{"text": "[GOAL]\na b : \u2115\nh : 0 < b\n\u22a2 a + 0 < a + b\n[PROOFSTEP]\napply Nat.add_lt_add_left\n[GOAL]\ncase h\na b : \u2115\nh : 0 < b\n\u22a2 0 < b\n[PROOFSTEP]\nassumption\n[GOAL]\na : \u2115\nthis : 0 < a + 1\n\u22a2 0 < 1 + a\n[PROOFSTEP]\nsimp [Nat.add_comm]\n[GOAL]\na : \u2115\nthis : 0 < a + 1\n\u22a2 0 < a + 1\n[PROOFSTEP]\nassumption\n", "meta": {"mathlib_filename": "Mathlib.Init.Meta.WellFoundedTactics", "llama_tokens": 168, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9136765234137297, "lm_q2_score": 0.8080672112416737, "lm_q1q2_score": 0.7383120402519203}}
{"text": "[GOAL]\na : \u2115\nha : a \u2264 pred a\n\u22a2 IsMin a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase zero\nha : zero \u2264 pred zero\n\u22a2 IsMin zero\n[PROOFSTEP]\nexact isMin_bot\n[GOAL]\ncase succ\nn\u271d : \u2115\nha : succ n\u271d \u2264 pred (succ n\u271d)\n\u22a2 IsMin (succ n\u271d)\n[PROOFSTEP]\nexact (not_succ_le_self _ ha).elim\n[GOAL]\na b : \u2115\nh : a < b\n\u22a2 a \u2264 pred b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase zero\na : \u2115\nh : a < zero\n\u22a2 a \u2264 pred zero\n[PROOFSTEP]\nexact (a.not_lt_zero h).elim\n[GOAL]\ncase succ\na n\u271d : \u2115\nh : a < succ n\u271d\n\u22a2 a \u2264 pred (succ n\u271d)\n[PROOFSTEP]\nexact le_of_succ_le_succ h\n[GOAL]\na b : \u2115\nh : pred a < b\n\u22a2 a \u2264 b\n[PROOFSTEP]\ncases a\n[GOAL]\ncase zero\nb : \u2115\nh : pred zero < b\n\u22a2 zero \u2264 b\n[PROOFSTEP]\nexact b.zero_le\n[GOAL]\ncase succ\nb n\u271d : \u2115\nh : pred (succ n\u271d) < b\n\u22a2 succ n\u271d \u2264 b\n[PROOFSTEP]\nexact h\n[GOAL]\na n : \u2115\n\u22a2 succ^[n + 1] a = a + (n + 1)\n[PROOFSTEP]\nrw [Function.iterate_succ', add_succ]\n[GOAL]\na n : \u2115\n\u22a2 (succ \u2218 succ^[n]) a = succ (a + n)\n[PROOFSTEP]\nexact congr_arg _ (succ_iterate a n)\n[GOAL]\na n : \u2115\n\u22a2 pred^[n + 1] a = a - (n + 1)\n[PROOFSTEP]\nrw [Function.iterate_succ', sub_succ]\n[GOAL]\na n : \u2115\n\u22a2 (pred \u2218 pred^[n]) a = pred (a - n)\n[PROOFSTEP]\nexact congr_arg _ (pred_iterate a n)\n[GOAL]\na b : \u2115\nh : a \u2264 b\n\u22a2 Order.succ^[b - a] a = b\n[PROOFSTEP]\nrw [succ_eq_succ, succ_iterate, add_tsub_cancel_of_le h]\n[GOAL]\na b : \u2115\nh : a \u2264 b\n\u22a2 Order.pred^[b - a] b = a\n[PROOFSTEP]\nrw [pred_eq_pred, pred_iterate, tsub_tsub_cancel_of_le h]\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.SuccPred", "llama_tokens": 761, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9073122113355092, "lm_q2_score": 0.8104789086703225, "lm_q1q2_score": 0.7353574108664604}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonUnitalNonAssocRing R\na b : R\nh : Commute a b\n\u22a2 a * a - b * b = (a + b) * (a - b)\n[PROOFSTEP]\nrw [add_mul, mul_sub, mul_sub, h.eq, sub_add_sub_cancel]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonUnitalNonAssocRing R\na b : R\nh : Commute a b\n\u22a2 a * a - b * b = (a - b) * (a + b)\n[PROOFSTEP]\nrw [mul_add, sub_mul, sub_mul, h.eq, sub_add_sub_cancel]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : NonUnitalNonAssocRing R\ninst\u271d : NoZeroDivisors R\na b : R\nh : Commute a b\n\u22a2 a * a = b * b \u2194 a = b \u2228 a = -b\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, h.mul_self_sub_mul_self_eq, mul_eq_zero, or_comm, sub_eq_zero, add_eq_zero_iff_eq_neg]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonAssocRing R\na : R\n\u22a2 a * a - 1 = (a + 1) * (a - 1)\n[PROOFSTEP]\nrw [\u2190 (Commute.one_right a).mul_self_sub_mul_self_eq, mul_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : NoZeroDivisors R\na : R\n\u22a2 a * a = 1 \u2194 a = 1 \u2228 a = -1\n[PROOFSTEP]\nrw [\u2190 (Commute.one_right a).mul_self_eq_mul_self_iff, mul_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : Ring R\ninst\u271d : NoZeroDivisors R\nu : R\u02e3\n\u22a2 u\u207b\u00b9 = u \u2194 u = 1 \u2228 u = -1\n[PROOFSTEP]\nrw [inv_eq_iff_mul_eq_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : Ring R\ninst\u271d : NoZeroDivisors R\nu : R\u02e3\n\u22a2 u * u = 1 \u2194 u = 1 \u2228 u = -1\n[PROOFSTEP]\nsimp only [ext_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : Ring R\ninst\u271d : NoZeroDivisors R\nu : R\u02e3\n\u22a2 \u2191(u * u) = \u21911 \u2194 \u2191u = \u21911 \u2228 \u2191u = \u2191(-1)\n[PROOFSTEP]\npush_cast\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : Ring R\ninst\u271d : NoZeroDivisors R\nu : R\u02e3\n\u22a2 \u2191u * \u2191u = 1 \u2194 \u2191u = 1 \u2228 \u2191u = -1\n[PROOFSTEP]\nexact mul_self_eq_one_iff\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Ring.Commute", "llama_tokens": 942, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513786759491, "lm_q2_score": 0.8152324983301567, "lm_q1q2_score": 0.7306532505698412}}
{"text": "[GOAL]\n\u03b9 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\n\u03b1 : Type u_5\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype p\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : AddCommMonoidWithOne R\n\u22a2 trace 1 = \u2191(Fintype.card n)\n[PROOFSTEP]\nsimp_rw [trace, diag_one, Pi.one_def, Finset.sum_const, nsmul_one, Finset.card_univ]\n[GOAL]\n\u03b9 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\n\u03b1 : Type u_5\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype p\ninst\u271d\u00b9 : AddCommMonoid R\ninst\u271d : CommSemigroup R\nA : Matrix m n R\nB : Matrix n m R\n\u22a2 trace (A * B) = trace (B * A)\n[PROOFSTEP]\nrw [\u2190 trace_transpose, \u2190 trace_transpose_mul, transpose_mul]\n[GOAL]\n\u03b9 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\n\u03b1 : Type u_5\nR : Type u_6\nS : Type u_7\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype p\ninst\u271d : NonUnitalCommSemiring R\nA : Matrix m n R\nB : Matrix n p R\nC : Matrix p m R\n\u22a2 trace (A * B * C) = trace (C * A * B)\n[PROOFSTEP]\nrw [trace_mul_comm, Matrix.mul_assoc]\n[GOAL]\n\u03b9 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\n\u03b1 : Type u_5\nR : Type u_6\nS : Type u_7\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype p\ninst\u271d : NonUnitalCommSemiring R\nA : Matrix m n R\nB : Matrix n p R\nC : Matrix p m R\n\u22a2 trace (A * (B * C)) = trace (C * (A * B))\n[PROOFSTEP]\nrw [\u2190 Matrix.mul_assoc, trace_mul_comm]\n[GOAL]\n\u03b9 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\n\u03b1 : Type u_5\nR : Type u_6\nS : Type u_7\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype p\ninst\u271d : NonUnitalNonAssocSemiring R\na b : n \u2192 R\n\u22a2 trace (col a * row b) = a \u2b1d\u1d65 b\n[PROOFSTEP]\napply Finset.sum_congr rfl\n[GOAL]\n\u03b9 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\n\u03b1 : Type u_5\nR : Type u_6\nS : Type u_7\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype p\ninst\u271d : NonUnitalNonAssocSemiring R\na b : n \u2192 R\n\u22a2 \u2200 (x : n), x \u2208 Finset.univ \u2192 diag (col a * row b) x = a x * b x\n[PROOFSTEP]\nsimp [mul_apply]\n[GOAL]\n\u03b9 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\n\u03b1 : Type u_5\nR : Type u_6\nS : Type u_7\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype p\ninst\u271d : AddCommMonoid R\nA : Matrix (Fin 3) (Fin 3) R\n\u22a2 trace A = A 0 0 + A 1 1 + A 2 2\n[PROOFSTEP]\nrw [\u2190 add_zero (A 2 2), add_assoc]\n[GOAL]\n\u03b9 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\n\u03b1 : Type u_5\nR : Type u_6\nS : Type u_7\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype p\ninst\u271d : AddCommMonoid R\nA : Matrix (Fin 3) (Fin 3) R\n\u22a2 trace A = A 0 0 + (A 1 1 + (A 2 2 + 0))\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Matrix.Trace", "llama_tokens": 1287, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9032942145139149, "lm_q2_score": 0.8080672227971211, "lm_q1q2_score": 0.7299224472909661}}
{"text": "[GOAL]\nn : \u2115\nsrc\u271d : Mul (Fin n) := inferInstanceAs (Mul (Fin n))\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : Fin n\na : \u2115\nha : a < n\nb : \u2115\nhb : b < n\nc : \u2115\nhc : c < n\n\u22a2 a * b * c \u2261 a * (b * c) [MOD n]\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\nn : \u2115\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : Fin n\na : \u2115\nha : a < n\nb : \u2115\nhb : b < n\nc : \u2115\nhc : c < n\n\u22a2 a * (b + c) \u2261 a * b + a * c [MOD n]\n[PROOFSTEP]\nrw [mul_add]\n[GOAL]\nn : \u2115\nsrc\u271d\u00b9 : AddCommSemigroup (Fin n) := addCommSemigroup n\nsrc\u271d : CommSemigroup (Fin n) := instCommSemigroup n\na b c : Fin n\n\u22a2 (a + b) * c = a * c + b * c\n[PROOFSTEP]\nrw [mul_comm, left_distrib_aux, mul_comm _ b, mul_comm]\n[GOAL]\n\u22a2 Repr (ZMod 0)\n[PROOFSTEP]\ndsimp [ZMod]\n[GOAL]\n\u22a2 Repr \u2124\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nn : \u2115\n\u22a2 Repr (ZMod (n + 1))\n[PROOFSTEP]\ndsimp [ZMod]\n[GOAL]\nn : \u2115\n\u22a2 Repr (Fin (n + 1))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nn : \u2115\ninst\u271d : Fintype (ZMod n)\n\u22a2 Fintype.card (ZMod n) = n\n[PROOFSTEP]\ncases n with\n| zero => exact (not_finite (ZMod 0)).elim\n| succ n => convert Fintype.card_fin (n + 1) using 2\n[GOAL]\nn : \u2115\ninst\u271d : Fintype (ZMod n)\n\u22a2 Fintype.card (ZMod n) = n\n[PROOFSTEP]\ncases n with\n| zero => exact (not_finite (ZMod 0)).elim\n| succ n => convert Fintype.card_fin (n + 1) using 2\n[GOAL]\ncase zero\ninst\u271d : Fintype (ZMod Nat.zero)\n\u22a2 Fintype.card (ZMod Nat.zero) = Nat.zero\n[PROOFSTEP]\n\n| zero => exact (not_finite (ZMod 0)).elim\n[GOAL]\ncase zero\ninst\u271d : Fintype (ZMod Nat.zero)\n\u22a2 Fintype.card (ZMod Nat.zero) = Nat.zero\n[PROOFSTEP]\nexact (not_finite (ZMod 0)).elim\n[GOAL]\ncase succ\nn : \u2115\ninst\u271d : Fintype (ZMod (Nat.succ n))\n\u22a2 Fintype.card (ZMod (Nat.succ n)) = Nat.succ n\n[PROOFSTEP]\n\n| succ n => convert Fintype.card_fin (n + 1) using 2\n[GOAL]\ncase succ\nn : \u2115\ninst\u271d : Fintype (ZMod (Nat.succ n))\n\u22a2 Fintype.card (ZMod (Nat.succ n)) = Nat.succ n\n[PROOFSTEP]\nconvert Fintype.card_fin (n + 1) using 2\n", "meta": {"mathlib_filename": "Mathlib.Data.ZMod.Defs", "llama_tokens": 928, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9124361557147439, "lm_q2_score": 0.7981867825403177, "lm_q1q2_score": 0.7282944794034077}}
{"text": "[GOAL]\nn : \u2115\n\u22a2 range (succ n) = n ::\u2098 range n\n[PROOFSTEP]\nrw [range, List.range_succ, \u2190 coe_add, add_comm]\n[GOAL]\nn : \u2115\n\u22a2 \u2191[n] + \u2191(List.range n) = n ::\u2098 range n\n[PROOFSTEP]\nrfl\n[GOAL]\na : \u2115\nm : Multiset \u2115\n\u22a2 Disjoint (range a) (map (fun x => a + x) m)\n[PROOFSTEP]\nintro x hxa hxb\n[GOAL]\na : \u2115\nm : Multiset \u2115\nx : \u2115\nhxa : x \u2208 range a\nhxb : x \u2208 map (fun x => a + x) m\n\u22a2 False\n[PROOFSTEP]\nrw [range, mem_coe, List.mem_range] at hxa \n[GOAL]\na : \u2115\nm : Multiset \u2115\nx : \u2115\nhxa : x < a\nhxb : x \u2208 map (fun x => a + x) m\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8c, _, rfl\u27e9 := mem_map.1 hxb\n[GOAL]\ncase intro.intro\na : \u2115\nm : Multiset \u2115\nc : \u2115\nleft\u271d : c \u2208 m\nhxa : a + c < a\nhxb : a + c \u2208 map (fun x => a + x) m\n\u22a2 False\n[PROOFSTEP]\nexact (self_le_add_right _ _).not_lt hxa\n[GOAL]\na b : \u2115\n\u22a2 range (a + b) = range a \u222a map (fun x => a + x) (range b)\n[PROOFSTEP]\nrw [range_add, add_eq_union_iff_disjoint]\n[GOAL]\na b : \u2115\n\u22a2 Disjoint (range a) (map (fun x => a + x) (range b))\n[PROOFSTEP]\napply range_disjoint_map_add\n", "meta": {"mathlib_filename": "Mathlib.Data.Multiset.Range", "llama_tokens": 519, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9032942119105695, "lm_q2_score": 0.8031737940012418, "lm_q1q2_score": 0.7255022392795738}}
{"text": "[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf g : \u03b1 \u2192CO \u03b2\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\nobtain \u27e8\u27e8_, _\u27e9, _\u27e9 := f\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ng : \u03b1 \u2192CO \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\ncontinuous_toFun\u271d : Continuous toFun\u271d\nmap_open'\u271d : IsOpenMap (ContinuousMap.mk toFun\u271d).toFun\nh : (fun f => f.toFun) { toContinuousMap := ContinuousMap.mk toFun\u271d, map_open' := map_open'\u271d } = (fun f => f.toFun) g\n\u22a2 { toContinuousMap := ContinuousMap.mk toFun\u271d, map_open' := map_open'\u271d } = g\n[PROOFSTEP]\nobtain \u27e8\u27e8_, _\u27e9, _\u27e9 := g\n[GOAL]\ncase mk.mk.mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\ncontinuous_toFun\u271d\u00b9 : Continuous toFun\u271d\u00b9\nmap_open'\u271d\u00b9 : IsOpenMap (ContinuousMap.mk toFun\u271d\u00b9).toFun\ntoFun\u271d : \u03b1 \u2192 \u03b2\ncontinuous_toFun\u271d : Continuous toFun\u271d\nmap_open'\u271d : IsOpenMap (ContinuousMap.mk toFun\u271d).toFun\nh :\n  (fun f => f.toFun) { toContinuousMap := ContinuousMap.mk toFun\u271d\u00b9, map_open' := map_open'\u271d\u00b9 } =\n    (fun f => f.toFun) { toContinuousMap := ContinuousMap.mk toFun\u271d, map_open' := map_open'\u271d }\n\u22a2 { toContinuousMap := ContinuousMap.mk toFun\u271d\u00b9, map_open' := map_open'\u271d\u00b9 } =\n    { toContinuousMap := ContinuousMap.mk toFun\u271d, map_open' := map_open'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ng : \u03b2 \u2192CO \u03b3\nf\u2081 f\u2082 : \u03b1 \u2192CO \u03b2\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\na : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 a) = \u2191g (\u2191f\u2082 a)\n[PROOFSTEP]\nrw [\u2190 comp_apply, h, comp_apply]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Hom.Open", "llama_tokens": 970, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.894789468908171, "lm_q2_score": 0.8104789178257654, "lm_q1q2_score": 0.7252080004425858}}
{"text": "[GOAL]\nm : \u2115\n\u22a2 \u2191(toPNat' (m + 1)) = if 0 < m + 1 then m + 1 else 1\n[PROOFSTEP]\nrw [if_pos (succ_pos m)]\n[GOAL]\nm : \u2115\n\u22a2 \u2191(toPNat' (m + 1)) = m + 1\n[PROOFSTEP]\nrfl\n[GOAL]\nm k : \u2115+\n\u22a2 \u2191(mod m k) = if \u2191m % \u2191k = 0 then \u2191k else \u2191m % \u2191k\n[PROOFSTEP]\ndsimp [mod, modDiv]\n[GOAL]\nm k : \u2115+\n\u22a2 \u2191(modDivAux k (\u2191m % \u2191k) (\u2191m / \u2191k)).fst = if \u2191m % \u2191k = 0 then \u2191k else \u2191m % \u2191k\n[PROOFSTEP]\ncases (m : \u2115) % (k : \u2115) with\n| zero =>\n  rw [if_pos rfl]\n  rfl\n| succ n =>\n  rw [if_neg n.succ_ne_zero]\n  rfl\n[GOAL]\nm k : \u2115+\nx\u271d : \u2115\n\u22a2 \u2191(modDivAux k x\u271d (\u2191m / \u2191k)).fst = if x\u271d = 0 then \u2191k else x\u271d\n[PROOFSTEP]\ncases (m : \u2115) % (k : \u2115) with\n| zero =>\n  rw [if_pos rfl]\n  rfl\n| succ n =>\n  rw [if_neg n.succ_ne_zero]\n  rfl\n[GOAL]\ncase zero\nm k : \u2115+\n\u22a2 \u2191(modDivAux k zero (\u2191m / \u2191k)).fst = if zero = 0 then \u2191k else zero\n[PROOFSTEP]\n\n| zero =>\n  rw [if_pos rfl]\n  rfl\n[GOAL]\ncase zero\nm k : \u2115+\n\u22a2 \u2191(modDivAux k zero (\u2191m / \u2191k)).fst = if zero = 0 then \u2191k else zero\n[PROOFSTEP]\nrw [if_pos rfl]\n[GOAL]\ncase zero\nm k : \u2115+\n\u22a2 \u2191(modDivAux k zero (\u2191m / \u2191k)).fst = \u2191k\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nm k : \u2115+\nn : \u2115\n\u22a2 \u2191(modDivAux k (succ n) (\u2191m / \u2191k)).fst = if succ n = 0 then \u2191k else succ n\n[PROOFSTEP]\n\n| succ n =>\n  rw [if_neg n.succ_ne_zero]\n  rfl\n[GOAL]\ncase succ\nm k : \u2115+\nn : \u2115\n\u22a2 \u2191(modDivAux k (succ n) (\u2191m / \u2191k)).fst = if succ n = 0 then \u2191k else succ n\n[PROOFSTEP]\nrw [if_neg n.succ_ne_zero]\n[GOAL]\ncase succ\nm k : \u2115+\nn : \u2115\n\u22a2 \u2191(modDivAux k (succ n) (\u2191m / \u2191k)).fst = succ n\n[PROOFSTEP]\nrfl\n[GOAL]\nm k : \u2115+\n\u22a2 div m k = if \u2191m % \u2191k = 0 then pred (\u2191m / \u2191k) else \u2191m / \u2191k\n[PROOFSTEP]\ndsimp [div, modDiv]\n[GOAL]\nm k : \u2115+\n\u22a2 (modDivAux k (\u2191m % \u2191k) (\u2191m / \u2191k)).snd = if \u2191m % \u2191k = 0 then pred (\u2191m / \u2191k) else \u2191m / \u2191k\n[PROOFSTEP]\ncases (m : \u2115) % (k : \u2115) with\n| zero =>\n  rw [if_pos rfl]\n  rfl\n| succ n =>\n  rw [if_neg n.succ_ne_zero]\n  rfl\n[GOAL]\nm k : \u2115+\nx\u271d : \u2115\n\u22a2 (modDivAux k x\u271d (\u2191m / \u2191k)).snd = if x\u271d = 0 then pred (\u2191m / \u2191k) else \u2191m / \u2191k\n[PROOFSTEP]\ncases (m : \u2115) % (k : \u2115) with\n| zero =>\n  rw [if_pos rfl]\n  rfl\n| succ n =>\n  rw [if_neg n.succ_ne_zero]\n  rfl\n[GOAL]\ncase zero\nm k : \u2115+\n\u22a2 (modDivAux k zero (\u2191m / \u2191k)).snd = if zero = 0 then pred (\u2191m / \u2191k) else \u2191m / \u2191k\n[PROOFSTEP]\n\n| zero =>\n  rw [if_pos rfl]\n  rfl\n[GOAL]\ncase zero\nm k : \u2115+\n\u22a2 (modDivAux k zero (\u2191m / \u2191k)).snd = if zero = 0 then pred (\u2191m / \u2191k) else \u2191m / \u2191k\n[PROOFSTEP]\nrw [if_pos rfl]\n[GOAL]\ncase zero\nm k : \u2115+\n\u22a2 (modDivAux k zero (\u2191m / \u2191k)).snd = pred (\u2191m / \u2191k)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nm k : \u2115+\nn : \u2115\n\u22a2 (modDivAux k (succ n) (\u2191m / \u2191k)).snd = if succ n = 0 then pred (\u2191m / \u2191k) else \u2191m / \u2191k\n[PROOFSTEP]\n\n| succ n =>\n  rw [if_neg n.succ_ne_zero]\n  rfl\n[GOAL]\ncase succ\nm k : \u2115+\nn : \u2115\n\u22a2 (modDivAux k (succ n) (\u2191m / \u2191k)).snd = if succ n = 0 then pred (\u2191m / \u2191k) else \u2191m / \u2191k\n[PROOFSTEP]\nrw [if_neg n.succ_ne_zero]\n[GOAL]\ncase succ\nm k : \u2115+\nn : \u2115\n\u22a2 (modDivAux k (succ n) (\u2191m / \u2191k)).snd = \u2191m / \u2191k\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2124\nhn : 0 < n\n\u22a2 \u2191\u2191(Nat.toPNat' (natAbs n)) = n\n[PROOFSTEP]\nrw [Nat.toPNat'_coe, if_pos (Int.natAbs_pos.2 hn.ne'), Int.natAbs_of_nonneg hn.le]\n", "meta": {"mathlib_filename": "Mathlib.Data.PNat.Defs", "llama_tokens": 1740, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894745194283, "lm_q2_score": 0.8080672227971211, "lm_q1q2_score": 0.7230500456630097}}
{"text": "[GOAL]\n\ud835\udd5c\u271d : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d : NonUnitalSeminormedRing \ud835\udd5c\na\u271d b\u271d : \ud835\udd5c\nhx : a\u271d \u2208 ball 0 1\nhy : b\u271d \u2208 ball 0 1\n\u22a2 a\u271d * b\u271d \u2208 ball 0 1\n[PROOFSTEP]\nrw [mem_ball_zero_iff] at *\n[GOAL]\n\ud835\udd5c\u271d : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d : NonUnitalSeminormedRing \ud835\udd5c\na\u271d b\u271d : \ud835\udd5c\nhx : \u2016a\u271d\u2016 < 1\nhy : \u2016b\u271d\u2016 < 1\n\u22a2 \u2016a\u271d * b\u271d\u2016 < 1\n[PROOFSTEP]\nexact (norm_mul_le _ _).trans_lt (mul_lt_one_of_nonneg_of_lt_one_left (norm_nonneg _) hx hy.le)\n[GOAL]\n\ud835\udd5c\u271d : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d : NonUnitalSeminormedRing \ud835\udd5c\na\u271d b\u271d : \ud835\udd5c\nhx : a\u271d \u2208 closedBall 0 1\nhy : b\u271d \u2208 closedBall 0 1\n\u22a2 a\u271d * b\u271d \u2208 closedBall 0 1\n[PROOFSTEP]\nrw [mem_closedBall_zero_iff] at *\n[GOAL]\n\ud835\udd5c\u271d : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d : NonUnitalSeminormedRing \ud835\udd5c\na\u271d b\u271d : \ud835\udd5c\nhx : \u2016a\u271d\u2016 \u2264 1\nhy : \u2016b\u271d\u2016 \u2264 1\n\u22a2 \u2016a\u271d * b\u271d\u2016 \u2264 1\n[PROOFSTEP]\nexact (norm_mul_le _ _).trans (mul_le_one hx (norm_nonneg _) hy)\n[GOAL]\n\ud835\udd5c\u271d : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d : NormedDivisionRing \ud835\udd5c\na\u271d b\u271d : \ud835\udd5c\nhx : a\u271d \u2208 sphere 0 1\nhy : b\u271d \u2208 sphere 0 1\n\u22a2 a\u271d * b\u271d \u2208 sphere 0 1\n[PROOFSTEP]\nrw [mem_sphere_zero_iff_norm] at *\n[GOAL]\n\ud835\udd5c\u271d : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d : NormedDivisionRing \ud835\udd5c\na\u271d b\u271d : \ud835\udd5c\nhx : \u2016a\u271d\u2016 = 1\nhy : \u2016b\u271d\u2016 = 1\n\u22a2 \u2016a\u271d * b\u271d\u2016 = 1\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : NormedDivisionRing \ud835\udd5c\nx : \u2191(sphere 0 1)\n\u22a2 \u2016(\u2191x)\u207b\u00b9\u2016 = 1\n[PROOFSTEP]\nrw [norm_inv, mem_sphere_zero_iff_norm.1 x.coe_prop, inv_one]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : NormedDivisionRing \ud835\udd5c\nx y : \u2191(sphere 0 1)\n\u22a2 \u2016\u2191x / \u2191y\u2016 = 1\n[PROOFSTEP]\nrw [norm_div, mem_sphere_zero_iff_norm.1 x.coe_prop, mem_sphere_zero_iff_norm.1 y.coe_prop, div_one]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : NormedDivisionRing \ud835\udd5c\nx : \u2191(sphere 0 1)\nn : \u2124\n\u22a2 \u2191x ^ n \u2208 sphere 0 1\n[PROOFSTEP]\nrw [mem_sphere_zero_iff_norm, norm_zpow, mem_sphere_zero_iff_norm.1 x.coe_prop, one_zpow]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : NormedDivisionRing \ud835\udd5c\nx y : \u2191(sphere 0 1)\nh : \u2191(unitSphereToUnits \ud835\udd5c) x = \u2191(unitSphereToUnits \ud835\udd5c) y\n\u22a2 \u2191x = \u2191y\n[PROOFSTEP]\nconvert congr_arg Units.val h\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Normed.Field.UnitBall", "llama_tokens": 1155, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.8080672066194946, "lm_q1q2_score": 0.7194175636676402}}
{"text": "[GOAL]\nR : Type u_1\nA : Type u_2\na\u271d b\u271d : A\nr : R\ninst\u271d\u00b3 : OrderedCommRing R\ninst\u271d\u00b2 : OrderedRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : OrderedSMul R A\na b : R\nh : a \u2264 b\n\u22a2 \u2191(algebraMap R A) a \u2264 \u2191(algebraMap R A) b\n[PROOFSTEP]\nrw [Algebra.algebraMap_eq_smul_one, Algebra.algebraMap_eq_smul_one, \u2190 sub_nonneg, \u2190 sub_smul]\n[GOAL]\nR : Type u_1\nA : Type u_2\na\u271d b\u271d : A\nr : R\ninst\u271d\u00b3 : OrderedCommRing R\ninst\u271d\u00b2 : OrderedRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : OrderedSMul R A\na b : R\nh : a \u2264 b\n\u22a2 0 \u2264 (b - a) \u2022 1\n[PROOFSTEP]\ntrans (b - a) \u2022 (0 : A)\n[GOAL]\nR : Type u_1\nA : Type u_2\na\u271d b\u271d : A\nr : R\ninst\u271d\u00b3 : OrderedCommRing R\ninst\u271d\u00b2 : OrderedRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : OrderedSMul R A\na b : R\nh : a \u2264 b\n\u22a2 0 \u2264 (b - a) \u2022 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nA : Type u_2\na\u271d b\u271d : A\nr : R\ninst\u271d\u00b3 : OrderedCommRing R\ninst\u271d\u00b2 : OrderedRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : OrderedSMul R A\na b : R\nh : a \u2264 b\n\u22a2 (b - a) \u2022 0 \u2264 (b - a) \u2022 1\n[PROOFSTEP]\nexact smul_le_smul_of_nonneg zero_le_one (sub_nonneg.mpr h)\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Algebra", "llama_tokens": 565, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9005297807787536, "lm_q2_score": 0.7981867873410141, "lm_q1q2_score": 0.718790972624701}}
{"text": "[GOAL]\np : Prop\nh : Decidable p\nq : Prop\n\u22a2 (if p then q else True) = (\u00acp \u2228 q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\np : Prop\nh : Decidable p\nq : Prop\n\u22a2 (if p then q else True) = (\u00acp \u2228 q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\ncase pos\np : Prop\nh\u271d : Decidable p\nq : Prop\nh : p\n\u22a2 (if p then q else True) = (\u00acp \u2228 q)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\np : Prop\nh\u271d : Decidable p\nq : Prop\nh : \u00acp\n\u22a2 (if p then q else True) = (\u00acp \u2228 q)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\np : Prop\nh : Decidable p\nq : Prop\n\u22a2 (if p then True else q) = (p \u2228 q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\np : Prop\nh : Decidable p\nq : Prop\n\u22a2 (if p then True else q) = (p \u2228 q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\ncase pos\np : Prop\nh\u271d : Decidable p\nq : Prop\nh : p\n\u22a2 (if p then True else q) = (p \u2228 q)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\np : Prop\nh\u271d : Decidable p\nq : Prop\nh : \u00acp\n\u22a2 (if p then True else q) = (p \u2228 q)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\np : Prop\nh : Decidable p\nq : Prop\n\u22a2 (if p then q else False) = (p \u2227 q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\np : Prop\nh : Decidable p\nq : Prop\n\u22a2 (if p then q else False) = (p \u2227 q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\ncase pos\np : Prop\nh\u271d : Decidable p\nq : Prop\nh : p\n\u22a2 (if p then q else False) = (p \u2227 q)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\np : Prop\nh\u271d : Decidable p\nq : Prop\nh : \u00acp\n\u22a2 (if p then q else False) = (p \u2227 q)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\np : Prop\nh : Decidable p\nq : Prop\n\u22a2 (if p then False else q) = (\u00acp \u2227 q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\np : Prop\nh : Decidable p\nq : Prop\n\u22a2 (if p then False else q) = (\u00acp \u2227 q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\ncase pos\np : Prop\nh\u271d : Decidable p\nq : Prop\nh : p\n\u22a2 (if p then False else q) = (\u00acp \u2227 q)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\np : Prop\nh\u271d : Decidable p\nq : Prop\nh : \u00acp\n\u22a2 (if p then False else q) = (\u00acp \u2227 q)\n[PROOFSTEP]\nsimp [h]\n", "meta": {"mathlib_filename": "Mathlib.Init.IteSimp", "llama_tokens": 880, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314738181876, "lm_q2_score": 0.8104788995148791, "lm_q1q2_score": 0.7177856222759051}}
{"text": "[GOAL]\na k : \u2115\nh0 : 0 < a\nh1 : a < k\nn : \u2115\n\u22a2 \u03c0' (k + n) \u2264 card (filter Prime (range k)) + card (filter Prime (Ico k (k + n)))\n[PROOFSTEP]\nrw [primeCounting', count_eq_card_filter_range, range_eq_Ico, \u2190 Ico_union_Ico_eq_Ico (zero_le k) le_self_add,\n  filter_union]\n[GOAL]\na k : \u2115\nh0 : 0 < a\nh1 : a < k\nn : \u2115\n\u22a2 card (filter Prime (Ico 0 k) \u222a filter Prime (Ico k (k + n))) \u2264\n    card (filter Prime (Ico 0 k)) + card (filter Prime (Ico k (k + n)))\n[PROOFSTEP]\napply card_union_le\n[GOAL]\na k : \u2115\nh0 : 0 < a\nh1 : a < k\nn : \u2115\n\u22a2 card (filter Prime (range k)) + card (filter Prime (Ico k (k + n))) \u2264 \u03c0' k + card (filter Prime (Ico k (k + n)))\n[PROOFSTEP]\nrw [primeCounting', count_eq_card_filter_range]\n[GOAL]\na k : \u2115\nh0 : 0 < a\nh1 : a < k\nn : \u2115\n\u22a2 \u03c0' k + card (filter Prime (Ico k (k + n))) \u2264 \u03c0' k + card (filter (coprime a) (Ico k (k + n)))\n[PROOFSTEP]\nrefine' add_le_add_left (card_le_of_subset _) k.primeCounting'\n[GOAL]\na k : \u2115\nh0 : 0 < a\nh1 : a < k\nn : \u2115\n\u22a2 filter Prime (Ico k (k + n)) \u2286 filter (coprime a) (Ico k (k + n))\n[PROOFSTEP]\nsimp only [subset_iff, and_imp, mem_filter, mem_Ico]\n[GOAL]\na k : \u2115\nh0 : 0 < a\nh1 : a < k\nn : \u2115\n\u22a2 \u2200 \u2983x : \u2115\u2984, k \u2264 x \u2192 x < k + n \u2192 Prime x \u2192 (k \u2264 x \u2227 x < k + n) \u2227 coprime a x\n[PROOFSTEP]\nintro p succ_k_le_p p_lt_n p_prime\n[GOAL]\na k : \u2115\nh0 : 0 < a\nh1 : a < k\nn p : \u2115\nsucc_k_le_p : k \u2264 p\np_lt_n : p < k + n\np_prime : Prime p\n\u22a2 (k \u2264 p \u2227 p < k + n) \u2227 coprime a p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\na k : \u2115\nh0 : 0 < a\nh1 : a < k\nn p : \u2115\nsucc_k_le_p : k \u2264 p\np_lt_n : p < k + n\np_prime : Prime p\n\u22a2 k \u2264 p \u2227 p < k + n\n[PROOFSTEP]\nexact \u27e8succ_k_le_p, p_lt_n\u27e9\n[GOAL]\ncase right\na k : \u2115\nh0 : 0 < a\nh1 : a < k\nn p : \u2115\nsucc_k_le_p : k \u2264 p\np_lt_n : p < k + n\np_prime : Prime p\n\u22a2 coprime a p\n[PROOFSTEP]\nrw [coprime_comm]\n[GOAL]\ncase right\na k : \u2115\nh0 : 0 < a\nh1 : a < k\nn p : \u2115\nsucc_k_le_p : k \u2264 p\np_lt_n : p < k + n\np_prime : Prime p\n\u22a2 coprime p a\n[PROOFSTEP]\nexact coprime_of_lt_prime h0 (gt_of_ge_of_gt succ_k_le_p h1) p_prime\n[GOAL]\na k : \u2115\nh0 : 0 < a\nh1 : a < k\nn : \u2115\n\u22a2 \u03c0' k + card (filter (coprime a) (Ico k (k + n))) \u2264 \u03c0' k + \u03c6 a * (n / a + 1)\n[PROOFSTEP]\nrw [add_le_add_iff_left]\n[GOAL]\na k : \u2115\nh0 : 0 < a\nh1 : a < k\nn : \u2115\n\u22a2 card (filter (coprime a) (Ico k (k + n))) \u2264 \u03c6 a * (n / a + 1)\n[PROOFSTEP]\nexact Ico_filter_coprime_le k n h0\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.PrimeCounting", "llama_tokens": 1230, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.894789468908171, "lm_q2_score": 0.8006919949619792, "lm_q1q2_score": 0.7164507649310533}}
{"text": "[GOAL]\nk x y : \u2115\nh : x \u2264 y / k\n\u22a2 x * k \u2264 y\n[PROOFSTEP]\nby_cases hk : k = 0\n[GOAL]\ncase pos\nk x y : \u2115\nh : x \u2264 y / k\nhk : k = 0\n\u22a2 x * k \u2264 y\ncase neg k x y : \u2115 h : x \u2264 y / k hk : \u00ack = 0 \u22a2 x * k \u2264 y\n[PROOFSTEP]\ncase pos => rw [hk, mul_zero]; exact zero_le _\n[GOAL]\nk x y : \u2115\nh : x \u2264 y / k\nhk : k = 0\n\u22a2 x * k \u2264 y\n[PROOFSTEP]\ncase pos => rw [hk, mul_zero]; exact zero_le _\n[GOAL]\nk x y : \u2115\nh : x \u2264 y / k\nhk : k = 0\n\u22a2 x * k \u2264 y\n[PROOFSTEP]\nrw [hk, mul_zero]\n[GOAL]\nk x y : \u2115\nh : x \u2264 y / k\nhk : k = 0\n\u22a2 0 \u2264 y\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase neg\nk x y : \u2115\nh : x \u2264 y / k\nhk : \u00ack = 0\n\u22a2 x * k \u2264 y\n[PROOFSTEP]\ncase neg => rwa [\u2190 le_div_iff_mul_le (pos_iff_ne_zero.2 hk)]\n[GOAL]\nk x y : \u2115\nh : x \u2264 y / k\nhk : \u00ack = 0\n\u22a2 x * k \u2264 y\n[PROOFSTEP]\ncase neg => rwa [\u2190 le_div_iff_mul_le (pos_iff_ne_zero.2 hk)]\n[GOAL]\nk x y : \u2115\nh : x \u2264 y / k\nhk : \u00ack = 0\n\u22a2 x * k \u2264 y\n[PROOFSTEP]\nrwa [\u2190 le_div_iff_mul_le (pos_iff_ne_zero.2 hk)]\n[GOAL]\na b c d : \u2115\n\u22a2 a / b * (c / d) \u2264 a * c / (b * d)\n[PROOFSTEP]\nby_cases hb : b = 0\n[GOAL]\ncase pos\na b c d : \u2115\nhb : b = 0\n\u22a2 a / b * (c / d) \u2264 a * c / (b * d)\ncase neg a b c d : \u2115 hb : \u00acb = 0 \u22a2 a / b * (c / d) \u2264 a * c / (b * d)\n[PROOFSTEP]\ncase pos => simp [hb]\n[GOAL]\na b c d : \u2115\nhb : b = 0\n\u22a2 a / b * (c / d) \u2264 a * c / (b * d)\n[PROOFSTEP]\ncase pos => simp [hb]\n[GOAL]\na b c d : \u2115\nhb : b = 0\n\u22a2 a / b * (c / d) \u2264 a * c / (b * d)\n[PROOFSTEP]\nsimp [hb]\n[GOAL]\ncase neg\na b c d : \u2115\nhb : \u00acb = 0\n\u22a2 a / b * (c / d) \u2264 a * c / (b * d)\n[PROOFSTEP]\nby_cases hd : d = 0\n[GOAL]\ncase pos\na b c d : \u2115\nhb : \u00acb = 0\nhd : d = 0\n\u22a2 a / b * (c / d) \u2264 a * c / (b * d)\ncase neg a b c d : \u2115 hb : \u00acb = 0 hd : \u00acd = 0 \u22a2 a / b * (c / d) \u2264 a * c / (b * d)\n[PROOFSTEP]\ncase pos => simp [hd]\n[GOAL]\na b c d : \u2115\nhb : \u00acb = 0\nhd : d = 0\n\u22a2 a / b * (c / d) \u2264 a * c / (b * d)\n[PROOFSTEP]\ncase pos => simp [hd]\n[GOAL]\na b c d : \u2115\nhb : \u00acb = 0\nhd : d = 0\n\u22a2 a / b * (c / d) \u2264 a * c / (b * d)\n[PROOFSTEP]\nsimp [hd]\n[GOAL]\ncase neg\na b c d : \u2115\nhb : \u00acb = 0\nhd : \u00acd = 0\n\u22a2 a / b * (c / d) \u2264 a * c / (b * d)\n[PROOFSTEP]\nhave hbd : b * d \u2260 0 := mul_ne_zero hb hd\n[GOAL]\ncase neg\na b c d : \u2115\nhb : \u00acb = 0\nhd : \u00acd = 0\nhbd : b * d \u2260 0\n\u22a2 a / b * (c / d) \u2264 a * c / (b * d)\n[PROOFSTEP]\nrw [le_div_iff_mul_le (Nat.pos_of_ne_zero hbd)]\n[GOAL]\ncase neg\na b c d : \u2115\nhb : \u00acb = 0\nhd : \u00acd = 0\nhbd : b * d \u2260 0\n\u22a2 a / b * (c / d) * (b * d) \u2264 a * c\n[PROOFSTEP]\ntransitivity ((a / b) * b) * ((c / d) * d)\n[GOAL]\ncase neg.a\na b c d : \u2115\nhb : \u00acb = 0\nhd : \u00acd = 0\nhbd : b * d \u2260 0\n\u22a2 a / b * (c / d) * (b * d) \u2264 a / b * b * (c / d * d)\n[PROOFSTEP]\napply le_of_eq\n[GOAL]\ncase neg.a.a\na b c d : \u2115\nhb : \u00acb = 0\nhd : \u00acd = 0\nhbd : b * d \u2260 0\n\u22a2 a / b * (c / d) * (b * d) = a / b * b * (c / d * d)\n[PROOFSTEP]\nsimp only [mul_assoc, mul_left_comm]\n[GOAL]\ncase neg.a\na b c d : \u2115\nhb : \u00acb = 0\nhd : \u00acd = 0\nhbd : b * d \u2260 0\n\u22a2 a / b * b * (c / d * d) \u2264 a * c\n[PROOFSTEP]\napply Nat.mul_le_mul\n[GOAL]\ncase neg.a.h\u2081\na b c d : \u2115\nhb : \u00acb = 0\nhd : \u00acd = 0\nhbd : b * d \u2260 0\n\u22a2 a / b * b \u2264 a\n[PROOFSTEP]\napply div_mul_le_self\n[GOAL]\ncase neg.a.h\u2082\na b c d : \u2115\nhb : \u00acb = 0\nhd : \u00acd = 0\nhbd : b * d \u2260 0\n\u22a2 c / d * d \u2264 c\n[PROOFSTEP]\napply div_mul_le_self\n[GOAL]\nn k : \u2115\n\u22a2 let iter_next := fun n guess => (guess + n / guess) / 2;\n  sqrt.iter n k \u2264 iter_next n (sqrt.iter n k)\n[PROOFSTEP]\nintro iter_next\n[GOAL]\nn k : \u2115\niter_next : \u2115 \u2192 \u2115 \u2192 \u2115 := fun n guess => (guess + n / guess) / 2\n\u22a2 sqrt.iter n k \u2264 iter_next n (sqrt.iter n k)\n[PROOFSTEP]\nunfold sqrt.iter\n[GOAL]\nn k : \u2115\niter_next : \u2115 \u2192 \u2115 \u2192 \u2115 := fun n guess => (guess + n / guess) / 2\n\u22a2 (let next := (k + n / k) / 2;\n    if _h : next < k then sqrt.iter n next else k) \u2264\n    iter_next n\n      (let next := (k + n / k) / 2;\n      if _h : next < k then sqrt.iter n next else k)\n[PROOFSTEP]\nby_cases h : (k + n / k) / 2 < k\n[GOAL]\ncase pos\nn k : \u2115\niter_next : \u2115 \u2192 \u2115 \u2192 \u2115 := fun n guess => (guess + n / guess) / 2\nh : (k + n / k) / 2 < k\n\u22a2 (let next := (k + n / k) / 2;\n    if _h : next < k then sqrt.iter n next else k) \u2264\n    iter_next n\n      (let next := (k + n / k) / 2;\n      if _h : next < k then sqrt.iter n next else k)\ncase neg\nn k : \u2115\niter_next : \u2115 \u2192 \u2115 \u2192 \u2115 := fun n guess => (guess + n / guess) / 2\nh : \u00ac(k + n / k) / 2 < k\n\u22a2 (let next := (k + n / k) / 2;\n    if _h : next < k then sqrt.iter n next else k) \u2264\n    iter_next n\n      (let next := (k + n / k) / 2;\n      if _h : next < k then sqrt.iter n next else k)\n[PROOFSTEP]\ncase pos => simp [if_pos h]; exact iter_fp_bound _ _\n[GOAL]\nn k : \u2115\niter_next : \u2115 \u2192 \u2115 \u2192 \u2115 := fun n guess => (guess + n / guess) / 2\nh : (k + n / k) / 2 < k\n\u22a2 (let next := (k + n / k) / 2;\n    if _h : next < k then sqrt.iter n next else k) \u2264\n    iter_next n\n      (let next := (k + n / k) / 2;\n      if _h : next < k then sqrt.iter n next else k)\n[PROOFSTEP]\ncase pos => simp [if_pos h]; exact iter_fp_bound _ _\n[GOAL]\nn k : \u2115\niter_next : \u2115 \u2192 \u2115 \u2192 \u2115 := fun n guess => (guess + n / guess) / 2\nh : (k + n / k) / 2 < k\n\u22a2 (let next := (k + n / k) / 2;\n    if _h : next < k then sqrt.iter n next else k) \u2264\n    iter_next n\n      (let next := (k + n / k) / 2;\n      if _h : next < k then sqrt.iter n next else k)\n[PROOFSTEP]\nsimp [if_pos h]\n[GOAL]\nn k : \u2115\niter_next : \u2115 \u2192 \u2115 \u2192 \u2115 := fun n guess => (guess + n / guess) / 2\nh : (k + n / k) / 2 < k\n\u22a2 sqrt.iter n ((k + n / k) / 2) \u2264 (sqrt.iter n ((k + n / k) / 2) + n / sqrt.iter n ((k + n / k) / 2)) / 2\n[PROOFSTEP]\nexact iter_fp_bound _ _\n[GOAL]\ncase neg\nn k : \u2115\niter_next : \u2115 \u2192 \u2115 \u2192 \u2115 := fun n guess => (guess + n / guess) / 2\nh : \u00ac(k + n / k) / 2 < k\n\u22a2 (let next := (k + n / k) / 2;\n    if _h : next < k then sqrt.iter n next else k) \u2264\n    iter_next n\n      (let next := (k + n / k) / 2;\n      if _h : next < k then sqrt.iter n next else k)\n[PROOFSTEP]\ncase neg => simp [if_neg h]; exact Nat.le_of_not_lt h\n[GOAL]\nn k : \u2115\niter_next : \u2115 \u2192 \u2115 \u2192 \u2115 := fun n guess => (guess + n / guess) / 2\nh : \u00ac(k + n / k) / 2 < k\n\u22a2 (let next := (k + n / k) / 2;\n    if _h : next < k then sqrt.iter n next else k) \u2264\n    iter_next n\n      (let next := (k + n / k) / 2;\n      if _h : next < k then sqrt.iter n next else k)\n[PROOFSTEP]\ncase neg => simp [if_neg h]; exact Nat.le_of_not_lt h\n[GOAL]\nn k : \u2115\niter_next : \u2115 \u2192 \u2115 \u2192 \u2115 := fun n guess => (guess + n / guess) / 2\nh : \u00ac(k + n / k) / 2 < k\n\u22a2 (let next := (k + n / k) / 2;\n    if _h : next < k then sqrt.iter n next else k) \u2264\n    iter_next n\n      (let next := (k + n / k) / 2;\n      if _h : next < k then sqrt.iter n next else k)\n[PROOFSTEP]\nsimp [if_neg h]\n[GOAL]\nn k : \u2115\niter_next : \u2115 \u2192 \u2115 \u2192 \u2115 := fun n guess => (guess + n / guess) / 2\nh : \u00ac(k + n / k) / 2 < k\n\u22a2 k \u2264 (k + n / k) / 2\n[PROOFSTEP]\nexact Nat.le_of_not_lt h\n[GOAL]\nx\u271d : \u2115\n\u22a2 4 * 0 * x\u271d \u2264 (0 + x\u271d) * (0 + x\u271d)\n[PROOFSTEP]\nrw [mul_zero, zero_mul]\n[GOAL]\nx\u271d : \u2115\n\u22a2 0 \u2264 (0 + x\u271d) * (0 + x\u271d)\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\nx\u271d : \u2115\n\u22a2 4 * x\u271d * 0 \u2264 (x\u271d + 0) * (x\u271d + 0)\n[PROOFSTEP]\nrw [mul_zero]\n[GOAL]\nx\u271d : \u2115\n\u22a2 0 \u2264 (x\u271d + 0) * (x\u271d + 0)\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\na b : \u2115\n\u22a2 4 * (a + 1) * (b + 1) \u2264 (a + 1 + (b + 1)) * (a + 1 + (b + 1))\n[PROOFSTEP]\nhave ih := add_le_add_right (@AM_GM a b) 4\n[GOAL]\na b : \u2115\nih : 4 * a * b + 4 \u2264 (a + b) * (a + b) + 4\n\u22a2 4 * (a + 1) * (b + 1) \u2264 (a + 1 + (b + 1)) * (a + 1 + (b + 1))\n[PROOFSTEP]\nsimp only [mul_add, add_mul, show (4 : \u2115) = 1 + 1 + 1 + 1 from rfl, one_mul, mul_one] at ih \u22a2\n[GOAL]\na b : \u2115\nih : a * b + a * b + a * b + a * b + (1 + 1 + 1 + 1) \u2264 a * a + b * a + (a * b + b * b) + (1 + 1 + 1 + 1)\n\u22a2 a * b + a * b + a * b + a * b + (b + b + b + b) + (a + a + a + a + (1 + 1 + 1 + 1)) \u2264\n    a * a + a + (b * a + a) + (a + 1 + (b + 1)) + (a * b + b + (b * b + b) + (a + 1 + (b + 1)))\n[PROOFSTEP]\nsimp only [add_assoc, add_left_comm, add_le_add_iff_left] at ih \u22a2\n[GOAL]\na b : \u2115\nih :\n  a * b + (a * b + (a * b + (a * b + (1 + (1 + (1 + 1)))))) \u2264 a * a + (a * b + (b * a + (b * b + (1 + (1 + (1 + 1))))))\n\u22a2 a * b + (a * b + (a * b + (a * b + (1 + (1 + (1 + 1)))))) \u2264 a * a + (a * b + (b * a + (b * b + (1 + (1 + (1 + 1))))))\n[PROOFSTEP]\nexact ih\n[GOAL]\nn guess : \u2115\n\u22a2 iter n guess * iter n guess \u2264 n\n[PROOFSTEP]\nunfold sqrt.iter\n[GOAL]\nn guess : \u2115\n\u22a2 ((let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) *\n      let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) \u2264\n    n\n[PROOFSTEP]\nlet next := (guess + n / guess) / 2\n[GOAL]\nn guess : \u2115\nnext : \u2115 := (guess + n / guess) / 2\n\u22a2 ((let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) *\n      let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) \u2264\n    n\n[PROOFSTEP]\nby_cases h : next < guess\n[GOAL]\ncase pos\nn guess : \u2115\nnext : \u2115 := (guess + n / guess) / 2\nh : next < guess\n\u22a2 ((let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) *\n      let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) \u2264\n    n\ncase neg\nn guess : \u2115\nnext : \u2115 := (guess + n / guess) / 2\nh : \u00acnext < guess\n\u22a2 ((let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) *\n      let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) \u2264\n    n\n[PROOFSTEP]\ncase pos => simpa only [dif_pos h] using sqrt.iter_sq_le n next\n[GOAL]\nn guess : \u2115\nnext : \u2115 := (guess + n / guess) / 2\nh : next < guess\n\u22a2 ((let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) *\n      let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) \u2264\n    n\n[PROOFSTEP]\ncase pos => simpa only [dif_pos h] using sqrt.iter_sq_le n next\n[GOAL]\nn guess : \u2115\nnext : \u2115 := (guess + n / guess) / 2\nh : next < guess\n\u22a2 ((let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) *\n      let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) \u2264\n    n\n[PROOFSTEP]\nsimpa only [dif_pos h] using sqrt.iter_sq_le n next\n[GOAL]\ncase neg\nn guess : \u2115\nnext : \u2115 := (guess + n / guess) / 2\nh : \u00acnext < guess\n\u22a2 ((let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) *\n      let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) \u2264\n    n\n[PROOFSTEP]\ncase neg =>\n  simp only [dif_neg h]\n  apply Nat.mul_le_of_le_div\n  apply le_of_add_le_add_left (a := guess)\n  rw [\u2190 mul_two, \u2190 le_div_iff_mul_le]\n  \u00b7 exact le_of_not_lt h\n  \u00b7 exact zero_lt_two\n[GOAL]\nn guess : \u2115\nnext : \u2115 := (guess + n / guess) / 2\nh : \u00acnext < guess\n\u22a2 ((let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) *\n      let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) \u2264\n    n\n[PROOFSTEP]\ncase neg =>\n  simp only [dif_neg h]\n  apply Nat.mul_le_of_le_div\n  apply le_of_add_le_add_left (a := guess)\n  rw [\u2190 mul_two, \u2190 le_div_iff_mul_le]\n  \u00b7 exact le_of_not_lt h\n  \u00b7 exact zero_lt_two\n[GOAL]\nn guess : \u2115\nnext : \u2115 := (guess + n / guess) / 2\nh : \u00acnext < guess\n\u22a2 ((let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) *\n      let next := (guess + n / guess) / 2;\n      if _h : next < guess then iter n next else guess) \u2264\n    n\n[PROOFSTEP]\nsimp only [dif_neg h]\n[GOAL]\nn guess : \u2115\nnext : \u2115 := (guess + n / guess) / 2\nh : \u00acnext < guess\n\u22a2 guess * guess \u2264 n\n[PROOFSTEP]\napply Nat.mul_le_of_le_div\n[GOAL]\ncase h\nn guess : \u2115\nnext : \u2115 := (guess + n / guess) / 2\nh : \u00acnext < guess\n\u22a2 guess \u2264 n / guess\n[PROOFSTEP]\napply le_of_add_le_add_left (a := guess)\n[GOAL]\ncase h\nn guess : \u2115\nnext : \u2115 := (guess + n / guess) / 2\nh : \u00acnext < guess\n\u22a2 guess + guess \u2264 guess + n / guess\n[PROOFSTEP]\nrw [\u2190 mul_two, \u2190 le_div_iff_mul_le]\n[GOAL]\ncase h\nn guess : \u2115\nnext : \u2115 := (guess + n / guess) / 2\nh : \u00acnext < guess\n\u22a2 guess \u2264 (guess + n / guess) / 2\n[PROOFSTEP]\nexact le_of_not_lt h\n[GOAL]\ncase h\nn guess : \u2115\nnext : \u2115 := (guess + n / guess) / 2\nh : \u00acnext < guess\n\u22a2 0 < 2\n[PROOFSTEP]\nexact zero_lt_two\n[GOAL]\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\n\u22a2 n < (iter n guess + 1) * (iter n guess + 1)\n[PROOFSTEP]\nunfold sqrt.iter\n[GOAL]\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\n\u22a2 n <\n    ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1) *\n      ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1)\n[PROOFSTEP]\nlet m := (guess + n / guess) / 2\n[GOAL]\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\n\u22a2 n <\n    ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1) *\n      ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1)\n[PROOFSTEP]\nby_cases h : m < guess\n[GOAL]\ncase pos\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : m < guess\n\u22a2 n <\n    ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1) *\n      ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1)\ncase neg\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : \u00acm < guess\n\u22a2 n <\n    ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1) *\n      ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1)\n[PROOFSTEP]\ncase pos =>\n  suffices : n < (m + 1) * (m + 1)\n  \u00b7 simpa only [dif_pos h] using sqrt.lt_iter_succ_sq n m this\n  refine lt_of_mul_lt_mul_left ?_ (4 * (guess * guess)).zero_le\n  apply lt_of_le_of_lt AM_GM\n  rw [show (4 : \u2115) = 2 * 2 from rfl]\n  rw [mul_mul_mul_comm 2, mul_mul_mul_comm (2 * guess)]\n  refine mul_self_lt_mul_self (?_ : _ < _ * succ (_ / 2))\n  rw [\u2190 add_div_right _ (by decide), mul_comm 2, mul_assoc,\n    show guess + n / guess + 2 = (guess + n / guess + 1) + 1 from rfl]\n  have aux_lemma {a : \u2115} : a \u2264 2 * ((a + 1) / 2) := by\n    rw [mul_comm]\n    exact (add_le_add_iff_right 2).1 $ succ_le_of_lt $ @lt_div_mul_add (a + 1) 2 zero_lt_two\n  refine lt_of_lt_of_le ?_ (act_rel_act_of_rel _ aux_lemma)\n  rw [add_assoc, mul_add]\n  exact add_lt_add_left (lt_mul_div_succ _ (lt_of_le_of_lt (Nat.zero_le m) h)) _\n[GOAL]\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : m < guess\n\u22a2 n <\n    ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1) *\n      ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1)\n[PROOFSTEP]\ncase pos =>\n  suffices : n < (m + 1) * (m + 1)\n  \u00b7 simpa only [dif_pos h] using sqrt.lt_iter_succ_sq n m this\n  refine lt_of_mul_lt_mul_left ?_ (4 * (guess * guess)).zero_le\n  apply lt_of_le_of_lt AM_GM\n  rw [show (4 : \u2115) = 2 * 2 from rfl]\n  rw [mul_mul_mul_comm 2, mul_mul_mul_comm (2 * guess)]\n  refine mul_self_lt_mul_self (?_ : _ < _ * succ (_ / 2))\n  rw [\u2190 add_div_right _ (by decide), mul_comm 2, mul_assoc,\n    show guess + n / guess + 2 = (guess + n / guess + 1) + 1 from rfl]\n  have aux_lemma {a : \u2115} : a \u2264 2 * ((a + 1) / 2) := by\n    rw [mul_comm]\n    exact (add_le_add_iff_right 2).1 $ succ_le_of_lt $ @lt_div_mul_add (a + 1) 2 zero_lt_two\n  refine lt_of_lt_of_le ?_ (act_rel_act_of_rel _ aux_lemma)\n  rw [add_assoc, mul_add]\n  exact add_lt_add_left (lt_mul_div_succ _ (lt_of_le_of_lt (Nat.zero_le m) h)) _\n[GOAL]\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : m < guess\n\u22a2 n <\n    ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1) *\n      ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1)\n[PROOFSTEP]\nsuffices : n < (m + 1) * (m + 1)\n[GOAL]\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : m < guess\nthis : n < (m + 1) * (m + 1)\n\u22a2 n <\n    ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1) *\n      ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1)\n[PROOFSTEP]\nsimpa only [dif_pos h] using sqrt.lt_iter_succ_sq n m this\n[GOAL]\ncase this\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : m < guess\n\u22a2 n < (m + 1) * (m + 1)\n[PROOFSTEP]\nrefine lt_of_mul_lt_mul_left ?_ (4 * (guess * guess)).zero_le\n[GOAL]\ncase this\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : m < guess\n\u22a2 4 * (guess * guess) * n < 4 * (guess * guess) * ((m + 1) * (m + 1))\n[PROOFSTEP]\napply lt_of_le_of_lt AM_GM\n[GOAL]\ncase this\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : m < guess\n\u22a2 (guess * guess + n) * (guess * guess + n) < 4 * (guess * guess) * ((m + 1) * (m + 1))\n[PROOFSTEP]\nrw [show (4 : \u2115) = 2 * 2 from rfl]\n[GOAL]\ncase this\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : m < guess\n\u22a2 (guess * guess + n) * (guess * guess + n) < 2 * 2 * (guess * guess) * ((m + 1) * (m + 1))\n[PROOFSTEP]\nrw [mul_mul_mul_comm 2, mul_mul_mul_comm (2 * guess)]\n[GOAL]\ncase this\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : m < guess\n\u22a2 (guess * guess + n) * (guess * guess + n) < 2 * guess * (m + 1) * (2 * guess * (m + 1))\n[PROOFSTEP]\nrefine mul_self_lt_mul_self (?_ : _ < _ * succ (_ / 2))\n[GOAL]\ncase this\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : m < guess\n\u22a2 guess * guess + n < 2 * guess * succ ((guess + n / guess) / 2)\n[PROOFSTEP]\nrw [\u2190 add_div_right _ (by decide), mul_comm 2, mul_assoc,\n  show guess + n / guess + 2 = (guess + n / guess + 1) + 1 from rfl]\n[GOAL]\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : m < guess\n\u22a2 0 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\ncase this\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : m < guess\n\u22a2 guess * guess + n < guess * (2 * ((guess + n / guess + 1 + 1) / 2))\n[PROOFSTEP]\nhave aux_lemma {a : \u2115} : a \u2264 2 * ((a + 1) / 2) := by\n  rw [mul_comm]\n  exact (add_le_add_iff_right 2).1 $ succ_le_of_lt $ @lt_div_mul_add (a + 1) 2 zero_lt_two\n[GOAL]\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : m < guess\na : \u2115\n\u22a2 a \u2264 2 * ((a + 1) / 2)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : m < guess\na : \u2115\n\u22a2 a \u2264 (a + 1) / 2 * 2\n[PROOFSTEP]\nexact (add_le_add_iff_right 2).1 $ succ_le_of_lt $ @lt_div_mul_add (a + 1) 2 zero_lt_two\n[GOAL]\ncase this\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : m < guess\naux_lemma : \u2200 {a : \u2115}, a \u2264 2 * ((a + 1) / 2)\n\u22a2 guess * guess + n < guess * (2 * ((guess + n / guess + 1 + 1) / 2))\n[PROOFSTEP]\nrefine lt_of_lt_of_le ?_ (act_rel_act_of_rel _ aux_lemma)\n[GOAL]\ncase this\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : m < guess\naux_lemma : \u2200 {a : \u2115}, a \u2264 2 * ((a + 1) / 2)\n\u22a2 guess * guess + n < guess * (guess + n / guess + 1)\n[PROOFSTEP]\nrw [add_assoc, mul_add]\n[GOAL]\ncase this\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : m < guess\naux_lemma : \u2200 {a : \u2115}, a \u2264 2 * ((a + 1) / 2)\n\u22a2 guess * guess + n < guess * guess + guess * (n / guess + 1)\n[PROOFSTEP]\nexact add_lt_add_left (lt_mul_div_succ _ (lt_of_le_of_lt (Nat.zero_le m) h)) _\n[GOAL]\ncase neg\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : \u00acm < guess\n\u22a2 n <\n    ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1) *\n      ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1)\n[PROOFSTEP]\ncase neg => simpa only [dif_neg h] using hn\n[GOAL]\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : \u00acm < guess\n\u22a2 n <\n    ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1) *\n      ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1)\n[PROOFSTEP]\ncase neg => simpa only [dif_neg h] using hn\n[GOAL]\nn guess : \u2115\nhn : n < (guess + 1) * (guess + 1)\nm : \u2115 := (guess + n / guess) / 2\nh : \u00acm < guess\n\u22a2 n <\n    ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1) *\n      ((let next := (guess + n / guess) / 2;\n        if _h : next < guess then iter n next else guess) +\n        1)\n[PROOFSTEP]\nsimpa only [dif_neg h] using hn\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.ForSqrt", "llama_tokens": 10044, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9059898203834277, "lm_q2_score": 0.7905303186696747, "lm_q1q2_score": 0.7162124214191924}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedRing A\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b9 : CompleteSpace A\ninst\u271d : ProperSpace \ud835\udd5c\n\u22a2 CompactSpace \u2191(characterSpace \ud835\udd5c A)\n[PROOFSTEP]\nrw [\u2190 isCompact_iff_compactSpace]\n[GOAL]\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedRing A\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b9 : CompleteSpace A\ninst\u271d : ProperSpace \ud835\udd5c\n\u22a2 IsCompact (characterSpace \ud835\udd5c A)\n[PROOFSTEP]\nhave h : characterSpace \ud835\udd5c A \u2286 toNormedDual \u207b\u00b9' Metric.closedBall 0 \u2016(1 : A)\u2016 :=\n  by\n  intro \u03c6 h\u03c6\n  rw [Set.mem_preimage, mem_closedBall_zero_iff]\n  exact (norm_le_norm_one \u27e8\u03c6, \u27e8h\u03c6.1, h\u03c6.2\u27e9\u27e9 : _)\n[GOAL]\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedRing A\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b9 : CompleteSpace A\ninst\u271d : ProperSpace \ud835\udd5c\n\u22a2 characterSpace \ud835\udd5c A \u2286 \u2191toNormedDual \u207b\u00b9' Metric.closedBall 0 \u20161\u2016\n[PROOFSTEP]\nintro \u03c6 h\u03c6\n[GOAL]\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedRing A\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b9 : CompleteSpace A\ninst\u271d : ProperSpace \ud835\udd5c\n\u03c6 : WeakDual \ud835\udd5c A\nh\u03c6 : \u03c6 \u2208 characterSpace \ud835\udd5c A\n\u22a2 \u03c6 \u2208 \u2191toNormedDual \u207b\u00b9' Metric.closedBall 0 \u20161\u2016\n[PROOFSTEP]\nrw [Set.mem_preimage, mem_closedBall_zero_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedRing A\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b9 : CompleteSpace A\ninst\u271d : ProperSpace \ud835\udd5c\n\u03c6 : WeakDual \ud835\udd5c A\nh\u03c6 : \u03c6 \u2208 characterSpace \ud835\udd5c A\n\u22a2 \u2016\u2191toNormedDual \u03c6\u2016 \u2264 \u20161\u2016\n[PROOFSTEP]\nexact (norm_le_norm_one \u27e8\u03c6, \u27e8h\u03c6.1, h\u03c6.2\u27e9\u27e9 : _)\n[GOAL]\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedRing A\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b9 : CompleteSpace A\ninst\u271d : ProperSpace \ud835\udd5c\nh : characterSpace \ud835\udd5c A \u2286 \u2191toNormedDual \u207b\u00b9' Metric.closedBall 0 \u20161\u2016\n\u22a2 IsCompact (characterSpace \ud835\udd5c A)\n[PROOFSTEP]\nexact isCompact_of_isClosed_subset (isCompact_closedBall \ud835\udd5c 0 _) CharacterSpace.isClosed h\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.Algebra", "llama_tokens": 945, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970811069351, "lm_q2_score": 0.8128673223709251, "lm_q1q2_score": 0.7159711648715209}}
{"text": "[GOAL]\nn m : \u2115\n\u22a2 dist n m = dist m n\n[PROOFSTEP]\nsimp [dist.def, add_comm]\n[GOAL]\nn : \u2115\n\u22a2 dist n n = 0\n[PROOFSTEP]\nsimp [dist.def, tsub_self]\n[GOAL]\nn m : \u2115\nh : n = m\n\u22a2 dist n m = 0\n[PROOFSTEP]\nrw [h, dist_self]\n[GOAL]\nn m : \u2115\nh : n \u2264 m\n\u22a2 dist n m = m - n\n[PROOFSTEP]\nrw [dist.def, tsub_eq_zero_iff_le.mpr h, zero_add]\n[GOAL]\nn m : \u2115\nh : m \u2264 n\n\u22a2 dist n m = n - m\n[PROOFSTEP]\nrw [dist_comm]\n[GOAL]\nn m : \u2115\nh : m \u2264 n\n\u22a2 dist m n = n - m\n[PROOFSTEP]\napply dist_eq_sub_of_le h\n[GOAL]\nn m : \u2115\n\u22a2 m \u2264 n + dist n m\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\nn m : \u2115\n\u22a2 m \u2264 dist n m + n\n[PROOFSTEP]\napply dist_tri_left\n[GOAL]\nn m : \u2115\n\u22a2 n \u2264 dist n m + m\n[PROOFSTEP]\nrw [dist_comm]\n[GOAL]\nn m : \u2115\n\u22a2 n \u2264 dist m n + m\n[PROOFSTEP]\napply dist_tri_left\n[GOAL]\nn m : \u2115\n\u22a2 n \u2264 m + dist n m\n[PROOFSTEP]\nrw [dist_comm]\n[GOAL]\nn m : \u2115\n\u22a2 n \u2264 m + dist m n\n[PROOFSTEP]\napply dist_tri_right\n[GOAL]\nn k m : \u2115\n\u22a2 n + k - (m + k) + (m + k - (n + k)) = n - m + (m + k - (n + k))\n[PROOFSTEP]\nrw [@add_tsub_add_eq_tsub_right]\n[GOAL]\nn k m : \u2115\n\u22a2 n - m + (m + k - (n + k)) = n - m + (m - n)\n[PROOFSTEP]\nrw [@add_tsub_add_eq_tsub_right]\n[GOAL]\nk n m : \u2115\n\u22a2 dist (k + n) (k + m) = dist n m\n[PROOFSTEP]\nrw [add_comm k n, add_comm k m]\n[GOAL]\nk n m : \u2115\n\u22a2 dist (n + k) (m + k) = dist n m\n[PROOFSTEP]\napply dist_add_add_right\n[GOAL]\nn m k l : \u2115\nh : n + m = k + l\n\u22a2 dist n k = dist (n + m) (k + m)\n[PROOFSTEP]\nrw [dist_add_add_right]\n[GOAL]\nn m k l : \u2115\nh : n + m = k + l\n\u22a2 dist (n + m) (k + m) = dist (k + l) (k + m)\n[PROOFSTEP]\nrw [h]\n[GOAL]\nn m k l : \u2115\nh : n + m = k + l\n\u22a2 dist (k + l) (k + m) = dist l m\n[PROOFSTEP]\nrw [dist_add_add_left]\n[GOAL]\nn m k : \u2115\n\u22a2 dist n k \u2264 dist n m + dist m k\n[PROOFSTEP]\nhave : dist n m + dist m k = n - m + (m - k) + (k - m + (m - n)) := by\n  simp [dist.def, add_comm, add_left_comm, add_assoc]\n[GOAL]\nn m k : \u2115\n\u22a2 dist n m + dist m k = n - m + (m - k) + (k - m + (m - n))\n[PROOFSTEP]\nsimp [dist.def, add_comm, add_left_comm, add_assoc]\n[GOAL]\nn m k : \u2115\nthis : dist n m + dist m k = n - m + (m - k) + (k - m + (m - n))\n\u22a2 dist n k \u2264 dist n m + dist m k\n[PROOFSTEP]\nrw [this, dist.def]\n[GOAL]\nn m k : \u2115\nthis : dist n m + dist m k = n - m + (m - k) + (k - m + (m - n))\n\u22a2 n - k + (k - n) \u2264 n - m + (m - k) + (k - m + (m - n))\n[PROOFSTEP]\nexact add_le_add tsub_le_tsub_add_tsub tsub_le_tsub_add_tsub\n[GOAL]\nn k m : \u2115\n\u22a2 dist (n * k) (m * k) = dist n m * k\n[PROOFSTEP]\nrw [dist.def, dist.def, right_distrib, tsub_mul n, tsub_mul m]\n[GOAL]\nk n m : \u2115\n\u22a2 dist (k * n) (k * m) = k * dist n m\n[PROOFSTEP]\nrw [mul_comm k n, mul_comm k m, dist_mul_right, mul_comm]\n[GOAL]\ni j : \u2115\n\u22a2 i < j \u2192 dist i j = max i j - min i j\n[PROOFSTEP]\nintro h\n[GOAL]\ni j : \u2115\nh : i < j\n\u22a2 dist i j = max i j - min i j\n[PROOFSTEP]\nrw [max_eq_right_of_lt h, min_eq_left_of_lt h, dist_eq_sub_of_le (Nat.le_of_lt h)]\n[GOAL]\ni j : \u2115\n\u22a2 i \u2265 j \u2192 dist i j = max i j - min i j\n[PROOFSTEP]\nintro h\n[GOAL]\ni j : \u2115\nh : i \u2265 j\n\u22a2 dist i j = max i j - min i j\n[PROOFSTEP]\nrw [max_eq_left h, min_eq_right h, dist_eq_sub_of_le_right h]\n[GOAL]\ni j : \u2115\n\u22a2 dist (succ i) (succ j) = dist i j\n[PROOFSTEP]\nsimp [dist.def, succ_sub_succ]\n[GOAL]\ni j : \u2115\nhne : i \u2260 j\nh : i < j\n\u22a2 0 < dist i j\n[PROOFSTEP]\nrw [dist_eq_sub_of_le (le_of_lt h)]\n[GOAL]\ni j : \u2115\nhne : i \u2260 j\nh : i < j\n\u22a2 0 < j - i\n[PROOFSTEP]\napply tsub_pos_of_lt h\n[GOAL]\ni j : \u2115\nhne : i \u2260 j\nh : i = j\n\u22a2 0 < dist i j\n[PROOFSTEP]\ncontradiction\n[GOAL]\ni j : \u2115\nhne : i \u2260 j\nh : i > j\n\u22a2 0 < dist i j\n[PROOFSTEP]\nrw [dist_eq_sub_of_le_right (le_of_lt h)]\n[GOAL]\ni j : \u2115\nhne : i \u2260 j\nh : i > j\n\u22a2 0 < i - j\n[PROOFSTEP]\napply tsub_pos_of_lt h\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Dist", "llama_tokens": 1901, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392756357327, "lm_q2_score": 0.8080672135527632, "lm_q1q2_score": 0.7143631541341697}}
{"text": "[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderedAddCommGroup H\nf : G \u2192 H\nh\u2081 : \u2200 (x : G), f (-x) = -f x\nh\u2082 : StrictMonoOn f (Ici 0)\n\u22a2 StrictMono f\n[PROOFSTEP]\nrefine' StrictMonoOn.Iic_union_Ici (fun x hx y hy hxy => neg_lt_neg_iff.1 _) h\u2082\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderedAddCommGroup H\nf : G \u2192 H\nh\u2081 : \u2200 (x : G), f (-x) = -f x\nh\u2082 : StrictMonoOn f (Ici 0)\nx : G\nhx : x \u2208 Iic 0\ny : G\nhy : y \u2208 Iic 0\nhxy : x < y\n\u22a2 -f y < -f x\n[PROOFSTEP]\nrw [\u2190 h\u2081, \u2190 h\u2081]\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderedAddCommGroup H\nf : G \u2192 H\nh\u2081 : \u2200 (x : G), f (-x) = -f x\nh\u2082 : StrictMonoOn f (Ici 0)\nx : G\nhx : x \u2208 Iic 0\ny : G\nhy : y \u2208 Iic 0\nhxy : x < y\n\u22a2 f (-y) < f (-x)\n[PROOFSTEP]\nexact h\u2082 (neg_nonneg.2 hy) (neg_nonneg.2 hx) (neg_lt_neg hxy)\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderedAddCommGroup H\nf : G \u2192 H\nh\u2081 : \u2200 (x : G), f (-x) = -f x\nh\u2082 : MonotoneOn f (Ici 0)\n\u22a2 Monotone f\n[PROOFSTEP]\nrefine' MonotoneOn.Iic_union_Ici (fun x hx y hy hxy => neg_le_neg_iff.1 _) h\u2082\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderedAddCommGroup H\nf : G \u2192 H\nh\u2081 : \u2200 (x : G), f (-x) = -f x\nh\u2082 : MonotoneOn f (Ici 0)\nx : G\nhx : x \u2208 Iic 0\ny : G\nhy : y \u2208 Iic 0\nhxy : x \u2264 y\n\u22a2 -f y \u2264 -f x\n[PROOFSTEP]\nrw [\u2190 h\u2081, \u2190 h\u2081]\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderedAddCommGroup H\nf : G \u2192 H\nh\u2081 : \u2200 (x : G), f (-x) = -f x\nh\u2082 : MonotoneOn f (Ici 0)\nx : G\nhx : x \u2208 Iic 0\ny : G\nhy : y \u2208 Iic 0\nhxy : x \u2264 y\n\u22a2 f (-y) \u2264 f (-x)\n[PROOFSTEP]\nexact h\u2082 (neg_nonneg.2 hy) (neg_nonneg.2 hx) (neg_le_neg hxy)\n", "meta": {"mathlib_filename": "Mathlib.Order.Monotone.Odd", "llama_tokens": 927, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587846530938, "lm_q2_score": 0.8031738034238807, "lm_q1q2_score": 0.7138277733962111}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\ninst\u271d : ContinuousConstSMul \u03b1 \u03b1\ns : Set \u03b1\nh : IsUpperSet s\nx y : \u03b1\nhxy : x \u2264 y\nhx : x \u2208 closure s\n\u22a2 y \u2208 closure ((y / x) \u2022 s)\n[PROOFSTEP]\nrw [closure_smul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\ninst\u271d : ContinuousConstSMul \u03b1 \u03b1\ns : Set \u03b1\nh : IsUpperSet s\nx y : \u03b1\nhxy : x \u2264 y\nhx : x \u2208 closure s\n\u22a2 y \u2208 (y / x) \u2022 closure s\n[PROOFSTEP]\nexact \u27e8x, hx, div_mul_cancel' _ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\ninst\u271d : ContinuousConstSMul \u03b1 \u03b1\ns : Set \u03b1\nh : IsLowerSet s\nx y : \u03b1\nhxy : y \u2264 x\nhx : x \u2208 closure s\n\u22a2 y \u2208 closure ((y / x) \u2022 s)\n[PROOFSTEP]\nrw [closure_smul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\ninst\u271d : ContinuousConstSMul \u03b1 \u03b1\ns : Set \u03b1\nh : IsLowerSet s\nx y : \u03b1\nhxy : y \u2264 x\nhx : x \u2208 closure s\n\u22a2 y \u2208 (y / x) \u2022 closure s\n[PROOFSTEP]\nexact \u27e8x, hx, div_mul_cancel' _ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\ninst\u271d : ContinuousConstSMul \u03b1 \u03b1\ns : Set \u03b1\nhs : IsOpen s\n\u22a2 IsOpen \u2191(upperClosure s)\n[PROOFSTEP]\nrw [\u2190 mul_one s, \u2190 mul_upperClosure]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\ninst\u271d : ContinuousConstSMul \u03b1 \u03b1\ns : Set \u03b1\nhs : IsOpen s\n\u22a2 IsOpen (s * \u2191(upperClosure 1))\n[PROOFSTEP]\nexact hs.mul_right\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\ninst\u271d : ContinuousConstSMul \u03b1 \u03b1\ns : Set \u03b1\nhs : IsOpen s\n\u22a2 IsOpen \u2191(lowerClosure s)\n[PROOFSTEP]\nrw [\u2190 mul_one s, \u2190 mul_lowerClosure]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\ninst\u271d : ContinuousConstSMul \u03b1 \u03b1\ns : Set \u03b1\nhs : IsOpen s\n\u22a2 IsOpen (s * \u2191(lowerClosure 1))\n[PROOFSTEP]\nexact hs.mul_right\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : HasUpperLowerClosure \u03b1\ns : Set \u03b1\nh : IsUpperSet s\n\u22a2 IsUpperSet (interior s)\n[PROOFSTEP]\nrw [\u2190 isLowerSet_compl, \u2190 closure_compl]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : HasUpperLowerClosure \u03b1\ns : Set \u03b1\nh : IsUpperSet s\n\u22a2 IsLowerSet (closure s\u1d9c)\n[PROOFSTEP]\nexact h.compl.closure\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : HasUpperLowerClosure \u03b1\ns : Set \u03b1\nh : OrdConnected s\n\u22a2 OrdConnected (interior s)\n[PROOFSTEP]\nrw [\u2190 h.upperClosure_inter_lowerClosure, interior_inter]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : HasUpperLowerClosure \u03b1\ns : Set \u03b1\nh : OrdConnected s\n\u22a2 OrdConnected (interior \u2191(upperClosure s) \u2229 interior \u2191(lowerClosure s))\n[PROOFSTEP]\nexact (upperClosure s).upper.interior.ordConnected.inter (lowerClosure s).lower.interior.ordConnected\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Order.UpperLower", "llama_tokens": 1255, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505402422644, "lm_q2_score": 0.7879311956428946, "lm_q1q2_score": 0.712802381812078}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddMonoidWithOne \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 0 \u2264 2\n[PROOFSTEP]\nrw [\u2190 one_add_one_eq_two]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddMonoidWithOne \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 0 \u2264 1 + 1\n[PROOFSTEP]\nexact add_nonneg zero_le_one zero_le_one\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddMonoidWithOne \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 0 \u2264 3\n[PROOFSTEP]\nrw [\u2190 two_add_one_eq_three]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddMonoidWithOne \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 0 \u2264 2 + 1\n[PROOFSTEP]\nexact add_nonneg zero_le_two zero_le_one\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddMonoidWithOne \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 0 \u2264 4\n[PROOFSTEP]\nrw [\u2190 three_add_one_eq_four]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddMonoidWithOne \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 0 \u2264 3 + 1\n[PROOFSTEP]\nexact add_nonneg zero_le_three zero_le_one\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : AddMonoidWithOne \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : ZeroLEOneClass \u03b1\ninst\u271d\u00b9 : NeZero 1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 0 < 3\n[PROOFSTEP]\nrw [\u2190 two_add_one_eq_three]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : AddMonoidWithOne \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : ZeroLEOneClass \u03b1\ninst\u271d\u00b9 : NeZero 1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 0 < 2 + 1\n[PROOFSTEP]\nexact lt_add_of_lt_of_nonneg zero_lt_two zero_le_one\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : AddMonoidWithOne \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : ZeroLEOneClass \u03b1\ninst\u271d\u00b9 : NeZero 1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 0 < 4\n[PROOFSTEP]\nrw [\u2190 three_add_one_eq_four]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : AddMonoidWithOne \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : ZeroLEOneClass \u03b1\ninst\u271d\u00b9 : NeZero 1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 0 < 3 + 1\n[PROOFSTEP]\nexact lt_add_of_lt_of_nonneg zero_lt_three zero_le_one\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : AddMonoidWithOne \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : ZeroLEOneClass \u03b1\ninst\u271d\u00b9 : NeZero 1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\n\u22a2 1 < 2\n[PROOFSTEP]\nrw [\u2190 one_add_one_eq_two]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : AddMonoidWithOne \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : ZeroLEOneClass \u03b1\ninst\u271d\u00b9 : NeZero 1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\n\u22a2 1 < 1 + 1\n[PROOFSTEP]\nexact lt_add_one _\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Monoid.NatCast", "llama_tokens": 1484, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9032942145139149, "lm_q2_score": 0.7879311881731379, "lm_q1q2_score": 0.7117336837118703}}
{"text": "[GOAL]\nG : Type u\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Inv G\ninst\u271d : One G\nassoc : \u2200 (a b c : G), a * b * c = a * (b * c)\none_mul : \u2200 (a : G), 1 * a = a\nmul_left_inv : \u2200 (a : G), a\u207b\u00b9 * a = 1\na : G\n\u22a2 a * 1 = a\n[PROOFSTEP]\nhave mul_right_inv : \u2200 a, a * a\u207b\u00b9 = 1 := fun a =>\n  calc\n    a * a\u207b\u00b9 = 1 * (a * a\u207b\u00b9) := (one_mul _).symm\n    _ = ((a * a\u207b\u00b9)\u207b\u00b9 * (a * a\u207b\u00b9)) * (a * a\u207b\u00b9) := by rw [mul_left_inv]\n    _ = (a * a\u207b\u00b9)\u207b\u00b9 * (a * ((a\u207b\u00b9 * a) * a\u207b\u00b9)) := by simp only [assoc]\n    _ = 1 := by rw [mul_left_inv, one_mul, mul_left_inv]\n[GOAL]\nG : Type u\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Inv G\ninst\u271d : One G\nassoc : \u2200 (a b c : G), a * b * c = a * (b * c)\none_mul : \u2200 (a : G), 1 * a = a\nmul_left_inv : \u2200 (a : G), a\u207b\u00b9 * a = 1\na\u271d a : G\n\u22a2 1 * (a * a\u207b\u00b9) = (a * a\u207b\u00b9)\u207b\u00b9 * (a * a\u207b\u00b9) * (a * a\u207b\u00b9)\n[PROOFSTEP]\nrw [mul_left_inv]\n[GOAL]\nG : Type u\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Inv G\ninst\u271d : One G\nassoc : \u2200 (a b c : G), a * b * c = a * (b * c)\none_mul : \u2200 (a : G), 1 * a = a\nmul_left_inv : \u2200 (a : G), a\u207b\u00b9 * a = 1\na\u271d a : G\n\u22a2 (a * a\u207b\u00b9)\u207b\u00b9 * (a * a\u207b\u00b9) * (a * a\u207b\u00b9) = (a * a\u207b\u00b9)\u207b\u00b9 * (a * (a\u207b\u00b9 * a * a\u207b\u00b9))\n[PROOFSTEP]\nsimp only [assoc]\n[GOAL]\nG : Type u\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Inv G\ninst\u271d : One G\nassoc : \u2200 (a b c : G), a * b * c = a * (b * c)\none_mul : \u2200 (a : G), 1 * a = a\nmul_left_inv : \u2200 (a : G), a\u207b\u00b9 * a = 1\na\u271d a : G\n\u22a2 (a * a\u207b\u00b9)\u207b\u00b9 * (a * (a\u207b\u00b9 * a * a\u207b\u00b9)) = 1\n[PROOFSTEP]\nrw [mul_left_inv, one_mul, mul_left_inv]\n[GOAL]\nG : Type u\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Inv G\ninst\u271d : One G\nassoc : \u2200 (a b c : G), a * b * c = a * (b * c)\none_mul : \u2200 (a : G), 1 * a = a\nmul_left_inv : \u2200 (a : G), a\u207b\u00b9 * a = 1\na : G\nmul_right_inv : \u2200 (a : G), a * a\u207b\u00b9 = 1\n\u22a2 a * 1 = a\n[PROOFSTEP]\nrw [\u2190 mul_left_inv a, \u2190 assoc, mul_right_inv a, one_mul]\n[GOAL]\nG : Type u\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Inv G\ninst\u271d : One G\nassoc : \u2200 (a b c : G), a * b * c = a * (b * c)\nmul_one : \u2200 (a : G), a * 1 = a\nmul_right_inv : \u2200 (a : G), a * a\u207b\u00b9 = 1\na : G\n\u22a2 a\u207b\u00b9 * a * 1 = a\u207b\u00b9 * a * (a\u207b\u00b9 * a * (a\u207b\u00b9 * a)\u207b\u00b9)\n[PROOFSTEP]\nrw [mul_right_inv]\n[GOAL]\nG : Type u\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Inv G\ninst\u271d : One G\nassoc : \u2200 (a b c : G), a * b * c = a * (b * c)\nmul_one : \u2200 (a : G), a * 1 = a\nmul_right_inv : \u2200 (a : G), a * a\u207b\u00b9 = 1\na : G\n\u22a2 a\u207b\u00b9 * a * (a\u207b\u00b9 * a * (a\u207b\u00b9 * a)\u207b\u00b9) = a\u207b\u00b9 * (a * a\u207b\u00b9) * a * (a\u207b\u00b9 * a)\u207b\u00b9\n[PROOFSTEP]\nsimp only [assoc]\n[GOAL]\nG : Type u\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Inv G\ninst\u271d : One G\nassoc : \u2200 (a b c : G), a * b * c = a * (b * c)\nmul_one : \u2200 (a : G), a * 1 = a\nmul_right_inv : \u2200 (a : G), a * a\u207b\u00b9 = 1\na : G\n\u22a2 a\u207b\u00b9 * (a * a\u207b\u00b9) * a * (a\u207b\u00b9 * a)\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [mul_right_inv, mul_one, mul_right_inv]\n[GOAL]\nG : Type u\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Inv G\ninst\u271d : One G\nassoc : \u2200 (a b c : G), a * b * c = a * (b * c)\nmul_one : \u2200 (a : G), a * 1 = a\nmul_right_inv : \u2200 (a : G), a * a\u207b\u00b9 = 1\nmul_left_inv : \u2200 (a : G), a\u207b\u00b9 * a = 1\na : G\n\u22a2 1 * a = a\n[PROOFSTEP]\nrw [\u2190 mul_right_inv a, assoc, mul_left_inv, mul_one]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Group.MinimalAxioms", "llama_tokens": 1594, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513731336204, "lm_q2_score": 0.793105953629227, "lm_q1q2_score": 0.7108222999806441}}
{"text": "[GOAL]\n\u22a2 {0} \u222a range succ = univ\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nn : \u2115\n\u22a2 n \u2208 {0} \u222a range succ \u2194 n \u2208 univ\n[PROOFSTEP]\ncases n\n[GOAL]\ncase h.zero\n\u22a2 zero \u2208 {0} \u222a range succ \u2194 zero \u2208 univ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.succ\nn\u271d : \u2115\n\u22a2 succ n\u271d \u2208 {0} \u222a range succ \u2194 succ n\u271d \u2208 univ\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 range succ = {i | 0 < i}\n[PROOFSTEP]\next (_ | i)\n[GOAL]\ncase h.zero\n\u22a2 zero \u2208 range succ \u2194 zero \u2208 {i | 0 < i}\n[PROOFSTEP]\nsimp [succ_pos, succ_ne_zero, Set.mem_setOf]\n[GOAL]\ncase h.succ\ni : \u2115\n\u22a2 succ i \u2208 range succ \u2194 succ i \u2208 {i | 0 < i}\n[PROOFSTEP]\nsimp [succ_pos, succ_ne_zero, Set.mem_setOf]\n[GOAL]\n\u03b1 : Type u_1\nf : \u2115 \u2192 \u03b1\n\u22a2 {f 0} \u222a range (f \u2218 succ) = range f\n[PROOFSTEP]\nrw [\u2190 image_singleton, range_comp, \u2190 image_union, zero_union_range_succ, image_univ]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\nx : \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u03b1\n\u22a2 (range fun n => rec x f n) = {x} \u222a range fun n => rec (f 0 x) (f \u2218 succ) n\n[PROOFSTEP]\nconvert (range_of_succ (fun n => Nat.rec x f n : \u2115 \u2192 \u03b1)).symm using 4\n[GOAL]\ncase h.e'_3.h.e'_4.h.e'_3.h\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\nx : \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u03b1\nx\u271d : \u2115\n\u22a2 rec (f 0 x) (f \u2218 succ) x\u271d = ((fun n => rec x f n) \u2218 succ) x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.e'_3.h.e'_4.h.e'_3.h\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\nx : \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u03b1\nx\u271d : \u2115\n\u22a2 rec (f 0 x) (f \u2218 succ) x\u271d = f x\u271d (rec x f x\u271d)\n[PROOFSTEP]\nrename_i n\n[GOAL]\ncase h.e'_3.h.e'_4.h.e'_3.h\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\nx : \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u03b1\nn : \u2115\n\u22a2 rec (f 0 x) (f \u2218 succ) n = f n (rec x f n)\n[PROOFSTEP]\ninduction' n with n ihn\n[GOAL]\ncase h.e'_3.h.e'_4.h.e'_3.h.zero\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\nx : \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u03b1\n\u22a2 rec (f 0 x) (f \u2218 succ) zero = f zero (rec x f zero)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_3.h.e'_4.h.e'_3.h.succ\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\nx : \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u03b1\nn : \u2115\nihn : rec (f 0 x) (f \u2218 succ) n = f n (rec x f n)\n\u22a2 rec (f 0 x) (f \u2218 succ) (succ n) = f (succ n) (rec x f (succ n))\n[PROOFSTEP]\ndsimp at ihn \u22a2\n[GOAL]\ncase h.e'_3.h.e'_4.h.e'_3.h.succ\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\nx : \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u03b1\nn : \u2115\nihn : rec (f 0 x) (f \u2218 succ) n = f n (rec x f n)\n\u22a2 f (succ n) (rec (f 0 x) (f \u2218 succ) n) = f (succ n) (f n (rec x f n))\n[PROOFSTEP]\nrw [ihn]\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Set", "llama_tokens": 1204, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767874818408, "lm_q2_score": 0.8080672204860316, "lm_q1q2_score": 0.7090602287014633}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : Zero \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : OrderTopology \u03b1\na : \u03b1\nh : 0 < a\n\u22a2 ContinuousAt (\u2191SignType.sign) a\n[PROOFSTEP]\nrefine' (continuousAt_const : ContinuousAt (fun _ => (1 : SignType)) a).congr _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : Zero \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : OrderTopology \u03b1\na : \u03b1\nh : 0 < a\n\u22a2 (fun x => 1) =\u1da0[nhds a] \u2191SignType.sign\n[PROOFSTEP]\nrw [Filter.EventuallyEq, eventually_nhds_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : Zero \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : OrderTopology \u03b1\na : \u03b1\nh : 0 < a\n\u22a2 \u2203 t, (\u2200 (x : \u03b1), x \u2208 t \u2192 1 = \u2191SignType.sign x) \u2227 IsOpen t \u2227 a \u2208 t\n[PROOFSTEP]\nexact \u27e8{x | 0 < x}, fun x hx => (sign_pos hx).symm, isOpen_lt' 0, h\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : Zero \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : OrderTopology \u03b1\na : \u03b1\nh : a < 0\n\u22a2 ContinuousAt (\u2191SignType.sign) a\n[PROOFSTEP]\nrefine' (continuousAt_const : ContinuousAt (fun x => (-1 : SignType)) a).congr _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : Zero \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : OrderTopology \u03b1\na : \u03b1\nh : a < 0\n\u22a2 (fun x => -1) =\u1da0[nhds a] \u2191SignType.sign\n[PROOFSTEP]\nrw [Filter.EventuallyEq, eventually_nhds_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : Zero \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : OrderTopology \u03b1\na : \u03b1\nh : a < 0\n\u22a2 \u2203 t, (\u2200 (x : \u03b1), x \u2208 t \u2192 -1 = \u2191SignType.sign x) \u2227 IsOpen t \u2227 a \u2208 t\n[PROOFSTEP]\nexact \u27e8{x | x < 0}, fun x hx => (sign_neg hx).symm, isOpen_gt' 0, h\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderTopology \u03b1\na : \u03b1\nh : a \u2260 0\n\u22a2 ContinuousAt (\u2191SignType.sign) a\n[PROOFSTEP]\nrcases h.lt_or_lt with (h_neg | h_pos)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderTopology \u03b1\na : \u03b1\nh : a \u2260 0\nh_neg : a < 0\n\u22a2 ContinuousAt (\u2191SignType.sign) a\n[PROOFSTEP]\nexact continuousAt_sign_of_neg h_neg\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderTopology \u03b1\na : \u03b1\nh : a \u2260 0\nh_pos : 0 < a\n\u22a2 ContinuousAt (\u2191SignType.sign) a\n[PROOFSTEP]\nexact continuousAt_sign_of_pos h_pos\n", "meta": {"mathlib_filename": "Mathlib.Topology.Instances.Sign", "llama_tokens": 1215, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8376199795472731, "lm_q2_score": 0.8438950986284991, "lm_q1q2_score": 0.7068633952532474}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192 \u03b1\ng : \u03b1 \u2192 M\nm n : \u2115\nx : \u03b1\n\u22a2 birkhoffSum f g (m + n) x = birkhoffSum f g m x + birkhoffSum f g n (f^[m] x)\n[PROOFSTEP]\nsimp_rw [birkhoffSum, sum_range_add, add_comm m, iterate_add_apply]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nh : IsFixedPt f x\ng : \u03b1 \u2192 M\nn : \u2115\n\u22a2 birkhoffSum f g n x = n \u2022 g x\n[PROOFSTEP]\nsimp [birkhoffSum, (h.iterate _).eq]\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d : AddCommGroup G\nf : \u03b1 \u2192 \u03b1\ng : \u03b1 \u2192 G\nn : \u2115\nx : \u03b1\n\u22a2 birkhoffSum f g n (f x) - birkhoffSum f g n x = g (f^[n] x) - g x\n[PROOFSTEP]\nrw [\u2190 sub_eq_iff_eq_add.2 (birkhoffSum_succ f g n x), \u2190 sub_eq_iff_eq_add.2 (birkhoffSum_succ' f g n x), \u2190 sub_add, \u2190\n  sub_add, sub_add_comm]\n", "meta": {"mathlib_filename": "Mathlib.Dynamics.BirkhoffSum.Basic", "llama_tokens": 393, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9111797148356995, "lm_q2_score": 0.7745833841649233, "lm_q1q2_score": 0.7057846670998659}}
{"text": "[GOAL]\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : TopologicalSpace R\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : ContinuousSMul R M\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : ContinuousAdd M\ninst\u271d : IsSimpleModule R N\nl : M \u2192\u2097[R] N\n\u22a2 IsClosed \u2191(ker l) \u2228 Dense \u2191(ker l)\n[PROOFSTEP]\nrcases l.surjective_or_eq_zero with (hl | rfl)\n[GOAL]\ncase inl\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : TopologicalSpace R\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : ContinuousSMul R M\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : ContinuousAdd M\ninst\u271d : IsSimpleModule R N\nl : M \u2192\u2097[R] N\nhl : Function.Surjective \u2191l\n\u22a2 IsClosed \u2191(ker l) \u2228 Dense \u2191(ker l)\n[PROOFSTEP]\nexact l.ker.isClosed_or_dense_of_isCoatom (LinearMap.isCoatom_ker_of_surjective hl)\n[GOAL]\ncase inr\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : TopologicalSpace R\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : ContinuousSMul R M\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : ContinuousAdd M\ninst\u271d : IsSimpleModule R N\n\u22a2 IsClosed \u2191(ker 0) \u2228 Dense \u2191(ker 0)\n[PROOFSTEP]\nrw [LinearMap.ker_zero]\n[GOAL]\ncase inr\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : TopologicalSpace R\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : ContinuousSMul R M\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : ContinuousAdd M\ninst\u271d : IsSimpleModule R N\n\u22a2 IsClosed \u2191\u22a4 \u2228 Dense \u2191\u22a4\n[PROOFSTEP]\nleft\n[GOAL]\ncase inr.h\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : TopologicalSpace R\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : ContinuousSMul R M\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : ContinuousAdd M\ninst\u271d : IsSimpleModule R N\n\u22a2 IsClosed \u2191\u22a4\n[PROOFSTEP]\nexact isClosed_univ\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Module.Simple", "llama_tokens": 900, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505453836382, "lm_q2_score": 0.7799929002541068, "lm_q1q2_score": 0.7056210026102434}}
{"text": "[GOAL]\nC : Type u\u2081\nD : Type u\u2082\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : Category.{v\u2083, u\u2083} E\nF : C \u00d7 D \u2964 E\nW : C\nX Y Z : D\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 F.map (\ud835\udfd9 W, f \u226b g) = F.map (\ud835\udfd9 W, f) \u226b F.map (\ud835\udfd9 W, g)\n[PROOFSTEP]\nrw [\u2190 Functor.map_comp, prod_comp, Category.comp_id]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : Category.{v\u2083, u\u2083} E\nF : C \u00d7 D \u2964 E\nX Y Z : C\nW : D\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 F.map (f \u226b g, \ud835\udfd9 W) = F.map (f, \ud835\udfd9 W) \u226b F.map (g, \ud835\udfd9 W)\n[PROOFSTEP]\nrw [\u2190 Functor.map_comp, prod_comp, Category.comp_id]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : Category.{v\u2083, u\u2083} E\nF : C \u00d7 D \u2964 E\nX X' : C\nf : X \u27f6 X'\nY Y' : D\ng : Y \u27f6 Y'\n\u22a2 F.map (\ud835\udfd9 X, g) \u226b F.map (f, \ud835\udfd9 Y') = F.map (f, g)\n[PROOFSTEP]\nrw [\u2190 Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : Category.{v\u2083, u\u2083} E\nF : C \u00d7 D \u2964 E\nX X' : C\nf : X \u27f6 X'\nY Y' : D\ng : Y \u27f6 Y'\n\u22a2 F.map (f, \ud835\udfd9 Y) \u226b F.map (\ud835\udfd9 X', g) = F.map (f, g)\n[PROOFSTEP]\nrw [\u2190 Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Products.Bifunctor", "llama_tokens": 723, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9073122238669026, "lm_q2_score": 0.7772998663336158, "lm_q1q2_score": 0.7052536703345991}}
{"text": "[GOAL]\nn : Type u\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nR : Type v\ninst\u271d : CommRing R\nA : GL n R\n\u22a2 Matrix.det \u2191A * Matrix.det \u2191A\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [\u2190 det_mul, A.mul_inv, det_one]\n[GOAL]\nn : Type u\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nR : Type v\ninst\u271d : CommRing R\nA : GL n R\n\u22a2 Matrix.det \u2191A\u207b\u00b9 * Matrix.det \u2191A = 1\n[PROOFSTEP]\nrw [\u2190 det_mul, A.inv_mul, det_one]\n[GOAL]\nn : Type u\nR : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : LinearOrderedCommRing R\ninst\u271d : Fact (Even (Fintype.card n))\ng : { x // x \u2208 GLPos n R }\n\u22a2 -\u2191g \u2208 GLPos n R\n[PROOFSTEP]\nrw [mem_glpos, GeneralLinearGroup.det_apply_val, Units.val_neg, det_neg,\n  (Fact.out (p := Even <| Fintype.card n)).neg_one_pow, one_mul]\n[GOAL]\nn : Type u\nR : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : LinearOrderedCommRing R\ninst\u271d : Fact (Even (Fintype.card n))\ng : { x // x \u2208 GLPos n R }\n\u22a2 0 < det \u2191\u2191g\n[PROOFSTEP]\nexact g.prop\n[GOAL]\nR : Type ?u.1604470\ninst\u271d : Field R\na b : R\nhab : a ^ 2 + b ^ 2 \u2260 0\n\u22a2 det (\u2191of ![![a, -b], ![b, a]]) \u2260 0\n[PROOFSTEP]\nsimpa [det_fin_two, sq] using hab\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "llama_tokens": 557, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467548438126, "lm_q2_score": 0.8006920092299293, "lm_q1q2_score": 0.7039257815438644}}
{"text": "[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\none_mul : \u2200 (a : R), 1 * a = a\nmul_one : \u2200 (a : R), a * 1 = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : \u2200 (a b c : R), (a + b) * c = a * c + b * c\nthis : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\n\u22a2 \u2200 (a b : R), a + b = b + a\n[PROOFSTEP]\nintro a b\n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\none_mul : \u2200 (a : R), 1 * a = a\nmul_one : \u2200 (a : R), a * 1 = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : \u2200 (a b c : R), (a + b) * c = a * c + b * c\nthis : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\na b : R\n\u22a2 a + b = b + a\n[PROOFSTEP]\nhave h\u2081 : (1 + 1 : R) * (a + b) = a + (a + b) + b :=\n  by\n  rw [left_distrib]\n  simp only [right_distrib, one_mul, add_assoc]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\none_mul : \u2200 (a : R), 1 * a = a\nmul_one : \u2200 (a : R), a * 1 = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : \u2200 (a b c : R), (a + b) * c = a * c + b * c\nthis : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\na b : R\n\u22a2 (1 + 1) * (a + b) = a + (a + b) + b\n[PROOFSTEP]\nrw [left_distrib]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\none_mul : \u2200 (a : R), 1 * a = a\nmul_one : \u2200 (a : R), a * 1 = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : \u2200 (a b c : R), (a + b) * c = a * c + b * c\nthis : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\na b : R\n\u22a2 (1 + 1) * a + (1 + 1) * b = a + (a + b) + b\n[PROOFSTEP]\nsimp only [right_distrib, one_mul, add_assoc]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\none_mul : \u2200 (a : R), 1 * a = a\nmul_one : \u2200 (a : R), a * 1 = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : \u2200 (a b c : R), (a + b) * c = a * c + b * c\nthis : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\na b : R\nh\u2081 : (1 + 1) * (a + b) = a + (a + b) + b\n\u22a2 a + b = b + a\n[PROOFSTEP]\nhave h\u2082 : (1 + 1 : R) * (a + b) = a + (b + a) + b :=\n  by\n  rw [right_distrib]\n  simp only [left_distrib, one_mul, add_assoc]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\none_mul : \u2200 (a : R), 1 * a = a\nmul_one : \u2200 (a : R), a * 1 = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : \u2200 (a b c : R), (a + b) * c = a * c + b * c\nthis : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\na b : R\nh\u2081 : (1 + 1) * (a + b) = a + (a + b) + b\n\u22a2 (1 + 1) * (a + b) = a + (b + a) + b\n[PROOFSTEP]\nrw [right_distrib]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\none_mul : \u2200 (a : R), 1 * a = a\nmul_one : \u2200 (a : R), a * 1 = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : \u2200 (a b c : R), (a + b) * c = a * c + b * c\nthis : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\na b : R\nh\u2081 : (1 + 1) * (a + b) = a + (a + b) + b\n\u22a2 1 * (a + b) + 1 * (a + b) = a + (b + a) + b\n[PROOFSTEP]\nsimp only [left_distrib, one_mul, add_assoc]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\none_mul : \u2200 (a : R), 1 * a = a\nmul_one : \u2200 (a : R), a * 1 = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : \u2200 (a b c : R), (a + b) * c = a * c + b * c\nthis : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\na b : R\nh\u2081 : (1 + 1) * (a + b) = a + (a + b) + b\nh\u2082 : (1 + 1) * (a + b) = a + (b + a) + b\n\u22a2 a + b = b + a\n[PROOFSTEP]\nhave := h\u2081.symm.trans h\u2082\n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\none_mul : \u2200 (a : R), 1 * a = a\nmul_one : \u2200 (a : R), a * 1 = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : \u2200 (a b c : R), (a + b) * c = a * c + b * c\nthis\u271d : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\na b : R\nh\u2081 : (1 + 1) * (a + b) = a + (a + b) + b\nh\u2082 : (1 + 1) * (a + b) = a + (b + a) + b\nthis : a + (a + b) + b = a + (b + a) + b\n\u22a2 a + b = b + a\n[PROOFSTEP]\nrwa [add_left_inj, add_right_inj] at this \n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\none_mul : \u2200 (a : R), 1 * a = a\nmul_one : \u2200 (a : R), a * 1 = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : \u2200 (a b c : R), (a + b) * c = a * c + b * c\nthis : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\nadd_comm : \u2200 (a b : R), a + b = b + a\na : R\n\u22a2 0 * a = 0\n[PROOFSTEP]\nhave : 0 * a = 0 * a + 0 * a :=\n  calc\n    0 * a = (0 + 0) * a := by rw [zero_add]\n    _ = 0 * a + 0 * a := by rw [right_distrib]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\none_mul : \u2200 (a : R), 1 * a = a\nmul_one : \u2200 (a : R), a * 1 = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : \u2200 (a b c : R), (a + b) * c = a * c + b * c\nthis : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\nadd_comm : \u2200 (a b : R), a + b = b + a\na : R\n\u22a2 0 * a = (0 + 0) * a\n[PROOFSTEP]\nrw [zero_add]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\none_mul : \u2200 (a : R), 1 * a = a\nmul_one : \u2200 (a : R), a * 1 = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : \u2200 (a b c : R), (a + b) * c = a * c + b * c\nthis : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\nadd_comm : \u2200 (a b : R), a + b = b + a\na : R\n\u22a2 (0 + 0) * a = 0 * a + 0 * a\n[PROOFSTEP]\nrw [right_distrib]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\none_mul : \u2200 (a : R), 1 * a = a\nmul_one : \u2200 (a : R), a * 1 = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : \u2200 (a b c : R), (a + b) * c = a * c + b * c\nthis\u271d : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\nadd_comm : \u2200 (a b : R), a + b = b + a\na : R\nthis : 0 * a = 0 * a + 0 * a\n\u22a2 0 * a = 0\n[PROOFSTEP]\nrwa [self_eq_add_right] at this \n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\none_mul : \u2200 (a : R), 1 * a = a\nmul_one : \u2200 (a : R), a * 1 = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : \u2200 (a b c : R), (a + b) * c = a * c + b * c\nthis : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\nadd_comm : \u2200 (a b : R), a + b = b + a\nzero_mul : \u2200 (a : R), 0 * a = 0\na : R\n\u22a2 a * 0 = 0\n[PROOFSTEP]\nhave : a * 0 = a * 0 + a * 0 :=\n  calc\n    a * 0 = a * (0 + 0) := by rw [zero_add]\n    _ = a * 0 + a * 0 := by rw [left_distrib]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\none_mul : \u2200 (a : R), 1 * a = a\nmul_one : \u2200 (a : R), a * 1 = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : \u2200 (a b c : R), (a + b) * c = a * c + b * c\nthis : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\nadd_comm : \u2200 (a b : R), a + b = b + a\nzero_mul : \u2200 (a : R), 0 * a = 0\na : R\n\u22a2 a * 0 = a * (0 + 0)\n[PROOFSTEP]\nrw [zero_add]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\none_mul : \u2200 (a : R), 1 * a = a\nmul_one : \u2200 (a : R), a * 1 = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : \u2200 (a b c : R), (a + b) * c = a * c + b * c\nthis : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\nadd_comm : \u2200 (a b : R), a + b = b + a\nzero_mul : \u2200 (a : R), 0 * a = 0\na : R\n\u22a2 a * (0 + 0) = a * 0 + a * 0\n[PROOFSTEP]\nrw [left_distrib]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\none_mul : \u2200 (a : R), 1 * a = a\nmul_one : \u2200 (a : R), a * 1 = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : \u2200 (a b c : R), (a + b) * c = a * c + b * c\nthis\u271d : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\nadd_comm : \u2200 (a b : R), a + b = b + a\nzero_mul : \u2200 (a : R), 0 * a = 0\na : R\nthis : a * 0 = a * 0 + a * 0\n\u22a2 a * 0 = 0\n[PROOFSTEP]\nrwa [self_eq_add_right] at this \n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\nmul_comm : \u2200 (a b : R), a * b = b * a\none_mul : \u2200 (a : R), 1 * a = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\na : R\n\u22a2 a * 1 = a\n[PROOFSTEP]\nrw [mul_comm, one_mul]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : Add R\ninst\u271d\u00b3 : Mul R\ninst\u271d\u00b2 : Neg R\ninst\u271d\u00b9 : Zero R\ninst\u271d : One R\nadd_assoc : \u2200 (a b c : R), a + b + c = a + (b + c)\nzero_add : \u2200 (a : R), 0 + a = a\nadd_left_neg : \u2200 (a : R), -a + a = 0\nmul_assoc : \u2200 (a b c : R), a * b * c = a * (b * c)\nmul_comm : \u2200 (a b : R), a * b = b * a\none_mul : \u2200 (a : R), 1 * a = a\nleft_distrib : \u2200 (a b c : R), a * (b + c) = a * b + a * c\nmul_one : \u2200 (a : R), a * 1 = a\na b c : R\n\u22a2 (a + b) * c = a * c + b * c\n[PROOFSTEP]\nrw [mul_comm, left_distrib, mul_comm, mul_comm b c]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Ring.MinimalAxioms", "llama_tokens": 6446, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9324533163686646, "lm_q2_score": 0.7549149868676283, "lm_q1q2_score": 0.7039229830811269}}
{"text": "[GOAL]\nG : Type u_1\nH : Type u_2\nM : Type u_3\nN : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : MulAction G M\ninst\u271d\u00b9 : SMulCommClass G M M\ninst\u271d : IsScalarTower G M M\ng : G\nm : M\u02e3\n\u22a2 g \u2022 \u2191m * g\u207b\u00b9 \u2022 \u2191m\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [smul_mul_smul, Units.mul_inv, mul_right_inv, one_smul]\n[GOAL]\nG : Type u_1\nH : Type u_2\nM : Type u_3\nN : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : MulAction G M\ninst\u271d\u00b9 : SMulCommClass G M M\ninst\u271d : IsScalarTower G M M\ng : G\nm : M\u02e3\n\u22a2 g\u207b\u00b9 \u2022 \u2191m\u207b\u00b9 * g \u2022 \u2191m = 1\n[PROOFSTEP]\nrw [smul_mul_smul, Units.inv_mul, mul_left_inv, one_smul]\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.GroupAction.Units", "llama_tokens": 326, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9184802395624257, "lm_q2_score": 0.7662936430859597, "lm_q1q2_score": 0.7038255688767563}}
{"text": "[GOAL]\nm n : \u2124\n\u22a2 dist m n = \u2191|m - n|\n[PROOFSTEP]\nrw [dist_eq]\n[GOAL]\nm n : \u2124\n\u22a2 |\u2191m - \u2191n| = \u2191|m - n|\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u22a2 Pairwise fun m n => 1 \u2264 dist m n\n[PROOFSTEP]\nintro m n hne\n[GOAL]\nm n : \u2124\nhne : m \u2260 n\n\u22a2 1 \u2264 dist m n\n[PROOFSTEP]\nrw [dist_eq]\n[GOAL]\nm n : \u2124\nhne : m \u2260 n\n\u22a2 1 \u2264 |\u2191m - \u2191n|\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nm n : \u2124\nhne : m \u2260 n\n\u22a2 1 \u2264 |m - n|\n[PROOFSTEP]\nrwa [\u2190 zero_add (1 : \u2124), Int.add_one_le_iff, abs_pos, sub_ne_zero]\n[GOAL]\nx : \u2124\nr : \u211d\n\u22a2 ball x r = Ioo \u230a\u2191x - r\u230b \u2308\u2191x + r\u2309\n[PROOFSTEP]\nrw [\u2190 preimage_ball, Real.ball_eq_Ioo, preimage_Ioo]\n[GOAL]\nx : \u2124\nr : \u211d\n\u22a2 closedBall x r = Icc \u2308\u2191x - r\u2309 \u230a\u2191x + r\u230b\n[PROOFSTEP]\nrw [\u2190 preimage_closedBall, Real.closedBall_eq_Icc, preimage_Icc]\n[GOAL]\nx : \u2124\nr : \u211d\n\u22a2 IsCompact (closedBall x r)\n[PROOFSTEP]\nrw [closedBall_eq_Icc]\n[GOAL]\nx : \u2124\nr : \u211d\n\u22a2 IsCompact (Icc \u2308\u2191x - r\u2309 \u230a\u2191x + r\u230b)\n[PROOFSTEP]\nexact (Set.finite_Icc _ _).isCompact\n[GOAL]\n\u22a2 cocompact \u2124 = atBot \u2294 atTop\n[PROOFSTEP]\nsimp_rw [\u2190 comap_dist_right_atTop_eq_cocompact (0 : \u2124), dist_eq', sub_zero, \u2190 comap_abs_atTop, \u2190\n  @Int.comap_cast_atTop \u211d, comap_comap]\n[GOAL]\n\u22a2 comap (fun y => \u2191|y|) atTop = comap (Int.cast \u2218 abs) atTop\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 cofinite = atBot \u2294 atTop\n[PROOFSTEP]\nrw [\u2190 cocompact_eq_cofinite, cocompact_eq]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Instances.Int", "llama_tokens": 732, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240825770433, "lm_q2_score": 0.8128673269042767, "lm_q1q2_score": 0.7033123871776064}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nr n : \u2115\n\u22a2 \u2191(choose n r) \u2264 \u2191(n ^ r) / \u2191r !\n[PROOFSTEP]\nrw [le_div_iff']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nr n : \u2115\n\u22a2 \u2191r ! * \u2191(choose n r) \u2264 \u2191(n ^ r)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nr n : \u2115\n\u22a2 r ! * choose n r \u2264 n ^ r\n[PROOFSTEP]\nrw [\u2190 Nat.descFactorial_eq_factorial_mul_choose]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nr n : \u2115\n\u22a2 descFactorial n r \u2264 n ^ r\n[PROOFSTEP]\nexact n.descFactorial_le_pow r\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nr n : \u2115\n\u22a2 0 < \u2191r !\n[PROOFSTEP]\nexact_mod_cast r.factorial_pos\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nr n : \u2115\n\u22a2 \u2191((n + 1 - r) ^ r) / \u2191r ! \u2264 \u2191(choose n r)\n[PROOFSTEP]\nrw [div_le_iff']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nr n : \u2115\n\u22a2 \u2191((n + 1 - r) ^ r) \u2264 \u2191r ! * \u2191(choose n r)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nr n : \u2115\n\u22a2 (n + 1 - r) ^ r \u2264 r ! * choose n r\n[PROOFSTEP]\nrw [\u2190 Nat.descFactorial_eq_factorial_mul_choose]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nr n : \u2115\n\u22a2 (n + 1 - r) ^ r \u2264 descFactorial n r\n[PROOFSTEP]\nexact n.pow_sub_le_descFactorial r\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nr n : \u2115\n\u22a2 0 < \u2191r !\n[PROOFSTEP]\nexact_mod_cast r.factorial_pos\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Choose.Bounds", "llama_tokens": 690, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9073122113355091, "lm_q2_score": 0.7745833789613196, "lm_q1q2_score": 0.7027889584291255}}
{"text": "[GOAL]\nx y : \u2115\n\u22a2 dist \u2191x \u2191y = dist x y\n[PROOFSTEP]\nrw [\u2190 Nat.dist_cast_real, \u2190 Rat.dist_cast]\n[GOAL]\nx y : \u2115\n\u22a2 dist \u2191\u2191x \u2191\u2191y = dist \u2191x \u2191y\n[PROOFSTEP]\ncongr\n[GOAL]\n\u22a2 Pairwise fun x y => 1 \u2264 dist \u2191x \u2191y\n[PROOFSTEP]\nsimpa using Nat.pairwise_one_le_dist\n[GOAL]\n\u22a2 Pairwise fun x y => 1 \u2264 dist \u2191x \u2191y\n[PROOFSTEP]\nsimpa using Nat.pairwise_one_le_dist\n[GOAL]\nx y : \u2124\n\u22a2 dist \u2191x \u2191y = dist x y\n[PROOFSTEP]\nrw [\u2190 Int.dist_cast_real, \u2190 Rat.dist_cast]\n[GOAL]\nx y : \u2124\n\u22a2 dist \u2191\u2191x \u2191\u2191y = dist \u2191x \u2191y\n[PROOFSTEP]\ncongr\n[GOAL]\n\u22a2 Pairwise fun x y => 1 \u2264 dist \u2191x \u2191y\n[PROOFSTEP]\nsimpa using Int.pairwise_one_le_dist\n[GOAL]\n\u22a2 Pairwise fun x y => 1 \u2264 dist \u2191x \u2191y\n[PROOFSTEP]\nsimpa using Int.pairwise_one_le_dist\n[GOAL]\n\u22a2 UniformContinuous (Rat.cast \u2218 fun p => p.fst + p.snd)\n[PROOFSTEP]\nsimp only [(\u00b7 \u2218 \u00b7), Rat.cast_add]\n[GOAL]\n\u22a2 UniformContinuous fun x => \u2191x.fst + \u2191x.snd\n[PROOFSTEP]\nexact Real.uniformContinuous_add.comp (Rat.uniformContinuous_coe_real.prod_map Rat.uniformContinuous_coe_real)\n[GOAL]\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\na\u271d b\u271d : \u211a\nh : dist a\u271d b\u271d < \u03b5\n\u22a2 dist (-a\u271d) (-b\u271d) < \u03b5\n[PROOFSTEP]\nrw [dist_comm] at h \n[GOAL]\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\na\u271d b\u271d : \u211a\nh : dist b\u271d a\u271d < \u03b5\n\u22a2 dist (-a\u271d) (-b\u271d) < \u03b5\n[PROOFSTEP]\nsimpa only [dist_eq, cast_neg, neg_sub_neg] using h\n[GOAL]\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\na\u271d b\u271d : \u211a\nh : dist a\u271d b\u271d < \u03b5\n\u22a2 dist |a\u271d| |b\u271d| \u2264 dist a\u271d b\u271d\n[PROOFSTEP]\nsimpa [Rat.dist_eq] using abs_abs_sub_abs_le_abs_sub _ _\n[GOAL]\na b : \u211a\n\u22a2 TotallyBounded (Icc a b)\n[PROOFSTEP]\nsimpa only [preimage_cast_Icc] using\n  totallyBounded_preimage Rat.uniformEmbedding_coe_real (totallyBounded_Icc (a : \u211d) b)\n", "meta": {"mathlib_filename": "Mathlib.Topology.Instances.Rat", "llama_tokens": 855, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.888758793492457, "lm_q2_score": 0.787931190663057, "lm_q1q2_score": 0.7002807743687736}}