{
  "summary": {
    "total_with_gt": 273,
    "matched": 55,
    "accuracy": 0.20146520146520147
  },
  "details": [
    {
      "id": 0,
      "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": 1,
      "match": false,
      "ground_truth": "exact mul_le_mul_of_nonneg_left hyu ha",
      "model_output": "apply mul_le_mul_of_nonneg_left hyu (le_of_lt 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": "apply lipschitz_of_norm_gradient_le\n  rw [lipschitzWith_iff_norm_sub_le]\n  intro z₁ z₂\n  rw [← Prod.fst_sub, ← Prod.snd_sub, ← Prod.fst_sub, ← Prod.snd_sub]\n  apply LipschitzWith.prodMk (lipschitzWith_iff_norm_grad_le.mpr hl) hl\n  apply lipschitz_of_norm_gradient_le\n  rw [lipschitzWith_iff_norm_grad_le]\n  intro z₁ z₂\n  apply LipschitzWith.prodMk (lipschitzWith_iff_norm"
    },
    {
      "id": 5,
      "match": false,
      "ground_truth": "by rw [grad_snd_eq h, grad_snd_eq h]",
      "model_output": "rw [grad_eq_block_grad, grad_eq_block_grad]"
    },
    {
      "id": 6,
      "match": false,
      "ground_truth": "rw [grad_snd_eq h, grad_snd_eq h]",
      "model_output": "rw [grad_eq_block_grad, grad_eq_block_grad]"
    },
    {
      "id": 7,
      "match": true,
      "ground_truth": "apply prox_well_define f hc lbdf x",
      "model_output": "apply prox_well_define f hc lbdf x"
    },
    {
      "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_real, ← mul_two, ← sq, ← mul_two, ← sq]\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": "apply Orthonormal.mk\n  · intro i hi j hj hij\n    simp [restrict_apply, hi, hj, hij]\n    exact hv.1 i j hij\n  · intro i hi\n    simp [restrict_apply, hi]\n    exact hv.2 i hi"
    },
    {
      "id": 11,
      "match": false,
      "ground_truth": "by_cases hsf : ProperFunction s f",
      "model_output": "apply closure_isClosed"
    },
    {
      "id": 12,
      "match": false,
      "ground_truth": "· exact closure_is_closed_of_proper f hs",
      "model_output": "· apply closure_is_closed\n  exact hs"
    },
    {
      "id": 13,
      "match": false,
      "ground_truth": "exact closure_is_closed_of_proper f hs",
      "model_output": "apply closure_isClosed_of_closed\n  exact hs"
    },
    {
      "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.hx 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 h := hT.finrank_range_add_finrank_ker\n  have : finrank 𝕜 (LinearMap.ker T) = 0 := by\n    rw [← ker_eq_bot_of_range_eq_top hT.range_eq_top]\n    exact finrank_bot 𝕜 V\n  simp [this] at h\n  exact h"
    },
    {
      "id": 19,
      "match": false,
      "ground_truth": "apply iff_iff_implies_and_implies.mpr",
      "model_output": "rw [← closure_convex_support_eq hscl hscv, ← closure_convex_support_eq htcl htcv, h]"
    },
    {
      "id": 20,
      "match": false,
      "ground_truth": "apply convex_conjugate_ge_bot_univ'",
      "model_output": "apply convex_conjugate_ge_bot_univ s.convex_indicator x"
    },
    {
      "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": "apply LinearProgram_convex"
    },
    {
      "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 : (0 : E) ∈ S := h_zero_mem\n  have h2 := hS (0 : E)\n  simp at h2\n  exact h2.1"
    },
    {
      "id": 29,
      "match": false,
      "ground_truth": "have : (0 : 𝕜) < (1 : 𝕜) := one_pos",
      "model_output": "apply (neq_iff_lt_or_gt).mp neq"
    },
    {
      "id": 30,
      "match": false,
      "ground_truth": "rw [inner_add_left, real_inner_smul_left, real_inner_smul_left]",
      "model_output": "rw [inner_add_left, ← sub_sub, sub_le_iff_le_add]"
    },
    {
      "id": 31,
      "match": false,
      "ground_truth": "rw [← one_mul (-ε * ‖y - x‖), ← hab, add_mul]",
      "model_output": "apply add_nonneg (mul_nonneg ha (neg_mul_le_iff_mul_le_neg.mpr hyu))\n        (mul_nonneg hb (neg_mul_le_iff_mul_le_neg.mpr hyv))"
    },
    {
      "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    rw [ck k]\n    apply div_pos\n    · linarith\n    · apply mul_pos\n      · linarith\n      · linarith [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    rw [ck k]\n    apply div_pos\n    · linarith\n    · apply mul_pos\n      · linarith\n      · linarith [alg.lpos]"
    },
    {
      "id": 35,
      "match": true,
      "ground_truth": "by\n    apply prox_well_define f hc lbdf x",
      "model_output": "apply prox_well_define f hc lbdf x"
    },
    {
      "id": 36,
      "match": false,
      "ground_truth": "by\n  unfold colSubmodule\n  rw [finrank_span_eq_card]\n  simp [h]",
      "model_output": "apply finrank_colSubmodule_eq_card"
    },
    {
      "id": 37,
      "match": false,
      "ground_truth": "unfold colSubmodule\n  rw [finrank_span_eq_card]\n  simp [h]",
      "model_output": "apply finrank_colSubmodule_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 : 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 [mem_orthogonal'] at hx₂\n[ASSISTANT]\nhave : T x₂ = 0 := hker hx₂\n[ASSISTANT]\nrw [this, add_zero]"
    },
    {
      "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": "apply Orthonormal.mk\n  · intro i hi j hj hij\n    simp [restrict_apply, hi, hj, hij]\n    exact hv.1 i j hij\n  · intro i hi\n    simp [restrict_apply, hi]\n    exact hv.2 i hi"
    },
    {
      "id": 43,
      "match": false,
      "ground_truth": "simp [Orthonormal, Pairwise] at *\n  aesop",
      "model_output": "apply Orthonormal.mk\n  · intro i hi j hj hij\n    simp [restrict_apply, hi, hj, hij]\n    exact hv.1 i j hij\n  · intro i hi\n    simp [restrict_apply, hi]\n    exact hv.2 i hi"
    },
    {
      "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": "apply HasGradientAt.continuousAt\n  apply HasGradientAt_of_HasFDerivAt\n  apply HasFDerivAt_of_C1\n  apply ContDiffOn.contDiffWithinAt\n  apply hf\n  apply he\n  apply ball_mem_nhds x\n  apply mem_ball_self he"
    },
    {
      "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": "apply HasGradientAt.continuousAt\n  apply HasGradientAt_of_HasFDerivAt\n  apply HasFDerivAt_of_C1\n  apply ContDiffOn.contDiffWithinAt\n  apply hf\n  apply he\n  apply ball_mem_nhds x\n  apply mem_ball_self he"
    },
    {
      "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": "apply HasGradientAt.continuousAt\n  apply HasGradientAt.continuousAt\n  apply ContDiffOn.continuousOn\n  apply hf\n  apply he\n  apply s"
    },
    {
      "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": "apply HasGradientAt.continuousAt h"
    },
    {
      "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 closure_univ_eq_top"
    },
    {
      "id": 50,
      "match": false,
      "ground_truth": "simp [Function.closure]\n  intro h; exfalso\n  obtain h1 := h.2; simp at h1",
      "model_output": "apply closure_univ_eq_top"
    },
    {
      "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": "· rw [if_pos hf]\n    exact isClosed_univ_iff.mpr trivial"
    },
    {
      "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 isClosed_epi_bot_of_isClosed hs"
    },
    {
      "id": 54,
      "match": false,
      "ground_truth": "have h' : Epi ⊤ s = ∅ := by\n    ext ⟨x, _⟩; simp [Function.Epi]",
      "model_output": "apply isClosed_epi_of_closed_of_exist_top hs (not_proper_of_bot_not_in_image f hs hf)"
    },
    {
      "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 Banach_HasSubgradientAt HasSubgradientAt\n    specialize h z\n    rw[← inner_apply_apply (toDual ℝ E g) z]\n    exact le_of_eq (congrArg EReal.toEReal h)\n  · apply Banach_HasSubgradientAt_to_HasSubgradientAt"
    },
    {
      "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 Banach_HasSubgradientAt HasSubgradientAt\n    specialize h z\n    rw[← inner_apply_apply (toDual ℝ E g) z]\n    exact le_of_eq (congrArg EReal.toEReal h)\n  · apply Banach_HasSubgradientAt_to_HasSubgradientAt"
    },
    {
      "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 Banach_HasSubgradientAt\n    rw[hasSubgradientAt_iff_isLittleO] at h\n    apply Asymptotics.isLittleO_iff.mpr\n    intro ε εpos\n    specialize h (ε / 2) (half_pos εpos)\n    rcases h with ⟨δ, δpos, hδ⟩\n    use δ\n    constructor\n    · linarith\n    · intro y hy\n      rw[← sub_add, sub_self, sub_zero] at hy\n      rw[← sub_add, sub_self, sub_zero]\n      apply"
    },
    {
      "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 h z\n  rw [hasSubgradientAt_iff_le] at h\n  apply h z\n  simp\n  rw [← ContinuousLinearMap.map_sub]; simp\n  apply real_inner_le_norm"
    },
    {
      "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": "· exact i.hg"
    },
    {
      "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    rw[← EReal.coe_add, ← inner_add_left, ← inner_add_left, ← inner_add_left]\n    apply le_trans\n    · apply i.1\n    · apply i.2 x xs"
    },
    {
      "id": 61,
      "match": false,
      "ground_truth": "apply Filter.tendsto_iff_seq_tendsto.1 Seq_converge_zero₃",
      "model_output": "apply tendsto_zero_iff_norm_tendsto_zero.2\n   apply tendsto_zero_iff_norm_tendsto_zero.2\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\n         (φ₁ (n + 1)) ≥ n + 1 := StrictMono.id_le (hphi_StrictMono fullrank₁ fullrank₂) (n + 1)\n         _ > n := by linarith\n   apply this\n   apply Nat.cast_lt.mpr\n   apply this\n   apply nplusone_le_phi"
    },
    {
      "id": 63,
      "match": false,
      "ground_truth": "suffices 0 ≤ (c - a - b).toEReal by exact this",
      "model_output": "rw [← EReal.coe_add, ← EReal.coe_sub] at h"
    },
    {
      "id": 64,
      "match": false,
      "ground_truth": "have : a + b ≤ c := by exact EReal.coe_le_coe_iff.mp h",
      "model_output": "apply EReal.coe_nonneg.mpr"
    },
    {
      "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": "apply StrongConvexOn_of_ConvexOn hs\n  intro x xs y ys a b ha hb hab\n  specialize hfun xs ys ha hb hab\n  rw [← sub_le_sub_iff, sub_add, sub_add_cancel, sub_le_sub_iff] at hfun\n  linarith [hfun]"
    },
    {
      "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": "apply StrongConvexOn_of_ConvexOn hs\n  intro x xs y ys a b ha hb hab\n  specialize hfun xs ys ha hb hab\n  rw [← sub_le_sub_iff, sub_add, sub_add_cancel, sub_le_sub_iff] at hfun\n  linarith [hfun]"
    },
    {
      "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 h := hT.finrank_range_add_finrank_ker\n  have : finrank 𝕜 (LinearMap.ker T) = 0 := by\n    rw [← ker_eq_bot_of_range_eq_top hT.range_eq_top]\n    exact finrank_bot 𝕜 V\n  simp [this] at h\n  exact h"
    },
    {
      "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 h := hT.finrank_range_add_finrank_ker\n  have : finrank 𝕜 (LinearMap.ker T) = 0 := by\n    rw [← ker_eq_bot_of_range_eq_top hT.range_eq_top]\n    exact finrank_bot 𝕜 V\n  simp [this] at h\n  exact h"
    },
    {
      "id": 71,
      "match": false,
      "ground_truth": "simp [finrank_span_eq_card (b:=hT.w) hT.w_orthonormal.linearIndependent]",
      "model_output": "rw [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 zin\n      have : s.convex_support z = 0 := by\n        rw [← h z]; exact convex_support_of_mem zin\n      have : t.convex_support z = 0 := by rw [← h]; exact this\n      apply convex_support_eq_zero htcl hscv htcv; exact this\n      apply convex_conjugate_of_zero_support\n      apply convex_convex_support htcl hscv\n      exact zin"
    },
    {
      "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_eq_empty_iff]; exact hzs\n    have h2 := congrFun h hzs\n    rw [h1] at h2\n    exact (Set.convex_indicator_empty t).convex_support_eq_closure h2 htcl\n    apply subset_closure\n    trivial"
    },
    {
      "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    rw [← h z]\n    apply convex_support_subset hscl htcl zt"
    },
    {
      "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 0\n  simp [convex_conjugate]\n  rw[Set.mem_setOf_eq]\n  exact ⟨hx, by simp⟩"
    },
    {
      "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  · apply hs.convex_indicator_nonempty\n  · simp"
    },
    {
      "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_of_nonempty_convex hs convex_indicator_convex\n  apply convex_univ\n  apply convex_indicator_convex_convex_conjugate_nonempty hs convex_univ\n  apply convex_univ"
    },
    {
      "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]"
    },
    {
      "id": 84,
      "match": false,
      "ground_truth": "simp only [sub_eq_add_neg, gradient_add_const]",
      "model_output": "rw [gradient_sub_const c]"
    },
    {
      "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": "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_mulVec, sub_mulVec, sub_add, sub_add, sub_add, sub_add, sub_add]"
    },
    {
      "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_mulVec, ← sub_mulVec, ← sub_mulVec, ← sub_mulVec]"
    },
    {
      "id": 96,
      "match": false,
      "ground_truth": "have h_zero_mem : (0 : E) ∈ S := S.zero_mem",
      "model_output": "have h1 : ∀ i, (hS (B (Fin.natAdd n i))).1 = (hS (B (Fin.natAdd n i))).2 := by\n    intro i\n    simp [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 [Matrix.mulVec_zero] at h_zero_eq"
    },
    {
      "id": 100,
      "match": false,
      "ground_truth": "by\n  unfold sup_value\n  unfold FeasSet FeasPoint\n  simp [hp]",
      "model_output": "unfold sup_value\n  simp [hp]"
    },
    {
      "id": 101,
      "match": false,
      "ground_truth": "unfold sup_value\n  unfold FeasSet FeasPoint\n  simp [hp]",
      "model_output": "unfold sup_value\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, ← ContinuousLinearMap.adjoint_comp_apply]"
    },
    {
      "id": 108,
      "match": false,
      "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp",
      "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply, ← ContinuousLinearMap.adjoint_comp_apply]"
    },
    {
      "id": 109,
      "match": false,
      "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]",
      "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply, ← ContinuousLinearMap.adjoint_comp_apply]"
    },
    {
      "id": 110,
      "match": false,
      "ground_truth": "by rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp",
      "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply, ← ContinuousLinearMap.adjoint_comp_apply]"
    },
    {
      "id": 111,
      "match": false,
      "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp",
      "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply, ← ContinuousLinearMap.adjoint_comp_apply]"
    },
    {
      "id": 112,
      "match": false,
      "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]",
      "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply, ← ContinuousLinearMap.adjoint_comp_apply]"
    },
    {
      "id": 113,
      "match": false,
      "ground_truth": "by rw [← trace_transpose, ← trace_transpose_mul, transpose_mul]",
      "model_output": "rw [Matrix.mul_comm, Matrix.trace_mul_comm]"
    },
    {
      "id": 114,
      "match": false,
      "ground_truth": "rw [← trace_transpose, ← trace_transpose_mul, transpose_mul]",
      "model_output": "rw [Matrix.mul_comm, Matrix.trace_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 convex_indicator_convex hs"
    },
    {
      "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.subset this (fun x a ↦ a)"
    },
    {
      "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": "apply Subset.antisymm\n  · exact iIntersection_intrinsicClosure_sub_intrinsicClosure_iIntersection\n  exact intrinsicClosure_iIntersection_sub_iIntersection_intrinsicClosure h hinter"
    },
    {
      "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": "apply Subset.antisymm\n  · exact iIntersection_intrinsicClosure_sub_intrinsicClosure_iIntersection\n  exact intrinsicClosure_iIntersection_sub_iIntersection_intrinsicClosure h hinter"
    },
    {
      "id": 119,
      "match": false,
      "ground_truth": "· exact iIntersection_closure_sub_closure_iIntersection'' h hinter",
      "model_output": "· apply iIntersection_intrinsicClosure_sub_intrinsicClosure_iIntersection\n    · apply 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] at hx\n    exact fun i ↦ intrinsicClosure_mono (hx i)\n  · exact convex_iInter h\n  · exact hinter"
    },
    {
      "id": 122,
      "match": false,
      "ground_truth": "have eqone : a + (1 - a) = 1 := add_sub_cancel a 1",
      "model_output": "have : a * (f x - f y) < 0 := mul_neg_of_pos_of_neg lta (sub_neg_of_lt hy (hx trivial))"
    },
    {
      "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' (hx trivial) (hy trivial) neq lta (sub_pos_of_lt alt) eqone\n      apply lt_of_le_of_lt (le_of_lt lta) (le_of_lt alt)"
    },
    {
      "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 : a + (1 - a) > 0 := by linarith"
    },
    {
      "id": 125,
      "match": false,
      "ground_truth": "apply Eq.trans_gt (max_eq_right (hx trivial))",
      "model_output": "apply hf' (x := x) (y := y) neq lta lta' eqone"
    },
    {
      "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 [eventually_and]\n  rw [eventually_nhds_iff_ball] at huy hvy\n  intro y hy\n  specialize huy y hy\n  specialize hvy y hy\n  rw [inner_add_left, ← sub_sub, sub_le_sub_iff, ← sub_sub, sub_le_sub_iff] at huy hvy\n  rw [← sub_sub, sub_le_sub_iff, ← sub_sub, sub_le_sub_iff]\n  rw [← sub_sub,"
    },
    {
      "id": 127,
      "match": false,
      "ground_truth": "rw [ge_iff_le] at *",
      "model_output": "apply add_le_add (mul_le_mul_of_nonneg_left hyu (le_of_lt ha))\n        (mul_le_mul_of_nonneg_left hyv (le_of_lt hb))"
    },
    {
      "id": 128,
      "match": false,
      "ground_truth": "apply add_le_add",
      "model_output": "apply add_le_add (mul_le_mul_of_nonneg_left hyu ha) (mul_le_mul_of_nonneg_left hyv hb)"
    },
    {
      "id": 129,
      "match": false,
      "ground_truth": "apply div_pos",
      "model_output": "apply div_pos zero_lt_one (mul_pos _ alg.lpos)\n[· linarith]"
    },
    {
      "id": 130,
      "match": false,
      "ground_truth": "apply mul_pos",
      "model_output": "apply mul_pos _ alg.lpos\n[apply 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": true,
      "ground_truth": "· apply alg.lpos",
      "model_output": "apply alg.lpos"
    },
    {
      "id": 134,
      "match": true,
      "ground_truth": "apply alg.lpos",
      "model_output": "apply alg.lpos"
    },
    {
      "id": 135,
      "match": false,
      "ground_truth": "rw [mem_nhds_iff]",
      "model_output": "apply Metric.ball_mem_nhds\n    exact he"
    },
    {
      "id": 136,
      "match": false,
      "ground_truth": "simp [Function.Epi]",
      "model_output": "simp [Epi]"
    },
    {
      "id": 137,
      "match": true,
      "ground_truth": "simp [Function.Epi]",
      "model_output": "simp [Function.Epi]"
    },
    {
      "id": 138,
      "match": true,
      "ground_truth": "exact isClosed_empty",
      "model_output": "exact isClosed_empty"
    },
    {
      "id": 139,
      "match": false,
      "ground_truth": "constructor",
      "model_output": "constructor\n  · intro h z\n    unfold Banach_HasSubgradientAt HasSubgradientAt\n    specialize h z\n    rw[← inner_apply_apply (toDual ℝ E g) z]\n    exact le_of_eq (congrArg EReal.toEReal h)\n  · apply Banach_HasSubgradientAt_to_HasSubgradientAt"
    },
    {
      "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": true,
      "ground_truth": "constructor",
      "model_output": "constructor"
    },
    {
      "id": 143,
      "match": false,
      "ground_truth": "intro x _hx",
      "model_output": "intro x _ [H]\n  rw [← H]\n  rfl"
    },
    {
      "id": 144,
      "match": false,
      "ground_truth": "simp at this",
      "model_output": "rw [← this]"
    },
    {
      "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.top_sub_coe, EReal.top_sub_coe]"
    },
    {
      "id": 149,
      "match": false,
      "ground_truth": "rw [hc2]; simp",
      "model_output": "rw [hc2, EReal.top_sub_coe, EReal.coe_sub_top]"
    },
    {
      "id": 150,
      "match": false,
      "ground_truth": "by exact this",
      "model_output": "exact EReal.coe_le_coe_iff.2 this"
    },
    {
      "id": 151,
      "match": false,
      "ground_truth": "constructor",
      "model_output": "apply StrongConvexOn_of_ConvexOn hs\n  intro x xs y ys a b ha hb hab\n  specialize hfun xs ys ha hb hab\n  rw [← sub_le_sub_iff, sub_add, sub_add_cancel, sub_le_sub_iff] at hfun\n  linarith [hfun]"
    },
    {
      "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"
    },
    {
      "id": 155,
      "match": true,
      "ground_truth": "constructor",
      "model_output": "constructor"
    },
    {
      "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 hs 0"
    },
    {
      "id": 162,
      "match": false,
      "ground_truth": "constructor",
      "model_output": "constructor <;> simp [Convex] <;> tauto"
    },
    {
      "id": 163,
      "match": false,
      "ground_truth": "dsimp[EqOn]",
      "model_output": "intro x hx\n  rw[← convex_indicator_eq_zero hs] at hx\n  rw[← hx]\n  simp only [Pi.zero_apply, zero_eq_iff]"
    },
    {
      "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": "apply convex_set_of_norm_le_one"
    },
    {
      "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, map_add, norm_add_sq_real]"
    },
    {
      "id": 170,
      "match": false,
      "ground_truth": "apply ContDiffOn.contDiffAt hf",
      "model_output": "apply ContDiffOn.contDiffAt hf (Metric.ball x ε) (Metric.mem_ball_self he)"
    },
    {
      "id": 171,
      "match": false,
      "ground_truth": "simp [Function.closure]",
      "model_output": "apply closure_univ_eq_top"
    },
    {
      "id": 172,
      "match": false,
      "ground_truth": "simp [Function.closure, hsf]",
      "model_output": "simp [Function.IsClosed, hsf]"
    },
    {
      "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]\n    push_neg at hsf\n    simp [h]\n    exact isClosed_univ"
    },
    {
      "id": 174,
      "match": false,
      "ground_truth": "simp [Function.IsClosed]",
      "model_output": "apply isClosed_of_subset_closure\n  · exact fun x hx ↦ by simp\n  rcases hf with ⟨x, hx, hfx⟩\n  exact hfx ▸ hx"
    },
    {
      "id": 175,
      "match": false,
      "ground_truth": "simp [Function.IsClosed]",
      "model_output": "simp [Function.IsClosed, hs]"
    },
    {
      "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\n      apply StrictMono.strictMonoOn\n      apply hphi_StrictMono fullrank₁ fullrank₂\n      apply lt_add_one\n   apply this\n   apply Nat.cast_lt.mpr\n   apply this\n   apply nlinarith\n   apply Nat.le_add_left\n   apply le_add_of_le_add_left\n   apply le_add_of_le_add_right\n   apply le_add_of_le_add_left\n   apply le_add"
    },
    {
      "id": 177,
      "match": false,
      "ground_truth": "apply StrictMono.comp",
      "model_output": "apply StrictMono.comp\n    · apply hphi_StrictMono\n    · apply StrictMono.add\n      · apply strictMono_id\n      · apply strictMono_nat_of_lt_succ\n        intro n\n        simp only [Nat.one_add, Nat.add_one, Nat.lt_succ_self]"
    },
    {
      "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 < ⊤ := LT.lt.ne_top ha2"
    },
    {
      "id": 181,
      "match": false,
      "ground_truth": "lift c to ℝ using ⟨hc2, hc1⟩",
      "model_output": "lift c to ℝ using ⟨hc2, LT.lt.ne_bot hc1⟩\n  rw [← EReal.coe_add, ← EReal.coe_sub, EReal.coe_le_coe_iff]\n  rw [← EReal.coe_sub, EReal.coe_sub_le_coe_sub_iff]\n  exact h"
    },
    {
      "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  specialize hfun xs ys a ha b hb hab\n  simp at hfun\n  exact hfun"
    },
    {
      "id": 183,
      "match": false,
      "ground_truth": "· exact fun h => (by rw [h])",
      "model_output": "· intro eq; rw [eq]"
    },
    {
      "id": 184,
      "match": false,
      "ground_truth": "exact fun h => (by rw [h])",
      "model_output": "intro hst\n  rw [hst]"
    },
    {
      "id": 185,
      "match": false,
      "ground_truth": "simp [convex_support]",
      "model_output": "apply convex_support_nonempty hs x trivial"
    },
    {
      "id": 186,
      "match": false,
      "ground_truth": "simp [convex_support]",
      "model_output": "rcases hs with ⟨x, hx⟩\n  use x, trivial\n  rw [convex_support]\n  simp\n  exact hx.2"
    },
    {
      "id": 187,
      "match": false,
      "ground_truth": "simp [convex_indicator]",
      "model_output": "by_cases hy : y ∈ s\n  · rw [convex_indicator_of_mem hy]; simp\n  simp [convex_indicator, hy]"
    },
    {
      "id": 188,
      "match": false,
      "ground_truth": "split_ifs with hy <;> simp",
      "model_output": "by_cases h : y ∈ s\n    · simp [h]\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.1"
    },
    {
      "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_sum"
    },
    {
      "id": 192,
      "match": false,
      "ground_truth": "exact closed_conic σ b",
      "model_output": "apply cone_is_closed"
    },
    {
      "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_of_mem hx]"
    },
    {
      "id": 195,
      "match": false,
      "ground_truth": "apply ConvexOn.congr this eq",
      "model_output": "apply EqOn.convexOn eq this"
    },
    {
      "id": 196,
      "match": false,
      "ground_truth": "apply Subset.antisymm",
      "model_output": "apply Subset.antisymm\n  · exact iIntersection_intrinsicClosure_sub_intrinsicClosure_iIntersection\n  exact intrinsicClosure_iIntersection_sub_iIntersection_intrinsicClosure h hinter"
    },
    {
      "id": 197,
      "match": false,
      "ground_truth": "apply hf' neq lta lta' eqone",
      "model_output": "apply lt_max_iff.2"
    },
    {
      "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 (mem_univ (a • x + (1 - a) • y)))"
    },
    {
      "id": 199,
      "match": false,
      "ground_truth": "apply not_le_of_gt h hy",
      "model_output": "linarith [h, hy]"
    },
    {
      "id": 200,
      "match": false,
      "ground_truth": "intro u",
      "model_output": "apply convex_set_of_norm_le_one"
    },
    {
      "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 [eventually_and]\n  rw [eventually_nhds_iff_ball] at huy hvy\n  intro y hy\n  specialize huy y hy\n  specialize hvy y hy\n  rw [inner_add_left, ← sub_sub, sub_le_sub_iff, ← sub_sub, sub_le_sub_iff] at huy hvy\n  rw [← sub_sub, sub_le_sub_iff, ← sub_sub, sub_le_sub_iff]\n  rw [← sub_sub,"
    },
    {
      "id": 202,
      "match": false,
      "ground_truth": "intro z z'",
      "model_output": "intro x y\n    have h1 := hl x y\n    rw [← Prod.fst_sub, ← Prod.snd_sub, ← Prod.fst_sub, ← Prod.snd_sub] at h1\n    apply le_trans (fst_norm_le_prod_L2 _) at h1\n    apply le_trans (snd_norm_le_prod_L2 _) at h1\n    apply le_trans (fst_norm_le_prod_L2 _) at h1\n    apply le_trans (snd_norm_le_prod_L2 _) at h1\n    apply le_trans (fst_norm_le_prod_L2 _) at h1\n    apply le_trans"
    },
    {
      "id": 203,
      "match": false,
      "ground_truth": "intro z",
      "model_output": "intro x y\n    have h1 := hl x y\n    rw [← Prod.fst_sub, ← Prod.snd_sub, ← Prod.fst_sub, ← Prod.snd_sub] at h1\n    apply le_trans (fst_norm_le_prod_L2 _) at h1\n    apply le_trans (snd_norm_le_prod_L2 _) at h1\n    apply le_trans (fst_norm_le_prod_L2 _) at h1\n    apply le_trans (snd_norm_le_prod_L2 _) at h1\n    apply le_trans (fst_norm_le_prod_L2 _) at h1\n    apply le_trans"
    },
    {
      "id": 204,
      "match": false,
      "ground_truth": "intro k",
      "model_output": "intro k\n    rw [ck k]\n    apply div_pos\n    · linarith\n    · apply mul_pos\n      · linarith\n      · linarith [alg.lpos]"
    },
    {
      "id": 205,
      "match": true,
      "ground_truth": "rw [ck]",
      "model_output": "rw [ck]"
    },
    {
      "id": 206,
      "match": false,
      "ground_truth": "· norm_num",
      "model_output": "· linarith\n        apply mul_pos\n        · linarith\n        · simp; linarith [alg.lpos]"
    },
    {
      "id": 207,
      "match": false,
      "ground_truth": "norm_num",
      "model_output": "linarith"
    },
    {
      "id": 208,
      "match": false,
      "ground_truth": "simp [h]",
      "model_output": "apply linearIndependent_iff.mpr\n  intro x a\n  simp [colVec, Matrix.mulVec, dotProduct]\n  rw [← Matrix.mulVec, ← Matrix.mulVec, h] at a\n  exact a.1"
    },
    {
      "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  exact univ_lowersemicontinoushull_top_eq_top h"
    },
    {
      "id": 211,
      "match": false,
      "ground_truth": "exfalso",
      "model_output": "apply le_antisymm_iff.mpr\n    constructor\n    · exact lowerSemicontinuousHull_le_top\n    exact fun a ↦ by\n      rw [← closure_eq_top_iff, ← closure_epi_eq_top_iff_lowerSemicontinuousHull_eq_top]\n      apply closure_epi_eq_top_of_properFunction h"
    },
    {
      "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": "simp [Function.Epi]"
    },
    {
      "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 [ProperFunction] at hsf"
    },
    {
      "id": 218,
      "match": true,
      "ground_truth": "rw [h']",
      "model_output": "rw [h']"
    },
    {
      "id": 219,
      "match": false,
      "ground_truth": "intro h z",
      "model_output": "intro h z\n  rw [hasSubgradientAt_iff_le] at h\n  apply h z\n  simp\n  rw [← ContinuousLinearMap.map_sub]; simp\n  apply real_inner_le_norm"
    },
    {
      "id": 220,
      "match": false,
      "ground_truth": "intro h",
      "model_output": "intro h z\n  rw [hasSubgradientAt_iff_le] at h\n  apply h z\n  simp\n  rw [← ContinuousLinearMap.map_sub]; simp\n  apply real_inner_le_norm"
    },
    {
      "id": 221,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp [toDual_apply]"
    },
    {
      "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 hz\n    rw [← mem_gradientSet_iff_hasSubgradientAt, ← Banach_gradientSet_iff] at h\n    rcases h with ⟨_, h⟩\n    exact h z hz"
    },
    {
      "id": 224,
      "match": false,
      "ground_truth": "intro h",
      "model_output": "intro h z hz\n    rw [← mem_gradientSet_iff_hasSubgradientAt, ← Banach_gradientSet_iff] at h\n    rcases h with ⟨_, h⟩\n    exact h z hz"
    },
    {
      "id": 225,
      "match": false,
      "ground_truth": "simp at hz",
      "model_output": "simp [toDual_apply] 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": "apply And.intro"
    },
    {
      "id": 230,
      "match": false,
      "ground_truth": "intro x hx",
      "model_output": "intro x xs\n    rw[← EReal.coe_add, ← inner_add_left, ← inner_add_left, ← inner_add_left]\n    apply le_trans\n    · apply i.1\n    · apply i.2 x xs"
    },
    {
      "id": 231,
      "match": false,
      "ground_truth": "intro x",
      "model_output": "intro x xs\n    rw[← EReal.coe_add, ← inner_add_left, ← inner_add_left, ← inner_add_left]\n    apply le_trans\n    · apply i.1\n    · apply i.2 x xs"
    },
    {
      "id": 232,
      "match": false,
      "ground_truth": "intro x",
      "model_output": "intro x _ [H]\n  rw [← H]\n  rfl"
    },
    {
      "id": 233,
      "match": false,
      "ground_truth": "ext n",
      "model_output": "ext n\n   congr\n   apply Nat.sub_one_add_one_eq_of_pos (φ_is_positive (n + 1))"
    },
    {
      "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.top_sub_coe, EReal.coe_sub_top]"
    },
    {
      "id": 237,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp [← EReal.coe_add] at *"
    },
    {
      "id": 238,
      "match": false,
      "ground_truth": "exact this",
      "model_output": "exact EReal.coe_le_coe_iff.2 this"
    },
    {
      "id": 239,
      "match": false,
      "ground_truth": "linarith",
      "model_output": "linarith [this]"
    },
    {
      "id": 240,
      "match": false,
      "ground_truth": "· exact hs",
      "model_output": "apply hs"
    },
    {
      "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  specialize hfun xs ys a ha b hb hab\n  simp at hfun\n  exact hfun"
    },
    {
      "id": 243,
      "match": false,
      "ground_truth": "dsimp",
      "model_output": "simp [← sub_nonneg] at hfun ⊢ [← mul_nonneg] at hab ⊢ [← ha] at hab ⊢ [← hb] at hab ⊢\n  rw [← hab, sub_mul, sub_le_sub_iff_left, mul_comm, mul_comm (a * b)]\n  apply hfun"
    },
    {
      "id": 244,
      "match": false,
      "ground_truth": "ring_nf",
      "model_output": "linarith"
    },
    {
      "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 h1 := fun x a ↦ (convex_support_eq_closure h x a).1\n  have h2 := fun x a ↦ (convex_support_eq_closure h x a).2\n  apply Subset.antisymm\n  · intro x hx\n    rw [← closure_convex_support_eq hscl hscv, ← closure_convex_support_eq htcl htcv, h]\n    exact h1 x hx\n  · intro x hx\n    rw [← closure_convex_support_eq hscl hscv, ← closure_convex_support_eq htcl htcv"
    },
    {
      "id": 249,
      "match": false,
      "ground_truth": "ext z",
      "model_output": "rw [← closure_convex_support_eq s hscl, ← closure_convex_support_eq t 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, trivial\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\n  exact convex_indicator_ge_zero hs y"
    },
    {
      "id": 254,
      "match": false,
      "ground_truth": "intro y",
      "model_output": "intro y\n  by_cases hy : y ∈ s\n  · rw [convex_indicator_of_mem hy]; simp\n  · simp [convex_indicator, hy]"
    },
    {
      "id": 255,
      "match": true,
      "ground_truth": "simp",
      "model_output": "simp"
    },
    {
      "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": false,
      "ground_truth": "ext x",
      "model_output": "ext x\n  exact Eq.symm (h 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'.1 h)\n  intro ⟨a, b, h⟩\n  exact (IsLinear_iff'.2 (fun x y ↦ by simp [h]; exact AddMonoidHomClass.map_add _ _ _))"
    },
    {
      "id": 264,
      "match": false,
      "ground_truth": "rfl",
      "model_output": "constructor\n  · intro h\n    exact (IsLinear_iff'.1 h)\n  intro ⟨a, b, h⟩\n  exact (IsLinear_iff'.2 (fun x y ↦ by simp [h]; exact AddMonoidHomClass.map_add _ _ _))"
    },
    {
      "id": 265,
      "match": true,
      "ground_truth": "rw [h]",
      "model_output": "rw [h]"
    },
    {
      "id": 266,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp [ContinuousLinearMap.adjoint]"
    },
    {
      "id": 267,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp [ContinuousLinearMap.adjoint]"
    },
    {
      "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 [smul_zero, add_zero]"
    },
    {
      "id": 271,
      "match": false,
      "ground_truth": "intro x hx",
      "model_output": "intro x hx\n    rw[convex_indicator_of_mem hx]"
    },
    {
      "id": 272,
      "match": false,
      "ground_truth": "intro x",
      "model_output": "intro x hx\n    rw[convex_indicator_of_mem hx]"
    }
  ]
}