{
  "summary": {
    "total_with_gt": 273,
    "matched": 57,
    "accuracy": 0.2087912087912088
  },
  "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  rw [← lipschitzOnWith_univ] at hl ⊢\n  have h1 : LipschitzOnWith l (fun z : E × F => (gradient H) z) := by\n    rw [lipschitzOnWith_iff_norm_sub_le]\n    intro x _ y _\n    apply hl\n    simp\n  have h2 : LipschitzOnWith l (fun z : E × F => (gradient H) z) := by\n    exact lipschitzOnWith_of_fderivWithin"
    },
    {
      "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]"
    },
    {
      "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]"
    },
    {
      "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": true,
      "ground_truth": "exact EReal.coe_le_coe_iff.mp h",
      "model_output": "exact EReal.coe_le_coe_iff.mp 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 (Fin r) = r := by simp\n    rw [← h, ← finrank_eq_card_basis hT.orthogonalBasis_range]\n    simp [hT.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    · exact conjugate_ge_zero univ s.convex_indicator x trivial"
    },
    {
      "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; simp [hS, A.col_apply, add_comm]"
    },
    {
      "id": 29,
      "match": false,
      "ground_truth": "have : (0 : 𝕜) < (1 : 𝕜) := one_pos",
      "model_output": "have : x ∈ univ := by simp"
    },
    {
      "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 (by linarith)) (mul_nonneg hb (by linarith))"
    },
    {
      "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\n    specialize ck k\n    rw [ck]\n    apply div_pos\n    · norm_num\n    apply mul_pos\n    · linarith [hγ]\n    · 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\n    specialize ck k\n    rw [ck]\n    apply div_pos\n    · norm_num\n    apply mul_pos\n    · linarith [hγ]\n    · 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_eq_span_cols, finrank_span_eq_card]"
    },
    {
      "id": 37,
      "match": false,
      "ground_truth": "unfold colSubmodule\n  rw [finrank_span_eq_card]\n  simp [h]",
      "model_output": "rw [colSubmodule_eq_span_cols, finrank_span_eq_card]"
    },
    {
      "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\n      apply ContDiffOn.zero; simp\n      exact hf\n    have h := contDiffOn_succ_iff_derivWithin.1 this\n    apply h.2\n    · apply he\n    · exact hf"
    },
    {
      "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\n      apply ContDiffOn.zero; simp\n      exact hf\n    have h := contDiffOn_succ_iff_derivWithin.1 this\n    apply h.2\n    · apply he\n    · exact hf"
    },
    {
      "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 1 f (Metric.ball x ε) := hf\n    apply HasGradientAt.continuousAt\n    apply ContDiffOn.continuousOn\n    exact this\n    apply 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": "apply epi_closed_of_bot\n    exact 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) (i.hs 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.comp\n    · apply hphi_StrictMono₁\n    · apply StrictMono.add\n      · apply strictMono_nat_of_lt_succ\n        intro n\n        use 1\n        rw [Nat.sub_add_cancel]\n        apply hphi₁_Lipschitz\n      · apply fullrank₁"
    },
    {
      "id": 63,
      "match": false,
      "ground_truth": "suffices 0 ≤ (c - a - b).toEReal by exact this",
      "model_output": "rw [← EReal.coe_sub, ← EReal.coe_sub] at h ⊢\n      apply EReal.coe_le_coe_iff.2\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 (Fin r) = r := by simp\n    rw [← h, ← finrank_eq_card_basis hT.orthogonalBasis_range]\n    simp [hT.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 (Fin r) = r := by simp\n    rw [← h, ← finrank_eq_card_basis hT.orthogonalBasis_range]\n    simp [hT.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, finrank_span_eq_card]"
    },
    {
      "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 [← closure_eq_iff_isClosed.mpr hscl z hz]\n      apply subset_closure; exact (h z).subset_right hz"
    },
    {
      "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] at htcl\n    exact htcl z h2"
    },
    {
      "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 : ∀ x, x ∈ s.convex_support → x ∈ t.convex_support := by\n      intro x hx\n      rw [← h x]\n      exact hx\n    contrapose! h1\n    exact fun x xs ↦ h1 x (by rw [← h x]; exact xs)"
    },
    {
      "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 [convex_indicator, hs]"
    },
    {
      "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[· apply convexOn_const 0; 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      have : intrinsicClosure ℝ (⋂ i, s i) ⊆ intrinsicClosure ℝ (s i) :=\n        intrinsicClosure_mono (iInter_subset_of_subset (by simp) i)\n      apply this hx\n    · exact convex_intrinsicClosure ℝ (⋂ i, h i) hinter\n    · exact convex_intrinsicClosure ℝ (⋂ i, h i) 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' (by assumption) (by assumption) lta alt (by norm_num)"
    },
    {
      "id": 123,
      "match": false,
      "ground_truth": "have lta' : 0 < 1 - a := sub_pos_of_lt alt",
      "model_output": "have : f (a • x + (1 - a) • y) < f x := by\n      apply hf' (by assumption) (by assumption) (by linarith) (by linarith) (by assumption)"
    },
    {
      "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) < f x := by\n      apply hf' (by assumption) (by assumption) lta lta' (by linarith) eqone\n      exact Ne.lt_of_le' neq (IsMinOn_univ_iff.mp hx)"
    },
    {
      "id": 125,
      "match": false,
      "ground_truth": "apply Eq.trans_gt (max_eq_right (hx trivial))",
      "model_output": "apply hf' (by assumption) (by assumption) lta lta' (by ring)"
    },
    {
      "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_left, real_inner_smul_left]\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 linarith)\n        · apply mul_le_mul_of_nonneg_left hyv (by linarith)"
    },
    {
      "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 simp [ha])\n    · apply mul_le_mul_of_nonneg_left hyv (by simp [hb])"
    },
    {
      "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]\nlinarith [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": true,
      "ground_truth": "· simp [StrictMono]",
      "model_output": "· simp [StrictMono]"
    },
    {
      "id": 146,
      "match": false,
      "ground_truth": "simp [StrictMono]",
      "model_output": "· simp [StrictMono]"
    },
    {
      "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 [← bi_convex_conjugate_eq_self hscl, ← bi_convex_conjugate_eq_self htcl, 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 <;> exact h"
    },
    {
      "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": "apply isClosed_eq (by simp) hs"
    },
    {
      "id": 175,
      "match": true,
      "ground_truth": "simp [Function.IsClosed]",
      "model_output": "simp [Function.IsClosed]"
    },
    {
      "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.Ioo_subset this\n         constructor\n         · simp\n         · linarith\n   apply this\n   apply Nat.succ_pos"
    },
    {
      "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      simp\n      exact rfl\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.mpr\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    have h : a = 1 - b := by linarith\n    rw [h]\n    simp\n    specialize hfun xs ys hb ha (by linarith)\n    linarith [hfun]"
    },
    {
      "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    apply convex_support_lt_top 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": "rw [← closure_affineSubspace_eq]"
    },
    {
      "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 x := by\n      apply lt_of_lt_of_le h (hy (by trivial))"
    },
    {
      "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_left, real_inner_smul_left]\n    linarith [huy, hvy]"
    },
    {
      "id": 202,
      "match": false,
      "ground_truth": "intro z z'",
      "model_output": "intro z z' [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] ["
    },
    {
      "id": 203,
      "match": false,
      "ground_truth": "intro z",
      "model_output": "intro z z' [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] [h] [hl] ["
    },
    {
      "id": 204,
      "match": false,
      "ground_truth": "intro k",
      "model_output": "intro k\n    specialize ck k\n    rw [ck]\n    apply div_pos\n    · norm_num\n    apply mul_pos\n    · linarith [hγ]\n    · 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 hb) <;> exact hv.2 a b 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": "apply epi_bot_eq"
    },
    {
      "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 z\n    unfold Banach_HasSubgradientAt at h\n    obtain hz := h z\n    simp; rw [← ContinuousLinearMap.map_sub]; linarith"
    },
    {
      "id": 224,
      "match": false,
      "ground_truth": "intro h",
      "model_output": "intro h z\n    unfold Banach_HasSubgradientAt at h\n    obtain hz := h z\n    simp; rw [← ContinuousLinearMap.map_sub]; linarith"
    },
    {
      "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]; exact 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": true,
      "ground_truth": "linarith",
      "model_output": "linarith"
    },
    {
      "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    have h : a = 1 - b := by linarith\n    rw [h]\n    simp\n    specialize hfun xs ys hb ha (by linarith)\n    linarith [hfun]"
    },
    {
      "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_assoc a b (m / 2), mul_assoc a (m / 2) 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 := by\n      apply subset_antisymm\n      · exact convex_support_le\n      rw [epi_eq_closure_epi_linear]\n      apply isClosed_closure\n    have teq : t = t.convex_support := by\n      apply subset_antisymm\n      · exact convex_support_le\n      rw [epi_eq_closure_epi_linear]\n      apply isClosed_closure\n    rw [heq, teq]\n    apply h"
    },
    {
      "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;rcases hs with ⟨x ,hx⟩\n        use x\n        rw[convex_support]\n        simp\n        exact Ne.lt_top' (id (Ne.symm 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, hx]"
    },
    {
      "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' h).mp\n  intro h\n  exact (IsLinear_iff' h).mpr"
    },
    {
      "id": 264,
      "match": false,
      "ground_truth": "rfl",
      "model_output": "constructor\n  · intro h\n    exact (IsLinear_iff' h).mp\n  intro h\n  exact (IsLinear_iff' h).mpr"
    },
    {
      "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 _"
    }
  ]
}