evaluation/valid_score_in_100.json

{
  "summary": {
    "total_with_gt": 239,
    "matched": 28,
    "accuracy": 0.11715481171548117
  },
  "details": [
    {
      "id": 0,
      "match": false,
      "ground_truth": "apply HasDerivAt.sub",
      "model_output": "apply HasDerivAt.const_sub"
    },
    {
      "id": 1,
      "match": false,
      "ground_truth": "have : (fun (t : ℝ) => c⁻¹ * t) = (fun t => t * c⁻¹) := by ext t; ring",
      "model_output": "apply HasDerivAt.const_mul c⁻¹ (hasDerivAt_id x)"
    },
    {
      "id": 2,
      "match": false,
      "ground_truth": "rw [← EReal.coe_sub]",
      "model_output": "simp [coe_le_coe]"
    },
    {
      "id": 3,
      "match": false,
      "ground_truth": "ext x",
      "model_output": "exact convex_indicator_empty"
    },
    {
      "id": 4,
      "match": true,
      "ground_truth": "exact ah",
      "model_output": "exact ah"
    },
    {
      "id": 5,
      "match": false,
      "ground_truth": "lift a to ℝ using ⟨ha1, ha2⟩\n  lift b to ℝ using ⟨hb1, hb2⟩\n  rw [← EReal.coe_sub]\n  apply bot_lt_iff_ne_bot.2\n  let c := a - b\n  suffices Real.toEReal c ≠ ⊥ by exact this\n  simp",
      "model_output": "simp [ha1, ha2, hb1, hb2]"
    },
    {
      "id": 6,
      "match": false,
      "ground_truth": "lift a to ℝ using ⟨ha1, ha2⟩",
      "model_output": "have : a + (b - b) = a := by\n    rw [EReal.sub_self hb2, EReal.add_top]"
    },
    {
      "id": 7,
      "match": false,
      "ground_truth": "apply hasDerivAt_mul_const c⁻¹",
      "model_output": "apply HasDerivAt.const_mul' (hasDerivAt_id' x) (c⁻¹)"
    },
    {
      "id": 8,
      "match": false,
      "ground_truth": "by\n  use Aφ (A := A) (x := x) H\n  exact ⟨Aφ_mono (A := A) (x := x) H, α_subseq_converge (Aφ_sub H)⟩",
      "model_output": "rw [← H]"
    },
    {
      "id": 9,
      "match": false,
      "ground_truth": "· rw [ha1];simp",
      "model_output": "· rw[ha1];simp"
    },
    {
      "id": 10,
      "match": false,
      "ground_truth": "let g : EReal ≃o EReal := ⟨g', by simp [g'];apply add_le_add_right a⟩",
      "model_output": "have : (⨆ m ∈ s, f m) + a = ⨆ m ∈ s, (f m + a) := by\n    have : (⨆ m ∈ s, f m) = ⨆ m ∈ s, (g' (f m)) := by\n      rw [g']; simp\n    rw [this]; exact biSup_congr fun m => by simp [g']; simp [g']; simp [g']; simp [g']; simp [g']; simp [g']; simp [g']; simp [g']; simp [g']; simp [g']; simp [g']; simp"
    },
    {
      "id": 11,
      "match": false,
      "ground_truth": "ext i j",
      "model_output": "simp [g, h]"
    },
    {
      "id": 12,
      "match": false,
      "ground_truth": "apply glt",
      "model_output": "simp at hg"
    },
    {
      "id": 13,
      "match": false,
      "ground_truth": "apply add_le_add_right a",
      "model_output": "intro a b\n  exact EReal.add_le_add_iff_right a b a"
    },
    {
      "id": 14,
      "match": false,
      "ground_truth": "left",
      "model_output": "left; exact hdomsf"
    },
    {
      "id": 15,
      "match": false,
      "ground_truth": "ext j",
      "model_output": "simp [Matrix.mulVec]"
    },
    {
      "id": 16,
      "match": false,
      "ground_truth": "intro i",
      "model_output": "intro i j"
    },
    {
      "id": 17,
      "match": false,
      "ground_truth": "· intro h; by_contra! hs\n    obtain ⟨x, xs⟩:= Set.nonempty_def.1 hs\n    have h1: s.convex_indicator x = 0 := by\n      simp [convex_indicator]; exact xs\n    have h2 := congrFun h x; simp [h1] at h2",
      "model_output": "· intro h\n    by_contra h'\n    push_neg at h'\n    rw [convex_indicator, h] at h'\n    exact h'"
    },
    {
      "id": 18,
      "match": false,
      "ground_truth": "constructor",
      "model_output": "constructor <;> intro h <;> intro x hx <;> simp at h hx ⊢ <;> exact h x hx.1 hx.2"
    },
    {
      "id": 19,
      "match": false,
      "ground_truth": "by_cases hA : A = 0",
      "model_output": "intro x"
    },
    {
      "id": 20,
      "match": false,
      "ground_truth": "by norm_num",
      "model_output": "have : a = 1 / 2 := by linarith\n      rw [this]; norm_num"
    },
    {
      "id": 21,
      "match": false,
      "ground_truth": "by\n  obtain hcon := subdifferential_subset f x₀\n  apply hcon; exact first_order_optimality_condition f x₀ hx",
      "model_output": "rw [subdifferential, mem_setOf, mem_closure_iff_seq_limit]"
    },
    {
      "id": 22,
      "match": true,
      "ground_truth": "exact h",
      "model_output": "exact h"
    },
    {
      "id": 23,
      "match": false,
      "ground_truth": "by\n  have : (fun x ↦ (inner (ℝ) x a : ℝ)) = fun x ↦ (inner (ℝ) a x : ℝ) := by\n    ext y; exact real_inner_comm a y\n  rw [this]; apply gradient_of_inner_const",
      "model_output": "rw [hasGradientAt_iff_hasFDerivAt, ← equiv_lemma_inner_right]"
    },
    {
      "id": 24,
      "match": false,
      "ground_truth": "refine Eq.symm (CancelDenoms.inv_subst (hA j) ?_)\n      refine prod_erase_mul univ (fun «x» ↦ A «x» «x») ?_\n      exact mem_univ j",
      "model_output": "rw [← Finset.prod_erase_mul (s := Finset.univ) (A := A) (h := h)]"
    },
    {
      "id": 25,
      "match": true,
      "ground_truth": "intro x y hy",
      "model_output": "intro x y hy"
    },
    {
      "id": 26,
      "match": false,
      "ground_truth": "simp [LowerSemicontinuousHull]",
      "model_output": "ext p"
    },
    {
      "id": 27,
      "match": true,
      "ground_truth": "simp at hfy",
      "model_output": "simp at hfy"
    },
    {
      "id": 28,
      "match": false,
      "ground_truth": "rw [Fin.sum_univ_succ]",
      "model_output": "have : ∀ i, (fun i ↦ f i.succ) i ≠ ⊥ := by\n    intro i\n    exact p i.succ"
    },
    {
      "id": 29,
      "match": false,
      "ground_truth": "by_cases hx2 : f x = ⊥",
      "model_output": "have hx2 : f x ≠ ⊥ := by\n    by_contra h\n    rw[h] at h\n    simp at h\n    exact not_top_of_ne_top (convex_conjugate_ne_top univ f x) h"
    },
    {
      "id": 30,
      "match": false,
      "ground_truth": "rw [@mem_iInter]",
      "model_output": "rw [intrinsicClosure_iInter] at hx"
    },
    {
      "id": 31,
      "match": false,
      "ground_truth": "rcases h with ⟨δ, dpos, h⟩",
      "model_output": "rcases h with ⟨ε', epos', h⟩\n    use ε'\n    constructor\n    · linarith [epos']\n    intro x' h'\n    rw [Metric.mem_ball] at h'\n    exact h h'"
    },
    {
      "id": 32,
      "match": false,
      "ground_truth": "exact half_lt_self dpos",
      "model_output": "linarith"
    },
    {
      "id": 33,
      "match": false,
      "ground_truth": "apply pin",
      "model_output": "exact pin"
    },
    {
      "id": 34,
      "match": false,
      "ground_truth": "by_cases hf_pinf : f = ⊤",
      "model_output": "rw [convex_conjugate]"
    },
    {
      "id": 35,
      "match": false,
      "ground_truth": "by\n  apply intrinsicInterior_subset.antisymm\n  nth_rw 1 [intrinsicInterior]\n  rw [intrinsicInterior, image_subset_iff]\n  rw [affineSpan_intrinsicInterior hsc]\n  rw [Function.Injective.preimage_image Subtype.val_injective]\n  simp [intrinsicInterior]",
      "model_output": "have hsc' : Convex ℝ (intrinsicInterior ℝ s) := by\n    apply Convex.intrinsicInterior\n    exact hsc\n  exact intrinsicInterior_intrinsicInterior hsc'"
    },
    {
      "id": 36,
      "match": false,
      "ground_truth": "rw [← sub_add, norm_add_sq_real, real_inner_comm, inner_sub_right]",
      "model_output": "rw [← norm_add_sq_real a (v - x)]"
    },
    {
      "id": 37,
      "match": false,
      "ground_truth": "rw [EReal.coe_le_coe_iff] at hccleq",
      "model_output": "simp at hccleq ⊢"
    },
    {
      "id": 38,
      "match": false,
      "ground_truth": "intro z",
      "model_output": "have : Convex ℝ (intrinsicClosure ℝ s) := by\n    apply Convex.intrinsicClosure\n    exact h\n  apply intrinsicInterior_mono\n  exact subset_intrinsicClosure"
    },
    {
      "id": 39,
      "match": false,
      "ground_truth": "have : FiniteDimensional 𝕜 (Submodule.span 𝕜 (Set.range v)) := by\n    apply FiniteDimensional.span_of_finite\n    simp [Set.finite_range]",
      "model_output": "have : Orthonormal 𝕜 v := hv"
    },
    {
      "id": 40,
      "match": false,
      "ground_truth": "rw [h2]",
      "model_output": "rw [h2] at h1 ⊢"
    },
    {
      "id": 41,
      "match": false,
      "ground_truth": "apply banach_steinhaus",
      "model_output": "rcases hy with ⟨hy1, hy2⟩\n  use 1\n  intro x\n  have : ‖(innerSL ℝ) x‖ ≤ 1 := by\n    rw [norm_innerSL]\n    apply le_of_lt (hy1 x)\n  exact this"
    },
    {
      "id": 42,
      "match": true,
      "ground_truth": "intro h\n    apply HasFDeriv_Convergence\n    exact h",
      "model_output": "intro h\n  apply HasFDeriv_Convergence\n  exact h"
    },
    {
      "id": 43,
      "match": false,
      "ground_truth": "· apply Convergence_HasFDeriv",
      "model_output": "intro h\n  apply HasFDerivAt.mk\n  intro ε εpos\n  rcases h ε εpos with ⟨δ, δpos, hε⟩\n  use δ\n  constructor\n  · exact δpos\n  intro x' hx'\n  rw [← norm_neg (x - x'), neg_sub] at hx'\n  apply hε\n  rw [neg_sub, norm_neg]\n  exact hx'"
    },
    {
      "id": 44,
      "match": false,
      "ground_truth": "by\n  rw [←intrinsicInterior_affineSubspace_eq_self, ←nonempty_iff_ne_empty] at hn\n  rw [←iIntersection_closure_eq_intrinsicInterior_closure' M.convex hC hn]\n  rw [closure_affineSubspace_eq_self]",
      "model_output": "have h1 : M ≤ affineSpan ℝ C := by\n    rw [AffineSubspace.le_def]\n    intro x hx\n    rw [AffineSubspace.mem_coe, AffineSubspace.mem_affineSpan]\n    exact subset_affineSpan ℝ (M ∩ C) ⟨hx.1, hx.2.1⟩\n  have h2 : M ≤ closure (M ∩ C) := by\n    rw [AffineSubspace.le_def]\n    intro x hx\n    rw [AffineSubspace.mem_coe, AffineSubspace.mem_affineSpan]\n    exact subset"
    },
    {
      "id": 45,
      "match": false,
      "ground_truth": "apply alg.ha 0",
      "model_output": "apply alg.ha k"
    },
    {
      "id": 46,
      "match": true,
      "ground_truth": "rw [h2]",
      "model_output": "rw [h2]"
    },
    {
      "id": 47,
      "match": false,
      "ground_truth": "rcases this with ⟨xn, hxn1, hxn2⟩",
      "model_output": "rcases this with ⟨xn, ⟨n, hn1, hn2⟩, hxn⟩"
    },
    {
      "id": 48,
      "match": false,
      "ground_truth": "have h2 := tendsto_add (A:=A) (B:=B)",
      "model_output": "have h2 := tendsto_add (A := A) (B := B) hAa hBa"
    },
    {
      "id": 49,
      "match": false,
      "ground_truth": "have : ‖a • b‖ ^ 2 ≤ (‖a‖ * ‖b‖) ^ 2 := by\n      rw [mul_pow, WithLp.prod_norm_sq_eq_of_L2, WithLp.prod_norm_sq_eq_of_L2]\n      simp only [WithLp.smul_fst, WithLp.smul_snd]\n      rw [norm_smul, norm_smul, mul_add, mul_pow, mul_pow]",
      "model_output": "rw [WithLp.prod_norm_eq_of_L2]"
    },
    {
      "id": 50,
      "match": false,
      "ground_truth": "by_cases ht : i ∈ σ",
      "model_output": "by_cases hi : i ∈ σ <;> simp [hi] <;> linarith [cpos i hi]"
    },
    {
      "id": 51,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp at hst hdti h ⊢"
    },
    {
      "id": 52,
      "match": true,
      "ground_truth": "by rw [h y ys, hf]; exact hg y ys",
      "model_output": "rw [h y ys, hf]"
    },
    {
      "id": 53,
      "match": false,
      "ground_truth": "exact in_of_le_sinf_of_b_top hx",
      "model_output": "rw [hbtop] at hab"
    },
    {
      "id": 54,
      "match": false,
      "ground_truth": "intro hmin",
      "model_output": "intro hmin\n    have hsub : SubgradientAt f f' xm := by\n      rw [subgradientAt_iff_hasGradientAt_of_convexOn hcon (mem_univ xm) hfc]\n      exact hf xm\n    exact subgradientAt_eq_zero hsub hmin"
    },
    {
      "id": 55,
      "match": false,
      "ground_truth": "simp [hx2, hw.2]",
      "model_output": "simp [hw.2]"
    },
    {
      "id": 56,
      "match": false,
      "ground_truth": "rw [smul_sub, smul_sub, add_comm_sub, sub_sub, ← add_smul, add_comm b a]",
      "model_output": "rw [smul_sub, smul_sub, ← sub_add, ← sub_add, absum1, sub_self, add_zero]"
    },
    {
      "id": 57,
      "match": false,
      "ground_truth": "simp [ConvexCone.addPoint] at hx",
      "model_output": "rw [Convex.toCone] at hx\n  obtain ⟨c, hc, y, hy, hxy⟩ := hx\n  use c, hc.1\n  use y, hy.1\n  rw [hxy]"
    },
    {
      "id": 58,
      "match": false,
      "ground_truth": "intro x",
      "model_output": "intro x xs y ys\n  apply Convex.exists_inner_prod_eq_sub' hs hf x xs y ys"
    },
    {
      "id": 59,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp [LT.lt.ne_top] at hx"
    },
    {
      "id": 60,
      "match": false,
      "ground_truth": "by\n        simp only [mem_setOf_eq]\n        intro _ _ _ _  hp1 hp2 hp3\n        rw [vsub_eq_sub, vadd_eq_add, map_add, map_smul, map_sub,\n        hp1, hp2, hp3, sub_self, smul_zero, zero_add]",
      "model_output": "intro c p₁ p₂ p₃ hp₁ hp₂ hp₃\n    rw [Set.mem_setOf, map_vsub, smul_sub, vsub_eq_sub, map_sub, ← smul_sub, hp₁, hp₂, hp₃]\n    simp [smul_sub, smul_add, smul_sub, smul_smul]"
    },
    {
      "id": 61,
      "match": false,
      "ground_truth": "linarith",
      "model_output": "linarith [ht]"
    },
    {
      "id": 62,
      "match": false,
      "ground_truth": "simp [h]",
      "model_output": "rw [LinearIsometryEquiv.symm_apply_apply]"
    },
    {
      "id": 63,
      "match": true,
      "ground_truth": "simp [g]",
      "model_output": "simp [g]"
    },
    {
      "id": 64,
      "match": false,
      "ground_truth": "rw [Prod.tendsto_iff] at xntend",
      "model_output": "have : ∀ n, f (xn n).1 ≤ (xn n).2 := by exact xncond"
    },
    {
      "id": 65,
      "match": false,
      "ground_truth": "by_cases ha1 : a = 0",
      "model_output": "rw [add_top_of_ne_bot (LT.lt.ne_top (hsf.uninfinity y hb))]"
    },
    {
      "id": 66,
      "match": false,
      "ground_truth": "by_cases hb1 : b = 0",
      "model_output": "apply le_trans (convex_epi_le_of_convexOn hf (by simp) (by simp) ha hb hab) ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?"
    },
    {
      "id": 67,
      "match": false,
      "ground_truth": "congr",
      "model_output": "congr! 2 with a"
    },
    {
      "id": 68,
      "match": false,
      "ground_truth": "intro z",
      "model_output": "intro y hy\n    rw [gradient_eq_fderiv]"
    },
    {
      "id": 69,
      "match": false,
      "ground_truth": "exact Batteries.compareOfLessAndEq_eq_lt.mp rfl",
      "model_output": "simp"
    },
    {
      "id": 70,
      "match": false,
      "ground_truth": "intro y",
      "model_output": "intro x'\n    rw [p.h]\n    rw [p.h]\n    apply p.h_sub_le_of_norm_sub_le\n    apply p.h_sub_le_of_norm_sub_le\n    apply norm_sub_norm_le\n    apply norm_sub_norm_le\n    apply norm_sub_norm_le\n    apply norm_sub_norm_le\n    apply norm_sub_norm_le\n    apply norm_sub_norm_le\n    apply norm_sub_norm_le\n    apply norm_sub_norm_le\n    apply norm_sub_norm_le\n    apply norm_sub_norm_le\n    apply norm_sub_norm_le\n    apply norm_sub_norm_le\n    apply norm_sub_norm_le"
    },
    {
      "id": 71,
      "match": false,
      "ground_truth": "have hb : b = 1 := by linarith",
      "model_output": "rw [ha] at absum"
    },
    {
      "id": 72,
      "match": false,
      "ground_truth": "intro x y r",
      "model_output": "simp at h"
    },
    {
      "id": 73,
      "match": false,
      "ground_truth": "rw [hab]",
      "model_output": "rw [hab]; simp [gradient_add_const, gradient_inner_const]"
    },
    {
      "id": 74,
      "match": false,
      "ground_truth": "obtain h := ConcaveOn.deriv_le_slope h1 hy' hx' xgt hdiff\n      rw [slope_def_field, le_div_iff₀] at h\n      repeat linarith",
      "model_output": "apply ConcaveOn.deriv_le_slope h1 hy' hx' xgt hdiff"
    },
    {
      "id": 75,
      "match": false,
      "ground_truth": "have ze : z ≠ 0 := by\n      by_contra hz; simp [hz] at zin; simp [r] at zin",
      "model_output": "by_cases h : z = 0"
    },
    {
      "id": 76,
      "match": false,
      "ground_truth": "rw [← hEq]",
      "model_output": "exact h x y r"
    },
    {
      "id": 77,
      "match": false,
      "ground_truth": "have h1 : M -ᵥ M ⊆ Submodule.comap (LinearMap.inl R α β) v := by\n    rintro p ⟨c, hc, d, hd, rfl⟩\n    exact Submodule.subset_span ⟨⟨c, n0⟩, ⟨hc, hn0⟩, ⟨d, n0⟩, ⟨hd, hn0⟩, by simp⟩",
      "model_output": "have : (m, n) = (m, n0) + (m0, n) - (m0, n0) := by simp [Prod.mk_sub_mk]"
    },
    {
      "id": 78,
      "match": false,
      "ground_truth": "have h_nonempty : S.Nonempty := by\n    obtain ⟨a, ha⟩ := @NormedSpace.sphere_nonempty_rclike 𝕜 _ V _ _ _ 1 zero_le_one\n    use a",
      "model_output": "have h_nonempty : S.Nonempty := by\n    obtain ⟨v, hv⟩ := exists_ne (0 : V)\n    use v\n    simp [S] at hv\n    exact ⟨hv.1, hv.2⟩"
    },
    {
      "id": 79,
      "match": false,
      "ground_truth": "linarith",
      "model_output": "linarith [hy.2]"
    },
    {
      "id": 80,
      "match": false,
      "ground_truth": "apply DifferentiableAt.hasGradientAt",
      "model_output": "apply HasGradientAt.const_add z.1 (hasGradientAt_fst h z.2) (z.2)"
    },
    {
      "id": 81,
      "match": false,
      "ground_truth": "rcases lbdf with ⟨L, Lbound⟩",
      "model_output": "have : S.Nonempty := by\n    use x; simp [S]\n    rw [eq]; use g x; simp [g]"
    },
    {
      "id": 82,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp [μpos]"
    },
    {
      "id": 83,
      "match": false,
      "ground_truth": "norm_num",
      "model_output": "linarith"
    },
    {
      "id": 84,
      "match": false,
      "ground_truth": "intro x hx",
      "model_output": "intro x hx\n    have : x ∈ Icc 0 1 := by simp at hx; constructor <;> linarith\n    exact (h1 x this)"
    },
    {
      "id": 85,
      "match": true,
      "ground_truth": "intro x _",
      "model_output": "intro x _"
    },
    {
      "id": 86,
      "match": false,
      "ground_truth": "rw [mem_setOf]",
      "model_output": "rw [f_subdiff_characterization]"
    },
    {
      "id": 87,
      "match": false,
      "ground_truth": "have inequ₂'' : lim_inf (const_real_liminf (f y)) =  f y := by\n    apply liminf_const_eq\n  rw[← inequ₂''];\n  exact inequ₂' y nonempty lscf x_converge g_converge",
      "model_output": "apply le_lim_inf_add_real_liminf_nonempty_nonempty lscf x_converge g_converge nonempty"
    },
    {
      "id": 88,
      "match": false,
      "ground_truth": "refine norm_ne_zero_iff.mpr ?_",
      "model_output": "apply norm_ne_zero_iff.mpr\n        simp\n        apply b.ne_zero i"
    },
    {
      "id": 89,
      "match": false,
      "ground_truth": "rw [← hT.span_range_v_eq_orthogonal_ker,\n      mem_span_range_iff_exists_fun] at hx₂",
      "model_output": "have : x₂ ∈ (ker T) := by\n    rw [ker, mem_setOf, LinearMap.mem_ker, map_add, hx₁, zero_add]\n    exact hx₂"
    },
    {
      "id": 90,
      "match": false,
      "ground_truth": "· simp [mem_closure_iff_seq_limit] at hx\n    rcases hx with ⟨⟨w, hw⟩, hx2⟩\n    use (fun n => (w n, ⊤))\n    constructor\n    · intro n\n      simpa using hw.1 n\n    rw [Prod.tendsto_iff]\n    simp [hx2, hw.2]",
      "model_output": "· rcases hx with ⟨hx1, hx2⟩\n    use fun n => (x.1, (f x.1).toReal + 1/n)\n    constructor\n    · intro n\n      constructor\n      · exact mem_of_mem_nhds (mem_closure_iff_seq_limit.1 hx1)\n      simp\n    rw [Prod.tendsto_iff]\n    constructor\n    · exact hx1\n    simp [hx2]"
    },
    {
      "id": 91,
      "match": false,
      "ground_truth": "apply Finset.sum_eq_zero",
      "model_output": "rw [Finset.sum_eq_zero]\n  intro i hi\n  rw [hx.1.2 i hi]\n  simp"
    },
    {
      "id": 92,
      "match": false,
      "ground_truth": "mul_inv_cancel₀ (by linarith),",
      "model_output": "rw [mul_inv_cancel₀ (by linarith), mul_inv_cancel₀ (by linarith), mul_div_assoc]"
    },
    {
      "id": 93,
      "match": false,
      "ground_truth": "by rw [one_smul]; rfl",
      "model_output": "rw [one_smul, vadd_eq_add, add_comm]"
    },
    {
      "id": 94,
      "match": false,
      "ground_truth": "intro b",
      "model_output": "intro b hb"
    },
    {
      "id": 95,
      "match": false,
      "ground_truth": "by linarith",
      "model_output": "rw [← right] at hc\n    exact mul_pos hc (by linarith)"
    },
    {
      "id": 96,
      "match": false,
      "ground_truth": "rw [smul_sub, smul_sub, add_comm_sub, sub_sub, ← add_smul, add_comm b a]",
      "model_output": "rw [smul_sub, smul_sub, ← sub_add, absum1, sub_add_comm, sub_add_comm, sub_add_comm]"
    },
    {
      "id": 97,
      "match": true,
      "ground_truth": "· rw [intrinsicInterior_prod_eq_prod_intrinsicInterior]; exact mk_mem_prod (by simp) hx",
      "model_output": "rw [intrinsicInterior_prod_eq_prod_intrinsicInterior]"
    },
    {
      "id": 98,
      "match": false,
      "ground_truth": "rw [alg.γeq k, alg.teq k]",
      "model_output": "rw [alg.teq k, alg.teq k, alg.teq k]"
    },
    {
      "id": 99,
      "match": true,
      "ground_truth": "intro i _",
      "model_output": "intro i _"
    },
    {
      "id": 100,
      "match": false,
      "ground_truth": "rcases vt with ⟨c, d, ⟨a, ha⟩, ⟨vt1, vt2⟩⟩",
      "model_output": "rcases vt with ⟨c, d, ⟨⟨a, ha⟩, ⟨cinf, cinf'⟩⟩⟩"
    },
    {
      "id": 101,
      "match": false,
      "ground_truth": "exact hu.2.2 (mem_inter hu.2.1 hx)",
      "model_output": "have : u ∩ s = s := by\n    ext y\n    constructor\n    · intro hy\n      exact ⟨hu.2.1 hy, hy.2⟩\n    intro hy\n    exact ⟨hu.2.1 hy, hy.2⟩\n    exact this"
    },
    {
      "id": 102,
      "match": false,
      "ground_truth": "have h1 :A x j + ε + ∑ x_1 with ¬x_1 = j, A x x_1 = ε + A x j + ∑ x_1\n              with ¬x_1 = j, A x x_1 := by ring_nf",
      "model_output": "rw [add_comm, add_assoc, add_comm (A x j), add_assoc, add_comm (A x j)]"
    },
    {
      "id": 103,
      "match": true,
      "ground_truth": "intro x",
      "model_output": "intro x"
    },
    {
      "id": 104,
      "match": false,
      "ground_truth": "apply LinearMap.congr_fun\n      ((LinearEquiv.symm_apply_eq (Module.dualProdDualEquivDual ℝ E ℝ)).1 (by rfl)) a",
      "model_output": "simp [Module.dualProdDualEquivDual_apply_apply]"
    },
    {
      "id": 105,
      "match": false,
      "ground_truth": "rw[Metric.isOpen_iff] at isopen",
      "model_output": "rcases ht with ⟨ht1, ht2⟩\n    have : ConvexOn ℝ (t ∩ s) f := by\n      apply ConvexOn.inter hf (convex_of_convexOn_convex_subset hf ht2.1)\n    apply lipschitzOnWith_of_lipschitzOnWith_of_subset\n    · apply lipschitzOnWith_of_lipschitzOnWith_of_subset\n      · apply lipschitzOnWith_of_lipschitzOnWith_of_subset\n        · apply lipschitzOnWith_of_lipschitzOnWith_of_subset\n          · apply lipsch"
    },
    {
      "id": 106,
      "match": false,
      "ground_truth": "rw [deriv_of_const_mul_func (f u - f x)]",
      "model_output": "have h1 : deriv φ (f u - f x) > 0 := by\n      rw [deriv_sub_const]\n      apply deriv_pos_of_deriv_pos_of_hasDerivAt\n      · apply HasDerivAt.const_sub\n        · apply hasDerivAt_inv\n        · simp\n      · apply HasDerivAt.const_sub\n        · apply hasDerivAt_id'\n        · simp"
    },
    {
      "id": 107,
      "match": false,
      "ground_truth": "rcases exists_seq_tendsto_sInf hs hs' with ⟨fx, _, cfx, fxs⟩",
      "model_output": "rcases hs with ⟨z, hz⟩\n  use z\n  intro x _\n  exact le_csInf hs' hz x"
    },
    {
      "id": 108,
      "match": false,
      "ground_truth": "induction n with",
      "model_output": "have h₁ : Φ n ≤ C := Φ_le_C n\n    have h₂ : Φ n ≥ 0 := Φ_nonneg n\n    linarith [h₁, h₂]"
    },
    {
      "id": 109,
      "match": false,
      "ground_truth": "simp at ht",
      "model_output": "simp at ht [Icc]"
    },
    {
      "id": 110,
      "match": false,
      "ground_truth": "· intro i; simp [c1];\n      by_cases ht : i ∈ σ\n      · simp [ht];\n        specialize cpos i ht;\n        exact cpos\n      simp [ht]",
      "model_output": "· intro i\n    simp [c1]\n    apply cpos i i.2"
    },
    {
      "id": 111,
      "match": false,
      "ground_truth": "rw [subdiff_gradient g g' hg gradcon]",
      "model_output": "rw [HasGradientAt_iff_Subdiff]"
    },
    {
      "id": 112,
      "match": false,
      "ground_truth": "rw [div_mul_cancel₀ _ hne, abs_of_nonneg (by positivity), mul_comm] at hd",
      "model_output": "rw [abs_of_nonneg (by positivity)] at hd\n            rw [mul_div_assoc, div_mul_cancel₀ _ hne] at hd\n            apply le_of_mul_le_mul_of_pos hd (by positivity) (by positivity) (by positivity)"
    },
    {
      "id": 113,
      "match": false,
      "ground_truth": "simp [phi_add] at this",
      "model_output": "rw [phi_add] at this"
    },
    {
      "id": 114,
      "match": false,
      "ground_truth": "have h_eq : {z | f z ≤ α} = f ⁻¹' Iic α ∩ univ := by\n      ext z\n      simp [Set.mem_setOf_eq, Set.mem_preimage, Set.mem_Iic]",
      "model_output": "simp at hu_eq"
    },
    {
      "id": 115,
      "match": false,
      "ground_truth": "lift t to ℝ using ⟨LT.lt.ne_top fx_le_top, LT.lt.ne_bot fx_ge_bot⟩",
      "model_output": "rw [← EReal.coe_toReal fx_ge_bot fx_le_top]"
    },
    {
      "id": 116,
      "match": false,
      "ground_truth": "rcases zin with ⟨u,hux,hufx,hv⟩",
      "model_output": "rcases zin with ⟨u, ⟨uconv, uconv2, uconv3⟩⟩"
    },
    {
      "id": 117,
      "match": false,
      "ground_truth": "by linarith",
      "model_output": "linarith [h.1]"
    },
    {
      "id": 118,
      "match": false,
      "ground_truth": "apply convexHull_min (union_subset h1 h2) cone_conv_s_t.convex",
      "model_output": "apply convexHull_min ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?"
    },
    {
      "id": 119,
      "match": false,
      "ground_truth": "have hΦ : ∃ r_Φ, range Φ ⊆ Metric.ball 0 r_Φ := by apply Φ_isBounded'",
      "model_output": "rcases admm_kkt with ⟨_, h₁, h₂⟩\n    use max (‖A₁‖ + ‖A₂‖) 1\n    intro x\n    rw [range_subset_iff]\n    intro x\n    rw [Function.comp_apply, Function.comp_apply]\n    apply add_mem_ball_zero\n    apply add_mem_ball_zero\n    · apply h₁\n    · apply h₂"
    },
    {
      "id": 120,
      "match": true,
      "ground_truth": "· simp",
      "model_output": "simp"
    },
    {
      "id": 121,
      "match": false,
      "ground_truth": "have : Tendsto (fun n => ‖A₁ (e₁ n) + A₂ (e₂ n)‖ ^ 2) atTop (𝓝 0) := by\n      apply Summable.tendsto_atTop_zero Summable₁\n   apply square_converge_zero₁ this",
      "model_output": "have h₁ : Tendsto (fun n => ‖A₁ (e₁ n) + A₂ (e₂ n)‖) atTop (𝓝 ‖A₁ (e₁) + A₂ (e₂)‖) := by\n      apply Continuous.tendsto; apply Continuous.norm; apply Continuous.add\n      · apply ContinuousLinearMap.continuous\n      · apply ContinuousLinearMap.continuous\n    have h₂ : ‖A₁ (e₁) + A₂ (e₂)‖ = 0 := by\n      rw [e₁_converge_zero, e"
    },
    {
      "id": 122,
      "match": false,
      "ground_truth": "rw[← EReal.coe_toReal (x := f x) hfx (LT.lt.ne_bot (hsf.uninfinity x hx))]",
      "model_output": "rw [← EReal.coe_toReal (hsf.ne_bot_of_convex hf hfcl hs x hx) hfx]"
    },
    {
      "id": 123,
      "match": false,
      "ground_truth": "by_cases u₁_eq_0 : u₁ = 0",
      "model_output": "have : u₁ ∈ (span 𝕜 (Set.range hT.v))ᗮ := by\n    rw [hu₁]; apply Finsupp.sum_mem_span; intro x _; apply smul_mem"
    },
    {
      "id": 124,
      "match": false,
      "ground_truth": "apply mul_pos",
      "model_output": "linarith [kpos]"
    },
    {
      "id": 125,
      "match": true,
      "ground_truth": "exact this",
      "model_output": "exact this"
    },
    {
      "id": 126,
      "match": false,
      "ground_truth": "intro y",
      "model_output": "intro y hy_neq\n    simp [C, hy_neq]"
    },
    {
      "id": 127,
      "match": true,
      "ground_truth": "calc\n      _= inner ℝ (u (n) - (- A₁† y')) (x₁ (n) - x₁') := by simp\n      _≥ (0 : ℝ) := by\n         apply subgradientAt_mono\n         · apply u_inthesubgradient\n         · exact admm_kkt.h.subgrad₁",
      "model_output": "apply subgradientAt_mono\n        · apply u_inthesubgradient\n        · exact admm_kkt.h.subgrad₁"
    },
    {
      "id": 128,
      "match": false,
      "ground_truth": "simp [grad_fun_comp,grad_comp] at h",
      "model_output": "simp [grad_fun_comp, grad_snd] at h ⊢"
    },
    {
      "id": 129,
      "match": true,
      "ground_truth": "exact hu₂",
      "model_output": "exact hu₂"
    },
    {
      "id": 130,
      "match": false,
      "ground_truth": "apply Finset.card_eq_zero.2",
      "model_output": "have h1 : ∀ i, ∑ j, A i j = 1 := h_row_eq\n  have h2 : ∀ j, ∑ i, A i j = 1 := h_col_eq\n  simp [h1, h2]"
    },
    {
      "id": 131,
      "match": true,
      "ground_truth": "exact False.elim (by simp at this)",
      "model_output": "simp at this"
    },
    {
      "id": 132,
      "match": false,
      "ground_truth": "simp",
      "model_output": "linarith [sq_nonneg b]"
    },
    {
      "id": 133,
      "match": false,
      "ground_truth": "rw [← add_assoc, ← add_assoc]",
      "model_output": "rw [add_add_add_comm, add_comm (a • i.1), add_comm (b • j.1)]"
    },
    {
      "id": 134,
      "match": false,
      "ground_truth": "simp",
      "model_output": "rw [Metric.mem_ball, dist_zero_right]"
    },
    {
      "id": 135,
      "match": false,
      "ground_truth": "intro y yin z zin a b anneg bnneg absum1",
      "model_output": "intro x1 hx1 y hy a b ha hb hab\n    have h1 := hfun (by simp) (by simp) ha hb hab\n    have h2 := hfun (by simp) (by simp) ha hb hab\n    have h3 := hfun (by simp) (by simp) ha hb hab\n    have h4 := hfun (by simp) (by simp) ha hb hab\n    have h5 := hfun (by simp) (by simp) ha hb hab\n    have h6 := hfun (by simp) (by simp) ha hb hab"
    },
    {
      "id": 136,
      "match": false,
      "ground_truth": "gcongr",
      "model_output": "apply mul_lt_mul_of_pos_left _ (by simp) (by positivity) (by positivity)"
    },
    {
      "id": 137,
      "match": false,
      "ground_truth": "have hzd : ∀ n, d n = z n - loc := fun _ => eq_sub_of_add_eq' rfl",
      "model_output": "have hz : ∀ n, z n ∈ s1 := by\n    intro n\n    simp [s1]; apply εball\n    constructor\n    · simp [Metric.mem_ball]; linarith [Real.dist_eq.1 (Metric.mem_ball.1 (ha n (le_of_lt (by linarith))))]\n    · simp [s1]; apply ha n (le_of_lt (by linarith))"
    },
    {
      "id": 138,
      "match": false,
      "ground_truth": "rfl",
      "model_output": "have h1 : r1 = ‖A₂‖ := by apply norm_A₂\n      have h2 : r2 = ‖A₁‖ := by apply norm_A₁\n      rw[h1, h2]; exact Eq.symm (add_comm ‖A₁‖ ‖A₂‖)"
    },
    {
      "id": 139,
      "match": false,
      "ground_truth": "rw [mul_one] at htp",
      "model_output": "linarith [mul_one b] at htp"
    },
    {
      "id": 140,
      "match": false,
      "ground_truth": "apply Continuous.tendsto",
      "model_output": "apply Continuous.tendsto\n      apply ContinuousLinearMap.continuous"
    },
    {
      "id": 141,
      "match": false,
      "ground_truth": "by\n   have h₁: Tendsto A₂ (𝓝 0) (𝓝 (A₂ 0)) := by\n      apply Continuous.tendsto; apply ContinuousLinearMap.continuous\n   simp at h₁; apply Filter.tendsto_iff_seq_tendsto.1 h₁; apply e₂_subseq_converge_zero",
      "model_output": "have : (fun n => A₂ (e₂ (φ n))) = (fun n => (A₂ ∘ e₂) (φ n)) := by rfl\n    rw[this]\n    apply Filter.tendsto_iff_seq_tendsto.1 (A₂.continuous.tendsto 0)\n    apply e₂_subseq_converge_zero fullrank₁ fullrank₂\n    apply Filter.tendsto_iff_seq_tendsto.1 (φ_subseq_converge_zero fullrank₁ fullrank₂)\n    apply Filter.tendsto_iff_seq_tendsto.1 (A₂.contin"
    },
    {
      "id": 142,
      "match": false,
      "ground_truth": "apply A₁e₁_converge_zero",
      "model_output": "have h₁ : Tendsto A₁ (𝓝 0) (𝓝 (A₁ 0)) := by\n      apply Continuous.tendsto; apply ContinuousLinearMap.continuous\n    simp at h₁; apply Filter.tendsto_iff_seq_tendsto.1 h₁; apply e₁_subseq_converge_zero\n    apply Filter.tendsto_iff_seq_tendsto.1 h₁; apply e₁_subseq_converge_zero\n    apply Filter.tendsto_iff_seq_tendsto.1 h₁; apply e₁_subseq_converge_zero\n    apply Filter"
    },
    {
      "id": 143,
      "match": true,
      "ground_truth": "intro j",
      "model_output": "intro j"
    },
    {
      "id": 144,
      "match": false,
      "ground_truth": "rw[this]",
      "model_output": "rw [this]"
    },
    {
      "id": 145,
      "match": false,
      "ground_truth": "· simp; linarith [μpos]",
      "model_output": "apply inv_nonneg.mpr μpos.le"
    },
    {
      "id": 146,
      "match": false,
      "ground_truth": "ext x",
      "model_output": "ext x\n            rw [mul_div_assoc, div_mul_comm, mul_div_assoc, div_self, mul_one, mul_div_assoc]\n            rw [mul_div_assoc, div_self, mul_one]\n            ring_nf"
    },
    {
      "id": 147,
      "match": false,
      "ground_truth": "have xBall_in_U : xBall ⊆ U := by\n    apply l1Ball_sub_convexHull r_pos hn",
      "model_output": "have xBall_in_U : xBall ⊆ U := by\n    apply subset_interior_of_mem_nhds\n    exact mem_nhdsWithin.mpr ⟨xBall, xBall_isopen, x_in_xBall⟩"
    },
    {
      "id": 148,
      "match": false,
      "ground_truth": "rw [← zero_pow]",
      "model_output": "have : Tendsto (fun n => ‖A₁ (e₁ (φ n)) + A₂ (e₂ (φ n))‖ ^ 2) atTop (𝓝 (‖A₁ (e₁ (φ)) + A₂ (e₂ (φ))‖ ^ 2)) := by\n      apply Filter.tendsto_pow\n      apply tendsto_norm_add_A₁e₁_A₂e₂_subseq_converge_zero fullrank₁ fullrank₂\n    rw [this]\n    apply Filter.tendsto_iff_seq_tendsto.1 (norm_subseq_converge_zero"
    },
    {
      "id": 149,
      "match": false,
      "ground_truth": "apply mul_nonneg",
      "model_output": "apply mul_nonneg\n    · apply mul_nonneg\n      · apply le_of_lt h1\n      · apply le_of_lt h2\n    · apply sq_nonneg"
    },
    {
      "id": 150,
      "match": false,
      "ground_truth": "by simp at h; simpa [h]",
      "model_output": "simp at h ⊢"
    },
    {
      "id": 151,
      "match": false,
      "ground_truth": "rw [← mul_neg]",
      "model_output": "rw [← neg_inj, neg_neg, ← mul_neg, neg_inj, neg_neg, neg_inj, neg_neg]"
    },
    {
      "id": 152,
      "match": true,
      "ground_truth": "exact h",
      "model_output": "exact h"
    },
    {
      "id": 153,
      "match": false,
      "ground_truth": "by rw [z]; simp; left; rw [z]; simp",
      "model_output": "rw [z]; simp; left; rw [z]; simp; left; rw [z]; simp; left; rw [z]; simp; left; simp [dk k]"
    },
    {
      "id": 154,
      "match": false,
      "ground_truth": "· simp\n      apply neg_le.mpr\n      · have pos: - 0 ≤ ε * ‖y - x‖:=by\n          simp\n          rw [mul_comm]\n          apply (mul_nonneg_iff_left_nonneg_of_pos εpos).mpr\n          simp\n        apply pos",
      "model_output": "apply neg_mul_le_zero.mpr"
    },
    {
      "id": 155,
      "match": false,
      "ground_truth": "rcases this with ⟨z_, _, α, ⟨monoa, conv⟩⟩",
      "model_output": "rcases this with ⟨z_,_,α,⟨monotoneα,converge⟩⟩"
    },
    {
      "id": 156,
      "match": false,
      "ground_truth": "intro n x",
      "model_output": "intro n x\n  apply ADMM_iter_process₂' n x\n  apply ADMM_iter_process₁' n x\n  apply ADMM_iter_process₂' n x\n  apply ADMM_iter_process₁' n x\n  apply ADMM_iter_process₂' n x\n  apply ADMM_iter_process₁' n x\n  apply ADMM_iter_process₂' n x\n  apply ADMM_iter_process₁' n x\n  apply ADMM_iter_process₂' n x\n  apply ADMM_iter_process₁' n x\n  apply ADMM_iter_process₂' n x\n  apply AD"
    },
    {
      "id": 157,
      "match": false,
      "ground_truth": "simp [coef.mem_support_toFun]; rfl",
      "model_output": "simp [Finset.mem_filter, Ne]"
    },
    {
      "id": 158,
      "match": false,
      "ground_truth": "simp only [Set.mem_union, Set.mem_iUnion, Set.mem_singleton_iff]",
      "model_output": "by_cases h : i = n + 1"
    },
    {
      "id": 159,
      "match": false,
      "ground_truth": "have h2 : ‖x + xx - xx‖ ≤ ‖x + xx‖ + ‖xx‖ := by apply norm_sub_le",
      "model_output": "rw [add_sub, add_sub_assoc, add_sub_cancel, add_comm, add_sub, add_sub_assoc, add_sub_cancel] at h1"
    },
    {
      "id": 160,
      "match": true,
      "ground_truth": "apply eq4 u v",
      "model_output": "apply eq4 u v"
    },
    {
      "id": 161,
      "match": false,
      "ground_truth": "have : IsClosed ((MM x).carrier ∩ (f.Epi univ)) := by\n    rw [ht]; exact IsClosed.prod isClosed_singleton isClosed_Ici",
      "model_output": "have h14 : (MM x).carrier ∩ intrinsicInterior (dom univ f) ≠ ∅ := by\n    rw [intrinsicInterior_epi_eq_epi_intrinsicInterior hc hx]\n    exact hi'"
    },
    {
      "id": 162,
      "match": false,
      "ground_truth": "simp [z]",
      "model_output": "rw [convex_iff_segment_subset] at hs\n    apply hs x xin y yin\n    simp; constructor\n    · linarith\n    · linarith\n    · simp; linarith [absum1]"
    },
    {
      "id": 163,
      "match": false,
      "ground_truth": "by\n   apply Tendsto.norm\n   apply Tendsto.sub_const\n   apply Tendsto.add\n   · apply Satisfying_equational_constraint1\n   · apply Satisfying_equational_constraint2",
      "model_output": "apply Filter.tendsto_iff_seq_tendsto.1\n      (Continuous.tendsto (fun x => ‖A₁ x₁ + A₂ x₂ - b‖) (𝓝 (A₁ x₁ + A₂ x₂ - b)) _)\n      (x₁_subseq_converge x₂_subseq_converge)"
    },
    {
      "id": 164,
      "match": false,
      "ground_truth": "· have h0 : ∑ j, C x j = ∑ j, A x j := by\n              dsimp [C]\n              simp [h]\n            rw [← h0]\n            exact h_sum_lt",
      "model_output": "have h1 : ∑ j, C x j = ∑ j, A x j := by\n        have h2 : x ≠ i := h\n        have h3 : ∑ j, C x j = ∑ j, A x j := by\n          apply Finset.sum_congr\n          · simp\n          · intro y _\n            exact hC_DoublySubStochastic.1 y x h3\n        exact h3\n      rw [h1] at h_sum_lt\n      exact h_sum_lt"
    },
    {
      "id": 165,
      "match": false,
      "ground_truth": "dsimp [ey_diff, t_Ax_Ax_n1, t_A_diff_x', A2_e2_n1, A2_e2_n]\n            linarith [Φ_isdescending_inequ5 n]",
      "model_output": "rw [ey_diff, t_Ax_Ax_n1, t_A_diff_x', A2_e2_n1, A2_e2_n]"
    },
    {
      "id": 166,
      "match": false,
      "ground_truth": "intro n",
      "model_output": "intro n\n  apply ADMM_iter_process₁ n"
    },
    {
      "id": 167,
      "match": false,
      "ground_truth": "· apply ContinuousOn.comp (g := f₁) (f := fun x ↦ x + x₀) (t := univ) hcon\n        · apply ContinuousOn.add continuousOn_id continuousOn_const\n        · simp",
      "model_output": "· apply ContinuousOn.comp hcon (continuousOn_id.add continuousOn_const) fun x _ => x₀"
    },
    {
      "id": 168,
      "match": false,
      "ground_truth": "simp only [one_div, mul_inv_rev, add_sub_cancel_right]",
      "model_output": "rw [smul_smul, ← mul_assoc, mul_div_assoc, mul_div_assoc, mul_div_assoc, mul_div_assoc]\n        simp; field_simp; ring_nf; simp; field_simp; ring_nf; simp; field_simp; ring_nf; simp; field_simp; ring_nf; simp; field_simp; ring_nf; simp; field_simp; ring_nf; simp; field_simp; ring_nf; simp; field_simp; ring_nf; simp; field_simp; ring_nf; simp; field_simp; ring_nf; simp; field_simp; ring_nf; simp"
    },
    {
      "id": 169,
      "match": false,
      "ground_truth": "suffices 0 ≤ (a1 • tx + b1 • ty) by exact this",
      "model_output": "intro i\n    simp [tx, ty]\n    apply add_nonneg\n    · apply smul_nonneg\n      · linarith [txcond i]\n      · simp\n    · apply smul_nonneg\n      · linarith [bnng]\n      · simp [tycond i]"
    },
    {
      "id": 170,
      "match": false,
      "ground_truth": "by\n        rw [sub_right_comm]",
      "model_output": "rw [sub_right_comm, norm_sub_rev]"
    },
    {
      "id": 171,
      "match": false,
      "ground_truth": "intro i hi",
      "model_output": "intro x hx\n    apply iSup₂_le\n    intro y hy\n    exact ht1 y hy"
    },
    {
      "id": 172,
      "match": false,
      "ground_truth": "apply le_trans (Nat.le_add_left ax N)",
      "model_output": "apply le_trans (Nat.le_add_left ax (φ n)) (hax (φ n) (hmφ n))"
    },
    {
      "id": 173,
      "match": false,
      "ground_truth": "rw [hdomain]; trivial",
      "model_output": "rw [hdomain]; simp [z]"
    },
    {
      "id": 174,
      "match": false,
      "ground_truth": "rfl",
      "model_output": "simp [xeq]"
    },
    {
      "id": 175,
      "match": false,
      "ground_truth": "exact trivial",
      "model_output": "simp [Finset.mem_erase] at *"
    },
    {
      "id": 176,
      "match": false,
      "ground_truth": "simp",
      "model_output": "apply sq_nonneg"
    },
    {
      "id": 177,
      "match": false,
      "ground_truth": "intro _ epos",
      "model_output": "intro ε εpos"
    },
    {
      "id": 178,
      "match": true,
      "ground_truth": "intro n",
      "model_output": "intro n"
    },
    {
      "id": 179,
      "match": false,
      "ground_truth": "simp",
      "model_output": "rw [abs_of_nonneg (by positivity), mul_pow, mul_pow, mul_assoc, mul_assoc, mul_assoc, mul_assoc]\n        field_simp [hσ_zero, hu_zero]"
    },
    {
      "id": 180,
      "match": false,
      "ground_truth": "· simp",
      "model_output": "simp [Metric.tendsto_nhds]"
    },
    {
      "id": 181,
      "match": false,
      "ground_truth": "simp at hu",
      "model_output": "apply h_Ω1 u (hO1 hu.1) hu.2.1 hu.2.2"
    },
    {
      "id": 182,
      "match": false,
      "ground_truth": "lift ⨆ u, Real.toEReal (inner ℝ u x) - f u to ℝ with c",
      "model_output": "apply add_le_add supa supb"
    },
    {
      "id": 183,
      "match": false,
      "ground_truth": "simp at vt1",
      "model_output": "have vt2 : ∀ᶠ a : ℕ in atTop, ‖1 / c a‖ ≤ ‖v‖ + ε := by\n      apply Filter.Eventually.mp (tendsto_one_div_atTop_nhds_zero_nat.eventually (gt_mem_nhds εpos)) _\n      simp [Metric.mem_nhds_iff]"
    },
    {
      "id": 184,
      "match": false,
      "ground_truth": "rw [pow_two]",
      "model_output": "rw [mul_assoc, ← mul_div_assoc, mul_div_assoc, mul_div_assoc, mul_div_assoc, ← mul_div_assoc]\n        simp; field_simp; linarith [alg.tpos]"
    },
    {
      "id": 185,
      "match": false,
      "ground_truth": "rcases exist_open_sub_finite_cover with ⟨t, ht1, ht2, ht3⟩",
      "model_output": "rcases exist_open_sub_finite_cover with ⟨b', h1, h2, h3⟩"
    },
    {
      "id": 186,
      "match": false,
      "ground_truth": "intro b _",
      "model_output": "intro b hb"
    },
    {
      "id": 187,
      "match": false,
      "ground_truth": "ring",
      "model_output": "rw [mul_assoc,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,mul_comm,m"
    },
    {
      "id": 188,
      "match": true,
      "ground_truth": "exact hcf",
      "model_output": "exact hcf"
    },
    {
      "id": 189,
      "match": false,
      "ground_truth": "ring_nf",
      "model_output": "rw [add_comm, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub,"
    },
    {
      "id": 190,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp [affSpan_iff]"
    },
    {
      "id": 191,
      "match": false,
      "ground_truth": "simp [FeasSet, FeasPoint] at xf",
      "model_output": "rw [hab]"
    },
    {
      "id": 192,
      "match": false,
      "ground_truth": "· simp [hx]",
      "model_output": "· left; right; exact hx"
    },
    {
      "id": 193,
      "match": false,
      "ground_truth": "have eq0 : inner ℝ (f' (z + d • (y - z)) - f' (x + c • (z - x))) (y - x) = (0 : ℝ) := by\n    contrapose! eq3\n    rw [mul_ne_zero_iff]\n    constructor\n    · rw [mul_ne_zero_iff]; constructor <;> linarith\n    · exact eq3",
      "model_output": "have : a • b • (y - x) = b • (y - x) := by\n      rw [smul_comm, smul_comm b]\n      exact smul_left_cancel₀ (by linarith) (by simp) (by simp)"
    },
    {
      "id": 194,
      "match": false,
      "ground_truth": "by ring",
      "model_output": "rw [one_div, mul_comm, mul_div_assoc, mul_div_assoc, mul_div_assoc, mul_div_assoc]"
    },
    {
      "id": 195,
      "match": false,
      "ground_truth": "by\n          rw[sign_mul_abs]\n          field_simp [bi_pos i]",
      "model_output": "rw [mul_div_assoc, div_self, mul_div_assoc, div_self, mul_div_assoc, div_self, mul_one]"
    },
    {
      "id": 196,
      "match": false,
      "ground_truth": "have : (fun n => Φ (φ n)) = (fun n => (1 / (τ * ρ)) * ‖ey (φ n)‖^2 +\n       ρ * ‖A₂ (e₂ (φ n))‖^2 +\n       ((max (1-τ) (1-(1/τ)))*ρ) * ‖A₁ (e₁ (φ n)) + A₂ (e₂ (φ n))‖^2) := by\n      funext n; rw [Φ, Ψ]",
      "model_output": "have h_add'' := Filter.tendsto_add h_add' tendsto_const_nhds"
    },
    {
      "id": 197,
      "match": true,
      "ground_truth": "· simp [V₂]",
      "model_output": "· simp [V₂]"
    },
    {
      "id": 198,
      "match": false,
      "ground_truth": "intro s",
      "model_output": "intro s g hs hsum i"
    },
    {
      "id": 199,
      "match": false,
      "ground_truth": "rw[zero_add]",
      "model_output": "rw [zero_add]"
    },
    {
      "id": 200,
      "match": false,
      "ground_truth": "linarith",
      "model_output": "linarith [this, kleaw]"
    },
    {
      "id": 201,
      "match": false,
      "ground_truth": "repeat apply mul_nonneg",
      "model_output": "· apply mul_nonneg\n        · apply mul_nonneg\n          · linarith [inv_pos.mpr h1]\n          · apply sq_nonneg\n        · apply sq_nonneg"
    },
    {
      "id": 202,
      "match": false,
      "ground_truth": "exact Eq.symm (Nat.add_sub_of_le kge)",
      "model_output": "rw [add_comm, Nat.sub_add_cancel]"
    },
    {
      "id": 203,
      "match": true,
      "ground_truth": "rw [Filter.tendsto_congr' this]",
      "model_output": "rw [Filter.tendsto_congr' this]"
    },
    {
      "id": 204,
      "match": false,
      "ground_truth": "intro c1",
      "model_output": "intro c1 hc1\n    apply Eventually.mp (tendsto_zero_iff_norm_tendsto_zero.mp (t5)) hc1"
    },
    {
      "id": 205,
      "match": false,
      "ground_truth": "exact min_le_right δ₀ δ₂",
      "model_output": "apply min_le_right δ₀ δ₂"
    },
    {
      "id": 206,
      "match": false,
      "ground_truth": "simp",
      "model_output": "rw [alg.step₁ k, alg.step₁ k]"
    },
    {
      "id": 207,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp [hσ_pos'.1]"
    },
    {
      "id": 208,
      "match": false,
      "ground_truth": "rw[← mul_div]",
      "model_output": "ring_nf"
    },
    {
      "id": 209,
      "match": false,
      "ground_truth": "simp",
      "model_output": "apply Nat.zero_le"
    },
    {
      "id": 210,
      "match": false,
      "ground_truth": "intro c",
      "model_output": "intro c hc"
    },
    {
      "id": 211,
      "match": false,
      "ground_truth": "intro i",
      "model_output": "intro b hb"
    },
    {
      "id": 212,
      "match": false,
      "ground_truth": "dsimp [M]",
      "model_output": "rw [M]"
    },
    {
      "id": 213,
      "match": false,
      "ground_truth": "intro v",
      "model_output": "intro v\n    simp [ContinuousLinearMap.coe_toLinearMap]"
    },
    {
      "id": 214,
      "match": false,
      "ground_truth": "by exact this",
      "model_output": "simp [u, v]"
    },
    {
      "id": 215,
      "match": true,
      "ground_truth": "apply lt_min δ₀pos (lt_min δ₄pos δ₅pos)",
      "model_output": "apply lt_min δ₀pos (lt_min δ₄pos δ₅pos)"
    },
    {
      "id": 216,
      "match": false,
      "ground_truth": "apply εpos",
      "model_output": "linarith [εpos]"
    },
    {
      "id": 217,
      "match": false,
      "ground_truth": "linarith [eq1]",
      "model_output": "linarith [alg.γbound k]"
    },
    {
      "id": 218,
      "match": false,
      "ground_truth": "· apply Finset.sum_pos\n            · intro i _; apply (alg.ha i)\n            · simp",
      "model_output": "apply Finset.sum_pos\n    · intro i _; apply (alg.ha i)\n    · simp [alg.ha]"
    },
    {
      "id": 219,
      "match": false,
      "ground_truth": "linarith [a1leab]",
      "model_output": "linarith [monoa (by linarith [Nat.lt_of_add_one_le a1leab])]"
    },
    {
      "id": 220,
      "match": false,
      "ground_truth": "rw [←sub_add,norm_add_sq_real,←sub_add,norm_add_sq_real] at ieq",
      "model_output": "rw [← add_assoc] at ieq"
    },
    {
      "id": 221,
      "match": false,
      "ground_truth": "simp at din",
      "model_output": "constructor <;> linarith [din.1, din.2]"
    },
    {
      "id": 222,
      "match": false,
      "ground_truth": "have : (alg.x (α q -1),alg.y (α q -1)) ∈ Metric.closedBall (0 : WithLp 2 (E × F)) C1 := by\n        apply mem_closedBall_iff_norm.mpr; simp\n        apply inin (alg.x (α q -1),alg.y (α q -1))\n        exact mem_image_of_mem alg.z trivial",
      "model_output": "apply sq (alg.z (α q - 1)) _"
    },
    {
      "id": 223,
      "match": false,
      "ground_truth": "by\n          rw [WithLp.prod_norm_eq_of_L2]; simp\n          exact (Real.le_sqrt (norm_nonneg _) (Left.add_nonneg (sq_nonneg _) (sq_nonneg _))).mpr\n            (le_add_of_nonneg_right (sq_nonneg _))",
      "model_output": "apply norm_fst_le_prod"
    },
    {
      "id": 224,
      "match": false,
      "ground_truth": "intro a t hat IH w hw_pos' hw_sum' hmem'",
      "model_output": "intro a s _ h1 h2 h3 h4 h5\n    have h1' : a ∉ s := by assumption\n    have h2' : ∀ i ∈ s, 0 ≤ w i := by\n      intro i hi\n      apply h2\n      simp\n      exact hi\n    have h3' : ∑ i ∈ s, w i = 1 := by\n      apply h3\n      simp\n    have h4' : ∀ i ∈ s, i ∈ (convexHull ℝ) ↑u' := by\n      intro i hi\n      apply h4"
    },
    {
      "id": 225,
      "match": false,
      "ground_truth": "by\n        apply add_lt_add\n        · apply ha₁ b hba₁\n        · apply ha₂ b hba₂",
      "model_output": "apply add_lt_add (ha₁ b hba₁) (ha₂ b hba₂)"
    },
    {
      "id": 226,
      "match": false,
      "ground_truth": "exact inv_sub_inv l' h₁",
      "model_output": "rw [inv_sub_inv l' h₁]"
    },
    {
      "id": 227,
      "match": true,
      "ground_truth": "apply hpos'",
      "model_output": "apply hpos'"
    },
    {
      "id": 228,
      "match": false,
      "ground_truth": "apply Set.add_subset_add_right",
      "model_output": "apply add_subset_add_right\n    (subset_iUnion₂_of_subset i₀ (subset_iUnion_of_subset i₀ (subset_iUnion_of_subset\n    (subset_iUnion_of_subset i₀ (subset_iUnion_of_subset i₀ (subset_iUnion_of_subset\n    (subset_iUnion_of_subset i₀ (subset_iUnion_of_subset i₀ (subset_iUnion_of_subset\n    (subset_iUnion_of_subset i₀ (subset_iUnion_of_subset i₀ (subset_iUnion_of_subset\n    (subset_iUnion_of_subset i₀ (subset_iUnion_of_subset i₀ (subset_iUnion_of_subset"
    },
    {
      "id": 229,
      "match": false,
      "ground_truth": "rw [y1, y2]",
      "model_output": "rw [y1, y2] at h5"
    },
    {
      "id": 230,
      "match": true,
      "ground_truth": "linarith [hγ]",
      "model_output": "linarith [hγ]"
    },
    {
      "id": 231,
      "match": true,
      "ground_truth": "apply norm_sub_le",
      "model_output": "apply norm_sub_le"
    },
    {
      "id": 232,
      "match": false,
      "ground_truth": "have h1 := hk (fun q ↦ alg.x (α q) - alg.x (α q - 1)) diflte1 (ε / (γ * l) / 3) (by positivity)",
      "model_output": "apply Eventually.mp (diflte1 ε epos) (1 / (γ * l)) (by positivity)"
    },
    {
      "id": 233,
      "match": false,
      "ground_truth": "intro n",
      "model_output": "intro n\n    apply And.intro\n    · exact (allh' (n + 1) (Nn n)).1\n    apply Filter.tendsto_iff_seq_tendsto.2\n    intro k\n    apply (allh' (n + 1) (Nn n)).2.2\n    apply Filter.tendsto_iff_seq_tendsto.1 (vnkt (n + 1))\n    apply Filter.tendsto_iff_seq_tendsto.1 (vnkt n)\n    apply Filter.tendsto_iff_seq_tendsto.1 (xnkt (n + 1"
    },
    {
      "id": 234,
      "match": false,
      "ground_truth": "by\n        apply zp5",
      "model_output": "rw [zp5]"
    },
    {
      "id": 235,
      "match": false,
      "ground_truth": "by\n        apply Simplifying₂\n        apply h₁\n        apply norm_nonneg",
      "model_output": "rw [mul_assoc, mul_assoc, mul_comm, mul_assoc, mul_comm, mul_assoc, mul_comm, mul_comm]"
    },
    {
      "id": 236,
      "match": false,
      "ground_truth": "refine LICQ_injM z m Z Ax ?_ mlen ?_ eq2 eq1 ⟨heq1, heq2⟩",
      "model_output": "have : Zᵀ *ᵥ z = 0 := heq2\n    have : Zᵀ *ᵥ z = 0 := this\n    have : Zᵀ *ᵥ z = 0 := this\n    have : Zᵀ *ᵥ z = 0 := this\n    have : Zᵀ *ᵥ z = 0 := this\n    have : Zᵀ *ᵥ z = 0 := this\n    have : Zᵀ *ᵥ z = 0 := this\n    have : Zᵀ *ᵥ z = 0 := this\n    have : Zᵀ *ᵥ z = 0 := this"
    },
    {
      "id": 237,
      "match": false,
      "ground_truth": "apply le_trans",
      "model_output": "apply le_trans Nge (by linarith)"
    },
    {
      "id": 238,
      "match": false,
      "ground_truth": "apply mul_eq_mul_left_iff.mpr",
      "model_output": "rw [norm_smul, norm_div, norm_one, one_mul]"
    }
  ]
}