evaluation/valid_score_in_200.json

{
  "summary": {
    "total_with_gt": 239,
    "matched": 46,
    "accuracy": 0.19246861924686193
  },
  "details": [
    {
      "id": 0,
      "match": false,
      "ground_truth": "apply HasDerivAt.sub",
      "model_output": "apply HasDerivAt.sub\n    · exact hasDerivAt_const t 1\n    exact hasDerivAt_id' t"
    },
    {
      "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": true,
      "ground_truth": "rw [← EReal.coe_sub]",
      "model_output": "rw [← EReal.coe_sub]"
    },
    {
      "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": "lift a to ℝ using ⟨ha1, ha2⟩\n  lift b to ℝ using ⟨hb1, hb2⟩\n  exact bot_lt_iff_ne_bot.2 (sub_ne_bot_of_real ha1 hb1)"
    },
    {
      "id": 6,
      "match": false,
      "ground_truth": "lift a to ℝ using ⟨ha1, ha2⟩",
      "model_output": "have : b - b = 0 := by exact EReal.coe_sub_coe_eq_zero_iff.mpr rfl\n      rw [this]; simp"
    },
    {
      "id": 7,
      "match": true,
      "ground_truth": "apply hasDerivAt_mul_const c⁻¹",
      "model_output": "apply hasDerivAt_mul_const"
    },
    {
      "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": "have := 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 : (g' (⨆ m ∈ s, f m)) = ⨆ m ∈ s, g' (f m) := by\n      apply OrderIso.map_biSup g' s f\n      exact ⟨fun x => x, fun x => x, fun x y => by simp⟩\n      exact ⟨fun x => x, fun x => x, fun x y => by simp⟩\n      exact ⟨fun x => x, fun x => x, fun x y => by simp⟩\n      exact ⟨fun x => x, fun x => x, fun x y => by 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": "have : g = 1 := by\n    simp at hg\n    linarith [hg]"
    },
    {
      "id": 13,
      "match": true,
      "ground_truth": "apply add_le_add_right a",
      "model_output": "apply add_le_add_right 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\n  simp only [one_apply_eq, mul_apply]\n  split_ifs with h\n  · exact (hdiag i).choose_spec\n  · exact False.elim (h (A i i))"
    },
    {
      "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 hs\n    simp [convex_indicator] at h\n    rw [h] at hs\n    exact hs rfl"
    },
    {
      "id": 18,
      "match": false,
      "ground_truth": "constructor",
      "model_output": "constructor\n  · intro h x hx\n    simp at *\n    exact ⟨hx.1, le_trans (h x.1 hx.1) hx.2⟩\n  intro h x hx\n  specialize h ⟨hx, le_refl (g x)⟩\n  simp at h\n  exact h.2"
    },
    {
      "id": 19,
      "match": false,
      "ground_truth": "by_cases hA : A = 0",
      "model_output": "intro x\n  apply HasGradientAt.const_mul\n  apply HasGradientAt.dotProduct\n  · apply HasGradientAt.const_mul\n    exact hasGradientAt_mulVec x\n  · exact hasGradientAt_mulVec x"
    },
    {
      "id": 20,
      "match": false,
      "ground_truth": "by norm_num",
      "model_output": "norm_num [a]"
    },
    {
      "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": "apply first_order_optimality_condition f x₀ hx"
    },
    {
      "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": "apply HasGradient_iff_Convergence_Point.mpr\n  intro ε εpos\n  use a\n  intro y\n  have : ⟪y, a⟫_ℝ - ⟪x, a⟫_ℝ = ⟪y - x, a⟫_ℝ := by rw [inner_sub_left]\n  rw [this]\n  apply real_inner_le_norm\n  apply le_of_lt εpos"
    },
    {
      "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": "refine (prod_erase_mul univ (fun i => A i i) (by exact mem_univ j)).symm"
    },
    {
      "id": 25,
      "match": false,
      "ground_truth": "intro x y hy",
      "model_output": "intro x y hy\n    by_cases hx : x ∈ C\n    · simp [hx] at hy; simp [hx]; exact eventually_of_forall (fun x' => hy.trans_le (if_le_right (by simp [hx])) )\n    simp [hx] at hy; simp [hx]; exact eventually_of_forall (fun x' => hy.trans_le (if_le_right (by simp [hx])) )"
    },
    {
      "id": 26,
      "match": true,
      "ground_truth": "simp [LowerSemicontinuousHull]",
      "model_output": "simp [LowerSemicontinuousHull]"
    },
    {
      "id": 27,
      "match": false,
      "ground_truth": "simp at hfy",
      "model_output": "have : f y = ⊤ := by\n      by_contra hfy'\n      exact (lt_of_le_of_ne hfy hfy').ne rfl"
    },
    {
      "id": 28,
      "match": false,
      "ground_truth": "rw [Fin.sum_univ_succ]",
      "model_output": "have : ∀ i, f i.succ ≠ ⊥ := fun i ↦ p i.succ"
    },
    {
      "id": 29,
      "match": true,
      "ground_truth": "by_cases hx2 : f x = ⊥",
      "model_output": "by_cases hx2 : f x = ⊥"
    },
    {
      "id": 30,
      "match": false,
      "ground_truth": "rw [@mem_iInter]",
      "model_output": "rw [mem_iInter] at hx ⊢\n  intro i\n  apply intrinsicClosure_mono (fun x a ↦ a i) hx"
    },
    {
      "id": 31,
      "match": false,
      "ground_truth": "rcases h with ⟨δ, dpos, h⟩",
      "model_output": "rcases h with ⟨δ, ⟨dpos, h⟩⟩\n  use δ; constructor\n  · exact dpos\n  intro x'; simp at h\n  rw [← dist_eq_norm]\n  apply h; simp; exact le_of_eq rfl"
    },
    {
      "id": 32,
      "match": false,
      "ground_truth": "exact half_lt_self dpos",
      "model_output": "linarith [dpos]"
    },
    {
      "id": 33,
      "match": true,
      "ground_truth": "apply pin",
      "model_output": "apply pin"
    },
    {
      "id": 34,
      "match": false,
      "ground_truth": "by_cases hf_pinf : f = ⊤",
      "model_output": "apply convex_conjugate_is_convex"
    },
    {
      "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 : intrinsicInterior ℝ (intrinsicInterior ℝ s) = intrinsicInterior ℝ (interior ℝ s) := by\n      rw [intrinsicInterior_equiv s]"
    },
    {
      "id": 36,
      "match": false,
      "ground_truth": "rw [← sub_add, norm_add_sq_real, real_inner_comm, inner_sub_right]",
      "model_output": "rw [← add_sub, norm_add_sq_real, inner_sub_right, real_inner_comm, inner_sub_right]"
    },
    {
      "id": 37,
      "match": false,
      "ground_truth": "rw [EReal.coe_le_coe_iff] at hccleq",
      "model_output": "apply EReal.coe_le_coe_iff.mpr"
    },
    {
      "id": 38,
      "match": false,
      "ground_truth": "intro z",
      "model_output": "apply intrinsicInterior_sub_intrinsicClosure_intrinsicInterior ℝ h hn"
    },
    {
      "id": 39,
      "match": true,
      "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 : FiniteDimensional 𝕜 (Submodule.span 𝕜 (Set.range v)) := by\n    apply FiniteDimensional.span_of_finite\n    simp [Set.finite_range]"
    },
    {
      "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": "have : ∀ x : s, ‖(innerSL ℝ) x‖ ≤ 1 := by\n      intro x\n      have : ‖(innerSL ℝ) x‖ ≤ ‖(innerSL ℝ)‖ * ‖x‖ := by\n        apply ContinuousLinearMap.le_opNorm\n      have : ‖x‖ ≤ 1 := by\n        have : x.1 ∈ s.convex_support x := by\n          rw [convex_support]\n          exact (mem_setOf_eq (by simp)).mpr (lt_top_iff_ne_top.mp (hy x).2)\n        rw [mem"
    },
    {
      "id": 42,
      "match": false,
      "ground_truth": "intro h\n    apply HasFDeriv_Convergence\n    exact h",
      "model_output": "intro h\n  apply h.fderiv\n  intro ε εpos\n  specialize h ε εpos\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  specialize h (ε / 2) (by positivity)\n  rcases h with ⟨δ, ⟨δpos, h⟩⟩\n  use δ; constructor\n  · exact δpos\n  intro x' x'eq\n  specialize h x' (by rw [x'eq]; exact norm_nonneg (x - x))\n  rw [mul_comm] at h; linarith [h]"
    },
    {
      "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 : (M : Set V) ∩ intrinsicInterior ℝ C = (M ∩ C) ∩ intrinsicInterior ℝ C := by\n      rw [← Set.inter_assoc, Set.inter_eq_self_of_subset_left inter_subset_left]"
    },
    {
      "id": 45,
      "match": true,
      "ground_truth": "apply alg.ha 0",
      "model_output": "apply alg.ha 0"
    },
    {
      "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, ⟨hn, hxn⟩⟩, hlt⟩"
    },
    {
      "id": 48,
      "match": false,
      "ground_truth": "have h2 := tendsto_add (A:=A) (B:=B)",
      "model_output": "have h2 := lim_inf_def B 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": "apply norm_smul_le"
    },
    {
      "id": 50,
      "match": false,
      "ground_truth": "by_cases ht : i ∈ σ",
      "model_output": "by_cases hi : i ∈ σ\n    · rw [hi]; simp; exact cpos i hi\n    · simp [hi]"
    },
    {
      "id": 51,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp at hst"
    },
    {
      "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": "have : (x, ⊤) ∈ closure {p | p.1 ∈ s ∧ f p.1 ≤ p.2} := by\n      have : (x, b) ∈ closure {p | p.1 ∈ s ∧ f p.1 ≤ p.2} := by\n        rw [hbtop]; exact ha\n      rwa [hbtop] at this"
    },
    {
      "id": 54,
      "match": false,
      "ground_truth": "intro hmin",
      "model_output": "intro hmin\n    apply first_order_unconstrained hf hmin hfc\n    simp [ConvexOn] at hcon\n    apply hcon.1 xm (by simp)"
    },
    {
      "id": 55,
      "match": false,
      "ground_truth": "simp [hx2, hw.2]",
      "model_output": "simp [hx2]"
    },
    {
      "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, add_sub, ← sub_sub, sub_eq_iff_eq_add]"
    },
    {
      "id": 57,
      "match": false,
      "ground_truth": "simp [ConvexCone.addPoint] at hx",
      "model_output": "rcases (Convex.mem_toCone hs).mp hx with ⟨c, hc, hcy⟩\n  use c\n  exact ⟨le_of_lt hc, hcy⟩"
    },
    {
      "id": 58,
      "match": false,
      "ground_truth": "intro x",
      "model_output": "intro x hx y hy\n  apply lagrange hs hf x hx y hy\n  simp [Ioo]"
    },
    {
      "id": 59,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp 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    simp at hp₁ hp₂ hp₃ ⊢\n    rw [map_vsub, smul_vsub_vadd_eq_vsub, ← map_smul, ← map_smul, hp₁, hp₂, hp₃]\n    simp\n    exact hp₃"
    },
    {
      "id": 61,
      "match": false,
      "ground_truth": "linarith",
      "model_output": "exact ht"
    },
    {
      "id": 62,
      "match": false,
      "ground_truth": "simp [h]",
      "model_output": "rw [LinearIsometryEquiv.symm_apply_apply] at h\n      exact h"
    },
    {
      "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": "rcases underboundf with ⟨b, ubf⟩"
    },
    {
      "id": 65,
      "match": false,
      "ground_truth": "by_cases ha1 : a = 0",
      "model_output": "simp [coe_mul_top_of_pos ha, h1]"
    },
    {
      "id": 66,
      "match": true,
      "ground_truth": "by_cases hb1 : b = 0",
      "model_output": "by_cases hb1 : b = 0"
    },
    {
      "id": 67,
      "match": false,
      "ground_truth": "congr",
      "model_output": "congr!"
    },
    {
      "id": 68,
      "match": false,
      "ground_truth": "intro z",
      "model_output": "intro y hy\n    have : HasFDerivAt f (gradient f y) y := by apply hasGradientAt_iff_hasFDerivAt.mp\n          have : gradient f y = (toDual ℝ E) ((gradient f y): E →L[ℝ] ℝ) := by simp\n          rw [this]\n          exact HasFDerivAt.unique hu3 y hy"
    },
    {
      "id": 69,
      "match": false,
      "ground_truth": "exact Batteries.compareOfLessAndEq_eq_lt.mp rfl",
      "model_output": "exact bot_lt_one'"
    },
    {
      "id": 70,
      "match": false,
      "ground_truth": "intro y",
      "model_output": "intro x' xdiff"
    },
    {
      "id": 71,
      "match": false,
      "ground_truth": "have hb : b = 1 := by linarith",
      "model_output": "simp [ha] at absum ⊢"
    },
    {
      "id": 72,
      "match": false,
      "ground_truth": "intro x y r",
      "model_output": "intro x y r\n    rw [← hEq]\n    exact affine_map_representation A x y r"
    },
    {
      "id": 73,
      "match": true,
      "ground_truth": "rw [hab]",
      "model_output": "rw [hab]"
    },
    {
      "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 z0 : z = 0"
    },
    {
      "id": 76,
      "match": false,
      "ground_truth": "rw [← hEq]",
      "model_output": "rw [← hEq]\n        exact affine_map_representation A 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, sub_eq_add_neg]"
    },
    {
      "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    apply Set.nonempty_def.mp\n    exact exists_ne 0"
    },
    {
      "id": 79,
      "match": false,
      "ground_truth": "linarith",
      "model_output": "linarith [hy.2]"
    },
    {
      "id": 80,
      "match": false,
      "ground_truth": "apply DifferentiableAt.hasGradientAt",
      "model_output": "apply DifferentiableAt.hasGradientAt\n      have : DifferentiableAt ℝ (fun x => (x, z.2)) (z.1, z.2) := by\n        apply DifferentiableAt.prodMk (differentiableAt_id') (differentiableAt_const)\n      apply DifferentiableAt.comp (diff_from_l2 h) this\n      simp [grad_fst]"
    },
    {
      "id": 81,
      "match": false,
      "ground_truth": "rcases lbdf with ⟨L, Lbound⟩",
      "model_output": "have eqImS : ImS = f '' univ := by\n      simp [S]; rw [← image_univ, image_image, univ_inter, image_univ]"
    },
    {
      "id": 82,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp [μpos]"
    },
    {
      "id": 83,
      "match": true,
      "ground_truth": "norm_num",
      "model_output": "norm_num"
    },
    {
      "id": 84,
      "match": false,
      "ground_truth": "intro x hx",
      "model_output": "intro x hx\n    have : x ∈ Icc 0 1 := by simp at hx; simp; constructor <;> linarith\n    exact h1 x this"
    },
    {
      "id": 85,
      "match": false,
      "ground_truth": "intro x _",
      "model_output": "intro x _; exact (h1 x)"
    },
    {
      "id": 86,
      "match": false,
      "ground_truth": "rw [mem_setOf]",
      "model_output": "constructor <;> intro h"
    },
    {
      "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 ge_liminf_of_lowerSemicontinuous lscf x_converge g_converge y nonempty"
    },
    {
      "id": 88,
      "match": false,
      "ground_truth": "refine norm_ne_zero_iff.mpr ?_",
      "model_output": "apply norm_ne_zero_iff.mpr\n        exact Basis.ne_zero b 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 ∈ range (toContinuousLinearMap T) := by\n      rw [mem_orthogonal, mem_ker, mem_range, mem_range, mem_range, mem_range]\n      exact ⟨x₂, x₂, x₂, x₂, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl⟩"
    },
    {
      "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 ⟨hx, hx2⟩\n    use (fun n => (x.1, ⊤))\n    constructor\n    · intro n\n      simpa [hx2] using hx\n    · simp [hx2] at hx\n      rw [mem_closure_iff_seq_limit] at hx\n      rcases hx with ⟨w, hw, h2⟩\n      use w\n      constructor\n      · exact fun n => hw n\n      rw [Prod.tendsto_iff]\n      exact ⟨hw, h2⟩"
    },
    {
      "id": 91,
      "match": false,
      "ground_truth": "apply Finset.sum_eq_zero",
      "model_output": "apply Finset.sum_eq_zero\n  intro i _\n  rw [hx.1.2 i i.2]"
    },
    {
      "id": 92,
      "match": false,
      "ground_truth": "mul_inv_cancel₀ (by linarith),",
      "model_output": "rw [← mul_assoc, ← mul_assoc, mul_inv_cancel₀, one_mul, mul_div_assoc, mul_one]"
    },
    {
      "id": 93,
      "match": true,
      "ground_truth": "by rw [one_smul]; rfl",
      "model_output": "rw [one_smul]"
    },
    {
      "id": 94,
      "match": false,
      "ground_truth": "intro b",
      "model_output": "intro b hb; specialize h 1 zero_lt_one; filter_upwards [h] with y hy; simp [sa] at hy; linarith [hy]"
    },
    {
      "id": 95,
      "match": false,
      "ground_truth": "by linarith",
      "model_output": "apply mul_pos hc (by linarith)"
    },
    {
      "id": 96,
      "match": true,
      "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, add_comm_sub, sub_sub, ← add_smul, add_comm b a]"
    },
    {
      "id": 97,
      "match": false,
      "ground_truth": "· rw [intrinsicInterior_prod_eq_prod_intrinsicInterior]; exact mk_mem_prod (by simp) hx",
      "model_output": "apply intrinsicInterior_prod_eq_prod_intrinsicInterior\n      exact mk_mem_prod (by simp) hx"
    },
    {
      "id": 98,
      "match": false,
      "ground_truth": "rw [alg.γeq k, alg.teq k]",
      "model_output": "have : 0 < k + 2 := by linarith [k.zero_le]"
    },
    {
      "id": 99,
      "match": false,
      "ground_truth": "intro i _",
      "model_output": "intro i _; apply mul_nonneg (h1 i) (h32 i i.2)"
    },
    {
      "id": 100,
      "match": false,
      "ground_truth": "rcases vt with ⟨c, d, ⟨a, ha⟩, ⟨vt1, vt2⟩⟩",
      "model_output": "rcases vt with ⟨c, d, ⟨⟨a, ha⟩, ⟨ct, cv⟩⟩⟩⟩"
    },
    {
      "id": 101,
      "match": false,
      "ground_truth": "exact hu.2.2 (mem_inter hu.2.1 hx)",
      "model_output": "apply hu.2.1"
    },
    {
      "id": 102,
      "match": true,
      "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": "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"
    },
    {
      "id": 103,
      "match": false,
      "ground_truth": "intro x",
      "model_output": "intro v"
    },
    {
      "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 [φ]"
    },
    {
      "id": 105,
      "match": false,
      "ground_truth": "rw[Metric.isOpen_iff] at isopen",
      "model_output": "rcases ht with ⟨ht, ht_conv, ht_open, ht_bd⟩\n    apply Lipschitz_of_Bounded hf ht_conv ht_open ht_bd x_pos"
    },
    {
      "id": 106,
      "match": false,
      "ground_truth": "rw [deriv_of_const_mul_func (f u - f x)]",
      "model_output": "have h1 : ‖u - x‖ + ‖f u - f x‖ < c := by\n      rw [← add_halves c]\n      apply add_lt_add\n      · apply mem_ball_iff_norm.mp hu.1\n      · simp at *\n        rw [abs_eq_self.2] <;> linarith [hu.2.1, hu.2.2]"
    },
    {
      "id": 107,
      "match": false,
      "ground_truth": "rcases exists_seq_tendsto_sInf hs hs' with ⟨fx, _, cfx, fxs⟩",
      "model_output": "rcases hs with ⟨x, hx⟩\n  rcases hs' with ⟨w, hw⟩\n  use x\n  intro x1 _\n  by_contra h\n  have : f x < f x1 := by exact Ne.lt_of_le' (id (Ne.symm h)) (le_of_lt h)\n  have : f x ∉ f ⁻¹' Set.Iic y := by simp; exact this\n  have : f x ∉ f ⁻¹' Set.Iic y := by simp; exact this\n  contradiction"
    },
    {
      "id": 108,
      "match": false,
      "ground_truth": "induction n with",
      "model_output": "apply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply Φ_isdescending\napply �"
    },
    {
      "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": "apply mem_of_hasGradientAt g' hg"
    },
    {
      "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"
    },
    {
      "id": 113,
      "match": true,
      "ground_truth": "simp [phi_add] at this",
      "model_output": "simp [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 [ht] at fx_le_top fx_ge_bot ⊢\n      exact EReal.coe_le_coe_iff 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,ux,hf,hv⟩"
    },
    {
      "id": 117,
      "match": false,
      "ground_truth": "by linarith",
      "model_output": "apply h.1"
    },
    {
      "id": 118,
      "match": true,
      "ground_truth": "apply convexHull_min (union_subset h1 h2) cone_conv_s_t.convex",
      "model_output": "apply convexHull_min (union_subset h1 h2) cone_conv_s_t.convex"
    },
    {
      "id": 119,
      "match": false,
      "ground_truth": "have hΦ : ∃ r_Φ, range Φ ⊆ Metric.ball 0 r_Φ := by apply Φ_isBounded'",
      "model_output": "apply A₁e₁_A₂e₂_isBounded'"
    },
    {
      "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": "apply Satisfying_equational_constraint1\n   · apply Satisfying_equational_constraint1\n      · apply Satisfying_equational_constraint2\n      · apply Satisfying_equational_constraint3\n   · apply Satisfying_equational_constraint4\n   · apply Satisfying_equational_constraint5\n   · apply Satisfying_equational_constraint6\n   · apply Satisfying_equational_constraint7\n   · apply Satisfying_equational_constraint8\n   · apply Satisfying_equational_constraint9\n   · apply Satisfying_equational_constraint10\n   · apply Satisfying_equational"
    },
    {
      "id": 122,
      "match": false,
      "ground_truth": "rw[← EReal.coe_toReal (x := f x) hfx (LT.lt.ne_bot (hsf.uninfinity x hx))]",
      "model_output": "have h1 : (-(φ l.1).1 x + t) / u l.1 ≤ f x := by\n      have h2 : (-(φ l.1).1 x + t) / u l.1 ≤ (f x).toReal := by\n        rw [← EReal.coe_toReal hfx]\n        apply (div_le_iff₀ (EReal.coe_lt_top (f x)).ne (EReal.coe_lt_top (f x)).ne_top) |>.mpr\n        rw [← EReal.coe_sub, ← EReal.coe_mul]"
    },
    {
      "id": 123,
      "match": false,
      "ground_truth": "by_cases u₁_eq_0 : u₁ = 0",
      "model_output": "have : u₁ = ∑ x ∈ coef.support, coef x • hT.v x := by exact hu₁"
    },
    {
      "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_ne"
    },
    {
      "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_comp, fderiv_prod_const, sub_add_sub_comm, sub_sub_sub_eq_add_sub] 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} = ∅ := by\n      ext i\n      simp\n      intro hlt\n      have h1 : ∑ j, A i j = 1 := h_row_eq i\n      linarith [h1]"
    },
    {
      "id": 131,
      "match": false,
      "ground_truth": "exact False.elim (by simp at this)",
      "model_output": "have : r = 0 := by omega"
    },
    {
      "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_assoc, add_comm (b • j.1), ← add_assoc]"
    },
    {
      "id": 134,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp [Metric.ball] at *"
    },
    {
      "id": 135,
      "match": false,
      "ground_truth": "intro y yin z zin a b anneg bnneg absum1",
      "model_output": "intro x1 _ y1 _ a b ha hb hab\n    specialize hfun (by simp) (by simp) ha hb hab\n    rw [add_div, add_sub, add_sub, add_sub, add_sub, add_sub, add_sub]\n    simp; rw [← add_sub, sub_add, sub_add, sub_add, sub_add, sub_add, sub_add]\n    simp; rw [← add_sub, sub_add, sub_add, sub_add, sub_add, sub_add]\n    simp; rw [← add_sub, sub_add, sub_add, sub_add, sub_add]\n    simp;"
    },
    {
      "id": 136,
      "match": false,
      "ground_truth": "gcongr",
      "model_output": "apply mul_lt_mul_of_pos_left _ hc'\n[ASSISTANT]\napply lt_of_lt_of_le zin antil"
    },
    {
      "id": 137,
      "match": false,
      "ground_truth": "have hzd : ∀ n, d n = z n - loc := fun _ => eq_sub_of_add_eq' rfl",
      "model_output": "have h1 : ∀ n, z n ∈ s1 := by\n      intro n\n      simp [z]\n      apply mem_inter (Metric.ball_mem_nhds _ (by positivity))\n      apply ha n (by simp)"
    },
    {
      "id": 138,
      "match": true,
      "ground_truth": "rfl",
      "model_output": "rfl"
    },
    {
      "id": 139,
      "match": false,
      "ground_truth": "rw [mul_one] at htp",
      "model_output": "simp 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": "apply Filter.tendsto_iff_seq_tendsto.1 (A₂e₂_subseq_converge_zero fullrank₁ fullrank₂)"
    },
    {
      "id": 142,
      "match": false,
      "ground_truth": "apply A₁e₁_converge_zero",
      "model_output": "apply Satisfying_equational_constraint1\n   · apply A₁e₁_converge_zero\n   · apply A₂e₂_converge_zero\n   · apply A₁e₁_converge_zero\n   · apply A₂e₂_converge_zero\n   · apply fullrank₁\n   · apply fullrank₂\n   · apply admm_kkt₁\n   · apply admm_kkt₂\n   · apply admm_kkt₃\n   · apply admm_kkt₄\n   · apply admm_kkt₅\n   · apply admm_kkt₆"
    },
    {
      "id": 143,
      "match": false,
      "ground_truth": "intro j",
      "model_output": "intro j\n    have h4 : ∑ i, A i j ≤ 1 := hA_col_sum j\n    have h5 : ∑ i, A i j ≥ 1 := by\n      by_contra hlt\n      push_neg at hlt\n      have h6 : ∑ i, A i j < 1 := by linarith [hlt]\n      have h7 : j ∈ Finset.univ.filter fun j => ∑ i, A i j < 1 := by\n        simp; exact h6\n      have h8 : Finset.card {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; linarith [μpos]"
    },
    {
      "id": 146,
      "match": false,
      "ground_truth": "ext x",
      "model_output": "ext x\n            simp only [c]\n            rw [add_sub_right_comm, add_sub_right_comm A₁ x (A₂ x₂ n.natPred)]\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 x_in_t : x ∈ t := by\n      simp[t];apply mem_inter x_in_xBall hr₀.1"
    },
    {
      "id": 148,
      "match": false,
      "ground_truth": "rw [← zero_pow]",
      "model_output": "apply squeeze_zero_norm\n   · have : ∀ n, ‖A₁ (e₁ (φ n)) + A₂ (e₂ (φ n))‖ ^ 2 ≤ 2 * (‖A₁ (e₁ (φ n))‖ ^ 2 + ‖A₂ (e₂ (φ n))‖ ^ 2) := by\n      intro n; apply pow_two_nonneg (‖A₁ (e₁ (φ n)) + A₂ (e₂ (φ n))‖)\n   · apply A₁e₁_A₂e₂_converge_zero₂\n   · apply"
    },
    {
      "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 admm.hrho\n    · apply pow_two_nonneg"
    },
    {
      "id": 150,
      "match": true,
      "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 [← mul_neg,neg_sub,← map_neg (A₂ (e₂ (n+1)))]"
    },
    {
      "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 ; rfl"
    },
    {
      "id": 154,
      "match": true,
      "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_le.mpr"
    },
    {
      "id": 155,
      "match": false,
      "ground_truth": "rcases this with ⟨z_, _, α, ⟨monoa, conv⟩⟩",
      "model_output": "rcases this with ⟨z_, α, ⟨monoa, conv⟩⟩"
    },
    {
      "id": 156,
      "match": false,
      "ground_truth": "intro n x",
      "model_output": "apply ADMM_iter_process₁'_eq3_3'"
    },
    {
      "id": 157,
      "match": true,
      "ground_truth": "simp [coef.mem_support_toFun]; rfl",
      "model_output": "simp [coef.mem_support_toFun]"
    },
    {
      "id": 158,
      "match": false,
      "ground_truth": "simp only [Set.mem_union, Set.mem_iUnion, Set.mem_singleton_iff]",
      "model_output": "by_cases h1 : i = n + 1"
    },
    {
      "id": 159,
      "match": false,
      "ground_truth": "have h2 : ‖x + xx - xx‖ ≤ ‖x + xx‖ + ‖xx‖ := by apply norm_sub_le",
      "model_output": "have h2: ‖x + xx - xx‖ ≤ ‖x + xx‖ + ‖xx‖ := by apply norm_sub_le"
    },
    {
      "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 h2 : (MM x).carrier ∩ (f.Epi univ) = (MM x).carrier ∩ (f.Epi univ) := rfl"
    },
    {
      "id": 162,
      "match": false,
      "ground_truth": "simp [z]",
      "model_output": "apply Convex.add_smul_sub_mem hs xin yin a b 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 Tendsto.norm\n   apply Satisfying_equational_constraint2\n   apply Satisfying_equational_constraint1\n   apply Satisfying_equational_constraint3\n   apply Satisfying_equational_constraint4\n   apply Satisfying_equational_constraint5\n   apply Satisfying_equational_constraint6\n   apply Satisfying_equational_constraint7\n   apply Satisfying_equational_constraint8\n   apply Satisfying_equational_constraint9\n   apply Satisfying_equational_constraint10\n   apply Satisfying_equational_constraint11\n   apply Satisfying_equational_constraint12"
    },
    {
      "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 h0 : ∑ j, C x j = ∑ j, A x j :=      dsimp [C]\n              simp [h]"
    },
    {
      "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": "apply le_trans [ey_diff_le] [t_Ax_Ax_n1_le]"
    },
    {
      "id": 166,
      "match": false,
      "ground_truth": "intro n",
      "model_output": "intro n\n   apply ADMM_iter_process₁'_eq2_3'"
    },
    {
      "id": 167,
      "match": true,
      "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 (g := f₁) (f := fun x ↦ x + x₀) (t := univ) hcon\n        · apply ContinuousOn.add continuousOn_id continuousOn_const\n        · simp"
    },
    {
      "id": 168,
      "match": false,
      "ground_truth": "simp only [one_div, mul_inv_rev, add_sub_cancel_right]",
      "model_output": "rw [smul_smul, one_div, inv_eq_one_div, div_mul_cancel₀, add_comm, add_sub, sub_self, add_zero]"
    },
    {
      "id": 169,
      "match": false,
      "ground_truth": "suffices 0 ≤ (a1 • tx + b1 • ty) by exact this",
      "model_output": "intro i\n      simp\n      apply add_nonneg\n      · apply mul_nonneg\n        · linarith [anng]\n        · apply txcond i\n      · apply mul_nonneg\n        · linarith [bnng]\n        · apply 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 i hi\n    apply iSup₂_le\n    intro y hy\n    apply ht1 y hy\n    exact hi"
    },
    {
      "id": 172,
      "match": false,
      "ground_truth": "apply le_trans (Nat.le_add_left ax N)",
      "model_output": "apply le_trans _ (hax _ (Nat.le_add_left ax (φ 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 [xsub']"
    },
    {
      "id": 175,
      "match": true,
      "ground_truth": "exact trivial",
      "model_output": "exact trivial"
    },
    {
      "id": 176,
      "match": false,
      "ground_truth": "simp",
      "model_output": "apply pow_two_nonneg"
    },
    {
      "id": 177,
      "match": false,
      "ground_truth": "intro _ epos",
      "model_output": "intro _ _"
    },
    {
      "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, one_mul, mul_assoc, mul_assoc, mul_comm, mul_comm ‖⟪T e, T u⟫_𝕜‖ ^ 2]"
    },
    {
      "id": 180,
      "match": false,
      "ground_truth": "· simp",
      "model_output": "apply tendsto_const_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": "lift ⨆ u, Real.toEReal (inner ℝ u x) - f u to ℝ with c₁\n      lift ⨆ u, Real.toEReal (inner ℝ u y) - f u to ℝ with c₂\n      rw [← EReal.coe_add, ← EReal.coe_add, ← EReal.coe_mul, ← EReal.coe_mul]\n      apply EReal.coe_le_coe_iff.mpr\n      apply add_le_add\n      · exact supa\n      · exact supb"
    },
    {
      "id": 183,
      "match": false,
      "ground_truth": "simp at vt1",
      "model_output": "have : ∀ᶠ (a : ℕ) in atTop, ‖1 / c a‖ ≤ ‖c a‖⁻¹ := by\n      apply Filter.Eventually.mp (tendsto_one_div_atTop_nhds_zero_nat.eventually hcp)\n      apply Filter.Eventually.of_forall; intro a ha; simp at ha; simp; exact ha.1"
    },
    {
      "id": 184,
      "match": false,
      "ground_truth": "rw [pow_two]",
      "model_output": "rw [← mul_assoc, ← mul_assoc, ← mul_assoc, ← mul_assoc, ← mul_assoc, ← mul_assoc]\n        rw [← mul_assoc, ← mul_assoc, ← mul_assoc, ← mul_assoc, ← mul_assoc]\n        rw [← mul_assoc, ← mul_assoc, ← mul_assoc, ← mul_assoc, ← mul_assoc]\n        rw [← mul_assoc, ← mul_assoc, ← mul_assoc, ← mul_assoc, ← mul_assoc]\n        rw [← mul_assoc, ← mul_assoc, ← mul_assoc, ← mul_assoc, ← mul_assoc]\n        rw [← mul_assoc, ← mul_assoc"
    },
    {
      "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', _, b'finite, subb'⟩"
    },
    {
      "id": 186,
      "match": true,
      "ground_truth": "intro b _",
      "model_output": "intro b _"
    },
    {
      "id": 187,
      "match": false,
      "ground_truth": "ring",
      "model_output": "rw [← mul_add,← mul_assoc,mul_comm _ (1 - τ),mul_assoc,mul_comm _ (1 - τ)]"
    },
    {
      "id": 188,
      "match": true,
      "ground_truth": "exact hcf",
      "model_output": "exact hcf"
    },
    {
      "id": 189,
      "match": true,
      "ground_truth": "ring_nf",
      "model_output": "ring_nf"
    },
    {
      "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": "· rw [hx]; simp [ha.ne']"
    },
    {
      "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 * ⟪f' (z + d • (y - z)) - f' (x + c • (z - x)), y - x⟫_ℝ = a * b * (0 : ℝ) := by\n      rw [← e2, ← e1, ← this1, ← this2, inner_sub_left, ← sub_sub, sub_self, inner_zero_right]"
    },
    {
      "id": 194,
      "match": false,
      "ground_truth": "by ring",
      "model_output": "rw [one_div_mul_cancel]"
    },
    {
      "id": 195,
      "match": false,
      "ground_truth": "by\n          rw[sign_mul_abs]\n          field_simp [bi_pos i]",
      "model_output": "have : ‖(b).equivFun (x₀ - x) i‖ ≠ 0 := by\n      exact norm_ne_zero_iff.mpr (b.equivFun_apply_ne_zero i)"
    },
    {
      "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": "apply h_add'"
    },
    {
      "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 cond sumeq i"
    },
    {
      "id": 199,
      "match": true,
      "ground_truth": "rw[zero_add]",
      "model_output": "rw[zero_add]"
    },
    {
      "id": 200,
      "match": false,
      "ground_truth": "linarith",
      "model_output": "linarith [kleaw, this]"
    },
    {
      "id": 201,
      "match": false,
      "ground_truth": "repeat apply mul_nonneg",
      "model_output": "· apply mul_nonneg\n      · apply mul_nonneg\n        · apply inv_nonneg_of_nonneg\n          apply le_of_lt admm.hrho\n        · apply pow_two_nonneg\n      · apply pow_two_nonneg"
    },
    {
      "id": 202,
      "match": false,
      "ground_truth": "exact Eq.symm (Nat.add_sub_of_le kge)",
      "model_output": "rw [Nat.add_sub_of_le kge]"
    },
    {
      "id": 203,
      "match": false,
      "ground_truth": "rw [Filter.tendsto_congr' this]",
      "model_output": "apply Filter.Tendsto.congr' this"
    },
    {
      "id": 204,
      "match": false,
      "ground_truth": "intro c1",
      "model_output": "intro c1 hc1\n    filter_upwards [t5] with n hn\n    rw [inner_sub_left, inner_smul_right]\n    apply le_of_lt\n    apply mul_lt_mul_of_pos_right\n    · exact hn\n    · apply norm_pos_iff.mpr\n      simp\n      exact inv_ne_zero (ne_of_gt 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 [abs_of_nonneg (alg.step₁ k)]"
    },
    {
      "id": 207,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp [h]"
    },
    {
      "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 i"
    },
    {
      "id": 210,
      "match": false,
      "ground_truth": "intro c",
      "model_output": "intro c cin"
    },
    {
      "id": 211,
      "match": false,
      "ground_truth": "intro i",
      "model_output": "intro i ieqN"
    },
    {
      "id": 212,
      "match": false,
      "ground_truth": "dsimp [M]",
      "model_output": "rw [M]"
    },
    {
      "id": 213,
      "match": false,
      "ground_truth": "intro v",
      "model_output": "simp [LinearMap.toContinuousLinearMap_apply]"
    },
    {
      "id": 214,
      "match": true,
      "ground_truth": "by exact this",
      "model_output": "exact this"
    },
    {
      "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": "· apply εpos"
    },
    {
      "id": 217,
      "match": true,
      "ground_truth": "linarith [eq1]",
      "model_output": "linarith [eq1]"
    },
    {
      "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 _\n          apply (alg.ha i)\n        · simp"
    },
    {
      "id": 219,
      "match": true,
      "ground_truth": "linarith [a1leab]",
      "model_output": "linarith [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_div] 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.x (α q - 1), alg.y (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 prod_norm_ge_of_first\n        apply norm_nonneg"
    },
    {
      "id": 224,
      "match": false,
      "ground_truth": "intro a t hat IH w hw_pos' hw_sum' hmem'",
      "model_output": "intro a s has IH w hw_nonneg hw_sum' hw_in_U'"
    },
    {
      "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\n    · apply ha₁ b (le_trans (le_max_left a₁ a₂) hba₁)\n    · apply ha₂ b (le_trans (le_max_right a₁ a₂) hba₂)"
    },
    {
      "id": 226,
      "match": false,
      "ground_truth": "exact inv_sub_inv l' h₁",
      "model_output": "rw [inv_eq_one_div, inv_eq_one_div, one_div, one_div, sub_div, div_mul_eq_div_div]"
    },
    {
      "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    · apply Finset.sum_subset\n      intro i hi\n      apply smul_subset_smul_left\n      · apply intrinsicInterior_subset\n      · simp [ι']\n      · exact hi\n    · exact subset_refl (l i₀ • intrinsicInterior ℝ (C i₀))"
    },
    {
      "id": 229,
      "match": true,
      "ground_truth": "rw [y1, y2]",
      "model_output": "rw [y1, y2]"
    },
    {
      "id": 230,
      "match": true,
      "ground_truth": "linarith [hγ]",
      "model_output": "linarith [hγ]"
    },
    {
      "id": 231,
      "match": false,
      "ground_truth": "apply norm_sub_le",
      "model_output": "apply norm_sub_norm_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": "have h1:= hk (fun q ↦ alg.x (α q) - alg.x (α q - 1)) diflte1 (ε / (γ * l) / 2) (by positivity)"
    },
    {
      "id": 233,
      "match": false,
      "ground_truth": "intro n",
      "model_output": "intro n\n    specialize allh' (n + 1) (Nn n)\n    simp at allh'\n    exact allh'"
    },
    {
      "id": 234,
      "match": false,
      "ground_truth": "by\n        apply zp5",
      "model_output": "exact zp5"
    },
    {
      "id": 235,
      "match": false,
      "ground_truth": "by\n        apply Simplifying₂\n        apply h₁\n        apply norm_nonneg",
      "model_output": "rw [mul_div_assoc, mul_div_assoc, mul_div_assoc, mul_div_assoc, mul_div_assoc]"
    },
    {
      "id": 236,
      "match": false,
      "ground_truth": "refine LICQ_injM z m Z Ax ?_ mlen ?_ eq2 eq1 ⟨heq1, heq2⟩",
      "model_output": "have h1 : Ax *ᵥ z = 0 := heq1"
    },
    {
      "id": 237,
      "match": false,
      "ground_truth": "apply le_trans",
      "model_output": "apply le_trans Nge (Nat.cast_le.mpr (Nat.le_add_right N n))"
    },
    {
      "id": 238,
      "match": false,
      "ground_truth": "apply mul_eq_mul_left_iff.mpr",
      "model_output": "rw [mul_eq_mul_left_iff]"
    }
  ]
}