evaluation/valid_score_in_300.json

{
  "summary": {
    "total_with_gt": 239,
    "matched": 63,
    "accuracy": 0.26359832635983266
  },
  "details": [
    {
      "id": 0,
      "match": true,
      "ground_truth": "apply HasDerivAt.sub",
      "model_output": "apply HasDerivAt.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 x"
    },
    {
      "id": 2,
      "match": true,
      "ground_truth": "rw [← EReal.coe_sub]",
      "model_output": "rw [← EReal.coe_sub]"
    },
    {
      "id": 3,
      "match": true,
      "ground_truth": "ext x",
      "model_output": "ext x"
    },
    {
      "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  rw [← EReal.coe_sub]\n  apply bot_lt_iff_ne_bot.2\n  exact sub_ne_bot a b"
    },
    {
      "id": 6,
      "match": false,
      "ground_truth": "lift a to ℝ using ⟨ha1, ha2⟩",
      "model_output": "lift a to ℝ using ⟨ha1, ha2⟩\n  lift b to ℝ using ⟨hb1, hb2⟩\n  rw [← EReal.coe_sub, ← EReal.coe_add]\n  apply EReal.coe_le_coe_iff.mpr\n  apply le_add_sub_self"
    },
    {
      "id": 7,
      "match": false,
      "ground_truth": "apply hasDerivAt_mul_const c⁻¹",
      "model_output": "apply hasDerivAt_mul_const c⁻¹ x"
    },
    {
      "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 := lim_inf_def A\n  rw[← H] at this\n  exact (liminf_equiv A).mp this"
    },
    {
      "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 s g' f\n  rw [← this, ← OrderIso.map_add g']\n  apply OrderIso.map_biSup s g' f"
    },
    {
      "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 [g] at 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": "ext i\n[USER]\nsimp [Matrix.mulVec]"
    },
    {
      "id": 16,
      "match": false,
      "ground_truth": "intro i",
      "model_output": "intro i j\n  by_cases h1 : i = j\n  · rw [h1]\n    simp only [one_apply_eq, mul_apply]\n    refine IsUnit.inv_mul_cancel ?_\n    exact diagpos_to_nonzero A hdiag i\n  · simp only [one_apply_ne, mul_apply, negsqrtdiagPart_apply, sqrtdiagPart_apply]\n    split_ifs\n    rw [← mul_assoc]\n    rw [← mul_assoc]\n    rw [← mul_assoc]\n    rw [← mul_assoc]\n    rw [← mul_assoc]\n    rw [← mul_assoc]"
    },
    {
      "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    have ⟨x, hx⟩ := Set.nonempty_def.1 hs\n    have h1 := congrFun h x\n    simp [convex_indicator] at h1\n    exact (lt_self_iff_false 0).mp h1.2"
    },
    {
      "id": 18,
      "match": false,
      "ground_truth": "constructor",
      "model_output": "constructor\n  · intro h x hx\n    simp at *\n    exact ⟨hx.left, le_trans (h x.1 hx.left) hx.right⟩\n  intro h x hx\n  have : (x, g x) ∈ {p | p.1 ∈ s ∧ g p.1 ≤ p.2} := by simpa\n  have := h this\n  simp at this\n  apply this.2"
    },
    {
      "id": 19,
      "match": false,
      "ground_truth": "by_cases hA : A = 0",
      "model_output": "intro x\n  apply hasGradientAt_iff_isLittleO.mpr\n  rw [Matrix.mulVec_eq_mul, Matrix.mulVec_eq_mul, ← Matrix.mul_assoc, ← Matrix.mul_assoc]\n  apply Matrix.mulVec_isLittleO\n  apply Matrix.mulVec_isLittleO\n  apply Matrix.mulVec_isLittleO\n  apply Matrix.mulVec_isLittleO\n  apply Matrix.mulVec_isLittleO\n  apply Matrix.mulVec_isLittleO\n  apply Matrix.mulVec_isLittleO\n  apply Matrix.mulVec_isLittleO\n  apply Matrix.mulVec_isLittleO\n  apply Matrix.mulVec_isLittle"
    },
    {
      "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 subdifferential_nonempty_of_isLocalMin 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 gradient_of_inner_const x a"
    },
    {
      "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 (IsUnit.prod_erase_mul univ (fun i ↦ A i i) (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; exact eventually_of_forall fun x' ↦ hy\n    simp [hx] at hy; exact eventually_of_forall fun x' ↦ hy"
    },
    {
      "id": 26,
      "match": true,
      "ground_truth": "simp [LowerSemicontinuousHull]",
      "model_output": "simp [LowerSemicontinuousHull]"
    },
    {
      "id": 27,
      "match": true,
      "ground_truth": "simp at hfy",
      "model_output": "simp at hfy"
    },
    {
      "id": 28,
      "match": true,
      "ground_truth": "rw [Fin.sum_univ_succ]",
      "model_output": "rw [Fin.sum_univ_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]"
    },
    {
      "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' h'\n  rw [← dist_eq_norm, dist_comm] at h'\n  exact le_of_lt (h h')"
    },
    {
      "id": 32,
      "match": false,
      "ground_truth": "exact half_lt_self dpos",
      "model_output": "linarith"
    },
    {
      "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": "apply intrinsicInterior_subset.antisymm\n  intro x hx\n  rw [mem_intrinsicInterior, affineSpan_intrinsicInterior hsc]\n  exact (mem_intrinsicInterior hsc).1 hx"
    },
    {
      "id": 36,
      "match": false,
      "ground_truth": "rw [← sub_add, norm_add_sq_real, real_inner_comm, inner_sub_right]",
      "model_output": "rw [← sub_add, norm_add_sq_real, inner_sub_left, inner_sub_right, inner_sub_left, 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.2 at hccleq"
    },
    {
      "id": 38,
      "match": false,
      "ground_truth": "intro z",
      "model_output": "apply intrinsicInterior_intrinsicClosure_sub_intrinsicInterior 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": "apply bounded_linear_functional_of_normed_add_comm_group"
    },
    {
      "id": 42,
      "match": false,
      "ground_truth": "intro h\n    apply HasFDeriv_Convergence\n    exact h",
      "model_output": "apply HasFDeriv_Convergence\n[USER]\napply this"
    },
    {
      "id": 43,
      "match": true,
      "ground_truth": "· apply Convergence_HasFDeriv",
      "model_output": "apply Convergence_HasFDeriv"
    },
    {
      "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": "rw [←intrinsicInterior_affineSubspace_eq_self, ←nonempty_iff_ne_empty] at hn\n  apply closure_affineSubspace_eq_self\n  exact hn"
    },
    {
      "id": 45,
      "match": false,
      "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⟩"
    },
    {
      "id": 48,
      "match": false,
      "ground_truth": "have h2 := tendsto_add (A:=A) (B:=B)",
      "model_output": "have h2 := tendsto_atTop.1 hAa"
    },
    {
      "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 hs : i ∈ σ\n    · simp [hs]; apply cpos\n    · simp [hs]"
    },
    {
      "id": 51,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp at hst hdti ⊢"
    },
    {
      "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": "apply in_of_le_sinf_of_b_top hx"
    },
    {
      "id": 54,
      "match": false,
      "ground_truth": "intro hmin",
      "model_output": "intro hmin\n  apply first_order_unconstrained hf hmin hfc\n  apply convex_univ\n  apply hcon\n  apply hmin"
    },
    {
      "id": 55,
      "match": false,
      "ground_truth": "simp [hx2, hw.2]",
      "model_output": "constructor\n        · exact hw.2\n        rw [hx2]\n        exact tendsto_const_nhds"
    },
    {
      "id": 56,
      "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": 57,
      "match": false,
      "ground_truth": "simp [ConvexCone.addPoint] at hx",
      "model_output": "by_cases hx0 : x = 0"
    },
    {
      "id": 58,
      "match": false,
      "ground_truth": "intro x",
      "model_output": "intro x xs y ys\n  apply lagrange hs hf x xs y ys\n  apply interior_nonempty.2\n  apply convex_segment_interior hs hf x xs y ys\n  apply interior_nonempty.2\n  apply convex_segment_interior hs hf x xs y ys\n  apply interior_nonempty.2\n  apply convex_segment_interior hs hf x xs y ys\n  apply interior_nonempty.2\n  apply convex_segment_interior hs hf x xs y ys\n  apply interior_nonempty.2\n  apply convex_segment_interior hs hf x xs y ys\n  apply interior_nonempty."
    },
    {
      "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 h₁ h₂ h₃\n    simp only [mem_setOf_eq]\n    rw [map_vsub, map_smul, vadd_eq_add, smul_add, ← eq_sub_iff_add_eq']\n    rw [h₁, h₂, h₃, ← smul_sub, smul_eq_mul, sub_self, mul_zero, sub_zero]"
    },
    {
      "id": 61,
      "match": false,
      "ground_truth": "linarith",
      "model_output": "apply ht"
    },
    {
      "id": 62,
      "match": true,
      "ground_truth": "simp [h]",
      "model_output": "simp [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": "apply le_of_tendsto (continuousOn_const (f p.1)).tendsto_atTop xntend xncond"
    },
    {
      "id": 65,
      "match": false,
      "ground_truth": "by_cases ha1 : a = 0",
      "model_output": "simp [LT.lt.ne_bot (hsf.uninfinity y (mem_univ y) b hb hab)]"
    },
    {
      "id": 66,
      "match": false,
      "ground_truth": "by_cases hb1 : b = 0",
      "model_output": "simp [coe_smul_top_of_pos hb, h2]"
    },
    {
      "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 : HasGradientAt f (f' y) y := by\n    exact HasFDerivAt.hasGradientAt (hu3 y hy)\n  rw [gradient_eq_toDual_symm] at this\n  exact this hy"
    },
    {
      "id": 69,
      "match": true,
      "ground_truth": "exact Batteries.compareOfLessAndEq_eq_lt.mp rfl",
      "model_output": "exact Batteries.compareOfLessAndEq_eq_lt.mp rfl"
    },
    {
      "id": 70,
      "match": false,
      "ground_truth": "intro y",
      "model_output": "intro x' hε\n    rw [p.heq]\n    rw [← Matrix.mulVec_sub, ← sub_add, sub_add_eq_add_sub, sub_add_cancel]\n    rw [← Matrix.mulVec_sub, ← sub_add, ← sub_add, sub_add_eq_add_sub, sub_add_cancel]\n    rw [← Matrix.mulVec_sub, ← sub_add, ← sub_add, sub_add_eq_add_sub, sub_add_cancel]\n    rw [← Matrix.mulVec_sub, ← sub_add, ← sub_add, sub_add_eq_add_sub, sub_add_cancel]\n    rw [← Matrix.mulVec_sub, ← sub"
    },
    {
      "id": 71,
      "match": false,
      "ground_truth": "have hb : b = 1 := by linarith",
      "model_output": "simp [ha, bpos, absum]"
    },
    {
      "id": 72,
      "match": true,
      "ground_truth": "intro x y r",
      "model_output": "intro 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) = (m0, n0) + (m - m0, n - n0) := by simp [Prod.eq_iff_fst_eq_snd_eq]"
    },
    {
      "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    exact ⟨v, mem_sphere_zero_iff_norm.mpr (by exact norm_ne_zero_iff.mpr hv)⟩"
    },
    {
      "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[apply diff_prod₁ h]"
    },
    {
      "id": 81,
      "match": false,
      "ground_truth": "rcases lbdf with ⟨L, Lbound⟩",
      "model_output": "have bddImS : BddBelow ImS := by\n    use g x\n    intro gy y\n    simp [ImS]\n    rcases gy with ⟨y1, _, h2⟩\n    exact h2"
    },
    {
      "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      exact (h1 x (by constructor <;> linarith))"
    },
    {
      "id": 85,
      "match": true,
      "ground_truth": "intro x _",
      "model_output": "intro x _"
    },
    {
      "id": 86,
      "match": false,
      "ground_truth": "rw [mem_setOf]",
      "model_output": "constructor\n  · intro h\n    rw [has_f_subdiff_iff]\n    exact h\n  intro h\n  rw [mem_setOf_eq]\n  rcases h with ⟨u, ⟨ux, ⟨fu⟩⟩⟩, v, hv⟩\n  apply mem_f_subdiff_of_tendsto hconv ⟨u, ⟨ux, ⟨fu⟩⟩⟩ v hv"
    },
    {
      "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 inequ₃ y nonempty lscf x_converge g_converge"
    },
    {
      "id": 88,
      "match": false,
      "ground_truth": "refine norm_ne_zero_iff.mpr ?_",
      "model_output": "apply norm_ne_zero_iff.mpr\n[apply Module.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 = x - x₁ := by simp [xeq]"
    },
    {
      "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, ⊤))\n    constructor\n    · intro n\n      simpa using hx1\n    rw [Prod.tendsto_iff]\n    simp [hx2, hx1]"
    },
    {
      "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.2, mul_zero]]"
    },
    {
      "id": 92,
      "match": false,
      "ground_truth": "mul_inv_cancel₀ (by linarith),",
      "model_output": "rw [mul_inv_cancel₀ (by linarith), mul_div_assoc, ← mul_assoc]"
    },
    {
      "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\n  by_contra h1\n  push_neg at h1\n  specialize h (-b) (neg_pos.mpr h1)\n  simp only [gt_iff_lt, neg_mul, neg_lt_iff_pos_add']\n  apply Filter.Eventually.mp h\n  apply Filter.Eventually.of_forall\n  intro z hz\n  rw [sub_neg_eq_add, ← sub_eq_add_neg, sub_le_iff_le_add] at hz\n  exact le_trans hz (mem_upperBounds.mp hb z hz)"
    },
    {
      "id": 95,
      "match": false,
      "ground_truth": "by linarith",
      "model_output": "linarith [hc]"
    },
    {
      "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 (A x) hC).2 hx"
    },
    {
      "id": 98,
      "match": false,
      "ground_truth": "rw [alg.γeq k, alg.teq k]",
      "model_output": "have : alg.t k > 0 := (alg.tbound k).1\n[ASSISTANT]\nhave : alg.γ k > 0 := (alg.γbound k).1\n[ASSISTANT]\napply (div_le_div_iff₀ (by linarith) (by linarith)).mpr\n[ASSISTANT]\napply (sq_le_sq_iff_abs_le_abs.mpr (abs_le_of_pos_of_le (alg.γbound k).1\n        (le_of_lt (alg.γbound k).2))).mpr\n[ASSISTANT]\napply (abs_le_abs.mpr (le_of"
    },
    {
      "id": 99,
      "match": false,
      "ground_truth": "intro i _",
      "model_output": "intro i _ [h4]"
    },
    {
      "id": 100,
      "match": false,
      "ground_truth": "rcases vt with ⟨c, d, ⟨a, ha⟩, ⟨vt1, vt2⟩⟩",
      "model_output": "rcases vt with ⟨c, d, ⟨a, ha⟩, ⟨vt1, vt2⟩⟩\n  apply ge_of_tendsto (tendsto_order.1 vt2) (fun n ↦ inner_nonneg_of_norm_le_mul_norm)\n  · apply vt1\n  · apply ha\n  · apply FeasSet\n  · apply cpos_of_tendsto_atTop vt1\n  · apply dpos_of_tendsto_atTop vt1\n  · apply apos_of_tendsto_atTop vt1\n  · apply tendsto_nhds_nhds.2 vt2"
    },
    {
      "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 x\n  rw [← div_le_iff₀ (norm_pos_iff.mpr (by exact exists_ne_zero_of_nontrivial 𝕜 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 [Module.dualProdDualEquivDual_apply_apply ℝ E ℝ]"
    },
    {
      "id": 105,
      "match": false,
      "ground_truth": "rw[Metric.isOpen_iff] at isopen",
      "model_output": "rcases (Metric.isOpen_iff_ball_subset x_isopen).1 isopen x x_pos with ⟨ε, εpos, εball⟩"
    },
    {
      "id": 106,
      "match": false,
      "ground_truth": "rw [deriv_of_const_mul_func (f u - f x)]",
      "model_output": "rw [deriv_of_const_mul_func]"
    },
    {
      "id": 107,
      "match": false,
      "ground_truth": "rcases exists_seq_tendsto_sInf hs hs' with ⟨fx, _, cfx, fxs⟩",
      "model_output": "have : ∃ x ∈ f ⁻¹' Iic y, f x = sInf {f x | x ∈ f ⁻¹' Iic y} := by\n    apply exists_eq_sInf_set (hs) (hs')"
    },
    {
      "id": 108,
      "match": false,
      "ground_truth": "induction n with",
      "model_output": "apply Φ_isdescending\napply Ψ_isdescending\napply Θ_isdescending\napply Φ_le_Ψ\napply Φ_le_Θ\napply Ψ_le_Θ\napply Φ_le_Φ\napply Ψ_le_Φ\napply Θ_le_Φ\napply Φ_le_Ψ\napply Φ_le_Θ\napply Ψ_le_Θ\napply Φ_isdescending\napply Ψ_isdescending\napply Θ_isdescending\napply Φ_le_Ψ\napply Φ_le_Θ\napply Ψ_le_Θ\napply Φ_le_Φ\napply"
    },
    {
      "id": 109,
      "match": true,
      "ground_truth": "simp at ht",
      "model_output": "simp at ht"
    },
    {
      "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_subdiff_of_fderiv g g' hg gradcon x"
    },
    {
      "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 : 0 ≤ δ / 2)] 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": "have : {z | f z ≤ α} = f ⁻¹' Iic α ∩ univ := by\n        ext z\n        simp\n        exact and_comm"
    },
    {
      "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 (LT.lt.ne_top fx_le_top) (LT.lt.ne_bot fx_ge_bot), EReal.coe_le_coe_iff]"
    },
    {
      "id": 116,
      "match": false,
      "ground_truth": "rcases zin with ⟨u,hux,hufx,hv⟩",
      "model_output": "rcases zin with ⟨u,ux,xu,fxu⟩\n  rcases fxu with ⟨v,vu,fxu⟩\n  have : v ∈ subdifferential f x := by\n    rw [subdifferential,mem_setOf]\n    apply fxu\n  have : v ∈ subdifferential g x := by\n    rw [subdifferential,mem_setOf]\n    apply fxu\n  use v; constructor\n  · assumption\n  use v; constructor\n  · assumption\n  simp; exact (add_comm v (g' x)).symm"
    },
    {
      "id": 117,
      "match": false,
      "ground_truth": "by linarith",
      "model_output": "apply 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 h1 h2 (union_subset (subset_trans hs1 h1) (subset_trans ht1 h2))"
    },
    {
      "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 tendsto_zero_iff_norm_tendsto_zero.1\n   apply A₁e₁_A₂e₂_converge_zero"
    },
    {
      "id": 122,
      "match": true,
      "ground_truth": "rw[← EReal.coe_toReal (x := f x) hfx (LT.lt.ne_bot (hsf.uninfinity x hx))]",
      "model_output": "rw[← EReal.coe_toReal (x := f x) hfx (LT.lt.ne_bot (hsf.uninfinity x hx))]"
    },
    {
      "id": 123,
      "match": false,
      "ground_truth": "by_cases u₁_eq_0 : u₁ = 0",
      "model_output": "have : (Finset.sum (Finset.univ.filter (fun x ↦ x = i)) (fun x ↦ coef x • hT.v x)) = 0 := by\n      simp\n      rw [← hu₁]\n      apply Finsupp.mem_span_of_mem\n      simp [coef.mem_support_toFun]"
    },
    {
      "id": 124,
      "match": false,
      "ground_truth": "apply mul_pos",
      "model_output": "· linarith"
    },
    {
      "id": 125,
      "match": false,
      "ground_truth": "exact this",
      "model_output": "apply this"
    },
    {
      "id": 126,
      "match": false,
      "ground_truth": "intro y",
      "model_output": "intro y hy_ne"
    },
    {
      "id": 127,
      "match": false,
      "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_u₁\n         · apply u_inthesubgradient\n         · apply admm_kkt.h.subgrad₁"
    },
    {
      "id": 128,
      "match": true,
      "ground_truth": "simp [grad_fun_comp,grad_comp] at h",
      "model_output": "simp [grad_fun_comp,grad_comp] at h"
    },
    {
      "id": 129,
      "match": true,
      "ground_truth": "exact hu₂",
      "model_output": "exact hu₂"
    },
    {
      "id": 130,
      "match": true,
      "ground_truth": "apply Finset.card_eq_zero.2",
      "model_output": "apply Finset.card_eq_zero.2"
    },
    {
      "id": 131,
      "match": true,
      "ground_truth": "exact False.elim (by simp at this)",
      "model_output": "exact False.elim (by simp at this)"
    },
    {
      "id": 132,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp [sq]"
    },
    {
      "id": 133,
      "match": true,
      "ground_truth": "rw [← add_assoc, ← add_assoc]",
      "model_output": "rw [← add_assoc, ← add_assoc]"
    },
    {
      "id": 134,
      "match": false,
      "ground_truth": "simp",
      "model_output": "rw [Metric.ball, Set.mem_setOf, dist_zero_right]"
    },
    {
      "id": 135,
      "match": false,
      "ground_truth": "intro y yin z zin a b anneg bnneg absum1",
      "model_output": "intro y _ z _ a b anneg bnneg _"
    },
    {
      "id": 136,
      "match": false,
      "ground_truth": "gcongr",
      "model_output": "apply mul_lt_mul_of_pos_left zin hc'"
    },
    {
      "id": 137,
      "match": false,
      "ground_truth": "have hzd : ∀ n, d n = z n - loc := fun _ => eq_sub_of_add_eq' rfl",
      "model_output": "have hz1 : ∀ n, z n ∈ s1 := by\n      intro n\n      simp [s1, z]\n      exact ⟨mem_ball_of_norm (by simp [z]) εpos, ha n (a_le n)⟩"
    },
    {
      "id": 138,
      "match": true,
      "ground_truth": "rfl",
      "model_output": "rfl"
    },
    {
      "id": 139,
      "match": false,
      "ground_truth": "rw [mul_one] at htp",
      "model_output": "linarith [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 tendsto_zero_iff_norm_tendsto_zero.2\n      apply A₂e₂_subseq_converge_zero"
    },
    {
      "id": 142,
      "match": false,
      "ground_truth": "apply A₁e₁_converge_zero",
      "model_output": "apply tendsto_zero_iff_norm_tendsto_zero.1\n   apply A₁e₁_converge_zero"
    },
    {
      "id": 143,
      "match": false,
      "ground_truth": "intro j",
      "model_output": "have h4 : ∀ j, ∑ i, A i j = 1 := by\n      intro j\n      have col_sum_le : ∑ i, A i j ≤ 1 := hA_col_sum j\n      have col_sum_pos : ∑ i, A i j ≥ 0 := Finset.sum_nonneg hA_pos\n      linarith [col_sum_pos, col_sum_le]"
    },
    {
      "id": 144,
      "match": false,
      "ground_truth": "rw[this]",
      "model_output": "rw [← this]"
    },
    {
      "id": 145,
      "match": false,
      "ground_truth": "· simp; linarith [μpos]",
      "model_output": "simp [μpos]"
    },
    {
      "id": 146,
      "match": false,
      "ground_truth": "ext x",
      "model_output": "ext x\n         simp only [c]\n         rw [← add_sub, sub_self, add_zero]"
    },
    {
      "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      apply mem_interior_iff_mem_nhds.mpr\n      apply mem_nhdsWithin_of_mem_nhds\n      apply x_in_xBall\n      apply t_isopen"
    },
    {
      "id": 148,
      "match": false,
      "ground_truth": "rw [← zero_pow]",
      "model_output": "apply Summable.tendsto_atTop_zero Summable₁"
    },
    {
      "id": 149,
      "match": false,
      "ground_truth": "apply mul_nonneg",
      "model_output": "apply mul_nonneg\n        apply mul_nonneg\n        apply le_of_lt admm.hrho\n        apply pow_two_nonneg"
    },
    {
      "id": 150,
      "match": false,
      "ground_truth": "by simp at h; simpa [h]",
      "model_output": "simp at h; simp; exact h.1"
    },
    {
      "id": 151,
      "match": true,
      "ground_truth": "rw [← mul_neg]",
      "model_output": "rw [← mul_neg]"
    },
    {
      "id": 152,
      "match": true,
      "ground_truth": "exact h",
      "model_output": "exact h"
    },
    {
      "id": 153,
      "match": true,
      "ground_truth": "by rw [z]; simp; left; rw [z]; simp",
      "model_output": "rw [z]"
    },
    {
      "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": true,
      "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": "intro n x\n  apply ADMM_iter_process₁'_eq3_2' n x\n  apply ADMM_iter_process₁'_eq3_1' n x\n  apply ADMM_iter_process₁'_eq3_1'_nabla n x\n  apply ADMM_iter_process₁'_eq3_1'_nabla' n x\n  apply ADMM_iter_process₁'_eq3_1'_nabla'' n x\n  apply ADMM_iter_process₁'_eq3_1'_nabla''' n x\n  apply ADMM_iter_process₁'_eq3_1'_"
    },
    {
      "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 h : i = n + 1"
    },
    {
      "id": 159,
      "match": true,
      "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 : (MM x).carrier ∩ (f.Epi univ) = (MM x).carrier ∩ (f.Epi univ) := by rfl"
    },
    {
      "id": 162,
      "match": true,
      "ground_truth": "simp [z]",
      "model_output": "simp [z]"
    },
    {
      "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[USER]\napply Tendsto.add\n[ASSISTANT]\napply Tendsto.const_mul\n[USER]\napply tendsto_zero_iff_norm_tendsto_zero.1\n[ASSISTANT]\napply tendsto_zero_iff_norm_tendsto_zero.2\napply A₁x₁_A₂x₂_sub_b_subseq_converge_zero fullrank₁ fullrank₂\n[USER]\napply Tendsto.const_mul\napply tendsto_zero_iff_norm_tendsto_zero.1\napply A₁x₁_A₂x₂_sub_b_subseq_converge_zero fullrank₁"
    },
    {
      "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]\n            rw [← h0]\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": "apply Φ_isdescending_inequ5 n"
    },
    {
      "id": 166,
      "match": false,
      "ground_truth": "intro n",
      "model_output": "intro n\n   apply Φ_isdescending_inequ3\n   apply Φ_isdescending_inequ2\n   apply Φ_isdescending_inequ1\n   apply Φ_isdescending_inequ0\n   apply Φ_isdescending_inequ\n   apply Φ_isdescending_inequ4\n   apply Φ_isdescending_inequ5\n   apply Φ_isdescending_inequ6\n   apply Φ_isdescending_inequ7\n   apply Φ_isdescending_inequ8\n   apply Φ_isdescending_inequ9\n   apply Φ_is"
    },
    {
      "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]"
    },
    {
      "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 smul_nonneg\n  apply le_of_lt anng\n  apply txcond i\n  apply smul_nonneg\n  apply le_of_lt bnng\n  apply tycond i"
    },
    {
      "id": 170,
      "match": true,
      "ground_truth": "by\n        rw [sub_right_comm]",
      "model_output": "rw [sub_right_comm]"
    },
    {
      "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 (Nat.le_add_left ax N) (Nat.le_add_right N ax)"
    },
    {
      "id": 173,
      "match": true,
      "ground_truth": "rw [hdomain]; trivial",
      "model_output": "rw [hdomain]"
    },
    {
      "id": 174,
      "match": false,
      "ground_truth": "rfl",
      "model_output": "simp [xeq]"
    },
    {
      "id": 175,
      "match": true,
      "ground_truth": "exact trivial",
      "model_output": "exact trivial"
    },
    {
      "id": 176,
      "match": false,
      "ground_truth": "simp",
      "model_output": "apply sq_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_pos (by positivity : 0 < 1 / (σ ^ 2 * ‖u‖ ^ 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": "apply add_le_add supa supb"
    },
    {
      "id": 183,
      "match": false,
      "ground_truth": "simp at vt1",
      "model_output": "rw [Filter.tendsto_atTop] at hzt"
    },
    {
      "id": 184,
      "match": false,
      "ground_truth": "rw [pow_two]",
      "model_output": "rw [mul_assoc, ← pow_two, mul_le_mul_iff_right₀ 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', _, b'finite, sub_cover⟩\n    have b'nonempty : b'.Nonempty := by\n      rw [Set.nonempty_iff_ne_empty]\n      by_contra b'empty\n      have : Ω = ∅ := subset_eq_empty sub_cover b'empty\n      have : Ω ≠ ∅ := Set.Nonempty.ne_empty h_nonempty\n      contradiction\n    rcases b' with ⟨x, hx⟩\n    use (η x), (hη2 x hx), (φ x), (hφ x hx)"
    },
    {
      "id": 186,
      "match": true,
      "ground_truth": "intro b _",
      "model_output": "intro b _"
    },
    {
      "id": 187,
      "match": false,
      "ground_truth": "ring",
      "model_output": "linarith"
    },
    {
      "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 [FeasSet, FeasPoint] at xf"
    },
    {
      "id": 192,
      "match": true,
      "ground_truth": "· simp [hx]",
      "model_output": "· simp [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 : inner ℝ (f' (z + d • (y - z)) - f' (x + c • (z - x))) (y - x) = 0 := by\n      apply mul_eq_zero.mp eq3\n      linarith [a, b, absum1]"
    },
    {
      "id": 194,
      "match": false,
      "ground_truth": "by ring",
      "model_output": "rw [one_div, one_div]"
    },
    {
      "id": 195,
      "match": false,
      "ground_truth": "by\n          rw[sign_mul_abs]\n          field_simp [bi_pos i]",
      "model_output": "rw[sign_mul_abs]\n            field_simp [bi_pos i]\n            simp only [ne_eq, Nat.cast_eq_zero, SignType.sign_eq_one_iff, one_mul, abs_eq_self,\n              zero_lt_one, ↓reduceDIte, mul_one, not_false_eq_true, zero_ne_one, h₁, h₂]"
    },
    {
      "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": false,
      "ground_truth": "· simp [V₂]",
      "model_output": "· rfl"
    },
    {
      "id": 198,
      "match": false,
      "ground_truth": "intro s",
      "model_output": "intro s g cond sumeq i hi\n  specialize cond i hi\n  rw [← Matrix.toLin'_apply'] at cond\n  rw [← Matrix.toLin'_apply'] at sumeq\n  rw [← Matrix.toLin'_apply'] at cond\n  rw [← Matrix.toLin'_apply'] at sumeq\n  rw [← Matrix.toLin'_apply'] at cond\n  rw [← Matrix.toLin'_apply'] at sumeq\n  rw [← Matrix.toLin'_apply'] at cond\n  rw [← Matrix.toLin'_apply'] at sumeq\n  rw [← Matrix"
    },
    {
      "id": 199,
      "match": true,
      "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_nonneg.mpr (le_of_lt admm.hrho)]\n           · linarith [sq_nonneg (‖ey n‖)]\n         · linarith [sq_nonneg (‖ey n‖)]"
    },
    {
      "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": 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 ε εpos\n  filter_upwards [t5] with n hn\n  rw [inner_smul_right, norm_smul, norm_norm, mul_comm]\n  apply le_of_lt\n  apply mul_pos hn (norm_sub_pos_iff.mpr (ne_of_gt (one_div_pos.mpr εpos)))"
    },
    {
      "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 [norm_mul, norm_sq, norm_eq_abs, abs_of_pos alg.step₁]"
    },
    {
      "id": 207,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp [σ₁]"
    },
    {
      "id": 208,
      "match": false,
      "ground_truth": "rw[← mul_div]",
      "model_output": "ring_nf"
    },
    {
      "id": 209,
      "match": false,
      "ground_truth": "simp",
      "model_output": "simp [Finset.mem_range] at hi"
    },
    {
      "id": 210,
      "match": false,
      "ground_truth": "intro c",
      "model_output": "intro c hc [hφc] [hφc'] [hφc''] [hφc'''] [hφc''] [hφc'] [hφc] [hφc'] [hφc] [hφc'] [hφc] [hφc'] [hφc] [hφc'] [hφc] [hφc'] [hφc] [hφc'] [hφc] [hφc'] [hφc] [hφc'] [hφc] [hφc'] [h"
    },
    {
      "id": 211,
      "match": false,
      "ground_truth": "intro i",
      "model_output": "intro i _"
    },
    {
      "id": 212,
      "match": false,
      "ground_truth": "dsimp [M]",
      "model_output": "rw [M]"
    },
    {
      "id": 213,
      "match": false,
      "ground_truth": "intro v",
      "model_output": "intro v [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·] [·]"
    },
    {
      "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": true,
      "ground_truth": "apply εpos",
      "model_output": "apply εpos"
    },
    {
      "id": 217,
      "match": false,
      "ground_truth": "linarith [eq1]",
      "model_output": "linarith [(alg.γbound k).2]"
    },
    {
      "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 [← sub_add, norm_add_sq_real, ← sub_add, norm_add_sq_real] at ieq"
    },
    {
      "id": 221,
      "match": false,
      "ground_truth": "simp at din",
      "model_output": "simp at din [le_of_lt]"
    },
    {
      "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)) _ [mem_closedBall_iff_norm]"
    },
    {
      "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_L2_norm\n          apply le_max_left"
    },
    {
      "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_pos' hw_sum' hw_in'"
    },
    {
      "id": 225,
      "match": true,
      "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 hba₁\n    · apply ha₂ b hba₂"
    },
    {
      "id": 226,
      "match": false,
      "ground_truth": "exact inv_sub_inv l' h₁",
      "model_output": "apply 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 _ (l i₀ • intrinsicInterior ℝ (C i₀))"
    },
    {
      "id": 229,
      "match": false,
      "ground_truth": "rw [y1, y2]",
      "model_output": "rw [y1, y2] at h5 ⊢\n            exact 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": "have : ∀ᶠ (q : ℕ) in atTop, |(1 / (γ * ↑l)) •\n        grad_fst H (alg.y (α q - 1)) (alg.x (α q - 1))| ≤ ε := by\n      apply hk (fun q ↦ (1 / (γ * ↑l)) • grad_fst H (alg.y (α q - 1)) (alg.x (α q - 1))) diflte2\n      apply abs_le_of_norm_le\n      apply le_of_tendsto (continuous_iff_seq_tendsto.mp (continuous_norm"
    },
    {
      "id": 233,
      "match": false,
      "ground_truth": "intro n",
      "model_output": "intro n\n        specialize allh' (n + 1) (Nn n)\n        constructor\n        · exact allh'.left\n        apply tendsto_atTop_nhds.mpr\n        apply (tendsto_congr' (eventually_of_forall (fun x ↦ (allh' x).right))).mpr\n        apply (tendsto_atTop_nhds.mpr (allh' (n + 1)).right.right)"
    },
    {
      "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": "apply mul_eq_mul_right_iff.mpr\n        left\n        rw [← mul_assoc, ← mul_assoc, mul_comm, mul_assoc, mul_comm ‖grad‖, mul_comm ‖grad‖ ‖x' - x‖]\n        apply mul_eq_mul_right_iff.mpr\n        left\n        rw [← div_mul, div_mul_cancel₀, div_self, div_one]\n        apply ne_of_gt\n        apply mul_pos\n        apply div_pos\n        apply ε2pos\n        apply mul_pos\n        apply norm_sub_pos_iff.mpr\n        apply h₁\n        apply norm_sub_pos_iff"
    },
    {
      "id": 236,
      "match": false,
      "ground_truth": "refine LICQ_injM z m Z Ax ?_ mlen ?_ eq2 eq1 ⟨heq1, heq2⟩",
      "model_output": "apply LIx.eq_zero_of_orthogonal_span_eq_zero heq1 heq2"
    },
    {
      "id": 237,
      "match": false,
      "ground_truth": "apply le_trans",
      "model_output": "apply le_trans Nge\n          apply Nat.cast_le.mpr\n          apply le_trans\n          apply nge\n          apply le_add_of_nonneg_right\n          apply zero_le_one"
    },
    {
      "id": 238,
      "match": false,
      "ground_truth": "apply mul_eq_mul_left_iff.mpr",
      "model_output": "rw [mul_eq_mul_left_iff]"
    }
  ]
}