{
  "summary": {
    "total_with_gt": 273,
    "matched": 38,
    "accuracy": 0.1391941391941392
  },
  "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 (by linarith) (by linarith) (by positivity)"
    },
    {
      "id": 2,
      "match": false,
      "ground_truth": "· exact mul_le_mul_of_nonneg_left hyv hb",
      "model_output": "apply mul_le_mul_of_nonneg hb hyv (by linarith) (by positivity)"
    },
    {
      "id": 3,
      "match": false,
      "ground_truth": "exact mul_le_mul_of_nonneg_left hyv hb",
      "model_output": "apply mul_le_mul_of_nonneg hb hyv (by linarith) (by positivity)"
    },
    {
      "id": 4,
      "match": false,
      "ground_truth": "rw [lipschitzWith_iff_norm_sub_le] at *",
      "model_output": "rw [lipschitzWith_iff_norm_sub_le] at hl ⊢\n  intro x y\n  apply hl\n  simp [gradient]"
    },
    {
      "id": 5,
      "match": false,
      "ground_truth": "by rw [grad_snd_eq h, grad_snd_eq h]",
      "model_output": "simp [grad_snd]"
    },
    {
      "id": 6,
      "match": false,
      "ground_truth": "rw [grad_snd_eq h, grad_snd_eq h]",
      "model_output": "simp [grad_snd]"
    },
    {
      "id": 7,
      "match": false,
      "ground_truth": "apply prox_well_define f hc lbdf x",
      "model_output": "have : ∃ y, f y = inf (f '' univ) := by\n    apply LowerSemicontinuous.exists_inf_of_bddBelow hc lbdf\n  rcases this with ⟨y, hy⟩\n  use y\n  rw [hy]\n  exact prox_set_eq_inf f x y"
    },
    {
      "id": 8,
      "match": false,
      "ground_truth": "refine ⟨x₁, hx₁, x₂, hx₂, hx, ?_⟩",
      "model_output": "refine ⟨x₁, hx₁, ?_⟩"
    },
    {
      "id": 9,
      "match": false,
      "ground_truth": "exact LinearMap.mem_ker.mp (hker hx₂)",
      "model_output": "have : x₂ ∈ LinearMap.ker T := hker hx₂\n      exact this"
    },
    {
      "id": 10,
      "match": false,
      "ground_truth": "simp [Orthonormal, Pairwise] at *",
      "model_output": "rw [orthonormal_iff_ite] at hv ⊢\n  constructor\n  · intro i j hij\n    by_cases h : i ∈ s\n    · exact hv.1 i j hij\n    simp [h]\n  intro i\n  by_cases h : i ∈ s\n  · exact hv.2 i h\n  simp [h]"
    },
    {
      "id": 11,
      "match": false,
      "ground_truth": "by_cases hsf : ProperFunction s f",
      "model_output": "rw [Function.closure_def]"
    },
    {
      "id": 12,
      "match": false,
      "ground_truth": "· exact closure_is_closed_of_proper f hs",
      "model_output": "· exact is_closed_of_closure f hs"
    },
    {
      "id": 13,
      "match": false,
      "ground_truth": "exact closure_is_closed_of_proper f hs",
      "model_output": "rw [Function.IsClosed]"
    },
    {
      "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": "rw [i.h]"
    },
    {
      "id": 16,
      "match": false,
      "ground_truth": "rw [φ₁_equ fullrank₁ fullrank₂ n]",
      "model_output": "simp [φ, φ₁]"
    },
    {
      "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 h1 : r = finrank 𝕜 (range (T.toContinuousLinearMap)) := by\n    rw [← hT.range_eq_range, hT.finrank_range_eq]"
    },
    {
      "id": 19,
      "match": false,
      "ground_truth": "apply iff_iff_implies_and_implies.mpr",
      "model_output": "rw [← convexHull_eq_self_of_closed hscl, ← convexHull_eq_self_of_closed htcl]"
    },
    {
      "id": 20,
      "match": false,
      "ground_truth": "apply convex_conjugate_ge_bot_univ'",
      "model_output": "apply lt_of_le_of_ne\n    · apply convex_conjugate_ge_bot\n    · intro h\n      rw [h] at hs\n      exact hs.1"
    },
    {
      "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": false,
      "ground_truth": "exact HasGradientAtFilter.const_add hf c",
      "model_output": "exact HasGradientAt.const_add hf c"
    },
    {
      "id": 24,
      "match": false,
      "ground_truth": "exact HasGradientAtFilter.const_sub hf c",
      "model_output": "exact HasGradientAt.const_sub hf c"
    },
    {
      "id": 25,
      "match": false,
      "ground_truth": "apply convex_on_p_top\n  ext x; exact h x",
      "model_output": "apply convexOn_of_const h"
    },
    {
      "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 : c = (fun j : Fin m => (B.repr 0) (Fin.castAdd m j)) := by\n    ext j; rw [hS]; simp [h_zero_mem]"
    },
    {
      "id": 29,
      "match": false,
      "ground_truth": "have : (0 : 𝕜) < (1 : 𝕜) := one_pos",
      "model_output": "apply hf' x y 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, ← smul_sub, inner_smul_left, inner_smul_left, ← smul_sub]"
    },
    {
      "id": 31,
      "match": false,
      "ground_truth": "rw [← one_mul (-ε * ‖y - x‖), ← hab, add_mul]",
      "model_output": "have : a * (f y - f x - inner ℝ u (y - x)) + b * (f y - f x - inner ℝ v (y - x)) ≥ -a * ε * ‖y - x‖ - b * ε * ‖y - x‖ := by\n    apply add_le_add (mul_le_mul_of_nonneg_left hyu (by linarith)) (mul_le_mul_of_nonneg_left hyv (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 : ‖gradient H z - gradient H z'‖ ≤ l * ‖z - z'‖ := 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": false,
      "ground_truth": "by\n    apply prox_well_define f hc lbdf x",
      "model_output": "have : ∃ y, f y = inf (f '' univ) := by\n    apply LowerSemicontinuous.exists_inf_of_bddBelow hc lbdf\n  rcases this with ⟨y, hy⟩\n  use y\n  rw [hy]\n  exact prox_set_eq_inf f x y"
    },
    {
      "id": 36,
      "match": false,
      "ground_truth": "by\n  unfold colSubmodule\n  rw [finrank_span_eq_card]\n  simp [h]",
      "model_output": "have : Aᴴ * A = 1 := h\n    have : Aᴴ = A† := by rfl\n    rw [this, ← conjTranspose_mul, conjTranspose_conjTranspose] at h\n    exact h"
    },
    {
      "id": 37,
      "match": false,
      "ground_truth": "unfold colSubmodule\n  rw [finrank_span_eq_card]\n  simp [h]",
      "model_output": "have : Aᴴ * A = 1 := h\n    have : Aᴴ = A† := by rfl\n    rw [this, ← conjTranspose_mul, conjTranspose_conjTranspose] at h\n    exact h"
    },
    {
      "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 = (LinearMap.ker T).lift (LinearMap.quotKerEquivKer T) x := by\n    exact (LinearEquiv.symm_apply_eq (LinearMap.quotKerEquivKer T)).mp rfl\n  rw [this]\n  use (LinearMap.ker T).lift (LinearMap.quotKerEquivKer T) x\n  constructor\n  · exact Submodule.mem_span_of_mem (LinearMap.mem_ker.1 (hker (LinearMap.quotKerEquivKer T x)))"
    },
    {
      "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 [LinearMap.map_add]"
    },
    {
      "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  constructor\n  · intro i j hij\n    by_cases h : i ∈ s\n    · exact hv.1 i j hij\n    simp [h]\n  intro i\n  by_cases h : i ∈ s\n  · exact hv.2 i h\n  simp [h]"
    },
    {
      "id": 43,
      "match": false,
      "ground_truth": "simp [Orthonormal, Pairwise] at *\n  aesop",
      "model_output": "rw [orthonormal_iff_ite] at hv ⊢\n  constructor\n  · intro i j hij\n    by_cases h : i ∈ s\n    · exact hv.1 i j hij\n    simp [h]\n  intro i\n  by_cases h : i ∈ s\n  · exact hv.2 i h\n  simp [h]"
    },
    {
      "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 1 f (Metric.ball x ε) := hf\n  have : ContDiffOn ℝ 0 f (Metric.ball x ε) := by\n    apply ContDiffOn.zero_of_le\n    linarith\n  apply HasGradientAt.continuousAt\n  apply HasGradientAt.continuousAt\n  apply HasGradientAt.const_smul\n  · apply HasGradientAt.const_add\n    · apply HasGradientAt.const_mul\n      · apply HasGradientAt.const_add\n      · apply HasGradientAt.const_mul\n      · apply HasGradientAt.const_add\n      · apply Has"
    },
    {
      "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 1 f (Metric.ball x ε) := hf\n  have : ContDiffOn ℝ 0 f (Metric.ball x ε) := by\n    apply ContDiffOn.zero_of_le\n    linarith\n  apply HasGradientAt.continuousAt\n  apply HasGradientAt.continuousAt\n  apply HasGradientAt.const_smul\n  · apply HasGradientAt.const_add\n    · apply HasGradientAt.const_mul\n      · apply HasGradientAt.const_add\n      · apply HasGradientAt.const_mul\n      · apply HasGradientAt.const_add\n      · apply Has"
    },
    {
      "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  have : ContDiffOn ℝ 0 f (Metric.ball x ε) := by\n    apply ContDiff.contDiffOn; apply ContDiff.continuous; apply ContDiff.contDiffAt\n    apply HasGradientAt.contDiffAt; apply HasGradientAt.contDiffAt; apply HasGradientAt.contDiffAt\n    apply HasGradientAt.contDiffAt; apply HasGradientAt.contDiffAt; apply HasGradientAt.contDiffAt\n    apply HasGradientAt.contDiffAt; apply HasGradientAt.contDiffAt; apply Has"
    },
    {
      "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_iff_contDiffAt_nhds] at h\n  exact HasGradientAt.continuousAt h.hasGradientAt"
    },
    {
      "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": "rw [Function.closure_top]"
    },
    {
      "id": 50,
      "match": false,
      "ground_truth": "simp [Function.closure]\n  intro h; exfalso\n  obtain h1 := h.2; simp at h1",
      "model_output": "rw [Function.closure_top]"
    },
    {
      "id": 51,
      "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": 52,
      "match": true,
      "ground_truth": "simp [hf]; simp [Function.IsClosed]\n    have h1 : Epi ⊥ s = s ×ˢ (univ : Set ℝ) := by\n      ext ⟨x, _⟩\n      simp [Function.Epi]\n    rw [h1]\n    apply IsClosed.prod hs isClosed_univ",
      "model_output": "simp [hf]"
    },
    {
      "id": 53,
      "match": false,
      "ground_truth": "have h1 : Epi ⊥ s = s ×ˢ (univ : Set ℝ) := by\n      ext ⟨x, _⟩\n      simp [Function.Epi]",
      "model_output": "simp [Epi]"
    },
    {
      "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  · exact HasSubgradientAt_to_Banach_HasSubgradientAt\n  exact 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  · exact HasSubgradientAt_to_Banach_HasSubgradientAt\n  exact 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\n    apply HasSubgradientAt.banach_hasSubgradientAt\n    exact h"
    },
    {
      "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\n  apply HasSubgradientAt_to_Banach_HasSubgradientAt\n  exact h"
    },
    {
      "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    exact i.hg 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 _"
    },
    {
      "id": 61,
      "match": false,
      "ground_truth": "apply Filter.tendsto_iff_seq_tendsto.1 Seq_converge_zero₃",
      "model_output": "have : (fun n => A₂ (x₂ (φ₁ n + 1) - x₂ (φ₁ n))) = (fun n => A₂ (x₂ (φ₁ n) + (1 : ℕ) - x₂ (φ₁ n))) := by\n        ext n; rw [Nat.cast_add, Nat.cast_one, add_sub_assoc, add_sub_cancel_left]\n      have h₁ : (fun n => A₂ (x₂ (φ₁ n) + (1 : ℕ) - x₂ (φ₁ n))) = (fun n => A₂ (1 : ℕ)) :="
    },
    {
      "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"
    },
    {
      "id": 63,
      "match": false,
      "ground_truth": "suffices 0 ≤ (c - a - b).toEReal by exact this",
      "model_output": "rw [← 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": "rw [← EReal.coe_sub] at h"
    },
    {
      "id": 65,
      "match": false,
      "ground_truth": "refine EReal.coe_nonneg.mpr ?intro.intro.a",
      "model_output": "rw [← EReal.coe_sub] at this"
    },
    {
      "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 h1 : r = finrank 𝕜 (range (T.toContinuousLinearMap)) := by\n    rw [← hT.range_eq_range, hT.finrank_range_eq]"
    },
    {
      "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 h1 : r = finrank 𝕜 (range (T.toContinuousLinearMap)) := by\n    rw [← hT.range_eq_range, hT.finrank_range_eq]"
    },
    {
      "id": 71,
      "match": false,
      "ground_truth": "simp [finrank_span_eq_card (b:=hT.w) hT.w_orthonormal.linearIndependent]",
      "model_output": "rw [hT.finrank_range_w]"
    },
    {
      "id": 72,
      "match": false,
      "ground_truth": "· exact fun a ↦ set_inclusion_convex_support_eq s t htcv htcl a h",
      "model_output": "· intro hzs\n    rw [← mem_convex_iff_support_le] at hzs\n    rw [← h z] at hzs\n    exact mem_convex_iff_support_le.1 hzs z hzs"
    },
    {
      "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 hzst : z ∈ s.convex_support := by\n      rw [h z]; exact mem_convex_support_of_mem htcv hscv htscl htcl hzs\n    exact mem_of_mem_convex_support hzst htcl"
    },
    {
      "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 hzt\n    rw [← h z]\n    apply convex_support_subset_convex_support hscv htcv htcl hz"
    },
    {
      "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\n  rw [convex_conjugate_indicator_eq_indicator_convex_conjugate]\n  simp [hx]"
    },
    {
      "id": 78,
      "match": false,
      "ground_truth": "refine (conjugate_le_zero_iff s.convex_indicator).mpr ?_",
      "model_output": "rw [convex_conjugate_indicator_eq_indicator_convex_conjugate hs]"
    },
    {
      "id": 79,
      "match": false,
      "ground_truth": "exact lt_of_le_of_lt (convex_conjugate_at_zero) (zero_lt_top)",
      "model_output": "apply lt_of_le_of_ne convex_conjugate_at_zero ?_"
    },
    {
      "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": false,
      "ground_truth": "by\n  exact HasGradientAtFilter.const_add hf c",
      "model_output": "exact HasGradientAt.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]"
    },
    {
      "id": 84,
      "match": false,
      "ground_truth": "simp only [sub_eq_add_neg, gradient_add_const]",
      "model_output": "rw [gradient_sub_const]"
    },
    {
      "id": 85,
      "match": false,
      "ground_truth": "by\n  exact HasGradientAtFilter.const_sub hf c",
      "model_output": "exact HasGradientAt.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 convexOn_of_const h"
    },
    {
      "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 _"
    },
    {
      "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 [← Matrix.mulVec_sub, ← Matrix.mulVec_sub, Matrix.mul_assoc, Matrix.mul_assoc, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix.mulVec_sub, Matrix"
    },
    {
      "id": 95,
      "match": false,
      "ground_truth": "rw [← dot_mul_eq_transpose_mul_dot,← Matrix.mulVec_sub, ← norm2eq_dot]",
      "model_output": "rw [← Matrix.mulVec_sub, transpose_mul_self_nonneg]"
    },
    {
      "id": 96,
      "match": false,
      "ground_truth": "have h_zero_mem : (0 : E) ∈ S := S.zero_mem",
      "model_output": "have h1 : ∀ (v : E), v ∈ S ↔ (B.repr v) (Fin.natAdd n) = c := by\n    intro v\n    have h1 := hS v\n    simp at h1\n    exact h1"
    },
    {
      "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": "rw [Matrix.mulVec_zero, add_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": "simp [sup_value]"
    },
    {
      "id": 101,
      "match": false,
      "ground_truth": "unfold sup_value\n  unfold FeasSet FeasPoint\n  simp [hp]",
      "model_output": "simp [sup_value]"
    },
    {
      "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]\n        simp [A.symm_comp_self]"
    },
    {
      "id": 108,
      "match": false,
      "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp",
      "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply]\n        simp [A.symm_comp_self]"
    },
    {
      "id": 109,
      "match": false,
      "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]",
      "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply]\n        simp [A.symm_comp_self]"
    },
    {
      "id": 110,
      "match": false,
      "ground_truth": "by rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp",
      "model_output": "rw [← ContinuousLinearMap.adjoint_comp]\n        simp [A.toContinuousLinearMap, A.symm_toContinuousLinearMap, A.toContinuousLinearEquiv]"
    },
    {
      "id": 111,
      "match": false,
      "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp",
      "model_output": "rw [← ContinuousLinearMap.adjoint_comp]\n        simp [A.toContinuousLinearMap, A.symm_toContinuousLinearMap, A.toContinuousLinearEquiv]"
    },
    {
      "id": 112,
      "match": false,
      "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]",
      "model_output": "rw [← ContinuousLinearMap.adjoint_comp]\n        simp [A.toContinuousLinearMap, A.symm_toContinuousLinearMap, A.toContinuousLinearEquiv]"
    },
    {
      "id": 113,
      "match": false,
      "ground_truth": "by rw [← trace_transpose, ← trace_transpose_mul, transpose_mul]",
      "model_output": "rw [Matrix.trace_mul_comm]"
    },
    {
      "id": 114,
      "match": false,
      "ground_truth": "rw [← trace_transpose, ← trace_transpose_mul, transpose_mul]",
      "model_output": "rw [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 ConvexOn.convex_indicator (convex_univ X) (convex_univ X) hs (by simp) (by simp)"
    },
    {
      "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.convex_indicator this (fun x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _"
    },
    {
      "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  rw [intrinsicClosure_iInter_eq_iInter_intrinsicClosure_of_forall_exists_intrinsicInterior 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  rw [intrinsicClosure_iInter_eq_iInter_intrinsicClosure_of_forall_exists_intrinsicInterior h hx]"
    },
    {
      "id": 119,
      "match": false,
      "ground_truth": "· exact iIntersection_closure_sub_closure_iIntersection'' h hinter",
      "model_output": "· apply iInter_intrinsicClosure_subset_intrinsicClosure_iInter\n    exact h"
    },
    {
      "id": 120,
      "match": false,
      "ground_truth": "exact iIntersection_closure_sub_closure_iIntersection'' h hinter",
      "model_output": "rcases hinter with ⟨x, hx⟩\n  rw [intrinsicClosure_iInter_eq_iInter_intrinsicClosure_of_exists_intrinsicInterior h hx]\n  apply iInter_mono\n  intro i\n  apply intrinsicClosure_mono\n  exact subset_iInter (fun i => intrinsicInterior_subset) i"
    },
    {
      "id": 121,
      "match": false,
      "ground_truth": "exact closure_iIntersection_sub_iIntersection_closure'",
      "model_output": "rw [intrinsicClosure_iInter_eq_iInter_intrinsicClosure_of_intrinsicInterior_nonempty]"
    },
    {
      "id": 122,
      "match": false,
      "ground_truth": "have eqone : a + (1 - a) = 1 := add_sub_cancel a 1",
      "model_output": "have : a ∈ Set.Ioo 0 1 := ⟨lta, alt⟩\n  have : ∃ c : E, c ∈ Set.Ioo 0 1 ∧ f c < min (f x) (f y) := by\n    apply strict_inequ_of_strong_quasi hf' this\n  obtain ⟨c, hc⟩ := this\n  have : f c < min (f x) (f y) := hc.2\n  have : f c < f x := by\n    apply lt_of_lt_of_eq this (min_eq_left (hx trivial))\n  have :"
    },
    {
      "id": 123,
      "match": false,
      "ground_truth": "have lta' : 0 < 1 - a := sub_pos_of_lt alt",
      "model_output": "have : a + (1 - a) = 1 := by ring_nf"
    },
    {
      "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) = 1 := by ring_nf"
    },
    {
      "id": 125,
      "match": false,
      "ground_truth": "apply Eq.trans_gt (max_eq_right (hx trivial))",
      "model_output": "apply hf' (a • x + (1 - a) • y) y lta' lta (by linarith) (by linarith) (by linarith)"
    },
    {
      "id": 126,
      "match": false,
      "ground_truth": "intro ε εpos",
      "model_output": "intro ε εpos\n  rw [add_comm] at hab\n  rw [add_comm] at hv\n  rw [add_comm] at hu\n  filter_upwards [hu ε εpos, hv ε εpos] with y hy hy'\n  rw [add_comm] at hy'\n  rw [add_comm] at hy\n  rw [add_comm] at hab\n  rw [add_comm] at hab\n  rw [add_comm] at hab\n  rw [add_comm] at hab\n  rw [add_comm] at hab\n  rw [add_comm] at hab\n  rw [add"
    },
    {
      "id": 127,
      "match": false,
      "ground_truth": "rw [ge_iff_le] at *",
      "model_output": "apply add_le_add hyu hyv"
    },
    {
      "id": 128,
      "match": false,
      "ground_truth": "apply add_le_add",
      "model_output": "apply add_le_add hyu hyv"
    },
    {
      "id": 129,
      "match": false,
      "ground_truth": "apply div_pos",
      "model_output": "apply div_pos _ (mul_pos (by linarith) (alg.lpos))"
    },
    {
      "id": 130,
      "match": false,
      "ground_truth": "apply mul_pos",
      "model_output": "apply mul_pos _ alg.lpos\n    linarith [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": "linarith [alg.lpos]"
    },
    {
      "id": 134,
      "match": false,
      "ground_truth": "apply alg.lpos",
      "model_output": "linarith [alg.lpos]"
    },
    {
      "id": 135,
      "match": false,
      "ground_truth": "rw [mem_nhds_iff]",
      "model_output": "exact 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  · exact HasSubgradientAt_to_Banach_HasSubgradientAt\n  exact 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": true,
      "ground_truth": "intro x _hx",
      "model_output": "intro x _"
    },
    {
      "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": false,
      "ground_truth": "by_cases hc2 : c = ⊤",
      "model_output": "by_cases ha3 : a = ⊤"
    },
    {
      "id": 148,
      "match": false,
      "ground_truth": "· rw [hc2]; simp",
      "model_output": "· rw [hc2]; simp; exact le_top"
    },
    {
      "id": 149,
      "match": false,
      "ground_truth": "rw [hc2]; simp",
      "model_output": "rw [hc2] at h ⊢\n  simp at h ⊢\n  linarith"
    },
    {
      "id": 150,
      "match": false,
      "ground_truth": "by exact this",
      "model_output": "rw [← 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"
    },
    {
      "id": 153,
      "match": false,
      "ground_truth": "constructor",
      "model_output": "constructor <;> intro h <;> rw [h] <;> exact Eq.symm (convex_support_eq_convex_cl hscv)"
    },
    {
      "id": 154,
      "match": false,
      "ground_truth": "simp [funext_iff]",
      "model_output": "intro hst"
    },
    {
      "id": 155,
      "match": true,
      "ground_truth": "constructor",
      "model_output": "constructor"
    },
    {
      "id": 156,
      "match": false,
      "ground_truth": "· apply convex_univ",
      "model_output": "apply Convex.univ"
    },
    {
      "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": false,
      "ground_truth": "exact h_zero_eq.symm",
      "model_output": "exact Eq.symm h_zero_eq"
    },
    {
      "id": 161,
      "match": false,
      "ground_truth": "dsimp [ConvexOn]",
      "model_output": "apply ConvexOn.const_zero"
    },
    {
      "id": 162,
      "match": true,
      "ground_truth": "constructor",
      "model_output": "constructor"
    },
    {
      "id": 163,
      "match": false,
      "ground_truth": "dsimp[EqOn]",
      "model_output": "intro x hx\n  rw [convex_indicator_of_mem (hs x hx) (fun x _ => 0) (by simp)]"
    },
    {
      "id": 164,
      "match": false,
      "ground_truth": "rw [if_pos hx]",
      "model_output": "simp [hx]"
    },
    {
      "id": 165,
      "match": false,
      "ground_truth": "by_contra neq",
      "model_output": "have h1 : f x ≤ f y := by\n    apply hx (mem_univ y)"
    },
    {
      "id": 166,
      "match": false,
      "ground_truth": "intro u hu v hv a b ha hb hab",
      "model_output": "rw [convex_iff_forall_pos]\n  intro y1 hy1 y2 hy2 a b ha hb hab\n  rw [f_subdiff_subset_iff_convex]\n  intro z hz\n  rw [f_subdiff_subset_iff_convex]\n  intro w hw\n  rw [f_subdiff_subset_iff_convex]\n  intro u hu\n  rw [f_subdiff_subset_iff_convex]\n  intro v hv\n  rw [f_subdiff_subset_iff_convex]\n  intro p hp\n  rw [f_subdiff_subset_iff_convex]\n  intro q hq\n  rw [f_sub"
    },
    {
      "id": 167,
      "match": false,
      "ground_truth": "rw [has_f_subdiff_iff] at *",
      "model_output": "rw [f_subdiff] at *"
    },
    {
      "id": 168,
      "match": false,
      "ground_truth": "rw [finrank_span_eq_card]",
      "model_output": "have : (Aᴴ * A).colVec = Aᴴ.colVec := by\n    ext i\n    rw [Matrix.mul_apply]\n    exact Eq.symm (Matrix.mulVec_colVec Aᴴ A i)"
    },
    {
      "id": 169,
      "match": false,
      "ground_truth": "rw [hx, LinearMap.map_add]",
      "model_output": "rw [hx, LinearMap.map_add, norm_add_sq_real]"
    },
    {
      "id": 170,
      "match": false,
      "ground_truth": "apply ContDiffOn.contDiffAt hf",
      "model_output": "have : ContDiffOn ℝ 1 f (Metric.ball x ε) := hf\n  have : ContDiffOn ℝ 1 f (Metric.ball x ε) := this\n  have : ContDiffAt ℝ 1 f x := ContDiffOn.contDiffAt this (Metric.mem_ball_self he)\n  exact this"
    },
    {
      "id": 171,
      "match": false,
      "ground_truth": "simp [Function.closure]",
      "model_output": "rw [Function.closure_top]"
    },
    {
      "id": 172,
      "match": false,
      "ground_truth": "simp [Function.closure, hsf]",
      "model_output": "simp [Function.IsClosed, Function.closure, hs]"
    },
    {
      "id": 173,
      "match": false,
      "ground_truth": "by_cases hf : ∃ x ∈ s, f x = ⊥",
      "model_output": "by_cases h : ∃ x ∈ s, f x = ⊥"
    },
    {
      "id": 174,
      "match": false,
      "ground_truth": "simp [Function.IsClosed]",
      "model_output": "rw [Function.IsClosed]"
    },
    {
      "id": 175,
      "match": false,
      "ground_truth": "simp [Function.IsClosed]",
      "model_output": "exact hs"
    },
    {
      "id": 176,
      "match": false,
      "ground_truth": "apply StrictMono.tendsto_atTop",
      "model_output": "apply Filter.tendsto_coe_nhds_infty"
    },
    {
      "id": 177,
      "match": false,
      "ground_truth": "apply StrictMono.comp",
      "model_output": "apply StrictMono.add_const"
    },
    {
      "id": 178,
      "match": false,
      "ground_truth": "· apply hphi_StrictMono",
      "model_output": "· apply strictMono_nat_of_lt_succ\n    intro n\n    apply φ_strict_mono' n"
    },
    {
      "id": 179,
      "match": false,
      "ground_truth": "apply hphi_StrictMono",
      "model_output": "apply StrictMono.addMonotone strictMono_id strictMono_const"
    },
    {
      "id": 180,
      "match": false,
      "ground_truth": "lift a to ℝ using ⟨ha2, ha1⟩",
      "model_output": "have hc2 : c ≠ ⊤ := by\n    by_contra h\n    rw [h] at hc1\n    simp at hc1"
    },
    {
      "id": 181,
      "match": false,
      "ground_truth": "lift c to ℝ using ⟨hc2, hc1⟩",
      "model_output": "rw [← EReal.coe_add] at h\n  rw [← EReal.coe_sub] at h\n  rw [← EReal.coe_sub] at *\n  rw [← EReal.coe_add]\n  exact EReal.coe_le_coe_iff.mpr (le_add_of_nonneg_left (by simp) h)"
    },
    {
      "id": 182,
      "match": false,
      "ground_truth": "intro x hx y hy a b ha hb hab",
      "model_output": "intro x hx y hy a b ha hb hab\n    have h1 : a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2 = a • f x + b • f y - (a * b * (m / 2) * ‖x - y‖ ^ 2) := by\n      rw [smul_eq_mul, smul_eq_mul, mul_assoc, mul_assoc, mul_comm b, mul_assoc]\n    rw [h1]\n    apply hfun hx hy ha hb hab"
    },
    {
      "id": 183,
      "match": false,
      "ground_truth": "· exact fun h => (by rw [h])",
      "model_output": "· intro h; rw [h]"
    },
    {
      "id": 184,
      "match": false,
      "ground_truth": "exact fun h => (by rw [h])",
      "model_output": "intro hst\n    rw [hst]"
    },
    {
      "id": 185,
      "match": false,
      "ground_truth": "simp [convex_support]",
      "model_output": "rw [convex_support]"
    },
    {
      "id": 186,
      "match": false,
      "ground_truth": "simp [convex_support]",
      "model_output": "rcases hs with ⟨x, hx⟩\n    use x\n    constructor\n    · simp\n    rw [convex_support]\n    exact lt_top_iff_ne_top.mpr (convex_conjugate_ne_top_of_ne_top x)"
    },
    {
      "id": 187,
      "match": false,
      "ground_truth": "simp [convex_indicator]",
      "model_output": "rw [convex_indicator]"
    },
    {
      "id": 188,
      "match": false,
      "ground_truth": "split_ifs with hy <;> simp",
      "model_output": "by_cases h : y ∈ s <;> simp [h] <;> simp [h] <;> linarith [hs]"
    },
    {
      "id": 189,
      "match": false,
      "ground_truth": "rcases h1 with ⟨x, hx⟩",
      "model_output": "rcases h1 with ⟨x, hx⟩\n  use x\n  exact hx"
    },
    {
      "id": 190,
      "match": false,
      "ground_truth": "apply convex_on_p_top",
      "model_output": "apply convexOn_of_const h"
    },
    {
      "id": 191,
      "match": false,
      "ground_truth": "rw [polyhedra_iff_cone b]",
      "model_output": "simp [IsClosed]"
    },
    {
      "id": 192,
      "match": false,
      "ground_truth": "exact closed_conic σ b",
      "model_output": "rw [cone_eq_iInter]\n  apply isClosed_iInter\n  intro i\n  apply IsClosed.inter\n  · apply isClosed_iInter\n    intro j\n    apply isClosed_iInter\n    intro k\n    apply isClosed_iInter\n    intro x\n    apply isClosed_iInter\n    intro h\n    apply isClosed_iInter\n    intro h'\n    apply isClosed_iInter\n    intro h''\n    apply isClosed_iInter\n    intro h'''\n    apply isClosed_iInter\n    intro h''''\n    apply isClosed_iInter\n    intro h''''\n    apply"
    },
    {
      "id": 193,
      "match": false,
      "ground_truth": "intro x _ y _ a b _ _ _",
      "model_output": "simp [add_comm]"
    },
    {
      "id": 194,
      "match": false,
      "ground_truth": "dsimp [Set.convex_indicator]",
      "model_output": "rw [convex_indicator_of_mem] <;> simp [hx]"
    },
    {
      "id": 195,
      "match": false,
      "ground_truth": "apply ConvexOn.congr this eq",
      "model_output": "rw [convex_indicator_eq_self] at eq"
    },
    {
      "id": 196,
      "match": false,
      "ground_truth": "apply Subset.antisymm",
      "model_output": "rcases hinter with ⟨x, hx⟩\n  rw [intrinsicClosure_iInter_eq_iInter_intrinsicClosure_of_forall_exists_intrinsicInterior 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_smul, one_smul, add_sub, add_comm, add_sub_assoc]"
    },
    {
      "id": 198,
      "match": false,
      "ground_truth": "simp only [isMinOn_iff] at hy",
      "model_output": "have : f y ≤ f (a • x + (1 - a) • y) := by\n      apply hy (a • x + (1 - a) • y) (by simp) (by simp) (by linarith) (by linarith) (by linarith)"
    },
    {
      "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": "rw [convex_iff_forall_pos]\n  intro y1 hy1 y2 hy2 a b ha hb hab\n  rw [f_subdiff_subset_iff_convex]\n  intro z hz\n  rw [f_subdiff_subset_iff_convex]\n  intro w hw\n  rw [f_subdiff_subset_iff_convex]\n  intro u hu\n  rw [f_subdiff_subset_iff_convex]\n  intro v hv\n  rw [f_subdiff_subset_iff_convex]\n  intro p hp\n  rw [f_subdiff_subset_iff_convex]\n  intro q hq\n  rw [f_sub"
    },
    {
      "id": 201,
      "match": false,
      "ground_truth": "intro ε",
      "model_output": "intro ε εpos\n  rw [add_comm] at hab\n  rw [add_comm] at hv\n  rw [add_comm] at hu\n  filter_upwards [hu ε εpos, hv ε εpos] with y hy hy'\n  rw [add_comm] at hy'\n  rw [add_comm] at hy\n  rw [add_comm] at hab\n  rw [add_comm] at hab\n  rw [add_comm] at hab\n  rw [add_comm] at hab\n  rw [add_comm] at hab\n  rw [add_comm] at hab\n  rw [add"
    },
    {
      "id": 202,
      "match": false,
      "ground_truth": "intro z z'",
      "model_output": "intro x y"
    },
    {
      "id": 203,
      "match": false,
      "ground_truth": "intro z",
      "model_output": "intro x y"
    },
    {
      "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": false,
      "ground_truth": "rw [ck]",
      "model_output": "rw [ck]; apply div_pos; constructor; linarith; linarith [alg.lpos]"
    },
    {
      "id": 206,
      "match": false,
      "ground_truth": "· norm_num",
      "model_output": "linarith [hγ]"
    },
    {
      "id": 207,
      "match": false,
      "ground_truth": "norm_num",
      "model_output": "linarith"
    },
    {
      "id": 208,
      "match": false,
      "ground_truth": "simp [h]",
      "model_output": "rw [← rowVec_transpose, ← colVec_transpose] at h"
    },
    {
      "id": 209,
      "match": false,
      "ground_truth": "aesop",
      "model_output": "constructor <;> intro a _ b _ hab <;> exact hv.1 a <;> exact hv.2 a b hab"
    },
    {
      "id": 210,
      "match": false,
      "ground_truth": "intro h",
      "model_output": "intro h\n    exact ProperFunction.lowerSemicontinuousHull_eq_top h"
    },
    {
      "id": 211,
      "match": false,
      "ground_truth": "exfalso",
      "model_output": "simp [LowerSemicontinuousHull]"
    },
    {
      "id": 212,
      "match": false,
      "ground_truth": "simp at h1",
      "model_output": "rcases h1 with h1 | h1"
    },
    {
      "id": 213,
      "match": true,
      "ground_truth": "simp [hf]",
      "model_output": "simp [hf]"
    },
    {
      "id": 214,
      "match": false,
      "ground_truth": "ext ⟨x, _⟩",
      "model_output": "simp [epi, hf]"
    },
    {
      "id": 215,
      "match": false,
      "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": false,
      "ground_truth": "rw [h']",
      "model_output": "rw[h']; apply isClosed_empty"
    },
    {
      "id": 219,
      "match": false,
      "ground_truth": "intro h z",
      "model_output": "intro h\n  apply HasSubgradientAt_to_Banach_HasSubgradientAt\n  exact h"
    },
    {
      "id": 220,
      "match": false,
      "ground_truth": "intro h",
      "model_output": "intro h\n  apply HasSubgradientAt_to_Banach_HasSubgradientAt\n  exact h"
    },
    {
      "id": 221,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp [toDual_apply]"
    },
    {
      "id": 222,
      "match": false,
      "ground_truth": "linarith",
      "model_output": "exact hz"
    },
    {
      "id": 223,
      "match": false,
      "ground_truth": "intro h z",
      "model_output": "intro h\n  rw [HasSubgradientAt] ; intro y\n  rw [← ContinuousLinearMap.map_sub]; apply h y; simp"
    },
    {
      "id": 224,
      "match": false,
      "ground_truth": "intro h",
      "model_output": "intro h\n  rw [HasSubgradientAt] ; intro y\n  rw [← ContinuousLinearMap.map_sub]; apply h y; simp"
    },
    {
      "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": "exact 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": "constructor"
    },
    {
      "id": 230,
      "match": false,
      "ground_truth": "intro x hx",
      "model_output": "intro x _"
    },
    {
      "id": 231,
      "match": false,
      "ground_truth": "intro x",
      "model_output": "intro x _"
    },
    {
      "id": 232,
      "match": false,
      "ground_truth": "intro x",
      "model_output": "intro x _"
    },
    {
      "id": 233,
      "match": false,
      "ground_truth": "ext n",
      "model_output": "ext n\n            simp [φ, φ₁]\n            congr! 1\n            exact Nat.add_sub_cancel' (n + 1)"
    },
    {
      "id": 234,
      "match": false,
      "ground_truth": "simp",
      "model_output": "apply Nat.toNat_of_nonneg"
    },
    {
      "id": 235,
      "match": true,
      "ground_truth": "rw [this]",
      "model_output": "rw [this]"
    },
    {
      "id": 236,
      "match": false,
      "ground_truth": "rw [hc2]",
      "model_output": "rw [hc2] at h ⊢\n  simp at h ⊢\n  linarith"
    },
    {
      "id": 237,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp [hc2] at h"
    },
    {
      "id": 238,
      "match": false,
      "ground_truth": "exact this",
      "model_output": "rw [← EReal.coe_sub] at this"
    },
    {
      "id": 239,
      "match": false,
      "ground_truth": "linarith",
      "model_output": "linarith [this]"
    },
    {
      "id": 240,
      "match": true,
      "ground_truth": "· exact hs",
      "model_output": "· exact 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 hx y hy a b ha hb hab\n    have h1 : a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2 = a • f x + b • f y - (a * b * (m / 2) * ‖x - y‖ ^ 2) := by\n      rw [smul_eq_mul, smul_eq_mul, mul_assoc, mul_assoc, mul_comm b, mul_assoc]\n    rw [h1]\n    apply hfun hx hy ha hb hab"
    },
    {
      "id": 243,
      "match": false,
      "ground_truth": "dsimp",
      "model_output": "simp [hab] at hfun ⊢"
    },
    {
      "id": 244,
      "match": true,
      "ground_truth": "ring_nf",
      "model_output": "ring_nf"
    },
    {
      "id": 245,
      "match": true,
      "ground_truth": "exact hfun",
      "model_output": "exact 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 hst\n  have hssub : s ⊆ t := by\n    rw [← bi_convex_support_eq_convex_convex_closed hscl hscv]\n    rw [← bi_convex_support_eq_convex_convex_closed htcl htcv]\n    exact fun x _ => (congrFun hst x).ge\n  have htsub : t ⊆ s := by\n    rw [← bi_convex_support_eq_convex_convex_closed htcl htcv]\n    rw [← bi_convex_support_eq_convex_convex_closed hscl hscv]\n    exact"
    },
    {
      "id": 249,
      "match": false,
      "ground_truth": "ext z",
      "model_output": "rw [← convex_hull_eq_self_of_closed hscl, ← convex_hull_eq_self_of_closed htcl]"
    },
    {
      "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; constructor\n        · trivial\n        rw [convex_support, convex_conjugate]\n        apply convex_indicator_univ_convex_conjugate_eq_indicator_univ_convex_conjugate\n        exact hs.convex_indicator_univ_convex_conjugate_eq_indicator_univ_convex_conjugate hs"
    },
    {
      "id": 253,
      "match": false,
      "ground_truth": "simp",
      "model_output": "intro y\n  rw [inner_zero_left, EReal.coe_zero]\n  apply le_convex_indicator_of_mem\n  exact hs"
    },
    {
      "id": 254,
      "match": false,
      "ground_truth": "intro y",
      "model_output": "intro y\n  rw [convex_indicator]\n  by_cases h : y ∈ s\n  · simp [h]\n  · simp [h]"
    },
    {
      "id": 255,
      "match": false,
      "ground_truth": "simp",
      "model_output": "linarith"
    },
    {
      "id": 256,
      "match": false,
      "ground_truth": "simp at h1",
      "model_output": "rcases h1 with h1 | h1"
    },
    {
      "id": 257,
      "match": false,
      "ground_truth": "simp [hx]",
      "model_output": "simp [dom]"
    },
    {
      "id": 258,
      "match": true,
      "ground_truth": "ext x",
      "model_output": "ext x"
    },
    {
      "id": 259,
      "match": true,
      "ground_truth": "exact h x",
      "model_output": "exact h x"
    },
    {
      "id": 260,
      "match": false,
      "ground_truth": "intro x",
      "model_output": "intro x _ y _"
    },
    {
      "id": 261,
      "match": false,
      "ground_truth": "simp [hp]",
      "model_output": "rw [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_to_linear f h\n  intro h\n  exact linear_to_IsLinear f h"
    },
    {
      "id": 264,
      "match": false,
      "ground_truth": "rfl",
      "model_output": "constructor\n  · intro h\n    exact IsLinear_to_linear f h\n  intro h\n  exact linear_to_IsLinear f h"
    },
    {
      "id": 265,
      "match": true,
      "ground_truth": "rw [h]",
      "model_output": "rw [h]"
    },
    {
      "id": 266,
      "match": false,
      "ground_truth": "simp",
      "model_output": "rw [ContinuousLinearMap.adjoint_comp]\n    simp [A.toContinuousLinearMap, A.symm_toContinuousLinearMap]"
    },
    {
      "id": 267,
      "match": false,
      "ground_truth": "simp",
      "model_output": "rw [ContinuousLinearMap.adjoint_comp]\n        rw [ContinuousLinearMap.adjoint_coe]\n        rw [ContinuousLinearMap.adjoint_coe]\n        simp\n        exact A.toContinuousLinearEquiv.symm_apply_apply x"
    },
    {
      "id": 268,
      "match": false,
      "ground_truth": "exact hs",
      "model_output": "apply hs"
    },
    {
      "id": 269,
      "match": false,
      "ground_truth": "intro x",
      "model_output": "simp [add_comm]"
    },
    {
      "id": 270,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp [add_zero, smul_zero]"
    },
    {
      "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 _"
    }
  ]
}