{"text": "[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\ninst\u271d : GCDMonoid R\np\u271d q\u271d p q : R\nhp : p \u2260 0\n\u22a2 p / GCDMonoid.gcd p q \u2260 0\n[PROOFSTEP]\nobtain \u27e8r, hr\u27e9 := GCDMonoid.gcd_dvd_left p q\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\ninst\u271d : GCDMonoid R\np\u271d q\u271d p q : R\nhp : p \u2260 0\nr : R\nhr : p = GCDMonoid.gcd p q * r\n\u22a2 p / GCDMonoid.gcd p q \u2260 0\n[PROOFSTEP]\nobtain \u27e8pq0, r0\u27e9 : GCDMonoid.gcd p q \u2260 0 \u2227 r \u2260 0 := mul_ne_zero_iff.mp (hr \u25b8 hp)\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\ninst\u271d : GCDMonoid R\np\u271d q\u271d p q : R\nhp : p \u2260 0\nr : R\nhr : p = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q \u2260 0\nr0 : r \u2260 0\n\u22a2 p / GCDMonoid.gcd p q \u2260 0\n[PROOFSTEP]\nnth_rw 1 [hr]\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\ninst\u271d : GCDMonoid R\np\u271d q\u271d p q : R\nhp : p \u2260 0\nr : R\nhr : p = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q \u2260 0\nr0 : r \u2260 0\n\u22a2 GCDMonoid.gcd p q * r / GCDMonoid.gcd p q \u2260 0\n[PROOFSTEP]\nrw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\ninst\u271d : GCDMonoid R\np\u271d q\u271d p q : R\nhp : p \u2260 0\nr : R\nhr : p = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q \u2260 0\nr0 : r \u2260 0\n\u22a2 r \u2260 0\n[PROOFSTEP]\nexact r0\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\ninst\u271d : GCDMonoid R\np\u271d q\u271d p q : R\nhq : q \u2260 0\n\u22a2 q / GCDMonoid.gcd p q \u2260 0\n[PROOFSTEP]\nobtain \u27e8r, hr\u27e9 := GCDMonoid.gcd_dvd_right p q\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\ninst\u271d : GCDMonoid R\np\u271d q\u271d p q : R\nhq : q \u2260 0\nr : R\nhr : q = GCDMonoid.gcd p q * r\n\u22a2 q / GCDMonoid.gcd p q \u2260 0\n[PROOFSTEP]\nobtain \u27e8pq0, r0\u27e9 : GCDMonoid.gcd p q \u2260 0 \u2227 r \u2260 0 := mul_ne_zero_iff.mp (hr \u25b8 hq)\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\ninst\u271d : GCDMonoid R\np\u271d q\u271d p q : R\nhq : q \u2260 0\nr : R\nhr : q = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q \u2260 0\nr0 : r \u2260 0\n\u22a2 q / GCDMonoid.gcd p q \u2260 0\n[PROOFSTEP]\nnth_rw 1 [hr]\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\ninst\u271d : GCDMonoid R\np\u271d q\u271d p q : R\nhq : q \u2260 0\nr : R\nhr : q = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q \u2260 0\nr0 : r \u2260 0\n\u22a2 GCDMonoid.gcd p q * r / GCDMonoid.gcd p q \u2260 0\n[PROOFSTEP]\nrw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\ninst\u271d : GCDMonoid R\np\u271d q\u271d p q : R\nhq : q \u2260 0\nr : R\nhr : q = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q \u2260 0\nr0 : r \u2260 0\n\u22a2 r \u2260 0\n[PROOFSTEP]\nexact r0\n[GOAL]\nR : Type ?u.6736\ninst\u271d\u00b9 : EuclideanDomain R\ninst\u271d : DecidableEq R\na b : R\n\u22a2 Associated (gcd a b * lcm a b) (a * b)\n[PROOFSTEP]\nrw [EuclideanDomain.gcd_mul_lcm]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.EuclideanDomain", "llama_tokens": 1494, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9099070133672955, "lm_q2_score": 0.8519528076067262, "lm_q1q2_score": 0.7751978346993184}}
{"text": "[GOAL]\nn : \u2115\n\u22a2 (n + 1)\u203c = (n + 1) * (n - 1)\u203c\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u22a2 (zero + 1)\u203c = (zero + 1) * (zero - 1)\u203c\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nn\u271d : \u2115\n\u22a2 (succ n\u271d + 1)\u203c = (succ n\u271d + 1) * (succ n\u271d - 1)\u203c\n[PROOFSTEP]\nrfl\n[GOAL]\nk : \u2115\n\u22a2 (k + 1 + 1)! = (k + 1 + 1)\u203c * (k + 1)\u203c\n[PROOFSTEP]\nrw [doubleFactorial_add_two, factorial, factorial_eq_mul_doubleFactorial _, mul_comm _ k\u203c, mul_assoc]\n[GOAL]\nn : \u2115\n\u22a2 (2 * (n + 1))\u203c = 2 ^ (n + 1) * (n + 1)!\n[PROOFSTEP]\nrw [mul_add, mul_one, doubleFactorial_add_two, factorial, pow_succ, doubleFactorial_two_mul _, succ_eq_add_one]\n[GOAL]\nn : \u2115\n\u22a2 (2 * n + 2) * (2 ^ n * n !) = 2 ^ n * 2 * ((n + 1) * n !)\n[PROOFSTEP]\nring\n[GOAL]\nn : \u2115\n\u22a2 (2 * (n + 1))\u203c = \u220f i in Finset.range (n + 1), 2 * (i + 1)\n[PROOFSTEP]\nrw [Finset.prod_range_succ, \u2190 doubleFactorial_eq_prod_even _, mul_comm (2 * n)\u203c, (by ring : 2 * (n + 1) = 2 * n + 2)]\n[GOAL]\nn : \u2115\n\u22a2 2 * (n + 1) = 2 * n + 2\n[PROOFSTEP]\nring\n[GOAL]\nn : \u2115\n\u22a2 (2 * n + 2)\u203c = (2 * n + 2) * (2 * n)\u203c\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\n\u22a2 (2 * (n + 1) + 1)\u203c = \u220f i in Finset.range (n + 1), (2 * (i + 1) + 1)\n[PROOFSTEP]\nrw [Finset.prod_range_succ, \u2190 doubleFactorial_eq_prod_odd _, mul_comm (2 * n + 1)\u203c,\n  (by ring : 2 * (n + 1) + 1 = 2 * n + 1 + 2)]\n[GOAL]\nn : \u2115\n\u22a2 2 * (n + 1) + 1 = 2 * n + 1 + 2\n[PROOFSTEP]\nring\n[GOAL]\nn : \u2115\n\u22a2 (2 * n + 1 + 2)\u203c = (2 * n + 1 + 2) * (2 * n + 1)\u203c\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Factorial.DoubleFactorial", "llama_tokens": 850, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.933430812881347, "lm_q2_score": 0.8221891327004132, "lm_q1q2_score": 0.7674566704787563}}
{"text": "[GOAL]\nx\u271d y x : \u211d\n\u22a2 exp (arsinh x) = x + sqrt (1 + x ^ 2)\n[PROOFSTEP]\napply exp_log\n[GOAL]\ncase hx\nx\u271d y x : \u211d\n\u22a2 0 < x + sqrt (1 + x ^ 2)\n[PROOFSTEP]\nrw [\u2190 neg_lt_iff_pos_add']\n[GOAL]\ncase hx\nx\u271d y x : \u211d\n\u22a2 -x < sqrt (1 + x ^ 2)\n[PROOFSTEP]\napply lt_sqrt_of_sq_lt\n[GOAL]\ncase hx.h\nx\u271d y x : \u211d\n\u22a2 (-x) ^ 2 < 1 + x ^ 2\n[PROOFSTEP]\nsimp\n[GOAL]\nx y : \u211d\n\u22a2 arsinh 0 = 0\n[PROOFSTEP]\nsimp [arsinh]\n[GOAL]\nx\u271d y x : \u211d\n\u22a2 arsinh (-x) = -arsinh x\n[PROOFSTEP]\nrw [\u2190 exp_eq_exp, exp_arsinh, exp_neg, exp_arsinh]\n[GOAL]\nx\u271d y x : \u211d\n\u22a2 -x + sqrt (1 + (-x) ^ 2) = (x + sqrt (1 + x ^ 2))\u207b\u00b9\n[PROOFSTEP]\napply eq_inv_of_mul_eq_one_left\n[GOAL]\ncase h\nx\u271d y x : \u211d\n\u22a2 (-x + sqrt (1 + (-x) ^ 2)) * (x + sqrt (1 + x ^ 2)) = 1\n[PROOFSTEP]\nrw [neg_sq, neg_add_eq_sub, add_comm x, mul_comm, \u2190 sq_sub_sq, sq_sqrt, add_sub_cancel]\n[GOAL]\ncase h\nx\u271d y x : \u211d\n\u22a2 0 \u2264 1 + x ^ 2\n[PROOFSTEP]\nexact add_nonneg zero_le_one (sq_nonneg _)\n[GOAL]\nx\u271d y x : \u211d\n\u22a2 sinh (arsinh x) = x\n[PROOFSTEP]\nrw [sinh_eq, \u2190 arsinh_neg, exp_arsinh, exp_arsinh, neg_sq]\n[GOAL]\nx\u271d y x : \u211d\n\u22a2 (x + sqrt (1 + x ^ 2) - (-x + sqrt (1 + x ^ 2))) / 2 = x\n[PROOFSTEP]\nfield_simp\n[GOAL]\nx\u271d y x : \u211d\n\u22a2 cosh (arsinh x) = sqrt (1 + x ^ 2)\n[PROOFSTEP]\nrw [\u2190 sqrt_sq (cosh_pos _).le, cosh_sq', sinh_arsinh]\n[GOAL]\nx y : \u211d\n\u22a2 0 \u2264 arsinh x \u2194 0 \u2264 x\n[PROOFSTEP]\nrw [\u2190 sinh_le_sinh, sinh_zero, sinh_arsinh]\n[GOAL]\nx y : \u211d\n\u22a2 arsinh x \u2264 0 \u2194 x \u2264 0\n[PROOFSTEP]\nrw [\u2190 sinh_le_sinh, sinh_zero, sinh_arsinh]\n[GOAL]\nx\u271d y x : \u211d\n\u22a2 HasStrictDerivAt arsinh (sqrt (1 + x ^ 2))\u207b\u00b9 x\n[PROOFSTEP]\nconvert\n  sinhHomeomorph.toLocalHomeomorph.hasStrictDerivAt_symm (mem_univ x) (cosh_pos _).ne' (hasStrictDerivAt_sinh _) using 2\n[GOAL]\ncase h.e'_7.h.e'_3\nx\u271d y x : \u211d\n\u22a2 sqrt (1 + x ^ 2) = cosh (\u2191(LocalHomeomorph.symm (Homeomorph.toLocalHomeomorph sinhHomeomorph)) x)\n[PROOFSTEP]\nexact (cosh_arsinh _).symm\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.Arsinh", "llama_tokens": 1034, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026528034425, "lm_q2_score": 0.8354835452961427, "lm_q1q2_score": 0.7663912724737768}}
{"text": "[GOAL]\na : \u211d\n\u22a2 interior {z | z.re \u2264 a} = {z | z.re < a}\n[PROOFSTEP]\nsimpa only [interior_Iic] using interior_preimage_re (Iic a)\n[GOAL]\na : \u211d\n\u22a2 interior {z | z.im \u2264 a} = {z | z.im < a}\n[PROOFSTEP]\nsimpa only [interior_Iic] using interior_preimage_im (Iic a)\n[GOAL]\na : \u211d\n\u22a2 interior {z | a \u2264 z.re} = {z | a < z.re}\n[PROOFSTEP]\nsimpa only [interior_Ici] using interior_preimage_re (Ici a)\n[GOAL]\na : \u211d\n\u22a2 interior {z | a \u2264 z.im} = {z | a < z.im}\n[PROOFSTEP]\nsimpa only [interior_Ici] using interior_preimage_im (Ici a)\n[GOAL]\na : \u211d\n\u22a2 closure {z | z.re < a} = {z | z.re \u2264 a}\n[PROOFSTEP]\nsimpa only [closure_Iio] using closure_preimage_re (Iio a)\n[GOAL]\na : \u211d\n\u22a2 closure {z | z.im < a} = {z | z.im \u2264 a}\n[PROOFSTEP]\nsimpa only [closure_Iio] using closure_preimage_im (Iio a)\n[GOAL]\na : \u211d\n\u22a2 closure {z | a < z.re} = {z | a \u2264 z.re}\n[PROOFSTEP]\nsimpa only [closure_Ioi] using closure_preimage_re (Ioi a)\n[GOAL]\na : \u211d\n\u22a2 closure {z | a < z.im} = {z | a \u2264 z.im}\n[PROOFSTEP]\nsimpa only [closure_Ioi] using closure_preimage_im (Ioi a)\n[GOAL]\na : \u211d\n\u22a2 frontier {z | z.re \u2264 a} = {z | z.re = a}\n[PROOFSTEP]\nsimpa only [frontier_Iic] using frontier_preimage_re (Iic a)\n[GOAL]\na : \u211d\n\u22a2 frontier {z | z.im \u2264 a} = {z | z.im = a}\n[PROOFSTEP]\nsimpa only [frontier_Iic] using frontier_preimage_im (Iic a)\n[GOAL]\na : \u211d\n\u22a2 frontier {z | a \u2264 z.re} = {z | z.re = a}\n[PROOFSTEP]\nsimpa only [frontier_Ici] using frontier_preimage_re (Ici a)\n[GOAL]\na : \u211d\n\u22a2 frontier {z | a \u2264 z.im} = {z | z.im = a}\n[PROOFSTEP]\nsimpa only [frontier_Ici] using frontier_preimage_im (Ici a)\n[GOAL]\na : \u211d\n\u22a2 frontier {z | z.re < a} = {z | z.re = a}\n[PROOFSTEP]\nsimpa only [frontier_Iio] using frontier_preimage_re (Iio a)\n[GOAL]\na : \u211d\n\u22a2 frontier {z | z.im < a} = {z | z.im = a}\n[PROOFSTEP]\nsimpa only [frontier_Iio] using frontier_preimage_im (Iio a)\n[GOAL]\na : \u211d\n\u22a2 frontier {z | a < z.re} = {z | z.re = a}\n[PROOFSTEP]\nsimpa only [frontier_Ioi] using frontier_preimage_re (Ioi a)\n[GOAL]\na : \u211d\n\u22a2 frontier {z | a < z.im} = {z | z.im = a}\n[PROOFSTEP]\nsimpa only [frontier_Ioi] using frontier_preimage_im (Ioi a)\n[GOAL]\ns t : Set \u211d\n\u22a2 closure (s \u00d7\u2102 t) = closure s \u00d7\u2102 closure t\n[PROOFSTEP]\nsimpa only [\u2190 preimage_eq_preimage equivRealProdClm.symm.toHomeomorph.surjective,\n  equivRealProdClm.symm.toHomeomorph.preimage_closure] using @closure_prod_eq _ _ _ _ s t\n[GOAL]\ns t : Set \u211d\n\u22a2 interior (s \u00d7\u2102 t) = interior s \u00d7\u2102 interior t\n[PROOFSTEP]\nrw [Set.reProdIm, Set.reProdIm, interior_inter, interior_preimage_re, interior_preimage_im]\n[GOAL]\ns t : Set \u211d\n\u22a2 frontier (s \u00d7\u2102 t) = closure s \u00d7\u2102 frontier t \u222a frontier s \u00d7\u2102 closure t\n[PROOFSTEP]\nsimpa only [\u2190 preimage_eq_preimage equivRealProdClm.symm.toHomeomorph.surjective,\n  equivRealProdClm.symm.toHomeomorph.preimage_frontier] using frontier_prod_eq s t\n[GOAL]\na b : \u211d\n\u22a2 frontier {z | a \u2264 z.re \u2227 b \u2264 z.im} = {z | a \u2264 z.re \u2227 z.im = b \u2228 z.re = a \u2227 b \u2264 z.im}\n[PROOFSTEP]\nsimpa only [closure_Ici, frontier_Ici] using frontier_reProdIm (Ici a) (Ici b)\n[GOAL]\na b : \u211d\n\u22a2 frontier {z | a \u2264 z.re \u2227 z.im \u2264 b} = {z | a \u2264 z.re \u2227 z.im = b \u2228 z.re = a \u2227 z.im \u2264 b}\n[PROOFSTEP]\nsimpa only [closure_Ici, closure_Iic, frontier_Ici, frontier_Iic] using frontier_reProdIm (Ici a) (Iic b)\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Complex.ReImTopology", "llama_tokens": 1521, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9149009526726544, "lm_q2_score": 0.8376199694135332, "lm_q1q2_score": 0.7663393079940811}}
{"text": "[GOAL]\na : \u2124\nn : \u2115\n\u22a2 succ^[n + 1] a = a + \u2191(n + 1)\n[PROOFSTEP]\nrw [Function.iterate_succ', Int.ofNat_succ, \u2190 add_assoc]\n[GOAL]\na : \u2124\nn : \u2115\n\u22a2 (succ \u2218 succ^[n]) a = a + \u2191n + 1\n[PROOFSTEP]\nexact congr_arg _ (succ_iterate a n)\n[GOAL]\na : \u2124\nn : \u2115\n\u22a2 pred^[n + 1] a = a - \u2191(n + 1)\n[PROOFSTEP]\nrw [Function.iterate_succ', Int.ofNat_succ, \u2190 sub_sub]\n[GOAL]\na : \u2124\nn : \u2115\n\u22a2 (pred \u2218 pred^[n]) a = a - \u2191n - 1\n[PROOFSTEP]\nexact congr_arg _ (pred_iterate a n)\n[GOAL]\na b : \u2124\nh : a \u2264 b\n\u22a2 Order.succ^[toNat (b - a)] a = b\n[PROOFSTEP]\nrw [succ_eq_succ, succ_iterate, toNat_sub_of_le h, \u2190 add_sub_assoc, add_sub_cancel']\n[GOAL]\na b : \u2124\nh : a \u2264 b\n\u22a2 Order.pred^[toNat (b - a)] b = a\n[PROOFSTEP]\nrw [pred_eq_pred, pred_iterate, toNat_sub_of_le h, sub_sub_cancel]\n[GOAL]\nz : \u2124\n\u22a2 z - 1 \u22d6 z\n[PROOFSTEP]\nrw [Int.covby_iff_succ_eq, sub_add_cancel]\n[GOAL]\na b : \u2115\n\u22a2 \u2191a \u22d6 \u2191b \u2194 a \u22d6 b\n[PROOFSTEP]\nrw [Nat.covby_iff_succ_eq, Int.covby_iff_succ_eq]\n[GOAL]\na b : \u2115\n\u22a2 \u2191a + 1 = \u2191b \u2194 a + 1 = b\n[PROOFSTEP]\nexact Int.coe_nat_inj'\n", "meta": {"mathlib_filename": "Mathlib.Data.Int.SuccPred", "llama_tokens": 585, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9207896715436483, "lm_q2_score": 0.8311430562234878, "lm_q1q2_score": 0.7653079417458094}}
{"text": "[GOAL]\n\u03b1 : Type u_1\np\u271d : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\u271d\nl : List \u03b1\na : \u03b1\np : \u03b1 \u2192 Bool\n\u22a2 all l p = true \u2194 \u2200 (a : \u03b1), a \u2208 l \u2192 p a = true\n[PROOFSTEP]\ninduction' l with a l ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\np\u271d : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\u271d\nl : List \u03b1\na : \u03b1\np : \u03b1 \u2192 Bool\n\u22a2 all [] p = true \u2194 \u2200 (a : \u03b1), a \u2208 [] \u2192 p a = true\n[PROOFSTEP]\nexact iff_of_true rfl (forall_mem_nil _)\n[GOAL]\ncase cons\n\u03b1 : Type u_1\np\u271d : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\u271d\nl\u271d : List \u03b1\na\u271d : \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\nih : all l p = true \u2194 \u2200 (a : \u03b1), a \u2208 l \u2192 p a = true\n\u22a2 all (a :: l) p = true \u2194 \u2200 (a_1 : \u03b1), a_1 \u2208 a :: l \u2192 p a_1 = true\n[PROOFSTEP]\nsimp only [all_cons, Bool.and_coe_iff, ih, forall_mem_cons]\n[GOAL]\n\u03b1 : Type u_1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nl : List \u03b1\na : \u03b1\n\u22a2 (all l fun a => decide (p a)) = true \u2194 \u2200 (a : \u03b1), a \u2208 l \u2192 p a\n[PROOFSTEP]\nsimp only [all_iff_forall, Bool.of_decide_iff]\n[GOAL]\n\u03b1 : Type u_1\np\u271d : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\u271d\nl : List \u03b1\na : \u03b1\np : \u03b1 \u2192 Bool\n\u22a2 any l p = true \u2194 \u2203 a, a \u2208 l \u2227 p a = true\n[PROOFSTEP]\ninduction' l with a l ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\np\u271d : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\u271d\nl : List \u03b1\na : \u03b1\np : \u03b1 \u2192 Bool\n\u22a2 any [] p = true \u2194 \u2203 a, a \u2208 [] \u2227 p a = true\n[PROOFSTEP]\nexact iff_of_false Bool.not_false' (not_exists_mem_nil _)\n[GOAL]\ncase cons\n\u03b1 : Type u_1\np\u271d : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\u271d\nl\u271d : List \u03b1\na\u271d : \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nl : List \u03b1\nih : any l p = true \u2194 \u2203 a, a \u2208 l \u2227 p a = true\n\u22a2 any (a :: l) p = true \u2194 \u2203 a_1, a_1 \u2208 a :: l \u2227 p a_1 = true\n[PROOFSTEP]\nsimp only [any_cons, Bool.or_coe_iff, ih, exists_mem_cons_iff]\n[GOAL]\n\u03b1 : Type u_1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nl : List \u03b1\na : \u03b1\n\u22a2 (any l fun a => decide (p a)) = true \u2194 \u2203 a, a \u2208 l \u2227 p a\n[PROOFSTEP]\nsimp [any_iff_exists]\n", "meta": {"mathlib_filename": "Mathlib.Data.Bool.AllAny", "llama_tokens": 892, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942261220292, "lm_q2_score": 0.8438950986284991, "lm_q1q2_score": 0.7513149337616332}}