data/code/lean_proofsteps/0.70-0.75.jsonl

{"text": "[GOAL]\na b : ℕ\nh : 0 < b\n⊢ a + 0 < a + b\n[PROOFSTEP]\napply Nat.add_lt_add_left\n[GOAL]\ncase h\na b : ℕ\nh : 0 < b\n⊢ 0 < b\n[PROOFSTEP]\nassumption\n[GOAL]\na : ℕ\nthis : 0 < a + 1\n⊢ 0 < 1 + a\n[PROOFSTEP]\nsimp [Nat.add_comm]\n[GOAL]\na : ℕ\nthis : 0 < a + 1\n⊢ 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 : ℕ\nha : a ≤ pred a\n⊢ IsMin a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase zero\nha : zero ≤ pred zero\n⊢ IsMin zero\n[PROOFSTEP]\nexact isMin_bot\n[GOAL]\ncase succ\nn✝ : ℕ\nha : succ n✝ ≤ pred (succ n✝)\n⊢ IsMin (succ n✝)\n[PROOFSTEP]\nexact (not_succ_le_self _ ha).elim\n[GOAL]\na b : ℕ\nh : a < b\n⊢ a ≤ pred b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase zero\na : ℕ\nh : a < zero\n⊢ a ≤ pred zero\n[PROOFSTEP]\nexact (a.not_lt_zero h).elim\n[GOAL]\ncase succ\na n✝ : ℕ\nh : a < succ n✝\n⊢ a ≤ pred (succ n✝)\n[PROOFSTEP]\nexact le_of_succ_le_succ h\n[GOAL]\na b : ℕ\nh : pred a < b\n⊢ a ≤ b\n[PROOFSTEP]\ncases a\n[GOAL]\ncase zero\nb : ℕ\nh : pred zero < b\n⊢ zero ≤ b\n[PROOFSTEP]\nexact b.zero_le\n[GOAL]\ncase succ\nb n✝ : ℕ\nh : pred (succ n✝) < b\n⊢ succ n✝ ≤ b\n[PROOFSTEP]\nexact h\n[GOAL]\na n : ℕ\n⊢ succ^[n + 1] a = a + (n + 1)\n[PROOFSTEP]\nrw [Function.iterate_succ', add_succ]\n[GOAL]\na n : ℕ\n⊢ (succ ∘ succ^[n]) a = succ (a + n)\n[PROOFSTEP]\nexact congr_arg _ (succ_iterate a n)\n[GOAL]\na n : ℕ\n⊢ pred^[n + 1] a = a - (n + 1)\n[PROOFSTEP]\nrw [Function.iterate_succ', sub_succ]\n[GOAL]\na n : ℕ\n⊢ (pred ∘ pred^[n]) a = pred (a - n)\n[PROOFSTEP]\nexact congr_arg _ (pred_iterate a n)\n[GOAL]\na b : ℕ\nh : a ≤ b\n⊢ Order.succ^[b - a] a = b\n[PROOFSTEP]\nrw [succ_eq_succ, succ_iterate, add_tsub_cancel_of_le h]\n[GOAL]\na b : ℕ\nh : a ≤ b\n⊢ 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]\nα : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝ : NonUnitalNonAssocRing R\na b : R\nh : Commute a b\n⊢ 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]\nα : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝ : NonUnitalNonAssocRing R\na b : R\nh : Commute a b\n⊢ 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]\nα : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝¹ : NonUnitalNonAssocRing R\ninst✝ : NoZeroDivisors R\na b : R\nh : Commute a b\n⊢ a * a = b * b ↔ a = b ∨ a = -b\n[PROOFSTEP]\nrw [← 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]\nα : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝ : NonAssocRing R\na : R\n⊢ a * a - 1 = (a + 1) * (a - 1)\n[PROOFSTEP]\nrw [← (Commute.one_right a).mul_self_sub_mul_self_eq, mul_one]\n[GOAL]\nα : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝¹ : NonAssocRing R\ninst✝ : NoZeroDivisors R\na : R\n⊢ a * a = 1 ↔ a = 1 ∨ a = -1\n[PROOFSTEP]\nrw [← (Commute.one_right a).mul_self_eq_mul_self_iff, mul_one]\n[GOAL]\nα : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nu : Rˣ\n⊢ u⁻¹ = u ↔ u = 1 ∨ u = -1\n[PROOFSTEP]\nrw [inv_eq_iff_mul_eq_one]\n[GOAL]\nα : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nu : Rˣ\n⊢ u * u = 1 ↔ u = 1 ∨ u = -1\n[PROOFSTEP]\nsimp only [ext_iff]\n[GOAL]\nα : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nu : Rˣ\n⊢ ↑(u * u) = ↑1 ↔ ↑u = ↑1 ∨ ↑u = ↑(-1)\n[PROOFSTEP]\npush_cast\n[GOAL]\nα : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nu : Rˣ\n⊢ ↑u * ↑u = 1 ↔ ↑u = 1 ∨ ↑u = -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]\nι : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nα : Type u_5\nR : Type u_6\nS : Type u_7\ninst✝⁴ : Fintype m\ninst✝³ : Fintype n\ninst✝² : Fintype p\ninst✝¹ : DecidableEq n\ninst✝ : AddCommMonoidWithOne R\n⊢ trace 1 = ↑(Fintype.card n)\n[PROOFSTEP]\nsimp_rw [trace, diag_one, Pi.one_def, Finset.sum_const, nsmul_one, Finset.card_univ]\n[GOAL]\nι : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nα : Type u_5\nR : Type u_6\nS : Type u_7\ninst✝⁴ : Fintype m\ninst✝³ : Fintype n\ninst✝² : Fintype p\ninst✝¹ : AddCommMonoid R\ninst✝ : CommSemigroup R\nA : Matrix m n R\nB : Matrix n m R\n⊢ trace (A * B) = trace (B * A)\n[PROOFSTEP]\nrw [← trace_transpose, ← trace_transpose_mul, transpose_mul]\n[GOAL]\nι : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nα : Type u_5\nR : Type u_6\nS : Type u_7\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Fintype p\ninst✝ : NonUnitalCommSemiring R\nA : Matrix m n R\nB : Matrix n p R\nC : Matrix p m R\n⊢ trace (A * B * C) = trace (C * A * B)\n[PROOFSTEP]\nrw [trace_mul_comm, Matrix.mul_assoc]\n[GOAL]\nι : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nα : Type u_5\nR : Type u_6\nS : Type u_7\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Fintype p\ninst✝ : NonUnitalCommSemiring R\nA : Matrix m n R\nB : Matrix n p R\nC : Matrix p m R\n⊢ trace (A * (B * C)) = trace (C * (A * B))\n[PROOFSTEP]\nrw [← Matrix.mul_assoc, trace_mul_comm]\n[GOAL]\nι : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nα : Type u_5\nR : Type u_6\nS : Type u_7\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Fintype p\ninst✝ : NonUnitalNonAssocSemiring R\na b : n → R\n⊢ trace (col a * row b) = a ⬝ᵥ b\n[PROOFSTEP]\napply Finset.sum_congr rfl\n[GOAL]\nι : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nα : Type u_5\nR : Type u_6\nS : Type u_7\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Fintype p\ninst✝ : NonUnitalNonAssocSemiring R\na b : n → R\n⊢ ∀ (x : n), x ∈ Finset.univ → diag (col a * row b) x = a x * b x\n[PROOFSTEP]\nsimp [mul_apply]\n[GOAL]\nι : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nα : Type u_5\nR : Type u_6\nS : Type u_7\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Fintype p\ninst✝ : AddCommMonoid R\nA : Matrix (Fin 3) (Fin 3) R\n⊢ trace A = A 0 0 + A 1 1 + A 2 2\n[PROOFSTEP]\nrw [← add_zero (A 2 2), add_assoc]\n[GOAL]\nι : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nα : Type u_5\nR : Type u_6\nS : Type u_7\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Fintype p\ninst✝ : AddCommMonoid R\nA : Matrix (Fin 3) (Fin 3) R\n⊢ 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 : ℕ\nsrc✝ : Mul (Fin n) := inferInstanceAs (Mul (Fin n))\nx✝² x✝¹ x✝ : Fin n\na : ℕ\nha : a < n\nb : ℕ\nhb : b < n\nc : ℕ\nhc : c < n\n⊢ a * b * c ≡ a * (b * c) [MOD n]\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\nn : ℕ\nx✝² x✝¹ x✝ : Fin n\na : ℕ\nha : a < n\nb : ℕ\nhb : b < n\nc : ℕ\nhc : c < n\n⊢ a * (b + c) ≡ a * b + a * c [MOD n]\n[PROOFSTEP]\nrw [mul_add]\n[GOAL]\nn : ℕ\nsrc✝¹ : AddCommSemigroup (Fin n) := addCommSemigroup n\nsrc✝ : CommSemigroup (Fin n) := instCommSemigroup n\na b c : Fin n\n⊢ (a + b) * c = a * c + b * c\n[PROOFSTEP]\nrw [mul_comm, left_distrib_aux, mul_comm _ b, mul_comm]\n[GOAL]\n⊢ Repr (ZMod 0)\n[PROOFSTEP]\ndsimp [ZMod]\n[GOAL]\n⊢ Repr ℤ\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nn : ℕ\n⊢ Repr (ZMod (n + 1))\n[PROOFSTEP]\ndsimp [ZMod]\n[GOAL]\nn : ℕ\n⊢ Repr (Fin (n + 1))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nn : ℕ\ninst✝ : Fintype (ZMod n)\n⊢ 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 : ℕ\ninst✝ : Fintype (ZMod n)\n⊢ 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✝ : Fintype (ZMod Nat.zero)\n⊢ Fintype.card (ZMod Nat.zero) = Nat.zero\n[PROOFSTEP]\n\n| zero => exact (not_finite (ZMod 0)).elim\n[GOAL]\ncase zero\ninst✝ : Fintype (ZMod Nat.zero)\n⊢ Fintype.card (ZMod Nat.zero) = Nat.zero\n[PROOFSTEP]\nexact (not_finite (ZMod 0)).elim\n[GOAL]\ncase succ\nn : ℕ\ninst✝ : Fintype (ZMod (Nat.succ n))\n⊢ 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 : ℕ\ninst✝ : Fintype (ZMod (Nat.succ n))\n⊢ 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 : ℕ\n⊢ range (succ n) = n ::ₘ range n\n[PROOFSTEP]\nrw [range, List.range_succ, ← coe_add, add_comm]\n[GOAL]\nn : ℕ\n⊢ ↑[n] + ↑(List.range n) = n ::ₘ range n\n[PROOFSTEP]\nrfl\n[GOAL]\na : ℕ\nm : Multiset ℕ\n⊢ Disjoint (range a) (map (fun x => a + x) m)\n[PROOFSTEP]\nintro x hxa hxb\n[GOAL]\na : ℕ\nm : Multiset ℕ\nx : ℕ\nhxa : x ∈ range a\nhxb : x ∈ map (fun x => a + x) m\n⊢ False\n[PROOFSTEP]\nrw [range, mem_coe, List.mem_range] at hxa \n[GOAL]\na : ℕ\nm : Multiset ℕ\nx : ℕ\nhxa : x < a\nhxb : x ∈ map (fun x => a + x) m\n⊢ False\n[PROOFSTEP]\nobtain ⟨c, _, rfl⟩ := mem_map.1 hxb\n[GOAL]\ncase intro.intro\na : ℕ\nm : Multiset ℕ\nc : ℕ\nleft✝ : c ∈ m\nhxa : a + c < a\nhxb : a + c ∈ map (fun x => a + x) m\n⊢ False\n[PROOFSTEP]\nexact (self_le_add_right _ _).not_lt hxa\n[GOAL]\na b : ℕ\n⊢ range (a + b) = range a ∪ map (fun x => a + x) (range b)\n[PROOFSTEP]\nrw [range_add, add_eq_union_iff_disjoint]\n[GOAL]\na b : ℕ\n⊢ 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\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf g : α →CO β\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n⊢ f = g\n[PROOFSTEP]\nobtain ⟨⟨_, _⟩, _⟩ := f\n[GOAL]\ncase mk.mk\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ng : α →CO β\ntoFun✝ : α → β\ncontinuous_toFun✝ : Continuous toFun✝\nmap_open'✝ : IsOpenMap (ContinuousMap.mk toFun✝).toFun\nh : (fun f => f.toFun) { toContinuousMap := ContinuousMap.mk toFun✝, map_open' := map_open'✝ } = (fun f => f.toFun) g\n⊢ { toContinuousMap := ContinuousMap.mk toFun✝, map_open' := map_open'✝ } = g\n[PROOFSTEP]\nobtain ⟨⟨_, _⟩, _⟩ := g\n[GOAL]\ncase mk.mk.mk.mk\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ntoFun✝¹ : α → β\ncontinuous_toFun✝¹ : Continuous toFun✝¹\nmap_open'✝¹ : IsOpenMap (ContinuousMap.mk toFun✝¹).toFun\ntoFun✝ : α → β\ncontinuous_toFun✝ : Continuous toFun✝\nmap_open'✝ : IsOpenMap (ContinuousMap.mk toFun✝).toFun\nh :\n  (fun f => f.toFun) { toContinuousMap := ContinuousMap.mk toFun✝¹, map_open' := map_open'✝¹ } =\n    (fun f => f.toFun) { toContinuousMap := ContinuousMap.mk toFun✝, map_open' := map_open'✝ }\n⊢ { toContinuousMap := ContinuousMap.mk toFun✝¹, map_open' := map_open'✝¹ } =\n    { toContinuousMap := ContinuousMap.mk toFun✝, map_open' := map_open'✝ }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ng : β →CO γ\nf₁ f₂ : α →CO β\nhg : Injective ↑g\nh : comp g f₁ = comp g f₂\na : α\n⊢ ↑g (↑f₁ a) = ↑g (↑f₂ a)\n[PROOFSTEP]\nrw [← 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 : ℕ\n⊢ ↑(toPNat' (m + 1)) = if 0 < m + 1 then m + 1 else 1\n[PROOFSTEP]\nrw [if_pos (succ_pos m)]\n[GOAL]\nm : ℕ\n⊢ ↑(toPNat' (m + 1)) = m + 1\n[PROOFSTEP]\nrfl\n[GOAL]\nm k : ℕ+\n⊢ ↑(mod m k) = if ↑m % ↑k = 0 then ↑k else ↑m % ↑k\n[PROOFSTEP]\ndsimp [mod, modDiv]\n[GOAL]\nm k : ℕ+\n⊢ ↑(modDivAux k (↑m % ↑k) (↑m / ↑k)).fst = if ↑m % ↑k = 0 then ↑k else ↑m % ↑k\n[PROOFSTEP]\ncases (m : ℕ) % (k : ℕ) 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 : ℕ+\nx✝ : ℕ\n⊢ ↑(modDivAux k x✝ (↑m / ↑k)).fst = if x✝ = 0 then ↑k else x✝\n[PROOFSTEP]\ncases (m : ℕ) % (k : ℕ) 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 : ℕ+\n⊢ ↑(modDivAux k zero (↑m / ↑k)).fst = if zero = 0 then ↑k else zero\n[PROOFSTEP]\n\n| zero =>\n  rw [if_pos rfl]\n  rfl\n[GOAL]\ncase zero\nm k : ℕ+\n⊢ ↑(modDivAux k zero (↑m / ↑k)).fst = if zero = 0 then ↑k else zero\n[PROOFSTEP]\nrw [if_pos rfl]\n[GOAL]\ncase zero\nm k : ℕ+\n⊢ ↑(modDivAux k zero (↑m / ↑k)).fst = ↑k\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nm k : ℕ+\nn : ℕ\n⊢ ↑(modDivAux k (succ n) (↑m / ↑k)).fst = if succ n = 0 then ↑k 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 : ℕ+\nn : ℕ\n⊢ ↑(modDivAux k (succ n) (↑m / ↑k)).fst = if succ n = 0 then ↑k else succ n\n[PROOFSTEP]\nrw [if_neg n.succ_ne_zero]\n[GOAL]\ncase succ\nm k : ℕ+\nn : ℕ\n⊢ ↑(modDivAux k (succ n) (↑m / ↑k)).fst = succ n\n[PROOFSTEP]\nrfl\n[GOAL]\nm k : ℕ+\n⊢ div m k = if ↑m % ↑k = 0 then pred (↑m / ↑k) else ↑m / ↑k\n[PROOFSTEP]\ndsimp [div, modDiv]\n[GOAL]\nm k : ℕ+\n⊢ (modDivAux k (↑m % ↑k) (↑m / ↑k)).snd = if ↑m % ↑k = 0 then pred (↑m / ↑k) else ↑m / ↑k\n[PROOFSTEP]\ncases (m : ℕ) % (k : ℕ) 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 : ℕ+\nx✝ : ℕ\n⊢ (modDivAux k x✝ (↑m / ↑k)).snd = if x✝ = 0 then pred (↑m / ↑k) else ↑m / ↑k\n[PROOFSTEP]\ncases (m : ℕ) % (k : ℕ) 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 : ℕ+\n⊢ (modDivAux k zero (↑m / ↑k)).snd = if zero = 0 then pred (↑m / ↑k) else ↑m / ↑k\n[PROOFSTEP]\n\n| zero =>\n  rw [if_pos rfl]\n  rfl\n[GOAL]\ncase zero\nm k : ℕ+\n⊢ (modDivAux k zero (↑m / ↑k)).snd = if zero = 0 then pred (↑m / ↑k) else ↑m / ↑k\n[PROOFSTEP]\nrw [if_pos rfl]\n[GOAL]\ncase zero\nm k : ℕ+\n⊢ (modDivAux k zero (↑m / ↑k)).snd = pred (↑m / ↑k)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nm k : ℕ+\nn : ℕ\n⊢ (modDivAux k (succ n) (↑m / ↑k)).snd = if succ n = 0 then pred (↑m / ↑k) else ↑m / ↑k\n[PROOFSTEP]\n\n| succ n =>\n  rw [if_neg n.succ_ne_zero]\n  rfl\n[GOAL]\ncase succ\nm k : ℕ+\nn : ℕ\n⊢ (modDivAux k (succ n) (↑m / ↑k)).snd = if succ n = 0 then pred (↑m / ↑k) else ↑m / ↑k\n[PROOFSTEP]\nrw [if_neg n.succ_ne_zero]\n[GOAL]\ncase succ\nm k : ℕ+\nn : ℕ\n⊢ (modDivAux k (succ n) (↑m / ↑k)).snd = ↑m / ↑k\n[PROOFSTEP]\nrfl\n[GOAL]\nn : ℤ\nhn : 0 < n\n⊢ ↑↑(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]\n𝕜✝ : Type u_1\n𝕜 : Type u_2\ninst✝ : NonUnitalSeminormedRing 𝕜\na✝ b✝ : 𝕜\nhx : a✝ ∈ ball 0 1\nhy : b✝ ∈ ball 0 1\n⊢ a✝ * b✝ ∈ ball 0 1\n[PROOFSTEP]\nrw [mem_ball_zero_iff] at *\n[GOAL]\n𝕜✝ : Type u_1\n𝕜 : Type u_2\ninst✝ : NonUnitalSeminormedRing 𝕜\na✝ b✝ : 𝕜\nhx : ‖a✝‖ < 1\nhy : ‖b✝‖ < 1\n⊢ ‖a✝ * b✝‖ < 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]\n𝕜✝ : Type u_1\n𝕜 : Type u_2\ninst✝ : NonUnitalSeminormedRing 𝕜\na✝ b✝ : 𝕜\nhx : a✝ ∈ closedBall 0 1\nhy : b✝ ∈ closedBall 0 1\n⊢ a✝ * b✝ ∈ closedBall 0 1\n[PROOFSTEP]\nrw [mem_closedBall_zero_iff] at *\n[GOAL]\n𝕜✝ : Type u_1\n𝕜 : Type u_2\ninst✝ : NonUnitalSeminormedRing 𝕜\na✝ b✝ : 𝕜\nhx : ‖a✝‖ ≤ 1\nhy : ‖b✝‖ ≤ 1\n⊢ ‖a✝ * b✝‖ ≤ 1\n[PROOFSTEP]\nexact (norm_mul_le _ _).trans (mul_le_one hx (norm_nonneg _) hy)\n[GOAL]\n𝕜✝ : Type u_1\n𝕜 : Type u_2\ninst✝ : NormedDivisionRing 𝕜\na✝ b✝ : 𝕜\nhx : a✝ ∈ sphere 0 1\nhy : b✝ ∈ sphere 0 1\n⊢ a✝ * b✝ ∈ sphere 0 1\n[PROOFSTEP]\nrw [mem_sphere_zero_iff_norm] at *\n[GOAL]\n𝕜✝ : Type u_1\n𝕜 : Type u_2\ninst✝ : NormedDivisionRing 𝕜\na✝ b✝ : 𝕜\nhx : ‖a✝‖ = 1\nhy : ‖b✝‖ = 1\n⊢ ‖a✝ * b✝‖ = 1\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n𝕜 : Type u_1\ninst✝ : NormedDivisionRing 𝕜\nx : ↑(sphere 0 1)\n⊢ ‖(↑x)⁻¹‖ = 1\n[PROOFSTEP]\nrw [norm_inv, mem_sphere_zero_iff_norm.1 x.coe_prop, inv_one]\n[GOAL]\n𝕜 : Type u_1\ninst✝ : NormedDivisionRing 𝕜\nx y : ↑(sphere 0 1)\n⊢ ‖↑x / ↑y‖ = 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]\n𝕜 : Type u_1\ninst✝ : NormedDivisionRing 𝕜\nx : ↑(sphere 0 1)\nn : ℤ\n⊢ ↑x ^ n ∈ 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]\n𝕜 : Type u_1\ninst✝ : NormedDivisionRing 𝕜\nx y : ↑(sphere 0 1)\nh : ↑(unitSphereToUnits 𝕜) x = ↑(unitSphereToUnits 𝕜) y\n⊢ ↑x = ↑y\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✝ b✝ : A\nr : R\ninst✝³ : OrderedCommRing R\ninst✝² : OrderedRing A\ninst✝¹ : Algebra R A\ninst✝ : OrderedSMul R A\na b : R\nh : a ≤ b\n⊢ ↑(algebraMap R A) a ≤ ↑(algebraMap R A) b\n[PROOFSTEP]\nrw [Algebra.algebraMap_eq_smul_one, Algebra.algebraMap_eq_smul_one, ← sub_nonneg, ← sub_smul]\n[GOAL]\nR : Type u_1\nA : Type u_2\na✝ b✝ : A\nr : R\ninst✝³ : OrderedCommRing R\ninst✝² : OrderedRing A\ninst✝¹ : Algebra R A\ninst✝ : OrderedSMul R A\na b : R\nh : a ≤ b\n⊢ 0 ≤ (b - a) • 1\n[PROOFSTEP]\ntrans (b - a) • (0 : A)\n[GOAL]\nR : Type u_1\nA : Type u_2\na✝ b✝ : A\nr : R\ninst✝³ : OrderedCommRing R\ninst✝² : OrderedRing A\ninst✝¹ : Algebra R A\ninst✝ : OrderedSMul R A\na b : R\nh : a ≤ b\n⊢ 0 ≤ (b - a) • 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nA : Type u_2\na✝ b✝ : A\nr : R\ninst✝³ : OrderedCommRing R\ninst✝² : OrderedRing A\ninst✝¹ : Algebra R A\ninst✝ : OrderedSMul R A\na b : R\nh : a ≤ b\n⊢ (b - a) • 0 ≤ (b - a) • 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⊢ (if p then q else True) = (¬p ∨ q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\np : Prop\nh : Decidable p\nq : Prop\n⊢ (if p then q else True) = (¬p ∨ q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\ncase pos\np : Prop\nh✝ : Decidable p\nq : Prop\nh : p\n⊢ (if p then q else True) = (¬p ∨ q)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\np : Prop\nh✝ : Decidable p\nq : Prop\nh : ¬p\n⊢ (if p then q else True) = (¬p ∨ q)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\np : Prop\nh : Decidable p\nq : Prop\n⊢ (if p then True else q) = (p ∨ q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\np : Prop\nh : Decidable p\nq : Prop\n⊢ (if p then True else q) = (p ∨ q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\ncase pos\np : Prop\nh✝ : Decidable p\nq : Prop\nh : p\n⊢ (if p then True else q) = (p ∨ q)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\np : Prop\nh✝ : Decidable p\nq : Prop\nh : ¬p\n⊢ (if p then True else q) = (p ∨ q)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\np : Prop\nh : Decidable p\nq : Prop\n⊢ (if p then q else False) = (p ∧ q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\np : Prop\nh : Decidable p\nq : Prop\n⊢ (if p then q else False) = (p ∧ q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\ncase pos\np : Prop\nh✝ : Decidable p\nq : Prop\nh : p\n⊢ (if p then q else False) = (p ∧ q)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\np : Prop\nh✝ : Decidable p\nq : Prop\nh : ¬p\n⊢ (if p then q else False) = (p ∧ q)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\np : Prop\nh : Decidable p\nq : Prop\n⊢ (if p then False else q) = (¬p ∧ q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\np : Prop\nh : Decidable p\nq : Prop\n⊢ (if p then False else q) = (¬p ∧ q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\ncase pos\np : Prop\nh✝ : Decidable p\nq : Prop\nh : p\n⊢ (if p then False else q) = (¬p ∧ q)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\np : Prop\nh✝ : Decidable p\nq : Prop\nh : ¬p\n⊢ (if p then False else q) = (¬p ∧ 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 : ℕ\nh0 : 0 < a\nh1 : a < k\nn : ℕ\n⊢ π' (k + n) ≤ card (filter Prime (range k)) + card (filter Prime (Ico k (k + n)))\n[PROOFSTEP]\nrw [primeCounting', count_eq_card_filter_range, range_eq_Ico, ← Ico_union_Ico_eq_Ico (zero_le k) le_self_add,\n  filter_union]\n[GOAL]\na k : ℕ\nh0 : 0 < a\nh1 : a < k\nn : ℕ\n⊢ card (filter Prime (Ico 0 k) ∪ filter Prime (Ico k (k + n))) ≤\n    card (filter Prime (Ico 0 k)) + card (filter Prime (Ico k (k + n)))\n[PROOFSTEP]\napply card_union_le\n[GOAL]\na k : ℕ\nh0 : 0 < a\nh1 : a < k\nn : ℕ\n⊢ card (filter Prime (range k)) + card (filter Prime (Ico k (k + n))) ≤ π' k + card (filter Prime (Ico k (k + n)))\n[PROOFSTEP]\nrw [primeCounting', count_eq_card_filter_range]\n[GOAL]\na k : ℕ\nh0 : 0 < a\nh1 : a < k\nn : ℕ\n⊢ π' k + card (filter Prime (Ico k (k + n))) ≤ π' 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 : ℕ\nh0 : 0 < a\nh1 : a < k\nn : ℕ\n⊢ filter Prime (Ico k (k + n)) ⊆ filter (coprime a) (Ico k (k + n))\n[PROOFSTEP]\nsimp only [subset_iff, and_imp, mem_filter, mem_Ico]\n[GOAL]\na k : ℕ\nh0 : 0 < a\nh1 : a < k\nn : ℕ\n⊢ ∀ ⦃x : ℕ⦄, k ≤ x → x < k + n → Prime x → (k ≤ x ∧ x < k + n) ∧ coprime a x\n[PROOFSTEP]\nintro p succ_k_le_p p_lt_n p_prime\n[GOAL]\na k : ℕ\nh0 : 0 < a\nh1 : a < k\nn p : ℕ\nsucc_k_le_p : k ≤ p\np_lt_n : p < k + n\np_prime : Prime p\n⊢ (k ≤ p ∧ p < k + n) ∧ coprime a p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\na k : ℕ\nh0 : 0 < a\nh1 : a < k\nn p : ℕ\nsucc_k_le_p : k ≤ p\np_lt_n : p < k + n\np_prime : Prime p\n⊢ k ≤ p ∧ p < k + n\n[PROOFSTEP]\nexact ⟨succ_k_le_p, p_lt_n⟩\n[GOAL]\ncase right\na k : ℕ\nh0 : 0 < a\nh1 : a < k\nn p : ℕ\nsucc_k_le_p : k ≤ p\np_lt_n : p < k + n\np_prime : Prime p\n⊢ coprime a p\n[PROOFSTEP]\nrw [coprime_comm]\n[GOAL]\ncase right\na k : ℕ\nh0 : 0 < a\nh1 : a < k\nn p : ℕ\nsucc_k_le_p : k ≤ p\np_lt_n : p < k + n\np_prime : Prime p\n⊢ 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 : ℕ\nh0 : 0 < a\nh1 : a < k\nn : ℕ\n⊢ π' k + card (filter (coprime a) (Ico k (k + n))) ≤ π' k + φ a * (n / a + 1)\n[PROOFSTEP]\nrw [add_le_add_iff_left]\n[GOAL]\na k : ℕ\nh0 : 0 < a\nh1 : a < k\nn : ℕ\n⊢ card (filter (coprime a) (Ico k (k + n))) ≤ φ 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 : ℕ\nh : x ≤ y / k\n⊢ x * k ≤ y\n[PROOFSTEP]\nby_cases hk : k = 0\n[GOAL]\ncase pos\nk x y : ℕ\nh : x ≤ y / k\nhk : k = 0\n⊢ x * k ≤ y\ncase neg k x y : ℕ h : x ≤ y / k hk : ¬k = 0 ⊢ x * k ≤ y\n[PROOFSTEP]\ncase pos => rw [hk, mul_zero]; exact zero_le _\n[GOAL]\nk x y : ℕ\nh : x ≤ y / k\nhk : k = 0\n⊢ x * k ≤ y\n[PROOFSTEP]\ncase pos => rw [hk, mul_zero]; exact zero_le _\n[GOAL]\nk x y : ℕ\nh : x ≤ y / k\nhk : k = 0\n⊢ x * k ≤ y\n[PROOFSTEP]\nrw [hk, mul_zero]\n[GOAL]\nk x y : ℕ\nh : x ≤ y / k\nhk : k = 0\n⊢ 0 ≤ y\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase neg\nk x y : ℕ\nh : x ≤ y / k\nhk : ¬k = 0\n⊢ x * k ≤ y\n[PROOFSTEP]\ncase neg => rwa [← le_div_iff_mul_le (pos_iff_ne_zero.2 hk)]\n[GOAL]\nk x y : ℕ\nh : x ≤ y / k\nhk : ¬k = 0\n⊢ x * k ≤ y\n[PROOFSTEP]\ncase neg => rwa [← le_div_iff_mul_le (pos_iff_ne_zero.2 hk)]\n[GOAL]\nk x y : ℕ\nh : x ≤ y / k\nhk : ¬k = 0\n⊢ x * k ≤ y\n[PROOFSTEP]\nrwa [← le_div_iff_mul_le (pos_iff_ne_zero.2 hk)]\n[GOAL]\na b c d : ℕ\n⊢ a / b * (c / d) ≤ a * c / (b * d)\n[PROOFSTEP]\nby_cases hb : b = 0\n[GOAL]\ncase pos\na b c d : ℕ\nhb : b = 0\n⊢ a / b * (c / d) ≤ a * c / (b * d)\ncase neg a b c d : ℕ hb : ¬b = 0 ⊢ a / b * (c / d) ≤ a * c / (b * d)\n[PROOFSTEP]\ncase pos => simp [hb]\n[GOAL]\na b c d : ℕ\nhb : b = 0\n⊢ a / b * (c / d) ≤ a * c / (b * d)\n[PROOFSTEP]\ncase pos => simp [hb]\n[GOAL]\na b c d : ℕ\nhb : b = 0\n⊢ a / b * (c / d) ≤ a * c / (b * d)\n[PROOFSTEP]\nsimp [hb]\n[GOAL]\ncase neg\na b c d : ℕ\nhb : ¬b = 0\n⊢ a / b * (c / d) ≤ a * c / (b * d)\n[PROOFSTEP]\nby_cases hd : d = 0\n[GOAL]\ncase pos\na b c d : ℕ\nhb : ¬b = 0\nhd : d = 0\n⊢ a / b * (c / d) ≤ a * c / (b * d)\ncase neg a b c d : ℕ hb : ¬b = 0 hd : ¬d = 0 ⊢ a / b * (c / d) ≤ a * c / (b * d)\n[PROOFSTEP]\ncase pos => simp [hd]\n[GOAL]\na b c d : ℕ\nhb : ¬b = 0\nhd : d = 0\n⊢ a / b * (c / d) ≤ a * c / (b * d)\n[PROOFSTEP]\ncase pos => simp [hd]\n[GOAL]\na b c d : ℕ\nhb : ¬b = 0\nhd : d = 0\n⊢ a / b * (c / d) ≤ a * c / (b * d)\n[PROOFSTEP]\nsimp [hd]\n[GOAL]\ncase neg\na b c d : ℕ\nhb : ¬b = 0\nhd : ¬d = 0\n⊢ a / b * (c / d) ≤ a * c / (b * d)\n[PROOFSTEP]\nhave hbd : b * d ≠ 0 := mul_ne_zero hb hd\n[GOAL]\ncase neg\na b c d : ℕ\nhb : ¬b = 0\nhd : ¬d = 0\nhbd : b * d ≠ 0\n⊢ a / b * (c / d) ≤ 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 : ℕ\nhb : ¬b = 0\nhd : ¬d = 0\nhbd : b * d ≠ 0\n⊢ a / b * (c / d) * (b * d) ≤ a * c\n[PROOFSTEP]\ntransitivity ((a / b) * b) * ((c / d) * d)\n[GOAL]\ncase neg.a\na b c d : ℕ\nhb : ¬b = 0\nhd : ¬d = 0\nhbd : b * d ≠ 0\n⊢ a / b * (c / d) * (b * d) ≤ a / b * b * (c / d * d)\n[PROOFSTEP]\napply le_of_eq\n[GOAL]\ncase neg.a.a\na b c d : ℕ\nhb : ¬b = 0\nhd : ¬d = 0\nhbd : b * d ≠ 0\n⊢ 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 : ℕ\nhb : ¬b = 0\nhd : ¬d = 0\nhbd : b * d ≠ 0\n⊢ a / b * b * (c / d * d) ≤ a * c\n[PROOFSTEP]\napply Nat.mul_le_mul\n[GOAL]\ncase neg.a.h₁\na b c d : ℕ\nhb : ¬b = 0\nhd : ¬d = 0\nhbd : b * d ≠ 0\n⊢ a / b * b ≤ a\n[PROOFSTEP]\napply div_mul_le_self\n[GOAL]\ncase neg.a.h₂\na b c d : ℕ\nhb : ¬b = 0\nhd : ¬d = 0\nhbd : b * d ≠ 0\n⊢ c / d * d ≤ c\n[PROOFSTEP]\napply div_mul_le_self\n[GOAL]\nn k : ℕ\n⊢ let iter_next := fun n guess => (guess + n / guess) / 2;\n  sqrt.iter n k ≤ iter_next n (sqrt.iter n k)\n[PROOFSTEP]\nintro iter_next\n[GOAL]\nn k : ℕ\niter_next : ℕ → ℕ → ℕ := fun n guess => (guess + n / guess) / 2\n⊢ sqrt.iter n k ≤ iter_next n (sqrt.iter n k)\n[PROOFSTEP]\nunfold sqrt.iter\n[GOAL]\nn k : ℕ\niter_next : ℕ → ℕ → ℕ := fun n guess => (guess + n / guess) / 2\n⊢ (let next := (k + n / k) / 2;\n    if _h : next < k then sqrt.iter n next else k) ≤\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 : ℕ\niter_next : ℕ → ℕ → ℕ := fun n guess => (guess + n / guess) / 2\nh : (k + n / k) / 2 < k\n⊢ (let next := (k + n / k) / 2;\n    if _h : next < k then sqrt.iter n next else k) ≤\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 : ℕ\niter_next : ℕ → ℕ → ℕ := fun n guess => (guess + n / guess) / 2\nh : ¬(k + n / k) / 2 < k\n⊢ (let next := (k + n / k) / 2;\n    if _h : next < k then sqrt.iter n next else k) ≤\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 : ℕ\niter_next : ℕ → ℕ → ℕ := fun n guess => (guess + n / guess) / 2\nh : (k + n / k) / 2 < k\n⊢ (let next := (k + n / k) / 2;\n    if _h : next < k then sqrt.iter n next else k) ≤\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 : ℕ\niter_next : ℕ → ℕ → ℕ := fun n guess => (guess + n / guess) / 2\nh : (k + n / k) / 2 < k\n⊢ (let next := (k + n / k) / 2;\n    if _h : next < k then sqrt.iter n next else k) ≤\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 : ℕ\niter_next : ℕ → ℕ → ℕ := fun n guess => (guess + n / guess) / 2\nh : (k + n / k) / 2 < k\n⊢ sqrt.iter n ((k + n / k) / 2) ≤ (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 : ℕ\niter_next : ℕ → ℕ → ℕ := fun n guess => (guess + n / guess) / 2\nh : ¬(k + n / k) / 2 < k\n⊢ (let next := (k + n / k) / 2;\n    if _h : next < k then sqrt.iter n next else k) ≤\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 : ℕ\niter_next : ℕ → ℕ → ℕ := fun n guess => (guess + n / guess) / 2\nh : ¬(k + n / k) / 2 < k\n⊢ (let next := (k + n / k) / 2;\n    if _h : next < k then sqrt.iter n next else k) ≤\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 : ℕ\niter_next : ℕ → ℕ → ℕ := fun n guess => (guess + n / guess) / 2\nh : ¬(k + n / k) / 2 < k\n⊢ (let next := (k + n / k) / 2;\n    if _h : next < k then sqrt.iter n next else k) ≤\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 : ℕ\niter_next : ℕ → ℕ → ℕ := fun n guess => (guess + n / guess) / 2\nh : ¬(k + n / k) / 2 < k\n⊢ k ≤ (k + n / k) / 2\n[PROOFSTEP]\nexact Nat.le_of_not_lt h\n[GOAL]\nx✝ : ℕ\n⊢ 4 * 0 * x✝ ≤ (0 + x✝) * (0 + x✝)\n[PROOFSTEP]\nrw [mul_zero, zero_mul]\n[GOAL]\nx✝ : ℕ\n⊢ 0 ≤ (0 + x✝) * (0 + x✝)\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\nx✝ : ℕ\n⊢ 4 * x✝ * 0 ≤ (x✝ + 0) * (x✝ + 0)\n[PROOFSTEP]\nrw [mul_zero]\n[GOAL]\nx✝ : ℕ\n⊢ 0 ≤ (x✝ + 0) * (x✝ + 0)\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\na b : ℕ\n⊢ 4 * (a + 1) * (b + 1) ≤ (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 : ℕ\nih : 4 * a * b + 4 ≤ (a + b) * (a + b) + 4\n⊢ 4 * (a + 1) * (b + 1) ≤ (a + 1 + (b + 1)) * (a + 1 + (b + 1))\n[PROOFSTEP]\nsimp only [mul_add, add_mul, show (4 : ℕ) = 1 + 1 + 1 + 1 from rfl, one_mul, mul_one] at ih ⊢\n[GOAL]\na b : ℕ\nih : a * b + a * b + a * b + a * b + (1 + 1 + 1 + 1) ≤ a * a + b * a + (a * b + b * b) + (1 + 1 + 1 + 1)\n⊢ a * b + a * b + a * b + a * b + (b + b + b + b) + (a + a + a + a + (1 + 1 + 1 + 1)) ≤\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 ⊢\n[GOAL]\na b : ℕ\nih :\n  a * b + (a * b + (a * b + (a * b + (1 + (1 + (1 + 1)))))) ≤ a * a + (a * b + (b * a + (b * b + (1 + (1 + (1 + 1))))))\n⊢ a * b + (a * b + (a * b + (a * b + (1 + (1 + (1 + 1)))))) ≤ a * a + (a * b + (b * a + (b * b + (1 + (1 + (1 + 1))))))\n[PROOFSTEP]\nexact ih\n[GOAL]\nn guess : ℕ\n⊢ iter n guess * iter n guess ≤ n\n[PROOFSTEP]\nunfold sqrt.iter\n[GOAL]\nn guess : ℕ\n⊢ ((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) ≤\n    n\n[PROOFSTEP]\nlet next := (guess + n / guess) / 2\n[GOAL]\nn guess : ℕ\nnext : ℕ := (guess + n / guess) / 2\n⊢ ((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) ≤\n    n\n[PROOFSTEP]\nby_cases h : next < guess\n[GOAL]\ncase pos\nn guess : ℕ\nnext : ℕ := (guess + n / guess) / 2\nh : next < guess\n⊢ ((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) ≤\n    n\ncase neg\nn guess : ℕ\nnext : ℕ := (guess + n / guess) / 2\nh : ¬next < guess\n⊢ ((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) ≤\n    n\n[PROOFSTEP]\ncase pos => simpa only [dif_pos h] using sqrt.iter_sq_le n next\n[GOAL]\nn guess : ℕ\nnext : ℕ := (guess + n / guess) / 2\nh : next < guess\n⊢ ((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) ≤\n    n\n[PROOFSTEP]\ncase pos => simpa only [dif_pos h] using sqrt.iter_sq_le n next\n[GOAL]\nn guess : ℕ\nnext : ℕ := (guess + n / guess) / 2\nh : next < guess\n⊢ ((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) ≤\n    n\n[PROOFSTEP]\nsimpa only [dif_pos h] using sqrt.iter_sq_le n next\n[GOAL]\ncase neg\nn guess : ℕ\nnext : ℕ := (guess + n / guess) / 2\nh : ¬next < guess\n⊢ ((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) ≤\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 [← mul_two, ← le_div_iff_mul_le]\n  · exact le_of_not_lt h\n  · exact zero_lt_two\n[GOAL]\nn guess : ℕ\nnext : ℕ := (guess + n / guess) / 2\nh : ¬next < guess\n⊢ ((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) ≤\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 [← mul_two, ← le_div_iff_mul_le]\n  · exact le_of_not_lt h\n  · exact zero_lt_two\n[GOAL]\nn guess : ℕ\nnext : ℕ := (guess + n / guess) / 2\nh : ¬next < guess\n⊢ ((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) ≤\n    n\n[PROOFSTEP]\nsimp only [dif_neg h]\n[GOAL]\nn guess : ℕ\nnext : ℕ := (guess + n / guess) / 2\nh : ¬next < guess\n⊢ guess * guess ≤ n\n[PROOFSTEP]\napply Nat.mul_le_of_le_div\n[GOAL]\ncase h\nn guess : ℕ\nnext : ℕ := (guess + n / guess) / 2\nh : ¬next < guess\n⊢ guess ≤ n / guess\n[PROOFSTEP]\napply le_of_add_le_add_left (a := guess)\n[GOAL]\ncase h\nn guess : ℕ\nnext : ℕ := (guess + n / guess) / 2\nh : ¬next < guess\n⊢ guess + guess ≤ guess + n / guess\n[PROOFSTEP]\nrw [← mul_two, ← le_div_iff_mul_le]\n[GOAL]\ncase h\nn guess : ℕ\nnext : ℕ := (guess + n / guess) / 2\nh : ¬next < guess\n⊢ guess ≤ (guess + n / guess) / 2\n[PROOFSTEP]\nexact le_of_not_lt h\n[GOAL]\ncase h\nn guess : ℕ\nnext : ℕ := (guess + n / guess) / 2\nh : ¬next < guess\n⊢ 0 < 2\n[PROOFSTEP]\nexact zero_lt_two\n[GOAL]\nn guess : ℕ\nhn : n < (guess + 1) * (guess + 1)\n⊢ n < (iter n guess + 1) * (iter n guess + 1)\n[PROOFSTEP]\nunfold sqrt.iter\n[GOAL]\nn guess : ℕ\nhn : n < (guess + 1) * (guess + 1)\n⊢ 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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\n⊢ 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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : m < guess\n⊢ 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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : ¬m < guess\n⊢ 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  · 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 : ℕ) = 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 [← 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 : ℕ} : a ≤ 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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : m < guess\n⊢ 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  · 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 : ℕ) = 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 [← 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 : ℕ} : a ≤ 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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : m < guess\n⊢ 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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : m < guess\nthis : n < (m + 1) * (m + 1)\n⊢ 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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : m < guess\n⊢ 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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : m < guess\n⊢ 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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : m < guess\n⊢ (guess * guess + n) * (guess * guess + n) < 4 * (guess * guess) * ((m + 1) * (m + 1))\n[PROOFSTEP]\nrw [show (4 : ℕ) = 2 * 2 from rfl]\n[GOAL]\ncase this\nn guess : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : m < guess\n⊢ (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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : m < guess\n⊢ (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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : m < guess\n⊢ guess * guess + n < 2 * guess * succ ((guess + n / guess) / 2)\n[PROOFSTEP]\nrw [← 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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : m < guess\n⊢ 0 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\ncase this\nn guess : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : m < guess\n⊢ guess * guess + n < guess * (2 * ((guess + n / guess + 1 + 1) / 2))\n[PROOFSTEP]\nhave aux_lemma {a : ℕ} : a ≤ 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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : m < guess\na : ℕ\n⊢ a ≤ 2 * ((a + 1) / 2)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nn guess : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : m < guess\na : ℕ\n⊢ a ≤ (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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : m < guess\naux_lemma : ∀ {a : ℕ}, a ≤ 2 * ((a + 1) / 2)\n⊢ 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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : m < guess\naux_lemma : ∀ {a : ℕ}, a ≤ 2 * ((a + 1) / 2)\n⊢ guess * guess + n < guess * (guess + n / guess + 1)\n[PROOFSTEP]\nrw [add_assoc, mul_add]\n[GOAL]\ncase this\nn guess : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : m < guess\naux_lemma : ∀ {a : ℕ}, a ≤ 2 * ((a + 1) / 2)\n⊢ 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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : ¬m < guess\n⊢ 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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : ¬m < guess\n⊢ 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 : ℕ\nhn : n < (guess + 1) * (guess + 1)\nm : ℕ := (guess + n / guess) / 2\nh : ¬m < guess\n⊢ 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]\n𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedRing A\ninst✝² : NormedAlgebra 𝕜 A\ninst✝¹ : CompleteSpace A\ninst✝ : ProperSpace 𝕜\n⊢ CompactSpace ↑(characterSpace 𝕜 A)\n[PROOFSTEP]\nrw [← isCompact_iff_compactSpace]\n[GOAL]\n𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedRing A\ninst✝² : NormedAlgebra 𝕜 A\ninst✝¹ : CompleteSpace A\ninst✝ : ProperSpace 𝕜\n⊢ IsCompact (characterSpace 𝕜 A)\n[PROOFSTEP]\nhave h : characterSpace 𝕜 A ⊆ toNormedDual ⁻¹' Metric.closedBall 0 ‖(1 : A)‖ :=\n  by\n  intro φ hφ\n  rw [Set.mem_preimage, mem_closedBall_zero_iff]\n  exact (norm_le_norm_one ⟨φ, ⟨hφ.1, hφ.2⟩⟩ : _)\n[GOAL]\n𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedRing A\ninst✝² : NormedAlgebra 𝕜 A\ninst✝¹ : CompleteSpace A\ninst✝ : ProperSpace 𝕜\n⊢ characterSpace 𝕜 A ⊆ ↑toNormedDual ⁻¹' Metric.closedBall 0 ‖1‖\n[PROOFSTEP]\nintro φ hφ\n[GOAL]\n𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedRing A\ninst✝² : NormedAlgebra 𝕜 A\ninst✝¹ : CompleteSpace A\ninst✝ : ProperSpace 𝕜\nφ : WeakDual 𝕜 A\nhφ : φ ∈ characterSpace 𝕜 A\n⊢ φ ∈ ↑toNormedDual ⁻¹' Metric.closedBall 0 ‖1‖\n[PROOFSTEP]\nrw [Set.mem_preimage, mem_closedBall_zero_iff]\n[GOAL]\n𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedRing A\ninst✝² : NormedAlgebra 𝕜 A\ninst✝¹ : CompleteSpace A\ninst✝ : ProperSpace 𝕜\nφ : WeakDual 𝕜 A\nhφ : φ ∈ characterSpace 𝕜 A\n⊢ ‖↑toNormedDual φ‖ ≤ ‖1‖\n[PROOFSTEP]\nexact (norm_le_norm_one ⟨φ, ⟨hφ.1, hφ.2⟩⟩ : _)\n[GOAL]\n𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedRing A\ninst✝² : NormedAlgebra 𝕜 A\ninst✝¹ : CompleteSpace A\ninst✝ : ProperSpace 𝕜\nh : characterSpace 𝕜 A ⊆ ↑toNormedDual ⁻¹' Metric.closedBall 0 ‖1‖\n⊢ IsCompact (characterSpace 𝕜 A)\n[PROOFSTEP]\nexact isCompact_of_isClosed_subset (isCompact_closedBall 𝕜 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 : ℕ\n⊢ dist n m = dist m n\n[PROOFSTEP]\nsimp [dist.def, add_comm]\n[GOAL]\nn : ℕ\n⊢ dist n n = 0\n[PROOFSTEP]\nsimp [dist.def, tsub_self]\n[GOAL]\nn m : ℕ\nh : n = m\n⊢ dist n m = 0\n[PROOFSTEP]\nrw [h, dist_self]\n[GOAL]\nn m : ℕ\nh : n ≤ m\n⊢ dist n m = m - n\n[PROOFSTEP]\nrw [dist.def, tsub_eq_zero_iff_le.mpr h, zero_add]\n[GOAL]\nn m : ℕ\nh : m ≤ n\n⊢ dist n m = n - m\n[PROOFSTEP]\nrw [dist_comm]\n[GOAL]\nn m : ℕ\nh : m ≤ n\n⊢ dist m n = n - m\n[PROOFSTEP]\napply dist_eq_sub_of_le h\n[GOAL]\nn m : ℕ\n⊢ m ≤ n + dist n m\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\nn m : ℕ\n⊢ m ≤ dist n m + n\n[PROOFSTEP]\napply dist_tri_left\n[GOAL]\nn m : ℕ\n⊢ n ≤ dist n m + m\n[PROOFSTEP]\nrw [dist_comm]\n[GOAL]\nn m : ℕ\n⊢ n ≤ dist m n + m\n[PROOFSTEP]\napply dist_tri_left\n[GOAL]\nn m : ℕ\n⊢ n ≤ m + dist n m\n[PROOFSTEP]\nrw [dist_comm]\n[GOAL]\nn m : ℕ\n⊢ n ≤ m + dist m n\n[PROOFSTEP]\napply dist_tri_right\n[GOAL]\nn k m : ℕ\n⊢ 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 : ℕ\n⊢ n - m + (m + k - (n + k)) = n - m + (m - n)\n[PROOFSTEP]\nrw [@add_tsub_add_eq_tsub_right]\n[GOAL]\nk n m : ℕ\n⊢ dist (k + n) (k + m) = dist n m\n[PROOFSTEP]\nrw [add_comm k n, add_comm k m]\n[GOAL]\nk n m : ℕ\n⊢ dist (n + k) (m + k) = dist n m\n[PROOFSTEP]\napply dist_add_add_right\n[GOAL]\nn m k l : ℕ\nh : n + m = k + l\n⊢ dist n k = dist (n + m) (k + m)\n[PROOFSTEP]\nrw [dist_add_add_right]\n[GOAL]\nn m k l : ℕ\nh : n + m = k + l\n⊢ dist (n + m) (k + m) = dist (k + l) (k + m)\n[PROOFSTEP]\nrw [h]\n[GOAL]\nn m k l : ℕ\nh : n + m = k + l\n⊢ dist (k + l) (k + m) = dist l m\n[PROOFSTEP]\nrw [dist_add_add_left]\n[GOAL]\nn m k : ℕ\n⊢ dist n k ≤ 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 : ℕ\n⊢ 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 : ℕ\nthis : dist n m + dist m k = n - m + (m - k) + (k - m + (m - n))\n⊢ dist n k ≤ dist n m + dist m k\n[PROOFSTEP]\nrw [this, dist.def]\n[GOAL]\nn m k : ℕ\nthis : dist n m + dist m k = n - m + (m - k) + (k - m + (m - n))\n⊢ n - k + (k - n) ≤ 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 : ℕ\n⊢ 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 : ℕ\n⊢ 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 : ℕ\n⊢ i < j → dist i j = max i j - min i j\n[PROOFSTEP]\nintro h\n[GOAL]\ni j : ℕ\nh : i < j\n⊢ 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 : ℕ\n⊢ i ≥ j → dist i j = max i j - min i j\n[PROOFSTEP]\nintro h\n[GOAL]\ni j : ℕ\nh : i ≥ j\n⊢ 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 : ℕ\n⊢ dist (succ i) (succ j) = dist i j\n[PROOFSTEP]\nsimp [dist.def, succ_sub_succ]\n[GOAL]\ni j : ℕ\nhne : i ≠ j\nh : i < j\n⊢ 0 < dist i j\n[PROOFSTEP]\nrw [dist_eq_sub_of_le (le_of_lt h)]\n[GOAL]\ni j : ℕ\nhne : i ≠ j\nh : i < j\n⊢ 0 < j - i\n[PROOFSTEP]\napply tsub_pos_of_lt h\n[GOAL]\ni j : ℕ\nhne : i ≠ j\nh : i = j\n⊢ 0 < dist i j\n[PROOFSTEP]\ncontradiction\n[GOAL]\ni j : ℕ\nhne : i ≠ j\nh : i > j\n⊢ 0 < dist i j\n[PROOFSTEP]\nrw [dist_eq_sub_of_le_right (le_of_lt h)]\n[GOAL]\ni j : ℕ\nhne : i ≠ j\nh : i > j\n⊢ 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✝¹ : LinearOrderedAddCommGroup G\ninst✝ : OrderedAddCommGroup H\nf : G → H\nh₁ : ∀ (x : G), f (-x) = -f x\nh₂ : StrictMonoOn f (Ici 0)\n⊢ StrictMono f\n[PROOFSTEP]\nrefine' StrictMonoOn.Iic_union_Ici (fun x hx y hy hxy => neg_lt_neg_iff.1 _) h₂\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : OrderedAddCommGroup H\nf : G → H\nh₁ : ∀ (x : G), f (-x) = -f x\nh₂ : StrictMonoOn f (Ici 0)\nx : G\nhx : x ∈ Iic 0\ny : G\nhy : y ∈ Iic 0\nhxy : x < y\n⊢ -f y < -f x\n[PROOFSTEP]\nrw [← h₁, ← h₁]\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : OrderedAddCommGroup H\nf : G → H\nh₁ : ∀ (x : G), f (-x) = -f x\nh₂ : StrictMonoOn f (Ici 0)\nx : G\nhx : x ∈ Iic 0\ny : G\nhy : y ∈ Iic 0\nhxy : x < y\n⊢ f (-y) < f (-x)\n[PROOFSTEP]\nexact h₂ (neg_nonneg.2 hy) (neg_nonneg.2 hx) (neg_lt_neg hxy)\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : OrderedAddCommGroup H\nf : G → H\nh₁ : ∀ (x : G), f (-x) = -f x\nh₂ : MonotoneOn f (Ici 0)\n⊢ Monotone f\n[PROOFSTEP]\nrefine' MonotoneOn.Iic_union_Ici (fun x hx y hy hxy => neg_le_neg_iff.1 _) h₂\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : OrderedAddCommGroup H\nf : G → H\nh₁ : ∀ (x : G), f (-x) = -f x\nh₂ : MonotoneOn f (Ici 0)\nx : G\nhx : x ∈ Iic 0\ny : G\nhy : y ∈ Iic 0\nhxy : x ≤ y\n⊢ -f y ≤ -f x\n[PROOFSTEP]\nrw [← h₁, ← h₁]\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst✝¹ : LinearOrderedAddCommGroup G\ninst✝ : OrderedAddCommGroup H\nf : G → H\nh₁ : ∀ (x : G), f (-x) = -f x\nh₂ : MonotoneOn f (Ici 0)\nx : G\nhx : x ∈ Iic 0\ny : G\nhy : y ∈ Iic 0\nhxy : x ≤ y\n⊢ f (-y) ≤ f (-x)\n[PROOFSTEP]\nexact h₂ (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]\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderedCommGroup α\ninst✝ : ContinuousConstSMul α α\ns : Set α\nh : IsUpperSet s\nx y : α\nhxy : x ≤ y\nhx : x ∈ closure s\n⊢ y ∈ closure ((y / x) • s)\n[PROOFSTEP]\nrw [closure_smul]\n[GOAL]\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderedCommGroup α\ninst✝ : ContinuousConstSMul α α\ns : Set α\nh : IsUpperSet s\nx y : α\nhxy : x ≤ y\nhx : x ∈ closure s\n⊢ y ∈ (y / x) • closure s\n[PROOFSTEP]\nexact ⟨x, hx, div_mul_cancel' _ _⟩\n[GOAL]\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderedCommGroup α\ninst✝ : ContinuousConstSMul α α\ns : Set α\nh : IsLowerSet s\nx y : α\nhxy : y ≤ x\nhx : x ∈ closure s\n⊢ y ∈ closure ((y / x) • s)\n[PROOFSTEP]\nrw [closure_smul]\n[GOAL]\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderedCommGroup α\ninst✝ : ContinuousConstSMul α α\ns : Set α\nh : IsLowerSet s\nx y : α\nhxy : y ≤ x\nhx : x ∈ closure s\n⊢ y ∈ (y / x) • closure s\n[PROOFSTEP]\nexact ⟨x, hx, div_mul_cancel' _ _⟩\n[GOAL]\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderedCommGroup α\ninst✝ : ContinuousConstSMul α α\ns : Set α\nhs : IsOpen s\n⊢ IsOpen ↑(upperClosure s)\n[PROOFSTEP]\nrw [← mul_one s, ← mul_upperClosure]\n[GOAL]\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderedCommGroup α\ninst✝ : ContinuousConstSMul α α\ns : Set α\nhs : IsOpen s\n⊢ IsOpen (s * ↑(upperClosure 1))\n[PROOFSTEP]\nexact hs.mul_right\n[GOAL]\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderedCommGroup α\ninst✝ : ContinuousConstSMul α α\ns : Set α\nhs : IsOpen s\n⊢ IsOpen ↑(lowerClosure s)\n[PROOFSTEP]\nrw [← mul_one s, ← mul_lowerClosure]\n[GOAL]\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderedCommGroup α\ninst✝ : ContinuousConstSMul α α\ns : Set α\nhs : IsOpen s\n⊢ IsOpen (s * ↑(lowerClosure 1))\n[PROOFSTEP]\nexact hs.mul_right\n[GOAL]\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : HasUpperLowerClosure α\ns : Set α\nh : IsUpperSet s\n⊢ IsUpperSet (interior s)\n[PROOFSTEP]\nrw [← isLowerSet_compl, ← closure_compl]\n[GOAL]\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : HasUpperLowerClosure α\ns : Set α\nh : IsUpperSet s\n⊢ IsLowerSet (closure sᶜ)\n[PROOFSTEP]\nexact h.compl.closure\n[GOAL]\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : HasUpperLowerClosure α\ns : Set α\nh : OrdConnected s\n⊢ OrdConnected (interior s)\n[PROOFSTEP]\nrw [← h.upperClosure_inter_lowerClosure, interior_inter]\n[GOAL]\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : HasUpperLowerClosure α\ns : Set α\nh : OrdConnected s\n⊢ OrdConnected (interior ↑(upperClosure s) ∩ interior ↑(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]\nα : Type u_1\ninst✝³ : AddMonoidWithOne α\ninst✝² : Preorder α\ninst✝¹ : ZeroLEOneClass α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\n⊢ 0 ≤ 2\n[PROOFSTEP]\nrw [← one_add_one_eq_two]\n[GOAL]\nα : Type u_1\ninst✝³ : AddMonoidWithOne α\ninst✝² : Preorder α\ninst✝¹ : ZeroLEOneClass α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\n⊢ 0 ≤ 1 + 1\n[PROOFSTEP]\nexact add_nonneg zero_le_one zero_le_one\n[GOAL]\nα : Type u_1\ninst✝³ : AddMonoidWithOne α\ninst✝² : Preorder α\ninst✝¹ : ZeroLEOneClass α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\n⊢ 0 ≤ 3\n[PROOFSTEP]\nrw [← two_add_one_eq_three]\n[GOAL]\nα : Type u_1\ninst✝³ : AddMonoidWithOne α\ninst✝² : Preorder α\ninst✝¹ : ZeroLEOneClass α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\n⊢ 0 ≤ 2 + 1\n[PROOFSTEP]\nexact add_nonneg zero_le_two zero_le_one\n[GOAL]\nα : Type u_1\ninst✝³ : AddMonoidWithOne α\ninst✝² : Preorder α\ninst✝¹ : ZeroLEOneClass α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\n⊢ 0 ≤ 4\n[PROOFSTEP]\nrw [← three_add_one_eq_four]\n[GOAL]\nα : Type u_1\ninst✝³ : AddMonoidWithOne α\ninst✝² : Preorder α\ninst✝¹ : ZeroLEOneClass α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\n⊢ 0 ≤ 3 + 1\n[PROOFSTEP]\nexact add_nonneg zero_le_three zero_le_one\n[GOAL]\nα : Type u_1\ninst✝⁴ : AddMonoidWithOne α\ninst✝³ : PartialOrder α\ninst✝² : ZeroLEOneClass α\ninst✝¹ : NeZero 1\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\n⊢ 0 < 3\n[PROOFSTEP]\nrw [← two_add_one_eq_three]\n[GOAL]\nα : Type u_1\ninst✝⁴ : AddMonoidWithOne α\ninst✝³ : PartialOrder α\ninst✝² : ZeroLEOneClass α\ninst✝¹ : NeZero 1\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\n⊢ 0 < 2 + 1\n[PROOFSTEP]\nexact lt_add_of_lt_of_nonneg zero_lt_two zero_le_one\n[GOAL]\nα : Type u_1\ninst✝⁴ : AddMonoidWithOne α\ninst✝³ : PartialOrder α\ninst✝² : ZeroLEOneClass α\ninst✝¹ : NeZero 1\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\n⊢ 0 < 4\n[PROOFSTEP]\nrw [← three_add_one_eq_four]\n[GOAL]\nα : Type u_1\ninst✝⁴ : AddMonoidWithOne α\ninst✝³ : PartialOrder α\ninst✝² : ZeroLEOneClass α\ninst✝¹ : NeZero 1\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\n⊢ 0 < 3 + 1\n[PROOFSTEP]\nexact lt_add_of_lt_of_nonneg zero_lt_three zero_le_one\n[GOAL]\nα : Type u_1\ninst✝⁴ : AddMonoidWithOne α\ninst✝³ : PartialOrder α\ninst✝² : ZeroLEOneClass α\ninst✝¹ : NeZero 1\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1\n⊢ 1 < 2\n[PROOFSTEP]\nrw [← one_add_one_eq_two]\n[GOAL]\nα : Type u_1\ninst✝⁴ : AddMonoidWithOne α\ninst✝³ : PartialOrder α\ninst✝² : ZeroLEOneClass α\ninst✝¹ : NeZero 1\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1\n⊢ 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✝² : Mul G\ninst✝¹ : Inv G\ninst✝ : One G\nassoc : ∀ (a b c : G), a * b * c = a * (b * c)\none_mul : ∀ (a : G), 1 * a = a\nmul_left_inv : ∀ (a : G), a⁻¹ * a = 1\na : G\n⊢ a * 1 = a\n[PROOFSTEP]\nhave mul_right_inv : ∀ a, a * a⁻¹ = 1 := fun a =>\n  calc\n    a * a⁻¹ = 1 * (a * a⁻¹) := (one_mul _).symm\n    _ = ((a * a⁻¹)⁻¹ * (a * a⁻¹)) * (a * a⁻¹) := by rw [mul_left_inv]\n    _ = (a * a⁻¹)⁻¹ * (a * ((a⁻¹ * a) * a⁻¹)) := by simp only [assoc]\n    _ = 1 := by rw [mul_left_inv, one_mul, mul_left_inv]\n[GOAL]\nG : Type u\ninst✝² : Mul G\ninst✝¹ : Inv G\ninst✝ : One G\nassoc : ∀ (a b c : G), a * b * c = a * (b * c)\none_mul : ∀ (a : G), 1 * a = a\nmul_left_inv : ∀ (a : G), a⁻¹ * a = 1\na✝ a : G\n⊢ 1 * (a * a⁻¹) = (a * a⁻¹)⁻¹ * (a * a⁻¹) * (a * a⁻¹)\n[PROOFSTEP]\nrw [mul_left_inv]\n[GOAL]\nG : Type u\ninst✝² : Mul G\ninst✝¹ : Inv G\ninst✝ : One G\nassoc : ∀ (a b c : G), a * b * c = a * (b * c)\none_mul : ∀ (a : G), 1 * a = a\nmul_left_inv : ∀ (a : G), a⁻¹ * a = 1\na✝ a : G\n⊢ (a * a⁻¹)⁻¹ * (a * a⁻¹) * (a * a⁻¹) = (a * a⁻¹)⁻¹ * (a * (a⁻¹ * a * a⁻¹))\n[PROOFSTEP]\nsimp only [assoc]\n[GOAL]\nG : Type u\ninst✝² : Mul G\ninst✝¹ : Inv G\ninst✝ : One G\nassoc : ∀ (a b c : G), a * b * c = a * (b * c)\none_mul : ∀ (a : G), 1 * a = a\nmul_left_inv : ∀ (a : G), a⁻¹ * a = 1\na✝ a : G\n⊢ (a * a⁻¹)⁻¹ * (a * (a⁻¹ * a * a⁻¹)) = 1\n[PROOFSTEP]\nrw [mul_left_inv, one_mul, mul_left_inv]\n[GOAL]\nG : Type u\ninst✝² : Mul G\ninst✝¹ : Inv G\ninst✝ : One G\nassoc : ∀ (a b c : G), a * b * c = a * (b * c)\none_mul : ∀ (a : G), 1 * a = a\nmul_left_inv : ∀ (a : G), a⁻¹ * a = 1\na : G\nmul_right_inv : ∀ (a : G), a * a⁻¹ = 1\n⊢ a * 1 = a\n[PROOFSTEP]\nrw [← mul_left_inv a, ← assoc, mul_right_inv a, one_mul]\n[GOAL]\nG : Type u\ninst✝² : Mul G\ninst✝¹ : Inv G\ninst✝ : One G\nassoc : ∀ (a b c : G), a * b * c = a * (b * c)\nmul_one : ∀ (a : G), a * 1 = a\nmul_right_inv : ∀ (a : G), a * a⁻¹ = 1\na : G\n⊢ a⁻¹ * a * 1 = a⁻¹ * a * (a⁻¹ * a * (a⁻¹ * a)⁻¹)\n[PROOFSTEP]\nrw [mul_right_inv]\n[GOAL]\nG : Type u\ninst✝² : Mul G\ninst✝¹ : Inv G\ninst✝ : One G\nassoc : ∀ (a b c : G), a * b * c = a * (b * c)\nmul_one : ∀ (a : G), a * 1 = a\nmul_right_inv : ∀ (a : G), a * a⁻¹ = 1\na : G\n⊢ a⁻¹ * a * (a⁻¹ * a * (a⁻¹ * a)⁻¹) = a⁻¹ * (a * a⁻¹) * a * (a⁻¹ * a)⁻¹\n[PROOFSTEP]\nsimp only [assoc]\n[GOAL]\nG : Type u\ninst✝² : Mul G\ninst✝¹ : Inv G\ninst✝ : One G\nassoc : ∀ (a b c : G), a * b * c = a * (b * c)\nmul_one : ∀ (a : G), a * 1 = a\nmul_right_inv : ∀ (a : G), a * a⁻¹ = 1\na : G\n⊢ a⁻¹ * (a * a⁻¹) * a * (a⁻¹ * a)⁻¹ = 1\n[PROOFSTEP]\nrw [mul_right_inv, mul_one, mul_right_inv]\n[GOAL]\nG : Type u\ninst✝² : Mul G\ninst✝¹ : Inv G\ninst✝ : One G\nassoc : ∀ (a b c : G), a * b * c = a * (b * c)\nmul_one : ∀ (a : G), a * 1 = a\nmul_right_inv : ∀ (a : G), a * a⁻¹ = 1\nmul_left_inv : ∀ (a : G), a⁻¹ * a = 1\na : G\n⊢ 1 * a = a\n[PROOFSTEP]\nrw [← 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⊢ {0} ∪ range succ = univ\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nn : ℕ\n⊢ n ∈ {0} ∪ range succ ↔ n ∈ univ\n[PROOFSTEP]\ncases n\n[GOAL]\ncase h.zero\n⊢ zero ∈ {0} ∪ range succ ↔ zero ∈ univ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.succ\nn✝ : ℕ\n⊢ succ n✝ ∈ {0} ∪ range succ ↔ succ n✝ ∈ univ\n[PROOFSTEP]\nsimp\n[GOAL]\n⊢ range succ = {i | 0 < i}\n[PROOFSTEP]\next (_ | i)\n[GOAL]\ncase h.zero\n⊢ zero ∈ range succ ↔ zero ∈ {i | 0 < i}\n[PROOFSTEP]\nsimp [succ_pos, succ_ne_zero, Set.mem_setOf]\n[GOAL]\ncase h.succ\ni : ℕ\n⊢ succ i ∈ range succ ↔ succ i ∈ {i | 0 < i}\n[PROOFSTEP]\nsimp [succ_pos, succ_ne_zero, Set.mem_setOf]\n[GOAL]\nα : Type u_1\nf : ℕ → α\n⊢ {f 0} ∪ range (f ∘ succ) = range f\n[PROOFSTEP]\nrw [← image_singleton, range_comp, ← image_union, zero_union_range_succ, image_univ]\n[GOAL]\nα✝ : Type u_1\nα : Type u_2\nx : α\nf : ℕ → α → α\n⊢ (range fun n => rec x f n) = {x} ∪ range fun n => rec (f 0 x) (f ∘ succ) n\n[PROOFSTEP]\nconvert (range_of_succ (fun n => Nat.rec x f n : ℕ → α)).symm using 4\n[GOAL]\ncase h.e'_3.h.e'_4.h.e'_3.h\nα✝ : Type u_1\nα : Type u_2\nx : α\nf : ℕ → α → α\nx✝ : ℕ\n⊢ rec (f 0 x) (f ∘ succ) x✝ = ((fun n => rec x f n) ∘ succ) x✝\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.e'_3.h.e'_4.h.e'_3.h\nα✝ : Type u_1\nα : Type u_2\nx : α\nf : ℕ → α → α\nx✝ : ℕ\n⊢ rec (f 0 x) (f ∘ succ) x✝ = f x✝ (rec x f x✝)\n[PROOFSTEP]\nrename_i n\n[GOAL]\ncase h.e'_3.h.e'_4.h.e'_3.h\nα✝ : Type u_1\nα : Type u_2\nx : α\nf : ℕ → α → α\nn : ℕ\n⊢ rec (f 0 x) (f ∘ 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\nα✝ : Type u_1\nα : Type u_2\nx : α\nf : ℕ → α → α\n⊢ rec (f 0 x) (f ∘ 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\nα✝ : Type u_1\nα : Type u_2\nx : α\nf : ℕ → α → α\nn : ℕ\nihn : rec (f 0 x) (f ∘ succ) n = f n (rec x f n)\n⊢ rec (f 0 x) (f ∘ succ) (succ n) = f (succ n) (rec x f (succ n))\n[PROOFSTEP]\ndsimp at ihn ⊢\n[GOAL]\ncase h.e'_3.h.e'_4.h.e'_3.h.succ\nα✝ : Type u_1\nα : Type u_2\nx : α\nf : ℕ → α → α\nn : ℕ\nihn : rec (f 0 x) (f ∘ succ) n = f n (rec x f n)\n⊢ f (succ n) (rec (f 0 x) (f ∘ 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]\nα : Type u_1\ninst✝⁴ : Zero α\ninst✝³ : TopologicalSpace α\ninst✝² : PartialOrder α\ninst✝¹ : DecidableRel fun x x_1 => x < x_1\ninst✝ : OrderTopology α\na : α\nh : 0 < a\n⊢ ContinuousAt (↑SignType.sign) a\n[PROOFSTEP]\nrefine' (continuousAt_const : ContinuousAt (fun _ => (1 : SignType)) a).congr _\n[GOAL]\nα : Type u_1\ninst✝⁴ : Zero α\ninst✝³ : TopologicalSpace α\ninst✝² : PartialOrder α\ninst✝¹ : DecidableRel fun x x_1 => x < x_1\ninst✝ : OrderTopology α\na : α\nh : 0 < a\n⊢ (fun x => 1) =ᶠ[nhds a] ↑SignType.sign\n[PROOFSTEP]\nrw [Filter.EventuallyEq, eventually_nhds_iff]\n[GOAL]\nα : Type u_1\ninst✝⁴ : Zero α\ninst✝³ : TopologicalSpace α\ninst✝² : PartialOrder α\ninst✝¹ : DecidableRel fun x x_1 => x < x_1\ninst✝ : OrderTopology α\na : α\nh : 0 < a\n⊢ ∃ t, (∀ (x : α), x ∈ t → 1 = ↑SignType.sign x) ∧ IsOpen t ∧ a ∈ t\n[PROOFSTEP]\nexact ⟨{x | 0 < x}, fun x hx => (sign_pos hx).symm, isOpen_lt' 0, h⟩\n[GOAL]\nα : Type u_1\ninst✝⁴ : Zero α\ninst✝³ : TopologicalSpace α\ninst✝² : PartialOrder α\ninst✝¹ : DecidableRel fun x x_1 => x < x_1\ninst✝ : OrderTopology α\na : α\nh : a < 0\n⊢ ContinuousAt (↑SignType.sign) a\n[PROOFSTEP]\nrefine' (continuousAt_const : ContinuousAt (fun x => (-1 : SignType)) a).congr _\n[GOAL]\nα : Type u_1\ninst✝⁴ : Zero α\ninst✝³ : TopologicalSpace α\ninst✝² : PartialOrder α\ninst✝¹ : DecidableRel fun x x_1 => x < x_1\ninst✝ : OrderTopology α\na : α\nh : a < 0\n⊢ (fun x => -1) =ᶠ[nhds a] ↑SignType.sign\n[PROOFSTEP]\nrw [Filter.EventuallyEq, eventually_nhds_iff]\n[GOAL]\nα : Type u_1\ninst✝⁴ : Zero α\ninst✝³ : TopologicalSpace α\ninst✝² : PartialOrder α\ninst✝¹ : DecidableRel fun x x_1 => x < x_1\ninst✝ : OrderTopology α\na : α\nh : a < 0\n⊢ ∃ t, (∀ (x : α), x ∈ t → -1 = ↑SignType.sign x) ∧ IsOpen t ∧ a ∈ t\n[PROOFSTEP]\nexact ⟨{x | x < 0}, fun x hx => (sign_neg hx).symm, isOpen_gt' 0, h⟩\n[GOAL]\nα : Type u_1\ninst✝³ : Zero α\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\nh : a ≠ 0\n⊢ ContinuousAt (↑SignType.sign) a\n[PROOFSTEP]\nrcases h.lt_or_lt with (h_neg | h_pos)\n[GOAL]\ncase inl\nα : Type u_1\ninst✝³ : Zero α\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\nh : a ≠ 0\nh_neg : a < 0\n⊢ ContinuousAt (↑SignType.sign) a\n[PROOFSTEP]\nexact continuousAt_sign_of_neg h_neg\n[GOAL]\ncase inr\nα : Type u_1\ninst✝³ : Zero α\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\nh : a ≠ 0\nh_pos : 0 < a\n⊢ ContinuousAt (↑SignType.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]\nα : Type u_1\nM : Type u_2\ninst✝ : AddCommMonoid M\nf : α → α\ng : α → M\nm n : ℕ\nx : α\n⊢ 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]\nα : Type u_1\nM : Type u_2\ninst✝ : AddCommMonoid M\nf : α → α\nx : α\nh : IsFixedPt f x\ng : α → M\nn : ℕ\n⊢ birkhoffSum f g n x = n • g x\n[PROOFSTEP]\nsimp [birkhoffSum, (h.iterate _).eq]\n[GOAL]\nα : Type u_1\nG : Type u_2\ninst✝ : AddCommGroup G\nf : α → α\ng : α → G\nn : ℕ\nx : α\n⊢ birkhoffSum f g n (f x) - birkhoffSum f g n x = g (f^[n] x) - g x\n[PROOFSTEP]\nrw [← sub_eq_iff_eq_add.2 (birkhoffSum_succ f g n x), ← sub_eq_iff_eq_add.2 (birkhoffSum_succ' f g n x), ← sub_add, ←\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✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Module R N\ninst✝¹ : ContinuousAdd M\ninst✝ : IsSimpleModule R N\nl : M →ₗ[R] N\n⊢ IsClosed ↑(ker l) ∨ Dense ↑(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✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Module R N\ninst✝¹ : ContinuousAdd M\ninst✝ : IsSimpleModule R N\nl : M →ₗ[R] N\nhl : Function.Surjective ↑l\n⊢ IsClosed ↑(ker l) ∨ Dense ↑(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✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Module R N\ninst✝¹ : ContinuousAdd M\ninst✝ : IsSimpleModule R N\n⊢ IsClosed ↑(ker 0) ∨ Dense ↑(ker 0)\n[PROOFSTEP]\nrw [LinearMap.ker_zero]\n[GOAL]\ncase inr\nR : Type u\nM : Type v\nN : Type w\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Module R N\ninst✝¹ : ContinuousAdd M\ninst✝ : IsSimpleModule R N\n⊢ IsClosed ↑⊤ ∨ Dense ↑⊤\n[PROOFSTEP]\nleft\n[GOAL]\ncase inr.h\nR : Type u\nM : Type v\nN : Type w\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Module R N\ninst✝¹ : ContinuousAdd M\ninst✝ : IsSimpleModule R N\n⊢ IsClosed ↑⊤\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 u₁\nD : Type u₂\nE : Type u₃\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : Category.{v₃, u₃} E\nF : C × D ⥤ E\nW : C\nX Y Z : D\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ F.map (𝟙 W, f ≫ g) = F.map (𝟙 W, f) ≫ F.map (𝟙 W, g)\n[PROOFSTEP]\nrw [← Functor.map_comp, prod_comp, Category.comp_id]\n[GOAL]\nC : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : Category.{v₃, u₃} E\nF : C × D ⥤ E\nX Y Z : C\nW : D\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ F.map (f ≫ g, 𝟙 W) = F.map (f, 𝟙 W) ≫ F.map (g, 𝟙 W)\n[PROOFSTEP]\nrw [← Functor.map_comp, prod_comp, Category.comp_id]\n[GOAL]\nC : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : Category.{v₃, u₃} E\nF : C × D ⥤ E\nX X' : C\nf : X ⟶ X'\nY Y' : D\ng : Y ⟶ Y'\n⊢ F.map (𝟙 X, g) ≫ F.map (f, 𝟙 Y') = F.map (f, g)\n[PROOFSTEP]\nrw [← Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id]\n[GOAL]\nC : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : Category.{v₃, u₃} E\nF : C × D ⥤ E\nX X' : C\nf : X ⟶ X'\nY Y' : D\ng : Y ⟶ Y'\n⊢ F.map (f, 𝟙 Y) ≫ F.map (𝟙 X', g) = F.map (f, g)\n[PROOFSTEP]\nrw [← 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✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type v\ninst✝ : CommRing R\nA : GL n R\n⊢ Matrix.det ↑A * Matrix.det ↑A⁻¹ = 1\n[PROOFSTEP]\nrw [← det_mul, A.mul_inv, det_one]\n[GOAL]\nn : Type u\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type v\ninst✝ : CommRing R\nA : GL n R\n⊢ Matrix.det ↑A⁻¹ * Matrix.det ↑A = 1\n[PROOFSTEP]\nrw [← det_mul, A.inv_mul, det_one]\n[GOAL]\nn : Type u\nR : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : LinearOrderedCommRing R\ninst✝ : Fact (Even (Fintype.card n))\ng : { x // x ∈ GLPos n R }\n⊢ -↑g ∈ 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✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : LinearOrderedCommRing R\ninst✝ : Fact (Even (Fintype.card n))\ng : { x // x ∈ GLPos n R }\n⊢ 0 < det ↑↑g\n[PROOFSTEP]\nexact g.prop\n[GOAL]\nR : Type ?u.1604470\ninst✝ : Field R\na b : R\nhab : a ^ 2 + b ^ 2 ≠ 0\n⊢ det (↑of ![![a, -b], ![b, a]]) ≠ 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✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\none_mul : ∀ (a : R), 1 * a = a\nmul_one : ∀ (a : R), a * 1 = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : ∀ (a b c : R), (a + b) * c = a * c + b * c\nthis : AddGroup R := AddGroup.ofLeftAxioms add_assoc zero_add add_left_neg\n⊢ ∀ (a b : R), a + b = b + a\n[PROOFSTEP]\nintro a b\n[GOAL]\nR : Type u\ninst✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\none_mul : ∀ (a : R), 1 * a = a\nmul_one : ∀ (a : R), a * 1 = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : ∀ (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⊢ a + b = b + a\n[PROOFSTEP]\nhave h₁ : (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✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\none_mul : ∀ (a : R), 1 * a = a\nmul_one : ∀ (a : R), a * 1 = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : ∀ (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⊢ (1 + 1) * (a + b) = a + (a + b) + b\n[PROOFSTEP]\nrw [left_distrib]\n[GOAL]\nR : Type u\ninst✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\none_mul : ∀ (a : R), 1 * a = a\nmul_one : ∀ (a : R), a * 1 = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : ∀ (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⊢ (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✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\none_mul : ∀ (a : R), 1 * a = a\nmul_one : ∀ (a : R), a * 1 = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : ∀ (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\nh₁ : (1 + 1) * (a + b) = a + (a + b) + b\n⊢ a + b = b + a\n[PROOFSTEP]\nhave h₂ : (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✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\none_mul : ∀ (a : R), 1 * a = a\nmul_one : ∀ (a : R), a * 1 = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : ∀ (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\nh₁ : (1 + 1) * (a + b) = a + (a + b) + b\n⊢ (1 + 1) * (a + b) = a + (b + a) + b\n[PROOFSTEP]\nrw [right_distrib]\n[GOAL]\nR : Type u\ninst✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\none_mul : ∀ (a : R), 1 * a = a\nmul_one : ∀ (a : R), a * 1 = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : ∀ (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\nh₁ : (1 + 1) * (a + b) = a + (a + b) + b\n⊢ 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✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\none_mul : ∀ (a : R), 1 * a = a\nmul_one : ∀ (a : R), a * 1 = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : ∀ (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\nh₁ : (1 + 1) * (a + b) = a + (a + b) + b\nh₂ : (1 + 1) * (a + b) = a + (b + a) + b\n⊢ a + b = b + a\n[PROOFSTEP]\nhave := h₁.symm.trans h₂\n[GOAL]\nR : Type u\ninst✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\none_mul : ∀ (a : R), 1 * a = a\nmul_one : ∀ (a : R), a * 1 = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : ∀ (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\nh₁ : (1 + 1) * (a + b) = a + (a + b) + b\nh₂ : (1 + 1) * (a + b) = a + (b + a) + b\nthis : a + (a + b) + b = a + (b + a) + b\n⊢ a + b = b + a\n[PROOFSTEP]\nrwa [add_left_inj, add_right_inj] at this \n[GOAL]\nR : Type u\ninst✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\none_mul : ∀ (a : R), 1 * a = a\nmul_one : ∀ (a : R), a * 1 = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : ∀ (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 : ∀ (a b : R), a + b = b + a\na : R\n⊢ 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✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\none_mul : ∀ (a : R), 1 * a = a\nmul_one : ∀ (a : R), a * 1 = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : ∀ (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 : ∀ (a b : R), a + b = b + a\na : R\n⊢ 0 * a = (0 + 0) * a\n[PROOFSTEP]\nrw [zero_add]\n[GOAL]\nR : Type u\ninst✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\none_mul : ∀ (a : R), 1 * a = a\nmul_one : ∀ (a : R), a * 1 = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : ∀ (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 : ∀ (a b : R), a + b = b + a\na : R\n⊢ (0 + 0) * a = 0 * a + 0 * a\n[PROOFSTEP]\nrw [right_distrib]\n[GOAL]\nR : Type u\ninst✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\none_mul : ∀ (a : R), 1 * a = a\nmul_one : ∀ (a : R), a * 1 = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : ∀ (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 : ∀ (a b : R), a + b = b + a\na : R\nthis : 0 * a = 0 * a + 0 * a\n⊢ 0 * a = 0\n[PROOFSTEP]\nrwa [self_eq_add_right] at this \n[GOAL]\nR : Type u\ninst✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\none_mul : ∀ (a : R), 1 * a = a\nmul_one : ∀ (a : R), a * 1 = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : ∀ (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 : ∀ (a b : R), a + b = b + a\nzero_mul : ∀ (a : R), 0 * a = 0\na : R\n⊢ 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✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\none_mul : ∀ (a : R), 1 * a = a\nmul_one : ∀ (a : R), a * 1 = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : ∀ (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 : ∀ (a b : R), a + b = b + a\nzero_mul : ∀ (a : R), 0 * a = 0\na : R\n⊢ a * 0 = a * (0 + 0)\n[PROOFSTEP]\nrw [zero_add]\n[GOAL]\nR : Type u\ninst✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\none_mul : ∀ (a : R), 1 * a = a\nmul_one : ∀ (a : R), a * 1 = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : ∀ (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 : ∀ (a b : R), a + b = b + a\nzero_mul : ∀ (a : R), 0 * a = 0\na : R\n⊢ a * (0 + 0) = a * 0 + a * 0\n[PROOFSTEP]\nrw [left_distrib]\n[GOAL]\nR : Type u\ninst✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\none_mul : ∀ (a : R), 1 * a = a\nmul_one : ∀ (a : R), a * 1 = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nright_distrib : ∀ (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 : ∀ (a b : R), a + b = b + a\nzero_mul : ∀ (a : R), 0 * a = 0\na : R\nthis : a * 0 = a * 0 + a * 0\n⊢ a * 0 = 0\n[PROOFSTEP]\nrwa [self_eq_add_right] at this \n[GOAL]\nR : Type u\ninst✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\nmul_comm : ∀ (a b : R), a * b = b * a\none_mul : ∀ (a : R), 1 * a = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\na : R\n⊢ a * 1 = a\n[PROOFSTEP]\nrw [mul_comm, one_mul]\n[GOAL]\nR : Type u\ninst✝⁴ : Add R\ninst✝³ : Mul R\ninst✝² : Neg R\ninst✝¹ : Zero R\ninst✝ : One R\nadd_assoc : ∀ (a b c : R), a + b + c = a + (b + c)\nzero_add : ∀ (a : R), 0 + a = a\nadd_left_neg : ∀ (a : R), -a + a = 0\nmul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)\nmul_comm : ∀ (a b : R), a * b = b * a\none_mul : ∀ (a : R), 1 * a = a\nleft_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c\nmul_one : ∀ (a : R), a * 1 = a\na b c : R\n⊢ (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\nα : Type u_5\ninst✝⁴ : Group G\ninst✝³ : Monoid M\ninst✝² : MulAction G M\ninst✝¹ : SMulCommClass G M M\ninst✝ : IsScalarTower G M M\ng : G\nm : Mˣ\n⊢ g • ↑m * g⁻¹ • ↑m⁻¹ = 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\nα : Type u_5\ninst✝⁴ : Group G\ninst✝³ : Monoid M\ninst✝² : MulAction G M\ninst✝¹ : SMulCommClass G M M\ninst✝ : IsScalarTower G M M\ng : G\nm : Mˣ\n⊢ g⁻¹ • ↑m⁻¹ * g • ↑m = 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 : ℤ\n⊢ dist m n = ↑|m - n|\n[PROOFSTEP]\nrw [dist_eq]\n[GOAL]\nm n : ℤ\n⊢ |↑m - ↑n| = ↑|m - n|\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n⊢ Pairwise fun m n => 1 ≤ dist m n\n[PROOFSTEP]\nintro m n hne\n[GOAL]\nm n : ℤ\nhne : m ≠ n\n⊢ 1 ≤ dist m n\n[PROOFSTEP]\nrw [dist_eq]\n[GOAL]\nm n : ℤ\nhne : m ≠ n\n⊢ 1 ≤ |↑m - ↑n|\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nm n : ℤ\nhne : m ≠ n\n⊢ 1 ≤ |m - n|\n[PROOFSTEP]\nrwa [← zero_add (1 : ℤ), Int.add_one_le_iff, abs_pos, sub_ne_zero]\n[GOAL]\nx : ℤ\nr : ℝ\n⊢ ball x r = Ioo ⌊↑x - r⌋ ⌈↑x + r⌉\n[PROOFSTEP]\nrw [← preimage_ball, Real.ball_eq_Ioo, preimage_Ioo]\n[GOAL]\nx : ℤ\nr : ℝ\n⊢ closedBall x r = Icc ⌈↑x - r⌉ ⌊↑x + r⌋\n[PROOFSTEP]\nrw [← preimage_closedBall, Real.closedBall_eq_Icc, preimage_Icc]\n[GOAL]\nx : ℤ\nr : ℝ\n⊢ IsCompact (closedBall x r)\n[PROOFSTEP]\nrw [closedBall_eq_Icc]\n[GOAL]\nx : ℤ\nr : ℝ\n⊢ IsCompact (Icc ⌈↑x - r⌉ ⌊↑x + r⌋)\n[PROOFSTEP]\nexact (Set.finite_Icc _ _).isCompact\n[GOAL]\n⊢ cocompact ℤ = atBot ⊔ atTop\n[PROOFSTEP]\nsimp_rw [← comap_dist_right_atTop_eq_cocompact (0 : ℤ), dist_eq', sub_zero, ← comap_abs_atTop, ←\n  @Int.comap_cast_atTop ℝ, comap_comap]\n[GOAL]\n⊢ comap (fun y => ↑|y|) atTop = comap (Int.cast ∘ abs) atTop\n[PROOFSTEP]\nrfl\n[GOAL]\n⊢ cofinite = atBot ⊔ atTop\n[PROOFSTEP]\nrw [← 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]\nα : Type u_1\ninst✝ : LinearOrderedSemifield α\nr n : ℕ\n⊢ ↑(choose n r) ≤ ↑(n ^ r) / ↑r !\n[PROOFSTEP]\nrw [le_div_iff']\n[GOAL]\nα : Type u_1\ninst✝ : LinearOrderedSemifield α\nr n : ℕ\n⊢ ↑r ! * ↑(choose n r) ≤ ↑(n ^ r)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nα : Type u_1\ninst✝ : LinearOrderedSemifield α\nr n : ℕ\n⊢ r ! * choose n r ≤ n ^ r\n[PROOFSTEP]\nrw [← Nat.descFactorial_eq_factorial_mul_choose]\n[GOAL]\nα : Type u_1\ninst✝ : LinearOrderedSemifield α\nr n : ℕ\n⊢ descFactorial n r ≤ n ^ r\n[PROOFSTEP]\nexact n.descFactorial_le_pow r\n[GOAL]\nα : Type u_1\ninst✝ : LinearOrderedSemifield α\nr n : ℕ\n⊢ 0 < ↑r !\n[PROOFSTEP]\nexact_mod_cast r.factorial_pos\n[GOAL]\nα : Type u_1\ninst✝ : LinearOrderedSemifield α\nr n : ℕ\n⊢ ↑((n + 1 - r) ^ r) / ↑r ! ≤ ↑(choose n r)\n[PROOFSTEP]\nrw [div_le_iff']\n[GOAL]\nα : Type u_1\ninst✝ : LinearOrderedSemifield α\nr n : ℕ\n⊢ ↑((n + 1 - r) ^ r) ≤ ↑r ! * ↑(choose n r)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nα : Type u_1\ninst✝ : LinearOrderedSemifield α\nr n : ℕ\n⊢ (n + 1 - r) ^ r ≤ r ! * choose n r\n[PROOFSTEP]\nrw [← Nat.descFactorial_eq_factorial_mul_choose]\n[GOAL]\nα : Type u_1\ninst✝ : LinearOrderedSemifield α\nr n : ℕ\n⊢ (n + 1 - r) ^ r ≤ descFactorial n r\n[PROOFSTEP]\nexact n.pow_sub_le_descFactorial r\n[GOAL]\nα : Type u_1\ninst✝ : LinearOrderedSemifield α\nr n : ℕ\n⊢ 0 < ↑r !\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 : ℕ\n⊢ dist ↑x ↑y = dist x y\n[PROOFSTEP]\nrw [← Nat.dist_cast_real, ← Rat.dist_cast]\n[GOAL]\nx y : ℕ\n⊢ dist ↑↑x ↑↑y = dist ↑x ↑y\n[PROOFSTEP]\ncongr\n[GOAL]\n⊢ Pairwise fun x y => 1 ≤ dist ↑x ↑y\n[PROOFSTEP]\nsimpa using Nat.pairwise_one_le_dist\n[GOAL]\n⊢ Pairwise fun x y => 1 ≤ dist ↑x ↑y\n[PROOFSTEP]\nsimpa using Nat.pairwise_one_le_dist\n[GOAL]\nx y : ℤ\n⊢ dist ↑x ↑y = dist x y\n[PROOFSTEP]\nrw [← Int.dist_cast_real, ← Rat.dist_cast]\n[GOAL]\nx y : ℤ\n⊢ dist ↑↑x ↑↑y = dist ↑x ↑y\n[PROOFSTEP]\ncongr\n[GOAL]\n⊢ Pairwise fun x y => 1 ≤ dist ↑x ↑y\n[PROOFSTEP]\nsimpa using Int.pairwise_one_le_dist\n[GOAL]\n⊢ Pairwise fun x y => 1 ≤ dist ↑x ↑y\n[PROOFSTEP]\nsimpa using Int.pairwise_one_le_dist\n[GOAL]\n⊢ UniformContinuous (Rat.cast ∘ fun p => p.fst + p.snd)\n[PROOFSTEP]\nsimp only [(· ∘ ·), Rat.cast_add]\n[GOAL]\n⊢ UniformContinuous fun x => ↑x.fst + ↑x.snd\n[PROOFSTEP]\nexact Real.uniformContinuous_add.comp (Rat.uniformContinuous_coe_real.prod_map Rat.uniformContinuous_coe_real)\n[GOAL]\nε : ℝ\nε0 : ε > 0\na✝ b✝ : ℚ\nh : dist a✝ b✝ < ε\n⊢ dist (-a✝) (-b✝) < ε\n[PROOFSTEP]\nrw [dist_comm] at h \n[GOAL]\nε : ℝ\nε0 : ε > 0\na✝ b✝ : ℚ\nh : dist b✝ a✝ < ε\n⊢ dist (-a✝) (-b✝) < ε\n[PROOFSTEP]\nsimpa only [dist_eq, cast_neg, neg_sub_neg] using h\n[GOAL]\nε : ℝ\nε0 : ε > 0\na✝ b✝ : ℚ\nh : dist a✝ b✝ < ε\n⊢ dist |a✝| |b✝| ≤ dist a✝ b✝\n[PROOFSTEP]\nsimpa [Rat.dist_eq] using abs_abs_sub_abs_le_abs_sub _ _\n[GOAL]\na b : ℚ\n⊢ TotallyBounded (Icc a b)\n[PROOFSTEP]\nsimpa only [preimage_cast_Icc] using\n  totallyBounded_preimage Rat.uniformEmbedding_coe_real (totallyBounded_Icc (a : ℝ) 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}}