| { |
| "summary": { |
| "total_with_gt": 273, |
| "matched": 53, |
| "accuracy": 0.19413919413919414 |
| }, |
| "details": [ |
| { |
| "id": 0, |
| "match": false, |
| "ground_truth": "· exact mul_le_mul_of_nonneg_left hyu ha", |
| "model_output": "· gcongr\n exact hyu" |
| }, |
| { |
| "id": 1, |
| "match": false, |
| "ground_truth": "exact mul_le_mul_of_nonneg_left hyu ha", |
| "model_output": "apply mul_le_mul_of_nonneg_left hyu ha" |
| }, |
| { |
| "id": 2, |
| "match": false, |
| "ground_truth": "· exact mul_le_mul_of_nonneg_left hyv hb", |
| "model_output": "apply mul_le_mul_of_nonneg_left hyv hb" |
| }, |
| { |
| "id": 3, |
| "match": false, |
| "ground_truth": "exact mul_le_mul_of_nonneg_left hyv hb", |
| "model_output": "apply mul_le_mul_of_nonneg_left hyv hb" |
| }, |
| { |
| "id": 4, |
| "match": false, |
| "ground_truth": "rw [lipschitzWith_iff_norm_sub_le] at *", |
| "model_output": "rw [lipschitzWith_iff_norm_sub_le] at hl ⊢\n simp at hl\n rw [← lipschitzOnWith_univ] at hl ⊢\n have h1 : LipschitzOnWith l (fun z : E × F => (gradient H) z) univ := by\n apply lipschitzOnWith_of_lipschitzWith\n exact hl\n have h2 : LipschitzOnWith l (fun z : E × F => (gradient H) z.1 z.2) univ := by\n apply LipschitzOn" |
| }, |
| { |
| "id": 5, |
| "match": false, |
| "ground_truth": "by rw [grad_snd_eq h, grad_snd_eq h]", |
| "model_output": "rw [grad_eq_block_grad, grad_fun_comp, grad_fun_comp, grad_fun_comp, grad_fun_comp]" |
| }, |
| { |
| "id": 6, |
| "match": false, |
| "ground_truth": "rw [grad_snd_eq h, grad_snd_eq h]", |
| "model_output": "rw [grad_eq_block_grad, grad_fun_comp, grad_fun_comp, grad_fun_comp, grad_fun_comp]" |
| }, |
| { |
| "id": 7, |
| "match": false, |
| "ground_truth": "apply prox_well_define f hc lbdf x", |
| "model_output": "rcases (prox_set_aux f x) with ⟨y, yprop⟩\n use y; simp [prox_set, yprop]" |
| }, |
| { |
| "id": 8, |
| "match": false, |
| "ground_truth": "refine ⟨x₁, hx₁, x₂, hx₂, hx, ?_⟩", |
| "model_output": "refine ⟨x₁, ⟨hx₁, ⟨x₂, ⟨hx₂, hx, ?_⟩⟩⟩⟩⟩\n rw [map_add, @norm_add_sq 𝕜 _ _ _ _ (T x₁) (T x₂)]\n simp [hker hx₂]" |
| }, |
| { |
| "id": 9, |
| "match": false, |
| "ground_truth": "exact LinearMap.mem_ker.mp (hker hx₂)", |
| "model_output": "apply hker hx₂" |
| }, |
| { |
| "id": 10, |
| "match": false, |
| "ground_truth": "simp [Orthonormal, Pairwise] at *", |
| "model_output": "rw [orthonormal_iff_ite] at hv ⊢\n simp [Set.restrict, hv]" |
| }, |
| { |
| "id": 11, |
| "match": false, |
| "ground_truth": "by_cases hsf : ProperFunction s f", |
| "model_output": "apply isClosed_closure" |
| }, |
| { |
| "id": 12, |
| "match": false, |
| "ground_truth": "· exact closure_is_closed_of_proper f hs", |
| "model_output": "· exact epi_is_closed_of_closure f hs" |
| }, |
| { |
| "id": 13, |
| "match": false, |
| "ground_truth": "exact closure_is_closed_of_proper f hs", |
| "model_output": "rw [Function.closure, hsf.1]" |
| }, |
| { |
| "id": 14, |
| "match": true, |
| "ground_truth": "apply IsClosed.prod hs isClosed_univ", |
| "model_output": "apply IsClosed.prod hs isClosed_univ" |
| }, |
| { |
| "id": 15, |
| "match": false, |
| "ground_truth": "simp [w, l, g, AffineMap.decomp']", |
| "model_output": "exact i.h_eq x _hx" |
| }, |
| { |
| "id": 16, |
| "match": true, |
| "ground_truth": "rw [φ₁_equ fullrank₁ fullrank₂ n]", |
| "model_output": "rw [φ₁_equ fullrank₁ fullrank₂ n]" |
| }, |
| { |
| "id": 17, |
| "match": false, |
| "ground_truth": "exact EReal.coe_le_coe_iff.mp h", |
| "model_output": "exact EReal.coe_le_coe_iff.1 h" |
| }, |
| { |
| "id": 18, |
| "match": false, |
| "ground_truth": "rw [← hT.span_range_w_eq_range]", |
| "model_output": "have := hT.finrank_range_eq_card_nonzero_singular_values\n have h : card (Finset.univ.filter (fun i => T.singular_value i ≠ 0)) = r := by\n simp [hT.singular_values]\n exact Nat.sub_add_cancel (Nat.le_of_succ_le hT.singular_values_card_pos)\n rw [← this, ← hT.finrank_range_eq_card_nonzero_singular_values]" |
| }, |
| { |
| "id": 19, |
| "match": false, |
| "ground_truth": "apply iff_iff_implies_and_implies.mpr", |
| "model_output": "rw [← bi_convex_support_eq_closure hscv hscl, ← bi_convex_support_eq_closure htcv htcl, h]" |
| }, |
| { |
| "id": 20, |
| "match": false, |
| "ground_truth": "apply convex_conjugate_ge_bot_univ'", |
| "model_output": "apply lt_of_le_of_lt\n · apply zero_le'\n · apply lt_of_le_of_lt'\n · apply convex_conjugate_ge_zero\n · simp [convex_indicator]\n apply hs" |
| }, |
| { |
| "id": 21, |
| "match": false, |
| "ground_truth": "exact HasGradientWithinAt.const_smul h c", |
| "model_output": "exact HasGradientAtFilter.const_smul h c" |
| }, |
| { |
| "id": 22, |
| "match": true, |
| "ground_truth": "exact HasGradientAtFilter.const_add hf c", |
| "model_output": "exact HasGradientAtFilter.const_add hf c" |
| }, |
| { |
| "id": 23, |
| "match": true, |
| "ground_truth": "exact HasGradientAtFilter.const_add hf c", |
| "model_output": "exact HasGradientAtFilter.const_add hf c" |
| }, |
| { |
| "id": 24, |
| "match": true, |
| "ground_truth": "exact HasGradientAtFilter.const_sub hf c", |
| "model_output": "exact HasGradientAtFilter.const_sub hf c" |
| }, |
| { |
| "id": 25, |
| "match": false, |
| "ground_truth": "apply convex_on_p_top\n ext x; exact h x", |
| "model_output": "apply convex_on_n_inf" |
| }, |
| { |
| "id": 26, |
| "match": false, |
| "ground_truth": "apply monotone_gradient_convex'", |
| "model_output": "rw [ConvexOn]" |
| }, |
| { |
| "id": 27, |
| "match": false, |
| "ground_truth": "exact (fun x _ => affine_sq_gradient x)", |
| "model_output": "intro x _" |
| }, |
| { |
| "id": 28, |
| "match": false, |
| "ground_truth": "have h_zero_eq := (hS 0).mp h_zero_mem", |
| "model_output": "have h1 : ∀ i : Fin m, (hS (A.col i)).1 = (hS 0).1 := by\n intro i\n simp only [map_zero, hS, col, row, Fin.natAdd_natSub_cancel, ↓reduceDIte,\n Fin.coe_castAdd, add_zero, eq_self]" |
| }, |
| { |
| "id": 29, |
| "match": false, |
| "ground_truth": "have : (0 : 𝕜) < (1 : 𝕜) := one_pos", |
| "model_output": "apply (hf' x y neq)" |
| }, |
| { |
| "id": 30, |
| "match": true, |
| "ground_truth": "rw [inner_add_left, real_inner_smul_left, real_inner_smul_left]", |
| "model_output": "rw [inner_add_left, real_inner_smul_left, real_inner_smul_left]" |
| }, |
| { |
| "id": 31, |
| "match": false, |
| "ground_truth": "rw [← one_mul (-ε * ‖y - x‖), ← hab, add_mul]", |
| "model_output": "apply add_nonneg (mul_nonneg ha (le_of_lt εpos)) (mul_nonneg hb (le_of_lt εpos))" |
| }, |
| { |
| "id": 32, |
| "match": false, |
| "ground_truth": "calc\n _ = ‖(gradient H z).2 - (gradient H z').2‖ := by rw [grad_snd_eq h, grad_snd_eq h]\n _ = ‖(gradient H z - gradient H z').2‖ := rfl\n _ ≤ ‖(gradient H z - gradient H z')‖ := snd_norm_le_prod_L2 _\n _ ≤ _ := hl z z'", |
| "model_output": "have h1 := hl z z'" |
| }, |
| { |
| "id": 33, |
| "match": false, |
| "ground_truth": "by\n intro k\n specialize ck k; rw [ck]\n apply div_pos\n · norm_num\n apply mul_pos\n · linarith [hγ]\n · apply alg.lpos", |
| "model_output": "intro k; specialize ck k; rw [ck]; apply div_pos; norm_num; apply mul_pos; linarith; apply alg.lpos" |
| }, |
| { |
| "id": 34, |
| "match": false, |
| "ground_truth": "intro k\n specialize ck k; rw [ck]\n apply div_pos\n · norm_num\n apply mul_pos\n · linarith [hγ]\n · apply alg.lpos", |
| "model_output": "intro k; specialize ck k; rw [ck]; apply div_pos; norm_num; apply mul_pos; linarith; apply alg.lpos" |
| }, |
| { |
| "id": 35, |
| "match": false, |
| "ground_truth": "by\n apply prox_well_define f hc lbdf x", |
| "model_output": "rcases (prox_set_aux f x) with ⟨y, yprop⟩\n use y; simp [prox_set, yprop]" |
| }, |
| { |
| "id": 36, |
| "match": false, |
| "ground_truth": "by\n unfold colSubmodule\n rw [finrank_span_eq_card]\n simp [h]", |
| "model_output": "rw [colSubmodule]" |
| }, |
| { |
| "id": 37, |
| "match": false, |
| "ground_truth": "unfold colSubmodule\n rw [finrank_span_eq_card]\n simp [h]", |
| "model_output": "rw [colSubmodule]" |
| }, |
| { |
| "id": 38, |
| "match": false, |
| "ground_truth": "let ⟨x₁, hx₁, x₂, hx₂, hx⟩ := exists_orthogonal_decompose s x\n refine ⟨x₁, hx₁, x₂, hx₂, hx, ?_⟩\n rw [hx, LinearMap.map_add]\n suffices T x₂ = 0 from by\n rw [this, add_zero, norm_sq_eq_re_inner (𝕜:=𝕜)]\n exact LinearMap.mem_ker.mp (hker hx₂)", |
| "model_output": "have h : x = (s.orthogonalProjection x) + (x - (s.orthogonalProjection x)) := by simp" |
| }, |
| { |
| "id": 39, |
| "match": false, |
| "ground_truth": "suffices T x₂ = 0 from by\n rw [this, add_zero, norm_sq_eq_re_inner (𝕜:=𝕜)]", |
| "model_output": "rw [map_add, norm_add_sq_real]" |
| }, |
| { |
| "id": 40, |
| "match": false, |
| "ground_truth": "by\n rw [this, add_zero, norm_sq_eq_re_inner (𝕜:=𝕜)]", |
| "model_output": "rw [this, add_zero]" |
| }, |
| { |
| "id": 41, |
| "match": false, |
| "ground_truth": "rw [this, add_zero, norm_sq_eq_re_inner (𝕜:=𝕜)]", |
| "model_output": "rw [this, add_zero]" |
| }, |
| { |
| "id": 42, |
| "match": false, |
| "ground_truth": "by\n simp [Orthonormal, Pairwise] at *\n aesop", |
| "model_output": "rw [orthonormal_iff_ite] at hv ⊢\n simp [Set.restrict, hv]" |
| }, |
| { |
| "id": 43, |
| "match": false, |
| "ground_truth": "simp [Orthonormal, Pairwise] at *\n aesop", |
| "model_output": "rw [orthonormal_iff_ite] at hv ⊢\n simp [Set.restrict, hv]" |
| }, |
| { |
| "id": 44, |
| "match": false, |
| "ground_truth": "by\n let s := Metric.ball x ε\n have h : ContDiffAt ℝ 1 f x := by\n apply ContDiffOn.contDiffAt hf\n rw [mem_nhds_iff]; use s\n exact ⟨Eq.subset rfl, ⟨Metric.isOpen_ball, Metric.mem_ball_self he⟩⟩\n exact gradient_continuous_of_contdiffat f h", |
| "model_output": "have : ContDiffOn ℝ 0 f (Metric.ball x ε) := by apply contDiffOn_zero.mpr; exact hf\n apply HasGradientAt.continuousAt\n have : HasGradientAt f (gradient f) x := by apply DifferentiableAt.hasGradientAt\n apply HasGradientAt.continuousAt this" |
| }, |
| { |
| "id": 45, |
| "match": false, |
| "ground_truth": "let s := Metric.ball x ε\n have h : ContDiffAt ℝ 1 f x := by\n apply ContDiffOn.contDiffAt hf\n rw [mem_nhds_iff]; use s\n exact ⟨Eq.subset rfl, ⟨Metric.isOpen_ball, Metric.mem_ball_self he⟩⟩\n exact gradient_continuous_of_contdiffat f h", |
| "model_output": "have : ContDiffOn ℝ 0 f (Metric.ball x ε) := by apply contDiffOn_zero.mpr; exact hf\n apply HasGradientAt.continuousAt\n have : HasGradientAt f (gradient f) x := by apply DifferentiableAt.hasGradientAt\n apply HasGradientAt.continuousAt this" |
| }, |
| { |
| "id": 46, |
| "match": false, |
| "ground_truth": "have h : ContDiffAt ℝ 1 f x := by\n apply ContDiffOn.contDiffAt hf\n rw [mem_nhds_iff]; use s\n exact ⟨Eq.subset rfl, ⟨Metric.isOpen_ball, Metric.mem_ball_self he⟩⟩", |
| "model_output": "have : ContDiffOn ℝ 0 f (Metric.ball x ε) := by apply contDiffOn_zero.mpr; exact hf\n apply (equicontinuousOn_iff_le_maximalSeminorm hf this).1\n apply (equicontinuousOn_iff_le_maximalSeminorm this hf).2\n exact continuousOn_maximalSeminorm he" |
| }, |
| { |
| "id": 47, |
| "match": false, |
| "ground_truth": "exact ⟨Eq.subset rfl, ⟨Metric.isOpen_ball, Metric.mem_ball_self he⟩⟩", |
| "model_output": "constructor" |
| }, |
| { |
| "id": 48, |
| "match": false, |
| "ground_truth": "exact gradient_continuous_of_contdiffat f h", |
| "model_output": "rw[contDiffAt_one_iff] at h\n exact h.2" |
| }, |
| { |
| "id": 49, |
| "match": false, |
| "ground_truth": "by\n simp [Function.closure]\n intro h; exfalso\n obtain h1 := h.2; simp at h1", |
| "model_output": "apply Function.closure_univ_eq_top.mpr\n simp [Function.closure]" |
| }, |
| { |
| "id": 50, |
| "match": false, |
| "ground_truth": "simp [Function.closure]\n intro h; exfalso\n obtain h1 := h.2; simp at h1", |
| "model_output": "apply Function.closure_univ_eq_top.mpr\n simp [Function.closure]" |
| }, |
| { |
| "id": 51, |
| "match": false, |
| "ground_truth": "· simp [hf]; simp [Function.IsClosed]\n have h1 : Epi ⊥ s = s ×ˢ (univ : Set ℝ) := by\n ext ⟨x, _⟩\n simp [Function.Epi]\n rw [h1]\n apply IsClosed.prod hs isClosed_univ", |
| "model_output": "· simp [hf]\n exact isClosed_univ" |
| }, |
| { |
| "id": 52, |
| "match": true, |
| "ground_truth": "simp [hf]; simp [Function.IsClosed]\n have h1 : Epi ⊥ s = s ×ˢ (univ : Set ℝ) := by\n ext ⟨x, _⟩\n simp [Function.Epi]\n rw [h1]\n apply IsClosed.prod hs isClosed_univ", |
| "model_output": "simp [hf]" |
| }, |
| { |
| "id": 53, |
| "match": false, |
| "ground_truth": "have h1 : Epi ⊥ s = s ×ˢ (univ : Set ℝ) := by\n ext ⟨x, _⟩\n simp [Function.Epi]", |
| "model_output": "simp [Epi]\n apply IsClosed.preimage (Continuous.add continuous_const continuous_id) hs" |
| }, |
| { |
| "id": 54, |
| "match": false, |
| "ground_truth": "have h' : Epi ⊤ s = ∅ := by\n ext ⟨x, _⟩; simp [Function.Epi]", |
| "model_output": "simp [Epi]" |
| }, |
| { |
| "id": 55, |
| "match": false, |
| "ground_truth": "by\n constructor\n · intro h z\n unfold HasSubgradientAt at h\n obtain hz := h z\n simp; linarith\n intro h z\n unfold Banach_HasSubgradientAt at h\n obtain hz := h z; simp at hz\n linarith", |
| "model_output": "constructor\n · intro h z\n unfold HasSubgradientAt at h\n obtain hz := h z\n simp; linarith\n intro h z\n unfold Banach_HasSubgradientAt at h\n obtain hz := h z\n simp; linarith" |
| }, |
| { |
| "id": 56, |
| "match": false, |
| "ground_truth": "constructor\n · intro h z\n unfold HasSubgradientAt at h\n obtain hz := h z\n simp; linarith\n intro h z\n unfold Banach_HasSubgradientAt at h\n obtain hz := h z; simp at hz\n linarith", |
| "model_output": "constructor\n · intro h z\n unfold HasSubgradientAt at h\n obtain hz := h z\n simp; linarith\n intro h z\n unfold Banach_HasSubgradientAt at h\n obtain hz := h z\n simp; linarith" |
| }, |
| { |
| "id": 57, |
| "match": false, |
| "ground_truth": "· intro h z\n unfold HasSubgradientAt at h\n obtain hz := h z\n simp; linarith", |
| "model_output": "· intro h z\n unfold HasSubgradientAt at h\n obtain hz := h z\n simp; rw [← ContinuousLinearMap.map_sub]; linarith" |
| }, |
| { |
| "id": 58, |
| "match": false, |
| "ground_truth": "intro h z\n unfold HasSubgradientAt at h\n obtain hz := h z\n simp; linarith", |
| "model_output": "intro hsf\n unfold HasSubgradientAt at hsf\n unfold Banach_HasSubgradientAt\n constructor\n · exact hsf\n · intro y; specialize hsf y; simp at hsf; exact hsf" |
| }, |
| { |
| "id": 59, |
| "match": false, |
| "ground_truth": "· intro x hx\n simpa [w, l, g, AffineMap.decomp', ← EReal.coe_sub, ← EReal.coe_add] using i.hlw x hx", |
| "model_output": "· intro x _\n rw [← EReal.coe_add, ← inner_add_left, ← EReal.coe_le_coe_iff]\n apply le_trans _ (i.hf x)\n apply le_add_of_nonneg_left\n simp; exact i.hl x" |
| }, |
| { |
| "id": 60, |
| "match": false, |
| "ground_truth": "intro x hx\n simpa [w, l, g, AffineMap.decomp', ← EReal.coe_sub, ← EReal.coe_add] using i.hlw x hx", |
| "model_output": "intro x xs\n have := i.2 x xs\n simp only [ContinuousLinearMap.coe_coe] at this\n rw [← EReal.coe_add, ← EReal.coe_le_coe_iff] at this\n exact this" |
| }, |
| { |
| "id": 61, |
| "match": false, |
| "ground_truth": "apply Filter.tendsto_iff_seq_tendsto.1 Seq_converge_zero₃", |
| "model_output": "apply Filter.tendsto_iff_seq_tendsto.1 (Satisfying_equational_constraint2' fullrank₁ fullrank₂)\n apply A₂_subseq_converge_zero" |
| }, |
| { |
| "id": 62, |
| "match": false, |
| "ground_truth": "have : (fun n => (Int.toNat (φ₁ n))) = (fun n => (φ (n+1))) := by\n ext n; rw [φ₁_equ fullrank₁ fullrank₂ n]; simp", |
| "model_output": "apply strictMono_nat_of_lt_succ\n intro n\n have : (φ₁ (n + 1)) > (φ₁ n) := by\n calc φ₁ (n + 1)\n _ ≥ n + 1 := StrictMono.id_le (hphi_StrictMono fullrank₁ fullrank₂) (n + 1)\n _ > n :=by linarith\n apply this\n simp only [Nat.ofNat_pos, gt_iff_lt, Nat.cast_add, Nat.cast_one, zero_lt_one, implies_true]" |
| }, |
| { |
| "id": 63, |
| "match": false, |
| "ground_truth": "suffices 0 ≤ (c - a - b).toEReal by exact this", |
| "model_output": "apply EReal.coe_nonneg.mpr\n simp at h\n linarith" |
| }, |
| { |
| "id": 64, |
| "match": false, |
| "ground_truth": "have : a + b ≤ c := by exact EReal.coe_le_coe_iff.mp h", |
| "model_output": "rw [← EReal.coe_add, ← EReal.coe_sub] at h" |
| }, |
| { |
| "id": 65, |
| "match": false, |
| "ground_truth": "refine EReal.coe_nonneg.mpr ?intro.intro.a", |
| "model_output": "apply EReal.coe_nonneg.mpr" |
| }, |
| { |
| "id": 66, |
| "match": false, |
| "ground_truth": "by\n constructor\n · exact hs\n intro x hx y hy a b ha hb hab\n specialize hfun hx hy ha hb hab\n dsimp\n have : m / 2 * a * b * ‖x - y‖ ^ 2 = a * b * (m / 2 * ‖x - y‖ ^ 2) := by ring_nf\n simp at this;\n rw [← this]; exact hfun", |
| "model_output": "rw [StrongConvexOn]" |
| }, |
| { |
| "id": 67, |
| "match": false, |
| "ground_truth": "constructor\n · exact hs\n intro x hx y hy a b ha hb hab\n specialize hfun hx hy ha hb hab\n dsimp\n have : m / 2 * a * b * ‖x - y‖ ^ 2 = a * b * (m / 2 * ‖x - y‖ ^ 2) := by ring_nf\n simp at this;\n rw [← this]; exact hfun", |
| "model_output": "rw [StrongConvexOn]" |
| }, |
| { |
| "id": 68, |
| "match": false, |
| "ground_truth": "have : m / 2 * a * b * ‖x - y‖ ^ 2 = a * b * (m / 2 * ‖x - y‖ ^ 2) := by ring_nf", |
| "model_output": "linarith [hfun]" |
| }, |
| { |
| "id": 69, |
| "match": false, |
| "ground_truth": "by\n rw [← hT.span_range_w_eq_range]\n simp [finrank_span_eq_card (b:=hT.w) hT.w_orthonormal.linearIndependent]", |
| "model_output": "have := hT.finrank_range_eq_card_nonzero_singular_values\n have h : card (Finset.univ.filter (fun i => T.singular_value i ≠ 0)) = r := by\n simp [hT.singular_values]\n exact Nat.sub_add_cancel (Nat.le_of_succ_le hT.singular_values_card_pos)\n rw [← this, ← hT.finrank_range_eq_card_nonzero_singular_values]" |
| }, |
| { |
| "id": 70, |
| "match": false, |
| "ground_truth": "rw [← hT.span_range_w_eq_range]\n simp [finrank_span_eq_card (b:=hT.w) hT.w_orthonormal.linearIndependent]", |
| "model_output": "have := hT.finrank_range_eq_card_nonzero_singular_values\n have h : card (Finset.univ.filter (fun i => T.singular_value i ≠ 0)) = r := by\n simp [hT.singular_values]\n exact Nat.sub_add_cancel (Nat.le_of_succ_le hT.singular_values_card_pos)\n rw [← this, ← hT.finrank_range_eq_card_nonzero_singular_values]" |
| }, |
| { |
| "id": 71, |
| "match": false, |
| "ground_truth": "simp [finrank_span_eq_card (b:=hT.w) hT.w_orthonormal.linearIndependent]", |
| "model_output": "rw[← hT.finrank_range_u_eq_r,← hT.finrank_range_v_eq_r]" |
| }, |
| { |
| "id": 72, |
| "match": false, |
| "ground_truth": "· exact fun a ↦ set_inclusion_convex_support_eq s t htcv htcl a h", |
| "model_output": "· intro hz; rw [← Set.mem_convex_support, ← h, Set.mem_convex_support] at hz\n exact hz htcl hscv htcv" |
| }, |
| { |
| "id": 73, |
| "match": false, |
| "ground_truth": "exact fun a ↦ set_inclusion_convex_support_eq s t htcv htcl a h", |
| "model_output": "intro hzs\n have h1 : s.convex_support z = ∅ := by\n rw [convex_support_empty_iff]\n exact hzs\n have h2 := congrFun h h1\n rw [h2]\n apply convex_support_empty_iff.1\n exact htcv" |
| }, |
| { |
| "id": 74, |
| "match": false, |
| "ground_truth": "exact fun a ↦ set_inclusion_convex_support_eq t s hscv hscl a (fun x => (h x).symm)", |
| "model_output": "intro zt\n have h1 : convex_support t z = ∅ := by\n rw [← h z]; simp [convex_support]\n exact convex_conjugate_eq_empty hscl hscv (by rfl) zt\n have h2 := convex_iff_convex_support_empty.1 htcl z h1\n exact h2" |
| }, |
| { |
| "id": 75, |
| "match": true, |
| "ground_truth": "by\n intro x _\n simp [convex_support]\n obtain := ConvexIndicator_is_proper_on_univ hs\n apply convex_conjugate_ge_bot_univ'", |
| "model_output": "intro x _" |
| }, |
| { |
| "id": 76, |
| "match": true, |
| "ground_truth": "intro x _\n simp [convex_support]\n obtain := ConvexIndicator_is_proper_on_univ hs\n apply convex_conjugate_ge_bot_univ'", |
| "model_output": "intro x _" |
| }, |
| { |
| "id": 77, |
| "match": false, |
| "ground_truth": "have convex_conjugate_at_zero :\n convex_conjugate univ (convex_indicator s) 0 ≤ 0 := by\n refine (conjugate_le_zero_iff s.convex_indicator).mpr ?_\n simp\n intro y\n simp [convex_indicator]\n split_ifs with hy <;> simp", |
| "model_output": "rcases hs with ⟨x, hx⟩\n use x; simp [convex_conjugate]\n apply lt_of_le_of_lt\n · apply convex_indicator_le\n · simp [hx]" |
| }, |
| { |
| "id": 78, |
| "match": false, |
| "ground_truth": "refine (conjugate_le_zero_iff s.convex_indicator).mpr ?_", |
| "model_output": "apply convex_conjugate_le_zero\n · simp\n · exact convex_indicator_convex hs.convex_to_nonconvex" |
| }, |
| { |
| "id": 79, |
| "match": false, |
| "ground_truth": "exact lt_of_le_of_lt (convex_conjugate_at_zero) (zero_lt_top)", |
| "model_output": "apply convex_conjugate_lt_top" |
| }, |
| { |
| "id": 80, |
| "match": false, |
| "ground_truth": "by\n exact HasGradientWithinAt.const_smul h c", |
| "model_output": "exact HasGradientAtFilter.const_smul h c" |
| }, |
| { |
| "id": 81, |
| "match": true, |
| "ground_truth": "by\n exact HasGradientAtFilter.const_add hf c", |
| "model_output": "exact HasGradientAtFilter.const_add hf c" |
| }, |
| { |
| "id": 82, |
| "match": true, |
| "ground_truth": "by\n exact HasGradientAtFilter.const_add hf c", |
| "model_output": "exact HasGradientAtFilter.const_add hf c" |
| }, |
| { |
| "id": 83, |
| "match": false, |
| "ground_truth": "by\n simp only [sub_eq_add_neg, gradient_add_const]", |
| "model_output": "rw [gradient_sub_const c, gradient_const]" |
| }, |
| { |
| "id": 84, |
| "match": false, |
| "ground_truth": "simp only [sub_eq_add_neg, gradient_add_const]", |
| "model_output": "rw [gradient_sub_const c, gradient_const]" |
| }, |
| { |
| "id": 85, |
| "match": true, |
| "ground_truth": "by\n exact HasGradientAtFilter.const_sub hf c", |
| "model_output": "exact HasGradientAtFilter.const_sub hf c" |
| }, |
| { |
| "id": 86, |
| "match": false, |
| "ground_truth": "by\n obtain h1 := h.2; simp at h1\n rcases h1 with ⟨x, hx⟩; use x; simp [hx]", |
| "model_output": "simp [dom]" |
| }, |
| { |
| "id": 87, |
| "match": false, |
| "ground_truth": "obtain h1 := h.2; simp at h1\n rcases h1 with ⟨x, hx⟩; use x; simp [hx]", |
| "model_output": "simp [dom]" |
| }, |
| { |
| "id": 88, |
| "match": false, |
| "ground_truth": "by\n apply convex_on_p_top\n ext x; exact h x", |
| "model_output": "apply convex_on_n_inf" |
| }, |
| { |
| "id": 89, |
| "match": false, |
| "ground_truth": "by\n rw [sub_eq_add_neg u v, mulVec_add, mulVec_neg, sub_eq_add_neg]", |
| "model_output": "rw [Matrix.mulVec_sub]" |
| }, |
| { |
| "id": 90, |
| "match": false, |
| "ground_truth": "rw [sub_eq_add_neg u v, mulVec_add, mulVec_neg, sub_eq_add_neg]", |
| "model_output": "rw [Matrix.mulVec_sub]" |
| }, |
| { |
| "id": 91, |
| "match": false, |
| "ground_truth": "· exact (fun x _ => affine_sq_gradient x)", |
| "model_output": "· intro x _\n apply linear_gradient" |
| }, |
| { |
| "id": 92, |
| "match": true, |
| "ground_truth": "· intro x _ y _\n rw [Matrix.mulVec_sub, Matrix.mulVec_sub, ← sub_add, sub_add_eq_add_sub, sub_add_cancel,\n ← Matrix.mulVec_sub, real_inner_eq_dot]\n rw [← dot_mul_eq_transpose_mul_dot,← Matrix.mulVec_sub, ← norm2eq_dot]\n apply sq_nonneg", |
| "model_output": "intro x _ y _" |
| }, |
| { |
| "id": 93, |
| "match": true, |
| "ground_truth": "intro x _ y _\n rw [Matrix.mulVec_sub, Matrix.mulVec_sub, ← sub_add, sub_add_eq_add_sub, sub_add_cancel,\n ← Matrix.mulVec_sub, real_inner_eq_dot]\n rw [← dot_mul_eq_transpose_mul_dot,← Matrix.mulVec_sub, ← norm2eq_dot]\n apply sq_nonneg", |
| "model_output": "intro x _ y _" |
| }, |
| { |
| "id": 94, |
| "match": false, |
| "ground_truth": "rw [Matrix.mulVec_sub, Matrix.mulVec_sub, ← sub_add, sub_add_eq_add_sub, sub_add_cancel,\n ← Matrix.mulVec_sub, real_inner_eq_dot]", |
| "model_output": "rw [← sub_add, sub_right_comm, sub_add, sub_self, sub_zero]" |
| }, |
| { |
| "id": 95, |
| "match": false, |
| "ground_truth": "rw [← dot_mul_eq_transpose_mul_dot,← Matrix.mulVec_sub, ← norm2eq_dot]", |
| "model_output": "rw [← Matrix.mulVec_sub, ← sub_add, sub_add_eq_add_sub, sub_add_cancel]" |
| }, |
| { |
| "id": 96, |
| "match": false, |
| "ground_truth": "have h_zero_mem : (0 : E) ∈ S := S.zero_mem", |
| "model_output": "have := hS 0\n simp at this\n have : (fun i ↦ (B.repr 0) (Fin.natAdd n i)) = 0 := by\n ext i; simp\n rw [this] at hS\n simp at hS\n exact hS" |
| }, |
| { |
| "id": 97, |
| "match": false, |
| "ground_truth": "have h_repr_zero : B.repr 0 = 0 := LinearEquiv.map_zero B.repr", |
| "model_output": "simp at h_zero_eq" |
| }, |
| { |
| "id": 98, |
| "match": false, |
| "ground_truth": "simp only [map_zero, Finsupp.coe_zero, Pi.zero_apply] at h_zero_eq", |
| "model_output": "simp [h_repr_zero] at h_zero_eq" |
| }, |
| { |
| "id": 99, |
| "match": false, |
| "ground_truth": "simp only [Matrix.mulVec_zero, add_zero] at h_zero_eq", |
| "model_output": "simp at h_zero_eq" |
| }, |
| { |
| "id": 100, |
| "match": true, |
| "ground_truth": "by\n unfold sup_value\n unfold FeasSet FeasPoint\n simp [hp]", |
| "model_output": "unfold sup_value\n unfold FeasSet FeasPoint\n simp [hp]" |
| }, |
| { |
| "id": 101, |
| "match": true, |
| "ground_truth": "unfold sup_value\n unfold FeasSet FeasPoint\n simp [hp]", |
| "model_output": "unfold sup_value\n unfold FeasSet FeasPoint\n simp [hp]" |
| }, |
| { |
| "id": 102, |
| "match": true, |
| "ground_truth": "by\n intro b; rw [polyhedra_iff_cone b]\n exact closed_conic σ b", |
| "model_output": "intro b" |
| }, |
| { |
| "id": 103, |
| "match": true, |
| "ground_truth": "intro b; rw [polyhedra_iff_cone b]\n exact closed_conic σ b", |
| "model_output": "intro b" |
| }, |
| { |
| "id": 104, |
| "match": true, |
| "ground_truth": "by\n rw [h]; apply (local_Minimum_TangentCone' loc hl hf)", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 105, |
| "match": true, |
| "ground_truth": "rw [h]; apply (local_Minimum_TangentCone' loc hl hf)", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 106, |
| "match": false, |
| "ground_truth": "apply (local_Minimum_TangentCone' loc hl hf)", |
| "model_output": "rw [← h]" |
| }, |
| { |
| "id": 107, |
| "match": false, |
| "ground_truth": "by rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp", |
| "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply, A.symm_comp, ContinuousLinearMap.adjoint_id]" |
| }, |
| { |
| "id": 108, |
| "match": false, |
| "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp", |
| "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply, A.symm_comp, ContinuousLinearMap.adjoint_id]" |
| }, |
| { |
| "id": 109, |
| "match": false, |
| "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]", |
| "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply, A.symm_comp, ContinuousLinearMap.adjoint_id]" |
| }, |
| { |
| "id": 110, |
| "match": false, |
| "ground_truth": "by rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp", |
| "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply, A.adjoint_comp_self]" |
| }, |
| { |
| "id": 111, |
| "match": false, |
| "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp", |
| "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply, A.adjoint_comp_self]" |
| }, |
| { |
| "id": 112, |
| "match": false, |
| "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]", |
| "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply, A.adjoint_comp_self]" |
| }, |
| { |
| "id": 113, |
| "match": false, |
| "ground_truth": "by rw [← trace_transpose, ← trace_transpose_mul, transpose_mul]", |
| "model_output": "rw [Matrix.mul_comm]" |
| }, |
| { |
| "id": 114, |
| "match": false, |
| "ground_truth": "rw [← trace_transpose, ← trace_transpose_mul, transpose_mul]", |
| "model_output": "rw [Matrix.mul_comm]" |
| }, |
| { |
| "id": 115, |
| "match": false, |
| "ground_truth": "have : ConvexOn ℝ X fun _ ↦ (0: EReal) := by\n dsimp [ConvexOn]\n constructor\n exact hs\n intro x _ y _ a b _ _ _\n simp", |
| "model_output": "apply ConvexOn.subset hs\n · apply convex_epigraph\n · simp [convex_indicator]" |
| }, |
| { |
| "id": 116, |
| "match": false, |
| "ground_truth": "have eq : EqOn (fun _ ↦ (0: EReal)) (Set.convex_indicator X) X := by\n dsimp[EqOn]\n intro x hx\n dsimp [Set.convex_indicator]\n rw [if_pos hx]", |
| "model_output": "apply ConvexOn.add this\n simp [ConvexOn] at hs\n exact hs.convex_indicator (fun _ _ _ _ _ ↦ by simp)" |
| }, |
| { |
| "id": 117, |
| "match": false, |
| "ground_truth": "by\n apply Subset.antisymm\n · exact iIntersection_closure_sub_closure_iIntersection'' h hinter\n exact closure_iIntersection_sub_iIntersection_closure'", |
| "model_output": "rcases hinter with ⟨x, hx⟩\n apply Subset.antisymm\n · exact iIntersection_intrinsicClosure_sub_intrinsicClosure_iIntersection\n exact intrinsicClosure_iIntersection_sub_iIntersection_intrinsicClosure h hx" |
| }, |
| { |
| "id": 118, |
| "match": false, |
| "ground_truth": "apply Subset.antisymm\n · exact iIntersection_closure_sub_closure_iIntersection'' h hinter\n exact closure_iIntersection_sub_iIntersection_closure'", |
| "model_output": "rcases hinter with ⟨x, hx⟩\n apply Subset.antisymm\n · exact iIntersection_intrinsicClosure_sub_intrinsicClosure_iIntersection\n exact intrinsicClosure_iIntersection_sub_iIntersection_intrinsicClosure h hx" |
| }, |
| { |
| "id": 119, |
| "match": false, |
| "ground_truth": "· exact iIntersection_closure_sub_closure_iIntersection'' h hinter", |
| "model_output": "· apply iIntersection_intrinsicClosure_sub_intrinsicClosure_iIntersection\n · exact h\n · exact hinter" |
| }, |
| { |
| "id": 120, |
| "match": false, |
| "ground_truth": "exact iIntersection_closure_sub_closure_iIntersection'' h hinter", |
| "model_output": "apply iIntersection_intrinsicClosure_sub_intrinsicClosure_iIntersection" |
| }, |
| { |
| "id": 121, |
| "match": false, |
| "ground_truth": "exact closure_iIntersection_sub_iIntersection_closure'", |
| "model_output": "apply intrinsicClosure_minimal\n · intro x hx\n rw [@mem_iInter]; intro i\n apply intrinsicClosure_mono (h i) (subset_iInter h)\n exact hx\n · exact hinter" |
| }, |
| { |
| "id": 122, |
| "match": false, |
| "ground_truth": "have eqone : a + (1 - a) = 1 := add_sub_cancel a 1", |
| "model_output": "have : f (a • x + (1 - a) • y) < f x := by\n apply hf' x y lta alt\n simp [neq]" |
| }, |
| { |
| "id": 123, |
| "match": false, |
| "ground_truth": "have lta' : 0 < 1 - a := sub_pos_of_lt alt", |
| "model_output": "have : f x ≤ f (a • x + (1 - a) • y) := by\n have : a • x + (1 - a) • y ∈ univ := by simp\n apply hx this\n simp [a]" |
| }, |
| { |
| "id": 124, |
| "match": false, |
| "ground_truth": "have h : f (a • x + (1 - a) • y) < f y := by\n apply Eq.trans_gt (max_eq_right (hx trivial))\n apply hf' neq lta lta' eqone", |
| "model_output": "have : f (a • x + (1 - a) • y) < max (f x) (f y) := by\n apply hf' (by simpa) (by simpa) lta lta' (by simp) eqone" |
| }, |
| { |
| "id": 125, |
| "match": false, |
| "ground_truth": "apply Eq.trans_gt (max_eq_right (hx trivial))", |
| "model_output": "apply hf' (by assumption) (by assumption) lta' hy (by assumption)" |
| }, |
| { |
| "id": 126, |
| "match": false, |
| "ground_truth": "intro ε εpos", |
| "model_output": "intro ε εpos\n filter_upwards [hu ε εpos, hv ε εpos] with y huy hvy\n rw [inner_add_left, real_inner_smul_right, real_inner_smul_right]\n linarith [huy, hvy]" |
| }, |
| { |
| "id": 127, |
| "match": false, |
| "ground_truth": "rw [ge_iff_le] at *", |
| "model_output": "apply add_le_add\n · apply mul_le_mul_of_nonneg_left hyu (by simpa)\n · apply mul_le_mul_of_nonneg_left hyv (by simpa)" |
| }, |
| { |
| "id": 128, |
| "match": false, |
| "ground_truth": "apply add_le_add", |
| "model_output": "apply add_le_add\n · apply mul_le_mul_of_nonneg_left hyu (by simpa)\n · apply mul_le_mul_of_nonneg_left hyv (by simpa)" |
| }, |
| { |
| "id": 129, |
| "match": false, |
| "ground_truth": "apply div_pos", |
| "model_output": "apply div_pos zero_lt_one\n apply mul_pos\n · linarith\n · apply alg.lpos" |
| }, |
| { |
| "id": 130, |
| "match": false, |
| "ground_truth": "apply mul_pos", |
| "model_output": "apply mul_pos _ alg.lpos\n[USER]\napply hγ" |
| }, |
| { |
| "id": 131, |
| "match": true, |
| "ground_truth": "· linarith [hγ]", |
| "model_output": "· linarith [hγ]" |
| }, |
| { |
| "id": 132, |
| "match": true, |
| "ground_truth": "linarith [hγ]", |
| "model_output": "linarith [hγ]" |
| }, |
| { |
| "id": 133, |
| "match": false, |
| "ground_truth": "· apply alg.lpos", |
| "model_output": "· exact alg.lpos" |
| }, |
| { |
| "id": 134, |
| "match": false, |
| "ground_truth": "apply alg.lpos", |
| "model_output": "· exact alg.lpos" |
| }, |
| { |
| "id": 135, |
| "match": false, |
| "ground_truth": "rw [mem_nhds_iff]", |
| "model_output": "apply Metric.ball_mem_nhds x he" |
| }, |
| { |
| "id": 136, |
| "match": false, |
| "ground_truth": "simp [Function.Epi]", |
| "model_output": "simp [Epi]" |
| }, |
| { |
| "id": 137, |
| "match": false, |
| "ground_truth": "simp [Function.Epi]", |
| "model_output": "simp [Epi]" |
| }, |
| { |
| "id": 138, |
| "match": false, |
| "ground_truth": "exact isClosed_empty", |
| "model_output": "simp" |
| }, |
| { |
| "id": 139, |
| "match": false, |
| "ground_truth": "constructor", |
| "model_output": "constructor\n · intro h z\n unfold HasSubgradientAt at h\n obtain hz := h z\n simp; linarith\n intro h z\n unfold Banach_HasSubgradientAt at h\n obtain hz := h z\n simp; linarith" |
| }, |
| { |
| "id": 140, |
| "match": true, |
| "ground_truth": "by\n ext g; rw [h]", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 141, |
| "match": true, |
| "ground_truth": "ext g; rw [h]", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 142, |
| "match": false, |
| "ground_truth": "constructor", |
| "model_output": "constructor <;> intro x hx <;> simp [i.h_eq x hx]" |
| }, |
| { |
| "id": 143, |
| "match": false, |
| "ground_truth": "intro x _hx", |
| "model_output": "intro x _ [H]\n rw [H]\n simp [h]" |
| }, |
| { |
| "id": 144, |
| "match": false, |
| "ground_truth": "simp at this", |
| "model_output": "rw [StrictMono]" |
| }, |
| { |
| "id": 145, |
| "match": false, |
| "ground_truth": "· simp [StrictMono]", |
| "model_output": "apply strictMono_add_const" |
| }, |
| { |
| "id": 146, |
| "match": false, |
| "ground_truth": "simp [StrictMono]", |
| "model_output": "apply strictMono_add_const" |
| }, |
| { |
| "id": 147, |
| "match": true, |
| "ground_truth": "by_cases hc2 : c = ⊤", |
| "model_output": "by_cases hc2 : c = ⊤" |
| }, |
| { |
| "id": 148, |
| "match": false, |
| "ground_truth": "· rw [hc2]; simp", |
| "model_output": "· rw [hc2, EReal.coe_sub]\n simp\n exact le_refl 0" |
| }, |
| { |
| "id": 149, |
| "match": false, |
| "ground_truth": "rw [hc2]; simp", |
| "model_output": "rw [hc2, EReal.coe_sub_top a]" |
| }, |
| { |
| "id": 150, |
| "match": false, |
| "ground_truth": "by exact this", |
| "model_output": "rw [← EReal.coe_sub, ← EReal.coe_sub] at this ⊢" |
| }, |
| { |
| "id": 151, |
| "match": false, |
| "ground_truth": "constructor", |
| "model_output": "rw [StrongConvexOn]" |
| }, |
| { |
| "id": 152, |
| "match": false, |
| "ground_truth": "rw [← this]", |
| "model_output": "rw [this] at hfun\n exact hfun" |
| }, |
| { |
| "id": 153, |
| "match": false, |
| "ground_truth": "constructor", |
| "model_output": "constructor\n · intro x\n rw[x]\n intro x\n rw[x]" |
| }, |
| { |
| "id": 154, |
| "match": false, |
| "ground_truth": "simp [funext_iff]", |
| "model_output": "intro h; rw [← bi_convex_support_eq_set hscv hscl, ← bi_convex_support_eq_set htcv htcl, h]" |
| }, |
| { |
| "id": 155, |
| "match": false, |
| "ground_truth": "constructor", |
| "model_output": "constructor <;> intro hz <;> contrapose hz <;> simp [convex_support] at h <;> specialize h z hz <;> simp at h <;> tauto" |
| }, |
| { |
| "id": 156, |
| "match": false, |
| "ground_truth": "· apply convex_univ", |
| "model_output": "apply convex_univ [f'']" |
| }, |
| { |
| "id": 157, |
| "match": true, |
| "ground_truth": "apply convex_univ", |
| "model_output": "apply convex_univ" |
| }, |
| { |
| "id": 158, |
| "match": true, |
| "ground_truth": "intro x _ y _", |
| "model_output": "intro x _ y _" |
| }, |
| { |
| "id": 159, |
| "match": true, |
| "ground_truth": "apply sq_nonneg", |
| "model_output": "apply sq_nonneg" |
| }, |
| { |
| "id": 160, |
| "match": true, |
| "ground_truth": "exact h_zero_eq.symm", |
| "model_output": "exact h_zero_eq.symm" |
| }, |
| { |
| "id": 161, |
| "match": false, |
| "ground_truth": "dsimp [ConvexOn]", |
| "model_output": "apply convexOn_const 0 hs" |
| }, |
| { |
| "id": 162, |
| "match": false, |
| "ground_truth": "constructor", |
| "model_output": "constructor <;> simp [Convex] at * <;> tauto" |
| }, |
| { |
| "id": 163, |
| "match": false, |
| "ground_truth": "dsimp[EqOn]", |
| "model_output": "intro x hx\n simp [convex_indicator] at hx\n simp [hx]" |
| }, |
| { |
| "id": 164, |
| "match": false, |
| "ground_truth": "rw [if_pos hx]", |
| "model_output": "rw[if_pos hx]" |
| }, |
| { |
| "id": 165, |
| "match": false, |
| "ground_truth": "by_contra neq", |
| "model_output": "by_contra hxy" |
| }, |
| { |
| "id": 166, |
| "match": false, |
| "ground_truth": "intro u hu v hv a b ha hb hab", |
| "model_output": "rw [convex_iff_forall_pos]" |
| }, |
| { |
| "id": 167, |
| "match": true, |
| "ground_truth": "rw [has_f_subdiff_iff] at *", |
| "model_output": "rw [has_f_subdiff_iff] at *" |
| }, |
| { |
| "id": 168, |
| "match": true, |
| "ground_truth": "rw [finrank_span_eq_card]", |
| "model_output": "rw [finrank_span_eq_card]" |
| }, |
| { |
| "id": 169, |
| "match": false, |
| "ground_truth": "rw [hx, LinearMap.map_add]", |
| "model_output": "rw [hx, (LinearMap.ker_eq_orthogonal_orthogonal hker).mp hx₂, add_zero]" |
| }, |
| { |
| "id": 170, |
| "match": false, |
| "ground_truth": "apply ContDiffOn.contDiffAt hf", |
| "model_output": "apply ContDiffOn.contDiffAt hf\n · apply mem_ball_self he" |
| }, |
| { |
| "id": 171, |
| "match": false, |
| "ground_truth": "simp [Function.closure]", |
| "model_output": "apply Function.closure_univ_eq_top.mpr\n simp [Function.closure]" |
| }, |
| { |
| "id": 172, |
| "match": false, |
| "ground_truth": "simp [Function.closure, hsf]", |
| "model_output": "simp [Function.IsClosed]\n exact epi_is_closed_of_closure f hs" |
| }, |
| { |
| "id": 173, |
| "match": false, |
| "ground_truth": "by_cases hf : ∃ x ∈ s, f x = ⊥", |
| "model_output": "by_cases h : ∃ x ∈ s, f x = ⊥\n · simp [h]; exact isClosed_univ\n simp [h]; exact isClosed_empty" |
| }, |
| { |
| "id": 174, |
| "match": false, |
| "ground_truth": "simp [Function.IsClosed]", |
| "model_output": "refine isClosed_eq ?h₁.hf ?h₁.hg\n exact continuous_const\n exact continuous_id'" |
| }, |
| { |
| "id": 175, |
| "match": false, |
| "ground_truth": "simp [Function.IsClosed]", |
| "model_output": "simp [Function.IsClosed, isClosed_preimage]" |
| }, |
| { |
| "id": 176, |
| "match": false, |
| "ground_truth": "apply StrictMono.tendsto_atTop", |
| "model_output": "apply StrictMono.tendsto_atTop\n apply strictMono_nat_of_lt_succ\n intro n\n have : φ₁ (n + 1) > φ₁ n := by apply StrictMono.strictMonoOn\n apply StrictMonoOn.mono this\n simp [φ₁]\n apply this\n apply φ₁_is_monotone\n apply Nat.succ_le_iff.mpr" |
| }, |
| { |
| "id": 177, |
| "match": false, |
| "ground_truth": "apply StrictMono.comp", |
| "model_output": "apply StrictMono.comp\n · apply hphi_StrictMono\n · apply StrictMono.add_const\n · exact strictMono_id\n · exact Nat.zero_le 1" |
| }, |
| { |
| "id": 178, |
| "match": true, |
| "ground_truth": "· apply hphi_StrictMono", |
| "model_output": "· apply hphi_StrictMono" |
| }, |
| { |
| "id": 179, |
| "match": true, |
| "ground_truth": "apply hphi_StrictMono", |
| "model_output": "apply hphi_StrictMono" |
| }, |
| { |
| "id": 180, |
| "match": false, |
| "ground_truth": "lift a to ℝ using ⟨ha2, ha1⟩", |
| "model_output": "have : a.toReal + b ≤ c.toReal := by\n lift a to ℝ using ⟨ha1, ha2⟩\n rw [← EReal.coe_add, ← EReal.coe_sub] at h\n exact h\n apply ha1\n apply ha2\n apply hc1\n simp\n simp\n simp\n exact h\n apply EReal.coe_le_coe_iff.1 this" |
| }, |
| { |
| "id": 181, |
| "match": false, |
| "ground_truth": "lift c to ℝ using ⟨hc2, hc1⟩", |
| "model_output": "lift c to ℝ using ⟨hc1, hc2⟩\n rw [← EReal.coe_sub, ← EReal.coe_add]\n apply EReal.coe_le_coe_iff.2\n apply le_sub_of_add_le\n simpa" |
| }, |
| { |
| "id": 182, |
| "match": false, |
| "ground_truth": "intro x hx y hy a b ha hb hab", |
| "model_output": "intro x xs y ys a b ha hb hab\n rw[smul_eq_mul,smul_eq_mul]\n linarith [hfun xs ys ha hb hab]" |
| }, |
| { |
| "id": 183, |
| "match": false, |
| "ground_truth": "· exact fun h => (by rw [h])", |
| "model_output": "· intro h; rw [h]" |
| }, |
| { |
| "id": 184, |
| "match": false, |
| "ground_truth": "exact fun h => (by rw [h])", |
| "model_output": "intro h; rw [h]" |
| }, |
| { |
| "id": 185, |
| "match": false, |
| "ground_truth": "simp [convex_support]", |
| "model_output": "rw [convex_support]" |
| }, |
| { |
| "id": 186, |
| "match": false, |
| "ground_truth": "simp [convex_support]", |
| "model_output": "rcases hs with ⟨x, hx⟩\n use x\n constructor\n · simp\n rw [convex_support]\n simp\n rw [iSup_apply, iSup_lt_top]\n exact fun b ↦ (convex_indicator s b).isProperOn_univ_of_mem hx" |
| }, |
| { |
| "id": 187, |
| "match": true, |
| "ground_truth": "simp [convex_indicator]", |
| "model_output": "simp [convex_indicator]" |
| }, |
| { |
| "id": 188, |
| "match": false, |
| "ground_truth": "split_ifs with hy <;> simp", |
| "model_output": "by_cases h : y ∈ s\n · rw [h]; simp\n simp [h]" |
| }, |
| { |
| "id": 189, |
| "match": false, |
| "ground_truth": "rcases h1 with ⟨x, hx⟩", |
| "model_output": "rcases h1 with ⟨x, hx⟩\n use x\n simp [dom]\n exact hx" |
| }, |
| { |
| "id": 190, |
| "match": false, |
| "ground_truth": "apply convex_on_p_top", |
| "model_output": "apply convex_on_n_inf" |
| }, |
| { |
| "id": 191, |
| "match": false, |
| "ground_truth": "rw [polyhedra_iff_cone b]", |
| "model_output": "apply isClosed_setOf_polyhedra" |
| }, |
| { |
| "id": 192, |
| "match": false, |
| "ground_truth": "exact closed_conic σ b", |
| "model_output": "rw [← closure_conic_eq]" |
| }, |
| { |
| "id": 193, |
| "match": false, |
| "ground_truth": "intro x _ y _ a b _ _ _", |
| "model_output": "simp" |
| }, |
| { |
| "id": 194, |
| "match": false, |
| "ground_truth": "dsimp [Set.convex_indicator]", |
| "model_output": "rw [convex_indicator]" |
| }, |
| { |
| "id": 195, |
| "match": false, |
| "ground_truth": "apply ConvexOn.congr this eq", |
| "model_output": "rw [convexOn_congr eq]" |
| }, |
| { |
| "id": 196, |
| "match": false, |
| "ground_truth": "apply Subset.antisymm", |
| "model_output": "rcases hinter with ⟨x, hx⟩\n apply Subset.antisymm\n · exact iIntersection_intrinsicClosure_sub_intrinsicClosure_iIntersection\n exact intrinsicClosure_iIntersection_sub_iIntersection_intrinsicClosure h hx" |
| }, |
| { |
| "id": 197, |
| "match": false, |
| "ground_truth": "apply hf' neq lta lta' eqone", |
| "model_output": "have : a • x + (1 - a) • y = a • x + (1 - a) • y := by simp only [add_sub_cancel]" |
| }, |
| { |
| "id": 198, |
| "match": false, |
| "ground_truth": "simp only [isMinOn_iff] at hy", |
| "model_output": "have : f (a • x + (1 - a) • y) < f (y) := h" |
| }, |
| { |
| "id": 199, |
| "match": false, |
| "ground_truth": "apply not_le_of_gt h hy", |
| "model_output": "linarith [hy, h]" |
| }, |
| { |
| "id": 200, |
| "match": false, |
| "ground_truth": "intro u", |
| "model_output": "rw [convex_iff_forall_pos]" |
| }, |
| { |
| "id": 201, |
| "match": false, |
| "ground_truth": "intro ε", |
| "model_output": "intro ε εpos\n filter_upwards [hu ε εpos, hv ε εpos] with y huy hvy\n rw [inner_add_left, real_inner_smul_right, real_inner_smul_right]\n linarith [huy, hvy]" |
| }, |
| { |
| "id": 202, |
| "match": false, |
| "ground_truth": "intro z z'", |
| "model_output": "intro z z' [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z" |
| }, |
| { |
| "id": 203, |
| "match": false, |
| "ground_truth": "intro z", |
| "model_output": "intro z z' [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z'] [hl z z" |
| }, |
| { |
| "id": 204, |
| "match": false, |
| "ground_truth": "intro k", |
| "model_output": "intro k; specialize ck k; rw [ck]; apply div_pos; norm_num; apply mul_pos; linarith; apply alg.lpos" |
| }, |
| { |
| "id": 205, |
| "match": true, |
| "ground_truth": "rw [ck]", |
| "model_output": "rw [ck]" |
| }, |
| { |
| "id": 206, |
| "match": true, |
| "ground_truth": "· norm_num", |
| "model_output": "· norm_num" |
| }, |
| { |
| "id": 207, |
| "match": true, |
| "ground_truth": "norm_num", |
| "model_output": "norm_num" |
| }, |
| { |
| "id": 208, |
| "match": false, |
| "ground_truth": "simp [h]", |
| "model_output": "rw [linearIndependent_iff_mul_conjTranspose_eq_one]" |
| }, |
| { |
| "id": 209, |
| "match": false, |
| "ground_truth": "aesop", |
| "model_output": "constructor <;> intro a ha b hb hab <;> exact hv.1 a ha <;> exact hv.2 a ha b hb hab" |
| }, |
| { |
| "id": 210, |
| "match": false, |
| "ground_truth": "intro h", |
| "model_output": "intro h\n unfold LowerSemicontinuousHull\n simp\n exact h.1" |
| }, |
| { |
| "id": 211, |
| "match": false, |
| "ground_truth": "exfalso", |
| "model_output": "apply lowersemicontinuoushull_eq_top_of_proper h.uninfinity" |
| }, |
| { |
| "id": 212, |
| "match": true, |
| "ground_truth": "simp at h1", |
| "model_output": "simp at h1" |
| }, |
| { |
| "id": 213, |
| "match": true, |
| "ground_truth": "simp [hf]", |
| "model_output": "simp [hf]" |
| }, |
| { |
| "id": 214, |
| "match": false, |
| "ground_truth": "ext ⟨x, _⟩", |
| "model_output": "rw [epi_eq_bot]" |
| }, |
| { |
| "id": 215, |
| "match": true, |
| "ground_truth": "rw [h1]", |
| "model_output": "rw [h1]" |
| }, |
| { |
| "id": 216, |
| "match": true, |
| "ground_truth": "simp [hf]", |
| "model_output": "simp [hf]" |
| }, |
| { |
| "id": 217, |
| "match": false, |
| "ground_truth": "ext ⟨x, _⟩", |
| "model_output": "simp [Epi]" |
| }, |
| { |
| "id": 218, |
| "match": true, |
| "ground_truth": "rw [h']", |
| "model_output": "rw [h']" |
| }, |
| { |
| "id": 219, |
| "match": false, |
| "ground_truth": "intro h z", |
| "model_output": "intro hsf\n unfold HasSubgradientAt at hsf\n unfold Banach_HasSubgradientAt\n constructor\n · exact hsf\n · intro y; specialize hsf y; simp at hsf; exact hsf" |
| }, |
| { |
| "id": 220, |
| "match": false, |
| "ground_truth": "intro h", |
| "model_output": "intro hsf\n unfold HasSubgradientAt at hsf\n unfold Banach_HasSubgradientAt\n constructor\n · exact hsf\n · intro y; specialize hsf y; simp at hsf; exact hsf" |
| }, |
| { |
| "id": 221, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "simp [toDual_apply] at hz ⊢\n exact hz" |
| }, |
| { |
| "id": 222, |
| "match": false, |
| "ground_truth": "linarith", |
| "model_output": "linarith [hz]" |
| }, |
| { |
| "id": 223, |
| "match": false, |
| "ground_truth": "intro h z", |
| "model_output": "intro h\n unfold Banach_HasSubgradientAt at h\n rcases h with ⟨C, hC⟩\n use C; rw [← hC]\n intro y; apply h y" |
| }, |
| { |
| "id": 224, |
| "match": false, |
| "ground_truth": "intro h", |
| "model_output": "intro h\n unfold Banach_HasSubgradientAt at h\n rcases h with ⟨C, hC⟩\n use C; rw [← hC]\n intro y; apply h y" |
| }, |
| { |
| "id": 225, |
| "match": false, |
| "ground_truth": "simp at hz", |
| "model_output": "simp [toDual] at hz" |
| }, |
| { |
| "id": 226, |
| "match": false, |
| "ground_truth": "linarith", |
| "model_output": "linarith [hz]" |
| }, |
| { |
| "id": 227, |
| "match": false, |
| "ground_truth": "ext g", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 228, |
| "match": true, |
| "ground_truth": "rw [h]", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 229, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "have := i.h_eq\n exact ⟨fun x a ↦ ?_, fun x a ↦ ?_⟩\n · simp\n rw [← EReal.coe_add, ← this]\n apply i.hf x a\n · simp [this]" |
| }, |
| { |
| "id": 230, |
| "match": false, |
| "ground_truth": "intro x hx", |
| "model_output": "intro x xs\n have := i.2 x xs\n simp only [ContinuousLinearMap.coe_coe] at this\n rw [← EReal.coe_add, ← EReal.coe_le_coe_iff] at this\n exact this" |
| }, |
| { |
| "id": 231, |
| "match": false, |
| "ground_truth": "intro x", |
| "model_output": "intro x xs\n have := i.2 x xs\n simp only [ContinuousLinearMap.coe_coe] at this\n rw [← EReal.coe_add, ← EReal.coe_le_coe_iff] at this\n exact this" |
| }, |
| { |
| "id": 232, |
| "match": false, |
| "ground_truth": "intro x", |
| "model_output": "intro x _ [H]\n rw [H]\n simp [h]" |
| }, |
| { |
| "id": 233, |
| "match": false, |
| "ground_truth": "ext n", |
| "model_output": "ext n [φ₁_equ fullrank₁ fullrank₂] [φ_equ fullrank₁ fullrank₂]" |
| }, |
| { |
| "id": 234, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "simp [φ]" |
| }, |
| { |
| "id": 235, |
| "match": true, |
| "ground_truth": "rw [this]", |
| "model_output": "rw [this]" |
| }, |
| { |
| "id": 236, |
| "match": false, |
| "ground_truth": "rw [hc2]", |
| "model_output": "rw [hc2, EReal.coe_sub_top a]" |
| }, |
| { |
| "id": 237, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "simp [← EReal.coe_add] at *" |
| }, |
| { |
| "id": 238, |
| "match": false, |
| "ground_truth": "exact this", |
| "model_output": "rw [← EReal.coe_sub, ← EReal.coe_sub] at this ⊢" |
| }, |
| { |
| "id": 239, |
| "match": false, |
| "ground_truth": "linarith", |
| "model_output": "linarith [this]" |
| }, |
| { |
| "id": 240, |
| "match": false, |
| "ground_truth": "· exact hs", |
| "model_output": "· apply hs\n apply hfun" |
| }, |
| { |
| "id": 241, |
| "match": false, |
| "ground_truth": "exact hs", |
| "model_output": "apply hs" |
| }, |
| { |
| "id": 242, |
| "match": false, |
| "ground_truth": "intro x", |
| "model_output": "intro x xs y ys a b ha hb hab\n rw[smul_eq_mul,smul_eq_mul]\n linarith [hfun xs ys ha hb hab]" |
| }, |
| { |
| "id": 243, |
| "match": false, |
| "ground_truth": "dsimp", |
| "model_output": "simp only [smul_eq_mul, mul_eq_mul_left_iff, or_self]" |
| }, |
| { |
| "id": 244, |
| "match": false, |
| "ground_truth": "ring_nf", |
| "model_output": "linarith [mul_comm (m / 2) a, mul_comm (m / 2) b, mul_assoc (m / 2) a b]" |
| }, |
| { |
| "id": 245, |
| "match": false, |
| "ground_truth": "exact hfun", |
| "model_output": "apply hfun" |
| }, |
| { |
| "id": 246, |
| "match": true, |
| "ground_truth": "by rw [h]", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 247, |
| "match": true, |
| "ground_truth": "rw [h]", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 248, |
| "match": false, |
| "ground_truth": "intro h", |
| "model_output": "intro h\n have heq : s = s.convex_support ∩ t.convex_support := by\n apply Subset.antisymm\n · exact fun x hx => by rw [← h x]; exact hx.1\n exact fun x hx => by rw [← h x]; exact ⟨hx.1, hx.2⟩\n have heq' : t = s.convex_support ∩ t.convex_support := by\n apply Subset.antisymm\n · exact fun x hx => by rw [← h x]; exact hx.2\n exact fun x hx => by rw" |
| }, |
| { |
| "id": 249, |
| "match": false, |
| "ground_truth": "ext z", |
| "model_output": "rw [← bi_convex_support_eq_closure hscv hscl, ← bi_convex_support_eq_closure htcv htcl, h]" |
| }, |
| { |
| "id": 250, |
| "match": true, |
| "ground_truth": "intro x _", |
| "model_output": "intro x _" |
| }, |
| { |
| "id": 251, |
| "match": false, |
| "ground_truth": "intro x", |
| "model_output": "intro x _" |
| }, |
| { |
| "id": 252, |
| "match": false, |
| "ground_truth": "right", |
| "model_output": "right\n rcases hs with ⟨x, hx⟩\n use x\n simp\n unfold convex_support\n rw [iSup_apply, iSup_eq_bot]\n intro b\n by_contra h\n simp at h\n have : s = ∅ := by simpa using h\n simp [this] at hx\n exact hx" |
| }, |
| { |
| "id": 253, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "intro y\n simp [convex_indicator]\n intro x _\n rw [inner_zero_left]\n simp" |
| }, |
| { |
| "id": 254, |
| "match": false, |
| "ground_truth": "intro y", |
| "model_output": "intro y\n simp [convex_indicator]\n by_cases h : y ∈ s\n · rw [if_pos h]; simp\n · rw [if_neg h]; simp [nonneg]" |
| }, |
| { |
| "id": 255, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "rfl" |
| }, |
| { |
| "id": 256, |
| "match": true, |
| "ground_truth": "simp at h1", |
| "model_output": "simp at h1" |
| }, |
| { |
| "id": 257, |
| "match": false, |
| "ground_truth": "simp [hx]", |
| "model_output": "simp [dom]" |
| }, |
| { |
| "id": 258, |
| "match": true, |
| "ground_truth": "ext x", |
| "model_output": "ext x" |
| }, |
| { |
| "id": 259, |
| "match": true, |
| "ground_truth": "exact h x", |
| "model_output": "exact h x" |
| }, |
| { |
| "id": 260, |
| "match": false, |
| "ground_truth": "intro x", |
| "model_output": "intro x _ y _" |
| }, |
| { |
| "id": 261, |
| "match": true, |
| "ground_truth": "simp [hp]", |
| "model_output": "simp [hp]" |
| }, |
| { |
| "id": 262, |
| "match": true, |
| "ground_truth": "intro b", |
| "model_output": "intro b" |
| }, |
| { |
| "id": 263, |
| "match": false, |
| "ground_truth": "by rfl", |
| "model_output": "constructor\n · intro h\n exact (isLinear_iff' f).mp h\n intro h\n exact (isLinear_iff' f).mpr h" |
| }, |
| { |
| "id": 264, |
| "match": false, |
| "ground_truth": "rfl", |
| "model_output": "constructor\n · intro h\n exact (isLinear_iff' f).mp h\n intro h\n exact (isLinear_iff' f).mpr h" |
| }, |
| { |
| "id": 265, |
| "match": true, |
| "ground_truth": "rw [h]", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 266, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "simp only [ContinuousLinearMap.coe_comp', comp_apply, LinearEquiv.coe_coe, LinearEquiv.symm_apply_apply]" |
| }, |
| { |
| "id": 267, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "simp only [ContinuousLinearMap.coe_comp', comp_apply, LinearEquiv.coe_coe, LinearEquiv.symm_apply_apply]" |
| }, |
| { |
| "id": 268, |
| "match": false, |
| "ground_truth": "exact hs", |
| "model_output": "· apply hs" |
| }, |
| { |
| "id": 269, |
| "match": false, |
| "ground_truth": "intro x", |
| "model_output": "simp" |
| }, |
| { |
| "id": 270, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "simp only [smul_zero, add_zero, ge_iff_le, zero_le_one, b]" |
| }, |
| { |
| "id": 271, |
| "match": false, |
| "ground_truth": "intro x hx", |
| "model_output": "intro x _" |
| }, |
| { |
| "id": 272, |
| "match": false, |
| "ground_truth": "intro x", |
| "model_output": "intro x _" |
| } |
| ] |
| } |
