec.obj (op R)))) x) s = 0\ncase h.refine'_3.refine_1\nX\u271d : Scheme\nX : Scheme\nhX\u271d : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns\u271d : \u2191(X.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X s\u271d = \u22a5\nR : CommRingCat\nhX : IsReduced (Scheme.Spec.obj (op R))\ns : \u2191((Scheme.Spec.obj (op R)).presheaf.obj (op \u22a4))\nx : { x // x \u2208 \u22a4 }\nhs : IsNilpotent (\u2191(Spec\u0393Identity.app R).inv (\u2191(Spec\u0393Identity.app R).hom s))\n\u22a2 _root_.IsReduced\n ((fun x => \u2191((Spec.toLocallyRingedSpace.rightOp \u22d9 LocallyRingedSpace.\u0393).obj R)) (\u2191(Spec\u0393Identity.app R).hom s))\n[PROOFSTEP]\nrw [Iso.hom_inv_id_apply] at hs \n[GOAL]\ncase h.refine'_3.refine_2\nX\u271d : Scheme\nX : Scheme\nhX\u271d : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns\u271d : \u2191(X.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X s\u271d = \u22a5\nR : CommRingCat\nhX : IsReduced (Scheme.Spec.obj (op R))\ns : \u2191((Scheme.Spec.obj (op R)).presheaf.obj (op \u22a4))\nx : { x // x \u2208 \u22a4 }\nhs : s = 0\n\u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf (Scheme.Spec.obj (op R)))) x) s = 0\ncase h.refine'_3.refine_1\nX\u271d : Scheme\nX : Scheme\nhX\u271d : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns\u271d : \u2191(X.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X s\u271d = \u22a5\nR : CommRingCat\nhX : IsReduced (Scheme.Spec.obj (op R))\ns : \u2191((Scheme.Spec.obj (op R)).presheaf.obj (op \u22a4))\nx : { x // x \u2208 \u22a4 }\nhs : IsNilpotent (\u2191(Spec\u0393Identity.app R).inv (\u2191(Spec\u0393Identity.app R).hom s))\n\u22a2 _root_.IsReduced\n ((fun x => \u2191((Spec.toLocallyRingedSpace.rightOp \u22d9 LocallyRingedSpace.\u0393).obj R)) (\u2191(Spec\u0393Identity.app R).hom s))\n[PROOFSTEP]\nrw [hs, map_zero]\n[GOAL]\ncase h.refine'_3.refine_1\nX\u271d : Scheme\nX : Scheme\nhX\u271d : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns\u271d : \u2191(X.presheaf.obj (op U))\nhs\u271d : Scheme.basicOpen X s\u271d = \u22a5\nR : CommRingCat\nhX : IsReduced (Scheme.Spec.obj (op R))\ns : \u2191((Scheme.Spec.obj (op R)).presheaf.obj (op \u22a4))\nx : { x // x \u2208 \u22a4 }\nhs : IsNilpotent (\u2191(Spec\u0393Identity.app R).inv (\u2191(Spec\u0393Identity.app R).hom s))\n\u22a2 _root_.IsReduced\n ((fun x => \u2191((Spec.toLocallyRingedSpace.rightOp \u22d9 LocallyRingedSpace.\u0393).obj R)) (\u2191(Spec\u0393Identity.app R).hom s))\n[PROOFSTEP]\nexact @IsReduced.component_reduced _ hX \u22a4\n[GOAL]\nX\u271d : Scheme\nX : Scheme\ninst\u271d : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns : \u2191(X.presheaf.obj (op U))\n\u22a2 Scheme.basicOpen X s = \u22a5 \u2194 s = 0\n[PROOFSTEP]\nrefine' \u27e8eq_zero_of_basicOpen_eq_bot s, _\u27e9\n[GOAL]\nX\u271d : Scheme\nX : Scheme\ninst\u271d : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\ns : \u2191(X.presheaf.obj (op U))\n\u22a2 s = 0 \u2192 Scheme.basicOpen X s = \u22a5\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nX\u271d : Scheme\nX : Scheme\ninst\u271d : IsReduced X\nU : Opens \u2191\u2191X.toPresheafedSpace\n\u22a2 Scheme.basicOpen X 0 = \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Scheme\nh : IsIntegral X\n\u22a2 Nonempty { x // x \u2208 \u22a4 }\n[PROOFSTEP]\nsimp only [Set.univ_nonempty, Opens.nonempty_coeSort, Opens.coe_top]\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\n\u22a2 IsReduced X\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase component_reduced\nX : Scheme\ninst\u271d : IsIntegral X\n\u22a2 autoParam (\u2200 (U : Opens \u2191\u2191X.toPresheafedSpace), _root_.IsReduced \u2191(X.presheaf.obj (op U))) _auto\u271d\n[PROOFSTEP]\nintro U\n[GOAL]\ncase component_reduced\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\n\u22a2 _root_.IsReduced \u2191(X.presheaf.obj (op U))\n[PROOFSTEP]\ncases' U.1.eq_empty_or_nonempty with h h\n[GOAL]\ncase component_reduced.inl\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nh : U.carrier = \u2205\n\u22a2 _root_.IsReduced \u2191(X.presheaf.obj (op U))\n[PROOFSTEP]\nhave : U = \u22a5 := SetLike.ext' h\n[GOAL]\ncase component_reduced.inl\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nh : U.carrier = \u2205\nthis : U = \u22a5\n\u22a2 _root_.IsReduced \u2191(X.presheaf.obj (op U))\n[PROOFSTEP]\nhaveI := CommRingCat.subsingleton_of_isTerminal (X.sheaf.isTerminalOfEqEmpty this)\n[GOAL]\ncase component_reduced.inl\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nh : U.carrier = \u2205\nthis\u271d : U = \u22a5\nthis : Subsingleton \u2191((Scheme.sheaf X).val.obj (op U))\n\u22a2 _root_.IsReduced \u2191(X.presheaf.obj (op U))\n[PROOFSTEP]\nchange _root_.IsReduced (X.sheaf.val.obj (op U))\n[GOAL]\ncase component_reduced.inl\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nh : U.carrier = \u2205\nthis\u271d : U = \u22a5\nthis : Subsingleton \u2191((Scheme.sheaf X).val.obj (op U))\n\u22a2 _root_.IsReduced \u2191((Scheme.sheaf X).val.obj (op U))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase component_reduced.inr\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nh : Set.Nonempty U.carrier\n\u22a2 _root_.IsReduced \u2191(X.presheaf.obj (op U))\n[PROOFSTEP]\nhaveI : Nonempty U := by simpa\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nh : Set.Nonempty U.carrier\n\u22a2 Nonempty { x // x \u2208 U }\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase component_reduced.inr\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nh : Set.Nonempty U.carrier\nthis : Nonempty { x // x \u2208 U }\n\u22a2 _root_.IsReduced \u2191(X.presheaf.obj (op U))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\n\u22a2 IrreducibleSpace \u2191\u2191X.toPresheafedSpace\n[PROOFSTEP]\nby_contra H\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nH : \u00acIrreducibleSpace \u2191\u2191X.toPresheafedSpace\n\u22a2 False\n[PROOFSTEP]\nreplace H : \u00acIsPreirreducible (\u22a4 : Set X.carrier) := fun h =>\n H { toPreirreducibleSpace := \u27e8h\u27e9\n toNonempty := inferInstance }\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nH : \u00acIsPreirreducible \u22a4\n\u22a2 False\n[PROOFSTEP]\nsimp_rw [isPreirreducible_iff_closed_union_closed, not_forall, not_or] at H \n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nH : \u2203 x x_1 h h h, \u00ac\u22a4 \u2286 x \u2227 \u00ac\u22a4 \u2286 x_1\n\u22a2 False\n[PROOFSTEP]\nrcases H with \u27e8S, T, hS, hT, h\u2081, h\u2082, h\u2083\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS : IsClosed S\nhT : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u00ac\u22a4 \u2286 S\nh\u2083 : \u00ac\u22a4 \u2286 T\n\u22a2 False\n[PROOFSTEP]\nerw [not_forall] at h\u2082 h\u2083 \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS : IsClosed S\nhT : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x, \u00ac(x \u2208 \u22a4 \u2192 x \u2208 S)\nh\u2083 : \u2203 x, \u00ac(x \u2208 \u22a4 \u2192 x \u2208 T)\n\u22a2 False\n[PROOFSTEP]\nsimp_rw [not_forall] at h\u2082 h\u2083 \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS : IsClosed S\nhT : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\n\u22a2 False\n[PROOFSTEP]\nhaveI : Nonempty (\u27e8S\u1d9c, hS.1\u27e9 : Opens X.carrier) := \u27e8\u27e8_, h\u2082.choose_spec.choose_spec\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS : IsClosed S\nhT : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\n\u22a2 False\n[PROOFSTEP]\nhaveI : Nonempty (\u27e8T\u1d9c, hT.1\u27e9 : Opens X.carrier) := \u27e8\u27e8_, h\u2083.choose_spec.choose_spec\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS : IsClosed S\nhT : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis\u271d : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\nthis : Nonempty { x // x \u2208 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\n\u22a2 False\n[PROOFSTEP]\nhaveI : Nonempty (\u27e8S\u1d9c, hS.1\u27e9 \u2294 \u27e8T\u1d9c, hT.1\u27e9 : Opens X.carrier) := \u27e8\u27e8_, Or.inl h\u2082.choose_spec.choose_spec\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS : IsClosed S\nhT : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis\u271d\u00b9 : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\nthis\u271d : Nonempty { x // x \u2208 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nthis :\n Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\n\u22a2 False\n[PROOFSTEP]\nlet e : X.presheaf.obj _ \u2245 CommRingCat.of _ :=\n (X.sheaf.isProductOfDisjoint \u27e8_, hS.1\u27e9 \u27e8_, hT.1\u27e9 ?_).conePointUniqueUpToIso (CommRingCat.prodFanIsLimit _ _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_2\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS : IsClosed S\nhT : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis\u271d\u00b9 : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\nthis\u271d : Nonempty { x // x \u2208 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nthis :\n Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\ne : X.presheaf.obj\n (op ({ carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) })) \u2245\n CommRingCat.of\n \u2191(CommRingCat.prodFan ((Scheme.sheaf X).val.obj (op { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) }))\n ((Scheme.sheaf X).val.obj (op { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) }))).1 :=\n IsLimit.conePointUniqueUpToIso\n (Sheaf.isProductOfDisjoint (Scheme.sheaf X) { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) }\n { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } ?intro.intro.intro.intro.intro.intro.refine_1)\n (CommRingCat.prodFanIsLimit ((Scheme.sheaf X).val.obj (op { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) }))\n ((Scheme.sheaf X).val.obj (op { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) })))\n\u22a2 False\ncase intro.intro.intro.intro.intro.intro.refine_1\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS : IsClosed S\nhT : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis\u271d\u00b9 : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\nthis\u271d : Nonempty { x // x \u2208 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nthis :\n Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\n\u22a2 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2293 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } = \u22a5\n[PROOFSTEP]\napply (config := { allowSynthFailures := true }) false_of_nontrivial_of_product_domain\n[GOAL]\ncase inst\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS : IsClosed S\nhT : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis\u271d\u00b9 : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\nthis\u271d : Nonempty { x // x \u2208 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nthis :\n Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\ne : X.presheaf.obj\n (op ({ carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) })) \u2245\n CommRingCat.of\n \u2191(CommRingCat.prodFan ((Scheme.sheaf X).val.obj (op { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) }))\n ((Scheme.sheaf X).val.obj (op { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) }))).1 :=\n IsLimit.conePointUniqueUpToIso\n (Sheaf.isProductOfDisjoint (Scheme.sheaf X) { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) }\n { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } ?intro.intro.intro.intro.intro.intro.refine_1)\n (CommRingCat.prodFanIsLimit ((Scheme.sheaf X).val.obj (op { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) }))\n ((Scheme.sheaf X).val.obj (op { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) })))\n\u22a2 IsDomain (?intro.intro.intro.intro.intro.intro.refine_2.R \u00d7 ?intro.intro.intro.intro.intro.intro.refine_2.S)\n[PROOFSTEP]\nexact e.symm.commRingCatIsoToRingEquiv.toMulEquiv.isDomain _\n[GOAL]\ncase inst\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS : IsClosed S\nhT : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis\u271d\u00b9 : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\nthis\u271d : Nonempty { x // x \u2208 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nthis :\n Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\ne : X.presheaf.obj\n (op ({ carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) })) \u2245\n CommRingCat.of\n \u2191(CommRingCat.prodFan ((Scheme.sheaf X).val.obj (op { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) }))\n ((Scheme.sheaf X).val.obj (op { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) }))).1 :=\n IsLimit.conePointUniqueUpToIso\n (Sheaf.isProductOfDisjoint (Scheme.sheaf X) { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) }\n { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } ?intro.intro.intro.intro.intro.intro.refine_1)\n (CommRingCat.prodFanIsLimit ((Scheme.sheaf X).val.obj (op { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) }))\n ((Scheme.sheaf X).val.obj (op { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) })))\n\u22a2 Nontrivial \u2191((Scheme.sheaf X).val.obj (op { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) }))\n[PROOFSTEP]\napply X.toLocallyRingedSpace.component_nontrivial\n[GOAL]\ncase inst\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS : IsClosed S\nhT : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis\u271d\u00b9 : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\nthis\u271d : Nonempty { x // x \u2208 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nthis :\n Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\ne : X.presheaf.obj\n (op ({ carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) })) \u2245\n CommRingCat.of\n \u2191(CommRingCat.prodFan ((Scheme.sheaf X).val.obj (op { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) }))\n ((Scheme.sheaf X).val.obj (op { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) }))).1 :=\n IsLimit.conePointUniqueUpToIso\n (Sheaf.isProductOfDisjoint (Scheme.sheaf X) { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) }\n { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } ?intro.intro.intro.intro.intro.intro.refine_1)\n (CommRingCat.prodFanIsLimit ((Scheme.sheaf X).val.obj (op { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) }))\n ((Scheme.sheaf X).val.obj (op { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) })))\n\u22a2 Nontrivial \u2191((Scheme.sheaf X).val.obj (op { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) }))\n[PROOFSTEP]\napply X.toLocallyRingedSpace.component_nontrivial\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_1\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS : IsClosed S\nhT : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis\u271d\u00b9 : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\nthis\u271d : Nonempty { x // x \u2208 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nthis :\n Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\n\u22a2 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2293 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } = \u22a5\n[PROOFSTEP]\next x\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_1.h.h\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS : IsClosed S\nhT : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis\u271d\u00b9 : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\nthis\u271d : Nonempty { x // x \u2208 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nthis :\n Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208 \u2191({ carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2293 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) }) \u2194 x \u2208 \u2191\u22a5\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_1.h.h.mp\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS : IsClosed S\nhT : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis\u271d\u00b9 : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\nthis\u271d : Nonempty { x // x \u2208 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nthis :\n Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208 \u2191({ carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2293 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) }) \u2192 x \u2208 \u2191\u22a5\n[PROOFSTEP]\nrintro \u27e8hS, hT\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_1.h.h.mp.intro\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS\u271d : IsClosed S\nhT\u271d : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis\u271d\u00b9 : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\nthis\u271d : Nonempty { x // x \u2208 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nthis :\n Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nx : \u2191\u2191X.toPresheafedSpace\nhS : x \u2208 \u2191{ carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) }\nhT : x \u2208 \u2191{ carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) }\n\u22a2 x \u2208 \u2191\u22a5\n[PROOFSTEP]\ncases' h\u2081 (show x \u2208 \u22a4 by trivial) with h h\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS\u271d : IsClosed S\nhT\u271d : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis\u271d\u00b9 : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\nthis\u271d : Nonempty { x // x \u2208 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nthis :\n Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nx : \u2191\u2191X.toPresheafedSpace\nhS : x \u2208 \u2191{ carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) }\nhT : x \u2208 \u2191{ carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) }\n\u22a2 x \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_1.h.h.mp.intro.inl\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS\u271d : IsClosed S\nhT\u271d : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis\u271d\u00b9 : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\nthis\u271d : Nonempty { x // x \u2208 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nthis :\n Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nx : \u2191\u2191X.toPresheafedSpace\nhS : x \u2208 \u2191{ carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) }\nhT : x \u2208 \u2191{ carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) }\nh : x \u2208 S\n\u22a2 x \u2208 \u2191\u22a5\ncase intro.intro.intro.intro.intro.intro.refine_1.h.h.mp.intro.inr\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS\u271d : IsClosed S\nhT\u271d : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis\u271d\u00b9 : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\nthis\u271d : Nonempty { x // x \u2208 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nthis :\n Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nx : \u2191\u2191X.toPresheafedSpace\nhS : x \u2208 \u2191{ carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) }\nhT : x \u2208 \u2191{ carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) }\nh : x \u2208 T\n\u22a2 x \u2208 \u2191\u22a5\n[PROOFSTEP]\nexacts [hS h, hT h]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_1.h.h.mpr\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS : IsClosed S\nhT : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis\u271d\u00b9 : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\nthis\u271d : Nonempty { x // x \u2208 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nthis :\n Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208 \u2191\u22a5 \u2192 x \u2208 \u2191({ carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2293 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) })\n[PROOFSTEP]\nintro x\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_1.h.h.mpr\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS : IsClosed S\nhT : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis\u271d\u00b9 : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\nthis\u271d : Nonempty { x // x \u2208 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nthis :\n Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nx\u271d : \u2191\u2191X.toPresheafedSpace\nx : x\u271d \u2208 \u2191\u22a5\n\u22a2 x\u271d \u2208 \u2191({ carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2293 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) })\n[PROOFSTEP]\nrefine' x.rec (by contradiction)\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nS T : Set \u2191\u2191X.toPresheafedSpace\nhS : IsClosed S\nhT : IsClosed T\nh\u2081 : \u22a4 \u2286 S \u222a T\nh\u2082 : \u2203 x x_1, \u00acx \u2208 S\nh\u2083 : \u2203 x x_1, \u00acx \u2208 T\nthis\u271d\u00b9 : Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } }\nthis\u271d : Nonempty { x // x \u2208 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nthis :\n Nonempty { x // x \u2208 { carrier := S\u1d9c, is_open' := (_ : IsOpen S\u1d9c) } \u2294 { carrier := T\u1d9c, is_open' := (_ : IsOpen T\u1d9c) } }\nx\u271d : \u2191\u2191X.toPresheafedSpace\nx : x\u271d \u2208 \u2191\u22a5\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\nX : Scheme\ninst\u271d : IsReduced X\nH : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\n\u22a2 IsIntegral X\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase nonempty\nX : Scheme\ninst\u271d : IsReduced X\nH : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\n\u22a2 autoParam (Nonempty \u2191\u2191X.toPresheafedSpace) _auto\u271d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase component_integral\nX : Scheme\ninst\u271d : IsReduced X\nH : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\n\u22a2 autoParam (\u2200 (U : Opens \u2191\u2191X.toPresheafedSpace) [inst : Nonempty { x // x \u2208 U }], IsDomain \u2191(X.presheaf.obj (op U)))\n _auto\u271d\n[PROOFSTEP]\nintro U hU\n[GOAL]\ncase component_integral\nX : Scheme\ninst\u271d : IsReduced X\nH : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\n\u22a2 IsDomain \u2191(X.presheaf.obj (op U))\n[PROOFSTEP]\nhaveI := (@LocallyRingedSpace.component_nontrivial X.toLocallyRingedSpace U hU).1\n[GOAL]\ncase component_integral\nX : Scheme\ninst\u271d : IsReduced X\nH : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis : \u2203 x y, x \u2260 y\n\u22a2 IsDomain \u2191(X.presheaf.obj (op U))\n[PROOFSTEP]\nhave : NoZeroDivisors (X.toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.obj (op U)) :=\n by\n refine' \u27e8fun {a b} e => _\u27e9\n simp_rw [\u2190 basicOpen_eq_bot_iff, \u2190 Opens.not_nonempty_iff_eq_bot]\n by_contra' h\n obtain \u27e8_, \u27e8x, hx\u2081, rfl\u27e9, \u27e8x, hx\u2082, e'\u27e9\u27e9 := nonempty_preirreducible_inter (X.basicOpen a).2 (X.basicOpen b).2 h.1 h.2\n replace e' := Subtype.eq e'\n subst e'\n replace e := congr_arg (X.presheaf.germ x) e\n rw [RingHom.map_mul, RingHom.map_zero] at e \n refine' zero_ne_one' (X.presheaf.stalk x.1) (isUnit_zero_iff.1 _)\n convert hx\u2081.mul hx\u2082\n exact e.symm\n[GOAL]\nX : Scheme\ninst\u271d : IsReduced X\nH : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis : \u2203 x y, x \u2260 y\n\u22a2 NoZeroDivisors \u2191(X.presheaf.obj (op U))\n[PROOFSTEP]\nrefine' \u27e8fun {a b} e => _\u27e9\n[GOAL]\nX : Scheme\ninst\u271d : IsReduced X\nH : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis : \u2203 x y, x \u2260 y\na b : \u2191(X.presheaf.obj (op U))\ne : a * b = 0\n\u22a2 a = 0 \u2228 b = 0\n[PROOFSTEP]\nsimp_rw [\u2190 basicOpen_eq_bot_iff, \u2190 Opens.not_nonempty_iff_eq_bot]\n[GOAL]\nX : Scheme\ninst\u271d : IsReduced X\nH : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis : \u2203 x y, x \u2260 y\na b : \u2191(X.presheaf.obj (op U))\ne : a * b = 0\n\u22a2 \u00acSet.Nonempty \u2191(Scheme.basicOpen X a) \u2228 \u00acSet.Nonempty \u2191(Scheme.basicOpen X b)\n[PROOFSTEP]\nby_contra' h\n[GOAL]\nX : Scheme\ninst\u271d : IsReduced X\nH : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis : \u2203 x y, x \u2260 y\na b : \u2191(X.presheaf.obj (op U))\ne : a * b = 0\nh : Set.Nonempty \u2191(Scheme.basicOpen X a) \u2227 Set.Nonempty \u2191(Scheme.basicOpen X b)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8_, \u27e8x, hx\u2081, rfl\u27e9, \u27e8x, hx\u2082, e'\u27e9\u27e9 := nonempty_preirreducible_inter (X.basicOpen a).2 (X.basicOpen b).2 h.1 h.2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsReduced X\nH : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis : \u2203 x y, x \u2260 y\na b : \u2191(X.presheaf.obj (op U))\ne : a * b = 0\nh : Set.Nonempty \u2191(Scheme.basicOpen X a) \u2227 Set.Nonempty \u2191(Scheme.basicOpen X b)\nx\u271d : { x // x \u2208 U }\nhx\u2081 :\n x\u271d \u2208\n {x |\n IsUnit\n (\u2191(Presheaf.germ (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf x) a)}\nx : { x // x \u2208 U }\nhx\u2082 :\n x \u2208\n {x |\n IsUnit\n (\u2191(Presheaf.germ (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf x) b)}\ne' : \u2191x = \u2191x\u271d\n\u22a2 False\n[PROOFSTEP]\nreplace e' := Subtype.eq e'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsReduced X\nH : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis : \u2203 x y, x \u2260 y\na b : \u2191(X.presheaf.obj (op U))\ne : a * b = 0\nh : Set.Nonempty \u2191(Scheme.basicOpen X a) \u2227 Set.Nonempty \u2191(Scheme.basicOpen X b)\nx\u271d : { x // x \u2208 U }\nhx\u2081 :\n x\u271d \u2208\n {x |\n IsUnit\n (\u2191(Presheaf.germ (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf x) a)}\nx : { x // x \u2208 U }\nhx\u2082 :\n x \u2208\n {x |\n IsUnit\n (\u2191(Presheaf.germ (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf x) b)}\ne' : x = x\u271d\n\u22a2 False\n[PROOFSTEP]\nsubst e'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsReduced X\nH : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis : \u2203 x y, x \u2260 y\na b : \u2191(X.presheaf.obj (op U))\ne : a * b = 0\nh : Set.Nonempty \u2191(Scheme.basicOpen X a) \u2227 Set.Nonempty \u2191(Scheme.basicOpen X b)\nx : { x // x \u2208 U }\nhx\u2082 :\n x \u2208\n {x |\n IsUnit\n (\u2191(Presheaf.germ (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf x) b)}\nhx\u2081 :\n x \u2208\n {x |\n IsUnit\n (\u2191(Presheaf.germ (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf x) a)}\n\u22a2 False\n[PROOFSTEP]\nreplace e := congr_arg (X.presheaf.germ x) e\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsReduced X\nH : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis : \u2203 x y, x \u2260 y\na b : \u2191(X.presheaf.obj (op U))\nh : Set.Nonempty \u2191(Scheme.basicOpen X a) \u2227 Set.Nonempty \u2191(Scheme.basicOpen X b)\nx : { x // x \u2208 U }\nhx\u2082 :\n x \u2208\n {x |\n IsUnit\n (\u2191(Presheaf.germ (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf x) b)}\nhx\u2081 :\n x \u2208\n {x |\n IsUnit\n (\u2191(Presheaf.germ (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf x) a)}\ne : \u2191(Presheaf.germ X.presheaf x) (a * b) = \u2191(Presheaf.germ X.presheaf x) 0\n\u22a2 False\n[PROOFSTEP]\nrw [RingHom.map_mul, RingHom.map_zero] at e \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsReduced X\nH : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis : \u2203 x y, x \u2260 y\na b : \u2191(X.presheaf.obj (op U))\nh : Set.Nonempty \u2191(Scheme.basicOpen X a) \u2227 Set.Nonempty \u2191(Scheme.basicOpen X b)\nx : { x // x \u2208 U }\nhx\u2082 :\n x \u2208\n {x |\n IsUnit\n (\u2191(Presheaf.germ (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf x) b)}\nhx\u2081 :\n x \u2208\n {x |\n IsUnit\n (\u2191(Presheaf.germ (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf x) a)}\ne : \u2191(Presheaf.germ X.presheaf x) a * \u2191(Presheaf.germ X.presheaf x) b = 0\n\u22a2 False\n[PROOFSTEP]\nrefine' zero_ne_one' (X.presheaf.stalk x.1) (isUnit_zero_iff.1 _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsReduced X\nH : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis : \u2203 x y, x \u2260 y\na b : \u2191(X.presheaf.obj (op U))\nh : Set.Nonempty \u2191(Scheme.basicOpen X a) \u2227 Set.Nonempty \u2191(Scheme.basicOpen X b)\nx : { x // x \u2208 U }\nhx\u2082 :\n x \u2208\n {x |\n IsUnit\n (\u2191(Presheaf.germ (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf x) b)}\nhx\u2081 :\n x \u2208\n {x |\n IsUnit\n (\u2191(Presheaf.germ (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf x) a)}\ne : \u2191(Presheaf.germ X.presheaf x) a * \u2191(Presheaf.germ X.presheaf x) b = 0\n\u22a2 IsUnit 0\n[PROOFSTEP]\nconvert hx\u2081.mul hx\u2082\n[GOAL]\ncase h.e'_3.h\nX : Scheme\ninst\u271d : IsReduced X\nH : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis : \u2203 x y, x \u2260 y\na b : \u2191(X.presheaf.obj (op U))\nh : Set.Nonempty \u2191(Scheme.basicOpen X a) \u2227 Set.Nonempty \u2191(Scheme.basicOpen X b)\nx : { x // x \u2208 U }\nhx\u2082 :\n x \u2208\n {x |\n IsUnit\n (\u2191(Presheaf.germ (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf x) b)}\nhx\u2081 :\n x \u2208\n {x |\n IsUnit\n (\u2191(Presheaf.germ (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf x) a)}\ne : \u2191(Presheaf.germ X.presheaf x) a * \u2191(Presheaf.germ X.presheaf x) b = 0\ne_1\u271d :\n \u2191(Presheaf.stalk X.presheaf \u2191x) =\n (fun x_1 =>\n (forget CommRingCat).obj\n (Presheaf.stalk (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf \u2191x))\n a\n\u22a2 0 =\n \u2191(Presheaf.germ (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf x) a *\n \u2191(Presheaf.germ (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf x) b\n[PROOFSTEP]\nexact e.symm\n[GOAL]\ncase component_integral\nX : Scheme\ninst\u271d : IsReduced X\nH : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis\u271d : \u2203 x y, x \u2260 y\nthis : NoZeroDivisors \u2191(X.presheaf.obj (op U))\n\u22a2 IsDomain \u2191(X.presheaf.obj (op U))\n[PROOFSTEP]\nexact NoZeroDivisors.to_isDomain _\n[GOAL]\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d\u00b9 : IsIntegral Y\ninst\u271d : Nonempty \u2191\u2191X.toPresheafedSpace\n\u22a2 IsIntegral X\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase nonempty\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d\u00b9 : IsIntegral Y\ninst\u271d : Nonempty \u2191\u2191X.toPresheafedSpace\n\u22a2 autoParam (Nonempty \u2191\u2191X.toPresheafedSpace) _auto\u271d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase component_integral\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d\u00b9 : IsIntegral Y\ninst\u271d : Nonempty \u2191\u2191X.toPresheafedSpace\n\u22a2 autoParam (\u2200 (U : Opens \u2191\u2191X.toPresheafedSpace) [inst : Nonempty { x // x \u2208 U }], IsDomain \u2191(X.presheaf.obj (op U)))\n _auto\u271d\n[PROOFSTEP]\nintro U hU\n[GOAL]\ncase component_integral\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d\u00b9 : IsIntegral Y\ninst\u271d : Nonempty \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\n\u22a2 IsDomain \u2191(X.presheaf.obj (op U))\n[PROOFSTEP]\nhave : U = (Opens.map f.1.base).obj (H.base_open.isOpenMap.functor.obj U) := by ext1;\n exact (Set.preimage_image_eq _ H.base_open.inj).symm\n[GOAL]\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d\u00b9 : IsIntegral Y\ninst\u271d : Nonempty \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\n\u22a2 U = (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f.val.base)).obj U)\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d\u00b9 : IsIntegral Y\ninst\u271d : Nonempty \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\n\u22a2 \u2191U = \u2191((Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f.val.base)).obj U))\n[PROOFSTEP]\nexact (Set.preimage_image_eq _ H.base_open.inj).symm\n[GOAL]\ncase component_integral\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d\u00b9 : IsIntegral Y\ninst\u271d : Nonempty \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis : U = (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f.val.base)).obj U)\n\u22a2 IsDomain \u2191(X.presheaf.obj (op U))\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase component_integral\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d\u00b9 : IsIntegral Y\ninst\u271d : Nonempty \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis : U = (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f.val.base)).obj U)\n\u22a2 IsDomain \u2191(X.presheaf.obj (op ((Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f.val.base)).obj U))))\n[PROOFSTEP]\nhave : IsDomain (Y.presheaf.obj (op (H.base_open.isOpenMap.functor.obj U))) :=\n by\n apply (config := { allowSynthFailures := true }) IsIntegral.component_integral\n refine' \u27e8\u27e8_, _, hU.some.prop, rfl\u27e9\u27e9\n[GOAL]\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d\u00b9 : IsIntegral Y\ninst\u271d : Nonempty \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis : U = (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f.val.base)).obj U)\n\u22a2 IsDomain \u2191(Y.presheaf.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f.val.base)).obj U)))\n[PROOFSTEP]\napply (config := { allowSynthFailures := true }) IsIntegral.component_integral\n[GOAL]\ncase inst\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d\u00b9 : IsIntegral Y\ninst\u271d : Nonempty \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis : U = (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f.val.base)).obj U)\n\u22a2 Nonempty { x // x \u2208 (IsOpenMap.functor (_ : IsOpenMap \u2191f.val.base)).obj U }\n[PROOFSTEP]\nrefine' \u27e8\u27e8_, _, hU.some.prop, rfl\u27e9\u27e9\n[GOAL]\ncase component_integral\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\ninst\u271d\u00b9 : IsIntegral Y\ninst\u271d : Nonempty \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : Nonempty { x // x \u2208 U }\nthis\u271d : U = (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f.val.base)).obj U)\nthis : IsDomain \u2191(Y.presheaf.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f.val.base)).obj U)))\n\u22a2 IsDomain \u2191(X.presheaf.obj (op ((Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f.val.base)).obj U))))\n[PROOFSTEP]\nexact\n (asIso <| f.1.c.app (op <| H.base_open.isOpenMap.functor.obj U) :\n Y.presheaf.obj _ \u2245 _).symm.commRingCatIsoToRingEquiv.toMulEquiv.isDomain\n _\n[GOAL]\nX : Scheme\nR : CommRingCat\nH : IsDomain \u2191R\n\u22a2 IrreducibleSpace \u2191\u2191(Scheme.Spec.obj (op R)).toPresheafedSpace\n[PROOFSTEP]\nconvert PrimeSpectrum.irreducibleSpace (R := R)\n[GOAL]\nX : Scheme\ninst\u271d\u00b9 : IsAffine X\ninst\u271d : Nonempty \u2191\u2191X.toPresheafedSpace\nh : IsDomain \u2191(X.presheaf.obj (op \u22a4))\n\u22a2 IsIntegral (Scheme.Spec.obj (op (Scheme.\u0393.obj (op X))))\n[PROOFSTEP]\nrw [affine_isIntegral_iff]\n[GOAL]\nX : Scheme\ninst\u271d\u00b9 : IsAffine X\ninst\u271d : Nonempty \u2191\u2191X.toPresheafedSpace\nh : IsDomain \u2191(X.presheaf.obj (op \u22a4))\n\u22a2 IsDomain \u2191(Scheme.\u0393.obj (op X))\n[PROOFSTEP]\nexact h\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nH : Nonempty { x // x \u2208 U }\n\u22a2 Function.Injective \u2191(X.presheaf.map i.op)\n[PROOFSTEP]\nrw [injective_iff_map_eq_zero]\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nH : Nonempty { x // x \u2208 U }\n\u22a2 \u2200 (a : (forget CommRingCat).obj (X.presheaf.obj (op V))), \u2191(X.presheaf.map i.op) a = 0 \u2192 a = 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nH : Nonempty { x // x \u2208 U }\nx : (forget CommRingCat).obj (X.presheaf.obj (op V))\nhx : \u2191(X.presheaf.map i.op) x = 0\n\u22a2 x = 0\n[PROOFSTEP]\nrw [\u2190 basicOpen_eq_bot_iff] at hx \u22a2\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nH : Nonempty { x // x \u2208 U }\nx : (forget CommRingCat).obj (X.presheaf.obj (op V))\nhx : Scheme.basicOpen X (\u2191(X.presheaf.map i.op) x) = \u22a5\n\u22a2 Scheme.basicOpen X x = \u22a5\n[PROOFSTEP]\nrw [Scheme.basicOpen_res] at hx \n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nH : Nonempty { x // x \u2208 U }\nx : (forget CommRingCat).obj (X.presheaf.obj (op V))\nhx : U \u2293 Scheme.basicOpen X x = \u22a5\n\u22a2 Scheme.basicOpen X x = \u22a5\n[PROOFSTEP]\nrevert hx\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nH : Nonempty { x // x \u2208 U }\nx : (forget CommRingCat).obj (X.presheaf.obj (op V))\n\u22a2 U \u2293 Scheme.basicOpen X x = \u22a5 \u2192 Scheme.basicOpen X x = \u22a5\n[PROOFSTEP]\ncontrapose!\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nH : Nonempty { x // x \u2208 U }\nx : (forget CommRingCat).obj (X.presheaf.obj (op V))\n\u22a2 Scheme.basicOpen X x \u2260 \u22a5 \u2192 U \u2293 Scheme.basicOpen X x \u2260 \u22a5\n[PROOFSTEP]\nsimp_rw [Ne.def, \u2190 Opens.not_nonempty_iff_eq_bot, Classical.not_not]\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nH : Nonempty { x // x \u2208 U }\nx : (forget CommRingCat).obj (X.presheaf.obj (op V))\n\u22a2 Set.Nonempty \u2191(Scheme.basicOpen X x) \u2192 Set.Nonempty \u2191(U \u2293 Scheme.basicOpen X x)\n[PROOFSTEP]\napply nonempty_preirreducible_inter U.isOpen (RingedSpace.basicOpen _ _).isOpen\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nH : Nonempty { x // x \u2208 U }\nx : (forget CommRingCat).obj (X.presheaf.obj (op V))\n\u22a2 Set.Nonempty \u2191U\n[PROOFSTEP]\nsimpa using H\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.Properties", "llama_tokens": 45231, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5544704796847396, "lm_q2_score": 0.4073334000459302, "lm_q1q2_score": 0.22585434571508287}} {"text": "[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\n\u22a2 LieRing L\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\n\u22a2 LieAlgebra R L\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra", "llama_tokens": 133, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5117166047041654, "lm_q2_score": 0.43398146480389854, "lm_q1q2_score": 0.2220755216739912}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\n\u22a2 Category.{?u.2036, u\u2081} (Skeleton C)\n[PROOFSTEP]\napply InducedCategory.category\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\n\u22a2 Full (fromSkeleton C)\n[PROOFSTEP]\napply InducedCategory.full\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\n\u22a2 Faithful (fromSkeleton C)\n[PROOFSTEP]\napply InducedCategory.faithful\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\n\u22a2 Skeletal (Skeleton C)\n[PROOFSTEP]\nrintro X Y \u27e8h\u27e9\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nX Y : Skeleton C\nh : X \u2245 Y\n\u22a2 X = Y\n[PROOFSTEP]\nhave : X.out \u2248 Y.out := \u27e8(fromSkeleton C).mapIso h\u27e9\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nX Y : Skeleton C\nh : X \u2245 Y\nthis : Quotient.out X \u2248 Quotient.out Y\n\u22a2 X = Y\n[PROOFSTEP]\nsimpa using Quotient.sound this\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\n\u22a2 \u2200 (a\u2081 b\u2081 a\u2082 b\u2082 : C), a\u2081 \u2248 a\u2082 \u2192 b\u2081 \u2248 b\u2082 \u2192 (fun X Y => Nonempty (X \u27f6 Y)) a\u2081 b\u2081 = (fun X Y => Nonempty (X \u27f6 Y)) a\u2082 b\u2082\n[PROOFSTEP]\nrintro _ _ _ _ \u27e8i\u2081\u27e9 \u27e8i\u2082\u27e9\n[GOAL]\ncase intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\na\u2081\u271d b\u2081\u271d a\u2082\u271d b\u2082\u271d : C\ni\u2081 : a\u2081\u271d \u2245 a\u2082\u271d\ni\u2082 : b\u2081\u271d \u2245 b\u2082\u271d\n\u22a2 (fun X Y => Nonempty (X \u27f6 Y)) a\u2081\u271d b\u2081\u271d = (fun X Y => Nonempty (X \u27f6 Y)) a\u2082\u271d b\u2082\u271d\n[PROOFSTEP]\nexact propext \u27e8Nonempty.map fun f => i\u2081.inv \u226b f \u226b i\u2082.hom, Nonempty.map fun f => i\u2081.hom \u226b f \u226b i\u2082.inv\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\n\u22a2 \u2200 (a : ThinSkeleton C), a \u2264 a\n[PROOFSTEP]\nrefine' Quotient.ind fun a => _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\na : C\n\u22a2 Quotient.mk (isIsomorphicSetoid C) a \u2264 Quotient.mk (isIsomorphicSetoid C) a\n[PROOFSTEP]\nexact \u27e8\ud835\udfd9 _\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nx\u271d\u00b9 x\u271d : ThinSkeleton C\n\u22a2 \u2200 (a b : x\u271d\u00b9 \u27f6 x\u271d), a = b\n[PROOFSTEP]\nrintro \u27e8\u27e8f\u2081\u27e9\u27e9 \u27e8\u27e8_\u27e9\u27e9\n[GOAL]\ncase up.up.up.up\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nx\u271d\u00b9 x\u271d : ThinSkeleton C\nf\u2081 down\u271d : x\u271d\u00b9 \u2264 x\u271d\n\u22a2 { down := { down := f\u2081 } } = { down := { down := down\u271d } }\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Quiver.IsThin C\nsrc\u271d : Preorder (ThinSkeleton C) := preorder C\n\u22a2 \u2200 (a b : C),\n Quotient.mk (isIsomorphicSetoid C) a \u2264 Quotient.mk (isIsomorphicSetoid C) b \u2192\n Quotient.mk (isIsomorphicSetoid C) b \u2264 Quotient.mk (isIsomorphicSetoid C) a \u2192\n Quotient.mk (isIsomorphicSetoid C) a = Quotient.mk (isIsomorphicSetoid C) b\n[PROOFSTEP]\nrintro _ _ \u27e8f\u27e9 \u27e8g\u27e9\n[GOAL]\ncase intro.intro\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Quiver.IsThin C\nsrc\u271d : Preorder (ThinSkeleton C) := preorder C\na\u271d b\u271d : C\nf : a\u271d \u27f6 b\u271d\ng : b\u271d \u27f6 a\u271d\n\u22a2 Quotient.mk (isIsomorphicSetoid C) a\u271d = Quotient.mk (isIsomorphicSetoid C) b\u271d\n[PROOFSTEP]\napply Quotient.sound (equiv_of_both_ways f g)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nR : D \u2964 C\nL : C \u2964 D\nh : L \u22a3 R\nX : ThinSkeleton C\n\u22a2 (\ud835\udfed (ThinSkeleton C)).obj X \u27f6 (map L \u22d9 map R).obj X\n[PROOFSTEP]\nletI := isIsomorphicSetoid C\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nR : D \u2964 C\nL : C \u2964 D\nh : L \u22a3 R\nX : ThinSkeleton C\nthis : Setoid C := isIsomorphicSetoid C\n\u22a2 (\ud835\udfed (ThinSkeleton C)).obj X \u27f6 (map L \u22d9 map R).obj X\n[PROOFSTEP]\nrefine' Quotient.recOnSubsingleton X fun x => homOfLE \u27e8h.unit.app x\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nR : D \u2964 C\nL : C \u2964 D\nh : L \u22a3 R\nX : ThinSkeleton D\n\u22a2 (map R \u22d9 map L).obj X \u27f6 (\ud835\udfed (ThinSkeleton D)).obj X\n[PROOFSTEP]\nletI := isIsomorphicSetoid D\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nR : D \u2964 C\nL : C \u2964 D\nh : L \u22a3 R\nX : ThinSkeleton D\nthis : Setoid D := isIsomorphicSetoid D\n\u22a2 (map R \u22d9 map L).obj X \u27f6 (\ud835\udfed (ThinSkeleton D)).obj X\n[PROOFSTEP]\nrefine' Quotient.recOnSubsingleton X fun x => homOfLE \u27e8h.counit.app x\u27e9\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Skeletal", "llama_tokens": 2729, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.3702253925955866, "lm_q1q2_score": 0.21802170963379436}} {"text": "[GOAL]\nS : Type u\ninst\u271d : State S\ns t : S\nm : t \u2208 l s\n\u22a2 turnBound s \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\nS : Type u\ninst\u271d : State S\ns t : S\nm : t \u2208 l s\nh : turnBound s = 0\n\u22a2 False\n[PROOFSTEP]\nhave t := left_bound m\n[GOAL]\nS : Type u\ninst\u271d : State S\ns t\u271d : S\nm : t\u271d \u2208 l s\nh : turnBound s = 0\nt : turnBound t\u271d < turnBound s\n\u22a2 False\n[PROOFSTEP]\nrw [h] at t \n[GOAL]\nS : Type u\ninst\u271d : State S\ns t\u271d : S\nm : t\u271d \u2208 l s\nh : turnBound s = 0\nt : turnBound t\u271d < 0\n\u22a2 False\n[PROOFSTEP]\nexact Nat.not_succ_le_zero _ t\n[GOAL]\nS : Type u\ninst\u271d : State S\ns t : S\nm : t \u2208 r s\n\u22a2 turnBound s \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\nS : Type u\ninst\u271d : State S\ns t : S\nm : t \u2208 r s\nh : turnBound s = 0\n\u22a2 False\n[PROOFSTEP]\nhave t := right_bound m\n[GOAL]\nS : Type u\ninst\u271d : State S\ns t\u271d : S\nm : t\u271d \u2208 r s\nh : turnBound s = 0\nt : turnBound t\u271d < turnBound s\n\u22a2 False\n[PROOFSTEP]\nrw [h] at t \n[GOAL]\nS : Type u\ninst\u271d : State S\ns t\u271d : S\nm : t\u271d \u2208 r s\nh : turnBound s = 0\nt : turnBound t\u271d < 0\n\u22a2 False\n[PROOFSTEP]\nexact Nat.not_succ_le_zero _ t\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 0\nt : { t // t \u2208 l s }\n\u22a2 PGame\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase h\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 0\nt : { t // t \u2208 l s }\n\u22a2 False\n[PROOFSTEP]\nexact turnBound_ne_zero_of_left_move t.2 (nonpos_iff_eq_zero.mp h)\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 0\nt : { t // t \u2208 r s }\n\u22a2 PGame\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase h\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 0\nt : { t // t \u2208 r s }\n\u22a2 False\n[PROOFSTEP]\nexact turnBound_ne_zero_of_right_move t.2 (nonpos_iff_eq_zero.mp h)\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nhn hm : turnBound s \u2264 0\n\u22a2 ofStateAux 0 s hn \u2261r ofStateAux 0 s hm\n[PROOFSTEP]\ndsimp [PGame.ofStateAux]\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nhn hm : turnBound s \u2264 0\n\u22a2 (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False)) \u2261r\n mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False)\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase L\nS : Type u\ninst\u271d : State S\ns : S\nhn hm : turnBound s \u2264 0\n\u22a2 LeftMoves (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False)) \u2243\n LeftMoves (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\ncase R\nS : Type u\ninst\u271d : State S\ns : S\nhn hm : turnBound s \u2264 0\n\u22a2 RightMoves (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False)) \u2243\n RightMoves (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nhn hm : turnBound s \u2264 0\n\u22a2 (i :\n LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))) \u2192\n moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n i \u2261r\n moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191?L i)\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nhn hm : turnBound s \u2264 0\n\u22a2 (j :\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))) \u2192\n moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n j \u2261r\n moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191?R j)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase R\nS : Type u\ninst\u271d : State S\ns : S\nhn hm : turnBound s \u2264 0\n\u22a2 RightMoves (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False)) \u2243\n RightMoves (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nhn hm : turnBound s \u2264 0\n\u22a2 (i :\n LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))) \u2192\n moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n i \u2261r\n moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191(Equiv.refl\n (LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t =>\n False.elim (_ : False))))\n i)\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nhn hm : turnBound s \u2264 0\n\u22a2 (j :\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))) \u2192\n moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n j \u2261r\n moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191?R j)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nhn hm : turnBound s \u2264 0\n\u22a2 (i :\n LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))) \u2192\n moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n i \u2261r\n moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191(Equiv.refl\n (LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t =>\n False.elim (_ : False))))\n i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nhn hm : turnBound s \u2264 0\ni : LeftMoves (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n\u22a2 moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False)) i \u2261r\n moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191(Equiv.refl\n (LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))))\n i)\n[PROOFSTEP]\ndsimp at i \n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nhn hm : turnBound s \u2264 0\ni : { t // t \u2208 l s }\n\u22a2 moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False)) i \u2261r\n moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191(Equiv.refl\n (LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))))\n i)\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase a.h\nS : Type u\ninst\u271d : State S\ns : S\nhn hm : turnBound s \u2264 0\ni : { t // t \u2208 l s }\n\u22a2 False\n[PROOFSTEP]\nexact turnBound_ne_zero_of_left_move i.2 (nonpos_iff_eq_zero.mp hn)\n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nhn hm : turnBound s \u2264 0\n\u22a2 (j :\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))) \u2192\n moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n j \u2261r\n moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191(Equiv.refl\n (RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t =>\n False.elim (_ : False))))\n j)\n[PROOFSTEP]\nintro j\n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nhn hm : turnBound s \u2264 0\nj : RightMoves (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n\u22a2 moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n j \u2261r\n moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191(Equiv.refl\n (RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))))\n j)\n[PROOFSTEP]\ndsimp at j \n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nhn hm : turnBound s \u2264 0\nj : { t // t \u2208 r s }\n\u22a2 moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n j \u2261r\n moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191(Equiv.refl\n (RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))))\n j)\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase a.h\nS : Type u\ninst\u271d : State S\ns : S\nhn hm : turnBound s \u2264 0\nj : { t // t \u2208 r s }\n\u22a2 False\n[PROOFSTEP]\nexact turnBound_ne_zero_of_right_move j.2 (nonpos_iff_eq_zero.mp hm)\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nm : \u2115\nhn : turnBound s \u2264 0\nhm : turnBound s \u2264 m + 1\n\u22a2 ofStateAux 0 s hn \u2261r ofStateAux (m + 1) s hm\n[PROOFSTEP]\ndsimp [PGame.ofStateAux]\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nm : \u2115\nhn : turnBound s \u2264 0\nhm : turnBound s \u2264 m + 1\n\u22a2 (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False)) \u2261r\n mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase L\nS : Type u\ninst\u271d : State S\ns : S\nm : \u2115\nhn : turnBound s \u2264 0\nhm : turnBound s \u2264 m + 1\n\u22a2 LeftMoves (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False)) \u2243\n LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\ncase R\nS : Type u\ninst\u271d : State S\ns : S\nm : \u2115\nhn : turnBound s \u2264 0\nhm : turnBound s \u2264 m + 1\n\u22a2 RightMoves (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False)) \u2243\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nm : \u2115\nhn : turnBound s \u2264 0\nhm : turnBound s \u2264 m + 1\n\u22a2 (i :\n LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))) \u2192\n moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n i \u2261r\n moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191?L i)\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nm : \u2115\nhn : turnBound s \u2264 0\nhm : turnBound s \u2264 m + 1\n\u22a2 (j :\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))) \u2192\n moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n j \u2261r\n moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191?R j)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase R\nS : Type u\ninst\u271d : State S\ns : S\nm : \u2115\nhn : turnBound s \u2264 0\nhm : turnBound s \u2264 m + 1\n\u22a2 RightMoves (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False)) \u2243\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nm : \u2115\nhn : turnBound s \u2264 0\nhm : turnBound s \u2264 m + 1\n\u22a2 (i :\n LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))) \u2192\n moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n i \u2261r\n moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191(Equiv.refl\n (LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t =>\n False.elim (_ : False))))\n i)\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nm : \u2115\nhn : turnBound s \u2264 0\nhm : turnBound s \u2264 m + 1\n\u22a2 (j :\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))) \u2192\n moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n j \u2261r\n moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191?R j)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nm : \u2115\nhn : turnBound s \u2264 0\nhm : turnBound s \u2264 m + 1\n\u22a2 (i :\n LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))) \u2192\n moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n i \u2261r\n moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191(Equiv.refl\n (LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t =>\n False.elim (_ : False))))\n i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nm : \u2115\nhn : turnBound s \u2264 0\nhm : turnBound s \u2264 m + 1\ni : LeftMoves (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n\u22a2 moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False)) i \u2261r\n moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191(Equiv.refl\n (LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))))\n i)\n[PROOFSTEP]\ndsimp at i \n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nm : \u2115\nhn : turnBound s \u2264 0\nhm : turnBound s \u2264 m + 1\ni : { t // t \u2208 l s }\n\u22a2 moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False)) i \u2261r\n moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191(Equiv.refl\n (LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))))\n i)\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase a.h\nS : Type u\ninst\u271d : State S\ns : S\nm : \u2115\nhn : turnBound s \u2264 0\nhm : turnBound s \u2264 m + 1\ni : { t // t \u2208 l s }\n\u22a2 False\n[PROOFSTEP]\nexact turnBound_ne_zero_of_left_move i.2 (nonpos_iff_eq_zero.mp hn)\n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nm : \u2115\nhn : turnBound s \u2264 0\nhm : turnBound s \u2264 m + 1\n\u22a2 (j :\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))) \u2192\n moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n j \u2261r\n moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191(Equiv.refl\n (RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t =>\n False.elim (_ : False))))\n j)\n[PROOFSTEP]\nintro j\n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nm : \u2115\nhn : turnBound s \u2264 0\nhm : turnBound s \u2264 m + 1\nj : RightMoves (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n\u22a2 moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n j \u2261r\n moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191(Equiv.refl\n (RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))))\n j)\n[PROOFSTEP]\ndsimp at j \n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nm : \u2115\nhn : turnBound s \u2264 0\nhm : turnBound s \u2264 m + 1\nj : { t // t \u2208 r s }\n\u22a2 moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n j \u2261r\n moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191(Equiv.refl\n (RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))))\n j)\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase a.h\nS : Type u\ninst\u271d : State S\ns : S\nm : \u2115\nhn : turnBound s \u2264 0\nhm : turnBound s \u2264 m + 1\nj : { t // t \u2208 r s }\n\u22a2 False\n[PROOFSTEP]\nexact turnBound_ne_zero_of_right_move j.2 (nonpos_iff_eq_zero.mp hn)\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nn : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 0\n\u22a2 ofStateAux (n + 1) s hn \u2261r ofStateAux 0 s hm\n[PROOFSTEP]\ndsimp [PGame.ofStateAux]\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nn : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 0\n\u22a2 (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) \u2261r\n mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False)\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase L\nS : Type u\ninst\u271d : State S\ns : S\nn : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 0\n\u22a2 LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) \u2243\n LeftMoves (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\ncase R\nS : Type u\ninst\u271d : State S\ns : S\nn : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 0\n\u22a2 RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) \u2243\n RightMoves (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 0\n\u22a2 (i :\n LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))) \u2192\n moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n i \u2261r\n moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191?L i)\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 0\n\u22a2 (j :\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))) \u2192\n moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n j \u2261r\n moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191?R j)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase R\nS : Type u\ninst\u271d : State S\ns : S\nn : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 0\n\u22a2 RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) \u2243\n RightMoves (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 0\n\u22a2 (i :\n LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))) \u2192\n moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n i \u2261r\n moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191(Equiv.refl\n (LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))))\n i)\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 0\n\u22a2 (j :\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))) \u2192\n moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n j \u2261r\n moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191?R j)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 0\n\u22a2 (i :\n LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))) \u2192\n moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n i \u2261r\n moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191(Equiv.refl\n (LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))))\n i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 0\ni :\n LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n\u22a2 moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n i \u2261r\n moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191(Equiv.refl\n (LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))))\n i)\n[PROOFSTEP]\ndsimp at i \n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 0\ni : { t // t \u2208 l s }\n\u22a2 moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n i \u2261r\n moveLeft (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191(Equiv.refl\n (LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))))\n i)\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase a.h\nS : Type u\ninst\u271d : State S\ns : S\nn : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 0\ni : { t // t \u2208 l s }\n\u22a2 False\n[PROOFSTEP]\nexact turnBound_ne_zero_of_left_move i.2 (nonpos_iff_eq_zero.mp hm)\n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 0\n\u22a2 (j :\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))) \u2192\n moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n j \u2261r\n moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191(Equiv.refl\n (RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))))\n j)\n[PROOFSTEP]\nintro j\n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 0\nj :\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n\u22a2 moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n j \u2261r\n moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191(Equiv.refl\n (RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))))\n j)\n[PROOFSTEP]\ndsimp at j \n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 0\nj : { t // t \u2208 r s }\n\u22a2 moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n j \u2261r\n moveRight (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => False.elim (_ : False)) fun t => False.elim (_ : False))\n (\u2191(Equiv.refl\n (RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))))\n j)\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase a.h\nS : Type u\ninst\u271d : State S\ns : S\nn : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 0\nj : { t // t \u2208 r s }\n\u22a2 False\n[PROOFSTEP]\nexact turnBound_ne_zero_of_right_move j.2 (nonpos_iff_eq_zero.mp hm)\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nn m : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 m + 1\n\u22a2 ofStateAux (n + 1) s hn \u2261r ofStateAux (m + 1) s hm\n[PROOFSTEP]\ndsimp [PGame.ofStateAux]\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nn m : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 m + 1\n\u22a2 (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) \u2261r\n mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase L\nS : Type u\ninst\u271d : State S\ns : S\nn m : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 m + 1\n\u22a2 LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) \u2243\n LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\ncase R\nS : Type u\ninst\u271d : State S\ns : S\nn m : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 m + 1\n\u22a2 RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) \u2243\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn m : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 m + 1\n\u22a2 (i :\n LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))) \u2192\n moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n i \u2261r\n moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191?L i)\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn m : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 m + 1\n\u22a2 (j :\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))) \u2192\n moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n j \u2261r\n moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191?R j)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase R\nS : Type u\ninst\u271d : State S\ns : S\nn m : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 m + 1\n\u22a2 RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) \u2243\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn m : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 m + 1\n\u22a2 (i :\n LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))) \u2192\n moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n i \u2261r\n moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191(Equiv.refl\n (LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))))\n i)\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn m : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 m + 1\n\u22a2 (j :\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))) \u2192\n moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n j \u2261r\n moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191?R j)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn m : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 m + 1\n\u22a2 (i :\n LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))) \u2192\n moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n i \u2261r\n moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191(Equiv.refl\n (LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))))\n i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn m : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 m + 1\ni :\n LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n\u22a2 moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n i \u2261r\n moveLeft\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191(Equiv.refl\n (LeftMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))))\n i)\n[PROOFSTEP]\napply ofStateAuxRelabelling\n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn m : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 m + 1\n\u22a2 (j :\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))) \u2192\n moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n j \u2261r\n moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191(Equiv.refl\n (RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))))\n j)\n[PROOFSTEP]\nintro j\n[GOAL]\ncase a\nS : Type u\ninst\u271d : State S\ns : S\nn m : \u2115\nhn : turnBound s \u2264 n + 1\nhm : turnBound s \u2264 m + 1\nj :\n RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n\u22a2 moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))\n j \u2261r\n moveRight\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m)) fun t =>\n ofStateAux m \u2191t (_ : turnBound \u2191t \u2264 m))\n (\u2191(Equiv.refl\n (RightMoves\n (mk { t // t \u2208 l s } { t // t \u2208 r s } (fun t => ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n)) fun t =>\n ofStateAux n \u2191t (_ : turnBound \u2191t \u2264 n))))\n j)\n[PROOFSTEP]\napply ofStateAuxRelabelling\n[GOAL]\nS : Type u\ninst\u271d : State S\nn : \u2115\ns : S\nh : turnBound s \u2264 n\n\u22a2 LeftMoves (ofStateAux n s h) \u2243 { t // t \u2208 l s }\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 Nat.zero\n\u22a2 LeftMoves (ofStateAux Nat.zero s h) \u2243 { t // t \u2208 l s }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nS : Type u\ninst\u271d : State S\ns : S\nn\u271d : \u2115\nn_ih\u271d : (h : turnBound s \u2264 n\u271d) \u2192 LeftMoves (ofStateAux n\u271d s h) \u2243 { t // t \u2208 l s }\nh : turnBound s \u2264 Nat.succ n\u271d\n\u22a2 LeftMoves (ofStateAux (Nat.succ n\u271d) s h) \u2243 { t // t \u2208 l s }\n[PROOFSTEP]\nrfl\n[GOAL]\nS : Type u\ninst\u271d : State S\nn : \u2115\ns : S\nh : turnBound s \u2264 n\n\u22a2 RightMoves (ofStateAux n s h) \u2243 { t // t \u2208 r s }\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 Nat.zero\n\u22a2 RightMoves (ofStateAux Nat.zero s h) \u2243 { t // t \u2208 r s }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nS : Type u\ninst\u271d : State S\ns : S\nn\u271d : \u2115\nn_ih\u271d : (h : turnBound s \u2264 n\u271d) \u2192 RightMoves (ofStateAux n\u271d s h) \u2243 { t // t \u2208 r s }\nh : turnBound s \u2264 Nat.succ n\u271d\n\u22a2 RightMoves (ofStateAux (Nat.succ n\u271d) s h) \u2243 { t // t \u2208 r s }\n[PROOFSTEP]\nrfl\n[GOAL]\nS : Type u\ninst\u271d : State S\nn : \u2115\ns : S\nh : turnBound s \u2264 n\nt : LeftMoves (ofStateAux n s h)\n\u22a2 moveLeft (ofStateAux n s h) t \u2261r\n ofStateAux (n - 1) \u2191(\u2191(leftMovesOfStateAux n h) t) (_ : turnBound \u2191(\u2191(leftMovesOfStateAux n h) t) \u2264 n - 1)\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 Nat.zero\nt : LeftMoves (ofStateAux Nat.zero s h)\n\u22a2 moveLeft (ofStateAux Nat.zero s h) t \u2261r\n ofStateAux (Nat.zero - 1) \u2191(\u2191(leftMovesOfStateAux Nat.zero h) t)\n (_ : turnBound \u2191(\u2191(leftMovesOfStateAux Nat.zero h) t) \u2264 Nat.zero - 1)\n[PROOFSTEP]\nhave t' := (leftMovesOfStateAux 0 h) t\n[GOAL]\ncase zero\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 Nat.zero\nt : LeftMoves (ofStateAux Nat.zero s h)\nt' : (fun x => { t // t \u2208 l s }) t\n\u22a2 moveLeft (ofStateAux Nat.zero s h) t \u2261r\n ofStateAux (Nat.zero - 1) \u2191(\u2191(leftMovesOfStateAux Nat.zero h) t)\n (_ : turnBound \u2191(\u2191(leftMovesOfStateAux Nat.zero h) t) \u2264 Nat.zero - 1)\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase zero.h\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 Nat.zero\nt : LeftMoves (ofStateAux Nat.zero s h)\nt' : (fun x => { t // t \u2208 l s }) t\n\u22a2 False\n[PROOFSTEP]\nexact turnBound_ne_zero_of_left_move t'.2 (nonpos_iff_eq_zero.mp h)\n[GOAL]\ncase succ\nS : Type u\ninst\u271d : State S\ns : S\nn\u271d : \u2115\nn_ih\u271d :\n (h : turnBound s \u2264 n\u271d) \u2192\n (t : LeftMoves (ofStateAux n\u271d s h)) \u2192\n moveLeft (ofStateAux n\u271d s h) t \u2261r\n ofStateAux (n\u271d - 1) \u2191(\u2191(leftMovesOfStateAux n\u271d h) t) (_ : turnBound \u2191(\u2191(leftMovesOfStateAux n\u271d h) t) \u2264 n\u271d - 1)\nh : turnBound s \u2264 Nat.succ n\u271d\nt : LeftMoves (ofStateAux (Nat.succ n\u271d) s h)\n\u22a2 moveLeft (ofStateAux (Nat.succ n\u271d) s h) t \u2261r\n ofStateAux (Nat.succ n\u271d - 1) \u2191(\u2191(leftMovesOfStateAux (Nat.succ n\u271d) h) t)\n (_ : turnBound \u2191(\u2191(leftMovesOfStateAux (Nat.succ n\u271d) h) t) \u2264 Nat.succ n\u271d - 1)\n[PROOFSTEP]\nrfl\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nt : LeftMoves (ofState s)\n\u22a2 moveLeft (ofState s) t \u2261r ofState \u2191(Equiv.toFun (leftMovesOfState s) t)\n[PROOFSTEP]\ntrans\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nt : LeftMoves (ofState s)\n\u22a2 moveLeft (ofState s) t \u2261r ?m.9328\nS : Type u\ninst\u271d : State S\ns : S\nt : LeftMoves (ofState s)\n\u22a2 ?m.9328 \u2261r ofState \u2191(Equiv.toFun (leftMovesOfState s) t)\nS : Type u inst\u271d : State S s : S t : LeftMoves (ofState s) \u22a2 PGame\n[PROOFSTEP]\napply relabellingMoveLeftAux\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nt : LeftMoves (ofState s)\n\u22a2 ofStateAux (turnBound s - 1) \u2191(\u2191(leftMovesOfStateAux (turnBound s) (_ : turnBound s \u2264 turnBound s)) t)\n (_ : turnBound \u2191(\u2191(leftMovesOfStateAux (turnBound s) (_ : turnBound s \u2264 turnBound s)) t) \u2264 turnBound s - 1) \u2261r\n ofState \u2191(Equiv.toFun (leftMovesOfState s) t)\n[PROOFSTEP]\napply ofStateAuxRelabelling\n[GOAL]\nS : Type u\ninst\u271d : State S\nn : \u2115\ns : S\nh : turnBound s \u2264 n\nt : RightMoves (ofStateAux n s h)\n\u22a2 moveRight (ofStateAux n s h) t \u2261r\n ofStateAux (n - 1) \u2191(\u2191(rightMovesOfStateAux n h) t) (_ : turnBound \u2191(\u2191(rightMovesOfStateAux n h) t) \u2264 n - 1)\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 Nat.zero\nt : RightMoves (ofStateAux Nat.zero s h)\n\u22a2 moveRight (ofStateAux Nat.zero s h) t \u2261r\n ofStateAux (Nat.zero - 1) \u2191(\u2191(rightMovesOfStateAux Nat.zero h) t)\n (_ : turnBound \u2191(\u2191(rightMovesOfStateAux Nat.zero h) t) \u2264 Nat.zero - 1)\n[PROOFSTEP]\nhave t' := (rightMovesOfStateAux 0 h) t\n[GOAL]\ncase zero\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 Nat.zero\nt : RightMoves (ofStateAux Nat.zero s h)\nt' : (fun x => { t // t \u2208 r s }) t\n\u22a2 moveRight (ofStateAux Nat.zero s h) t \u2261r\n ofStateAux (Nat.zero - 1) \u2191(\u2191(rightMovesOfStateAux Nat.zero h) t)\n (_ : turnBound \u2191(\u2191(rightMovesOfStateAux Nat.zero h) t) \u2264 Nat.zero - 1)\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase zero.h\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 Nat.zero\nt : RightMoves (ofStateAux Nat.zero s h)\nt' : (fun x => { t // t \u2208 r s }) t\n\u22a2 False\n[PROOFSTEP]\nexact turnBound_ne_zero_of_right_move t'.2 (nonpos_iff_eq_zero.mp h)\n[GOAL]\ncase succ\nS : Type u\ninst\u271d : State S\ns : S\nn\u271d : \u2115\nn_ih\u271d :\n (h : turnBound s \u2264 n\u271d) \u2192\n (t : RightMoves (ofStateAux n\u271d s h)) \u2192\n moveRight (ofStateAux n\u271d s h) t \u2261r\n ofStateAux (n\u271d - 1) \u2191(\u2191(rightMovesOfStateAux n\u271d h) t) (_ : turnBound \u2191(\u2191(rightMovesOfStateAux n\u271d h) t) \u2264 n\u271d - 1)\nh : turnBound s \u2264 Nat.succ n\u271d\nt : RightMoves (ofStateAux (Nat.succ n\u271d) s h)\n\u22a2 moveRight (ofStateAux (Nat.succ n\u271d) s h) t \u2261r\n ofStateAux (Nat.succ n\u271d - 1) \u2191(\u2191(rightMovesOfStateAux (Nat.succ n\u271d) h) t)\n (_ : turnBound \u2191(\u2191(rightMovesOfStateAux (Nat.succ n\u271d) h) t) \u2264 Nat.succ n\u271d - 1)\n[PROOFSTEP]\nrfl\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nt : RightMoves (ofState s)\n\u22a2 moveRight (ofState s) t \u2261r ofState \u2191(Equiv.toFun (rightMovesOfState s) t)\n[PROOFSTEP]\ntrans\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nt : RightMoves (ofState s)\n\u22a2 moveRight (ofState s) t \u2261r ?m.11225\nS : Type u\ninst\u271d : State S\ns : S\nt : RightMoves (ofState s)\n\u22a2 ?m.11225 \u2261r ofState \u2191(Equiv.toFun (rightMovesOfState s) t)\nS : Type u inst\u271d : State S s : S t : RightMoves (ofState s) \u22a2 PGame\n[PROOFSTEP]\napply relabellingMoveRightAux\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nt : RightMoves (ofState s)\n\u22a2 ofStateAux (turnBound s - 1) \u2191(\u2191(rightMovesOfStateAux (turnBound s) (_ : turnBound s \u2264 turnBound s)) t)\n (_ : turnBound \u2191(\u2191(rightMovesOfStateAux (turnBound s) (_ : turnBound s \u2264 turnBound s)) t) \u2264 turnBound s - 1) \u2261r\n ofState \u2191(Equiv.toFun (rightMovesOfState s) t)\n[PROOFSTEP]\napply ofStateAuxRelabelling\n[GOAL]\nS : Type u\ninst\u271d : State S\nn : \u2115\ns : S\nh : turnBound s \u2264 n\n\u22a2 Fintype (LeftMoves (ofStateAux n s h))\n[PROOFSTEP]\napply Fintype.ofEquiv _ (leftMovesOfStateAux _ _).symm\n[GOAL]\nS : Type u\ninst\u271d : State S\nn : \u2115\ns : S\nh : turnBound s \u2264 n\n\u22a2 Fintype (RightMoves (ofStateAux n s h))\n[PROOFSTEP]\napply Fintype.ofEquiv _ (rightMovesOfStateAux _ _).symm\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 0\ni : LeftMoves (ofStateAux 0 s h)\n\u22a2 Short (moveLeft (ofStateAux 0 s h) i)\n[PROOFSTEP]\nhave i := (leftMovesOfStateAux _ _).toFun i\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 0\ni\u271d : LeftMoves (ofStateAux 0 s h)\ni : { t // t \u2208 l s }\n\u22a2 Short (moveLeft (ofStateAux 0 s h) i\u271d)\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase h\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 0\ni\u271d : LeftMoves (ofStateAux 0 s h)\ni : { t // t \u2208 l s }\n\u22a2 False\n[PROOFSTEP]\nexact turnBound_ne_zero_of_left_move i.2 (nonpos_iff_eq_zero.mp h)\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 0\nj : RightMoves (ofStateAux 0 s h)\n\u22a2 Short (moveRight (ofStateAux 0 s h) j)\n[PROOFSTEP]\nhave j := (rightMovesOfStateAux _ _).toFun j\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 0\nj\u271d : RightMoves (ofStateAux 0 s h)\nj : { t // t \u2208 r s }\n\u22a2 Short (moveRight (ofStateAux 0 s h) j\u271d)\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase h\nS : Type u\ninst\u271d : State S\ns : S\nh : turnBound s \u2264 0\nj\u271d : RightMoves (ofStateAux 0 s h)\nj : { t // t \u2208 r s }\n\u22a2 False\n[PROOFSTEP]\nexact turnBound_ne_zero_of_right_move j.2 (nonpos_iff_eq_zero.mp h)\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\n\u22a2 Short (ofState s)\n[PROOFSTEP]\ndsimp [PGame.ofState]\n[GOAL]\nS : Type u\ninst\u271d : State S\ns : S\n\u22a2 Short (ofStateAux (turnBound s) s (_ : turnBound s \u2264 turnBound s))\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.SetTheory.Game.State", "llama_tokens": 19930, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5039061705290804, "lm_q2_score": 0.42632159254749036, "lm_q1q2_score": 0.2148260811144648}} {"text": "[GOAL]\nG H : AddCommGroupCat\n\u22a2 \u2200 \u2983X Y : Discrete WalkingPair\u2984 (f : X \u27f6 Y),\n ((Functor.const (Discrete WalkingPair)).obj (of (\u2191G \u00d7 \u2191H))).map f \u226b\n (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst \u2191G \u2191H) (AddMonoidHom.snd \u2191G \u2191H))\n Y =\n (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst \u2191G \u2191H) (AddMonoidHom.snd \u2191G \u2191H)) X \u226b\n (pair G H).map f\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9\u27e9 \u27e8\u27e8\u27e9\u27e9 \u27e8\u27e8\u27e8\u27e9\u27e9\u27e9\n[GOAL]\ncase mk.left.mk.left.up.up.refl\nG H : AddCommGroupCat\n\u22a2 ((Functor.const (Discrete WalkingPair)).obj (of (\u2191G \u00d7 \u2191H))).map\n { down := { down := (_ : { as := WalkingPair.left }.as = { as := WalkingPair.left }.as) } } \u226b\n (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst \u2191G \u2191H) (AddMonoidHom.snd \u2191G \u2191H))\n { as := WalkingPair.left } =\n (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst \u2191G \u2191H) (AddMonoidHom.snd \u2191G \u2191H))\n { as := WalkingPair.left } \u226b\n (pair G H).map { down := { down := (_ : { as := WalkingPair.left }.as = { as := WalkingPair.left }.as) } }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.right.mk.right.up.up.refl\nG H : AddCommGroupCat\n\u22a2 ((Functor.const (Discrete WalkingPair)).obj (of (\u2191G \u00d7 \u2191H))).map\n { down := { down := (_ : { as := WalkingPair.right }.as = { as := WalkingPair.right }.as) } } \u226b\n (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst \u2191G \u2191H) (AddMonoidHom.snd \u2191G \u2191H))\n { as := WalkingPair.right } =\n (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst \u2191G \u2191H) (AddMonoidHom.snd \u2191G \u2191H))\n { as := WalkingPair.right } \u226b\n (pair G H).map { down := { down := (_ : { as := WalkingPair.right }.as = { as := WalkingPair.right }.as) } }\n[PROOFSTEP]\nrfl\n[GOAL]\nG H : AddCommGroupCat\n\u22a2 \u2200 (s : Cone (pair G H)) (j : Discrete WalkingPair),\n (fun s =>\n AddMonoidHom.prod (NatTrans.app s.\u03c0 { as := WalkingPair.left })\n (NatTrans.app s.\u03c0 { as := WalkingPair.right }))\n s \u226b\n NatTrans.app\n { pt := of (\u2191G \u00d7 \u2191H),\n \u03c0 :=\n NatTrans.mk fun j =>\n Discrete.casesOn j fun j =>\n WalkingPair.casesOn j (AddMonoidHom.fst \u2191G \u2191H) (AddMonoidHom.snd \u2191G \u2191H) }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\n[PROOFSTEP]\nrintro s (\u27e8\u27e9 | \u27e8\u27e9)\n[GOAL]\ncase mk.left\nG H : AddCommGroupCat\ns : Cone (pair G H)\n\u22a2 (fun s =>\n AddMonoidHom.prod (NatTrans.app s.\u03c0 { as := WalkingPair.left })\n (NatTrans.app s.\u03c0 { as := WalkingPair.right }))\n s \u226b\n NatTrans.app\n { pt := of (\u2191G \u00d7 \u2191H),\n \u03c0 :=\n NatTrans.mk fun j =>\n Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst \u2191G \u2191H) (AddMonoidHom.snd \u2191G \u2191H) }.\u03c0\n { as := WalkingPair.left } =\n NatTrans.app s.\u03c0 { as := WalkingPair.left }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.right\nG H : AddCommGroupCat\ns : Cone (pair G H)\n\u22a2 (fun s =>\n AddMonoidHom.prod (NatTrans.app s.\u03c0 { as := WalkingPair.left })\n (NatTrans.app s.\u03c0 { as := WalkingPair.right }))\n s \u226b\n NatTrans.app\n { pt := of (\u2191G \u00d7 \u2191H),\n \u03c0 :=\n NatTrans.mk fun j =>\n Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst \u2191G \u2191H) (AddMonoidHom.snd \u2191G \u2191H) }.\u03c0\n { as := WalkingPair.right } =\n NatTrans.app s.\u03c0 { as := WalkingPair.right }\n[PROOFSTEP]\nrfl\n[GOAL]\nG H : AddCommGroupCat\ns : Cone (pair G H)\nm :\n s.pt \u27f6\n { pt := of (\u2191G \u00d7 \u2191H),\n \u03c0 :=\n NatTrans.mk fun j =>\n Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst \u2191G \u2191H) (AddMonoidHom.snd \u2191G \u2191H) }.pt\nw :\n \u2200 (j : Discrete WalkingPair),\n m \u226b\n NatTrans.app\n { pt := of (\u2191G \u00d7 \u2191H),\n \u03c0 :=\n NatTrans.mk fun j =>\n Discrete.casesOn j fun j =>\n WalkingPair.casesOn j (AddMonoidHom.fst \u2191G \u2191H) (AddMonoidHom.snd \u2191G \u2191H) }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\n\u22a2 m =\n (fun s =>\n AddMonoidHom.prod (NatTrans.app s.\u03c0 { as := WalkingPair.left }) (NatTrans.app s.\u03c0 { as := WalkingPair.right }))\n s\n[PROOFSTEP]\nsimp_rw [\u2190 w \u27e8WalkingPair.left\u27e9, \u2190 w \u27e8WalkingPair.right\u27e9]\n[GOAL]\nG H : AddCommGroupCat\ns : Cone (pair G H)\nm :\n s.pt \u27f6\n { pt := of (\u2191G \u00d7 \u2191H),\n \u03c0 :=\n NatTrans.mk fun j =>\n Discrete.casesOn j fun j => WalkingPair.casesOn j (AddMonoidHom.fst \u2191G \u2191H) (AddMonoidHom.snd \u2191G \u2191H) }.pt\nw :\n \u2200 (j : Discrete WalkingPair),\n m \u226b\n NatTrans.app\n { pt := of (\u2191G \u00d7 \u2191H),\n \u03c0 :=\n NatTrans.mk fun j =>\n Discrete.casesOn j fun j =>\n WalkingPair.casesOn j (AddMonoidHom.fst \u2191G \u2191H) (AddMonoidHom.snd \u2191G \u2191H) }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\n\u22a2 m = AddMonoidHom.prod (m \u226b AddMonoidHom.fst \u2191G \u2191H) (m \u226b AddMonoidHom.snd \u2191G \u2191H)\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type w\nf : J \u2192 AddCommGroupCat\ns : Fan f\n\u22a2 (fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x) 0 = 0\n[PROOFSTEP]\nsimp only [Functor.const_obj_obj, map_zero]\n[GOAL]\nJ : Type w\nf : J \u2192 AddCommGroupCat\ns : Fan f\n\u22a2 (fun j => 0) = 0\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type w\nf : J \u2192 AddCommGroupCat\ns : Fan f\nx y : \u2191s.pt\n\u22a2 ZeroHom.toFun\n { toFun := fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x,\n map_zero' := (_ : (fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x) 0 = 0) }\n (x + y) =\n ZeroHom.toFun\n { toFun := fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x,\n map_zero' := (_ : (fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x) 0 = 0) }\n x +\n ZeroHom.toFun\n { toFun := fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x,\n map_zero' := (_ : (fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x) 0 = 0) }\n y\n[PROOFSTEP]\nsimp only [Functor.const_obj_obj, map_add]\n[GOAL]\nJ : Type w\nf : J \u2192 AddCommGroupCat\ns : Fan f\nx y : \u2191s.pt\n\u22a2 (fun j => \u2191(NatTrans.app s.\u03c0 { as := j }) x + \u2191(NatTrans.app s.\u03c0 { as := j }) y) =\n (fun j => \u2191(NatTrans.app s.\u03c0 { as := j }) x) + fun j => \u2191(NatTrans.app s.\u03c0 { as := j }) y\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type w\nf : J \u2192 AddCommGroupCat\ns : Cone (Discrete.functor f)\nm :\n s.pt \u27f6 { pt := of ((j : J) \u2192 \u2191(f j)), \u03c0 := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => \u2191(f j)) j.as }.pt\nw :\n \u2200 (j : Discrete J),\n m \u226b\n NatTrans.app\n { pt := of ((j : J) \u2192 \u2191(f j)), \u03c0 := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => \u2191(f j)) j.as }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\n\u22a2 m = lift f s\n[PROOFSTEP]\next x\n[GOAL]\ncase w\nJ : Type w\nf : J \u2192 AddCommGroupCat\ns : Cone (Discrete.functor f)\nm :\n s.pt \u27f6 { pt := of ((j : J) \u2192 \u2191(f j)), \u03c0 := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => \u2191(f j)) j.as }.pt\nw :\n \u2200 (j : Discrete J),\n m \u226b\n NatTrans.app\n { pt := of ((j : J) \u2192 \u2191(f j)), \u03c0 := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => \u2191(f j)) j.as }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\nx : \u2191s.pt\n\u22a2 \u2191m x = \u2191(lift f s) x\n[PROOFSTEP]\nfunext j\n[GOAL]\ncase w.h\nJ : Type w\nf : J \u2192 AddCommGroupCat\ns : Cone (Discrete.functor f)\nm :\n s.pt \u27f6 { pt := of ((j : J) \u2192 \u2191(f j)), \u03c0 := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => \u2191(f j)) j.as }.pt\nw :\n \u2200 (j : Discrete J),\n m \u226b\n NatTrans.app\n { pt := of ((j : J) \u2192 \u2191(f j)), \u03c0 := Discrete.natTrans fun j => Pi.evalAddMonoidHom (fun j => \u2191(f j)) j.as }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\nx : \u2191s.pt\nj : J\n\u22a2 \u2191m x j = \u2191(lift f s) x j\n[PROOFSTEP]\nexact congr_arg (fun g : s.pt \u27f6 f j => (g : s.pt \u2192 f j) x) (w \u27e8j\u27e9)\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.GroupCat.Biproducts", "llama_tokens": 3292, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.41111086923216805, "lm_q1q2_score": 0.21358086225236997}} {"text": "[GOAL]\nD : GlueData\ni j : D.J\n\u22a2 LocallyRingedSpace.IsOpenImmersion (f (toLocallyRingedSpaceGlueData D).toGlueData i j)\n[PROOFSTEP]\napply GlueData.f_open\n[GOAL]\nD : GlueData\ni j : D.J\n\u22a2 SheafedSpace.IsOpenImmersion\n (f (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)).toGlueData i j)\n[PROOFSTEP]\napply GlueData.f_open\n[GOAL]\nD : GlueData\ni j : D.J\n\u22a2 PresheafedSpace.IsOpenImmersion\n (f\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D))).toGlueData\n i j)\n[PROOFSTEP]\napply GlueData.f_open\n[GOAL]\nD : GlueData\ni : D.J\n\u22a2 LocallyRingedSpace.IsOpenImmersion (\u03b9 (toLocallyRingedSpaceGlueData D).toGlueData i)\n[PROOFSTEP]\napply LocallyRingedSpace.GlueData.\u03b9_isOpenImmersion\n[GOAL]\nD : GlueData\n\u22a2 Scheme\n[PROOFSTEP]\napply LocallyRingedSpace.IsOpenImmersion.scheme D.toLocallyRingedSpaceGlueData.toGlueData.glued\n[GOAL]\nD : GlueData\n\u22a2 \u2200 (x : \u2191(LocallyRingedSpace.toTopCat (glued (toLocallyRingedSpaceGlueData D).toGlueData))),\n \u2203 R f, x \u2208 Set.range \u2191f.val.base \u2227 LocallyRingedSpace.IsOpenImmersion f\n[PROOFSTEP]\nintro x\n[GOAL]\nD : GlueData\nx : \u2191(LocallyRingedSpace.toTopCat (glued (toLocallyRingedSpaceGlueData D).toGlueData))\n\u22a2 \u2203 R f, x \u2208 Set.range \u2191f.val.base \u2227 LocallyRingedSpace.IsOpenImmersion f\n[PROOFSTEP]\nobtain \u27e8i, y, rfl\u27e9 := D.toLocallyRingedSpaceGlueData.\u03b9_jointly_surjective x\n[GOAL]\ncase intro.intro\nD : GlueData\ni : (toLocallyRingedSpaceGlueData D).toGlueData.J\ny : \u2191(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))\n\u22a2 \u2203 R f,\n \u2191(\u03b9 (toLocallyRingedSpaceGlueData D).toGlueData i).val.base y \u2208 Set.range \u2191f.val.base \u2227\n LocallyRingedSpace.IsOpenImmersion f\n[PROOFSTEP]\nrefine' \u27e8_, _ \u226b D.toLocallyRingedSpaceGlueData.toGlueData.\u03b9 i, _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\nD : GlueData\ni : (toLocallyRingedSpaceGlueData D).toGlueData.J\ny : \u2191(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))\n\u22a2 CommRingCat\ncase intro.intro.refine'_2\nD : GlueData\ni : (toLocallyRingedSpaceGlueData D).toGlueData.J\ny : \u2191(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))\n\u22a2 Spec.toLocallyRingedSpace.obj (op ?intro.intro.refine'_1) \u27f6 U (toLocallyRingedSpaceGlueData D).toGlueData i\ncase intro.intro.refine'_3\nD : GlueData\ni : (toLocallyRingedSpaceGlueData D).toGlueData.J\ny : \u2191(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))\n\u22a2 \u2191(\u03b9 (toLocallyRingedSpaceGlueData D).toGlueData i).val.base y \u2208\n Set.range \u2191(?intro.intro.refine'_2 \u226b \u03b9 (toLocallyRingedSpaceGlueData D).toGlueData i).val.base \u2227\n LocallyRingedSpace.IsOpenImmersion (?intro.intro.refine'_2 \u226b \u03b9 (toLocallyRingedSpaceGlueData D).toGlueData i)\n[PROOFSTEP]\nswap\n[GOAL]\ncase intro.intro.refine'_2\nD : GlueData\ni : (toLocallyRingedSpaceGlueData D).toGlueData.J\ny : \u2191(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))\n\u22a2 Spec.toLocallyRingedSpace.obj (op ?intro.intro.refine'_1) \u27f6 U (toLocallyRingedSpaceGlueData D).toGlueData i\ncase intro.intro.refine'_1\nD : GlueData\ni : (toLocallyRingedSpaceGlueData D).toGlueData.J\ny : \u2191(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))\n\u22a2 CommRingCat\ncase intro.intro.refine'_3\nD : GlueData\ni : (toLocallyRingedSpaceGlueData D).toGlueData.J\ny : \u2191(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))\n\u22a2 \u2191(\u03b9 (toLocallyRingedSpaceGlueData D).toGlueData i).val.base y \u2208\n Set.range \u2191(?intro.intro.refine'_2 \u226b \u03b9 (toLocallyRingedSpaceGlueData D).toGlueData i).val.base \u2227\n LocallyRingedSpace.IsOpenImmersion (?intro.intro.refine'_2 \u226b \u03b9 (toLocallyRingedSpaceGlueData D).toGlueData i)\n[PROOFSTEP]\nexact (D.U i).affineCover.map y\n[GOAL]\ncase intro.intro.refine'_3\nD : GlueData\ni : (toLocallyRingedSpaceGlueData D).toGlueData.J\ny : \u2191(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))\n\u22a2 \u2191(\u03b9 (toLocallyRingedSpaceGlueData D).toGlueData i).val.base y \u2208\n Set.range\n \u2191(OpenCover.map (affineCover (U D.toGlueData i)) y \u226b \u03b9 (toLocallyRingedSpaceGlueData D).toGlueData i).val.base \u2227\n LocallyRingedSpace.IsOpenImmersion\n (OpenCover.map (affineCover (U D.toGlueData i)) y \u226b \u03b9 (toLocallyRingedSpaceGlueData D).toGlueData i)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.refine'_3.left\nD : GlueData\ni : (toLocallyRingedSpaceGlueData D).toGlueData.J\ny : \u2191(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))\n\u22a2 \u2191(\u03b9 (toLocallyRingedSpaceGlueData D).toGlueData i).val.base y \u2208\n Set.range\n \u2191(OpenCover.map (affineCover (U D.toGlueData i)) y \u226b \u03b9 (toLocallyRingedSpaceGlueData D).toGlueData i).val.base\n[PROOFSTEP]\ndsimp [-Set.mem_range]\n[GOAL]\ncase intro.intro.refine'_3.left\nD : GlueData\ni : (toLocallyRingedSpaceGlueData D).toGlueData.J\ny : \u2191(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))\n\u22a2 \u2191(\u03b9 (mapGlueData D.toGlueData forgetToLocallyRingedSpace) i).val.base y \u2208\n Set.range\n \u2191((OpenCover.map (affineCover (U D.toGlueData i)) y).val.base \u226b\n (\u03b9 (mapGlueData D.toGlueData forgetToLocallyRingedSpace) i).val.base)\n[PROOFSTEP]\nrw [coe_comp, Set.range_comp]\n[GOAL]\ncase intro.intro.refine'_3.left\nD : GlueData\ni : (toLocallyRingedSpaceGlueData D).toGlueData.J\ny : \u2191(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))\n\u22a2 \u2191(\u03b9 (mapGlueData D.toGlueData forgetToLocallyRingedSpace) i).val.base y \u2208\n \u2191(\u03b9 (mapGlueData D.toGlueData forgetToLocallyRingedSpace) i).val.base ''\n Set.range \u2191(OpenCover.map (affineCover (U D.toGlueData i)) y).val.base\n[PROOFSTEP]\nrefine' Set.mem_image_of_mem _ _\n[GOAL]\ncase intro.intro.refine'_3.left\nD : GlueData\ni : (toLocallyRingedSpaceGlueData D).toGlueData.J\ny : \u2191(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))\n\u22a2 y \u2208 Set.range \u2191(OpenCover.map (affineCover (U D.toGlueData i)) y).val.base\n[PROOFSTEP]\nexact (D.U i).affineCover.Covers y\n[GOAL]\ncase intro.intro.refine'_3.right\nD : GlueData\ni : (toLocallyRingedSpaceGlueData D).toGlueData.J\ny : \u2191(LocallyRingedSpace.toTopCat (U (toLocallyRingedSpaceGlueData D).toGlueData i))\n\u22a2 LocallyRingedSpace.IsOpenImmersion\n (OpenCover.map (affineCover (U D.toGlueData i)) y \u226b \u03b9 (toLocallyRingedSpaceGlueData D).toGlueData i)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nD : GlueData\ni : D.J\n\u22a2 IsOpenImmersion (CategoryTheory.GlueData.\u03b9 D.toGlueData i)\n[PROOFSTEP]\nrw [\u2190 D.\u03b9_isoLocallyRingedSpace_inv]\n[GOAL]\nD : GlueData\ni : D.J\n\u22a2 IsOpenImmersion\n (CategoryTheory.GlueData.\u03b9 (toLocallyRingedSpaceGlueData D).toGlueData i \u226b (isoLocallyRingedSpace D).inv)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nD : GlueData\ni j : D.J\n\u22a2 f D.toGlueData i j \u226b \u03b9 D i = (t D.toGlueData i j \u226b f D.toGlueData j i) \u226b \u03b9 D j\n[PROOFSTEP]\nsimp\n[GOAL]\nD : GlueData\n\u22a2 \u2191(glued D).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace \u2245\n CategoryTheory.GlueData.glued\n (GlueData.toTopGlueData\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData\n[PROOFSTEP]\nrefine (PresheafedSpace.forget _).mapIso ?_ \u226a\u226b GlueData.gluedIso _ (PresheafedSpace.forget.{_, _, u} _)\n[GOAL]\nD : GlueData\n\u22a2 (glued D).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace \u2245\n CategoryTheory.GlueData.glued\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D))).toGlueData\n[PROOFSTEP]\nrefine SheafedSpace.forgetToPresheafedSpace.mapIso ?_ \u226a\u226b SheafedSpace.GlueData.isoPresheafedSpace _\n[GOAL]\nD : GlueData\n\u22a2 (glued D).toLocallyRingedSpace.toSheafedSpace \u2245\n CategoryTheory.GlueData.glued\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)).toGlueData\n[PROOFSTEP]\nrefine LocallyRingedSpace.forgetToSheafedSpace.mapIso ?_ \u226a\u226b LocallyRingedSpace.GlueData.isoSheafedSpace _\n[GOAL]\nD : GlueData\n\u22a2 (glued D).toLocallyRingedSpace \u2245 CategoryTheory.GlueData.glued (toLocallyRingedSpaceGlueData D).toGlueData\n[PROOFSTEP]\nexact Scheme.GlueData.isoLocallyRingedSpace _\n[GOAL]\nD : GlueData\ni : D.J\n\u22a2 CategoryTheory.GlueData.\u03b9\n (GlueData.toTopGlueData\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData\n i \u226b\n (isoCarrier D).inv =\n (\u03b9 D i).val.base\n[PROOFSTEP]\ndelta isoCarrier\n[GOAL]\nD : GlueData\ni : D.J\n\u22a2 CategoryTheory.GlueData.\u03b9\n (GlueData.toTopGlueData\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData\n i \u226b\n ((PresheafedSpace.forget CommRingCat).mapIso\n (SheafedSpace.forgetToPresheafedSpace.mapIso\n (LocallyRingedSpace.forgetToSheafedSpace.mapIso (isoLocallyRingedSpace D) \u226a\u226b\n LocallyRingedSpace.GlueData.isoSheafedSpace (toLocallyRingedSpaceGlueData D)) \u226a\u226b\n SheafedSpace.GlueData.isoPresheafedSpace\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D))) \u226a\u226b\n gluedIso\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D))).toGlueData\n (PresheafedSpace.forget CommRingCat)).inv =\n (\u03b9 D i).val.base\n[PROOFSTEP]\nrw [Iso.trans_inv, GlueData.\u03b9_gluedIso_inv_assoc, Functor.mapIso_inv, Iso.trans_inv, Functor.mapIso_inv, Iso.trans_inv,\n SheafedSpace.forgetToPresheafedSpace_map, forget_map, forget_map, \u2190 comp_base, \u2190 Category.assoc,\n D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.\u03b9_isoPresheafedSpace_inv i]\n[GOAL]\nD : GlueData\ni : D.J\n\u22a2 (CategoryTheory.GlueData.\u03b9\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)).toGlueData i \u226b\n (LocallyRingedSpace.GlueData.isoSheafedSpace (toLocallyRingedSpaceGlueData D)).inv \u226b\n (LocallyRingedSpace.forgetToSheafedSpace.mapIso (isoLocallyRingedSpace D)).inv).base =\n (\u03b9 D i).val.base\n[PROOFSTEP]\nerw [\u2190 Category.assoc, D.toLocallyRingedSpaceGlueData.\u03b9_isoSheafedSpace_inv i]\n[GOAL]\nD : GlueData\ni : D.J\n\u22a2 ((CategoryTheory.GlueData.\u03b9 (toLocallyRingedSpaceGlueData D).toGlueData i).val \u226b\n (LocallyRingedSpace.forgetToSheafedSpace.mapIso (isoLocallyRingedSpace D)).inv).base =\n (\u03b9 D i).val.base\n[PROOFSTEP]\nchange (_ \u226b D.isoLocallyRingedSpace.inv).1.base = _\n[GOAL]\nD : GlueData\ni : D.J\n\u22a2 (CategoryTheory.GlueData.\u03b9 (toLocallyRingedSpaceGlueData D).toGlueData i \u226b (isoLocallyRingedSpace D).inv).val.base =\n (\u03b9 D i).val.base\n[PROOFSTEP]\nrw [D.\u03b9_isoLocallyRingedSpace_inv i]\n[GOAL]\nD : GlueData\ni j : D.J\nx : \u2191\u2191(U D.toGlueData i).toPresheafedSpace\ny : \u2191\u2191(U D.toGlueData j).toPresheafedSpace\n\u22a2 \u2191(CategoryTheory.GlueData.\u03b9 D.toGlueData i).val.base x = \u2191(CategoryTheory.GlueData.\u03b9 D.toGlueData j).val.base y \u2194\n Rel D { fst := i, snd := x } { fst := j, snd := y }\n[PROOFSTEP]\nrefine'\n Iff.trans _\n (TopCat.GlueData.\u03b9_eq_iff_rel\n D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData i j x y)\n[GOAL]\nD : GlueData\ni j : D.J\nx : \u2191\u2191(U D.toGlueData i).toPresheafedSpace\ny : \u2191\u2191(U D.toGlueData j).toPresheafedSpace\n\u22a2 \u2191(CategoryTheory.GlueData.\u03b9 D.toGlueData i).val.base x = \u2191(CategoryTheory.GlueData.\u03b9 D.toGlueData j).val.base y \u2194\n \u2191(CategoryTheory.GlueData.\u03b9\n (GlueData.toTopGlueData\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData\n i)\n x =\n \u2191(CategoryTheory.GlueData.\u03b9\n (GlueData.toTopGlueData\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData\n j)\n y\n[PROOFSTEP]\nrw [\u2190 ((TopCat.mono_iff_injective D.isoCarrier.inv).mp _).eq_iff]\n[GOAL]\nD : GlueData\ni j : D.J\nx : \u2191\u2191(U D.toGlueData i).toPresheafedSpace\ny : \u2191\u2191(U D.toGlueData j).toPresheafedSpace\n\u22a2 \u2191(CategoryTheory.GlueData.\u03b9 D.toGlueData i).val.base x = \u2191(CategoryTheory.GlueData.\u03b9 D.toGlueData j).val.base y \u2194\n \u2191(isoCarrier D).inv\n (\u2191(CategoryTheory.GlueData.\u03b9\n (GlueData.toTopGlueData\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData\n i)\n x) =\n \u2191(isoCarrier D).inv\n (\u2191(CategoryTheory.GlueData.\u03b9\n (GlueData.toTopGlueData\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData\n j)\n y)\n[PROOFSTEP]\nsimp_rw [\u2190 comp_apply, \u2190 D.\u03b9_isoCarrier_inv]\n[GOAL]\nD : GlueData\ni j : D.J\nx : \u2191\u2191(U D.toGlueData i).toPresheafedSpace\ny : \u2191\u2191(U D.toGlueData j).toPresheafedSpace\n\u22a2 \u2191(CategoryTheory.GlueData.\u03b9\n (mapGlueData\n (mapGlueData\n (mapGlueData (mapGlueData D.toGlueData forgetToLocallyRingedSpace)\n LocallyRingedSpace.forgetToSheafedSpace)\n SheafedSpace.forgetToPresheafedSpace)\n (PresheafedSpace.forget CommRingCat))\n i \u226b\n (isoCarrier D).inv)\n x =\n \u2191(CategoryTheory.GlueData.\u03b9\n (mapGlueData\n (mapGlueData\n (mapGlueData (mapGlueData D.toGlueData forgetToLocallyRingedSpace)\n LocallyRingedSpace.forgetToSheafedSpace)\n SheafedSpace.forgetToPresheafedSpace)\n (PresheafedSpace.forget CommRingCat))\n j \u226b\n (isoCarrier D).inv)\n y \u2194\n \u2191(CategoryTheory.GlueData.\u03b9\n (mapGlueData\n (mapGlueData\n (mapGlueData (mapGlueData D.toGlueData forgetToLocallyRingedSpace)\n LocallyRingedSpace.forgetToSheafedSpace)\n SheafedSpace.forgetToPresheafedSpace)\n (PresheafedSpace.forget CommRingCat))\n i \u226b\n (isoCarrier D).inv)\n x =\n \u2191(CategoryTheory.GlueData.\u03b9\n (mapGlueData\n (mapGlueData\n (mapGlueData (mapGlueData D.toGlueData forgetToLocallyRingedSpace)\n LocallyRingedSpace.forgetToSheafedSpace)\n SheafedSpace.forgetToPresheafedSpace)\n (PresheafedSpace.forget CommRingCat))\n j \u226b\n (isoCarrier D).inv)\n y\n[PROOFSTEP]\nrfl\n[GOAL]\nD : GlueData\ni j : D.J\nx : \u2191\u2191(U D.toGlueData i).toPresheafedSpace\ny : \u2191\u2191(U D.toGlueData j).toPresheafedSpace\n\u22a2 Mono (isoCarrier D).inv\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nD : GlueData\nU : Set \u2191\u2191(glued D).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n\u22a2 IsOpen U \u2194 \u2200 (i : D.J), IsOpen (\u2191(\u03b9 D i).val.base \u207b\u00b9' U)\n[PROOFSTEP]\nrw [\u2190 (TopCat.homeoOfIso D.isoCarrier.symm).isOpen_preimage]\n[GOAL]\nD : GlueData\nU : Set \u2191\u2191(glued D).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n\u22a2 IsOpen (\u2191(TopCat.homeoOfIso (isoCarrier D).symm) \u207b\u00b9' U) \u2194 \u2200 (i : D.J), IsOpen (\u2191(\u03b9 D i).val.base \u207b\u00b9' U)\n[PROOFSTEP]\nrw [TopCat.GlueData.isOpen_iff]\n[GOAL]\nD : GlueData\nU : Set \u2191\u2191(glued D).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n\u22a2 (\u2200\n (i :\n (GlueData.toTopGlueData\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData.J),\n IsOpen\n (\u2191(CategoryTheory.GlueData.\u03b9\n (GlueData.toTopGlueData\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData\n i) \u207b\u00b9'\n (\u2191(TopCat.homeoOfIso (isoCarrier D).symm) \u207b\u00b9' U))) \u2194\n \u2200 (i : D.J), IsOpen (\u2191(\u03b9 D i).val.base \u207b\u00b9' U)\n[PROOFSTEP]\napply forall_congr'\n[GOAL]\ncase h\nD : GlueData\nU : Set \u2191\u2191(glued D).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n\u22a2 \u2200\n (a :\n (GlueData.toTopGlueData\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData.J),\n IsOpen\n (\u2191(CategoryTheory.GlueData.\u03b9\n (GlueData.toTopGlueData\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData\n a) \u207b\u00b9'\n (\u2191(TopCat.homeoOfIso (isoCarrier D).symm) \u207b\u00b9' U)) \u2194\n IsOpen (\u2191(\u03b9 D a).val.base \u207b\u00b9' U)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h\nD : GlueData\nU : Set \u2191\u2191(glued D).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\ni :\n (GlueData.toTopGlueData\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData.J\n\u22a2 IsOpen\n (\u2191(CategoryTheory.GlueData.\u03b9\n (GlueData.toTopGlueData\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData\n i) \u207b\u00b9'\n (\u2191(TopCat.homeoOfIso (isoCarrier D).symm) \u207b\u00b9' U)) \u2194\n IsOpen (\u2191(\u03b9 D i).val.base \u207b\u00b9' U)\n[PROOFSTEP]\nerw [\u2190 Set.preimage_comp, \u2190 \u03b9_isoCarrier_inv]\n[GOAL]\ncase h\nD : GlueData\nU : Set \u2191\u2191(glued D).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\ni :\n (GlueData.toTopGlueData\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData.J\n\u22a2 IsOpen\n (\u2191(TopCat.homeoOfIso (isoCarrier D).symm) \u2218\n \u2191(CategoryTheory.GlueData.\u03b9\n (GlueData.toTopGlueData\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData\n i) \u207b\u00b9'\n U) \u2194\n IsOpen\n (\u2191(CategoryTheory.GlueData.\u03b9\n (GlueData.toTopGlueData\n (SheafedSpace.GlueData.toPresheafedSpaceGlueData\n (LocallyRingedSpace.GlueData.toSheafedSpaceGlueData (toLocallyRingedSpaceGlueData D)))).toGlueData\n i \u226b\n (isoCarrier D).inv) \u207b\u00b9'\n U)\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 pullback pullback.fst pullback.fst \u27f6 pullback pullback.fst pullback.fst\n[PROOFSTEP]\nrefine' (pullbackRightPullbackFstIso _ _ _).hom \u226b _\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 pullback (pullback.fst \u226b map \ud835\udcb0 x) (map \ud835\udcb0 z) \u27f6 pullback pullback.fst pullback.fst\n[PROOFSTEP]\nrefine' _ \u226b (pullbackSymmetry _ _).hom\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 pullback (pullback.fst \u226b map \ud835\udcb0 x) (map \ud835\udcb0 z) \u27f6 pullback pullback.fst pullback.fst\n[PROOFSTEP]\nrefine' _ \u226b (pullbackRightPullbackFstIso _ _ _).inv\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 pullback (pullback.fst \u226b map \ud835\udcb0 x) (map \ud835\udcb0 z) \u27f6 pullback (pullback.fst \u226b map \ud835\udcb0 y) (map \ud835\udcb0 z)\n[PROOFSTEP]\nrefine' pullback.map _ _ _ _ (pullbackSymmetry _ _).hom (\ud835\udfd9 _) (\ud835\udfd9 _) _ _\n[GOAL]\ncase refine'_1\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 (pullback.fst \u226b map \ud835\udcb0 x) \u226b \ud835\udfd9 X = (pullbackSymmetry (map \ud835\udcb0 x) (map \ud835\udcb0 y)).hom \u226b pullback.fst \u226b map \ud835\udcb0 y\n[PROOFSTEP]\nsimp [pullback.condition]\n[GOAL]\ncase refine'_2\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 map \ud835\udcb0 z \u226b \ud835\udfd9 X = \ud835\udfd9 (obj \ud835\udcb0 z) \u226b map \ud835\udcb0 z\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 gluedCoverT' \ud835\udcb0 x y z \u226b pullback.fst \u226b pullback.fst = pullback.fst \u226b pullback.snd\n[PROOFSTEP]\ndelta gluedCoverT'\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 ((pullbackRightPullbackFstIso (map \ud835\udcb0 x) (map \ud835\udcb0 z) pullback.fst).hom \u226b\n (pullback.map (pullback.fst \u226b map \ud835\udcb0 x) (map \ud835\udcb0 z) (pullback.fst \u226b map \ud835\udcb0 y) (map \ud835\udcb0 z)\n (pullbackSymmetry (map \ud835\udcb0 x) (map \ud835\udcb0 y)).hom (\ud835\udfd9 (obj \ud835\udcb0 z)) (\ud835\udfd9 X)\n (_ : (pullback.fst \u226b map \ud835\udcb0 x) \u226b \ud835\udfd9 X = (pullbackSymmetry (map \ud835\udcb0 x) (map \ud835\udcb0 y)).hom \u226b pullback.fst \u226b map \ud835\udcb0 y)\n (_ : map \ud835\udcb0 z \u226b \ud835\udfd9 X = \ud835\udfd9 (obj \ud835\udcb0 z) \u226b map \ud835\udcb0 z) \u226b\n (pullbackRightPullbackFstIso (map \ud835\udcb0 y) (map \ud835\udcb0 z) pullback.fst).inv) \u226b\n (pullbackSymmetry pullback.fst pullback.fst).hom) \u226b\n pullback.fst \u226b pullback.fst =\n pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 gluedCoverT' \ud835\udcb0 x y z \u226b pullback.fst \u226b pullback.snd = pullback.snd \u226b pullback.snd\n[PROOFSTEP]\ndelta gluedCoverT'\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 ((pullbackRightPullbackFstIso (map \ud835\udcb0 x) (map \ud835\udcb0 z) pullback.fst).hom \u226b\n (pullback.map (pullback.fst \u226b map \ud835\udcb0 x) (map \ud835\udcb0 z) (pullback.fst \u226b map \ud835\udcb0 y) (map \ud835\udcb0 z)\n (pullbackSymmetry (map \ud835\udcb0 x) (map \ud835\udcb0 y)).hom (\ud835\udfd9 (obj \ud835\udcb0 z)) (\ud835\udfd9 X)\n (_ : (pullback.fst \u226b map \ud835\udcb0 x) \u226b \ud835\udfd9 X = (pullbackSymmetry (map \ud835\udcb0 x) (map \ud835\udcb0 y)).hom \u226b pullback.fst \u226b map \ud835\udcb0 y)\n (_ : map \ud835\udcb0 z \u226b \ud835\udfd9 X = \ud835\udfd9 (obj \ud835\udcb0 z) \u226b map \ud835\udcb0 z) \u226b\n (pullbackRightPullbackFstIso (map \ud835\udcb0 y) (map \ud835\udcb0 z) pullback.fst).inv) \u226b\n (pullbackSymmetry pullback.fst pullback.fst).hom) \u226b\n pullback.fst \u226b pullback.snd =\n pullback.snd \u226b pullback.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 gluedCoverT' \ud835\udcb0 x y z \u226b pullback.snd \u226b pullback.fst = pullback.fst \u226b pullback.snd\n[PROOFSTEP]\ndelta gluedCoverT'\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 ((pullbackRightPullbackFstIso (map \ud835\udcb0 x) (map \ud835\udcb0 z) pullback.fst).hom \u226b\n (pullback.map (pullback.fst \u226b map \ud835\udcb0 x) (map \ud835\udcb0 z) (pullback.fst \u226b map \ud835\udcb0 y) (map \ud835\udcb0 z)\n (pullbackSymmetry (map \ud835\udcb0 x) (map \ud835\udcb0 y)).hom (\ud835\udfd9 (obj \ud835\udcb0 z)) (\ud835\udfd9 X)\n (_ : (pullback.fst \u226b map \ud835\udcb0 x) \u226b \ud835\udfd9 X = (pullbackSymmetry (map \ud835\udcb0 x) (map \ud835\udcb0 y)).hom \u226b pullback.fst \u226b map \ud835\udcb0 y)\n (_ : map \ud835\udcb0 z \u226b \ud835\udfd9 X = \ud835\udfd9 (obj \ud835\udcb0 z) \u226b map \ud835\udcb0 z) \u226b\n (pullbackRightPullbackFstIso (map \ud835\udcb0 y) (map \ud835\udcb0 z) pullback.fst).inv) \u226b\n (pullbackSymmetry pullback.fst pullback.fst).hom) \u226b\n pullback.snd \u226b pullback.fst =\n pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 gluedCoverT' \ud835\udcb0 x y z \u226b pullback.snd \u226b pullback.snd = pullback.fst \u226b pullback.fst\n[PROOFSTEP]\ndelta gluedCoverT'\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 ((pullbackRightPullbackFstIso (map \ud835\udcb0 x) (map \ud835\udcb0 z) pullback.fst).hom \u226b\n (pullback.map (pullback.fst \u226b map \ud835\udcb0 x) (map \ud835\udcb0 z) (pullback.fst \u226b map \ud835\udcb0 y) (map \ud835\udcb0 z)\n (pullbackSymmetry (map \ud835\udcb0 x) (map \ud835\udcb0 y)).hom (\ud835\udfd9 (obj \ud835\udcb0 z)) (\ud835\udfd9 X)\n (_ : (pullback.fst \u226b map \ud835\udcb0 x) \u226b \ud835\udfd9 X = (pullbackSymmetry (map \ud835\udcb0 x) (map \ud835\udcb0 y)).hom \u226b pullback.fst \u226b map \ud835\udcb0 y)\n (_ : map \ud835\udcb0 z \u226b \ud835\udfd9 X = \ud835\udfd9 (obj \ud835\udcb0 z) \u226b map \ud835\udcb0 z) \u226b\n (pullbackRightPullbackFstIso (map \ud835\udcb0 y) (map \ud835\udcb0 z) pullback.fst).inv) \u226b\n (pullbackSymmetry pullback.fst pullback.fst).hom) \u226b\n pullback.snd \u226b pullback.snd =\n pullback.fst \u226b pullback.fst\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 gluedCoverT' \ud835\udcb0 x y z \u226b gluedCoverT' \ud835\udcb0 y z x \u226b gluedCoverT' \ud835\udcb0 z x y \u226b pullback.fst = pullback.fst\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2080\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 (gluedCoverT' \ud835\udcb0 x y z \u226b gluedCoverT' \ud835\udcb0 y z x \u226b gluedCoverT' \ud835\udcb0 z x y \u226b pullback.fst) \u226b pullback.fst =\n pullback.fst \u226b pullback.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2081\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 (gluedCoverT' \ud835\udcb0 x y z \u226b gluedCoverT' \ud835\udcb0 y z x \u226b gluedCoverT' \ud835\udcb0 z x y \u226b pullback.fst) \u226b pullback.snd =\n pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 gluedCoverT' \ud835\udcb0 x y z \u226b gluedCoverT' \ud835\udcb0 y z x \u226b gluedCoverT' \ud835\udcb0 z x y \u226b pullback.snd = pullback.snd\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2080\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 (gluedCoverT' \ud835\udcb0 x y z \u226b gluedCoverT' \ud835\udcb0 y z x \u226b gluedCoverT' \ud835\udcb0 z x y \u226b pullback.snd) \u226b pullback.fst =\n pullback.snd \u226b pullback.fst\n[PROOFSTEP]\nsimp [pullback.condition]\n[GOAL]\ncase h\u2081\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 (gluedCoverT' \ud835\udcb0 x y z \u226b gluedCoverT' \ud835\udcb0 y z x \u226b gluedCoverT' \ud835\udcb0 z x y \u226b pullback.snd) \u226b pullback.snd =\n pullback.snd \u226b pullback.snd\n[PROOFSTEP]\nsimp [pullback.condition]\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 gluedCoverT' \ud835\udcb0 x y z \u226b gluedCoverT' \ud835\udcb0 y z x \u226b gluedCoverT' \ud835\udcb0 z x y = \ud835\udfd9 (pullback pullback.fst pullback.fst)\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2080\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 (gluedCoverT' \ud835\udcb0 x y z \u226b gluedCoverT' \ud835\udcb0 y z x \u226b gluedCoverT' \ud835\udcb0 z x y) \u226b pullback.fst =\n \ud835\udfd9 (pullback pullback.fst pullback.fst) \u226b pullback.fst\n[PROOFSTEP]\nsimp_rw [Category.id_comp, Category.assoc]\n[GOAL]\ncase h\u2081\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 (gluedCoverT' \ud835\udcb0 x y z \u226b gluedCoverT' \ud835\udcb0 y z x \u226b gluedCoverT' \ud835\udcb0 z x y) \u226b pullback.snd =\n \ud835\udfd9 (pullback pullback.fst pullback.fst) \u226b pullback.snd\n[PROOFSTEP]\nsimp_rw [Category.id_comp, Category.assoc]\n[GOAL]\ncase h\u2080\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 gluedCoverT' \ud835\udcb0 x y z \u226b gluedCoverT' \ud835\udcb0 y z x \u226b gluedCoverT' \ud835\udcb0 z x y \u226b pullback.fst = pullback.fst\ncase h\u2081\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 gluedCoverT' \ud835\udcb0 x y z \u226b gluedCoverT' \ud835\udcb0 y z x \u226b gluedCoverT' \ud835\udcb0 z x y \u226b pullback.snd = pullback.snd\n[PROOFSTEP]\napply glued_cover_cocycle_fst\n[GOAL]\ncase h\u2081\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 gluedCoverT' \ud835\udcb0 x y z \u226b gluedCoverT' \ud835\udcb0 y z x \u226b gluedCoverT' \ud835\udcb0 z x y \u226b pullback.snd = pullback.snd\n[PROOFSTEP]\napply glued_cover_cocycle_snd\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx : \ud835\udcb0.J\n\u22a2 (fun x y => (pullbackSymmetry (map \ud835\udcb0 x) (map \ud835\udcb0 y)).hom) x x =\n \ud835\udfd9\n ((fun x =>\n match x with\n | (x, y) => pullback (map \ud835\udcb0 x) (map \ud835\udcb0 y))\n (x, x))\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 (fun x y z => gluedCoverT' \ud835\udcb0 x y z) x y z \u226b pullback.snd =\n pullback.fst \u226b (fun x y => (pullbackSymmetry (map \ud835\udcb0 x) (map \ud835\udcb0 y)).hom) x y\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2080\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 ((fun x y z => gluedCoverT' \ud835\udcb0 x y z) x y z \u226b pullback.snd) \u226b pullback.fst =\n (pullback.fst \u226b (fun x y => (pullbackSymmetry (map \ud835\udcb0 x) (map \ud835\udcb0 y)).hom) x y) \u226b pullback.fst\n[PROOFSTEP]\nsimp\n -- The `cocycle` field could have been `by tidy` but lean timeouts.\n[GOAL]\ncase h\u2081\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y z : \ud835\udcb0.J\n\u22a2 ((fun x y z => gluedCoverT' \ud835\udcb0 x y z) x y z \u226b pullback.snd) \u226b pullback.snd =\n (pullback.fst \u226b (fun x y => (pullbackSymmetry (map \ud835\udcb0 x) (map \ud835\udcb0 y)).hom) x y) \u226b pullback.snd\n[PROOFSTEP]\nsimp\n -- The `cocycle` field could have been `by tidy` but lean timeouts.\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\n\u22a2 GlueData.glued (gluedCover \ud835\udcb0) \u27f6 X\n[PROOFSTEP]\nfapply Multicoequalizer.desc\n[GOAL]\ncase k\nX : Scheme\n\ud835\udcb0 : OpenCover X\n\u22a2 (b : (diagram (gluedCover \ud835\udcb0).toGlueData).R) \u2192 MultispanIndex.right (diagram (gluedCover \ud835\udcb0).toGlueData) b \u27f6 X\ncase h\nX : Scheme\n\ud835\udcb0 : OpenCover X\n\u22a2 \u2200 (a : (diagram (gluedCover \ud835\udcb0).toGlueData).L),\n MultispanIndex.fst (diagram (gluedCover \ud835\udcb0).toGlueData) a \u226b\n ?k (MultispanIndex.fstFrom (diagram (gluedCover \ud835\udcb0).toGlueData) a) =\n MultispanIndex.snd (diagram (gluedCover \ud835\udcb0).toGlueData) a \u226b\n ?k (MultispanIndex.sndFrom (diagram (gluedCover \ud835\udcb0).toGlueData) a)\n[PROOFSTEP]\nexact fun x => \ud835\udcb0.map x\n[GOAL]\ncase h\nX : Scheme\n\ud835\udcb0 : OpenCover X\n\u22a2 \u2200 (a : (diagram (gluedCover \ud835\udcb0).toGlueData).L),\n MultispanIndex.fst (diagram (gluedCover \ud835\udcb0).toGlueData) a \u226b\n map \ud835\udcb0 (MultispanIndex.fstFrom (diagram (gluedCover \ud835\udcb0).toGlueData) a) =\n MultispanIndex.snd (diagram (gluedCover \ud835\udcb0).toGlueData) a \u226b\n map \ud835\udcb0 (MultispanIndex.sndFrom (diagram (gluedCover \ud835\udcb0).toGlueData) a)\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y : (gluedCover \ud835\udcb0).toGlueData.J\n\u22a2 MultispanIndex.fst (diagram (gluedCover \ud835\udcb0).toGlueData) (x, y) \u226b\n map \ud835\udcb0 (MultispanIndex.fstFrom (diagram (gluedCover \ud835\udcb0).toGlueData) (x, y)) =\n MultispanIndex.snd (diagram (gluedCover \ud835\udcb0).toGlueData) (x, y) \u226b\n map \ud835\udcb0 (MultispanIndex.sndFrom (diagram (gluedCover \ud835\udcb0).toGlueData) (x, y))\n[PROOFSTEP]\nchange pullback.fst \u226b _ = ((pullbackSymmetry _ _).hom \u226b pullback.fst) \u226b _\n[GOAL]\ncase h.mk\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y : (gluedCover \ud835\udcb0).toGlueData.J\n\u22a2 pullback.fst \u226b map \ud835\udcb0 (MultispanIndex.fstFrom (diagram (gluedCover \ud835\udcb0).toGlueData) (x, y)) =\n ((pullbackSymmetry (map \ud835\udcb0 x) (map \ud835\udcb0 y)).hom \u226b pullback.fst) \u226b\n map \ud835\udcb0 (MultispanIndex.sndFrom (diagram (gluedCover \ud835\udcb0).toGlueData) (x, y))\n[PROOFSTEP]\nsimpa using pullback.condition\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\n\u22a2 Function.Injective \u2191(fromGlued \ud835\udcb0).val.base\n[PROOFSTEP]\nintro x y h\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx y :\n (CategoryTheory.forget TopCat).obj\n \u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nh : \u2191(fromGlued \ud835\udcb0).val.base x = \u2191(fromGlued \ud835\udcb0).val.base y\n\u22a2 x = y\n[PROOFSTEP]\nobtain \u27e8i, x, rfl\u27e9 := \ud835\udcb0.gluedCover.\u03b9_jointly_surjective x\n[GOAL]\ncase intro.intro\nX : Scheme\n\ud835\udcb0 : OpenCover X\ny :\n (CategoryTheory.forget TopCat).obj\n \u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nh : \u2191(fromGlued \ud835\udcb0).val.base (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base x) = \u2191(fromGlued \ud835\udcb0).val.base y\n\u22a2 \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base x = y\n[PROOFSTEP]\nobtain \u27e8j, y, rfl\u27e9 := \ud835\udcb0.gluedCover.\u03b9_jointly_surjective y\n[GOAL]\ncase intro.intro.intro.intro\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nj : (gluedCover \ud835\udcb0).toGlueData.J\ny : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData j).toPresheafedSpace\nh :\n \u2191(fromGlued \ud835\udcb0).val.base (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base x) =\n \u2191(fromGlued \ud835\udcb0).val.base (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) j).val.base y)\n\u22a2 \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base x = \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) j).val.base y\n[PROOFSTEP]\nsimp_rw [\u2190 comp_apply, \u2190 SheafedSpace.comp_base, \u2190 LocallyRingedSpace.comp_val] at h \n[GOAL]\ncase intro.intro.intro.intro\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nj : (gluedCover \ud835\udcb0).toGlueData.J\ny : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData j).toPresheafedSpace\nh : \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i \u226b fromGlued \ud835\udcb0).val.base x = \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) j \u226b fromGlued \ud835\udcb0).val.base y\n\u22a2 \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base x = \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) j).val.base y\n[PROOFSTEP]\nerw [\u03b9_fromGlued, \u03b9_fromGlued] at h \n[GOAL]\ncase intro.intro.intro.intro\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nj : (gluedCover \ud835\udcb0).toGlueData.J\ny : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData j).toPresheafedSpace\nh : \u2191(map \ud835\udcb0 i).val.base x = \u2191(map \ud835\udcb0 j).val.base y\n\u22a2 \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base x = \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) j).val.base y\n[PROOFSTEP]\nlet e :=\n (TopCat.pullbackConeIsLimit _ _).conePointUniqueUpToIso\n (isLimitOfHasPullbackOfPreservesLimit Scheme.forgetToTop (\ud835\udcb0.map i) (\ud835\udcb0.map j))\n[GOAL]\ncase intro.intro.intro.intro\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nj : (gluedCover \ud835\udcb0).toGlueData.J\ny : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData j).toPresheafedSpace\nh : \u2191(map \ud835\udcb0 i).val.base x = \u2191(map \ud835\udcb0 j).val.base y\ne : (TopCat.pullbackCone (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j))).pt \u2245\n (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)\n (_ :\n forgetToTop.map pullback.fst \u226b forgetToTop.map (map \ud835\udcb0 i) =\n forgetToTop.map pullback.snd \u226b forgetToTop.map (map \ud835\udcb0 j))).pt :=\n IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j)))\n (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map \ud835\udcb0 i) (map \ud835\udcb0 j))\n\u22a2 \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base x = \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) j).val.base y\n[PROOFSTEP]\nrw [\ud835\udcb0.gluedCover.\u03b9_eq_iff]\n[GOAL]\ncase intro.intro.intro.intro\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nj : (gluedCover \ud835\udcb0).toGlueData.J\ny : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData j).toPresheafedSpace\nh : \u2191(map \ud835\udcb0 i).val.base x = \u2191(map \ud835\udcb0 j).val.base y\ne : (TopCat.pullbackCone (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j))).pt \u2245\n (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)\n (_ :\n forgetToTop.map pullback.fst \u226b forgetToTop.map (map \ud835\udcb0 i) =\n forgetToTop.map pullback.snd \u226b forgetToTop.map (map \ud835\udcb0 j))).pt :=\n IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j)))\n (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map \ud835\udcb0 i) (map \ud835\udcb0 j))\n\u22a2 GlueData.Rel (gluedCover \ud835\udcb0) { fst := i, snd := x } { fst := j, snd := y }\n[PROOFSTEP]\nright\n[GOAL]\ncase intro.intro.intro.intro.h\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nj : (gluedCover \ud835\udcb0).toGlueData.J\ny : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData j).toPresheafedSpace\nh : \u2191(map \ud835\udcb0 i).val.base x = \u2191(map \ud835\udcb0 j).val.base y\ne : (TopCat.pullbackCone (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j))).pt \u2245\n (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)\n (_ :\n forgetToTop.map pullback.fst \u226b forgetToTop.map (map \ud835\udcb0 i) =\n forgetToTop.map pullback.snd \u226b forgetToTop.map (map \ud835\udcb0 j))).pt :=\n IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j)))\n (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map \ud835\udcb0 i) (map \ud835\udcb0 j))\n\u22a2 \u2203 x_1,\n \u2191(GlueData.f (gluedCover \ud835\udcb0).toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst).val.base x_1 =\n { fst := i, snd := x }.snd \u2227\n \u2191(t (gluedCover \ud835\udcb0).toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst \u226b\n GlueData.f (gluedCover \ud835\udcb0).toGlueData { fst := j, snd := y }.fst { fst := i, snd := x }.fst).val.base\n x_1 =\n { fst := j, snd := y }.snd\n[PROOFSTEP]\nuse e.hom \u27e8\u27e8x, y\u27e9, h\u27e9\n[GOAL]\ncase h\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nj : (gluedCover \ud835\udcb0).toGlueData.J\ny : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData j).toPresheafedSpace\nh : \u2191(map \ud835\udcb0 i).val.base x = \u2191(map \ud835\udcb0 j).val.base y\ne : (TopCat.pullbackCone (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j))).pt \u2245\n (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)\n (_ :\n forgetToTop.map pullback.fst \u226b forgetToTop.map (map \ud835\udcb0 i) =\n forgetToTop.map pullback.snd \u226b forgetToTop.map (map \ud835\udcb0 j))).pt :=\n IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j)))\n (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map \ud835\udcb0 i) (map \ud835\udcb0 j))\n\u22a2 \u2191(GlueData.f (gluedCover \ud835\udcb0).toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst).val.base\n (\u2191e.hom { val := (x, y), property := h }) =\n { fst := i, snd := x }.snd \u2227\n \u2191(t (gluedCover \ud835\udcb0).toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst \u226b\n GlueData.f (gluedCover \ud835\udcb0).toGlueData { fst := j, snd := y }.fst { fst := i, snd := x }.fst).val.base\n (\u2191e.hom { val := (x, y), property := h }) =\n { fst := j, snd := y }.snd\n[PROOFSTEP]\nconstructor\n -- Porting note: in the two subproofs below, added the `change` lines\n[GOAL]\ncase h.left\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nj : (gluedCover \ud835\udcb0).toGlueData.J\ny : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData j).toPresheafedSpace\nh : \u2191(map \ud835\udcb0 i).val.base x = \u2191(map \ud835\udcb0 j).val.base y\ne : (TopCat.pullbackCone (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j))).pt \u2245\n (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)\n (_ :\n forgetToTop.map pullback.fst \u226b forgetToTop.map (map \ud835\udcb0 i) =\n forgetToTop.map pullback.snd \u226b forgetToTop.map (map \ud835\udcb0 j))).pt :=\n IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j)))\n (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map \ud835\udcb0 i) (map \ud835\udcb0 j))\n\u22a2 \u2191(GlueData.f (gluedCover \ud835\udcb0).toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst).val.base\n (\u2191e.hom { val := (x, y), property := h }) =\n { fst := i, snd := x }.snd\n[PROOFSTEP]\nchange (e.hom \u226b _) \u27e8(x, y), h\u27e9 = x\n[GOAL]\ncase h.left\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nj : (gluedCover \ud835\udcb0).toGlueData.J\ny : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData j).toPresheafedSpace\nh : \u2191(map \ud835\udcb0 i).val.base x = \u2191(map \ud835\udcb0 j).val.base y\ne : (TopCat.pullbackCone (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j))).pt \u2245\n (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)\n (_ :\n forgetToTop.map pullback.fst \u226b forgetToTop.map (map \ud835\udcb0 i) =\n forgetToTop.map pullback.snd \u226b forgetToTop.map (map \ud835\udcb0 j))).pt :=\n IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j)))\n (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map \ud835\udcb0 i) (map \ud835\udcb0 j))\n\u22a2 \u2191(e.hom \u226b (GlueData.f (gluedCover \ud835\udcb0).toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst).val.base)\n { val := (x, y), property := h } =\n x\n[PROOFSTEP]\nerw [IsLimit.conePointUniqueUpToIso_hom_comp _ _ WalkingCospan.left]\n[GOAL]\ncase h.left\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nj : (gluedCover \ud835\udcb0).toGlueData.J\ny : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData j).toPresheafedSpace\nh : \u2191(map \ud835\udcb0 i).val.base x = \u2191(map \ud835\udcb0 j).val.base y\ne : (TopCat.pullbackCone (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j))).pt \u2245\n (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)\n (_ :\n forgetToTop.map pullback.fst \u226b forgetToTop.map (map \ud835\udcb0 i) =\n forgetToTop.map pullback.snd \u226b forgetToTop.map (map \ud835\udcb0 j))).pt :=\n IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j)))\n (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map \ud835\udcb0 i) (map \ud835\udcb0 j))\n\u22a2 \u2191(NatTrans.app (TopCat.pullbackCone (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j))).\u03c0 WalkingCospan.left)\n { val := (x, y), property := h } =\n x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.right\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nj : (gluedCover \ud835\udcb0).toGlueData.J\ny : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData j).toPresheafedSpace\nh : \u2191(map \ud835\udcb0 i).val.base x = \u2191(map \ud835\udcb0 j).val.base y\ne : (TopCat.pullbackCone (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j))).pt \u2245\n (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)\n (_ :\n forgetToTop.map pullback.fst \u226b forgetToTop.map (map \ud835\udcb0 i) =\n forgetToTop.map pullback.snd \u226b forgetToTop.map (map \ud835\udcb0 j))).pt :=\n IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j)))\n (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map \ud835\udcb0 i) (map \ud835\udcb0 j))\n\u22a2 \u2191(t (gluedCover \ud835\udcb0).toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst \u226b\n GlueData.f (gluedCover \ud835\udcb0).toGlueData { fst := j, snd := y }.fst { fst := i, snd := x }.fst).val.base\n (\u2191e.hom { val := (x, y), property := h }) =\n { fst := j, snd := y }.snd\n[PROOFSTEP]\nchange (e.hom \u226b ((gluedCover \ud835\udcb0).toGlueData.t i j \u226b (gluedCover \ud835\udcb0).toGlueData.f j i).val.base) \u27e8(x, y), h\u27e9 = y\n[GOAL]\ncase h.right\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nj : (gluedCover \ud835\udcb0).toGlueData.J\ny : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData j).toPresheafedSpace\nh : \u2191(map \ud835\udcb0 i).val.base x = \u2191(map \ud835\udcb0 j).val.base y\ne : (TopCat.pullbackCone (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j))).pt \u2245\n (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)\n (_ :\n forgetToTop.map pullback.fst \u226b forgetToTop.map (map \ud835\udcb0 i) =\n forgetToTop.map pullback.snd \u226b forgetToTop.map (map \ud835\udcb0 j))).pt :=\n IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j)))\n (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map \ud835\udcb0 i) (map \ud835\udcb0 j))\n\u22a2 \u2191(e.hom \u226b (t (gluedCover \ud835\udcb0).toGlueData i j \u226b GlueData.f (gluedCover \ud835\udcb0).toGlueData j i).val.base)\n { val := (x, y), property := h } =\n y\n[PROOFSTEP]\nerw [pullbackSymmetry_hom_comp_fst, IsLimit.conePointUniqueUpToIso_hom_comp _ _ WalkingCospan.right]\n[GOAL]\ncase h.right\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nj : (gluedCover \ud835\udcb0).toGlueData.J\ny : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData j).toPresheafedSpace\nh : \u2191(map \ud835\udcb0 i).val.base x = \u2191(map \ud835\udcb0 j).val.base y\ne : (TopCat.pullbackCone (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j))).pt \u2245\n (PullbackCone.mk (forgetToTop.map pullback.fst) (forgetToTop.map pullback.snd)\n (_ :\n forgetToTop.map pullback.fst \u226b forgetToTop.map (map \ud835\udcb0 i) =\n forgetToTop.map pullback.snd \u226b forgetToTop.map (map \ud835\udcb0 j))).pt :=\n IsLimit.conePointUniqueUpToIso (TopCat.pullbackConeIsLimit (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j)))\n (isLimitOfHasPullbackOfPreservesLimit forgetToTop (map \ud835\udcb0 i) (map \ud835\udcb0 j))\n\u22a2 \u2191(NatTrans.app (TopCat.pullbackCone (forgetToTop.map (map \ud835\udcb0 i)) (forgetToTop.map (map \ud835\udcb0 j))).\u03c0 WalkingCospan.right)\n { val := (x, y), property := h } =\n y\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx : \u2191\u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n\u22a2 IsIso (stalkMap (fromGlued \ud835\udcb0).val x)\n[PROOFSTEP]\nobtain \u27e8i, x, rfl\u27e9 := \ud835\udcb0.gluedCover.\u03b9_jointly_surjective x\n[GOAL]\ncase intro.intro\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\n\u22a2 IsIso (stalkMap (fromGlued \ud835\udcb0).val (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base x))\n[PROOFSTEP]\nhave := PresheafedSpace.stalkMap.congr_hom _ _ (congr_arg LocallyRingedSpace.Hom.val <| \ud835\udcb0.\u03b9_fromGlued i) x\n[GOAL]\ncase intro.intro\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nthis :\n stalkMap (GlueData.\u03b9 (gluedCover \ud835\udcb0) i \u226b fromGlued \ud835\udcb0).val x =\n eqToHom\n (_ :\n stalk X.toPresheafedSpace (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i \u226b fromGlued \ud835\udcb0).val.base x) =\n stalk X.toPresheafedSpace (\u2191(map \ud835\udcb0 i).val.base x)) \u226b\n stalkMap (map \ud835\udcb0 i).val x\n\u22a2 IsIso (stalkMap (fromGlued \ud835\udcb0).val (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base x))\n[PROOFSTEP]\nerw [PresheafedSpace.stalkMap.comp] at this \n[GOAL]\ncase intro.intro\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nthis :\n stalkMap (fromGlued \ud835\udcb0).val (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base x) \u226b\n stalkMap (GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val x =\n eqToHom\n (_ :\n stalk X.toPresheafedSpace (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i \u226b fromGlued \ud835\udcb0).val.base x) =\n stalk X.toPresheafedSpace (\u2191(map \ud835\udcb0 i).val.base x)) \u226b\n stalkMap (map \ud835\udcb0 i).val x\n\u22a2 IsIso (stalkMap (fromGlued \ud835\udcb0).val (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base x))\n[PROOFSTEP]\nrw [\u2190 IsIso.eq_comp_inv] at this \n[GOAL]\ncase intro.intro\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nthis :\n stalkMap (fromGlued \ud835\udcb0).val (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base x) =\n (eqToHom\n (_ :\n stalk X.toPresheafedSpace (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i \u226b fromGlued \ud835\udcb0).val.base x) =\n stalk X.toPresheafedSpace (\u2191(map \ud835\udcb0 i).val.base x)) \u226b\n stalkMap (map \ud835\udcb0 i).val x) \u226b\n inv (stalkMap (GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val x)\n\u22a2 IsIso (stalkMap (fromGlued \ud835\udcb0).val (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base x))\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase intro.intro\nX : Scheme\n\ud835\udcb0 : OpenCover X\ni : (gluedCover \ud835\udcb0).toGlueData.J\nx : \u2191\u2191(U (gluedCover \ud835\udcb0).toGlueData i).toPresheafedSpace\nthis :\n stalkMap (fromGlued \ud835\udcb0).val (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base x) =\n (eqToHom\n (_ :\n stalk X.toPresheafedSpace (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i \u226b fromGlued \ud835\udcb0).val.base x) =\n stalk X.toPresheafedSpace (\u2191(map \ud835\udcb0 i).val.base x)) \u226b\n stalkMap (map \ud835\udcb0 i).val x) \u226b\n inv (stalkMap (GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val x)\n\u22a2 IsIso\n ((eqToHom\n (_ :\n stalk X.toPresheafedSpace (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i \u226b fromGlued \ud835\udcb0).val.base x) =\n stalk X.toPresheafedSpace (\u2191(map \ud835\udcb0 i).val.base x)) \u226b\n stalkMap (map \ud835\udcb0 i).val x) \u226b\n inv (stalkMap (GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val x))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\n\u22a2 IsOpenMap \u2191(fromGlued \ud835\udcb0).val.base\n[PROOFSTEP]\nintro U hU\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nU :\n Set\n ((CategoryTheory.forget TopCat).obj\n \u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\nhU : _root_.IsOpen U\n\u22a2 _root_.IsOpen (\u2191(fromGlued \ud835\udcb0).val.base '' U)\n[PROOFSTEP]\nrw [isOpen_iff_forall_mem_open]\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nU :\n Set\n ((CategoryTheory.forget TopCat).obj\n \u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\nhU : _root_.IsOpen U\n\u22a2 \u2200 (x : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace),\n x \u2208 \u2191(fromGlued \ud835\udcb0).val.base '' U \u2192 \u2203 t, t \u2286 \u2191(fromGlued \ud835\udcb0).val.base '' U \u2227 _root_.IsOpen t \u2227 x \u2208 t\n[PROOFSTEP]\nintro x hx\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nU :\n Set\n ((CategoryTheory.forget TopCat).obj\n \u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\nhU : _root_.IsOpen U\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhx : x \u2208 \u2191(fromGlued \ud835\udcb0).val.base '' U\n\u22a2 \u2203 t, t \u2286 \u2191(fromGlued \ud835\udcb0).val.base '' U \u2227 _root_.IsOpen t \u2227 x \u2208 t\n[PROOFSTEP]\nrw [\ud835\udcb0.gluedCover.isOpen_iff] at hU \n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nU :\n Set\n ((CategoryTheory.forget TopCat).obj\n \u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\nhU : \u2200 (i : (gluedCover \ud835\udcb0).toGlueData.J), _root_.IsOpen (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base \u207b\u00b9' U)\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhx : x \u2208 \u2191(fromGlued \ud835\udcb0).val.base '' U\n\u22a2 \u2203 t, t \u2286 \u2191(fromGlued \ud835\udcb0).val.base '' U \u2227 _root_.IsOpen t \u2227 x \u2208 t\n[PROOFSTEP]\nuse \ud835\udcb0.fromGlued.val.base '' U \u2229 Set.range (\ud835\udcb0.map (\ud835\udcb0.f x)).1.base\n[GOAL]\ncase h\nX : Scheme\n\ud835\udcb0 : OpenCover X\nU :\n Set\n ((CategoryTheory.forget TopCat).obj\n \u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\nhU : \u2200 (i : (gluedCover \ud835\udcb0).toGlueData.J), _root_.IsOpen (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base \u207b\u00b9' U)\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhx : x \u2208 \u2191(fromGlued \ud835\udcb0).val.base '' U\n\u22a2 \u2191(fromGlued \ud835\udcb0).val.base '' U \u2229 Set.range \u2191(map \ud835\udcb0 (f \ud835\udcb0 x)).val.base \u2286 \u2191(fromGlued \ud835\udcb0).val.base '' U \u2227\n _root_.IsOpen (\u2191(fromGlued \ud835\udcb0).val.base '' U \u2229 Set.range \u2191(map \ud835\udcb0 (f \ud835\udcb0 x)).val.base) \u2227\n x \u2208 \u2191(fromGlued \ud835\udcb0).val.base '' U \u2229 Set.range \u2191(map \ud835\udcb0 (f \ud835\udcb0 x)).val.base\n[PROOFSTEP]\nuse Set.inter_subset_left _ _\n[GOAL]\ncase right\nX : Scheme\n\ud835\udcb0 : OpenCover X\nU :\n Set\n ((CategoryTheory.forget TopCat).obj\n \u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\nhU : \u2200 (i : (gluedCover \ud835\udcb0).toGlueData.J), _root_.IsOpen (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base \u207b\u00b9' U)\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhx : x \u2208 \u2191(fromGlued \ud835\udcb0).val.base '' U\n\u22a2 _root_.IsOpen (\u2191(fromGlued \ud835\udcb0).val.base '' U \u2229 Set.range \u2191(map \ud835\udcb0 (f \ud835\udcb0 x)).val.base) \u2227\n x \u2208 \u2191(fromGlued \ud835\udcb0).val.base '' U \u2229 Set.range \u2191(map \ud835\udcb0 (f \ud835\udcb0 x)).val.base\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase right.left\nX : Scheme\n\ud835\udcb0 : OpenCover X\nU :\n Set\n ((CategoryTheory.forget TopCat).obj\n \u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\nhU : \u2200 (i : (gluedCover \ud835\udcb0).toGlueData.J), _root_.IsOpen (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base \u207b\u00b9' U)\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhx : x \u2208 \u2191(fromGlued \ud835\udcb0).val.base '' U\n\u22a2 _root_.IsOpen (\u2191(fromGlued \ud835\udcb0).val.base '' U \u2229 Set.range \u2191(map \ud835\udcb0 (f \ud835\udcb0 x)).val.base)\n[PROOFSTEP]\nrw [\u2190 Set.image_preimage_eq_inter_range]\n[GOAL]\ncase right.left\nX : Scheme\n\ud835\udcb0 : OpenCover X\nU :\n Set\n ((CategoryTheory.forget TopCat).obj\n \u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\nhU : \u2200 (i : (gluedCover \ud835\udcb0).toGlueData.J), _root_.IsOpen (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base \u207b\u00b9' U)\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhx : x \u2208 \u2191(fromGlued \ud835\udcb0).val.base '' U\n\u22a2 _root_.IsOpen (\u2191(map \ud835\udcb0 (f \ud835\udcb0 x)).val.base '' (\u2191(map \ud835\udcb0 (f \ud835\udcb0 x)).val.base \u207b\u00b9' (\u2191(fromGlued \ud835\udcb0).val.base '' U)))\n[PROOFSTEP]\napply (show IsOpenImmersion (\ud835\udcb0.map (\ud835\udcb0.f x)) from inferInstance).base_open.isOpenMap\n[GOAL]\ncase right.left.a\nX : Scheme\n\ud835\udcb0 : OpenCover X\nU :\n Set\n ((CategoryTheory.forget TopCat).obj\n \u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\nhU : \u2200 (i : (gluedCover \ud835\udcb0).toGlueData.J), _root_.IsOpen (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base \u207b\u00b9' U)\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhx : x \u2208 \u2191(fromGlued \ud835\udcb0).val.base '' U\n\u22a2 _root_.IsOpen (\u2191(map \ud835\udcb0 (f \ud835\udcb0 x)).val.base \u207b\u00b9' (\u2191(fromGlued \ud835\udcb0).val.base '' U))\n[PROOFSTEP]\nconvert hU (\ud835\udcb0.f x) using 1\n[GOAL]\ncase h.e'_3.h\nX : Scheme\n\ud835\udcb0 : OpenCover X\nU :\n Set\n ((CategoryTheory.forget TopCat).obj\n \u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\nhU : \u2200 (i : (gluedCover \ud835\udcb0).toGlueData.J), _root_.IsOpen (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base \u207b\u00b9' U)\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhx : x \u2208 \u2191(fromGlued \ud835\udcb0).val.base '' U\ne_1\u271d :\n (CategoryTheory.forget TopCat).obj \u2191(obj \ud835\udcb0 (f \ud835\udcb0 x)).toPresheafedSpace =\n (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U (gluedCover \ud835\udcb0).toGlueData (f \ud835\udcb0 x)).toPresheafedSpace\n\u22a2 \u2191(map \ud835\udcb0 (f \ud835\udcb0 x)).val.base \u207b\u00b9' (\u2191(fromGlued \ud835\udcb0).val.base '' U) = \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) (f \ud835\udcb0 x)).val.base \u207b\u00b9' U\n[PROOFSTEP]\nrw [\u2190 \u03b9_fromGlued]\n[GOAL]\ncase h.e'_3.h\nX : Scheme\n\ud835\udcb0 : OpenCover X\nU :\n Set\n ((CategoryTheory.forget TopCat).obj\n \u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\nhU : \u2200 (i : (gluedCover \ud835\udcb0).toGlueData.J), _root_.IsOpen (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base \u207b\u00b9' U)\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhx : x \u2208 \u2191(fromGlued \ud835\udcb0).val.base '' U\ne_1\u271d :\n (CategoryTheory.forget TopCat).obj \u2191(obj \ud835\udcb0 (f \ud835\udcb0 x)).toPresheafedSpace =\n (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U (gluedCover \ud835\udcb0).toGlueData (f \ud835\udcb0 x)).toPresheafedSpace\n\u22a2 \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) (f \ud835\udcb0 x) \u226b fromGlued \ud835\udcb0).val.base \u207b\u00b9' (\u2191(fromGlued \ud835\udcb0).val.base '' U) =\n \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) (f \ud835\udcb0 x)).val.base \u207b\u00b9' U\n[PROOFSTEP]\nerw [coe_comp]\n[GOAL]\ncase h.e'_3.h\nX : Scheme\n\ud835\udcb0 : OpenCover X\nU :\n Set\n ((CategoryTheory.forget TopCat).obj\n \u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\nhU : \u2200 (i : (gluedCover \ud835\udcb0).toGlueData.J), _root_.IsOpen (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base \u207b\u00b9' U)\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhx : x \u2208 \u2191(fromGlued \ud835\udcb0).val.base '' U\ne_1\u271d :\n (CategoryTheory.forget TopCat).obj \u2191(obj \ud835\udcb0 (f \ud835\udcb0 x)).toPresheafedSpace =\n (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U (gluedCover \ud835\udcb0).toGlueData (f \ud835\udcb0 x)).toPresheafedSpace\n\u22a2 \u2191(fromGlued \ud835\udcb0).val.base \u2218 \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) (f \ud835\udcb0 x)).val.base \u207b\u00b9' (\u2191(fromGlued \ud835\udcb0).val.base '' U) =\n \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) (f \ud835\udcb0 x)).val.base \u207b\u00b9' U\n[PROOFSTEP]\nrw [Set.preimage_comp]\n[GOAL]\ncase h.e'_3.h\nX : Scheme\n\ud835\udcb0 : OpenCover X\nU :\n Set\n ((CategoryTheory.forget TopCat).obj\n \u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\nhU : \u2200 (i : (gluedCover \ud835\udcb0).toGlueData.J), _root_.IsOpen (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base \u207b\u00b9' U)\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhx : x \u2208 \u2191(fromGlued \ud835\udcb0).val.base '' U\ne_1\u271d :\n (CategoryTheory.forget TopCat).obj \u2191(obj \ud835\udcb0 (f \ud835\udcb0 x)).toPresheafedSpace =\n (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U (gluedCover \ud835\udcb0).toGlueData (f \ud835\udcb0 x)).toPresheafedSpace\n\u22a2 \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) (f \ud835\udcb0 x)).val.base \u207b\u00b9' (\u2191(fromGlued \ud835\udcb0).val.base \u207b\u00b9' (\u2191(fromGlued \ud835\udcb0).val.base '' U)) =\n \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) (f \ud835\udcb0 x)).val.base \u207b\u00b9' U\n[PROOFSTEP]\ncongr! 1\n[GOAL]\ncase h.e'_3.h.h.e'_4\nX : Scheme\n\ud835\udcb0 : OpenCover X\nU :\n Set\n ((CategoryTheory.forget TopCat).obj\n \u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\nhU : \u2200 (i : (gluedCover \ud835\udcb0).toGlueData.J), _root_.IsOpen (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base \u207b\u00b9' U)\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhx : x \u2208 \u2191(fromGlued \ud835\udcb0).val.base '' U\ne_1\u271d :\n (CategoryTheory.forget TopCat).obj \u2191(obj \ud835\udcb0 (f \ud835\udcb0 x)).toPresheafedSpace =\n (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U (gluedCover \ud835\udcb0).toGlueData (f \ud835\udcb0 x)).toPresheafedSpace\n\u22a2 \u2191(fromGlued \ud835\udcb0).val.base \u207b\u00b9' (\u2191(fromGlued \ud835\udcb0).val.base '' U) = U\n[PROOFSTEP]\nrefine' Set.preimage_image_eq _ \ud835\udcb0.fromGlued_injective\n[GOAL]\ncase right.right\nX : Scheme\n\ud835\udcb0 : OpenCover X\nU :\n Set\n ((CategoryTheory.forget TopCat).obj\n \u2191(GlueData.glued (gluedCover \ud835\udcb0)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\nhU : \u2200 (i : (gluedCover \ud835\udcb0).toGlueData.J), _root_.IsOpen (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) i).val.base \u207b\u00b9' U)\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhx : x \u2208 \u2191(fromGlued \ud835\udcb0).val.base '' U\n\u22a2 x \u2208 \u2191(fromGlued \ud835\udcb0).val.base '' U \u2229 Set.range \u2191(map \ud835\udcb0 (f \ud835\udcb0 x)).val.base\n[PROOFSTEP]\nexact \u27e8hx, \ud835\udcb0.Covers x\u27e9\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\n\u22a2 Epi (fromGlued \ud835\udcb0).val.base\n[PROOFSTEP]\nrw [TopCat.epi_iff_surjective]\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\n\u22a2 Function.Surjective \u2191(fromGlued \ud835\udcb0).val.base\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\n\u22a2 \u2203 a, \u2191(fromGlued \ud835\udcb0).val.base a = x\n[PROOFSTEP]\nobtain \u27e8y, h\u27e9 := \ud835\udcb0.Covers x\n[GOAL]\ncase intro\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\ny : (CategoryTheory.forget TopCat).obj \u2191(obj \ud835\udcb0 (f \ud835\udcb0 x)).toPresheafedSpace\nh : \u2191(map \ud835\udcb0 (f \ud835\udcb0 x)).val.base y = x\n\u22a2 \u2203 a, \u2191(fromGlued \ud835\udcb0).val.base a = x\n[PROOFSTEP]\nuse(\ud835\udcb0.gluedCover.\u03b9 (\ud835\udcb0.f x)).1.base y\n[GOAL]\ncase h\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\ny : (CategoryTheory.forget TopCat).obj \u2191(obj \ud835\udcb0 (f \ud835\udcb0 x)).toPresheafedSpace\nh : \u2191(map \ud835\udcb0 (f \ud835\udcb0 x)).val.base y = x\n\u22a2 \u2191(fromGlued \ud835\udcb0).val.base (\u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) (f \ud835\udcb0 x)).val.base y) = x\n[PROOFSTEP]\nrw [\u2190 comp_apply]\n[GOAL]\ncase h\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\ny : (CategoryTheory.forget TopCat).obj \u2191(obj \ud835\udcb0 (f \ud835\udcb0 x)).toPresheafedSpace\nh : \u2191(map \ud835\udcb0 (f \ud835\udcb0 x)).val.base y = x\n\u22a2 \u2191((GlueData.\u03b9 (gluedCover \ud835\udcb0) (f \ud835\udcb0 x)).val.base \u226b (fromGlued \ud835\udcb0).val.base) y = x\n[PROOFSTEP]\nrw [\u2190 \ud835\udcb0.\u03b9_fromGlued (\ud835\udcb0.f x)] at h \n[GOAL]\ncase h\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\ny : (CategoryTheory.forget TopCat).obj \u2191(obj \ud835\udcb0 (f \ud835\udcb0 x)).toPresheafedSpace\nh : \u2191(GlueData.\u03b9 (gluedCover \ud835\udcb0) (f \ud835\udcb0 x) \u226b fromGlued \ud835\udcb0).val.base y = x\n\u22a2 \u2191((GlueData.\u03b9 (gluedCover \ud835\udcb0) (f \ud835\udcb0 x)).val.base \u226b (fromGlued \ud835\udcb0).val.base) y = x\n[PROOFSTEP]\nexact h\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nF : Scheme \u2964 PresheafedSpace CommRingCat :=\n forgetToLocallyRingedSpace \u22d9 LocallyRingedSpace.forgetToSheafedSpace \u22d9 SheafedSpace.forgetToPresheafedSpace\n\u22a2 IsIso (F.map (fromGlued \ud835\udcb0))\n[PROOFSTEP]\nchange @IsIso (PresheafedSpace _) _ _ _ \ud835\udcb0.fromGlued.val\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nF : Scheme \u2964 PresheafedSpace CommRingCat :=\n forgetToLocallyRingedSpace \u22d9 LocallyRingedSpace.forgetToSheafedSpace \u22d9 SheafedSpace.forgetToPresheafedSpace\n\u22a2 IsIso (fromGlued \ud835\udcb0).val\n[PROOFSTEP]\napply PresheafedSpace.IsOpenImmersion.to_iso\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf : (x : \ud835\udcb0.J) \u2192 obj \ud835\udcb0 x \u27f6 Y\nhf : \u2200 (x y : \ud835\udcb0.J), pullback.fst \u226b f x = pullback.snd \u226b f y\n\u22a2 X \u27f6 Y\n[PROOFSTEP]\nrefine' inv \ud835\udcb0.fromGlued \u226b _\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf : (x : \ud835\udcb0.J) \u2192 obj \ud835\udcb0 x \u27f6 Y\nhf : \u2200 (x y : \ud835\udcb0.J), pullback.fst \u226b f x = pullback.snd \u226b f y\n\u22a2 GlueData.glued (gluedCover \ud835\udcb0) \u27f6 Y\n[PROOFSTEP]\nfapply Multicoequalizer.desc\n[GOAL]\ncase k\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf : (x : \ud835\udcb0.J) \u2192 obj \ud835\udcb0 x \u27f6 Y\nhf : \u2200 (x y : \ud835\udcb0.J), pullback.fst \u226b f x = pullback.snd \u226b f y\n\u22a2 (b : (diagram (gluedCover \ud835\udcb0).toGlueData).R) \u2192 MultispanIndex.right (diagram (gluedCover \ud835\udcb0).toGlueData) b \u27f6 Y\ncase h\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf : (x : \ud835\udcb0.J) \u2192 obj \ud835\udcb0 x \u27f6 Y\nhf : \u2200 (x y : \ud835\udcb0.J), pullback.fst \u226b f x = pullback.snd \u226b f y\n\u22a2 \u2200 (a : (diagram (gluedCover \ud835\udcb0).toGlueData).L),\n MultispanIndex.fst (diagram (gluedCover \ud835\udcb0).toGlueData) a \u226b\n ?k (MultispanIndex.fstFrom (diagram (gluedCover \ud835\udcb0).toGlueData) a) =\n MultispanIndex.snd (diagram (gluedCover \ud835\udcb0).toGlueData) a \u226b\n ?k (MultispanIndex.sndFrom (diagram (gluedCover \ud835\udcb0).toGlueData) a)\n[PROOFSTEP]\nexact f\n[GOAL]\ncase h\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf : (x : \ud835\udcb0.J) \u2192 obj \ud835\udcb0 x \u27f6 Y\nhf : \u2200 (x y : \ud835\udcb0.J), pullback.fst \u226b f x = pullback.snd \u226b f y\n\u22a2 \u2200 (a : (diagram (gluedCover \ud835\udcb0).toGlueData).L),\n MultispanIndex.fst (diagram (gluedCover \ud835\udcb0).toGlueData) a \u226b\n f (MultispanIndex.fstFrom (diagram (gluedCover \ud835\udcb0).toGlueData) a) =\n MultispanIndex.snd (diagram (gluedCover \ud835\udcb0).toGlueData) a \u226b\n f (MultispanIndex.sndFrom (diagram (gluedCover \ud835\udcb0).toGlueData) a)\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9\n[GOAL]\ncase h.mk\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf : (x : \ud835\udcb0.J) \u2192 obj \ud835\udcb0 x \u27f6 Y\nhf : \u2200 (x y : \ud835\udcb0.J), pullback.fst \u226b f x = pullback.snd \u226b f y\ni j : (gluedCover \ud835\udcb0).toGlueData.J\n\u22a2 MultispanIndex.fst (diagram (gluedCover \ud835\udcb0).toGlueData) (i, j) \u226b\n f (MultispanIndex.fstFrom (diagram (gluedCover \ud835\udcb0).toGlueData) (i, j)) =\n MultispanIndex.snd (diagram (gluedCover \ud835\udcb0).toGlueData) (i, j) \u226b\n f (MultispanIndex.sndFrom (diagram (gluedCover \ud835\udcb0).toGlueData) (i, j))\n[PROOFSTEP]\nchange pullback.fst \u226b f i = (_ \u226b _) \u226b f j\n[GOAL]\ncase h.mk\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf : (x : \ud835\udcb0.J) \u2192 obj \ud835\udcb0 x \u27f6 Y\nhf : \u2200 (x y : \ud835\udcb0.J), pullback.fst \u226b f x = pullback.snd \u226b f y\ni j : (gluedCover \ud835\udcb0).toGlueData.J\n\u22a2 pullback.fst \u226b f i = (t (gluedCover \ud835\udcb0).toGlueData i j \u226b GlueData.f (gluedCover \ud835\udcb0).toGlueData j i) \u226b f j\n[PROOFSTEP]\nerw [pullbackSymmetry_hom_comp_fst]\n[GOAL]\ncase h.mk\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf : (x : \ud835\udcb0.J) \u2192 obj \ud835\udcb0 x \u27f6 Y\nhf : \u2200 (x y : \ud835\udcb0.J), pullback.fst \u226b f x = pullback.snd \u226b f y\ni j : (gluedCover \ud835\udcb0).toGlueData.J\n\u22a2 pullback.fst \u226b f i = pullback.snd \u226b f j\n[PROOFSTEP]\nexact hf i j\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf : (x : \ud835\udcb0.J) \u2192 obj \ud835\udcb0 x \u27f6 Y\nhf : \u2200 (x y : \ud835\udcb0.J), pullback.fst \u226b f x = pullback.snd \u226b f y\nx : \ud835\udcb0.J\n\u22a2 map \ud835\udcb0 x \u226b glueMorphisms \ud835\udcb0 f hf = f x\n[PROOFSTEP]\nrw [\u2190 \u03b9_fromGlued, Category.assoc]\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf : (x : \ud835\udcb0.J) \u2192 obj \ud835\udcb0 x \u27f6 Y\nhf : \u2200 (x y : \ud835\udcb0.J), pullback.fst \u226b f x = pullback.snd \u226b f y\nx : \ud835\udcb0.J\n\u22a2 GlueData.\u03b9 (gluedCover \ud835\udcb0) x \u226b fromGlued \ud835\udcb0 \u226b glueMorphisms \ud835\udcb0 f hf = f x\n[PROOFSTEP]\nerw [IsIso.hom_inv_id_assoc, Multicoequalizer.\u03c0_desc]\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf\u2081 f\u2082 : X \u27f6 Y\nh : \u2200 (x : \ud835\udcb0.J), map \ud835\udcb0 x \u226b f\u2081 = map \ud835\udcb0 x \u226b f\u2082\n\u22a2 f\u2081 = f\u2082\n[PROOFSTEP]\nrw [\u2190 cancel_epi \ud835\udcb0.fromGlued]\n[GOAL]\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf\u2081 f\u2082 : X \u27f6 Y\nh : \u2200 (x : \ud835\udcb0.J), map \ud835\udcb0 x \u226b f\u2081 = map \ud835\udcb0 x \u226b f\u2082\n\u22a2 fromGlued \ud835\udcb0 \u226b f\u2081 = fromGlued \ud835\udcb0 \u226b f\u2082\n[PROOFSTEP]\napply Multicoequalizer.hom_ext\n[GOAL]\ncase h\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf\u2081 f\u2082 : X \u27f6 Y\nh : \u2200 (x : \ud835\udcb0.J), map \ud835\udcb0 x \u226b f\u2081 = map \ud835\udcb0 x \u226b f\u2082\n\u22a2 \u2200 (b : (diagram (gluedCover \ud835\udcb0).toGlueData).R),\n Multicoequalizer.\u03c0 (diagram (gluedCover \ud835\udcb0).toGlueData) b \u226b fromGlued \ud835\udcb0 \u226b f\u2081 =\n Multicoequalizer.\u03c0 (diagram (gluedCover \ud835\udcb0).toGlueData) b \u226b fromGlued \ud835\udcb0 \u226b f\u2082\n[PROOFSTEP]\nintro x\n[GOAL]\ncase h\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf\u2081 f\u2082 : X \u27f6 Y\nh : \u2200 (x : \ud835\udcb0.J), map \ud835\udcb0 x \u226b f\u2081 = map \ud835\udcb0 x \u226b f\u2082\nx : (diagram (gluedCover \ud835\udcb0).toGlueData).R\n\u22a2 Multicoequalizer.\u03c0 (diagram (gluedCover \ud835\udcb0).toGlueData) x \u226b fromGlued \ud835\udcb0 \u226b f\u2081 =\n Multicoequalizer.\u03c0 (diagram (gluedCover \ud835\udcb0).toGlueData) x \u226b fromGlued \ud835\udcb0 \u226b f\u2082\n[PROOFSTEP]\nerw [Multicoequalizer.\u03c0_desc_assoc]\n[GOAL]\ncase h\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf\u2081 f\u2082 : X \u27f6 Y\nh : \u2200 (x : \ud835\udcb0.J), map \ud835\udcb0 x \u226b f\u2081 = map \ud835\udcb0 x \u226b f\u2082\nx : (diagram (gluedCover \ud835\udcb0).toGlueData).R\n\u22a2 map \ud835\udcb0 x \u226b f\u2081 = Multicoequalizer.\u03c0 (diagram (gluedCover \ud835\udcb0).toGlueData) x \u226b fromGlued \ud835\udcb0 \u226b f\u2082\n[PROOFSTEP]\nerw [Multicoequalizer.\u03c0_desc_assoc]\n[GOAL]\ncase h\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf\u2081 f\u2082 : X \u27f6 Y\nh : \u2200 (x : \ud835\udcb0.J), map \ud835\udcb0 x \u226b f\u2081 = map \ud835\udcb0 x \u226b f\u2082\nx : (diagram (gluedCover \ud835\udcb0).toGlueData).R\n\u22a2 map \ud835\udcb0 x \u226b f\u2081 = map \ud835\udcb0 x \u226b f\u2082\n[PROOFSTEP]\nexact h x\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.Gluing", "llama_tokens": 31129, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.40733340004593027, "lm_q1q2_score": 0.21161838646710243}} {"text": "[GOAL]\nR : Type u\ninst\u271d : Ring R\nX\u271d Y\u271d : ModuleCat R\na\u2081\u271d a\u2082\u271d : X\u271d \u27f6 Y\u271d\nh :\n (CategoryTheory.Functor.mk { obj := fun R_1 => \u2191R_1, map := fun {X Y} f => f.toFun }).map a\u2081\u271d =\n (CategoryTheory.Functor.mk { obj := fun R_1 => \u2191R_1, map := fun {X Y} f => f.toFun }).map a\u2082\u271d\nx : \u2191X\u271d\n\u22a2 \u2191a\u2081\u271d x = \u2191a\u2082\u271d x\n[PROOFSTEP]\ndsimp at h \n[GOAL]\nR : Type u\ninst\u271d : Ring R\nX\u271d Y\u271d : ModuleCat R\na\u2081\u271d a\u2082\u271d : X\u271d \u27f6 Y\u271d\nh : \u2191a\u2081\u271d = \u2191a\u2082\u271d\nx : \u2191X\u271d\n\u22a2 \u2191a\u2081\u271d x = \u2191a\u2082\u271d x\n[PROOFSTEP]\nrw [h]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\nM : ModuleCat R\ninst\u271d : Subsingleton \u2191M\nX : ModuleCat R\nf : M \u27f6 X\n\u22a2 f = default\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Ring R\nM : ModuleCat R\ninst\u271d : Subsingleton \u2191M\nX : ModuleCat R\nf : M \u27f6 X\nx : \u2191M\n\u22a2 \u2191f x = \u2191default x\n[PROOFSTEP]\nrw [Subsingleton.elim x (0 : M)]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Ring R\nM : ModuleCat R\ninst\u271d : Subsingleton \u2191M\nX : ModuleCat R\nf : M \u27f6 X\nx : \u2191M\n\u22a2 \u2191f 0 = \u2191default 0\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Ring R\nM : ModuleCat R\ninst\u271d : Subsingleton \u2191M\nX : ModuleCat R\nf : M \u27f6 X\nx : \u2191M\n\u22a2 \u2191f 0 = \u21910 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\nM : ModuleCat R\ninst\u271d : Subsingleton \u2191M\nX : ModuleCat R\nf : X \u27f6 M\n\u22a2 f = default\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Ring R\nM : ModuleCat R\ninst\u271d : Subsingleton \u2191M\nX : ModuleCat R\nf : X \u27f6 M\nx : \u2191X\n\u22a2 \u2191f x = \u2191default x\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nX\u2081 X\u2082 : Type v\ng\u2081 : AddCommGroup X\u2081\ng\u2082 : AddCommGroup X\u2082\nm\u2081 : Module R X\u2081\nm\u2082 : Module R X\u2082\ne : X\u2081 \u2243\u2097[R] X\u2082\n\u22a2 \u2191e \u226b \u2191(symm e) = \ud835\udfd9 (ModuleCat.of R X\u2081)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\ninst\u271d : Ring R\nX\u2081 X\u2082 : Type v\ng\u2081 : AddCommGroup X\u2081\ng\u2082 : AddCommGroup X\u2082\nm\u2081 : Module R X\u2081\nm\u2082 : Module R X\u2082\ne : X\u2081 \u2243\u2097[R] X\u2082\nx\u271d : \u2191(ModuleCat.of R X\u2081)\n\u22a2 \u2191(\u2191e \u226b \u2191(symm e)) x\u271d = \u2191(\ud835\udfd9 (ModuleCat.of R X\u2081)) x\u271d\n[PROOFSTEP]\napply e.left_inv\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nX\u2081 X\u2082 : Type v\ng\u2081 : AddCommGroup X\u2081\ng\u2082 : AddCommGroup X\u2082\nm\u2081 : Module R X\u2081\nm\u2082 : Module R X\u2082\ne : X\u2081 \u2243\u2097[R] X\u2082\n\u22a2 \u2191(symm e) \u226b \u2191e = \ud835\udfd9 (ModuleCat.of R X\u2082)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\ninst\u271d : Ring R\nX\u2081 X\u2082 : Type v\ng\u2081 : AddCommGroup X\u2081\ng\u2082 : AddCommGroup X\u2082\nm\u2081 : Module R X\u2081\nm\u2082 : Module R X\u2082\ne : X\u2081 \u2243\u2097[R] X\u2082\nx\u271d : \u2191(ModuleCat.of R X\u2082)\n\u22a2 \u2191(\u2191(symm e) \u226b \u2191e) x\u271d = \u2191(\ud835\udfd9 (ModuleCat.of R X\u2082)) x\u271d\n[PROOFSTEP]\napply e.right_inv\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nX\u2081 X\u2082 : Type v\nX Y : ModuleCat R\ni : X \u2245 Y\n\u22a2 \u2200 (x y : \u2191X), \u2191i.hom (x + y) = \u2191i.hom x + \u2191i.hom y\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nX\u2081 X\u2082 : Type v\nX Y : ModuleCat R\ni : X \u2245 Y\n\u22a2 \u2200 (r : R) (x : \u2191X),\n AddHom.toFun { toFun := \u2191i.hom, map_add' := (_ : \u2200 (a a_1 : \u2191X), \u2191i.hom (a + a_1) = \u2191i.hom a + \u2191i.hom a_1) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun { toFun := \u2191i.hom, map_add' := (_ : \u2200 (a a_1 : \u2191X), \u2191i.hom (a + a_1) = \u2191i.hom a + \u2191i.hom a_1) } x\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nX\u2081 X\u2082 : Type v\nX Y : ModuleCat R\ni : X \u2245 Y\nx : \u2191X\n\u22a2 \u2191i.inv\n (AddHom.toFun\n { toAddHom := { toFun := \u2191i.hom, map_add' := (_ : \u2200 (a a_1 : \u2191X), \u2191i.hom (a + a_1) = \u2191i.hom a + \u2191i.hom a_1) },\n map_smul' := (_ : \u2200 (a : R) (a_1 : \u2191X), \u2191i.hom (a \u2022 a_1) = a \u2022 \u2191i.hom a_1) }.toAddHom\n x) =\n x\n[PROOFSTEP]\nchange (i.hom \u226b i.inv) x = x\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nX\u2081 X\u2082 : Type v\nX Y : ModuleCat R\ni : X \u2245 Y\nx : \u2191X\n\u22a2 \u2191(i.hom \u226b i.inv) x = x\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nX\u2081 X\u2082 : Type v\nX Y : ModuleCat R\ni : X \u2245 Y\nx : \u2191Y\n\u22a2 AddHom.toFun\n { toAddHom := { toFun := \u2191i.hom, map_add' := (_ : \u2200 (a a_1 : \u2191X), \u2191i.hom (a + a_1) = \u2191i.hom a + \u2191i.hom a_1) },\n map_smul' := (_ : \u2200 (a : R) (a_1 : \u2191X), \u2191i.hom (a \u2022 a_1) = a \u2022 \u2191i.hom a_1) }.toAddHom\n (\u2191i.inv x) =\n x\n[PROOFSTEP]\nchange (i.inv \u226b i.hom) x = x\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nX\u2081 X\u2082 : Type v\nX Y : ModuleCat R\ni : X \u2245 Y\nx : \u2191Y\n\u22a2 \u2191(i.inv \u226b i.hom) x = x\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u271d : Type u\ninst\u271d : Ring R\u271d\nX\u2081 X\u2082 : Type v\nP Q R : ModuleCat R\u271d\nf f' : P \u27f6 Q\ng : Q \u27f6 R\n\u22a2 (f + f') \u226b g = f \u226b g + f' \u226b g\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR\u271d : Type u\ninst\u271d : Ring R\u271d\nX\u2081 X\u2082 : Type v\nP Q R : ModuleCat R\u271d\nf f' : P \u27f6 Q\ng : Q \u27f6 R\nx\u271d : \u2191P\n\u22a2 \u2191((f + f') \u226b g) x\u271d = \u2191(f \u226b g + f' \u226b g) x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nR\u271d : Type u\ninst\u271d : Ring R\u271d\nX\u2081 X\u2082 : Type v\nP Q R : ModuleCat R\u271d\nf f' : P \u27f6 Q\ng : Q \u27f6 R\nx\u271d : \u2191P\n\u22a2 \u2191g (\u2191(f + f') x\u271d) = \u2191(f \u226b g + f' \u226b g) x\u271d\n[PROOFSTEP]\nerw [map_add]\n[GOAL]\ncase h\nR\u271d : Type u\ninst\u271d : Ring R\u271d\nX\u2081 X\u2082 : Type v\nP Q R : ModuleCat R\u271d\nf f' : P \u27f6 Q\ng : Q \u27f6 R\nx\u271d : \u2191P\n\u22a2 \u2191g (\u2191f x\u271d) + \u2191g (\u2191f' x\u271d) = \u2191(f \u226b g + f' \u226b g) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR\u271d : Type u\ninst\u271d : Ring R\u271d\nX\u2081 X\u2082 : Type v\nP Q R : ModuleCat R\u271d\nf : P \u27f6 Q\ng g' : Q \u27f6 R\n\u22a2 f \u226b (g + g') = f \u226b g + f \u226b g'\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR\u271d : Type u\ninst\u271d : Ring R\u271d\nX\u2081 X\u2082 : Type v\nP Q R : ModuleCat R\u271d\nf : P \u27f6 Q\ng g' : Q \u27f6 R\nx\u271d : \u2191P\n\u22a2 \u2191(f \u226b (g + g')) x\u271d = \u2191(f \u226b g + f \u226b g') x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\nX\u2081 X\u2082 : Type v\nS : Type u\ninst\u271d : CommRing S\n\u22a2 \u2200 (X Y Z : ModuleCat S) (r : S) (f : X \u27f6 Y) (g : Y \u27f6 Z), (r \u2022 f) \u226b g = r \u2022 f \u226b g\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\nX\u2081 X\u2082 : Type v\nS : Type u\ninst\u271d : CommRing S\nX\u271d Y\u271d Z\u271d : ModuleCat S\nr\u271d : S\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 (r\u271d \u2022 f\u271d) \u226b g\u271d = r\u271d \u2022 f\u271d \u226b g\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Ring R\nX\u2081 X\u2082 : Type v\nS : Type u\ninst\u271d : CommRing S\nX\u271d Y\u271d Z\u271d : ModuleCat S\nr\u271d : S\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\nx\u271d : \u2191X\u271d\n\u22a2 \u2191((r\u271d \u2022 f\u271d) \u226b g\u271d) x\u271d = \u2191(r\u271d \u2022 f\u271d \u226b g\u271d) x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Ring R\nX\u2081 X\u2082 : Type v\nS : Type u\ninst\u271d : CommRing S\nX\u271d Y\u271d Z\u271d : ModuleCat S\nr\u271d : S\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\nx\u271d : \u2191X\u271d\n\u22a2 \u2191g\u271d (\u2191(r\u271d \u2022 f\u271d) x\u271d) = \u2191(r\u271d \u2022 f\u271d \u226b g\u271d) x\u271d\n[PROOFSTEP]\nrw [LinearMap.smul_apply, LinearMap.smul_apply, map_smul]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Ring R\nX\u2081 X\u2082 : Type v\nS : Type u\ninst\u271d : CommRing S\nX\u271d Y\u271d Z\u271d : ModuleCat S\nr\u271d : S\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\nx\u271d : \u2191X\u271d\n\u22a2 r\u271d \u2022 \u2191g\u271d (\u2191f\u271d x\u271d) = r\u271d \u2022 \u2191(f\u271d \u226b g\u271d) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\nX\u2081 X\u2082 : Type v\nS : Type u\ninst\u271d : CommRing S\n\u22a2 \u2200 (X Y Z : ModuleCat S) (f : X \u27f6 Y) (r : S) (g : Y \u27f6 Z), f \u226b (r \u2022 g) = r \u2022 f \u226b g\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\nX\u2081 X\u2082 : Type v\nS : Type u\ninst\u271d : CommRing S\nX\u271d Y\u271d Z\u271d : ModuleCat S\nf\u271d : X\u271d \u27f6 Y\u271d\nr\u271d : S\ng\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 f\u271d \u226b (r\u271d \u2022 g\u271d) = r\u271d \u2022 f\u271d \u226b g\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : Ring R\nX\u2081 X\u2082 : Type v\nS : Type u\ninst\u271d : CommRing S\nX\u271d Y\u271d Z\u271d : ModuleCat S\nf\u271d : X\u271d \u27f6 Y\u271d\nr\u271d : S\ng\u271d : Y\u271d \u27f6 Z\u271d\nx\u271d : \u2191X\u271d\n\u22a2 \u2191(f\u271d \u226b (r\u271d \u2022 g\u271d)) x\u271d = \u2191(r\u271d \u2022 f\u271d \u226b g\u271d) x\u271d\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.ModuleCat.Basic", "llama_tokens": 4025, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5039061705290805, "lm_q2_score": 0.41869690935568665, "lm_q1q2_score": 0.21098395620578558}} {"text": "[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : SetLike S R\ninst\u271d : NonUnitalSubsemiringClass S R\ns : S\n\u22a2 \u2200 (x y : { x // x \u2208 s }), \u2191(x + y) = \u2191x + \u2191y\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u271d : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : NonUnitalNonAssocSemiring R\u271d\nM : Subsemigroup R\u271d\ninst\u271d\u2074 : SetLike S R\u271d\ninst\u271d\u00b3 : NonUnitalSubsemiringClass S R\u271d\ns : S\nR : Type ?u.22770\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : SetLike S R\ninst\u271d : NonUnitalSubsemiringClass S R\n\u22a2 \u2200 (x y : { x // x \u2208 s }), \u2191(x + y) = \u2191x + \u2191y\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u271d : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : NonUnitalNonAssocSemiring R\u271d\nM : Subsemigroup R\u271d\ninst\u271d\u2074 : SetLike S R\u271d\ninst\u271d\u00b3 : NonUnitalSubsemiringClass S R\u271d\ns : S\nR : Type ?u.28609\ninst\u271d\u00b2 : NonUnitalCommSemiring R\ninst\u271d\u00b9 : SetLike S R\ninst\u271d : NonUnitalSubsemiringClass S R\n\u22a2 \u2200 (x y : { x // x \u2208 s }), \u2191(x + y) = \u2191x + \u2191y\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\np q : NonUnitalSubsemiring R\nh : (fun s => s.carrier) p = (fun s => s.carrier) q\n\u22a2 p = q\n[PROOFSTEP]\ncases p\n[GOAL]\ncase mk\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\nq : NonUnitalSubsemiring R\ntoAddSubmonoid\u271d : AddSubmonoid R\nmul_mem'\u271d : \u2200 {a b : R}, a \u2208 toAddSubmonoid\u271d.carrier \u2192 b \u2208 toAddSubmonoid\u271d.carrier \u2192 a * b \u2208 toAddSubmonoid\u271d.carrier\nh : (fun s => s.carrier) { toAddSubmonoid := toAddSubmonoid\u271d, mul_mem' := mul_mem'\u271d } = (fun s => s.carrier) q\n\u22a2 { toAddSubmonoid := toAddSubmonoid\u271d, mul_mem' := mul_mem'\u271d } = q\n[PROOFSTEP]\ncases q\n[GOAL]\ncase mk.mk\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\ntoAddSubmonoid\u271d\u00b9 : AddSubmonoid R\nmul_mem'\u271d\u00b9 : \u2200 {a b : R}, a \u2208 toAddSubmonoid\u271d\u00b9.carrier \u2192 b \u2208 toAddSubmonoid\u271d\u00b9.carrier \u2192 a * b \u2208 toAddSubmonoid\u271d\u00b9.carrier\ntoAddSubmonoid\u271d : AddSubmonoid R\nmul_mem'\u271d : \u2200 {a b : R}, a \u2208 toAddSubmonoid\u271d.carrier \u2192 b \u2208 toAddSubmonoid\u271d.carrier \u2192 a * b \u2208 toAddSubmonoid\u271d.carrier\nh :\n (fun s => s.carrier) { toAddSubmonoid := toAddSubmonoid\u271d\u00b9, mul_mem' := mul_mem'\u271d\u00b9 } =\n (fun s => s.carrier) { toAddSubmonoid := toAddSubmonoid\u271d, mul_mem' := mul_mem'\u271d }\n\u22a2 { toAddSubmonoid := toAddSubmonoid\u271d\u00b9, mul_mem' := mul_mem'\u271d\u00b9 } =\n { toAddSubmonoid := toAddSubmonoid\u271d, mul_mem' := mul_mem'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_toAddSubmonoid\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\ntoAddSubmonoid\u271d\u00b9 : AddSubmonoid R\nmul_mem'\u271d\u00b9 : \u2200 {a b : R}, a \u2208 toAddSubmonoid\u271d\u00b9.carrier \u2192 b \u2208 toAddSubmonoid\u271d\u00b9.carrier \u2192 a * b \u2208 toAddSubmonoid\u271d\u00b9.carrier\ntoAddSubmonoid\u271d : AddSubmonoid R\nmul_mem'\u271d : \u2200 {a b : R}, a \u2208 toAddSubmonoid\u271d.carrier \u2192 b \u2208 toAddSubmonoid\u271d.carrier \u2192 a * b \u2208 toAddSubmonoid\u271d.carrier\nh :\n (fun s => s.carrier) { toAddSubmonoid := toAddSubmonoid\u271d\u00b9, mul_mem' := mul_mem'\u271d\u00b9 } =\n (fun s => s.carrier) { toAddSubmonoid := toAddSubmonoid\u271d, mul_mem' := mul_mem'\u271d }\n\u22a2 toAddSubmonoid\u271d\u00b9 = toAddSubmonoid\u271d\n[PROOFSTEP]\nexact SetLike.coe_injective' h\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\ns : Set R\nsg : Subsemigroup R\nhg : \u2191sg = s\nsa : AddSubmonoid R\nha : \u2191sa = s\n\u22a2 \u2200 {a b : R}, a \u2208 s \u2192 b \u2208 s \u2192 a + b \u2208 s\n[PROOFSTEP]\nsubst ha\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\nsg : Subsemigroup R\nsa : AddSubmonoid R\nhg : \u2191sg = \u2191sa\n\u22a2 \u2200 {a b : R}, a \u2208 \u2191sa \u2192 b \u2208 \u2191sa \u2192 a + b \u2208 \u2191sa\n[PROOFSTEP]\nexact sa.add_mem\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\ns : Set R\nsg : Subsemigroup R\nhg : \u2191sg = s\nsa : AddSubmonoid R\nha : \u2191sa = s\n\u22a2 0 \u2208 { carrier := s, add_mem' := (_ : \u2200 {a b : R}, a \u2208 s \u2192 b \u2208 s \u2192 a + b \u2208 s) }.carrier\n[PROOFSTEP]\nsubst ha\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\nsg : Subsemigroup R\nsa : AddSubmonoid R\nhg : \u2191sg = \u2191sa\n\u22a2 0 \u2208 { carrier := \u2191sa, add_mem' := (_ : \u2200 {a b : R}, a \u2208 \u2191sa \u2192 b \u2208 \u2191sa \u2192 a + b \u2208 \u2191sa) }.carrier\n[PROOFSTEP]\nexact sa.zero_mem\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\ns : Set R\nsg : Subsemigroup R\nhg : \u2191sg = s\nsa : AddSubmonoid R\nha : \u2191sa = s\n\u22a2 \u2200 {a b : R},\n a \u2208\n { toAddSubsemigroup := { carrier := s, add_mem' := (_ : \u2200 {a b : R}, a \u2208 s \u2192 b \u2208 s \u2192 a + b \u2208 s) },\n zero_mem' :=\n (_ :\n 0 \u2208\n { carrier := s,\n add_mem' :=\n (_ : \u2200 {a b : R}, a \u2208 s \u2192 b \u2208 s \u2192 a + b \u2208 s) }.carrier) }.toAddSubsemigroup.carrier \u2192\n b \u2208\n { toAddSubsemigroup := { carrier := s, add_mem' := (_ : \u2200 {a b : R}, a \u2208 s \u2192 b \u2208 s \u2192 a + b \u2208 s) },\n zero_mem' :=\n (_ :\n 0 \u2208\n { carrier := s,\n add_mem' :=\n (_ : \u2200 {a b : R}, a \u2208 s \u2192 b \u2208 s \u2192 a + b \u2208 s) }.carrier) }.toAddSubsemigroup.carrier \u2192\n a * b \u2208\n { toAddSubsemigroup := { carrier := s, add_mem' := (_ : \u2200 {a b : R}, a \u2208 s \u2192 b \u2208 s \u2192 a + b \u2208 s) },\n zero_mem' :=\n (_ :\n 0 \u2208\n { carrier := s,\n add_mem' :=\n (_ : \u2200 {a b : R}, a \u2208 s \u2192 b \u2208 s \u2192 a + b \u2208 s) }.carrier) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nsubst hg\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\nsg : Subsemigroup R\nsa : AddSubmonoid R\nha : \u2191sa = \u2191sg\n\u22a2 \u2200 {a b : R},\n a \u2208\n { toAddSubsemigroup := { carrier := \u2191sg, add_mem' := (_ : \u2200 {a b : R}, a \u2208 \u2191sg \u2192 b \u2208 \u2191sg \u2192 a + b \u2208 \u2191sg) },\n zero_mem' :=\n (_ :\n 0 \u2208\n { carrier := \u2191sg,\n add_mem' :=\n (_ : \u2200 {a b : R}, a \u2208 \u2191sg \u2192 b \u2208 \u2191sg \u2192 a + b \u2208 \u2191sg) }.carrier) }.toAddSubsemigroup.carrier \u2192\n b \u2208\n { toAddSubsemigroup := { carrier := \u2191sg, add_mem' := (_ : \u2200 {a b : R}, a \u2208 \u2191sg \u2192 b \u2208 \u2191sg \u2192 a + b \u2208 \u2191sg) },\n zero_mem' :=\n (_ :\n 0 \u2208\n { carrier := \u2191sg,\n add_mem' :=\n (_ : \u2200 {a b : R}, a \u2208 \u2191sg \u2192 b \u2208 \u2191sg \u2192 a + b \u2208 \u2191sg) }.carrier) }.toAddSubsemigroup.carrier \u2192\n a * b \u2208\n { toAddSubsemigroup := { carrier := \u2191sg, add_mem' := (_ : \u2200 {a b : R}, a \u2208 \u2191sg \u2192 b \u2208 \u2191sg \u2192 a + b \u2208 \u2191sg) },\n zero_mem' :=\n (_ :\n 0 \u2208\n { carrier := \u2191sg,\n add_mem' :=\n (_ : \u2200 {a b : R}, a \u2208 \u2191sg \u2192 b \u2208 \u2191sg \u2192 a + b \u2208 \u2191sg) }.carrier) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nexact sg.mul_mem\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2074 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring S\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring T\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b9 : NonUnitalRingHomClass F R S\ninst\u271d : NonUnitalRingHomClass G S T\ns\u271d : NonUnitalSubsemiring R\nf : F\ns : NonUnitalSubsemiring R\ny : S\n\u22a2 y \u2208 map f s \u2194 \u2203 x, x \u2208 s \u2227 \u2191f x = y\n[PROOFSTEP]\nconvert Set.mem_image_iff_bex\n[GOAL]\ncase h.e'_2.h.e'_2.h.a\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2074 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring S\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring T\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b9 : NonUnitalRingHomClass F R S\ninst\u271d : NonUnitalRingHomClass G S T\ns\u271d : NonUnitalSubsemiring R\nf : F\ns : NonUnitalSubsemiring R\ny : S\nx\u271d : R\n\u22a2 x\u271d \u2208 s \u2227 \u2191f x\u271d = y \u2194 \u2203 x, \u2191f x\u271d = y\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2074 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring S\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring T\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b9 : NonUnitalRingHomClass F R S\ninst\u271d : NonUnitalRingHomClass G S T\nf : F\ng : G\n\u22a2 srange f = map f \u22a4\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2074 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring S\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring T\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b9 : NonUnitalRingHomClass F R S\ninst\u271d : NonUnitalRingHomClass G S T\nf : F\ng : G\nx\u271d : S\n\u22a2 x\u271d \u2208 srange f \u2194 x\u271d \u2208 map f \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2074 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring S\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring T\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b9 : NonUnitalRingHomClass F R S\ninst\u271d : NonUnitalRingHomClass G S T\nf\u271d : F\ng\u271d : G\ng : S \u2192\u2099+* T\nf : R \u2192\u2099+* S\n\u22a2 map g (srange f) = srange (comp g f)\n[PROOFSTEP]\nsimpa only [srange_eq_map] using (\u22a4 : NonUnitalSubsemiring R).map_map g f\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\na\u271d b\u271d : R\nx\u271d\u00b9 : a\u271d \u2208 {0}\nx\u271d : b\u271d \u2208 {0}\n\u22a2 a\u271d + b\u271d \u2208 {0}\n[PROOFSTEP]\nsimp_all\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\na\u271d b\u271d : R\nx\u271d\u00b9 :\n a\u271d \u2208\n { toAddSubsemigroup := { carrier := {0}, add_mem' := (_ : \u2200 {a b : R}, a \u2208 {0} \u2192 b \u2208 {0} \u2192 a + b \u2208 {0}) },\n zero_mem' := (_ : 0 \u2208 {0}) }.toAddSubsemigroup.carrier\nx\u271d :\n b\u271d \u2208\n { toAddSubsemigroup := { carrier := {0}, add_mem' := (_ : \u2200 {a b : R}, a \u2208 {0} \u2192 b \u2208 {0} \u2192 a + b \u2208 {0}) },\n zero_mem' := (_ : 0 \u2208 {0}) }.toAddSubsemigroup.carrier\n\u22a2 a\u271d * b\u271d \u2208\n { toAddSubsemigroup := { carrier := {0}, add_mem' := (_ : \u2200 {a b : R}, a \u2208 {0} \u2192 b \u2208 {0} \u2192 a + b \u2208 {0}) },\n zero_mem' := (_ : 0 \u2208 {0}) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nsimp_all\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\ns : Set (NonUnitalSubsemiring R)\n\u22a2 \u2191(\u2a05 (t : NonUnitalSubsemiring R) (_ : t \u2208 s), toSubsemigroup t) = \u22c2 (t : NonUnitalSubsemiring R) (_ : t \u2208 s), \u2191t\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\ns : Set (NonUnitalSubsemiring R)\n\u22a2 \u2191(\u2a05 (t : NonUnitalSubsemiring R) (_ : t \u2208 s), t.toAddSubmonoid) = \u22c2 (t : NonUnitalSubsemiring R) (_ : t \u2208 s), \u2191t\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\nf : R \u2243+* S\nK : NonUnitalSubsemiring R\nx : S\n\u22a2 x \u2208 map (\u2191f) K \u2194 \u2191(RingEquiv.symm f) x \u2208 K\n[PROOFSTEP]\nconvert @Set.mem_image_equiv _ _ (\u2191K) f.toEquiv x\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\n\u22a2 nonUnitalSubsemiringClosure M = NonUnitalSubsemiring.closure \u2191M\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\nx\u271d : R\n\u22a2 x\u271d \u2208 nonUnitalSubsemiringClosure M \u2194 x\u271d \u2208 NonUnitalSubsemiring.closure \u2191M\n[PROOFSTEP]\nrefine \u27e8fun hx => ?_, fun hx => (NonUnitalSubsemiring.mem_closure.mp hx) M.nonUnitalSubsemiringClosure fun s sM => ?_\u27e9\n[GOAL]\ncase h.refine_1\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\nx\u271d : R\nhx : x\u271d \u2208 nonUnitalSubsemiringClosure M\n\u22a2 x\u271d \u2208 NonUnitalSubsemiring.closure \u2191M\n[PROOFSTEP]\nrintro - \u27e8H1, rfl\u27e9\n[GOAL]\ncase h.refine_2\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\nx\u271d : R\nhx : x\u271d \u2208 NonUnitalSubsemiring.closure \u2191M\ns : R\nsM : s \u2208 \u2191M\n\u22a2 s \u2208 \u2191(nonUnitalSubsemiringClosure M)\n[PROOFSTEP]\nrintro - \u27e8H1, rfl\u27e9\n[GOAL]\ncase h.refine_1.intro\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\nx\u271d : R\nhx : x\u271d \u2208 nonUnitalSubsemiringClosure M\nH1 : NonUnitalSubsemiring R\n\u22a2 x\u271d \u2208 (fun t => \u22c2 (_ : t \u2208 {S | \u2191M \u2286 \u2191S}), \u2191t) H1\n[PROOFSTEP]\nrintro - \u27e8H2, rfl\u27e9\n[GOAL]\ncase h.refine_2.intro\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\nx\u271d : R\nhx : x\u271d \u2208 NonUnitalSubsemiring.closure \u2191M\ns : R\nsM : s \u2208 \u2191M\nH1 : AddSubmonoid R\n\u22a2 s \u2208 (fun t => \u22c2 (_ : t \u2208 {S | \u2191M \u2286 \u2191S}), \u2191t) H1\n[PROOFSTEP]\nrintro - \u27e8H2, rfl\u27e9\n[GOAL]\ncase h.refine_1.intro.intro\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\nx\u271d : R\nhx : x\u271d \u2208 nonUnitalSubsemiringClosure M\nH1 : NonUnitalSubsemiring R\nH2 : H1 \u2208 {S | \u2191M \u2286 \u2191S}\n\u22a2 x\u271d \u2208 (fun h => \u2191H1) H2\n[PROOFSTEP]\nexact AddSubmonoid.mem_closure.mp hx H1.toAddSubmonoid H2\n[GOAL]\ncase h.refine_2.intro.intro\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\nx\u271d : R\nhx : x\u271d \u2208 NonUnitalSubsemiring.closure \u2191M\ns : R\nsM : s \u2208 \u2191M\nH1 : AddSubmonoid R\nH2 : H1 \u2208 {S | \u2191M \u2286 \u2191S}\n\u22a2 s \u2208 (fun h => \u2191H1) H2\n[PROOFSTEP]\nexact H2 sM\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\ns : Set R\n\u22a2 \u2191(closure s) = \u2191(AddSubmonoid.closure \u2191(Subsemigroup.closure s))\n[PROOFSTEP]\nsimp [\u2190 Subsemigroup.nonUnitalSubsemiringClosure_toAddSubmonoid, Subsemigroup.nonUnitalSubsemiringClosure_eq_closure]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\ns : Set R\n\u22a2 closure \u2191(AddSubmonoid.closure s) = closure s\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\ns : Set R\nx : R\n\u22a2 x \u2208 closure \u2191(AddSubmonoid.closure s) \u2194 x \u2208 closure s\n[PROOFSTEP]\nrefine' \u27e8fun hx => _, fun hx => closure_mono AddSubmonoid.subset_closure hx\u27e9\n[GOAL]\ncase h\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\ns : Set R\nx : R\nhx : x \u2208 closure \u2191(AddSubmonoid.closure s)\n\u22a2 x \u2208 closure s\n[PROOFSTEP]\nrintro - \u27e8H, rfl\u27e9\n[GOAL]\ncase h.intro\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\ns : Set R\nx : R\nhx : x \u2208 closure \u2191(AddSubmonoid.closure s)\nH : NonUnitalSubsemiring R\n\u22a2 x \u2208 (fun t => \u22c2 (_ : t \u2208 {S | s \u2286 \u2191S}), \u2191t) H\n[PROOFSTEP]\nrintro - \u27e8J, rfl\u27e9\n[GOAL]\ncase h.intro.intro\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\ns : Set R\nx : R\nhx : x \u2208 closure \u2191(AddSubmonoid.closure s)\nH : NonUnitalSubsemiring R\nJ : H \u2208 {S | s \u2286 \u2191S}\n\u22a2 x \u2208 (fun h => \u2191H) J\n[PROOFSTEP]\nrefine' (AddSubmonoid.mem_closure.mp (mem_closure_iff.mp hx)) H.toAddSubmonoid fun y hy => _\n[GOAL]\ncase h.intro.intro\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\ns : Set R\nx : R\nhx : x \u2208 closure \u2191(AddSubmonoid.closure s)\nH : NonUnitalSubsemiring R\nJ : H \u2208 {S | s \u2286 \u2191S}\ny : R\nhy : y \u2208 \u2191(Subsemigroup.closure \u2191(AddSubmonoid.closure s))\n\u22a2 y \u2208 \u2191H.toAddSubmonoid\n[PROOFSTEP]\nrefine' (Subsemigroup.mem_closure.mp hy) H.toSubsemigroup fun z hz => _\n[GOAL]\ncase h.intro.intro\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\ns : Set R\nx : R\nhx : x \u2208 closure \u2191(AddSubmonoid.closure s)\nH : NonUnitalSubsemiring R\nJ : H \u2208 {S | s \u2286 \u2191S}\ny : R\nhy : y \u2208 \u2191(Subsemigroup.closure \u2191(AddSubmonoid.closure s))\nz : R\nhz : z \u2208 \u2191(AddSubmonoid.closure s)\n\u22a2 z \u2208 \u2191(toSubsemigroup H)\n[PROOFSTEP]\nexact (AddSubmonoid.mem_closure.mp hz) H.toAddSubmonoid fun w hw => J hw\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring T\nF : Type u_1\ninst\u271d : NonUnitalRingHomClass F R S\ns : NonUnitalSubsemiring R\nx : R \u00d7 S\n\u22a2 x \u2208 prod s \u22a4 \u2194 x \u2208 comap (NonUnitalRingHom.fst R S) s\n[PROOFSTEP]\nsimp [mem_prod, MonoidHom.coe_fst]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring T\nF : Type u_1\ninst\u271d : NonUnitalRingHomClass F R S\ns : NonUnitalSubsemiring S\nx : R \u00d7 S\n\u22a2 x \u2208 prod \u22a4 s \u2194 x \u2208 comap (NonUnitalRingHom.snd R S) s\n[PROOFSTEP]\nsimp [mem_prod, MonoidHom.coe_snd]\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring T\nF : Type u_1\ninst\u271d : NonUnitalRingHomClass F R S\u271d\n\u03b9 : Sort u_2\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 NonUnitalSubsemiring R\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : R\n\u22a2 x \u2208 \u2a06 (i : \u03b9), S i \u2194 \u2203 i, x \u2208 S i\n[PROOFSTEP]\nrefine' \u27e8_, fun \u27e8i, hi\u27e9 => (SetLike.le_def.1 <| le_iSup S i) hi\u27e9\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring T\nF : Type u_1\ninst\u271d : NonUnitalRingHomClass F R S\u271d\n\u03b9 : Sort u_2\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 NonUnitalSubsemiring R\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : R\n\u22a2 x \u2208 \u2a06 (i : \u03b9), S i \u2192 \u2203 i, x \u2208 S i\n[PROOFSTEP]\nlet U : NonUnitalSubsemiring R :=\n NonUnitalSubsemiring.mk' (\u22c3 i, (S i : Set R)) (\u2a06 i, (S i).toSubsemigroup)\n (Subsemigroup.coe_iSup_of_directed <| hS.mono_comp _ fun _ _ => id) (\u2a06 i, (S i).toAddSubmonoid)\n (AddSubmonoid.coe_iSup_of_directed <| hS.mono_comp _ fun _ _ => id)\n -- Porting note `@this` doesn't work\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring T\nF : Type u_1\ninst\u271d : NonUnitalRingHomClass F R S\u271d\n\u03b9 : Sort u_2\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 NonUnitalSubsemiring R\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : R\nU : NonUnitalSubsemiring R :=\n NonUnitalSubsemiring.mk' (\u22c3 (i : \u03b9), \u2191(S i)) (\u2a06 (i : \u03b9), toSubsemigroup (S i))\n (_ : \u2191(\u2a06 (i : \u03b9), toSubsemigroup (S i)) = \u22c3 (i : \u03b9), \u2191(toSubsemigroup (S i))) (\u2a06 (i : \u03b9), (S i).toAddSubmonoid)\n (_ : \u2191(\u2a06 (i : \u03b9), (S i).toAddSubmonoid) = \u22c3 (i : \u03b9), \u2191(S i).toAddSubmonoid)\n\u22a2 x \u2208 \u2a06 (i : \u03b9), S i \u2192 \u2203 i, x \u2208 S i\n[PROOFSTEP]\nsuffices H : \u2a06 i, S i \u2264 U\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring T\nF : Type u_1\ninst\u271d : NonUnitalRingHomClass F R S\u271d\n\u03b9 : Sort u_2\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 NonUnitalSubsemiring R\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : R\nU : NonUnitalSubsemiring R :=\n NonUnitalSubsemiring.mk' (\u22c3 (i : \u03b9), \u2191(S i)) (\u2a06 (i : \u03b9), toSubsemigroup (S i))\n (_ : \u2191(\u2a06 (i : \u03b9), toSubsemigroup (S i)) = \u22c3 (i : \u03b9), \u2191(toSubsemigroup (S i))) (\u2a06 (i : \u03b9), (S i).toAddSubmonoid)\n (_ : \u2191(\u2a06 (i : \u03b9), (S i).toAddSubmonoid) = \u22c3 (i : \u03b9), \u2191(S i).toAddSubmonoid)\nH : \u2a06 (i : \u03b9), S i \u2264 U\n\u22a2 x \u2208 \u2a06 (i : \u03b9), S i \u2192 \u2203 i, x \u2208 S i\ncase H\nR : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring T\nF : Type u_1\ninst\u271d : NonUnitalRingHomClass F R S\u271d\n\u03b9 : Sort u_2\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 NonUnitalSubsemiring R\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : R\nU : NonUnitalSubsemiring R :=\n NonUnitalSubsemiring.mk' (\u22c3 (i : \u03b9), \u2191(S i)) (\u2a06 (i : \u03b9), toSubsemigroup (S i))\n (_ : \u2191(\u2a06 (i : \u03b9), toSubsemigroup (S i)) = \u22c3 (i : \u03b9), \u2191(toSubsemigroup (S i))) (\u2a06 (i : \u03b9), (S i).toAddSubmonoid)\n (_ : \u2191(\u2a06 (i : \u03b9), (S i).toAddSubmonoid) = \u22c3 (i : \u03b9), \u2191(S i).toAddSubmonoid)\n\u22a2 \u2a06 (i : \u03b9), S i \u2264 U\n[PROOFSTEP]\nsimpa using @H x\n[GOAL]\ncase H\nR : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring T\nF : Type u_1\ninst\u271d : NonUnitalRingHomClass F R S\u271d\n\u03b9 : Sort u_2\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 NonUnitalSubsemiring R\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : R\nU : NonUnitalSubsemiring R :=\n NonUnitalSubsemiring.mk' (\u22c3 (i : \u03b9), \u2191(S i)) (\u2a06 (i : \u03b9), toSubsemigroup (S i))\n (_ : \u2191(\u2a06 (i : \u03b9), toSubsemigroup (S i)) = \u22c3 (i : \u03b9), \u2191(toSubsemigroup (S i))) (\u2a06 (i : \u03b9), (S i).toAddSubmonoid)\n (_ : \u2191(\u2a06 (i : \u03b9), (S i).toAddSubmonoid) = \u22c3 (i : \u03b9), \u2191(S i).toAddSubmonoid)\n\u22a2 \u2a06 (i : \u03b9), S i \u2264 U\n[PROOFSTEP]\nexact iSup_le fun i x hx => Set.mem_iUnion.2 \u27e8i, hx\u27e9\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring T\nF : Type u_1\ninst\u271d : NonUnitalRingHomClass F R S\u271d\n\u03b9 : Sort u_2\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 NonUnitalSubsemiring R\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : R\n\u22a2 x \u2208 \u2191(\u2a06 (i : \u03b9), S i) \u2194 x \u2208 \u22c3 (i : \u03b9), \u2191(S i)\n[PROOFSTEP]\nsimp [mem_iSup_of_directed hS]\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring T\nF : Type u_1\ninst\u271d : NonUnitalRingHomClass F R S\u271d\nS : Set (NonUnitalSubsemiring R)\nSne : Set.Nonempty S\nhS : DirectedOn (fun x x_1 => x \u2264 x_1) S\nx : R\n\u22a2 x \u2208 sSup S \u2194 \u2203 s, s \u2208 S \u2227 x \u2208 s\n[PROOFSTEP]\nhaveI : Nonempty S := Sne.to_subtype\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring T\nF : Type u_1\ninst\u271d : NonUnitalRingHomClass F R S\u271d\nS : Set (NonUnitalSubsemiring R)\nSne : Set.Nonempty S\nhS : DirectedOn (fun x x_1 => x \u2264 x_1) S\nx : R\nthis : Nonempty \u2191S\n\u22a2 x \u2208 sSup S \u2194 \u2203 s, s \u2208 S \u2227 x \u2208 s\n[PROOFSTEP]\nsimp only [sSup_eq_iSup', mem_iSup_of_directed hS.directed_val, Subtype.exists, exists_prop]\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring T\nF : Type u_1\ninst\u271d : NonUnitalRingHomClass F R S\u271d\nS : Set (NonUnitalSubsemiring R)\nSne : Set.Nonempty S\nhS : DirectedOn (fun x x_1 => x \u2264 x_1) S\nx : R\n\u22a2 x \u2208 \u2191(sSup S) \u2194 x \u2208 \u22c3 (s : NonUnitalSubsemiring R) (_ : s \u2208 S), \u2191s\n[PROOFSTEP]\nsimp [mem_sSup_of_directedOn Sne hS]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2076 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring S\ninst\u271d\u2074 : NonUnitalNonAssocSemiring T\nF : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring T\ninst\u271d\u00b2 : NonUnitalRingHomClass F R S\nS' : Type u_2\ninst\u271d\u00b9 : SetLike S' S\ninst\u271d : NonUnitalSubsemiringClass S' S\ns : NonUnitalSubsemiring R\nf : F\n\u22a2 \u2191(srange f) = \u2191\u22a4 \u2194 Set.range \u2191f = Set.univ\n[PROOFSTEP]\nrw [coe_srange, coe_top]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring T\ns t : NonUnitalSubsemiring R\nF : Type u_1\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192 R\nf : F\nh : Function.LeftInverse g \u2191f\nsrc\u271d : R \u2192\u2099+* { x // x \u2208 srange f } := srangeRestrict f\nx : { x // x \u2208 srange f }\nx' : R\nhx' : \u2191f x' = \u2191x\n\u22a2 \u2191f (g \u2191x) = \u2191x\n[PROOFSTEP]\nrw [\u2190 hx', h x']\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "llama_tokens": 11939, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5736783928749127, "lm_q2_score": 0.3665897363221599, "lm_q1q2_score": 0.2103046107777347}} {"text": "[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\ni : \u03b9\n\u22a2 aeSeqSet hf p \u2286 {x | \u2200 (i : \u03b9), f i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x}\n[PROOFSTEP]\nrw [aeSeqSet, \u2190 compl_compl {x | \u2200 i, f i x = (hf i).mk (f i) x}, Set.compl_subset_compl]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\ni : \u03b9\n\u22a2 {x | \u2200 (i : \u03b9), f i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x}\u1d9c \u2286\n toMeasurable \u03bc {x | (\u2200 (i : \u03b9), f i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x) \u2227 p x fun n => f n x}\u1d9c\n[PROOFSTEP]\nrefine' Set.Subset.trans (Set.compl_subset_compl.mpr fun x h => _) (subset_toMeasurable _ _)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx\u271d : \u03b1\nhx : x\u271d \u2208 aeSeqSet hf p\ni : \u03b9\nx : \u03b1\nh : x \u2208 {x | (\u2200 (i : \u03b9), f i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x) \u2227 p x fun n => f n x}\n\u22a2 x \u2208 {x | \u2200 (i : \u03b9), f i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x}\n[PROOFSTEP]\nexact h.1\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\ni : \u03b9\n\u22a2 aeSeq hf p i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x\n[PROOFSTEP]\nsimp only [aeSeq, hx, if_true]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\ni : \u03b9\n\u22a2 aeSeq hf p i x = f i x\n[PROOFSTEP]\nsimp only [aeSeq_eq_mk_of_mem_aeSeqSet hf hx i, mk_eq_fun_of_mem_aeSeqSet hf hx i]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 p x fun n => aeSeq hf p n x\n[PROOFSTEP]\nsimp only [aeSeq, hx, if_true]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 p x fun n => AEMeasurable.mk (f n) (_ : AEMeasurable (f n)) x\n[PROOFSTEP]\nrw [funext fun n => mk_eq_fun_of_mem_aeSeqSet hf hx n]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 p x fun n => f n x\n[PROOFSTEP]\nhave h_ss : aeSeqSet hf p \u2286 {x | p x fun n => f n x} :=\n by\n rw [\u2190 compl_compl {x | p x fun n => f n x}, aeSeqSet, Set.compl_subset_compl]\n refine' Set.Subset.trans (Set.compl_subset_compl.mpr _) (subset_toMeasurable _ _)\n exact fun x hx => hx.2\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 aeSeqSet hf p \u2286 {x | p x fun n => f n x}\n[PROOFSTEP]\nrw [\u2190 compl_compl {x | p x fun n => f n x}, aeSeqSet, Set.compl_subset_compl]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 {x | p x fun n => f n x}\u1d9c \u2286\n toMeasurable \u03bc {x | (\u2200 (i : \u03b9), f i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x) \u2227 p x fun n => f n x}\u1d9c\n[PROOFSTEP]\nrefine' Set.Subset.trans (Set.compl_subset_compl.mpr _) (subset_toMeasurable _ _)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 {x | (\u2200 (i : \u03b9), f i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x) \u2227 p x fun n => f n x} \u2286\n {x | p x fun n => f n x}\n[PROOFSTEP]\nexact fun x hx => hx.2\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\nh_ss : aeSeqSet hf p \u2286 {x | p x fun n => f n x}\n\u22a2 p x fun n => f n x\n[PROOFSTEP]\nhave hx' := Set.mem_of_subset_of_mem h_ss hx\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\nh_ss : aeSeqSet hf p \u2286 {x | p x fun n => f n x}\nhx' : x \u2208 {x | p x fun n => f n x}\n\u22a2 p x fun n => f n x\n[PROOFSTEP]\nexact hx'\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 p x fun n => f n x\n[PROOFSTEP]\nhave h_eq : (fun n => f n x) = fun n => aeSeq hf p n x := funext fun n => (aeSeq_eq_fun_of_mem_aeSeqSet hf hx n).symm\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\nh_eq : (fun n => f n x) = fun n => aeSeq hf p n x\n\u22a2 p x fun n => f n x\n[PROOFSTEP]\nrw [h_eq]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\nh_eq : (fun n => f n x) = fun n => aeSeq hf p n x\n\u22a2 p x fun n => aeSeq hf p n x\n[PROOFSTEP]\nexact prop_of_mem_aeSeqSet hf hx\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\ninst\u271d : Countable \u03b9\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f n x\n\u22a2 \u2191\u2191\u03bc (aeSeqSet hf p)\u1d9c = 0\n[PROOFSTEP]\nrw [aeSeqSet, compl_compl, measure_toMeasurable]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\ninst\u271d : Countable \u03b9\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f n x\n\u22a2 \u2191\u2191\u03bc {x | (\u2200 (i : \u03b9), f i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x) \u2227 p x fun n => f n x}\u1d9c = 0\n[PROOFSTEP]\nhave hf_eq := fun i => (hf i).ae_eq_mk\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\ninst\u271d : Countable \u03b9\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f n x\nhf_eq : \u2200 (i : \u03b9), f i =\u1d50[\u03bc] AEMeasurable.mk (f i) (_ : AEMeasurable (f i))\n\u22a2 \u2191\u2191\u03bc {x | (\u2200 (i : \u03b9), f i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x) \u2227 p x fun n => f n x}\u1d9c = 0\n[PROOFSTEP]\nsimp_rw [Filter.EventuallyEq, \u2190 ae_all_iff] at hf_eq \n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\ninst\u271d : Countable \u03b9\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f n x\nhf_eq : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u03b9), f i a = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) a\n\u22a2 \u2191\u2191\u03bc {x | (\u2200 (i : \u03b9), f i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x) \u2227 p x fun n => f n x}\u1d9c = 0\n[PROOFSTEP]\nexact Filter.Eventually.and hf_eq hp\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\ninst\u271d : Countable \u03b9\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f n x\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\ni : \u03b9\n\u22a2 aeSeq hf p i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x\n[PROOFSTEP]\nsimp only [aeSeq, hx, if_true]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\ninst\u271d\u00b9 : CompleteLattice \u03b2\ninst\u271d : Countable \u03b9\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f n x\n\u22a2 \u2a06 (n : \u03b9), aeSeq hf p n =\u1d50[\u03bc] \u2a06 (n : \u03b9), f n\n[PROOFSTEP]\nsimp_rw [Filter.EventuallyEq, ae_iff, iSup_apply]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\ninst\u271d\u00b9 : CompleteLattice \u03b2\ninst\u271d : Countable \u03b9\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f n x\n\u22a2 \u2191\u2191\u03bc {a | \u00ac\u2a06 (i : \u03b9), aeSeq hf p i a = \u2a06 (i : \u03b9), f i a} = 0\n[PROOFSTEP]\nhave h_ss : aeSeqSet hf p \u2286 {a : \u03b1 | \u2a06 i : \u03b9, aeSeq hf p i a = \u2a06 i : \u03b9, f i a} :=\n by\n intro x hx\n congr\n exact funext fun i => aeSeq_eq_fun_of_mem_aeSeqSet hf hx i\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\ninst\u271d\u00b9 : CompleteLattice \u03b2\ninst\u271d : Countable \u03b9\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f n x\n\u22a2 aeSeqSet hf p \u2286 {a | \u2a06 (i : \u03b9), aeSeq hf p i a = \u2a06 (i : \u03b9), f i a}\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\ninst\u271d\u00b9 : CompleteLattice \u03b2\ninst\u271d : Countable \u03b9\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f n x\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 x \u2208 {a | \u2a06 (i : \u03b9), aeSeq hf p i a = \u2a06 (i : \u03b9), f i a}\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\ninst\u271d\u00b9 : CompleteLattice \u03b2\ninst\u271d : Countable \u03b9\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f n x\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 (fun i => aeSeq hf p i x) = fun i => f i x\n[PROOFSTEP]\nexact funext fun i => aeSeq_eq_fun_of_mem_aeSeqSet hf hx i\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\ninst\u271d\u00b9 : CompleteLattice \u03b2\ninst\u271d : Countable \u03b9\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f n x\nh_ss : aeSeqSet hf p \u2286 {a | \u2a06 (i : \u03b9), aeSeq hf p i a = \u2a06 (i : \u03b9), f i a}\n\u22a2 \u2191\u2191\u03bc {a | \u00ac\u2a06 (i : \u03b9), aeSeq hf p i a = \u2a06 (i : \u03b9), f i a} = 0\n[PROOFSTEP]\nexact measure_mono_null (Set.compl_subset_compl.mpr h_ss) (measure_compl_aeSeqSet_eq_zero hf hp)\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "llama_tokens": 6047, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.38121956625614994, "lm_q1q2_score": 0.20842728115526346}} {"text": "[GOAL]\n\u22a2 ConcreteCategory HeytAlgCat\n[PROOFSTEP]\ndsimp [HeytAlgCat]\n[GOAL]\n\u22a2 ConcreteCategory (Bundled HeytingAlgebra)\n[PROOFSTEP]\ninfer_instance\n -- Porting note: No idea why it does not find this instance...\n[GOAL]\n\u03b1 \u03b2 : HeytAlgCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\n\u22a2 {\n toLatticeHom :=\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) },\n map_bot' := (_ : \u2191e \u22a5 = \u22a5), map_himp' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u21e8 b) = \u2191e a \u21e8 \u2191e b) } \u226b\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) },\n map_bot' := (_ : \u2191(OrderIso.symm e) \u22a5 = \u22a5),\n map_himp' := (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u21e8 b) = \u2191(OrderIso.symm e) a \u21e8 \u2191(OrderIso.symm e) b) } =\n \ud835\udfd9 \u03b1\n[PROOFSTEP]\next\n[GOAL]\ncase w\n\u03b1 \u03b2 : HeytAlgCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\nx\u271d : (forget HeytAlgCat).obj \u03b1\n\u22a2 \u2191({\n toLatticeHom :=\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) },\n map_bot' := (_ : \u2191e \u22a5 = \u22a5), map_himp' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u21e8 b) = \u2191e a \u21e8 \u2191e b) } \u226b\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) },\n map_bot' := (_ : \u2191(OrderIso.symm e) \u22a5 = \u22a5),\n map_himp' := (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u21e8 b) = \u2191(OrderIso.symm e) a \u21e8 \u2191(OrderIso.symm e) b) })\n x\u271d =\n \u2191(\ud835\udfd9 \u03b1) x\u271d\n[PROOFSTEP]\nexact e.symm_apply_apply _\n[GOAL]\n\u03b1 \u03b2 : HeytAlgCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\n\u22a2 {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) },\n map_bot' := (_ : \u2191(OrderIso.symm e) \u22a5 = \u22a5),\n map_himp' := (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u21e8 b) = \u2191(OrderIso.symm e) a \u21e8 \u2191(OrderIso.symm e) b) } \u226b\n {\n toLatticeHom :=\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) },\n map_bot' := (_ : \u2191e \u22a5 = \u22a5), map_himp' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u21e8 b) = \u2191e a \u21e8 \u2191e b) } =\n \ud835\udfd9 \u03b2\n[PROOFSTEP]\next\n[GOAL]\ncase w\n\u03b1 \u03b2 : HeytAlgCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\nx\u271d : (forget HeytAlgCat).obj \u03b2\n\u22a2 \u2191({\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) },\n map_bot' := (_ : \u2191(OrderIso.symm e) \u22a5 = \u22a5),\n map_himp' :=\n (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u21e8 b) = \u2191(OrderIso.symm e) a \u21e8 \u2191(OrderIso.symm e) b) } \u226b\n {\n toLatticeHom :=\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) },\n map_bot' := (_ : \u2191e \u22a5 = \u22a5), map_himp' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u21e8 b) = \u2191e a \u21e8 \u2191e b) })\n x\u271d =\n \u2191(\ud835\udfd9 \u03b2) x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n", "meta": {"mathlib_filename": "Mathlib.Order.Category.HeytAlgCat", "llama_tokens": 2162, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.36296921930155557, "lm_q1q2_score": 0.20683897200071807}} {"text": "[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nU V : C\u1d52\u1d56\nx\u271d\u00b9 : U \u27f6 V\nx : F.obj U\nx\u271d : x \u2208 (fun U => \u22a4) U\n\u22a2 x \u2208 F.map x\u271d\u00b9 \u207b\u00b9' (fun U => \u22a4) V\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nX : C\u1d52\u1d56\n\u22a2 { obj := fun U => \u2191(obj G U),\n map := fun U V i x => { val := F.map i \u2191x, property := (_ : \u2191x \u2208 F.map i \u207b\u00b9' obj G V) } }.map\n (\ud835\udfd9 X) =\n \ud835\udfd9\n ({ obj := fun U => \u2191(obj G U),\n map := fun U V i x => { val := F.map i \u2191x, property := (_ : \u2191x \u2208 F.map i \u207b\u00b9' obj G V) } }.obj\n X)\n[PROOFSTEP]\next \u27e8x, _\u27e9\n[GOAL]\ncase h.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nX : C\u1d52\u1d56\nx : F.obj X\nproperty\u271d : x \u2208 obj G X\n\u22a2 { obj := fun U => \u2191(obj G U),\n map := fun U V i x => { val := F.map i \u2191x, property := (_ : \u2191x \u2208 F.map i \u207b\u00b9' obj G V) } }.map\n (\ud835\udfd9 X) { val := x, property := property\u271d } =\n \ud835\udfd9\n ({ obj := fun U => \u2191(obj G U),\n map := fun U V i x => { val := F.map i \u2191x, property := (_ : \u2191x \u2208 F.map i \u207b\u00b9' obj G V) } }.obj\n X)\n { val := x, property := property\u271d }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nX : C\u1d52\u1d56\nx : F.obj X\nproperty\u271d : x \u2208 obj G X\n\u22a2 { val := F.map (\ud835\udfd9 X) x, property := (_ : x \u2208 F.map (\ud835\udfd9 X) \u207b\u00b9' obj G X) } = { val := x, property := property\u271d }\n[PROOFSTEP]\nsimp only [FunctorToTypes.map_id_apply]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nX Y Z : C\u1d52\u1d56\ni : X \u27f6 Y\nj : Y \u27f6 Z\n\u22a2 { obj := fun U => \u2191(obj G U),\n map := fun U V i x => { val := F.map i \u2191x, property := (_ : \u2191x \u2208 F.map i \u207b\u00b9' obj G V) } }.map\n (i \u226b j) =\n { obj := fun U => \u2191(obj G U),\n map := fun U V i x => { val := F.map i \u2191x, property := (_ : \u2191x \u2208 F.map i \u207b\u00b9' obj G V) } }.map\n i \u226b\n { obj := fun U => \u2191(obj G U),\n map := fun U V i x => { val := F.map i \u2191x, property := (_ : \u2191x \u2208 F.map i \u207b\u00b9' obj G V) } }.map\n j\n[PROOFSTEP]\next \u27e8x, _\u27e9\n[GOAL]\ncase h.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nX Y Z : C\u1d52\u1d56\ni : X \u27f6 Y\nj : Y \u27f6 Z\nx : F.obj X\nproperty\u271d : x \u2208 obj G X\n\u22a2 { obj := fun U => \u2191(obj G U),\n map := fun U V i x => { val := F.map i \u2191x, property := (_ : \u2191x \u2208 F.map i \u207b\u00b9' obj G V) } }.map\n (i \u226b j) { val := x, property := property\u271d } =\n ({ obj := fun U => \u2191(obj G U),\n map := fun U V i x => { val := F.map i \u2191x, property := (_ : \u2191x \u2208 F.map i \u207b\u00b9' obj G V) } }.map\n i \u226b\n { obj := fun U => \u2191(obj G U),\n map := fun U V i x => { val := F.map i \u2191x, property := (_ : \u2191x \u2208 F.map i \u207b\u00b9' obj G V) } }.map\n j)\n { val := x, property := property\u271d }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nX Y Z : C\u1d52\u1d56\ni : X \u27f6 Y\nj : Y \u27f6 Z\nx : F.obj X\nproperty\u271d : x \u2208 obj G X\n\u22a2 { val := F.map (i \u226b j) x, property := (_ : x \u2208 F.map (i \u226b j) \u207b\u00b9' obj G Z) } =\n { val := F.map j (F.map i x), property := (_ : F.map i x \u2208 F.map j \u207b\u00b9' obj G Z) }\n[PROOFSTEP]\nsimp only [FunctorToTypes.map_comp_apply]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG\u271d G'\u271d G G' : Subpresheaf F\nh : G \u2264 G'\n\u22a2 homOfLe h \u226b \u03b9 G' = \u03b9 G\n[PROOFSTEP]\next\n[GOAL]\ncase w.h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG\u271d G'\u271d G G' : Subpresheaf F\nh : G \u2264 G'\nx\u271d : C\u1d52\u1d56\na\u271d : (toPresheaf G).obj x\u271d\n\u22a2 NatTrans.app (homOfLe h \u226b \u03b9 G') x\u271d a\u271d = NatTrans.app (\u03b9 G) x\u271d a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\n\u22a2 IsIso (Subpresheaf.\u03b9 \u22a4)\n[PROOFSTEP]\nrefine @NatIso.isIso_of_isIso_app _ _ _ _ _ _ _ ?_\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\n\u22a2 \u2200 (X : C\u1d52\u1d56), IsIso (NatTrans.app (Subpresheaf.\u03b9 \u22a4) X)\n[PROOFSTEP]\nintro X\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nX : C\u1d52\u1d56\n\u22a2 IsIso (NatTrans.app (Subpresheaf.\u03b9 \u22a4) X)\n[PROOFSTEP]\nrw [isIso_iff_bijective]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nX : C\u1d52\u1d56\n\u22a2 Function.Bijective (NatTrans.app (Subpresheaf.\u03b9 \u22a4) X)\n[PROOFSTEP]\nexact \u27e8Subtype.coe_injective, fun x => \u27e8\u27e8x, _root_.trivial\u27e9, rfl\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\n\u22a2 G = \u22a4 \u2194 IsIso (\u03b9 G)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\n\u22a2 G = \u22a4 \u2192 IsIso (\u03b9 G)\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mp\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG' : Subpresheaf F\n\u22a2 IsIso (\u03b9 \u22a4)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\n\u22a2 IsIso (\u03b9 G) \u2192 G = \u22a4\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nH : IsIso (\u03b9 G)\n\u22a2 G = \u22a4\n[PROOFSTEP]\next U x\n[GOAL]\ncase mpr.obj.h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nH : IsIso (\u03b9 G)\nU : C\u1d52\u1d56\nx : F.obj U\n\u22a2 x \u2208 obj G U \u2194 x \u2208 obj \u22a4 U\n[PROOFSTEP]\napply iff_true_iff.mpr\n[GOAL]\ncase mpr.obj.h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nH : IsIso (\u03b9 G)\nU : C\u1d52\u1d56\nx : F.obj U\n\u22a2 x \u2208 obj G U\n[PROOFSTEP]\nrw [\u2190 IsIso.inv_hom_id_apply (G.\u03b9.app U) x]\n[GOAL]\ncase mpr.obj.h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nH : IsIso (\u03b9 G)\nU : C\u1d52\u1d56\nx : F.obj U\n\u22a2 \u2191(NatTrans.app (\u03b9 G) U)\n (\u2191(Classical.choose\n (_ :\n \u2203 inv, NatTrans.app (\u03b9 G) U \u226b inv = \ud835\udfd9 ((toPresheaf G).obj U) \u2227 inv \u226b NatTrans.app (\u03b9 G) U = \ud835\udfd9 (F.obj U)))\n x) \u2208\n obj G U\n[PROOFSTEP]\nexact ((inv (G.\u03b9.app U)) x).2\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : F' \u27f6 F\nhf : \u2200 (U : C\u1d52\u1d56) (x : F'.obj U), NatTrans.app f U x \u2208 obj G U\n\u22a2 \u2200 \u2983X Y : C\u1d52\u1d56\u2984 (f_1 : X \u27f6 Y),\n F'.map f_1 \u226b (fun U x => { val := NatTrans.app f U x, property := (_ : NatTrans.app f U x \u2208 obj G U) }) Y =\n (fun U x => { val := NatTrans.app f U x, property := (_ : NatTrans.app f U x \u2208 obj G U) }) X \u226b\n (toPresheaf G).map f_1\n[PROOFSTEP]\nhave := elementwise_of% f.naturality\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : F' \u27f6 F\nhf : \u2200 (U : C\u1d52\u1d56) (x : F'.obj U), NatTrans.app f U x \u2208 obj G U\nthis : \u2200 \u2983X Y : C\u1d52\u1d56\u2984 (f_1 : X \u27f6 Y) (x : F'.obj X), NatTrans.app f Y (F'.map f_1 x) = F.map f_1 (NatTrans.app f X x)\n\u22a2 \u2200 \u2983X Y : C\u1d52\u1d56\u2984 (f_1 : X \u27f6 Y),\n F'.map f_1 \u226b (fun U x => { val := NatTrans.app f U x, property := (_ : NatTrans.app f U x \u2208 obj G U) }) Y =\n (fun U x => { val := NatTrans.app f U x, property := (_ : NatTrans.app f U x \u2208 obj G U) }) X \u226b\n (toPresheaf G).map f_1\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : F' \u27f6 F\nhf : \u2200 (U : C\u1d52\u1d56) (x : F'.obj U), NatTrans.app f U x \u2208 obj G U\nthis : \u2200 \u2983X Y : C\u1d52\u1d56\u2984 (f_1 : X \u27f6 Y) (x : F'.obj X), NatTrans.app f Y (F'.map f_1 x) = F.map f_1 (NatTrans.app f X x)\nX\u271d Y\u271d : C\u1d52\u1d56\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 F'.map f\u271d \u226b (fun U x => { val := NatTrans.app f U x, property := (_ : NatTrans.app f U x \u2208 obj G U) }) Y\u271d =\n (fun U x => { val := NatTrans.app f U x, property := (_ : NatTrans.app f U x \u2208 obj G U) }) X\u271d \u226b\n (toPresheaf G).map f\u271d\n[PROOFSTEP]\nrefine funext fun x => Subtype.ext ?_\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : F' \u27f6 F\nhf : \u2200 (U : C\u1d52\u1d56) (x : F'.obj U), NatTrans.app f U x \u2208 obj G U\nthis : \u2200 \u2983X Y : C\u1d52\u1d56\u2984 (f_1 : X \u27f6 Y) (x : F'.obj X), NatTrans.app f Y (F'.map f_1 x) = F.map f_1 (NatTrans.app f X x)\nX\u271d Y\u271d : C\u1d52\u1d56\nf\u271d : X\u271d \u27f6 Y\u271d\nx : F'.obj X\u271d\n\u22a2 \u2191((F'.map f\u271d \u226b (fun U x => { val := NatTrans.app f U x, property := (_ : NatTrans.app f U x \u2208 obj G U) }) Y\u271d) x) =\n \u2191(((fun U x => { val := NatTrans.app f U x, property := (_ : NatTrans.app f U x \u2208 obj G U) }) X\u271d \u226b\n (toPresheaf G).map f\u271d)\n x)\n[PROOFSTEP]\nsimp only [toPresheaf_obj, types_comp_apply]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : F' \u27f6 F\nhf : \u2200 (U : C\u1d52\u1d56) (x : F'.obj U), NatTrans.app f U x \u2208 obj G U\nthis : \u2200 \u2983X Y : C\u1d52\u1d56\u2984 (f_1 : X \u27f6 Y) (x : F'.obj X), NatTrans.app f Y (F'.map f_1 x) = F.map f_1 (NatTrans.app f X x)\nX\u271d Y\u271d : C\u1d52\u1d56\nf\u271d : X\u271d \u27f6 Y\u271d\nx : F'.obj X\u271d\n\u22a2 NatTrans.app f Y\u271d (F'.map f\u271d x) =\n \u2191((toPresheaf G).map f\u271d { val := NatTrans.app f X\u271d x, property := (_ : NatTrans.app f X\u271d x \u2208 obj G X\u271d) })\n[PROOFSTEP]\nexact this _ _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : F' \u27f6 F\nhf : \u2200 (U : C\u1d52\u1d56) (x : F'.obj U), NatTrans.app f U x \u2208 obj G U\n\u22a2 lift G f hf \u226b \u03b9 G = f\n[PROOFSTEP]\next\n[GOAL]\ncase w.h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : F' \u27f6 F\nhf : \u2200 (U : C\u1d52\u1d56) (x : F'.obj U), NatTrans.app f U x \u2208 obj G U\nx\u271d : C\u1d52\u1d56\na\u271d : F'.obj x\u271d\n\u22a2 NatTrans.app (lift G f hf \u226b \u03b9 G) x\u271d a\u271d = NatTrans.app f x\u271d a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nU : C\u1d52\u1d56\ns : F.obj U\nV W : C\ni : V \u27f6 U.unop\nhi : (fun V f => F.map f.op s \u2208 obj G (op V)) V i\nj : W \u27f6 V\n\u22a2 (fun V f => F.map f.op s \u2208 obj G (op V)) W (j \u226b i)\n[PROOFSTEP]\nsimp only [op_unop, op_comp, FunctorToTypes.map_comp_apply]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nU : C\u1d52\u1d56\ns : F.obj U\nV W : C\ni : V \u27f6 U.unop\nhi : (fun V f => F.map f.op s \u2208 obj G (op V)) V i\nj : W \u27f6 V\n\u22a2 F.map j.op (F.map i.op s) \u2208 obj G (op W)\n[PROOFSTEP]\nexact G.map _ hi\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nU : C\u1d52\u1d56\ns : F.obj U\n\u22a2 Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s)\n[PROOFSTEP]\nintro Y\u2081 Y\u2082 Z g\u2081 g\u2082 f\u2081 f\u2082 h\u2081 h\u2082 e\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nU : C\u1d52\u1d56\ns : F.obj U\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 U.unop\nf\u2082 : Y\u2082 \u27f6 U.unop\nh\u2081 : (sieveOfSection G s).arrows f\u2081\nh\u2082 : (sieveOfSection G s).arrows f\u2082\ne : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 (toPresheaf G).map g\u2081.op (familyOfElementsOfSection G s f\u2081 h\u2081) =\n (toPresheaf G).map g\u2082.op (familyOfElementsOfSection G s f\u2082 h\u2082)\n[PROOFSTEP]\nrefine\n Subtype.ext\n ?_\n -- port note: `ext1` does not work here\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nU : C\u1d52\u1d56\ns : F.obj U\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 U.unop\nf\u2082 : Y\u2082 \u27f6 U.unop\nh\u2081 : (sieveOfSection G s).arrows f\u2081\nh\u2082 : (sieveOfSection G s).arrows f\u2082\ne : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 \u2191((toPresheaf G).map g\u2081.op (familyOfElementsOfSection G s f\u2081 h\u2081)) =\n \u2191((toPresheaf G).map g\u2082.op (familyOfElementsOfSection G s f\u2082 h\u2082))\n[PROOFSTEP]\nchange F.map g\u2081.op (F.map f\u2081.op s) = F.map g\u2082.op (F.map f\u2082.op s)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nU : C\u1d52\u1d56\ns : F.obj U\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 U.unop\nf\u2082 : Y\u2082 \u27f6 U.unop\nh\u2081 : (sieveOfSection G s).arrows f\u2081\nh\u2082 : (sieveOfSection G s).arrows f\u2082\ne : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 F.map g\u2081.op (F.map f\u2081.op s) = F.map g\u2082.op (F.map f\u2082.op s)\n[PROOFSTEP]\nrw [\u2190 FunctorToTypes.map_comp_apply, \u2190 FunctorToTypes.map_comp_apply, \u2190 op_comp, \u2190 op_comp, e]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\n\u22a2 \u2200 {U V : C\u1d52\u1d56} (i : U \u27f6 V),\n (fun U => {s | sieveOfSection G s \u2208 sieves J U.unop}) U \u2286\n F.map i \u207b\u00b9' (fun U => {s | sieveOfSection G s \u2208 sieves J U.unop}) V\n[PROOFSTEP]\nrintro U V i s hs\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : F.obj U\nhs : s \u2208 (fun U => {s | sieveOfSection G s \u2208 sieves J U.unop}) U\n\u22a2 s \u2208 F.map i \u207b\u00b9' (fun U => {s | sieveOfSection G s \u2208 sieves J U.unop}) V\n[PROOFSTEP]\nrefine' J.superset_covering _ (J.pullback_stable i.unop hs)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : F.obj U\nhs : s \u2208 (fun U => {s | sieveOfSection G s \u2208 sieves J U.unop}) U\n\u22a2 Sieve.pullback i.unop (sieveOfSection G s) \u2264 sieveOfSection G (F.map i s)\n[PROOFSTEP]\nintro _ _ h\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : F.obj U\nhs : s \u2208 (fun U => {s | sieveOfSection G s \u2208 sieves J U.unop}) U\nY\u271d : C\nf\u271d : Y\u271d \u27f6 V.unop\nh : (Sieve.pullback i.unop (sieveOfSection G s)).arrows f\u271d\n\u22a2 (sieveOfSection G (F.map i s)).arrows f\u271d\n[PROOFSTEP]\ndsimp at h \u22a2\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : F.obj U\nhs : s \u2208 (fun U => {s | sieveOfSection G s \u2208 sieves J U.unop}) U\nY\u271d : C\nf\u271d : Y\u271d \u27f6 V.unop\nh : F.map (i \u226b f\u271d.op) s \u2208 obj G (op Y\u271d)\n\u22a2 F.map f\u271d.op (F.map i s) \u2208 obj G (op Y\u271d)\n[PROOFSTEP]\nrwa [\u2190 FunctorToTypes.map_comp_apply]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\n\u22a2 G \u2264 sheafify J G\n[PROOFSTEP]\nintro U s hs\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nU : C\u1d52\u1d56\ns : F.obj U\nhs : s \u2208 obj G U\n\u22a2 s \u2208 obj (sheafify J G) U\n[PROOFSTEP]\nchange _ \u2208 J _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nU : C\u1d52\u1d56\ns : F.obj U\nhs : s \u2208 obj G U\n\u22a2 sieveOfSection G s \u2208 sieves J U.unop\n[PROOFSTEP]\nconvert J.top_mem U.unop\n[GOAL]\ncase h.e'_4\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nU : C\u1d52\u1d56\ns : F.obj U\nhs : s \u2208 obj G U\n\u22a2 sieveOfSection G s = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff]\n[GOAL]\ncase h.e'_4\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nU : C\u1d52\u1d56\ns : F.obj U\nhs : s \u2208 obj G U\n\u22a2 \u22a4 \u2264 sieveOfSection G s\n[PROOFSTEP]\nrintro V i -\n[GOAL]\ncase h.e'_4\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nU : C\u1d52\u1d56\ns : F.obj U\nhs : s \u2208 obj G U\nV : C\ni : V \u27f6 U.unop\n\u22a2 (sieveOfSection G s).arrows i\n[PROOFSTEP]\nexact G.map i.op hs\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\n\u22a2 G = sheafify J G\n[PROOFSTEP]\napply (G.le_sheafify J).antisymm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\n\u22a2 sheafify J G \u2264 G\n[PROOFSTEP]\nintro U s hs\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\nU : C\u1d52\u1d56\ns : F.obj U\nhs : s \u2208 obj (sheafify J G) U\n\u22a2 s \u2208 obj G U\n[PROOFSTEP]\nsuffices ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).1 = s\n by\n rw [\u2190 this]\n exact ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).2\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\nU : C\u1d52\u1d56\ns : F.obj U\nhs : s \u2208 obj (sheafify J G) U\nthis :\n \u2191(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)\n (familyOfElementsOfSection G s) (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) =\n s\n\u22a2 s \u2208 obj G U\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\nU : C\u1d52\u1d56\ns : F.obj U\nhs : s \u2208 obj (sheafify J G) U\nthis :\n \u2191(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)\n (familyOfElementsOfSection G s) (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) =\n s\n\u22a2 \u2191(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)\n (familyOfElementsOfSection G s) (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) \u2208\n obj G U\n[PROOFSTEP]\nexact ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).2\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\nU : C\u1d52\u1d56\ns : F.obj U\nhs : s \u2208 obj (sheafify J G) U\n\u22a2 \u2191(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)\n (familyOfElementsOfSection G s) (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) =\n s\n[PROOFSTEP]\napply (h _ hs).isSeparatedFor.ext\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\nU : C\u1d52\u1d56\ns : F.obj U\nhs : s \u2208 obj (sheafify J G) U\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 U.unop\u2984,\n (sieveOfSection G s).arrows f \u2192\n F.map f.op\n \u2191(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)\n (familyOfElementsOfSection G s)\n (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) =\n F.map f.op s\n[PROOFSTEP]\nintro V i hi\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\nU : C\u1d52\u1d56\ns : F.obj U\nhs : s \u2208 obj (sheafify J G) U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G s).arrows i\n\u22a2 F.map i.op\n \u2191(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)\n (familyOfElementsOfSection G s) (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) =\n F.map i.op s\n[PROOFSTEP]\nexact (congr_arg Subtype.val ((hG _ hs).valid_glue (G.family_of_elements_compatible s) _ hi) : _)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\n\u22a2 Presieve.IsSheaf J (toPresheaf (sheafify J G))\n[PROOFSTEP]\nintro U S hS x hx\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nlet S' := Sieve.bind S fun Y f hf => G.sieveOfSection (x f hf).1\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nhave := fun (V) (i : V \u27f6 U) (hi : S' i) => hi\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nthis : \u2200 (V : C) (i : V \u27f6 U), S'.arrows i \u2192 S'.arrows i\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nchoose W i\u2081 i\u2082 hi\u2082 h\u2081 h\u2082 using this\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i),\n (fun Y f h => ((fun Y f hf => sieveOfSection G \u2191(x f hf)) Y f h).arrows) (W V i hi) (i\u2082 V i hi)\n (_ : S.arrows (i\u2082 V i hi)) (i\u2081 V i hi)\nh\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), i\u2081 V i hi \u226b i\u2082 V i hi = i\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\ndsimp [-Sieve.bind_apply] at *\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nlet x'' : Presieve.FamilyOfElements F S' := fun V i hi => F.map (i\u2081 V i hi).op (x _ (hi\u2082 V i hi))\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nhave H : \u2200 s, x.IsAmalgamation s \u2194 x''.IsAmalgamation s.1 :=\n by\n intro s\n constructor\n \u00b7 intro H V i hi\n dsimp only [show x'' = fun V i hi => F.map (i\u2081 V i hi).op (x _ (hi\u2082 V i hi)) from rfl]\n conv_lhs => rw [\u2190 h\u2082 _ _ hi]\n rw [\u2190 H _ (hi\u2082 _ _ hi)]\n exact FunctorToTypes.map_comp_apply F (i\u2082 _ _ hi).op (i\u2081 _ _ hi).op _\n \u00b7 intro H V i hi\n refine Subtype.ext ?_\n apply (hF _ (x i hi).2).isSeparatedFor.ext\n intro V' i' hi'\n have hi'' : S' (i' \u226b i) := \u27e8_, _, _, hi, hi', rfl\u27e9\n have := H _ hi''\n rw [op_comp, F.map_comp] at this \n refine' this.trans (congr_arg Subtype.val (hx _ _ (hi\u2082 _ _ hi'') hi (h\u2082 _ _ hi'')))\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\n\u22a2 \u2200 (s : (toPresheaf (sheafify J G)).obj (op U)),\n Presieve.FamilyOfElements.IsAmalgamation x s \u2194 Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\n[PROOFSTEP]\nintro s\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\ns : (toPresheaf (sheafify J G)).obj (op U)\n\u22a2 Presieve.FamilyOfElements.IsAmalgamation x s \u2194 Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\ns : (toPresheaf (sheafify J G)).obj (op U)\n\u22a2 Presieve.FamilyOfElements.IsAmalgamation x s \u2192 Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\n[PROOFSTEP]\nintro H V i hi\n[GOAL]\ncase mp\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\ns : (toPresheaf (sheafify J G)).obj (op U)\nH : Presieve.FamilyOfElements.IsAmalgamation x s\nV : C\ni : V \u27f6 U\nhi : S'.arrows i\n\u22a2 F.map i.op \u2191s = x'' i hi\n[PROOFSTEP]\ndsimp only [show x'' = fun V i hi => F.map (i\u2081 V i hi).op (x _ (hi\u2082 V i hi)) from rfl]\n[GOAL]\ncase mp\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\ns : (toPresheaf (sheafify J G)).obj (op U)\nH : Presieve.FamilyOfElements.IsAmalgamation x s\nV : C\ni : V \u27f6 U\nhi : S'.arrows i\n\u22a2 F.map i.op \u2191s = F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\n[PROOFSTEP]\nconv_lhs => rw [\u2190 h\u2082 _ _ hi]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\ns : (toPresheaf (sheafify J G)).obj (op U)\nH : Presieve.FamilyOfElements.IsAmalgamation x s\nV : C\ni : V \u27f6 U\nhi : S'.arrows i\n| F.map i.op \u2191s\n[PROOFSTEP]\nrw [\u2190 h\u2082 _ _ hi]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\ns : (toPresheaf (sheafify J G)).obj (op U)\nH : Presieve.FamilyOfElements.IsAmalgamation x s\nV : C\ni : V \u27f6 U\nhi : S'.arrows i\n| F.map i.op \u2191s\n[PROOFSTEP]\nrw [\u2190 h\u2082 _ _ hi]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\ns : (toPresheaf (sheafify J G)).obj (op U)\nH : Presieve.FamilyOfElements.IsAmalgamation x s\nV : C\ni : V \u27f6 U\nhi : S'.arrows i\n| F.map i.op \u2191s\n[PROOFSTEP]\nrw [\u2190 h\u2082 _ _ hi]\n[GOAL]\ncase mp\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\ns : (toPresheaf (sheafify J G)).obj (op U)\nH : Presieve.FamilyOfElements.IsAmalgamation x s\nV : C\ni : V \u27f6 U\nhi : S'.arrows i\n\u22a2 F.map (i\u2081 V i hi \u226b i\u2082 V i hi).op \u2191s = F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\n[PROOFSTEP]\nrw [\u2190 H _ (hi\u2082 _ _ hi)]\n[GOAL]\ncase mp\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\ns : (toPresheaf (sheafify J G)).obj (op U)\nH : Presieve.FamilyOfElements.IsAmalgamation x s\nV : C\ni : V \u27f6 U\nhi : S'.arrows i\n\u22a2 F.map (i\u2081 V i hi \u226b i\u2082 V i hi).op \u2191s = F.map (i\u2081 V i hi).op \u2191((toPresheaf (sheafify J G)).map (i\u2082 V i hi).op s)\n[PROOFSTEP]\nexact FunctorToTypes.map_comp_apply F (i\u2082 _ _ hi).op (i\u2081 _ _ hi).op _\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\ns : (toPresheaf (sheafify J G)).obj (op U)\n\u22a2 Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s \u2192 Presieve.FamilyOfElements.IsAmalgamation x s\n[PROOFSTEP]\nintro H V i hi\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\ns : (toPresheaf (sheafify J G)).obj (op U)\nH : Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\nV : C\ni : V \u27f6 U\nhi : S.arrows i\n\u22a2 (toPresheaf (sheafify J G)).map i.op s = x i hi\n[PROOFSTEP]\nrefine Subtype.ext ?_\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\ns : (toPresheaf (sheafify J G)).obj (op U)\nH : Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\nV : C\ni : V \u27f6 U\nhi : S.arrows i\n\u22a2 \u2191((toPresheaf (sheafify J G)).map i.op s) = \u2191(x i hi)\n[PROOFSTEP]\napply (hF _ (x i hi).2).isSeparatedFor.ext\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\ns : (toPresheaf (sheafify J G)).obj (op U)\nH : Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\nV : C\ni : V \u27f6 U\nhi : S.arrows i\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 (op V).unop\u2984,\n (sieveOfSection G \u2191(x i hi)).arrows f \u2192 F.map f.op \u2191((toPresheaf (sheafify J G)).map i.op s) = F.map f.op \u2191(x i hi)\n[PROOFSTEP]\nintro V' i' hi'\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\ns : (toPresheaf (sheafify J G)).obj (op U)\nH : Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\nV : C\ni : V \u27f6 U\nhi : S.arrows i\nV' : C\ni' : V' \u27f6 (op V).unop\nhi' : (sieveOfSection G \u2191(x i hi)).arrows i'\n\u22a2 F.map i'.op \u2191((toPresheaf (sheafify J G)).map i.op s) = F.map i'.op \u2191(x i hi)\n[PROOFSTEP]\nhave hi'' : S' (i' \u226b i) := \u27e8_, _, _, hi, hi', rfl\u27e9\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\ns : (toPresheaf (sheafify J G)).obj (op U)\nH : Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\nV : C\ni : V \u27f6 U\nhi : S.arrows i\nV' : C\ni' : V' \u27f6 (op V).unop\nhi' : (sieveOfSection G \u2191(x i hi)).arrows i'\nhi'' : S'.arrows (i' \u226b i)\n\u22a2 F.map i'.op \u2191((toPresheaf (sheafify J G)).map i.op s) = F.map i'.op \u2191(x i hi)\n[PROOFSTEP]\nhave := H _ hi''\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\ns : (toPresheaf (sheafify J G)).obj (op U)\nH : Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\nV : C\ni : V \u27f6 U\nhi : S.arrows i\nV' : C\ni' : V' \u27f6 (op V).unop\nhi' : (sieveOfSection G \u2191(x i hi)).arrows i'\nhi'' : S'.arrows (i' \u226b i)\nthis : F.map (i' \u226b i).op \u2191s = x'' (i' \u226b i) hi''\n\u22a2 F.map i'.op \u2191((toPresheaf (sheafify J G)).map i.op s) = F.map i'.op \u2191(x i hi)\n[PROOFSTEP]\nrw [op_comp, F.map_comp] at this \n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\ns : (toPresheaf (sheafify J G)).obj (op U)\nH : Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\nV : C\ni : V \u27f6 U\nhi : S.arrows i\nV' : C\ni' : V' \u27f6 (op V).unop\nhi' : (sieveOfSection G \u2191(x i hi)).arrows i'\nhi'' : S'.arrows (i' \u226b i)\nthis : (F.map i.op \u226b F.map i'.op) \u2191s = x'' (i' \u226b i) hi''\n\u22a2 F.map i'.op \u2191((toPresheaf (sheafify J G)).map i.op s) = F.map i'.op \u2191(x i hi)\n[PROOFSTEP]\nrefine' this.trans (congr_arg Subtype.val (hx _ _ (hi\u2082 _ _ hi'') hi (h\u2082 _ _ hi'')))\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\nH :\n \u2200 (s : (toPresheaf (sheafify J G)).obj (op U)),\n Presieve.FamilyOfElements.IsAmalgamation x s \u2194 Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nhave : x''.Compatible := by\n intro V\u2081 V\u2082 V\u2083 g\u2081 g\u2082 g\u2083 g\u2084 S\u2081 S\u2082 e\n rw [\u2190 FunctorToTypes.map_comp_apply, \u2190 FunctorToTypes.map_comp_apply]\n exact\n congr_arg Subtype.val\n (hx (g\u2081 \u226b i\u2081 _ _ S\u2081) (g\u2082 \u226b i\u2081 _ _ S\u2082) (hi\u2082 _ _ S\u2081) (hi\u2082 _ _ S\u2082) (by simp only [Category.assoc, h\u2082, e]))\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\nH :\n \u2200 (s : (toPresheaf (sheafify J G)).obj (op U)),\n Presieve.FamilyOfElements.IsAmalgamation x s \u2194 Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\n\u22a2 Presieve.FamilyOfElements.Compatible x''\n[PROOFSTEP]\nintro V\u2081 V\u2082 V\u2083 g\u2081 g\u2082 g\u2083 g\u2084 S\u2081 S\u2082 e\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\nH :\n \u2200 (s : (toPresheaf (sheafify J G)).obj (op U)),\n Presieve.FamilyOfElements.IsAmalgamation x s \u2194 Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\nV\u2081 V\u2082 V\u2083 : C\ng\u2081 : V\u2083 \u27f6 V\u2081\ng\u2082 : V\u2083 \u27f6 V\u2082\ng\u2083 : V\u2081 \u27f6 U\ng\u2084 : V\u2082 \u27f6 U\nS\u2081 : S'.arrows g\u2083\nS\u2082 : S'.arrows g\u2084\ne : g\u2081 \u226b g\u2083 = g\u2082 \u226b g\u2084\n\u22a2 F.map g\u2081.op (x'' g\u2083 S\u2081) = F.map g\u2082.op (x'' g\u2084 S\u2082)\n[PROOFSTEP]\nrw [\u2190 FunctorToTypes.map_comp_apply, \u2190 FunctorToTypes.map_comp_apply]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\nH :\n \u2200 (s : (toPresheaf (sheafify J G)).obj (op U)),\n Presieve.FamilyOfElements.IsAmalgamation x s \u2194 Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\nV\u2081 V\u2082 V\u2083 : C\ng\u2081 : V\u2083 \u27f6 V\u2081\ng\u2082 : V\u2083 \u27f6 V\u2082\ng\u2083 : V\u2081 \u27f6 U\ng\u2084 : V\u2082 \u27f6 U\nS\u2081 : S'.arrows g\u2083\nS\u2082 : S'.arrows g\u2084\ne : g\u2081 \u226b g\u2083 = g\u2082 \u226b g\u2084\n\u22a2 F.map ((i\u2081 V\u2081 g\u2083 S\u2081).op \u226b g\u2081.op) \u2191(x (i\u2082 V\u2081 g\u2083 S\u2081) (_ : S.arrows (i\u2082 V\u2081 g\u2083 S\u2081))) =\n F.map ((i\u2081 V\u2082 g\u2084 S\u2082).op \u226b g\u2082.op) \u2191(x (i\u2082 V\u2082 g\u2084 S\u2082) (_ : S.arrows (i\u2082 V\u2082 g\u2084 S\u2082)))\n[PROOFSTEP]\nexact\n congr_arg Subtype.val\n (hx (g\u2081 \u226b i\u2081 _ _ S\u2081) (g\u2082 \u226b i\u2081 _ _ S\u2082) (hi\u2082 _ _ S\u2081) (hi\u2082 _ _ S\u2082) (by simp only [Category.assoc, h\u2082, e]))\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\nH :\n \u2200 (s : (toPresheaf (sheafify J G)).obj (op U)),\n Presieve.FamilyOfElements.IsAmalgamation x s \u2194 Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\nV\u2081 V\u2082 V\u2083 : C\ng\u2081 : V\u2083 \u27f6 V\u2081\ng\u2082 : V\u2083 \u27f6 V\u2082\ng\u2083 : V\u2081 \u27f6 U\ng\u2084 : V\u2082 \u27f6 U\nS\u2081 : S'.arrows g\u2083\nS\u2082 : S'.arrows g\u2084\ne : g\u2081 \u226b g\u2083 = g\u2082 \u226b g\u2084\n\u22a2 (g\u2081 \u226b i\u2081 V\u2081 g\u2083 S\u2081) \u226b i\u2082 V\u2081 g\u2083 S\u2081 = (g\u2082 \u226b i\u2081 V\u2082 g\u2084 S\u2082) \u226b i\u2082 V\u2082 g\u2084 S\u2082\n[PROOFSTEP]\nsimp only [Category.assoc, h\u2082, e]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\nH :\n \u2200 (s : (toPresheaf (sheafify J G)).obj (op U)),\n Presieve.FamilyOfElements.IsAmalgamation x s \u2194 Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\nthis : Presieve.FamilyOfElements.Compatible x''\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nobtain \u27e8t, ht, ht'\u27e9 := hF _ (J.bind_covering hS fun V i hi => (x i hi).2) _ this\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\nH :\n \u2200 (s : (toPresheaf (sheafify J G)).obj (op U)),\n Presieve.FamilyOfElements.IsAmalgamation x s \u2194 Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\nthis : Presieve.FamilyOfElements.Compatible x''\nt : F.obj (op U)\nht : Presieve.FamilyOfElements.IsAmalgamation x'' t\nht' : \u2200 (y : F.obj (op U)), (fun t => Presieve.FamilyOfElements.IsAmalgamation x'' t) y \u2192 y = t\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nrefine' \u27e8\u27e8t, _\u27e9, (H \u27e8t, _\u27e9).mpr ht, fun y hy => Subtype.ext (ht' _ ((H _).mp hy))\u27e9\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\nH :\n \u2200 (s : (toPresheaf (sheafify J G)).obj (op U)),\n Presieve.FamilyOfElements.IsAmalgamation x s \u2194 Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\nthis : Presieve.FamilyOfElements.Compatible x''\nt : F.obj (op U)\nht : Presieve.FamilyOfElements.IsAmalgamation x'' t\nht' : \u2200 (y : F.obj (op U)), (fun t => Presieve.FamilyOfElements.IsAmalgamation x'' t) y \u2192 y = t\n\u22a2 t \u2208 obj (sheafify J G) (op U)\n[PROOFSTEP]\nrefine' J.superset_covering _ (J.bind_covering hS fun V i hi => (x i hi).2)\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\nH :\n \u2200 (s : (toPresheaf (sheafify J G)).obj (op U)),\n Presieve.FamilyOfElements.IsAmalgamation x s \u2194 Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\nthis : Presieve.FamilyOfElements.Compatible x''\nt : F.obj (op U)\nht : Presieve.FamilyOfElements.IsAmalgamation x'' t\nht' : \u2200 (y : F.obj (op U)), (fun t => Presieve.FamilyOfElements.IsAmalgamation x'' t) y \u2192 y = t\n\u22a2 (Sieve.bind S.arrows fun V i hi => sieveOfSection G \u2191(x i hi)) \u2264 sieveOfSection G t\n[PROOFSTEP]\nintro V i hi\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\nH :\n \u2200 (s : (toPresheaf (sheafify J G)).obj (op U)),\n Presieve.FamilyOfElements.IsAmalgamation x s \u2194 Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\nthis : Presieve.FamilyOfElements.Compatible x''\nt : F.obj (op U)\nht : Presieve.FamilyOfElements.IsAmalgamation x'' t\nht' : \u2200 (y : F.obj (op U)), (fun t => Presieve.FamilyOfElements.IsAmalgamation x'' t) y \u2192 y = t\nV : C\ni : V \u27f6 (op U).unop\nhi : (Sieve.bind S.arrows fun V i hi => sieveOfSection G \u2191(x i hi)).arrows i\n\u22a2 (sieveOfSection G t).arrows i\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\nH :\n \u2200 (s : (toPresheaf (sheafify J G)).obj (op U)),\n Presieve.FamilyOfElements.IsAmalgamation x s \u2194 Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\nthis : Presieve.FamilyOfElements.Compatible x''\nt : F.obj (op U)\nht : Presieve.FamilyOfElements.IsAmalgamation x'' t\nht' : \u2200 (y : F.obj (op U)), (fun t => Presieve.FamilyOfElements.IsAmalgamation x'' t) y \u2192 y = t\nV : C\ni : V \u27f6 (op U).unop\nhi : (Sieve.bind S.arrows fun V i hi => sieveOfSection G \u2191(x i hi)).arrows i\n\u22a2 F.map i.op t \u2208 obj G (op V)\n[PROOFSTEP]\nrw [ht _ hi]\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S \u2208 sieves J U\nx : Presieve.FamilyOfElements (toPresheaf (sheafify J G)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)\nW : (V : C) \u2192 (i : V \u27f6 U) \u2192 S'.arrows i \u2192 C\ni\u2081 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 V \u27f6 W V i hi\ni\u2082 : (V : C) \u2192 (i : V \u27f6 U) \u2192 (hi : S'.arrows i) \u2192 W V i hi \u27f6 U\nhi\u2082 : \u2200 (V : C) (i : V \u27f6 U) (hi : S'.arrows i), S.arrows (i\u2082 V i hi)\nh\u2081 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi))) \u2208 obj G (op V)\nh\u2082 :\n \u2200 (V : C) (i : V \u27f6 U) (hi : (Sieve.bind S.arrows fun Y f hf => sieveOfSection G \u2191(x f hf)).arrows i),\n i\u2081 V i hi \u226b i\u2082 V i hi = i\nx'' : Presieve.FamilyOfElements F S'.arrows :=\n fun V i hi => F.map (i\u2081 V i hi).op \u2191(x (i\u2082 V i hi) (_ : S.arrows (i\u2082 V i hi)))\nH :\n \u2200 (s : (toPresheaf (sheafify J G)).obj (op U)),\n Presieve.FamilyOfElements.IsAmalgamation x s \u2194 Presieve.FamilyOfElements.IsAmalgamation x'' \u2191s\nthis : Presieve.FamilyOfElements.Compatible x''\nt : F.obj (op U)\nht : Presieve.FamilyOfElements.IsAmalgamation x'' t\nht' : \u2200 (y : F.obj (op U)), (fun t => Presieve.FamilyOfElements.IsAmalgamation x'' t) y \u2192 y = t\nV : C\ni : V \u27f6 (op U).unop\nhi : (Sieve.bind S.arrows fun V i hi => sieveOfSection G \u2191(x i hi)).arrows i\n\u22a2 x'' i hi \u2208 obj G (op V)\n[PROOFSTEP]\nexact h\u2081 _ _ hi\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\n\u22a2 Presieve.IsSheaf J (toPresheaf G) \u2194 \u2200 (U : C\u1d52\u1d56) (s : F.obj U), sieveOfSection G s \u2208 sieves J U.unop \u2192 s \u2208 obj G U\n[PROOFSTEP]\nrw [\u2190 G.eq_sheafify_iff h]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\n\u22a2 G = sheafify J G \u2194 \u2200 (U : C\u1d52\u1d56) (s : F.obj U), sieveOfSection G s \u2208 sieves J U.unop \u2192 s \u2208 obj G U\n[PROOFSTEP]\nchange _ \u2194 G.sheafify J \u2264 G\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\n\u22a2 G = sheafify J G \u2194 sheafify J G \u2264 G\n[PROOFSTEP]\nexact \u27e8Eq.ge, (G.le_sheafify J).antisymm\u27e9\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\n\u22a2 \u2200 \u2983X Y : C\u1d52\u1d56\u2984 (f_1 : X \u27f6 Y),\n (toPresheaf (sheafify J G)).map f_1 \u226b\n (fun U s =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G \u2191s).arrows)\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))\n (_ :\n Presieve.FamilyOfElements.Compatible\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))))\n Y =\n (fun U s =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G \u2191s).arrows)\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))\n (_ :\n Presieve.FamilyOfElements.Compatible\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))))\n X \u226b\n F'.map f_1\n[PROOFSTEP]\nintro U V i\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU V : C\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 (toPresheaf (sheafify J G)).map i \u226b\n (fun U s =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G \u2191s).arrows)\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))\n (_ :\n Presieve.FamilyOfElements.Compatible\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))))\n V =\n (fun U s =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G \u2191s).arrows)\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))\n (_ :\n Presieve.FamilyOfElements.Compatible\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))))\n U \u226b\n F'.map i\n[PROOFSTEP]\next s\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : (toPresheaf (sheafify J G)).obj U\n\u22a2 ((toPresheaf (sheafify J G)).map i \u226b\n (fun U s =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G \u2191s).arrows)\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))\n (_ :\n Presieve.FamilyOfElements.Compatible\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))))\n V)\n s =\n ((fun U s =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G \u2191s).arrows)\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))\n (_ :\n Presieve.FamilyOfElements.Compatible\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))))\n U \u226b\n F'.map i)\n s\n[PROOFSTEP]\napply (h _ ((Subpresheaf.sheafify J G).toPresheaf.map i s).prop).isSeparatedFor.ext\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : (toPresheaf (sheafify J G)).obj U\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f_1 : Y \u27f6 V.unop\u2984,\n (sieveOfSection G \u2191((toPresheaf (sheafify J G)).map i s)).arrows f_1 \u2192\n F'.map f_1.op\n (((toPresheaf (sheafify J G)).map i \u226b\n (fun U s =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G \u2191s).arrows)\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))\n (_ :\n Presieve.FamilyOfElements.Compatible\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))))\n V)\n s) =\n F'.map f_1.op\n (((fun U s =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G \u2191s).arrows)\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))\n (_ :\n Presieve.FamilyOfElements.Compatible\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))))\n U \u226b\n F'.map i)\n s)\n[PROOFSTEP]\nintro W j hj\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : (toPresheaf (sheafify J G)).obj U\nW : C\nj : W \u27f6 V.unop\nhj : (sieveOfSection G \u2191((toPresheaf (sheafify J G)).map i s)).arrows j\n\u22a2 F'.map j.op\n (((toPresheaf (sheafify J G)).map i \u226b\n (fun U s =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G \u2191s).arrows)\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))\n (_ :\n Presieve.FamilyOfElements.Compatible\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))))\n V)\n s) =\n F'.map j.op\n (((fun U s =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G \u2191s).arrows)\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))\n (_ :\n Presieve.FamilyOfElements.Compatible\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))))\n U \u226b\n F'.map i)\n s)\n[PROOFSTEP]\nrefine'\n (Presieve.IsSheafFor.valid_glue (h _ ((G.sheafify J).toPresheaf.map i s).2)\n ((G.family_of_elements_compatible _).compPresheafMap _) _ hj).trans\n _\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : (toPresheaf (sheafify J G)).obj U\nW : C\nj : W \u27f6 V.unop\nhj : (sieveOfSection G \u2191((toPresheaf (sheafify J G)).map i s)).arrows j\n\u22a2 Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191((toPresheaf (sheafify J G)).map i s)) j\n hj =\n F'.map j.op\n (((fun U s =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G \u2191s).arrows)\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))\n (_ :\n Presieve.FamilyOfElements.Compatible\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))))\n U \u226b\n F'.map i)\n s)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : (toPresheaf (sheafify J G)).obj U\nW : C\nj : W \u27f6 V.unop\nhj : (sieveOfSection G \u2191((toPresheaf (sheafify J G)).map i s)).arrows j\n\u22a2 Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G (F.map i \u2191s)) j hj =\n F'.map j.op\n (F'.map i\n (Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G \u2191s).arrows)\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))\n (_ :\n Presieve.FamilyOfElements.Compatible\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s)))))\n[PROOFSTEP]\nconv_rhs => rw [\u2190 FunctorToTypes.map_comp_apply]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : (toPresheaf (sheafify J G)).obj U\nW : C\nj : W \u27f6 V.unop\nhj : (sieveOfSection G \u2191((toPresheaf (sheafify J G)).map i s)).arrows j\n| F'.map j.op\n (F'.map i\n (Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G \u2191s).arrows)\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))\n (_ :\n Presieve.FamilyOfElements.Compatible\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s)))))\n[PROOFSTEP]\nrw [\u2190 FunctorToTypes.map_comp_apply]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : (toPresheaf (sheafify J G)).obj U\nW : C\nj : W \u27f6 V.unop\nhj : (sieveOfSection G \u2191((toPresheaf (sheafify J G)).map i s)).arrows j\n| F'.map j.op\n (F'.map i\n (Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G \u2191s).arrows)\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))\n (_ :\n Presieve.FamilyOfElements.Compatible\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s)))))\n[PROOFSTEP]\nrw [\u2190 FunctorToTypes.map_comp_apply]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : (toPresheaf (sheafify J G)).obj U\nW : C\nj : W \u27f6 V.unop\nhj : (sieveOfSection G \u2191((toPresheaf (sheafify J G)).map i s)).arrows j\n| F'.map j.op\n (F'.map i\n (Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G \u2191s).arrows)\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))\n (_ :\n Presieve.FamilyOfElements.Compatible\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s)))))\n[PROOFSTEP]\nrw [\u2190 FunctorToTypes.map_comp_apply]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : (toPresheaf (sheafify J G)).obj U\nW : C\nj : W \u27f6 V.unop\nhj : (sieveOfSection G \u2191((toPresheaf (sheafify J G)).map i s)).arrows j\n\u22a2 Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G (F.map i \u2191s)) j hj =\n F'.map (i \u226b j.op)\n (Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G \u2191s).arrows)\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))\n (_ :\n Presieve.FamilyOfElements.Compatible\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))))\n[PROOFSTEP]\nchange _ = F'.map (j \u226b i.unop).op _\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : (toPresheaf (sheafify J G)).obj U\nW : C\nj : W \u27f6 V.unop\nhj : (sieveOfSection G \u2191((toPresheaf (sheafify J G)).map i s)).arrows j\n\u22a2 Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G (F.map i \u2191s)) j hj =\n F'.map (j \u226b i.unop).op\n (Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor F' (sieveOfSection G \u2191s).arrows)\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))\n (_ :\n Presieve.FamilyOfElements.Compatible\n (Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s))))\n[PROOFSTEP]\nrefine'\n Eq.trans _\n (Presieve.IsSheafFor.valid_glue (h _ s.2) ((G.family_of_elements_compatible s.1).compPresheafMap f) (j \u226b i.unop)\n _).symm\n[GOAL]\ncase h.refine'_1\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : (toPresheaf (sheafify J G)).obj U\nW : C\nj : W \u27f6 V.unop\nhj : (sieveOfSection G \u2191((toPresheaf (sheafify J G)).map i s)).arrows j\n\u22a2 Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G (F.map i \u2191s)) j hj =\n Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s) (j \u226b i.unop) ?h.refine'_2\ncase h.refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : (toPresheaf (sheafify J G)).obj U\nW : C\nj : W \u27f6 V.unop\nhj : (sieveOfSection G \u2191((toPresheaf (sheafify J G)).map i s)).arrows j\n\u22a2 (sieveOfSection G \u2191s).arrows (j \u226b i.unop)\n[PROOFSTEP]\nswap\n -- porting note: need to swap two goals otherwise the first goal needs to be proven\n -- inside the second goal any way\n[GOAL]\ncase h.refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : (toPresheaf (sheafify J G)).obj U\nW : C\nj : W \u27f6 V.unop\nhj : (sieveOfSection G \u2191((toPresheaf (sheafify J G)).map i s)).arrows j\n\u22a2 (sieveOfSection G \u2191s).arrows (j \u226b i.unop)\n[PROOFSTEP]\ndsimp [Presieve.FamilyOfElements.compPresheafMap] at hj \u22a2\n[GOAL]\ncase h.refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : (toPresheaf (sheafify J G)).obj U\nW : C\nj : W \u27f6 V.unop\nhj : F.map j.op (F.map i \u2191s) \u2208 obj G (op W)\n\u22a2 F.map (i \u226b j.op) \u2191s \u2208 obj G (op W)\n[PROOFSTEP]\nrwa [FunctorToTypes.map_comp_apply]\n[GOAL]\ncase h.refine'_1\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : (toPresheaf (sheafify J G)).obj U\nW : C\nj : W \u27f6 V.unop\nhj : (sieveOfSection G \u2191((toPresheaf (sheafify J G)).map i s)).arrows j\n\u22a2 Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G (F.map i \u2191s)) j hj =\n Presieve.FamilyOfElements.compPresheafMap f (familyOfElementsOfSection G \u2191s) (j \u226b i.unop)\n (_ : (sieveOfSection G \u2191s).arrows (j \u226b i.unop))\n[PROOFSTEP]\ndsimp [Presieve.FamilyOfElements.compPresheafMap]\n[GOAL]\ncase h.refine'_1\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU V : C\u1d52\u1d56\ni : U \u27f6 V\ns : (toPresheaf (sheafify J G)).obj U\nW : C\nj : W \u27f6 V.unop\nhj : (sieveOfSection G \u2191((toPresheaf (sheafify J G)).map i s)).arrows j\n\u22a2 NatTrans.app f (op W) (familyOfElementsOfSection G (F.map i \u2191s) j hj) =\n NatTrans.app f (op W) (familyOfElementsOfSection G (\u2191s) (j \u226b i.unop) (_ : F.map (i \u226b j.op) \u2191s \u2208 obj G (op W)))\n[PROOFSTEP]\nexact congr_arg _ (Subtype.ext (FunctorToTypes.map_comp_apply _ _ _ _).symm)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\n\u22a2 homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G f h = f\n[PROOFSTEP]\next U s\n[GOAL]\ncase w.h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU : C\u1d52\u1d56\ns : (toPresheaf G).obj U\n\u22a2 NatTrans.app (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G f h) U s = NatTrans.app f U s\n[PROOFSTEP]\napply (h _ ((Subpresheaf.homOfLe (G.le_sheafify J)).app U s).prop).isSeparatedFor.ext\n[GOAL]\ncase w.h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU : C\u1d52\u1d56\ns : (toPresheaf G).obj U\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f_1 : Y \u27f6 U.unop\u2984,\n (sieveOfSection G \u2191(NatTrans.app (homOfLe (_ : G \u2264 sheafify J G)) U s)).arrows f_1 \u2192\n F'.map f_1.op (NatTrans.app (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G f h) U s) =\n F'.map f_1.op (NatTrans.app f U s)\n[PROOFSTEP]\nintro V i hi\n[GOAL]\ncase w.h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU : C\u1d52\u1d56\ns : (toPresheaf G).obj U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G \u2191(NatTrans.app (homOfLe (_ : G \u2264 sheafify J G)) U s)).arrows i\n\u22a2 F'.map i.op (NatTrans.app (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G f h) U s) =\n F'.map i.op (NatTrans.app f U s)\n[PROOFSTEP]\nhave := elementwise_of% f.naturality\n[GOAL]\ncase w.h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU : C\u1d52\u1d56\ns : (toPresheaf G).obj U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G \u2191(NatTrans.app (homOfLe (_ : G \u2264 sheafify J G)) U s)).arrows i\nthis :\n \u2200 \u2983X Y : C\u1d52\u1d56\u2984 (f_1 : X \u27f6 Y) (x : (toPresheaf G).obj X),\n NatTrans.app f Y ((toPresheaf G).map f_1 x) = F'.map f_1 (NatTrans.app f X x)\n\u22a2 F'.map i.op (NatTrans.app (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G f h) U s) =\n F'.map i.op (NatTrans.app f U s)\n[PROOFSTEP]\nexact\n (Presieve.IsSheafFor.valid_glue (h _ ((homOfLe (_ : G \u2264 sheafify J G)).app U s).2)\n ((G.family_of_elements_compatible _).compPresheafMap _) _ hi).trans\n (this _ _)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl\u2081 l\u2082 : toPresheaf (sheafify J G) \u27f6 F'\ne : homOfLe (_ : G \u2264 sheafify J G) \u226b l\u2081 = homOfLe (_ : G \u2264 sheafify J G) \u226b l\u2082\n\u22a2 l\u2081 = l\u2082\n[PROOFSTEP]\next U \u27e8s, hs\u27e9\n[GOAL]\ncase w.h.h.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl\u2081 l\u2082 : toPresheaf (sheafify J G) \u27f6 F'\ne : homOfLe (_ : G \u2264 sheafify J G) \u226b l\u2081 = homOfLe (_ : G \u2264 sheafify J G) \u226b l\u2082\nU : C\u1d52\u1d56\ns : F.obj U\nhs : s \u2208 obj (sheafify J G) U\n\u22a2 NatTrans.app l\u2081 U { val := s, property := hs } = NatTrans.app l\u2082 U { val := s, property := hs }\n[PROOFSTEP]\napply (h _ hs).isSeparatedFor.ext\n[GOAL]\ncase w.h.h.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl\u2081 l\u2082 : toPresheaf (sheafify J G) \u27f6 F'\ne : homOfLe (_ : G \u2264 sheafify J G) \u226b l\u2081 = homOfLe (_ : G \u2264 sheafify J G) \u226b l\u2082\nU : C\u1d52\u1d56\ns : F.obj U\nhs : s \u2208 obj (sheafify J G) U\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 U.unop\u2984,\n (sieveOfSection G s).arrows f \u2192\n F'.map f.op (NatTrans.app l\u2081 U { val := s, property := hs }) =\n F'.map f.op (NatTrans.app l\u2082 U { val := s, property := hs })\n[PROOFSTEP]\nrintro V i hi\n[GOAL]\ncase w.h.h.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl\u2081 l\u2082 : toPresheaf (sheafify J G) \u27f6 F'\ne : homOfLe (_ : G \u2264 sheafify J G) \u226b l\u2081 = homOfLe (_ : G \u2264 sheafify J G) \u226b l\u2082\nU : C\u1d52\u1d56\ns : F.obj U\nhs : s \u2208 obj (sheafify J G) U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G s).arrows i\n\u22a2 F'.map i.op (NatTrans.app l\u2081 U { val := s, property := hs }) =\n F'.map i.op (NatTrans.app l\u2082 U { val := s, property := hs })\n[PROOFSTEP]\ndsimp at hi \n[GOAL]\ncase w.h.h.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl\u2081 l\u2082 : toPresheaf (sheafify J G) \u27f6 F'\ne : homOfLe (_ : G \u2264 sheafify J G) \u226b l\u2081 = homOfLe (_ : G \u2264 sheafify J G) \u226b l\u2082\nU : C\u1d52\u1d56\ns : F.obj U\nhs : s \u2208 obj (sheafify J G) U\nV : C\ni : V \u27f6 U.unop\nhi : F.map i.op s \u2208 obj G (op V)\n\u22a2 F'.map i.op (NatTrans.app l\u2081 U { val := s, property := hs }) =\n F'.map i.op (NatTrans.app l\u2082 U { val := s, property := hs })\n[PROOFSTEP]\nerw [\u2190 FunctorToTypes.naturality, \u2190 FunctorToTypes.naturality]\n[GOAL]\ncase w.h.h.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl\u2081 l\u2082 : toPresheaf (sheafify J G) \u27f6 F'\ne : homOfLe (_ : G \u2264 sheafify J G) \u226b l\u2081 = homOfLe (_ : G \u2264 sheafify J G) \u226b l\u2082\nU : C\u1d52\u1d56\ns : F.obj U\nhs : s \u2208 obj (sheafify J G) U\nV : C\ni : V \u27f6 U.unop\nhi : F.map i.op s \u2208 obj G (op V)\n\u22a2 NatTrans.app l\u2081 (op V) ((toPresheaf (sheafify J G)).map i.op { val := s, property := hs }) =\n NatTrans.app l\u2082 (op V) ((toPresheaf (sheafify J G)).map i.op { val := s, property := hs })\n[PROOFSTEP]\nexact (congr_fun (congr_app e <| op V) \u27e8_, hi\u27e9 : _)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\n\u22a2 sheafify J G \u2264 G'\n[PROOFSTEP]\nintro U x hx\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\n\u22a2 x \u2208 obj G' U\n[PROOFSTEP]\nconvert ((G.sheafifyLift (Subpresheaf.homOfLe h) hG').app U \u27e8x, hx\u27e9).2\n[GOAL]\ncase h.e'_4\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\n\u22a2 x = \u2191(NatTrans.app (sheafifyLift G (homOfLe h) hG') U { val := x, property := hx })\n[PROOFSTEP]\napply (hF _ hx).isSeparatedFor.ext\n[GOAL]\ncase h.e'_4\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 U.unop\u2984,\n (sieveOfSection G x).arrows f \u2192\n F.map f.op x = F.map f.op \u2191(NatTrans.app (sheafifyLift G (homOfLe h) hG') U { val := x, property := hx })\n[PROOFSTEP]\nintro V i hi\n[GOAL]\ncase h.e'_4\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G x).arrows i\n\u22a2 F.map i.op x = F.map i.op \u2191(NatTrans.app (sheafifyLift G (homOfLe h) hG') U { val := x, property := hx })\n[PROOFSTEP]\nhave :=\n congr_arg (fun f : G.toPresheaf \u27f6 G'.toPresheaf => (NatTrans.app f (op V) \u27e8_, hi\u27e9).1)\n (G.to_sheafifyLift (Subpresheaf.homOfLe h) hG')\n[GOAL]\ncase h.e'_4\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G x).arrows i\nthis :\n (fun f => \u2191(NatTrans.app f (op V) { val := F.map i.op x, property := hi }))\n (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G (homOfLe h) hG') =\n (fun f => \u2191(NatTrans.app f (op V) { val := F.map i.op x, property := hi })) (homOfLe h)\n\u22a2 F.map i.op x = F.map i.op \u2191(NatTrans.app (sheafifyLift G (homOfLe h) hG') U { val := x, property := hx })\n[PROOFSTEP]\nconvert this.symm\n[GOAL]\ncase h.e'_3\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G x).arrows i\nthis :\n (fun f => \u2191(NatTrans.app f (op V) { val := F.map i.op x, property := hi }))\n (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G (homOfLe h) hG') =\n (fun f => \u2191(NatTrans.app f (op V) { val := F.map i.op x, property := hi })) (homOfLe h)\n\u22a2 F.map i.op \u2191(NatTrans.app (sheafifyLift G (homOfLe h) hG') U { val := x, property := hx }) =\n (fun f => \u2191(NatTrans.app f (op V) { val := F.map i.op x, property := hi }))\n (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G (homOfLe h) hG')\n[PROOFSTEP]\nerw [\u2190 Subpresheaf.nat_trans_naturality]\n[GOAL]\ncase h.e'_3\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G x).arrows i\nthis :\n (fun f => \u2191(NatTrans.app f (op V) { val := F.map i.op x, property := hi }))\n (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G (homOfLe h) hG') =\n (fun f => \u2191(NatTrans.app f (op V) { val := F.map i.op x, property := hi })) (homOfLe h)\n\u22a2 \u2191(NatTrans.app (sheafifyLift G (homOfLe h) hG') (op V)\n ((toPresheaf (sheafify J G)).map i.op { val := x, property := hx })) =\n (fun f => \u2191(NatTrans.app f (op V) { val := F.map i.op x, property := hi }))\n (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G (homOfLe h) hG')\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : F' \u27f6 F\n\u22a2 \u2200 {U V : C\u1d52\u1d56} (i : U \u27f6 V),\n (fun U => Set.range (NatTrans.app f U)) U \u2286 F.map i \u207b\u00b9' (fun U => Set.range (NatTrans.app f U)) V\n[PROOFSTEP]\nrintro U V i _ \u27e8x, rfl\u27e9\n[GOAL]\ncase intro\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : F' \u27f6 F\nU V : C\u1d52\u1d56\ni : U \u27f6 V\nx : F'.obj U\n\u22a2 NatTrans.app f U x \u2208 F.map i \u207b\u00b9' (fun U => Set.range (NatTrans.app f U)) V\n[PROOFSTEP]\nhave := elementwise_of% f.naturality\n[GOAL]\ncase intro\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : F' \u27f6 F\nU V : C\u1d52\u1d56\ni : U \u27f6 V\nx : F'.obj U\nthis : \u2200 \u2983X Y : C\u1d52\u1d56\u2984 (f_1 : X \u27f6 Y) (x : F'.obj X), NatTrans.app f Y (F'.map f_1 x) = F.map f_1 (NatTrans.app f X x)\n\u22a2 NatTrans.app f U x \u2208 F.map i \u207b\u00b9' (fun U => Set.range (NatTrans.app f U)) V\n[PROOFSTEP]\nexact \u27e8_, this i x\u27e9\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\n\u22a2 imagePresheaf (\ud835\udfd9 F) = \u22a4\n[PROOFSTEP]\next\n[GOAL]\ncase obj.h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nx\u271d\u00b9 : C\u1d52\u1d56\nx\u271d : F.obj x\u271d\u00b9\n\u22a2 x\u271d \u2208 Subpresheaf.obj (imagePresheaf (\ud835\udfd9 F)) x\u271d\u00b9 \u2194 x\u271d \u2208 Subpresheaf.obj \u22a4 x\u271d\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : C\u1d52\u1d56 \u2964 TypeMax\nf : F \u27f6 F'\nhf : Mono f\n\u22a2 IsIso (toImagePresheaf f)\n[PROOFSTEP]\nhave : \u2200 (X : C\u1d52\u1d56), IsIso ((toImagePresheaf f).app X) :=\n by\n intro X\n rw [isIso_iff_bijective]\n constructor\n \u00b7 intro x y e\n have := (NatTrans.mono_iff_mono_app _ _).mp hf X\n rw [mono_iff_injective] at this \n exact this (congr_arg Subtype.val e : _)\n \u00b7 rintro \u27e8_, \u27e8x, rfl\u27e9\u27e9\n exact \u27e8x, rfl\u27e9\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : C\u1d52\u1d56 \u2964 TypeMax\nf : F \u27f6 F'\nhf : Mono f\n\u22a2 \u2200 (X : C\u1d52\u1d56), IsIso (NatTrans.app (toImagePresheaf f) X)\n[PROOFSTEP]\nintro X\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : C\u1d52\u1d56 \u2964 TypeMax\nf : F \u27f6 F'\nhf : Mono f\nX : C\u1d52\u1d56\n\u22a2 IsIso (NatTrans.app (toImagePresheaf f) X)\n[PROOFSTEP]\nrw [isIso_iff_bijective]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : C\u1d52\u1d56 \u2964 TypeMax\nf : F \u27f6 F'\nhf : Mono f\nX : C\u1d52\u1d56\n\u22a2 Function.Bijective (NatTrans.app (toImagePresheaf f) X)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : C\u1d52\u1d56 \u2964 TypeMax\nf : F \u27f6 F'\nhf : Mono f\nX : C\u1d52\u1d56\n\u22a2 Function.Injective (NatTrans.app (toImagePresheaf f) X)\n[PROOFSTEP]\nintro x y e\n[GOAL]\ncase left\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : C\u1d52\u1d56 \u2964 TypeMax\nf : F \u27f6 F'\nhf : Mono f\nX : C\u1d52\u1d56\nx y : F.obj X\ne : NatTrans.app (toImagePresheaf f) X x = NatTrans.app (toImagePresheaf f) X y\n\u22a2 x = y\n[PROOFSTEP]\nhave := (NatTrans.mono_iff_mono_app _ _).mp hf X\n[GOAL]\ncase left\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : C\u1d52\u1d56 \u2964 TypeMax\nf : F \u27f6 F'\nhf : Mono f\nX : C\u1d52\u1d56\nx y : F.obj X\ne : NatTrans.app (toImagePresheaf f) X x = NatTrans.app (toImagePresheaf f) X y\nthis : Mono (NatTrans.app f X)\n\u22a2 x = y\n[PROOFSTEP]\nrw [mono_iff_injective] at this \n[GOAL]\ncase left\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : C\u1d52\u1d56 \u2964 TypeMax\nf : F \u27f6 F'\nhf : Mono f\nX : C\u1d52\u1d56\nx y : F.obj X\ne : NatTrans.app (toImagePresheaf f) X x = NatTrans.app (toImagePresheaf f) X y\nthis : Function.Injective (NatTrans.app f X)\n\u22a2 x = y\n[PROOFSTEP]\nexact this (congr_arg Subtype.val e : _)\n[GOAL]\ncase right\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : C\u1d52\u1d56 \u2964 TypeMax\nf : F \u27f6 F'\nhf : Mono f\nX : C\u1d52\u1d56\n\u22a2 Function.Surjective (NatTrans.app (toImagePresheaf f) X)\n[PROOFSTEP]\nrintro \u27e8_, \u27e8x, rfl\u27e9\u27e9\n[GOAL]\ncase right.mk.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : C\u1d52\u1d56 \u2964 TypeMax\nf : F \u27f6 F'\nhf : Mono f\nX : C\u1d52\u1d56\nx : F.obj X\n\u22a2 \u2203 a,\n NatTrans.app (toImagePresheaf f) X a =\n { val := NatTrans.app f X x, property := (_ : \u2203 y, NatTrans.app f X y = NatTrans.app f X x) }\n[PROOFSTEP]\nexact \u27e8x, rfl\u27e9\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : C\u1d52\u1d56 \u2964 TypeMax\nf : F \u27f6 F'\nhf : Mono f\nthis : \u2200 (X : C\u1d52\u1d56), IsIso (NatTrans.app (toImagePresheaf f) X)\n\u22a2 IsIso (toImagePresheaf f)\n[PROOFSTEP]\napply NatIso.isIso_of_isIso_app\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\n\u22a2 Presheaf.IsSheaf J (Subpresheaf.toPresheaf (Subpresheaf.sheafify J (imagePresheaf f.val)))\n[PROOFSTEP]\nrw [isSheaf_iff_isSheaf_of_type]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\n\u22a2 Presieve.IsSheaf J (Subpresheaf.toPresheaf (Subpresheaf.sheafify J (imagePresheaf f.val)))\n[PROOFSTEP]\napply Subpresheaf.sheafify_isSheaf\n[GOAL]\ncase hF\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\n\u22a2 Presieve.IsSheaf J F'.val\n[PROOFSTEP]\nrw [\u2190 isSheaf_iff_isSheaf_of_type]\n[GOAL]\ncase hF\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\n\u22a2 Presheaf.IsSheaf J F'.val\n[PROOFSTEP]\nexact F'.2\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\n\u22a2 toImageSheaf f \u226b imageSheaf\u03b9 f = f\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\n\u22a2 (toImageSheaf f \u226b imageSheaf\u03b9 f).val = f.val\n[PROOFSTEP]\nsimp [toImagePresheafSheafify]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\n\u22a2 Mono ((sheafToPresheaf J (Type w)).map (imageSheaf\u03b9 f))\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\n\u22a2 Mono (Subpresheaf.\u03b9 (Subpresheaf.sheafify J (imagePresheaf f.val)))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\n\u22a2 Epi (toImageSheaf f)\n[PROOFSTEP]\nrefine' \u27e8@fun G' g\u2081 g\u2082 e => _\u27e9\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G'\u271d : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\nG' : Sheaf J (Type w)\ng\u2081 g\u2082 : imageSheaf f \u27f6 G'\ne : toImageSheaf f \u226b g\u2081 = toImageSheaf f \u226b g\u2082\n\u22a2 g\u2081 = g\u2082\n[PROOFSTEP]\next U \u27e8s, hx\u27e9\n[GOAL]\ncase h.w.h.h.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G'\u271d : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\nG' : Sheaf J (Type w)\ng\u2081 g\u2082 : imageSheaf f \u27f6 G'\ne : toImageSheaf f \u226b g\u2081 = toImageSheaf f \u226b g\u2082\nU : C\u1d52\u1d56\ns : F'.val.obj U\nhx : s \u2208 Subpresheaf.obj (Subpresheaf.sheafify J (imagePresheaf f.val)) U\n\u22a2 NatTrans.app g\u2081.val U { val := s, property := hx } = NatTrans.app g\u2082.val U { val := s, property := hx }\n[PROOFSTEP]\napply ((isSheaf_iff_isSheaf_of_type J _).mp G'.2 _ hx).isSeparatedFor.ext\n[GOAL]\ncase h.w.h.h.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G'\u271d : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\nG' : Sheaf J (Type w)\ng\u2081 g\u2082 : imageSheaf f \u27f6 G'\ne : toImageSheaf f \u226b g\u2081 = toImageSheaf f \u226b g\u2082\nU : C\u1d52\u1d56\ns : F'.val.obj U\nhx : s \u2208 Subpresheaf.obj (Subpresheaf.sheafify J (imagePresheaf f.val)) U\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f_1 : Y \u27f6 U.unop\u2984,\n (Subpresheaf.sieveOfSection (imagePresheaf f.val) s).arrows f_1 \u2192\n G'.val.map f_1.op (NatTrans.app g\u2081.val U { val := s, property := hx }) =\n G'.val.map f_1.op (NatTrans.app g\u2082.val U { val := s, property := hx })\n[PROOFSTEP]\nrintro V i \u27e8y, e'\u27e9\n[GOAL]\ncase h.w.h.h.mk.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G'\u271d : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\nG' : Sheaf J (Type w)\ng\u2081 g\u2082 : imageSheaf f \u27f6 G'\ne : toImageSheaf f \u226b g\u2081 = toImageSheaf f \u226b g\u2082\nU : C\u1d52\u1d56\ns : F'.val.obj U\nhx : s \u2208 Subpresheaf.obj (Subpresheaf.sheafify J (imagePresheaf f.val)) U\nV : C\ni : V \u27f6 U.unop\ny : F.val.obj (op V)\ne' : NatTrans.app f.val (op V) y = F'.val.map i.op s\n\u22a2 G'.val.map i.op (NatTrans.app g\u2081.val U { val := s, property := hx }) =\n G'.val.map i.op (NatTrans.app g\u2082.val U { val := s, property := hx })\n[PROOFSTEP]\nchange (g\u2081.val.app _ \u226b G'.val.map _) _ = (g\u2082.val.app _ \u226b G'.val.map _) _\n[GOAL]\ncase h.w.h.h.mk.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G'\u271d : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\nG' : Sheaf J (Type w)\ng\u2081 g\u2082 : imageSheaf f \u27f6 G'\ne : toImageSheaf f \u226b g\u2081 = toImageSheaf f \u226b g\u2082\nU : C\u1d52\u1d56\ns : F'.val.obj U\nhx : s \u2208 Subpresheaf.obj (Subpresheaf.sheafify J (imagePresheaf f.val)) U\nV : C\ni : V \u27f6 U.unop\ny : F.val.obj (op V)\ne' : NatTrans.app f.val (op V) y = F'.val.map i.op s\n\u22a2 (NatTrans.app g\u2081.val (op U.unop) \u226b G'.val.map i.op) { val := s, property := hx } =\n (NatTrans.app g\u2082.val (op U.unop) \u226b G'.val.map i.op) { val := s, property := hx }\n[PROOFSTEP]\nrw [\u2190 NatTrans.naturality, \u2190 NatTrans.naturality]\n[GOAL]\ncase h.w.h.h.mk.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G'\u271d : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\nG' : Sheaf J (Type w)\ng\u2081 g\u2082 : imageSheaf f \u27f6 G'\ne : toImageSheaf f \u226b g\u2081 = toImageSheaf f \u226b g\u2082\nU : C\u1d52\u1d56\ns : F'.val.obj U\nhx : s \u2208 Subpresheaf.obj (Subpresheaf.sheafify J (imagePresheaf f.val)) U\nV : C\ni : V \u27f6 U.unop\ny : F.val.obj (op V)\ne' : NatTrans.app f.val (op V) y = F'.val.map i.op s\n\u22a2 ((imageSheaf f).val.map i.op \u226b NatTrans.app g\u2081.val (op V)) { val := s, property := hx } =\n ((imageSheaf f).val.map i.op \u226b NatTrans.app g\u2082.val (op V)) { val := s, property := hx }\n[PROOFSTEP]\nhave E : (toImageSheaf f).val.app (op V) y = (imageSheaf f).val.map i.op \u27e8s, hx\u27e9 := Subtype.ext e'\n[GOAL]\ncase h.w.h.h.mk.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G'\u271d : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\nG' : Sheaf J (Type w)\ng\u2081 g\u2082 : imageSheaf f \u27f6 G'\ne : toImageSheaf f \u226b g\u2081 = toImageSheaf f \u226b g\u2082\nU : C\u1d52\u1d56\ns : F'.val.obj U\nhx : s \u2208 Subpresheaf.obj (Subpresheaf.sheafify J (imagePresheaf f.val)) U\nV : C\ni : V \u27f6 U.unop\ny : F.val.obj (op V)\ne' : NatTrans.app f.val (op V) y = F'.val.map i.op s\nE : NatTrans.app (toImageSheaf f).val (op V) y = (imageSheaf f).val.map i.op { val := s, property := hx }\n\u22a2 ((imageSheaf f).val.map i.op \u226b NatTrans.app g\u2081.val (op V)) { val := s, property := hx } =\n ((imageSheaf f).val.map i.op \u226b NatTrans.app g\u2082.val (op V)) { val := s, property := hx }\n[PROOFSTEP]\nhave := congr_arg (fun f : F \u27f6 G' => (Sheaf.Hom.val f).app _ y) e\n[GOAL]\ncase h.w.h.h.mk.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G'\u271d : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\nG' : Sheaf J (Type w)\ng\u2081 g\u2082 : imageSheaf f \u27f6 G'\ne : toImageSheaf f \u226b g\u2081 = toImageSheaf f \u226b g\u2082\nU : C\u1d52\u1d56\ns : F'.val.obj U\nhx : s \u2208 Subpresheaf.obj (Subpresheaf.sheafify J (imagePresheaf f.val)) U\nV : C\ni : V \u27f6 U.unop\ny : F.val.obj (op V)\ne' : NatTrans.app f.val (op V) y = F'.val.map i.op s\nE : NatTrans.app (toImageSheaf f).val (op V) y = (imageSheaf f).val.map i.op { val := s, property := hx }\nthis :\n (fun f => NatTrans.app f.val (op V) y) (toImageSheaf f \u226b g\u2081) =\n (fun f => NatTrans.app f.val (op V) y) (toImageSheaf f \u226b g\u2082)\n\u22a2 ((imageSheaf f).val.map i.op \u226b NatTrans.app g\u2081.val (op V)) { val := s, property := hx } =\n ((imageSheaf f).val.map i.op \u226b NatTrans.app g\u2082.val (op V)) { val := s, property := hx }\n[PROOFSTEP]\ndsimp at this \u22a2\n[GOAL]\ncase h.w.h.h.mk.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G'\u271d : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\nG' : Sheaf J (Type w)\ng\u2081 g\u2082 : imageSheaf f \u27f6 G'\ne : toImageSheaf f \u226b g\u2081 = toImageSheaf f \u226b g\u2082\nU : C\u1d52\u1d56\ns : F'.val.obj U\nhx : s \u2208 Subpresheaf.obj (Subpresheaf.sheafify J (imagePresheaf f.val)) U\nV : C\ni : V \u27f6 U.unop\ny : F.val.obj (op V)\ne' : NatTrans.app f.val (op V) y = F'.val.map i.op s\nE : NatTrans.app (toImageSheaf f).val (op V) y = (imageSheaf f).val.map i.op { val := s, property := hx }\nthis :\n NatTrans.app g\u2081.val (op V) (NatTrans.app (toImagePresheafSheafify J f.val) (op V) y) =\n NatTrans.app g\u2082.val (op V) (NatTrans.app (toImagePresheafSheafify J f.val) (op V) y)\n\u22a2 NatTrans.app g\u2081.val (op V)\n ((Subpresheaf.toPresheaf (Subpresheaf.sheafify J (imagePresheaf f.val))).map i.op { val := s, property := hx }) =\n NatTrans.app g\u2082.val (op V)\n ((Subpresheaf.toPresheaf (Subpresheaf.sheafify J (imagePresheaf f.val))).map i.op { val := s, property := hx })\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_2.h.e'_5\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G'\u271d : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\nG' : Sheaf J (Type w)\ng\u2081 g\u2082 : imageSheaf f \u27f6 G'\ne : toImageSheaf f \u226b g\u2081 = toImageSheaf f \u226b g\u2082\nU : C\u1d52\u1d56\ns : F'.val.obj U\nhx : s \u2208 Subpresheaf.obj (Subpresheaf.sheafify J (imagePresheaf f.val)) U\nV : C\ni : V \u27f6 U.unop\ny : F.val.obj (op V)\ne' : NatTrans.app f.val (op V) y = F'.val.map i.op s\nE : NatTrans.app (toImageSheaf f).val (op V) y = (imageSheaf f).val.map i.op { val := s, property := hx }\nthis :\n NatTrans.app g\u2081.val (op V) (NatTrans.app (toImagePresheafSheafify J f.val) (op V) y) =\n NatTrans.app g\u2082.val (op V) (NatTrans.app (toImagePresheafSheafify J f.val) (op V) y)\n\u22a2 (Subpresheaf.toPresheaf (Subpresheaf.sheafify J (imagePresheaf f.val))).map i.op { val := s, property := hx } =\n NatTrans.app (toImagePresheafSheafify J f.val) (op V) y\n[PROOFSTEP]\nexact E.symm\n[GOAL]\ncase h.e'_3.h.e'_5\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G'\u271d : Subpresheaf F\u271d\nF F' : Sheaf J (Type w)\nf : F \u27f6 F'\nG' : Sheaf J (Type w)\ng\u2081 g\u2082 : imageSheaf f \u27f6 G'\ne : toImageSheaf f \u226b g\u2081 = toImageSheaf f \u226b g\u2082\nU : C\u1d52\u1d56\ns : F'.val.obj U\nhx : s \u2208 Subpresheaf.obj (Subpresheaf.sheafify J (imagePresheaf f.val)) U\nV : C\ni : V \u27f6 U.unop\ny : F.val.obj (op V)\ne' : NatTrans.app f.val (op V) y = F'.val.map i.op s\nE : NatTrans.app (toImageSheaf f).val (op V) y = (imageSheaf f).val.map i.op { val := s, property := hx }\nthis :\n NatTrans.app g\u2081.val (op V) (NatTrans.app (toImagePresheafSheafify J f.val) (op V) y) =\n NatTrans.app g\u2082.val (op V) (NatTrans.app (toImagePresheafSheafify J f.val) (op V) y)\n\u22a2 (Subpresheaf.toPresheaf (Subpresheaf.sheafify J (imagePresheaf f.val))).map i.op { val := s, property := hx } =\n NatTrans.app (toImagePresheafSheafify J f.val) (op V) y\n[PROOFSTEP]\nexact E.symm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\n\u22a2 (imageMonoFactorization f).I \u27f6 I.I\n[PROOFSTEP]\nhaveI M := (Sheaf.Hom.mono_iff_presheaf_mono J TypeMax.{v, u} _).mp I.m_mono\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\nM : Mono I.m.val\n\u22a2 (imageMonoFactorization f).I \u27f6 I.I\n[PROOFSTEP]\nhaveI := isIso_toImagePresheaf I.m.1\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\nM : Mono I.m.val\nthis : IsIso (toImagePresheaf I.m.val)\n\u22a2 (imageMonoFactorization f).I \u27f6 I.I\n[PROOFSTEP]\nrefine' \u27e8Subpresheaf.homOfLe _ \u226b inv (toImagePresheaf I.m.1)\u27e9\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\nM : Mono I.m.val\nthis : IsIso (toImagePresheaf I.m.val)\n\u22a2 Subpresheaf.sheafify J (imagePresheaf f.val) \u2264 imagePresheaf I.m.val\n[PROOFSTEP]\napply Subpresheaf.sheafify_le\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\nM : Mono I.m.val\nthis : IsIso (toImagePresheaf I.m.val)\n\u22a2 imagePresheaf f.val \u2264 imagePresheaf I.m.val\n[PROOFSTEP]\nconv_lhs => rw [\u2190 I.fac]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\nM : Mono I.m.val\nthis : IsIso (toImagePresheaf I.m.val)\n| imagePresheaf f.val\n[PROOFSTEP]\nrw [\u2190 I.fac]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\nM : Mono I.m.val\nthis : IsIso (toImagePresheaf I.m.val)\n| imagePresheaf f.val\n[PROOFSTEP]\nrw [\u2190 I.fac]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\nM : Mono I.m.val\nthis : IsIso (toImagePresheaf I.m.val)\n| imagePresheaf f.val\n[PROOFSTEP]\nrw [\u2190 I.fac]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\nM : Mono I.m.val\nthis : IsIso (toImagePresheaf I.m.val)\n\u22a2 imagePresheaf (I.e \u226b I.m).val \u2264 imagePresheaf I.m.val\n[PROOFSTEP]\napply imagePresheaf_comp_le\n[GOAL]\ncase hF\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\nM : Mono I.m.val\nthis : IsIso (toImagePresheaf I.m.val)\n\u22a2 Presieve.IsSheaf J F'.val\n[PROOFSTEP]\nrw [\u2190 isSheaf_iff_isSheaf_of_type]\n[GOAL]\ncase hF\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\nM : Mono I.m.val\nthis : IsIso (toImagePresheaf I.m.val)\n\u22a2 Presheaf.IsSheaf J F'.val\n[PROOFSTEP]\nexact F'.2\n[GOAL]\ncase hG'\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\nM : Mono I.m.val\nthis : IsIso (toImagePresheaf I.m.val)\n\u22a2 Presieve.IsSheaf J (Subpresheaf.toPresheaf (imagePresheaf I.m.val))\n[PROOFSTEP]\napply Presieve.isSheaf_iso J (asIso <| toImagePresheaf I.m.1)\n[GOAL]\ncase hG'\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\nM : Mono I.m.val\nthis : IsIso (toImagePresheaf I.m.val)\n\u22a2 Presieve.IsSheaf J I.I.val\n[PROOFSTEP]\nrw [\u2190 isSheaf_iff_isSheaf_of_type]\n[GOAL]\ncase hG'\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\nM : Mono I.m.val\nthis : IsIso (toImagePresheaf I.m.val)\n\u22a2 Presheaf.IsSheaf J I.I.val\n[PROOFSTEP]\nexact I.I.2\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\n\u22a2 (fun I =>\n {\n val :=\n Subpresheaf.homOfLe (_ : Subpresheaf.sheafify J (imagePresheaf f.val) \u2264 imagePresheaf I.m.val) \u226b\n inv (toImagePresheaf I.m.val) })\n I \u226b\n I.m =\n (imageMonoFactorization f).m\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\n\u22a2 ((fun I =>\n {\n val :=\n Subpresheaf.homOfLe (_ : Subpresheaf.sheafify J (imagePresheaf f.val) \u2264 imagePresheaf I.m.val) \u226b\n inv (toImagePresheaf I.m.val) })\n I \u226b\n I.m).val =\n (imageMonoFactorization f).m.val\n[PROOFSTEP]\ndsimp [imageMonoFactorization]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\n\u22a2 (Subpresheaf.homOfLe (_ : Subpresheaf.sheafify J (imagePresheaf f.val) \u2264 imagePresheaf I.m.val) \u226b\n inv (toImagePresheaf I.m.val)) \u226b\n I.m.val =\n Subpresheaf.\u03b9 (Subpresheaf.sheafify J (imagePresheaf f.val))\n[PROOFSTEP]\ngeneralize_proofs h\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\nh : Subpresheaf.sheafify J (imagePresheaf f.val) \u2264 imagePresheaf I.m.val\nh\u271d : IsIso (toImagePresheaf I.m.val)\n\u22a2 (Subpresheaf.homOfLe (_ : Subpresheaf.sheafify J (imagePresheaf f.val) \u2264 imagePresheaf I.m.val) \u226b\n inv (toImagePresheaf I.m.val)) \u226b\n I.m.val =\n Subpresheaf.\u03b9 (Subpresheaf.sheafify J (imagePresheaf f.val))\n[PROOFSTEP]\nrw [\u2190 Subpresheaf.homOfLe_\u03b9 h, Category.assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\nh : Subpresheaf.sheafify J (imagePresheaf f.val) \u2264 imagePresheaf I.m.val\nh\u271d : IsIso (toImagePresheaf I.m.val)\n\u22a2 Subpresheaf.homOfLe (_ : Subpresheaf.sheafify J (imagePresheaf f.val) \u2264 imagePresheaf I.m.val) \u226b\n inv (toImagePresheaf I.m.val) \u226b I.m.val =\n Subpresheaf.homOfLe h \u226b Subpresheaf.\u03b9 (imagePresheaf I.m.val)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e_a\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : GrothendieckTopology C\nF\u271d F'\u271d F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\u271d\nF F' : Sheaf J TypeMax\nf : F \u27f6 F'\nI : Limits.MonoFactorisation f\nh : Subpresheaf.sheafify J (imagePresheaf f.val) \u2264 imagePresheaf I.m.val\nh\u271d : IsIso (toImagePresheaf I.m.val)\n\u22a2 inv (toImagePresheaf I.m.val) \u226b I.m.val = Subpresheaf.\u03b9 (imagePresheaf I.m.val)\n[PROOFSTEP]\nrw [IsIso.inv_comp_eq, toImagePresheaf_\u03b9]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.Subsheaf", "llama_tokens": 57161, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.3923368301671084, "lm_q1q2_score": 0.2053570805466392}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d : LE \u03b1\ns\u271d t\u271d s t : NonemptyInterval \u03b1\nh : s.toProd = t.toProd\n\u22a2 s = t\n[PROOFSTEP]\ncases s\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d : LE \u03b1\ns t\u271d t : NonemptyInterval \u03b1\ntoProd\u271d : \u03b1 \u00d7 \u03b1\nfst_le_snd\u271d : toProd\u271d.fst \u2264 toProd\u271d.snd\nh : { toProd := toProd\u271d, fst_le_snd := fst_le_snd\u271d }.toProd = t.toProd\n\u22a2 { toProd := toProd\u271d, fst_le_snd := fst_le_snd\u271d } = t\n[PROOFSTEP]\ncases t\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d : LE \u03b1\ns t : NonemptyInterval \u03b1\ntoProd\u271d\u00b9 : \u03b1 \u00d7 \u03b1\nfst_le_snd\u271d\u00b9 : toProd\u271d\u00b9.fst \u2264 toProd\u271d\u00b9.snd\ntoProd\u271d : \u03b1 \u00d7 \u03b1\nfst_le_snd\u271d : toProd\u271d.fst \u2264 toProd\u271d.snd\nh : { toProd := toProd\u271d\u00b9, fst_le_snd := fst_le_snd\u271d\u00b9 }.toProd = { toProd := toProd\u271d, fst_le_snd := fst_le_snd\u271d }.toProd\n\u22a2 { toProd := toProd\u271d\u00b9, fst_le_snd := fst_le_snd\u271d\u00b9 } = { toProd := toProd\u271d, fst_le_snd := fst_le_snd\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : Preorder \u03b3\ninst\u271d\u00b9 : Preorder \u03b4\ns : NonemptyInterval \u03b1\nx : \u03b1 \u00d7 \u03b1\na : \u03b1\ninst\u271d : Nonempty \u03b1\n\u22a2 Nonempty \u03b1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\ns t : NonemptyInterval \u03b1\nx : \u03b1 \u00d7 \u03b1\na b : \u03b1\n\u22a2 b \u2208 pure a \u2194 b = a\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, coe_pure, mem_singleton_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192o \u03b2\ns : Interval \u03b1\n\u22a2 \u2191dual (map f s) = map (\u2191OrderHom.dual f) (\u2191dual s)\n[PROOFSTEP]\ncases s\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192o \u03b2\n\u22a2 \u2191dual (map f none) = map (\u2191OrderHom.dual f) (\u2191dual none)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192o \u03b2\nval\u271d : NonemptyInterval \u03b1\n\u22a2 \u2191dual (map f (some val\u271d)) = map (\u2191OrderHom.dual f) (\u2191dual (some val\u271d))\n[PROOFSTEP]\nexact WithBot.map_comm rfl _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\ns\u271d t : Interval \u03b1\na b : \u03b1\ns : Interval \u03b1\n\u22a2 \u2191(\u2191dual s) = \u2191ofDual \u207b\u00b9' \u2191s\n[PROOFSTEP]\ncases s with\n| none => rfl\n| some s\u2080 => exact NonemptyInterval.coe_dual s\u2080\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\ns\u271d t : Interval \u03b1\na b : \u03b1\ns : Interval \u03b1\n\u22a2 \u2191(\u2191dual s) = \u2191ofDual \u207b\u00b9' \u2191s\n[PROOFSTEP]\ncases s with\n| none => rfl\n| some s\u2080 => exact NonemptyInterval.coe_dual s\u2080\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\ns t : Interval \u03b1\na b : \u03b1\n\u22a2 \u2191(\u2191dual none) = \u2191ofDual \u207b\u00b9' \u2191none\n[PROOFSTEP]\n\n| none => rfl\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\ns t : Interval \u03b1\na b : \u03b1\n\u22a2 \u2191(\u2191dual none) = \u2191ofDual \u207b\u00b9' \u2191none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\ns t : Interval \u03b1\na b : \u03b1\ns\u2080 : NonemptyInterval \u03b1\n\u22a2 \u2191(\u2191dual (some s\u2080)) = \u2191ofDual \u207b\u00b9' \u2191(some s\u2080)\n[PROOFSTEP]\n\n| some s\u2080 => exact NonemptyInterval.coe_dual s\u2080\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\ns t : Interval \u03b1\na b : \u03b1\ns\u2080 : NonemptyInterval \u03b1\n\u22a2 \u2191(\u2191dual (some s\u2080)) = \u2191ofDual \u207b\u00b9' \u2191(some s\u2080)\n[PROOFSTEP]\nexact NonemptyInterval.coe_dual s\u2080\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\ns t : Interval \u03b1\na b : \u03b1\nf : \u03b1 \u2192o \u03b2\n\u22a2 \u2191f '' \u2191\u22a5 \u2286 \u2191(map f \u22a5)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\ns t : Interval \u03b1\na b : \u03b1\n\u22a2 b \u2208 pure a \u2194 b = a\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, coe_pure, mem_singleton_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d : Interval \u03b1\ns t : NonemptyInterval \u03b1\n\u22a2 some s \u2293 some t \u2264 some s\n[PROOFSTEP]\nchange dite _ _ _ \u2264 _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d : Interval \u03b1\ns t : NonemptyInterval \u03b1\n\u22a2 (if h : s.fst \u2264 t.snd \u2227 t.fst \u2264 s.snd then\n some\n { toProd := (s.fst \u2294 t.fst, s.snd \u2293 t.snd),\n fst_le_snd := (_ : s.fst \u2294 t.fst \u2264 (s.fst \u2294 t.fst, s.snd \u2293 t.snd).snd) }\n else \u22a5) \u2264\n some s\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d : Interval \u03b1\ns t : NonemptyInterval \u03b1\nh\u271d : s.fst \u2264 t.snd \u2227 t.fst \u2264 s.snd\n\u22a2 some\n { toProd := (s.fst \u2294 t.fst, s.snd \u2293 t.snd),\n fst_le_snd := (_ : s.fst \u2294 t.fst \u2264 (s.fst \u2294 t.fst, s.snd \u2293 t.snd).snd) } \u2264\n some s\n[PROOFSTEP]\nexact WithBot.some_le_some.2 \u27e8le_sup_left, inf_le_left\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d : Interval \u03b1\ns t : NonemptyInterval \u03b1\nh\u271d : \u00ac(s.fst \u2264 t.snd \u2227 t.fst \u2264 s.snd)\n\u22a2 \u22a5 \u2264 some s\n[PROOFSTEP]\nexact bot_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d : Interval \u03b1\ns t : NonemptyInterval \u03b1\n\u22a2 some s \u2293 some t \u2264 some t\n[PROOFSTEP]\nchange dite _ _ _ \u2264 _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d : Interval \u03b1\ns t : NonemptyInterval \u03b1\n\u22a2 (if h : s.fst \u2264 t.snd \u2227 t.fst \u2264 s.snd then\n some\n { toProd := (s.fst \u2294 t.fst, s.snd \u2293 t.snd),\n fst_le_snd := (_ : s.fst \u2294 t.fst \u2264 (s.fst \u2294 t.fst, s.snd \u2293 t.snd).snd) }\n else \u22a5) \u2264\n some t\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d : Interval \u03b1\ns t : NonemptyInterval \u03b1\nh\u271d : s.fst \u2264 t.snd \u2227 t.fst \u2264 s.snd\n\u22a2 some\n { toProd := (s.fst \u2294 t.fst, s.snd \u2293 t.snd),\n fst_le_snd := (_ : s.fst \u2294 t.fst \u2264 (s.fst \u2294 t.fst, s.snd \u2293 t.snd).snd) } \u2264\n some t\n[PROOFSTEP]\nexact WithBot.some_le_some.2 \u27e8le_sup_right, inf_le_right\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d : Interval \u03b1\ns t : NonemptyInterval \u03b1\nh\u271d : \u00ac(s.fst \u2264 t.snd \u2227 t.fst \u2264 s.snd)\n\u22a2 \u22a5 \u2264 some t\n[PROOFSTEP]\nexact bot_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d c\u271d : Interval \u03b1\ns : NonemptyInterval \u03b1\nt c : Interval \u03b1\nhb : some s \u2264 t\nhc : some s \u2264 c\n\u22a2 some s \u2264 t \u2293 c\n[PROOFSTEP]\nlift t to NonemptyInterval \u03b1 using ne_bot_of_le_ne_bot WithBot.coe_ne_bot hb\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d c\u271d : Interval \u03b1\ns : NonemptyInterval \u03b1\nc : Interval \u03b1\nhc : some s \u2264 c\nt : NonemptyInterval \u03b1\nhb : some s \u2264 \u2191t\n\u22a2 some s \u2264 \u2191t \u2293 c\n[PROOFSTEP]\nlift c to NonemptyInterval \u03b1 using ne_bot_of_le_ne_bot WithBot.coe_ne_bot hc\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d c\u271d : Interval \u03b1\ns t : NonemptyInterval \u03b1\nhb : some s \u2264 \u2191t\nc : NonemptyInterval \u03b1\nhc : some s \u2264 \u2191c\n\u22a2 some s \u2264 \u2191t \u2293 \u2191c\n[PROOFSTEP]\nchange\n _ \u2264\n dite _ _\n _\n -- Porting note: was `simp only` but that fails to use the second lemma.\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d c\u271d : Interval \u03b1\ns t : NonemptyInterval \u03b1\nhb : some s \u2264 \u2191t\nc : NonemptyInterval \u03b1\nhc : some s \u2264 \u2191c\n\u22a2 some s \u2264\n if h : t.fst \u2264 c.snd \u2227 c.fst \u2264 t.snd then\n some\n { toProd := (t.fst \u2294 c.fst, t.snd \u2293 c.snd),\n fst_le_snd := (_ : t.fst \u2294 c.fst \u2264 (t.fst \u2294 c.fst, t.snd \u2293 c.snd).snd) }\n else \u22a5\n[PROOFSTEP]\nrw [WithBot.some_eq_coe, WithBot.coe_le_coe] at hb hc \n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d c\u271d : Interval \u03b1\ns t : NonemptyInterval \u03b1\nhb : s \u2264 t\nc : NonemptyInterval \u03b1\nhc : s \u2264 c\n\u22a2 some s \u2264\n if h : t.fst \u2264 c.snd \u2227 c.fst \u2264 t.snd then\n some\n { toProd := (t.fst \u2294 c.fst, t.snd \u2293 c.snd),\n fst_le_snd := (_ : t.fst \u2294 c.fst \u2264 (t.fst \u2294 c.fst, t.snd \u2293 c.snd).snd) }\n else \u22a5\n[PROOFSTEP]\nsimp only [WithBot.some_eq_coe, WithBot.coe_le_coe]\n -- at hb hc \u22a2\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d c\u271d : Interval \u03b1\ns t : NonemptyInterval \u03b1\nhb : s \u2264 t\nc : NonemptyInterval \u03b1\nhc : s \u2264 c\n\u22a2 \u2191s \u2264\n if h : t.fst \u2264 c.snd \u2227 c.fst \u2264 t.snd then\n \u2191{ toProd := (t.fst \u2294 c.fst, t.snd \u2293 c.snd),\n fst_le_snd := (_ : t.fst \u2294 c.fst \u2264 (t.fst \u2294 c.fst, t.snd \u2293 c.snd).snd) }\n else \u22a5\n[PROOFSTEP]\nrw [dif_pos, WithBot.coe_le_coe]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d c\u271d : Interval \u03b1\ns t : NonemptyInterval \u03b1\nhb : s \u2264 t\nc : NonemptyInterval \u03b1\nhc : s \u2264 c\n\u22a2 s \u2264\n { toProd := (t.fst \u2294 c.fst, t.snd \u2293 c.snd), fst_le_snd := (_ : t.fst \u2294 c.fst \u2264 (t.fst \u2294 c.fst, t.snd \u2293 c.snd).snd) }\ncase intro.intro.hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d c\u271d : Interval \u03b1\ns t : NonemptyInterval \u03b1\nhb : s \u2264 t\nc : NonemptyInterval \u03b1\nhc : s \u2264 c\n\u22a2 t.fst \u2264 c.snd \u2227 c.fst \u2264 t.snd\ncase intro.intro.hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d c\u271d : Interval \u03b1\ns t : NonemptyInterval \u03b1\nhb : s \u2264 t\nc : NonemptyInterval \u03b1\nhc : s \u2264 c\n\u22a2 t.fst \u2264 c.snd \u2227 c.fst \u2264 t.snd\n[PROOFSTEP]\nexact\n \u27e8sup_le hb.1 hc.1, le_inf hb.2 hc.2\u27e9\n -- Porting note: had to add the next 6 lines including the changes because\n -- it seems that lean cannot automatically turn `NonemptyInterval.toDualProd s`\n -- into `s.toProd` anymore.\n[GOAL]\ncase intro.intro.hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d c\u271d : Interval \u03b1\ns t : NonemptyInterval \u03b1\nhb : s \u2264 t\nc : NonemptyInterval \u03b1\nhc : s \u2264 c\n\u22a2 t.fst \u2264 c.snd \u2227 c.fst \u2264 t.snd\ncase intro.intro.hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d c\u271d : Interval \u03b1\ns t : NonemptyInterval \u03b1\nhb : s \u2264 t\nc : NonemptyInterval \u03b1\nhc : s \u2264 c\n\u22a2 t.fst \u2264 c.snd \u2227 c.fst \u2264 t.snd\n[PROOFSTEP]\nrcases hb with \u27e8hb\u2081, hb\u2082\u27e9\n[GOAL]\ncase intro.intro.hc.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d c\u271d : Interval \u03b1\ns t c : NonemptyInterval \u03b1\nhc : s \u2264 c\nhb\u2081 : (NonemptyInterval.toDualProd s).fst \u2264 (NonemptyInterval.toDualProd t).fst\nhb\u2082 : (NonemptyInterval.toDualProd s).snd \u2264 (NonemptyInterval.toDualProd t).snd\n\u22a2 t.fst \u2264 c.snd \u2227 c.fst \u2264 t.snd\n[PROOFSTEP]\nrcases hc with \u27e8hc\u2081, hc\u2082\u27e9\n[GOAL]\ncase intro.intro.hc.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d c\u271d : Interval \u03b1\ns t c : NonemptyInterval \u03b1\nhb\u2081 : (NonemptyInterval.toDualProd s).fst \u2264 (NonemptyInterval.toDualProd t).fst\nhb\u2082 : (NonemptyInterval.toDualProd s).snd \u2264 (NonemptyInterval.toDualProd t).snd\nhc\u2081 : (NonemptyInterval.toDualProd s).fst \u2264 (NonemptyInterval.toDualProd c).fst\nhc\u2082 : (NonemptyInterval.toDualProd s).snd \u2264 (NonemptyInterval.toDualProd c).snd\n\u22a2 t.fst \u2264 c.snd \u2227 c.fst \u2264 t.snd\n[PROOFSTEP]\nchange t.toProd.fst \u2264 s.toProd.fst at hb\u2081 \n[GOAL]\ncase intro.intro.hc.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d c\u271d : Interval \u03b1\ns t c : NonemptyInterval \u03b1\nhb\u2082 : (NonemptyInterval.toDualProd s).snd \u2264 (NonemptyInterval.toDualProd t).snd\nhc\u2081 : (NonemptyInterval.toDualProd s).fst \u2264 (NonemptyInterval.toDualProd c).fst\nhc\u2082 : (NonemptyInterval.toDualProd s).snd \u2264 (NonemptyInterval.toDualProd c).snd\nhb\u2081 : t.fst \u2264 s.fst\n\u22a2 t.fst \u2264 c.snd \u2227 c.fst \u2264 t.snd\n[PROOFSTEP]\nchange s.toProd.snd \u2264 t.toProd.snd at hb\u2082 \n[GOAL]\ncase intro.intro.hc.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d c\u271d : Interval \u03b1\ns t c : NonemptyInterval \u03b1\nhc\u2081 : (NonemptyInterval.toDualProd s).fst \u2264 (NonemptyInterval.toDualProd c).fst\nhc\u2082 : (NonemptyInterval.toDualProd s).snd \u2264 (NonemptyInterval.toDualProd c).snd\nhb\u2081 : t.fst \u2264 s.fst\nhb\u2082 : s.snd \u2264 t.snd\n\u22a2 t.fst \u2264 c.snd \u2227 c.fst \u2264 t.snd\n[PROOFSTEP]\nchange c.toProd.fst \u2264 s.toProd.fst at hc\u2081 \n[GOAL]\ncase intro.intro.hc.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d c\u271d : Interval \u03b1\ns t c : NonemptyInterval \u03b1\nhc\u2082 : (NonemptyInterval.toDualProd s).snd \u2264 (NonemptyInterval.toDualProd c).snd\nhb\u2081 : t.fst \u2264 s.fst\nhb\u2082 : s.snd \u2264 t.snd\nhc\u2081 : c.fst \u2264 s.fst\n\u22a2 t.fst \u2264 c.snd \u2227 c.fst \u2264 t.snd\n[PROOFSTEP]\nchange s.toProd.snd \u2264 c.toProd.snd at hc\u2082 \n[GOAL]\ncase intro.intro.hc.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : SemilatticeSup (Interval \u03b1) := semilatticeSup\ns\u271d t\u271d c\u271d : Interval \u03b1\ns t c : NonemptyInterval \u03b1\nhb\u2081 : t.fst \u2264 s.fst\nhb\u2082 : s.snd \u2264 t.snd\nhc\u2081 : c.fst \u2264 s.fst\nhc\u2082 : s.snd \u2264 c.snd\n\u22a2 t.fst \u2264 c.snd \u2227 c.fst \u2264 t.snd\n[PROOFSTEP]\nexact \u27e8hb\u2081.trans <| s.fst_le_snd.trans hc\u2082, hc\u2081.trans <| s.fst_le_snd.trans hb\u2082\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\ns t : Interval \u03b1\n\u22a2 \u2191(s \u2293 t) = \u2191s \u2229 \u2191t\n[PROOFSTEP]\ncases s with\n| none =>\n rw [WithBot.none_eq_bot, bot_inf_eq]\n exact (empty_inter _).symm\n| some s =>\n cases t with\n | none =>\n rw [WithBot.none_eq_bot, inf_bot_eq]\n exact (inter_empty _).symm\n |\n some t =>\n refine'\n (_ : setLike.coe (dite (s.toProd.fst \u2264 t.toProd.snd \u2227 t.toProd.fst \u2264 s.toProd.snd) _ _) = _).trans\n Icc_inter_Icc.symm\n split_ifs with h\n \u00b7 rfl\n \u00b7\n exact\n (Icc_eq_empty fun H =>\n h \u27e8le_sup_left.trans <| H.trans inf_le_right, le_sup_right.trans <| H.trans inf_le_left\u27e9).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\ns t : Interval \u03b1\n\u22a2 \u2191(s \u2293 t) = \u2191s \u2229 \u2191t\n[PROOFSTEP]\ncases s with\n| none =>\n rw [WithBot.none_eq_bot, bot_inf_eq]\n exact (empty_inter _).symm\n| some s =>\n cases t with\n | none =>\n rw [WithBot.none_eq_bot, inf_bot_eq]\n exact (inter_empty _).symm\n |\n some t =>\n refine'\n (_ : setLike.coe (dite (s.toProd.fst \u2264 t.toProd.snd \u2227 t.toProd.fst \u2264 s.toProd.snd) _ _) = _).trans\n Icc_inter_Icc.symm\n split_ifs with h\n \u00b7 rfl\n \u00b7\n exact\n (Icc_eq_empty fun H =>\n h \u27e8le_sup_left.trans <| H.trans inf_le_right, le_sup_right.trans <| H.trans inf_le_left\u27e9).symm\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nt : Interval \u03b1\n\u22a2 \u2191(none \u2293 t) = \u2191none \u2229 \u2191t\n[PROOFSTEP]\n\n| none =>\n rw [WithBot.none_eq_bot, bot_inf_eq]\n exact (empty_inter _).symm\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nt : Interval \u03b1\n\u22a2 \u2191(none \u2293 t) = \u2191none \u2229 \u2191t\n[PROOFSTEP]\nrw [WithBot.none_eq_bot, bot_inf_eq]\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nt : Interval \u03b1\n\u22a2 \u2191\u22a5 = \u2191\u22a5 \u2229 \u2191t\n[PROOFSTEP]\nexact (empty_inter _).symm\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nt : Interval \u03b1\ns : NonemptyInterval \u03b1\n\u22a2 \u2191(some s \u2293 t) = \u2191(some s) \u2229 \u2191t\n[PROOFSTEP]\n\n| some s =>\n cases t with\n | none =>\n rw [WithBot.none_eq_bot, inf_bot_eq]\n exact (inter_empty _).symm\n |\n some t =>\n refine'\n (_ : setLike.coe (dite (s.toProd.fst \u2264 t.toProd.snd \u2227 t.toProd.fst \u2264 s.toProd.snd) _ _) = _).trans\n Icc_inter_Icc.symm\n split_ifs with h\n \u00b7 rfl\n \u00b7\n exact\n (Icc_eq_empty fun H =>\n h \u27e8le_sup_left.trans <| H.trans inf_le_right, le_sup_right.trans <| H.trans inf_le_left\u27e9).symm\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nt : Interval \u03b1\ns : NonemptyInterval \u03b1\n\u22a2 \u2191(some s \u2293 t) = \u2191(some s) \u2229 \u2191t\n[PROOFSTEP]\ncases t with\n| none =>\n rw [WithBot.none_eq_bot, inf_bot_eq]\n exact (inter_empty _).symm\n|\n some t =>\n refine'\n (_ : setLike.coe (dite (s.toProd.fst \u2264 t.toProd.snd \u2227 t.toProd.fst \u2264 s.toProd.snd) _ _) = _).trans\n Icc_inter_Icc.symm\n split_ifs with h\n \u00b7 rfl\n \u00b7\n exact\n (Icc_eq_empty fun H =>\n h \u27e8le_sup_left.trans <| H.trans inf_le_right, le_sup_right.trans <| H.trans inf_le_left\u27e9).symm\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nt : Interval \u03b1\ns : NonemptyInterval \u03b1\n\u22a2 \u2191(some s \u2293 t) = \u2191(some s) \u2229 \u2191t\n[PROOFSTEP]\ncases t with\n| none =>\n rw [WithBot.none_eq_bot, inf_bot_eq]\n exact (inter_empty _).symm\n|\n some t =>\n refine'\n (_ : setLike.coe (dite (s.toProd.fst \u2264 t.toProd.snd \u2227 t.toProd.fst \u2264 s.toProd.snd) _ _) = _).trans\n Icc_inter_Icc.symm\n split_ifs with h\n \u00b7 rfl\n \u00b7\n exact\n (Icc_eq_empty fun H =>\n h \u27e8le_sup_left.trans <| H.trans inf_le_right, le_sup_right.trans <| H.trans inf_le_left\u27e9).symm\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\ns : NonemptyInterval \u03b1\n\u22a2 \u2191(some s \u2293 none) = \u2191(some s) \u2229 \u2191none\n[PROOFSTEP]\n\n| none =>\n rw [WithBot.none_eq_bot, inf_bot_eq]\n exact (inter_empty _).symm\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\ns : NonemptyInterval \u03b1\n\u22a2 \u2191(some s \u2293 none) = \u2191(some s) \u2229 \u2191none\n[PROOFSTEP]\nrw [WithBot.none_eq_bot, inf_bot_eq]\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\ns : NonemptyInterval \u03b1\n\u22a2 \u2191\u22a5 = \u2191(some s) \u2229 \u2191\u22a5\n[PROOFSTEP]\nexact (inter_empty _).symm\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\ns t : NonemptyInterval \u03b1\n\u22a2 \u2191(some s \u2293 some t) = \u2191(some s) \u2229 \u2191(some t)\n[PROOFSTEP]\n\n|\n some t =>\n refine'\n (_ : setLike.coe (dite (s.toProd.fst \u2264 t.toProd.snd \u2227 t.toProd.fst \u2264 s.toProd.snd) _ _) = _).trans\n Icc_inter_Icc.symm\n split_ifs with h\n \u00b7 rfl\n \u00b7\n exact\n (Icc_eq_empty fun H =>\n h \u27e8le_sup_left.trans <| H.trans inf_le_right, le_sup_right.trans <| H.trans inf_le_left\u27e9).symm\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\ns t : NonemptyInterval \u03b1\n\u22a2 \u2191(some s \u2293 some t) = \u2191(some s) \u2229 \u2191(some t)\n[PROOFSTEP]\nrefine'\n (_ : setLike.coe (dite (s.toProd.fst \u2264 t.toProd.snd \u2227 t.toProd.fst \u2264 s.toProd.snd) _ _) = _).trans Icc_inter_Icc.symm\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\ns t : NonemptyInterval \u03b1\n\u22a2 \u2191(if h : s.fst \u2264 t.snd \u2227 t.fst \u2264 s.snd then\n some\n { toProd := (s.fst \u2294 t.fst, s.snd \u2293 t.snd),\n fst_le_snd := (_ : s.fst \u2294 t.fst \u2264 (s.fst \u2294 t.fst, s.snd \u2293 t.snd).snd) }\n else \u22a5) =\n Icc (s.fst \u2294 t.fst) (s.snd \u2293 t.snd)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\ns t : NonemptyInterval \u03b1\nh : s.fst \u2264 t.snd \u2227 t.fst \u2264 s.snd\n\u22a2 \u2191(some\n { toProd := (s.fst \u2294 t.fst, s.snd \u2293 t.snd),\n fst_le_snd := (_ : s.fst \u2294 t.fst \u2264 (s.fst \u2294 t.fst, s.snd \u2293 t.snd).snd) }) =\n Icc (s.fst \u2294 t.fst) (s.snd \u2293 t.snd)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\ns t : NonemptyInterval \u03b1\nh : \u00ac(s.fst \u2264 t.snd \u2227 t.fst \u2264 s.snd)\n\u22a2 \u2191\u22a5 = Icc (s.fst \u2294 t.fst) (s.snd \u2293 t.snd)\n[PROOFSTEP]\nexact\n (Icc_eq_empty fun H => h \u27e8le_sup_left.trans <| H.trans inf_le_right, le_sup_right.trans <| H.trans inf_le_left\u27e9).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d : Lattice \u03b1\ns t : Interval \u03b1\n\u22a2 Disjoint \u2191s \u2191t \u2194 Disjoint s t\n[PROOFSTEP]\nclassical\nrw [disjoint_iff_inf_le, disjoint_iff_inf_le, \u2190 coe_subset_coe, coe_inf]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d : Lattice \u03b1\ns t : Interval \u03b1\n\u22a2 Disjoint \u2191s \u2191t \u2194 Disjoint s t\n[PROOFSTEP]\nrw [disjoint_iff_inf_le, disjoint_iff_inf_le, \u2190 coe_subset_coe, coe_inf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d : Lattice \u03b1\ns t : Interval \u03b1\n\u22a2 \u2191s \u2293 \u2191t \u2264 \u22a5 \u2194 \u2191s \u2229 \u2191t \u2286 \u2191\u22a5\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d : Preorder \u03b1\ns : NonemptyInterval \u03b1\na : \u03b1\n\u22a2 \u2191s = Interval.pure a \u2194 s = pure a\n[PROOFSTEP]\nrw [\u2190 Interval.coe_inj, coe_pure_interval]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\n\u22a2 CompleteLattice (Interval \u03b1)\n[PROOFSTEP]\nclassical exact\n { Interval.lattice,\n Interval.boundedOrder with\n sSup := fun S =>\n if h : S \u2286 {\u22a5} then \u22a5\n else\n some\n \u27e8\u27e8\u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst, \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd\u27e9,\n by\n obtain \u27e8s, hs, ha\u27e9 := not_subset.1 h\n lift s to NonemptyInterval \u03b1 using ha\n exact iInf\u2082_le_of_le s hs (le_iSup\u2082_of_le s hs s.fst_le_snd)\u27e9\n le_sSup := fun s s ha =>\n by\n dsimp only\n -- Porting note: added\n split_ifs with h\n \u00b7 exact (h ha).le\n cases s\n \u00b7 exact bot_le\n \u00b7\n -- Porting note: This case was\n -- `exact WithBot.some_le_some.2 \u27e8iInf\u2082_le _ ha, le_iSup\u2082_of_le _ ha le_rfl\u27e9`\n -- but there seems to be a defEq-problem at `iInf\u2082_le` that lean cannot resolve yet.\n apply WithBot.some_le_some.2\n constructor\n \u00b7 apply iInf\u2082_le\n exact ha\n \u00b7 exact le_iSup\u2082_of_le _ ha le_rfl\n sSup_le := fun s s ha =>\n by\n dsimp only\n -- Porting note: added\n split_ifs with h\n \u00b7 exact bot_le\n obtain \u27e8b, hs, hb\u27e9 := not_subset.1 h\n lift s to NonemptyInterval \u03b1 using ne_bot_of_le_ne_bot hb (ha _ hs)\n exact\n WithBot.coe_le_coe.2\n \u27e8le_iInf\u2082 fun c hc => (WithBot.coe_le_coe.1 <| ha _ hc).1,\n iSup\u2082_le fun c hc => (WithBot.coe_le_coe.1 <| ha _ hc).2\u27e9\n sInf := fun S =>\n if h : \u22a5 \u2209 S \u2227 \u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd then\n some\n \u27e8\u27e8\u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst, \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd\u27e9,\n iSup\u2082_le fun s hs => le_iInf\u2082 <| h.2 hs\u27e9\n else \u22a5\n sInf_le := fun s\u2081 s ha =>\n by\n dsimp only\n -- Porting note: added\n split_ifs with h\n \u00b7 lift s to NonemptyInterval \u03b1 using\n ne_of_mem_of_not_mem ha\n h.1\n -- Porting note: Lean failed to figure out the function `f` by itself,\n -- so I added it through manually\n let f := fun (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u2081) => s.toProd.fst\n exact WithBot.coe_le_coe.2 \u27e8le_iSup\u2082 (f := f) s ha, iInf\u2082_le s ha\u27e9\n \u00b7 exact bot_le\n le_sInf := by\n intro S s ha\n cases s with\n | none => exact bot_le\n |\n some s =>\n dsimp\n -- Porting note: added\n split_ifs with h\n \u00b7\n exact\n WithBot.some_le_some.2\n \u27e8iSup\u2082_le fun t hb => (WithBot.coe_le_coe.1 <| ha _ hb).1,\n le_iInf\u2082 fun t hb => (WithBot.coe_le_coe.1 <| ha _ hb).2\u27e9\n \u00b7 rw [not_and_or, not_not] at h \n rcases h with h | h\n \u00b7 exact ha _ h\n \u00b7\n -- Porting note: ungolfed, due to identification problems\n -- between `toProd` and `toDualProd`. Original mathport output:\n -- cases h fun t hb c hc =>\n -- (WithBot.coe_le_coe.1 <| ha _ hb).1.trans <|\n -- s.fst_le_snd.trans (WithBot.coe_le_coe.1 <| ha _ hc).2 }\n exfalso\n apply h\n intro b hb c hc\n have h\u2081 := (WithBot.coe_le_coe.1 <| ha _ hb).1\n repeat rw [NonemptyInterval.toDualProd_apply] at h\u2081 \n rw [OrderDual.toDual_le_toDual] at h\u2081 \n exact h\u2081.trans (s.fst_le_snd.trans (WithBot.coe_le_coe.1 <| ha _ hc).2) }\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\n\u22a2 CompleteLattice (Interval \u03b1)\n[PROOFSTEP]\nexact\n { Interval.lattice,\n Interval.boundedOrder with\n sSup := fun S =>\n if h : S \u2286 {\u22a5} then \u22a5\n else\n some\n \u27e8\u27e8\u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst, \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd\u27e9,\n by\n obtain \u27e8s, hs, ha\u27e9 := not_subset.1 h\n lift s to NonemptyInterval \u03b1 using ha\n exact iInf\u2082_le_of_le s hs (le_iSup\u2082_of_le s hs s.fst_le_snd)\u27e9\n le_sSup := fun s s ha =>\n by\n dsimp only\n -- Porting note: added\n split_ifs with h\n \u00b7 exact (h ha).le\n cases s\n \u00b7 exact bot_le\n \u00b7\n -- Porting note: This case was\n -- `exact WithBot.some_le_some.2 \u27e8iInf\u2082_le _ ha, le_iSup\u2082_of_le _ ha le_rfl\u27e9`\n -- but there seems to be a defEq-problem at `iInf\u2082_le` that lean cannot resolve yet.\n apply WithBot.some_le_some.2\n constructor\n \u00b7 apply iInf\u2082_le\n exact ha\n \u00b7 exact le_iSup\u2082_of_le _ ha le_rfl\n sSup_le := fun s s ha =>\n by\n dsimp only\n -- Porting note: added\n split_ifs with h\n \u00b7 exact bot_le\n obtain \u27e8b, hs, hb\u27e9 := not_subset.1 h\n lift s to NonemptyInterval \u03b1 using ne_bot_of_le_ne_bot hb (ha _ hs)\n exact\n WithBot.coe_le_coe.2\n \u27e8le_iInf\u2082 fun c hc => (WithBot.coe_le_coe.1 <| ha _ hc).1,\n iSup\u2082_le fun c hc => (WithBot.coe_le_coe.1 <| ha _ hc).2\u27e9\n sInf := fun S =>\n if h : \u22a5 \u2209 S \u2227 \u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd then\n some\n \u27e8\u27e8\u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst, \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd\u27e9,\n iSup\u2082_le fun s hs => le_iInf\u2082 <| h.2 hs\u27e9\n else \u22a5\n sInf_le := fun s\u2081 s ha =>\n by\n dsimp only\n -- Porting note: added\n split_ifs with h\n \u00b7 lift s to NonemptyInterval \u03b1 using\n ne_of_mem_of_not_mem ha\n h.1\n -- Porting note: Lean failed to figure out the function `f` by itself,\n -- so I added it through manually\n let f := fun (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u2081) => s.toProd.fst\n exact WithBot.coe_le_coe.2 \u27e8le_iSup\u2082 (f := f) s ha, iInf\u2082_le s ha\u27e9\n \u00b7 exact bot_le\n le_sInf := by\n intro S s ha\n cases s with\n | none => exact bot_le\n |\n some s =>\n dsimp\n -- Porting note: added\n split_ifs with h\n \u00b7\n exact\n WithBot.some_le_some.2\n \u27e8iSup\u2082_le fun t hb => (WithBot.coe_le_coe.1 <| ha _ hb).1,\n le_iInf\u2082 fun t hb => (WithBot.coe_le_coe.1 <| ha _ hb).2\u27e9\n \u00b7 rw [not_and_or, not_not] at h \n rcases h with h | h\n \u00b7 exact ha _ h\n \u00b7\n -- Porting note: ungolfed, due to identification problems\n -- between `toProd` and `toDualProd`. Original mathport output:\n -- cases h fun t hb c hc =>\n -- (WithBot.coe_le_coe.1 <| ha _ hb).1.trans <|\n -- s.fst_le_snd.trans (WithBot.coe_le_coe.1 <| ha _ hc).2 }\n exfalso\n apply h\n intro b hb c hc\n have h\u2081 := (WithBot.coe_le_coe.1 <| ha _ hb).1\n repeat rw [NonemptyInterval.toDualProd_apply] at h\u2081 \n rw [OrderDual.toDual_le_toDual] at h\u2081 \n exact h\u2081.trans (s.fst_le_snd.trans (WithBot.coe_le_coe.1 <| ha _ hc).2) }\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\nh : \u00acS \u2286 {\u22a5}\n\u22a2 (\u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst, \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd).fst \u2264\n (\u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst, \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd).snd\n[PROOFSTEP]\nobtain \u27e8s, hs, ha\u27e9 := not_subset.1 h\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\nh : \u00acS \u2286 {\u22a5}\ns : Interval \u03b1\nhs : s \u2208 S\nha : \u00acs \u2208 {\u22a5}\n\u22a2 (\u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst, \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd).fst \u2264\n (\u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst, \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd).snd\n[PROOFSTEP]\nlift s to NonemptyInterval \u03b1 using ha\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\nh : \u00acS \u2286 {\u22a5}\ns : NonemptyInterval \u03b1\nhs : \u2191s \u2208 S\n\u22a2 (\u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst, \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd).fst \u2264\n (\u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst, \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd).snd\n[PROOFSTEP]\nexact iInf\u2082_le_of_le s hs (le_iSup\u2082_of_le s hs s.fst_le_snd)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns\u271d : Set (Interval \u03b1)\ns : Interval \u03b1\nha : s \u2208 s\u271d\n\u22a2 s \u2264 sSup s\u271d\n[PROOFSTEP]\ndsimp only\n -- Porting note: added\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns\u271d : Set (Interval \u03b1)\ns : Interval \u03b1\nha : s \u2208 s\u271d\n\u22a2 s \u2264\n if h : s\u271d \u2286 {\u22a5} then \u22a5\n else\n some\n { toProd := (\u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.fst, \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.snd),\n fst_le_snd :=\n (_ : \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.fst \u2264 \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.snd) }\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns\u271d : Set (Interval \u03b1)\ns : Interval \u03b1\nha : s \u2208 s\u271d\nh : s\u271d \u2286 {\u22a5}\n\u22a2 s \u2264 \u22a5\n[PROOFSTEP]\nexact (h ha).le\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns\u271d : Set (Interval \u03b1)\ns : Interval \u03b1\nha : s \u2208 s\u271d\nh : \u00acs\u271d \u2286 {\u22a5}\n\u22a2 s \u2264\n some\n { toProd := (\u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.fst, \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.snd),\n fst_le_snd :=\n (_ : \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.fst \u2264 \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.snd) }\n[PROOFSTEP]\ncases s\n[GOAL]\ncase neg.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns : Set (Interval \u03b1)\nh : \u00acs \u2286 {\u22a5}\nha : none \u2208 s\n\u22a2 none \u2264\n some\n {\n toProd :=\n (\u2a05 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.fst, \u2a06 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.snd),\n fst_le_snd :=\n (_ :\n \u2a05 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.fst \u2264\n \u2a06 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.snd) }\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase neg.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns : Set (Interval \u03b1)\nh : \u00acs \u2286 {\u22a5}\nval\u271d : NonemptyInterval \u03b1\nha : some val\u271d \u2208 s\n\u22a2 some val\u271d \u2264\n some\n {\n toProd :=\n (\u2a05 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.fst, \u2a06 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.snd),\n fst_le_snd :=\n (_ :\n \u2a05 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.fst \u2264\n \u2a06 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.snd) }\n[PROOFSTEP]\napply WithBot.some_le_some.2\n[GOAL]\ncase neg.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns : Set (Interval \u03b1)\nh : \u00acs \u2286 {\u22a5}\nval\u271d : NonemptyInterval \u03b1\nha : some val\u271d \u2208 s\n\u22a2 val\u271d \u2264\n {\n toProd :=\n (\u2a05 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.fst, \u2a06 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.snd),\n fst_le_snd :=\n (_ :\n \u2a05 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.fst \u2264 \u2a06 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.snd) }\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase neg.some.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns : Set (Interval \u03b1)\nh : \u00acs \u2286 {\u22a5}\nval\u271d : NonemptyInterval \u03b1\nha : some val\u271d \u2208 s\n\u22a2 (NonemptyInterval.toDualProd val\u271d).fst \u2264\n (NonemptyInterval.toDualProd\n {\n toProd :=\n (\u2a05 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.fst,\n \u2a06 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.snd),\n fst_le_snd :=\n (_ :\n \u2a05 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.fst \u2264\n \u2a06 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.snd) }).fst\n[PROOFSTEP]\napply iInf\u2082_le\n[GOAL]\ncase neg.some.left.j\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns : Set (Interval \u03b1)\nh : \u00acs \u2286 {\u22a5}\nval\u271d : NonemptyInterval \u03b1\nha : some val\u271d \u2208 s\n\u22a2 \u2191val\u271d \u2208 s\n[PROOFSTEP]\nexact ha\n[GOAL]\ncase neg.some.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns : Set (Interval \u03b1)\nh : \u00acs \u2286 {\u22a5}\nval\u271d : NonemptyInterval \u03b1\nha : some val\u271d \u2208 s\n\u22a2 (NonemptyInterval.toDualProd val\u271d).snd \u2264\n (NonemptyInterval.toDualProd\n {\n toProd :=\n (\u2a05 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.fst,\n \u2a06 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.snd),\n fst_le_snd :=\n (_ :\n \u2a05 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.fst \u2264\n \u2a06 (s_1 : NonemptyInterval \u03b1) (_ : \u2191s_1 \u2208 s), s_1.snd) }).snd\n[PROOFSTEP]\nexact le_iSup\u2082_of_le _ ha le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns\u271d : Set (Interval \u03b1)\ns : Interval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 s\u271d \u2192 b \u2264 s\n\u22a2 sSup s\u271d \u2264 s\n[PROOFSTEP]\ndsimp only\n -- Porting note: added\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns\u271d : Set (Interval \u03b1)\ns : Interval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 s\u271d \u2192 b \u2264 s\n\u22a2 (if h : s\u271d \u2286 {\u22a5} then \u22a5\n else\n some\n { toProd := (\u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.fst, \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.snd),\n fst_le_snd :=\n (_ : \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.fst \u2264 \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.snd) }) \u2264\n s\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns\u271d : Set (Interval \u03b1)\ns : Interval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 s\u271d \u2192 b \u2264 s\nh : s\u271d \u2286 {\u22a5}\n\u22a2 \u22a5 \u2264 s\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns\u271d : Set (Interval \u03b1)\ns : Interval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 s\u271d \u2192 b \u2264 s\nh : \u00acs\u271d \u2286 {\u22a5}\n\u22a2 some\n { toProd := (\u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.fst, \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.snd),\n fst_le_snd :=\n (_ : \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.fst \u2264 \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.snd) } \u2264\n s\n[PROOFSTEP]\nobtain \u27e8b, hs, hb\u27e9 := not_subset.1 h\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns\u271d : Set (Interval \u03b1)\ns : Interval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 s\u271d \u2192 b \u2264 s\nh : \u00acs\u271d \u2286 {\u22a5}\nb : Interval \u03b1\nhs : b \u2208 s\u271d\nhb : \u00acb \u2208 {\u22a5}\n\u22a2 some\n { toProd := (\u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.fst, \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.snd),\n fst_le_snd :=\n (_ : \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.fst \u2264 \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.snd) } \u2264\n s\n[PROOFSTEP]\nlift s to NonemptyInterval \u03b1 using ne_bot_of_le_ne_bot hb (ha _ hs)\n[GOAL]\ncase neg.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns\u271d : Set (Interval \u03b1)\nh : \u00acs\u271d \u2286 {\u22a5}\nb : Interval \u03b1\nhs : b \u2208 s\u271d\nhb : \u00acb \u2208 {\u22a5}\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 s\u271d \u2192 b \u2264 \u2191s\n\u22a2 some\n { toProd := (\u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.fst, \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.snd),\n fst_le_snd :=\n (_ : \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.fst \u2264 \u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u271d), s.snd) } \u2264\n \u2191s\n[PROOFSTEP]\nexact\n WithBot.coe_le_coe.2\n \u27e8le_iInf\u2082 fun c hc => (WithBot.coe_le_coe.1 <| ha _ hc).1, iSup\u2082_le fun c hc => (WithBot.coe_le_coe.1 <| ha _ hc).2\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns\u2081 : Set (Interval \u03b1)\ns : Interval \u03b1\nha : s \u2208 s\u2081\n\u22a2 sInf s\u2081 \u2264 s\n[PROOFSTEP]\ndsimp only\n -- Porting note: added\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns\u2081 : Set (Interval \u03b1)\ns : Interval \u03b1\nha : s \u2208 s\u2081\n\u22a2 (if h : \u00ac\u22a5 \u2208 s\u2081 \u2227 \u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 s\u2081 \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 s\u2081 \u2192 s.fst \u2264 t.snd then\n some\n { toProd := (\u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u2081), s.fst, \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u2081), s.snd),\n fst_le_snd :=\n (_ : \u2a06 (i : NonemptyInterval \u03b1) (_ : \u2191i \u2208 s\u2081), i.fst \u2264 \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u2081), s.snd) }\n else \u22a5) \u2264\n s\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns\u2081 : Set (Interval \u03b1)\ns : Interval \u03b1\nha : s \u2208 s\u2081\nh : \u00ac\u22a5 \u2208 s\u2081 \u2227 \u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 s\u2081 \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 s\u2081 \u2192 s.fst \u2264 t.snd\n\u22a2 some\n { toProd := (\u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u2081), s.fst, \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u2081), s.snd),\n fst_le_snd :=\n (_ : \u2a06 (i : NonemptyInterval \u03b1) (_ : \u2191i \u2208 s\u2081), i.fst \u2264 \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u2081), s.snd) } \u2264\n s\n[PROOFSTEP]\nlift s to NonemptyInterval \u03b1 using\n ne_of_mem_of_not_mem ha\n h.1\n -- Porting note: Lean failed to figure out the function `f` by itself,\n -- so I added it through manually\n[GOAL]\ncase pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns\u2081 : Set (Interval \u03b1)\nh : \u00ac\u22a5 \u2208 s\u2081 \u2227 \u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 s\u2081 \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 s\u2081 \u2192 s.fst \u2264 t.snd\ns : NonemptyInterval \u03b1\nha : \u2191s \u2208 s\u2081\n\u22a2 some\n { toProd := (\u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u2081), s.fst, \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u2081), s.snd),\n fst_le_snd :=\n (_ : \u2a06 (i : NonemptyInterval \u03b1) (_ : \u2191i \u2208 s\u2081), i.fst \u2264 \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u2081), s.snd) } \u2264\n \u2191s\n[PROOFSTEP]\nlet f := fun (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u2081) => s.toProd.fst\n[GOAL]\ncase pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns\u2081 : Set (Interval \u03b1)\nh : \u00ac\u22a5 \u2208 s\u2081 \u2227 \u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 s\u2081 \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 s\u2081 \u2192 s.fst \u2264 t.snd\ns : NonemptyInterval \u03b1\nha : \u2191s \u2208 s\u2081\nf : (s : NonemptyInterval \u03b1) \u2192 \u2191s \u2208 s\u2081 \u2192 \u03b1 := fun s x => s.fst\n\u22a2 some\n { toProd := (\u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u2081), s.fst, \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u2081), s.snd),\n fst_le_snd :=\n (_ : \u2a06 (i : NonemptyInterval \u03b1) (_ : \u2191i \u2208 s\u2081), i.fst \u2264 \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 s\u2081), s.snd) } \u2264\n \u2191s\n[PROOFSTEP]\nexact WithBot.coe_le_coe.2 \u27e8le_iSup\u2082 (f := f) s ha, iInf\u2082_le s ha\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\ns\u2081 : Set (Interval \u03b1)\ns : Interval \u03b1\nha : s \u2208 s\u2081\nh : \u00ac(\u00ac\u22a5 \u2208 s\u2081 \u2227 \u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 s\u2081 \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 s\u2081 \u2192 s.fst \u2264 t.snd)\n\u22a2 \u22a5 \u2264 s\n[PROOFSTEP]\nexact bot_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\n\u22a2 \u2200 (s : Set (Interval \u03b1)) (a : Interval \u03b1), (\u2200 (b : Interval \u03b1), b \u2208 s \u2192 a \u2264 b) \u2192 a \u2264 sInf s\n[PROOFSTEP]\nintro S s ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : Interval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 s \u2264 b\n\u22a2 s \u2264 sInf S\n[PROOFSTEP]\ncases s with\n| none => exact bot_le\n|\n some s =>\n dsimp\n -- Porting note: added\n split_ifs with h\n \u00b7\n exact\n WithBot.some_le_some.2\n \u27e8iSup\u2082_le fun t hb => (WithBot.coe_le_coe.1 <| ha _ hb).1,\n le_iInf\u2082 fun t hb => (WithBot.coe_le_coe.1 <| ha _ hb).2\u27e9\n \u00b7 rw [not_and_or, not_not] at h \n rcases h with h | h\n \u00b7 exact ha _ h\n \u00b7\n -- Porting note: ungolfed, due to identification problems\n -- between `toProd` and `toDualProd`. Original mathport output:\n -- cases h fun t hb c hc =>\n -- (WithBot.coe_le_coe.1 <| ha _ hb).1.trans <|\n -- s.fst_le_snd.trans (WithBot.coe_le_coe.1 <| ha _ hc).2 }\n exfalso\n apply h\n intro b hb c hc\n have h\u2081 := (WithBot.coe_le_coe.1 <| ha _ hb).1\n repeat rw [NonemptyInterval.toDualProd_apply] at h\u2081 \n rw [OrderDual.toDual_le_toDual] at h\u2081 \n exact h\u2081.trans (s.fst_le_snd.trans (WithBot.coe_le_coe.1 <| ha _ hc).2)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : Interval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 s \u2264 b\n\u22a2 s \u2264 sInf S\n[PROOFSTEP]\ncases s with\n| none => exact bot_le\n|\n some s =>\n dsimp\n -- Porting note: added\n split_ifs with h\n \u00b7\n exact\n WithBot.some_le_some.2\n \u27e8iSup\u2082_le fun t hb => (WithBot.coe_le_coe.1 <| ha _ hb).1,\n le_iInf\u2082 fun t hb => (WithBot.coe_le_coe.1 <| ha _ hb).2\u27e9\n \u00b7 rw [not_and_or, not_not] at h \n rcases h with h | h\n \u00b7 exact ha _ h\n \u00b7\n -- Porting note: ungolfed, due to identification problems\n -- between `toProd` and `toDualProd`. Original mathport output:\n -- cases h fun t hb c hc =>\n -- (WithBot.coe_le_coe.1 <| ha _ hb).1.trans <|\n -- s.fst_le_snd.trans (WithBot.coe_le_coe.1 <| ha _ hc).2 }\n exfalso\n apply h\n intro b hb c hc\n have h\u2081 := (WithBot.coe_le_coe.1 <| ha _ hb).1\n repeat rw [NonemptyInterval.toDualProd_apply] at h\u2081 \n rw [OrderDual.toDual_le_toDual] at h\u2081 \n exact h\u2081.trans (s.fst_le_snd.trans (WithBot.coe_le_coe.1 <| ha _ hc).2)\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 none \u2264 b\n\u22a2 none \u2264 sInf S\n[PROOFSTEP]\n\n| none => exact bot_le\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 none \u2264 b\n\u22a2 none \u2264 sInf S\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 some s \u2264 b\n\u22a2 some s \u2264 sInf S\n[PROOFSTEP]\n\n|\n some s =>\n dsimp\n -- Porting note: added\n split_ifs with h\n \u00b7\n exact\n WithBot.some_le_some.2\n \u27e8iSup\u2082_le fun t hb => (WithBot.coe_le_coe.1 <| ha _ hb).1,\n le_iInf\u2082 fun t hb => (WithBot.coe_le_coe.1 <| ha _ hb).2\u27e9\n \u00b7 rw [not_and_or, not_not] at h \n rcases h with h | h\n \u00b7 exact ha _ h\n \u00b7\n -- Porting note: ungolfed, due to identification problems\n -- between `toProd` and `toDualProd`. Original mathport output:\n -- cases h fun t hb c hc =>\n -- (WithBot.coe_le_coe.1 <| ha _ hb).1.trans <|\n -- s.fst_le_snd.trans (WithBot.coe_le_coe.1 <| ha _ hc).2 }\n exfalso\n apply h\n intro b hb c hc\n have h\u2081 := (WithBot.coe_le_coe.1 <| ha _ hb).1\n repeat rw [NonemptyInterval.toDualProd_apply] at h\u2081 \n rw [OrderDual.toDual_le_toDual] at h\u2081 \n exact h\u2081.trans (s.fst_le_snd.trans (WithBot.coe_le_coe.1 <| ha _ hc).2)\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 some s \u2264 b\n\u22a2 some s \u2264 sInf S\n[PROOFSTEP]\ndsimp\n -- Porting note: added\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 some s \u2264 b\n\u22a2 some s \u2264\n if h : \u00ac\u22a5 \u2208 S \u2227 \u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd then\n some\n { toProd := (\u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst, \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd),\n fst_le_snd :=\n (_ : \u2a06 (i : NonemptyInterval \u03b1) (_ : \u2191i \u2208 S), i.fst \u2264 \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd) }\n else \u22a5\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 some s \u2264 b\nh : \u00ac\u22a5 \u2208 S \u2227 \u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\n\u22a2 some s \u2264\n some\n { toProd := (\u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst, \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd),\n fst_le_snd :=\n (_ : \u2a06 (i : NonemptyInterval \u03b1) (_ : \u2191i \u2208 S), i.fst \u2264 \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd) }\n[PROOFSTEP]\nexact\n WithBot.some_le_some.2\n \u27e8iSup\u2082_le fun t hb => (WithBot.coe_le_coe.1 <| ha _ hb).1, le_iInf\u2082 fun t hb => (WithBot.coe_le_coe.1 <| ha _ hb).2\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 some s \u2264 b\nh : \u00ac(\u00ac\u22a5 \u2208 S \u2227 \u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd)\n\u22a2 some s \u2264 \u22a5\n[PROOFSTEP]\nrw [not_and_or, not_not] at h \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 some s \u2264 b\nh : \u22a5 \u2208 S \u2228 \u00ac\u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\n\u22a2 some s \u2264 \u22a5\n[PROOFSTEP]\nrcases h with h | h\n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 some s \u2264 b\nh : \u22a5 \u2208 S\n\u22a2 some s \u2264 \u22a5\n[PROOFSTEP]\nexact ha _ h\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 some s \u2264 b\nh : \u00ac\u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\n\u22a2 some s \u2264 \u22a5\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase neg.inr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 some s \u2264 b\nh : \u00ac\u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\n\u22a2 False\n[PROOFSTEP]\napply h\n[GOAL]\ncase neg.inr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 some s \u2264 b\nh : \u00ac\u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\n\u22a2 \u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\n[PROOFSTEP]\nintro b hb c hc\n[GOAL]\ncase neg.inr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 some s \u2264 b\nh : \u00ac\u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\nb : NonemptyInterval \u03b1\nhb : \u2191b \u2208 S\nc : NonemptyInterval \u03b1\nhc : \u2191c \u2208 S\n\u22a2 b.fst \u2264 c.snd\n[PROOFSTEP]\nhave h\u2081 := (WithBot.coe_le_coe.1 <| ha _ hb).1\n[GOAL]\ncase neg.inr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 some s \u2264 b\nh : \u00ac\u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\nb : NonemptyInterval \u03b1\nhb : \u2191b \u2208 S\nc : NonemptyInterval \u03b1\nhc : \u2191c \u2208 S\nh\u2081 : (NonemptyInterval.toDualProd s).fst \u2264 (NonemptyInterval.toDualProd b).fst\n\u22a2 b.fst \u2264 c.snd\n[PROOFSTEP]\nrepeat rw [NonemptyInterval.toDualProd_apply] at h\u2081 \n[GOAL]\ncase neg.inr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 some s \u2264 b\nh : \u00ac\u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\nb : NonemptyInterval \u03b1\nhb : \u2191b \u2208 S\nc : NonemptyInterval \u03b1\nhc : \u2191c \u2208 S\nh\u2081 : (NonemptyInterval.toDualProd s).fst \u2264 (NonemptyInterval.toDualProd b).fst\n\u22a2 b.fst \u2264 c.snd\n[PROOFSTEP]\nrw [NonemptyInterval.toDualProd_apply] at h\u2081 \n[GOAL]\ncase neg.inr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 some s \u2264 b\nh : \u00ac\u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\nb : NonemptyInterval \u03b1\nhb : \u2191b \u2208 S\nc : NonemptyInterval \u03b1\nhc : \u2191c \u2208 S\nh\u2081 : (\u2191toDual s.fst, s.snd).fst \u2264 (NonemptyInterval.toDualProd b).fst\n\u22a2 b.fst \u2264 c.snd\n[PROOFSTEP]\nrw [NonemptyInterval.toDualProd_apply] at h\u2081 \n[GOAL]\ncase neg.inr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 some s \u2264 b\nh : \u00ac\u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\nb : NonemptyInterval \u03b1\nhb : \u2191b \u2208 S\nc : NonemptyInterval \u03b1\nhc : \u2191c \u2208 S\nh\u2081 : (\u2191toDual s.fst, s.snd).fst \u2264 (\u2191toDual b.fst, b.snd).fst\n\u22a2 b.fst \u2264 c.snd\n[PROOFSTEP]\nrw [NonemptyInterval.toDualProd_apply] at h\u2081 \n[GOAL]\ncase neg.inr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 some s \u2264 b\nh : \u00ac\u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\nb : NonemptyInterval \u03b1\nhb : \u2191b \u2208 S\nc : NonemptyInterval \u03b1\nhc : \u2191c \u2208 S\nh\u2081 : (\u2191toDual s.fst, s.snd).fst \u2264 (\u2191toDual b.fst, b.snd).fst\n\u22a2 b.fst \u2264 c.snd\n[PROOFSTEP]\nrw [OrderDual.toDual_le_toDual] at h\u2081 \n[GOAL]\ncase neg.inr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : Lattice (Interval \u03b1) := lattice\nsrc\u271d : BoundedOrder (Interval \u03b1) := boundedOrder\nS : Set (Interval \u03b1)\ns : NonemptyInterval \u03b1\nha : \u2200 (b : Interval \u03b1), b \u2208 S \u2192 some s \u2264 b\nh : \u00ac\u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\nb : NonemptyInterval \u03b1\nhb : \u2191b \u2208 S\nc : NonemptyInterval \u03b1\nhc : \u2191c \u2208 S\nh\u2081 : b.fst \u2264 s.fst\n\u22a2 b.fst \u2264 c.snd\n[PROOFSTEP]\nexact h\u2081.trans (s.fst_le_snd.trans (WithBot.coe_le_coe.1 <| ha _ hc).2)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nS : Set (Interval \u03b1)\n\u22a2 \u2191(sInf S) = \u22c2 (s : Interval \u03b1) (_ : s \u2208 S), \u2191s\n[PROOFSTEP]\nclassical\n -- Porting note: added\n -- Porting note: this `change` was\n -- change \u2191 (dite _ _ _) = _\nchange ((dite _ _ _ : Interval \u03b1) : Set \u03b1) = \u22c2 (s : Interval \u03b1) (_ : s \u2208 S), (s : Set \u03b1)\nsplit_ifs with h\n\u00b7 ext\n simp [WithBot.some_eq_coe, Interval.forall, h.1, \u2190 forall_and, \u2190 NonemptyInterval.mem_def]\nsimp_rw [not_and_or, Classical.not_not] at h \nrcases h with h | h\n\u00b7 refine' (eq_empty_of_subset_empty _).symm\n exact iInter\u2082_subset_of_subset _ h Subset.rfl\n\u00b7 refine' (not_nonempty_iff_eq_empty.1 _).symm\n rintro \u27e8x, hx\u27e9\n rw [mem_iInter\u2082] at hx \n exact h fun s ha t hb => (hx _ ha).1.trans (hx _ hb).2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nS : Set (Interval \u03b1)\n\u22a2 \u2191(sInf S) = \u22c2 (s : Interval \u03b1) (_ : s \u2208 S), \u2191s\n[PROOFSTEP]\nchange ((dite _ _ _ : Interval \u03b1) : Set \u03b1) = \u22c2 (s : Interval \u03b1) (_ : s \u2208 S), (s : Set \u03b1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nS : Set (Interval \u03b1)\n\u22a2 \u2191(if h : \u00ac\u22a5 \u2208 S \u2227 \u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd then\n some\n { toProd := (\u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst, \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd),\n fst_le_snd :=\n (_ :\n \u2a06 (i : NonemptyInterval \u03b1) (_ : \u2191i \u2208 S), i.fst \u2264\n (\u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst,\n \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd).snd) }\n else \u22a5) =\n \u22c2 (s : Interval \u03b1) (_ : s \u2208 S), \u2191s\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nS : Set (Interval \u03b1)\nh : \u00ac\u22a5 \u2208 S \u2227 \u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\n\u22a2 \u2191(some\n { toProd := (\u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst, \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd),\n fst_le_snd :=\n (_ :\n \u2a06 (i : NonemptyInterval \u03b1) (_ : \u2191i \u2208 S), i.fst \u2264\n (\u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst,\n \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd).snd) }) =\n \u22c2 (s : Interval \u03b1) (_ : s \u2208 S), \u2191s\n[PROOFSTEP]\next\n[GOAL]\ncase pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nS : Set (Interval \u03b1)\nh : \u00ac\u22a5 \u2208 S \u2227 \u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208\n \u2191(some\n { toProd := (\u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst, \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd),\n fst_le_snd :=\n (_ :\n \u2a06 (i : NonemptyInterval \u03b1) (_ : \u2191i \u2208 S), i.fst \u2264\n (\u2a06 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.fst,\n \u2a05 (s : NonemptyInterval \u03b1) (_ : \u2191s \u2208 S), s.snd).snd) }) \u2194\n x\u271d \u2208 \u22c2 (s : Interval \u03b1) (_ : s \u2208 S), \u2191s\n[PROOFSTEP]\nsimp [WithBot.some_eq_coe, Interval.forall, h.1, \u2190 forall_and, \u2190 NonemptyInterval.mem_def]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nS : Set (Interval \u03b1)\nh : \u00ac(\u00ac\u22a5 \u2208 S \u2227 \u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd)\n\u22a2 \u2191\u22a5 = \u22c2 (s : Interval \u03b1) (_ : s \u2208 S), \u2191s\n[PROOFSTEP]\nsimp_rw [not_and_or, Classical.not_not] at h \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nS : Set (Interval \u03b1)\nh : \u22a5 \u2208 S \u2228 \u00ac\u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\n\u22a2 \u2191\u22a5 = \u22c2 (s : Interval \u03b1) (_ : s \u2208 S), \u2191s\n[PROOFSTEP]\nrcases h with h | h\n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nS : Set (Interval \u03b1)\nh : \u22a5 \u2208 S\n\u22a2 \u2191\u22a5 = \u22c2 (s : Interval \u03b1) (_ : s \u2208 S), \u2191s\n[PROOFSTEP]\nrefine' (eq_empty_of_subset_empty _).symm\n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nS : Set (Interval \u03b1)\nh : \u22a5 \u2208 S\n\u22a2 \u22c2 (s : Interval \u03b1) (_ : s \u2208 S), \u2191s \u2286 \u2205\n[PROOFSTEP]\nexact iInter\u2082_subset_of_subset _ h Subset.rfl\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nS : Set (Interval \u03b1)\nh : \u00ac\u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\n\u22a2 \u2191\u22a5 = \u22c2 (s : Interval \u03b1) (_ : s \u2208 S), \u2191s\n[PROOFSTEP]\nrefine' (not_nonempty_iff_eq_empty.1 _).symm\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nS : Set (Interval \u03b1)\nh : \u00ac\u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\n\u22a2 \u00acSet.Nonempty (\u22c2 (s : Interval \u03b1) (_ : s \u2208 S), \u2191s)\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase neg.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nS : Set (Interval \u03b1)\nh : \u00ac\u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\nx : \u03b1\nhx : x \u2208 \u22c2 (s : Interval \u03b1) (_ : s \u2208 S), \u2191s\n\u22a2 False\n[PROOFSTEP]\nrw [mem_iInter\u2082] at hx \n[GOAL]\ncase neg.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nS : Set (Interval \u03b1)\nh : \u00ac\u2200 \u2983s : NonemptyInterval \u03b1\u2984, \u2191s \u2208 S \u2192 \u2200 \u2983t : NonemptyInterval \u03b1\u2984, \u2191t \u2208 S \u2192 s.fst \u2264 t.snd\nx : \u03b1\nhx : \u2200 (i : Interval \u03b1), i \u2208 S \u2192 x \u2208 \u2191i\n\u22a2 False\n[PROOFSTEP]\nexact h fun s ha t hb => (hx _ ha).1.trans (hx _ hb).2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nf : \u03b9 \u2192 Interval \u03b1\n\u22a2 \u2191(\u2a05 (i : \u03b9), f i) = \u22c2 (i : \u03b9), \u2191(f i)\n[PROOFSTEP]\nsimp [iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 Interval \u03b1\n\u22a2 \u2191(\u2a05 (i : \u03b9) (j : \u03ba i), f i j) = \u22c2 (i : \u03b9) (j : \u03ba i), \u2191(f i j)\n[PROOFSTEP]\nsimp_rw [coe_iInf]\n", "meta": {"mathlib_filename": "Mathlib.Order.Interval", "llama_tokens": 34154, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5312093733737563, "lm_q2_score": 0.3849121444839335, "lm_q1q2_score": 0.20446893907525907}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 \u2200 (x y : \u210d[DualNumber R]),\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n (x * y) =\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI,\n imJ := TrivSqZeroExt.fst q.imJ, imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n x *\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI,\n imJ := TrivSqZeroExt.fst q.imJ, imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n y\n[PROOFSTEP]\nrintro \u27e8\u27e8xr, xr\u03b5\u27e9, \u27e8xi, xi\u03b5\u27e9, \u27e8xj, xj\u03b5\u27e9, \u27e8xk, xk\u03b5\u27e9\u27e9\n[GOAL]\ncase mk.mk.mk.mk.mk\nR : Type u_1\ninst\u271d : CommRing R\nxr xr\u03b5 xi xi\u03b5 xj xj\u03b5 xk xk\u03b5 : R\n\u22a2 \u2200 (y : \u210d[DualNumber R]),\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n ({ re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } * y) =\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI,\n imJ := TrivSqZeroExt.fst q.imJ, imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n { re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } *\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI,\n imJ := TrivSqZeroExt.fst q.imJ, imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n y\n[PROOFSTEP]\nrintro \u27e8\u27e8yr, yr\u03b5\u27e9, \u27e8yi, yi\u03b5\u27e9, \u27e8yj, yj\u03b5\u27e9, \u27e8yk, yk\u03b5\u27e9\u27e9\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk.mk.mk\nR : Type u_1\ninst\u271d : CommRing R\nxr xr\u03b5 xi xi\u03b5 xj xj\u03b5 xk xk\u03b5 yr yr\u03b5 yi yi\u03b5 yj yj\u03b5 yk yk\u03b5 : R\n\u22a2 Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n ({ re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } *\n { re := (yr, yr\u03b5), imI := (yi, yi\u03b5), imJ := (yj, yj\u03b5), imK := (yk, yk\u03b5) }) =\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n { re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } *\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n { re := (yr, yr\u03b5), imI := (yi, yi\u03b5), imJ := (yj, yj\u03b5), imK := (yk, yk\u03b5) }\n[PROOFSTEP]\next : 1\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk.mk.mk.h1\nR : Type u_1\ninst\u271d : CommRing R\nxr xr\u03b5 xi xi\u03b5 xj xj\u03b5 xk xk\u03b5 yr yr\u03b5 yi yi\u03b5 yj yj\u03b5 yk yk\u03b5 : R\n\u22a2 TrivSqZeroExt.fst\n (Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n ({ re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } *\n { re := (yr, yr\u03b5), imI := (yi, yi\u03b5), imJ := (yj, yj\u03b5), imK := (yk, yk\u03b5) })) =\n TrivSqZeroExt.fst\n (Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI,\n imJ := TrivSqZeroExt.fst q.imJ, imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n { re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } *\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI,\n imJ := TrivSqZeroExt.fst q.imJ, imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n { re := (yr, yr\u03b5), imI := (yi, yi\u03b5), imJ := (yj, yj\u03b5), imK := (yk, yk\u03b5) })\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk.mk.mk.h2\nR : Type u_1\ninst\u271d : CommRing R\nxr xr\u03b5 xi xi\u03b5 xj xj\u03b5 xk xk\u03b5 yr yr\u03b5 yi yi\u03b5 yj yj\u03b5 yk yk\u03b5 : R\n\u22a2 TrivSqZeroExt.snd\n (Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n ({ re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } *\n { re := (yr, yr\u03b5), imI := (yi, yi\u03b5), imJ := (yj, yj\u03b5), imK := (yk, yk\u03b5) })) =\n TrivSqZeroExt.snd\n (Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI,\n imJ := TrivSqZeroExt.fst q.imJ, imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n { re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } *\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI,\n imJ := TrivSqZeroExt.fst q.imJ, imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n { re := (yr, yr\u03b5), imI := (yi, yi\u03b5), imJ := (yj, yj\u03b5), imK := (yk, yk\u03b5) })\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk.mk.mk.h2\nR : Type u_1\ninst\u271d : CommRing R\nxr xr\u03b5 xi xi\u03b5 xj xj\u03b5 xk xk\u03b5 yr yr\u03b5 yi yi\u03b5 yj yj\u03b5 yk yk\u03b5 : R\n\u22a2 {\n re :=\n TrivSqZeroExt.snd\n ({ re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } *\n { re := (yr, yr\u03b5), imI := (yi, yi\u03b5), imJ := (yj, yj\u03b5), imK := (yk, yk\u03b5) }).re,\n imI :=\n TrivSqZeroExt.snd\n ({ re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } *\n { re := (yr, yr\u03b5), imI := (yi, yi\u03b5), imJ := (yj, yj\u03b5), imK := (yk, yk\u03b5) }).imI,\n imJ :=\n TrivSqZeroExt.snd\n ({ re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } *\n { re := (yr, yr\u03b5), imI := (yi, yi\u03b5), imJ := (yj, yj\u03b5), imK := (yk, yk\u03b5) }).imJ,\n imK :=\n TrivSqZeroExt.snd\n ({ re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } *\n { re := (yr, yr\u03b5), imI := (yi, yi\u03b5), imJ := (yj, yj\u03b5), imK := (yk, yk\u03b5) }).imK } =\n { re := xr, imI := xi, imJ := xj, imK := xk } * { re := yr\u03b5, imI := yi\u03b5, imJ := yj\u03b5, imK := yk\u03b5 } +\n { re := xr\u03b5, imI := xi\u03b5, imJ := xj\u03b5, imK := xk\u03b5 } * { re := yr, imI := yi, imJ := yj, imK := yk }\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk.mk.mk.h2.e_re\nR : Type u_1\ninst\u271d : CommRing R\nxr xr\u03b5 xi xi\u03b5 xj xj\u03b5 xk xk\u03b5 yr yr\u03b5 yi yi\u03b5 yj yj\u03b5 yk yk\u03b5 : R\n\u22a2 TrivSqZeroExt.snd\n ({ re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } *\n { re := (yr, yr\u03b5), imI := (yi, yi\u03b5), imJ := (yj, yj\u03b5), imK := (yk, yk\u03b5) }).re =\n ({ re := xr, imI := xi, imJ := xj, imK := xk } * { re := yr\u03b5, imI := yi\u03b5, imJ := yj\u03b5, imK := yk\u03b5 }).re +\n ({ re := xr\u03b5, imI := xi\u03b5, imJ := xj\u03b5, imK := xk\u03b5 } * { re := yr, imI := yi, imJ := yj, imK := yk }).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk.mk.mk.h2.e_imI\nR : Type u_1\ninst\u271d : CommRing R\nxr xr\u03b5 xi xi\u03b5 xj xj\u03b5 xk xk\u03b5 yr yr\u03b5 yi yi\u03b5 yj yj\u03b5 yk yk\u03b5 : R\n\u22a2 TrivSqZeroExt.snd\n ({ re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } *\n { re := (yr, yr\u03b5), imI := (yi, yi\u03b5), imJ := (yj, yj\u03b5), imK := (yk, yk\u03b5) }).imI =\n ({ re := xr, imI := xi, imJ := xj, imK := xk } * { re := yr\u03b5, imI := yi\u03b5, imJ := yj\u03b5, imK := yk\u03b5 }).imI +\n ({ re := xr\u03b5, imI := xi\u03b5, imJ := xj\u03b5, imK := xk\u03b5 } * { re := yr, imI := yi, imJ := yj, imK := yk }).imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk.mk.mk.h2.e_imJ\nR : Type u_1\ninst\u271d : CommRing R\nxr xr\u03b5 xi xi\u03b5 xj xj\u03b5 xk xk\u03b5 yr yr\u03b5 yi yi\u03b5 yj yj\u03b5 yk yk\u03b5 : R\n\u22a2 TrivSqZeroExt.snd\n ({ re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } *\n { re := (yr, yr\u03b5), imI := (yi, yi\u03b5), imJ := (yj, yj\u03b5), imK := (yk, yk\u03b5) }).imJ =\n ({ re := xr, imI := xi, imJ := xj, imK := xk } * { re := yr\u03b5, imI := yi\u03b5, imJ := yj\u03b5, imK := yk\u03b5 }).imJ +\n ({ re := xr\u03b5, imI := xi\u03b5, imJ := xj\u03b5, imK := xk\u03b5 } * { re := yr, imI := yi, imJ := yj, imK := yk }).imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk.mk.mk.h2.e_imK\nR : Type u_1\ninst\u271d : CommRing R\nxr xr\u03b5 xi xi\u03b5 xj xj\u03b5 xk xk\u03b5 yr yr\u03b5 yi yi\u03b5 yj yj\u03b5 yk yk\u03b5 : R\n\u22a2 TrivSqZeroExt.snd\n ({ re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } *\n { re := (yr, yr\u03b5), imI := (yi, yi\u03b5), imJ := (yj, yj\u03b5), imK := (yk, yk\u03b5) }).imK =\n ({ re := xr, imI := xi, imJ := xj, imK := xk } * { re := yr\u03b5, imI := yi\u03b5, imJ := yj\u03b5, imK := yk\u03b5 }).imK +\n ({ re := xr\u03b5, imI := xi\u03b5, imJ := xj\u03b5, imK := xk\u03b5 } * { re := yr, imI := yi, imJ := yj, imK := yk }).imK\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk.mk.mk.h2.e_re\nR : Type u_1\ninst\u271d : CommRing R\nxr xr\u03b5 xi xi\u03b5 xj xj\u03b5 xk xk\u03b5 yr yr\u03b5 yi yi\u03b5 yj yj\u03b5 yk yk\u03b5 : R\n\u22a2 xr * yr\u03b5 + xr\u03b5 * yr - (xi * yi\u03b5 + xi\u03b5 * yi) - (xj * yj\u03b5 + xj\u03b5 * yj) - (xk * yk\u03b5 + xk\u03b5 * yk) =\n xr * yr\u03b5 - xi * yi\u03b5 - xj * yj\u03b5 - xk * yk\u03b5 + (xr\u03b5 * yr - xi\u03b5 * yi - xj\u03b5 * yj - xk\u03b5 * yk)\n[PROOFSTEP]\nring\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk.mk.mk.h2.e_imI\nR : Type u_1\ninst\u271d : CommRing R\nxr xr\u03b5 xi xi\u03b5 xj xj\u03b5 xk xk\u03b5 yr yr\u03b5 yi yi\u03b5 yj yj\u03b5 yk yk\u03b5 : R\n\u22a2 xr * yi\u03b5 + xr\u03b5 * yi + (xi * yr\u03b5 + xi\u03b5 * yr) + (xj * yk\u03b5 + xj\u03b5 * yk) - (xk * yj\u03b5 + xk\u03b5 * yj) =\n xr * yi\u03b5 + xi * yr\u03b5 + xj * yk\u03b5 - xk * yj\u03b5 + (xr\u03b5 * yi + xi\u03b5 * yr + xj\u03b5 * yk - xk\u03b5 * yj)\n[PROOFSTEP]\nring\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk.mk.mk.h2.e_imJ\nR : Type u_1\ninst\u271d : CommRing R\nxr xr\u03b5 xi xi\u03b5 xj xj\u03b5 xk xk\u03b5 yr yr\u03b5 yi yi\u03b5 yj yj\u03b5 yk yk\u03b5 : R\n\u22a2 xr * yj\u03b5 + xr\u03b5 * yj - (xi * yk\u03b5 + xi\u03b5 * yk) + (xj * yr\u03b5 + xj\u03b5 * yr) + (xk * yi\u03b5 + xk\u03b5 * yi) =\n xr * yj\u03b5 - xi * yk\u03b5 + xj * yr\u03b5 + xk * yi\u03b5 + (xr\u03b5 * yj - xi\u03b5 * yk + xj\u03b5 * yr + xk\u03b5 * yi)\n[PROOFSTEP]\nring\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk.mk.mk.h2.e_imK\nR : Type u_1\ninst\u271d : CommRing R\nxr xr\u03b5 xi xi\u03b5 xj xj\u03b5 xk xk\u03b5 yr yr\u03b5 yi yi\u03b5 yj yj\u03b5 yk yk\u03b5 : R\n\u22a2 xr * yk\u03b5 + xr\u03b5 * yk + (xi * yj\u03b5 + xi\u03b5 * yj) - (xj * yi\u03b5 + xj\u03b5 * yi) + (xk * yr\u03b5 + xk\u03b5 * yr) =\n xr * yk\u03b5 + xi * yj\u03b5 - xj * yi\u03b5 + xk * yr\u03b5 + (xr\u03b5 * yk + xi\u03b5 * yj - xj\u03b5 * yi + xk\u03b5 * yr)\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 \u2200 (x y : \u210d[DualNumber R]),\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n (x + y) =\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI,\n imJ := TrivSqZeroExt.fst q.imJ, imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n x +\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI,\n imJ := TrivSqZeroExt.fst q.imJ, imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n y\n[PROOFSTEP]\nrintro \u27e8\u27e8xr, xr\u03b5\u27e9, \u27e8xi, xi\u03b5\u27e9, \u27e8xj, xj\u03b5\u27e9, \u27e8xk, xk\u03b5\u27e9\u27e9\n[GOAL]\ncase mk.mk.mk.mk.mk\nR : Type u_1\ninst\u271d : CommRing R\nxr xr\u03b5 xi xi\u03b5 xj xj\u03b5 xk xk\u03b5 : R\n\u22a2 \u2200 (y : \u210d[DualNumber R]),\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n ({ re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } + y) =\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI,\n imJ := TrivSqZeroExt.fst q.imJ, imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n { re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } +\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI,\n imJ := TrivSqZeroExt.fst q.imJ, imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n y\n[PROOFSTEP]\nrintro \u27e8\u27e8yr, yr\u03b5\u27e9, \u27e8yi, yi\u03b5\u27e9, \u27e8yj, yj\u03b5\u27e9, \u27e8yk, yk\u03b5\u27e9\u27e9\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk.mk.mk\nR : Type u_1\ninst\u271d : CommRing R\nxr xr\u03b5 xi xi\u03b5 xj xj\u03b5 xk xk\u03b5 yr yr\u03b5 yi yi\u03b5 yj yj\u03b5 yk yk\u03b5 : R\n\u22a2 Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n ({ re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } +\n { re := (yr, yr\u03b5), imI := (yi, yi\u03b5), imJ := (yj, yj\u03b5), imK := (yk, yk\u03b5) }) =\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n { re := (xr, xr\u03b5), imI := (xi, xi\u03b5), imJ := (xj, xj\u03b5), imK := (xk, xk\u03b5) } +\n Equiv.toFun\n {\n toFun := fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }),\n invFun := fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) },\n left_inv :=\n (_ :\n \u2200 (x : \u210d[DualNumber R]),\n (fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n ((fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI,\n imJ := TrivSqZeroExt.snd q.imJ, imK := TrivSqZeroExt.snd q.imK }))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : DualNumber \u210d[R]),\n (fun q =>\n ({ re := TrivSqZeroExt.fst q.re, imI := TrivSqZeroExt.fst q.imI, imJ := TrivSqZeroExt.fst q.imJ,\n imK := TrivSqZeroExt.fst q.imK },\n { re := TrivSqZeroExt.snd q.re, imI := TrivSqZeroExt.snd q.imI, imJ := TrivSqZeroExt.snd q.imJ,\n imK := TrivSqZeroExt.snd q.imK }))\n ((fun d =>\n { re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),\n imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),\n imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),\n imK := ((TrivSqZeroExt.fst d).imK, (TrivSqZeroExt.snd d).imK) })\n x) =\n x) }\n { re := (yr, yr\u03b5), imI := (yi, yi\u03b5), imJ := (yj, yj\u03b5), imK := (yk, yk\u03b5) }\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.DualQuaternion", "llama_tokens": 28429, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.3775406687981454, "lm_q1q2_score": 0.2020214186165276}}