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{
"summary": {
"total_with_gt": 239,
"matched": 76,
"accuracy": 0.3179916317991632
},
"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 (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": "ext x; simp\n exact convex_indicator_empty x (by simp)"
},
{
"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⟩\nlift b to ℝ using ⟨hb1, hb2⟩\nrw [← EReal.coe_sub]\nsimp only [gt_iff_lt, EReal.coe_lt_coe_iff]\nexact bot_lt_iff_ne_bot.2 (sub_ne_bot.2 ⟨ha1, ha2, hb1, hb2⟩)"
},
{
"id": 6,
"match": false,
"ground_truth": "lift a to ℝ using ⟨ha1, ha2⟩",
"model_output": "have h1 : b - b = 0 := EReal.sub_eq_zero_of_eq hb2 hb1\n rw [h1]\n simp\n exact le_add_of_ge ha1 ha2 hb1 hb2"
},
{
"id": 7,
"match": true,
"ground_truth": "apply hasDerivAt_mul_const c⁻¹",
"model_output": "apply hasDerivAt_mul_const c⁻¹"
},
{
"id": 8,
"match": false,
"ground_truth": "by\n use Aφ (A := A) (x := x) H\n exact ⟨Aφ_mono (A := A) (x := x) H, α_subseq_converge (Aφ_sub H)⟩",
"model_output": "by\n use Aφ (A := A) (x := x) H\n exact ⟨Aφ_mono, Aφ_tend⟩"
},
{
"id": 9,
"match": true,
"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": "by\n have : OrderIso g' := by exact EReal.instOrderIsoEquiv\n apply OrderIso.map_biSup s g' rfl"
},
{
"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 [real_inner_eq_dot] at hg"
},
{
"id": 13,
"match": false,
"ground_truth": "apply add_le_add_right a",
"model_output": "by\n exact fun a b ↦ add_le_add_right a b a"
},
{
"id": 14,
"match": false,
"ground_truth": "left",
"model_output": "left; exact hdomsf"
},
{
"id": 15,
"match": false,
"ground_truth": "ext j",
"model_output": "rfl"
},
{
"id": 16,
"match": false,
"ground_truth": "intro i",
"model_output": "intro i j"
},
{
"id": 17,
"match": false,
"ground_truth": "· intro h; by_contra! hs\n obtain ⟨x, xs⟩:= Set.nonempty_def.1 hs\n have h1: s.convex_indicator x = 0 := by\n simp [convex_indicator]; exact xs\n have h2 := congrFun h x; simp [h1] at h2",
"model_output": "· intro h\n by_contra! 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\n exact False.elim (hs h2)"
},
{
"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.const_mul' 2\n apply HasGradientAt.dotProduct\n · apply HasGradientAt.const_mul' (Aᵀ *ᵥ A)\n apply gradient_of_inner\n · apply hasGradientAt_const"
},
{
"id": 20,
"match": false,
"ground_truth": "by norm_num",
"model_output": "by 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": "rw [subdifferential, Set.mem_setOf]"
},
{
"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 InnerProductSpace.hasGradientAt_inner_right"
},
{
"id": 24,
"match": true,
"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 Eq.symm (CancelDenoms.inv_subst (hA j) ?_)"
},
{
"id": 25,
"match": true,
"ground_truth": "intro x y hy",
"model_output": "intro x y hy"
},
{
"id": 26,
"match": false,
"ground_truth": "simp [LowerSemicontinuousHull]",
"model_output": "by\n simp [LowerSemicontinuousHull, iInter_setOf, iInter_and]"
},
{
"id": 27,
"match": true,
"ground_truth": "simp at hfy",
"model_output": "simp at hfy"
},
{
"id": 28,
"match": false,
"ground_truth": "rw [Fin.sum_univ_succ]",
"model_output": "have : ∀ (i : Fin (n + 1)), 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_intrinsicClosure] at 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' h'\n rw[← dist_eq_norm]\n rw[← Metric.ball, Set.mem_ball] at h'\n exact h h'"
},
{
"id": 32,
"match": false,
"ground_truth": "exact half_lt_self dpos",
"model_output": "by linarith"
},
{
"id": 33,
"match": false,
"ground_truth": "apply pin",
"model_output": "exact pin"
},
{
"id": 34,
"match": false,
"ground_truth": "by_cases hf_pinf : f = ⊤",
"model_output": "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": "by\n rw [intrinsicInterior]"
},
{
"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_right, ← add_sub, add_comm]"
},
{
"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": "intro x hx\n rw [intrinsicInterior_iff (convex_intrinsicClosure ℝ h) (nonempty_of_mem (intrinsicClosure ℝ s))] at hx\n exact (intrinsicInterior_subset _ _ ) hx"
},
{
"id": 39,
"match": false,
"ground_truth": "have : FiniteDimensional 𝕜 (Submodule.span 𝕜 (Set.range v)) := by\n apply FiniteDimensional.span_of_finite\n simp [Set.finite_range]",
"model_output": "have : FiniteDimensional 𝕜 (span 𝕜 (Set.range v)) := by\n apply FiniteDimensional.span_of_finite; simp\n exact Orthonormal.exists_orthonormalBasis_span hv"
},
{
"id": 40,
"match": false,
"ground_truth": "rw [h2]",
"model_output": "rw [h2] at h1 ⊢"
},
{
"id": 41,
"match": false,
"ground_truth": "apply banach_steinhaus",
"model_output": "have : ∀ x : s, ‖(innerSL ℝ) x‖ = ‖x‖ := by\n intro x\n simp [innerSL_apply_norm]"
},
{
"id": 42,
"match": false,
"ground_truth": "intro h\n apply HasFDeriv_Convergence\n exact h",
"model_output": "exact fun a ε ε0 ↦ HasFDeriv_Convergence a ε ε0"
},
{
"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] at hn"
},
{
"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": true,
"ground_truth": "rcases this with ⟨xn, hxn1, hxn2⟩",
"model_output": "rcases this with ⟨xn, hxn1, hxn2⟩"
},
{
"id": 48,
"match": false,
"ground_truth": "have h2 := tendsto_add (A:=A) (B:=B)",
"model_output": "have h2 := tendsto_atTop_ciInf (A_nonempty hAa) (B_nonempty hBa) h1"
},
{
"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": "by\n have h : ‖a • b‖ ^ 2 ≤ (‖a‖ * ‖b‖) ^ 2 := by\n rw [WithLp.smul_norm_sq_eq_inner, ← mul_pow]\n apply real_inner_le_norm\n apply nonneg_le_nonneg_of_sq_le_sq (norm_nonneg _) (mul_nonneg (norm_nonneg a) (norm_nonneg b))\n rwa [← pow_two, ← pow_two]"
},
{
"id": 50,
"match": false,
"ground_truth": "by_cases ht : i ∈ σ",
"model_output": "by_cases hs : i ∈ σ"
},
{
"id": 51,
"match": false,
"ground_truth": "simp",
"model_output": "simp; exact ⟨hd, fun x _ y _ a b _ _ _ => by simp⟩"
},
{
"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": true,
"ground_truth": "exact in_of_le_sinf_of_b_top hx",
"model_output": "exact in_of_le_sinf_of_b_top hx"
},
{
"id": 54,
"match": false,
"ground_truth": "intro hmin",
"model_output": "intro hmin\n exact first_order_unconstrained hf hmin hfc"
},
{
"id": 55,
"match": true,
"ground_truth": "simp [hx2, hw.2]",
"model_output": "simp [hx2, hw.2]"
},
{
"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": true,
"ground_truth": "simp [ConvexCone.addPoint] at hx",
"model_output": "simp [ConvexCone.addPoint] at hx"
},
{
"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 add_comm\n apply smul_sub\n apply inner_sub_left\n apply eq_sub_of_add_eq\n apply Eq.symm\n apply sub_eq_add_neg\n apply sub_eq_add_neg"
},
{
"id": 59,
"match": false,
"ground_truth": "simp",
"model_output": "simp [EReal.coe_mul_bot_of_pos] 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 _ _ _ _ _ hp1 hp2 hp3\n simp only [mem_setOf_eq]\n rw [A.map_vadd, A.map_vsub, vsub_eq_sub, map_smul, map_sub, hp1, hp2, hp3]\n simp only [vadd_eq_add, add_sub_cancel]"
},
{
"id": 61,
"match": false,
"ground_truth": "linarith",
"model_output": "linarith [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": "have xnunifcond : ∀ (n : ℕ), f (xn n).1 ≤ (xn n).2 := by\n exact fun n ↦ xncond n"
},
{
"id": 65,
"match": false,
"ground_truth": "by_cases ha1 : a = 0",
"model_output": "by_cases hb1 : b = 0\n[ASSISTANT]\n· rw [hb1]; simp; rw [one_smul]; simp at hab; rw [hab]; simp\n[ASSISTANT]\n· rw [EReal.smul_top_of_pos (by linarith)]; simp\n rw [EReal.smul_top_of_pos (by linarith)]; simp\n rw [EReal.smul_top_of_pos (by linarith)]; simp\n apply le_top"
},
{
"id": 66,
"match": false,
"ground_truth": "by_cases hb1 : b = 0",
"model_output": "simp [ConvexOn] at hf"
},
{
"id": 67,
"match": false,
"ground_truth": "congr",
"model_output": "congr!"
},
{
"id": 68,
"match": false,
"ground_truth": "intro z",
"model_output": "intro z hz\n have : HasFDerivAt f (f' z) z := by apply hu3; exact hz\n exact (hasGradientAt_iff_hasFDerivAt).mp this"
},
{
"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 y hy"
},
{
"id": 71,
"match": false,
"ground_truth": "have hb : b = 1 := by linarith",
"model_output": "simp [ha] at absum ⊢\n rw [absum]\n simp\n apply convex_conjugate_geq_zero"
},
{
"id": 72,
"match": false,
"ground_truth": "intro x y r",
"model_output": "have h' : A = (fun x => (1 : ℝ) • x) := by\n ext x; exact (AffineMap.map_smul A x (1 : ℝ)).symm"
},
{
"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.slope_le_deriv 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": true,
"ground_truth": "rw [← hEq]",
"model_output": "rw [← hEq]"
},
{
"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 exact unit_ball_nonempty 𝕜 V"
},
{
"id": 79,
"match": false,
"ground_truth": "linarith",
"model_output": "linarith [hy.2, hx₂]"
},
{
"id": 80,
"match": true,
"ground_truth": "apply DifferentiableAt.hasGradientAt",
"model_output": "apply DifferentiableAt.hasGradientAt"
},
{
"id": 81,
"match": false,
"ground_truth": "rcases lbdf with ⟨L, Lbound⟩",
"model_output": "have bddbelowImS : BddBelow ImS := by\n rcases lbdf with ⟨L, Lbound⟩\n use L; intro gx; simp [ImS]\n rcases gx with ⟨z, _, gzeq⟩\n rw [← gzeq]; exact Lbound z"
},
{
"id": 82,
"match": false,
"ground_truth": "simp",
"model_output": "simp; linarith [μpos]"
},
{
"id": 83,
"match": false,
"ground_truth": "norm_num",
"model_output": "by 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": true,
"ground_truth": "rw [mem_setOf]",
"model_output": "rw [mem_setOf]"
},
{
"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_of_liminf (f := fun n ↦ f (x n) + ⟪g n, y - x n⟫) (x := y) (by apply fx_lowerbd) (by apply fx_BddBelow) (by apply fx_BddAbove) (by apply fx_lowersemicontinuous) (by apply fx_lowersemicontinuous) (by apply fx_lowersemicontinuous) (by apply fx_lowersemicontinuous) (by apply fx_lowersemicontinuous) (by apply fx_lowersemicontinuous) (by apply fx_lowersemicontinuous) ("
},
{
"id": 88,
"match": false,
"ground_truth": "refine norm_ne_zero_iff.mpr ?_",
"model_output": "apply norm_ne_zero_iff.mpr\n[· exact 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 := hT.orthogonal_span_range_v_eq_orthogonal_ker"
},
{
"id": 90,
"match": true,
"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": "· 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]"
},
{
"id": 91,
"match": true,
"ground_truth": "apply Finset.sum_eq_zero",
"model_output": "apply Finset.sum_eq_zero"
},
{
"id": 92,
"match": false,
"ground_truth": "mul_inv_cancel₀ (by linarith),",
"model_output": "by\n field_simp; ring_nf; simp [norm_smul, mul_pow]"
},
{
"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"
},
{
"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": true,
"ground_truth": "· rw [intrinsicInterior_prod_eq_prod_intrinsicInterior]; exact mk_mem_prod (by simp) hx",
"model_output": "rw [intrinsicInterior_prod_eq_prod_intrinsicInterior]"
},
{
"id": 98,
"match": true,
"ground_truth": "rw [alg.γeq k, alg.teq k]",
"model_output": "rw [alg.γeq k, alg.teq k]"
},
{
"id": 99,
"match": false,
"ground_truth": "intro i _",
"model_output": "intro i _ [h4]"
},
{
"id": 100,
"match": true,
"ground_truth": "rcases vt with ⟨c, d, ⟨a, ha⟩, ⟨vt1, vt2⟩⟩",
"model_output": "rcases vt with ⟨c, d, ⟨a, ha⟩, ⟨vt1, 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": false,
"ground_truth": "have h1 :A x j + ε + ∑ x_1 with ¬x_1 = j, A x x_1 = ε + A x j + ∑ x_1\n with ¬x_1 = j, A x x_1 := by ring_nf",
"model_output": "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 [add_comm, add_left_comm]"
},
{
"id": 103,
"match": true,
"ground_truth": "intro x",
"model_output": "intro x"
},
{
"id": 104,
"match": false,
"ground_truth": "apply LinearMap.congr_fun\n ((LinearEquiv.symm_apply_eq (Module.dualProdDualEquivDual ℝ E ℝ)).1 (by rfl)) a",
"model_output": "simp [φ]"
},
{
"id": 105,
"match": false,
"ground_truth": "rw[Metric.isOpen_iff] at isopen",
"model_output": "rcases (mem_nhdsWithin.1 ht.1).2 with ⟨r, hr, ht1⟩\n use ‖toEuclidean (toDual ℝ α) (f := f)‖ * r\n use t ∩ s\n constructor\n · apply mem_nhdsWithin_of_mem_nhds\n apply mem_nhds_of_mem_ball\n exact ⟨r, hr, isopen, x_pos⟩\n · constructor\n · apply LipschitzOnWith.mono (convexOn_norm_toEuclidean α f) isopen.convex\n · apply LipschitzOn"
},
{
"id": 106,
"match": false,
"ground_truth": "rw [deriv_of_const_mul_func (f u - f x)]",
"model_output": "have : ‖u - x‖ + ‖f u - f x‖ < c := by\n rw [← add_halves c]\n apply add_lt_add\n · apply mem_ball_iff_norm.1 hu.left\n · simp at *\n rw [abs_eq_self.2] <;> linarith [hu.right]"
},
{
"id": 107,
"match": false,
"ground_truth": "rcases exists_seq_tendsto_sInf hs hs' with ⟨fx, _, cfx, fxs⟩",
"model_output": "have hs'' : ∃ x ∈ (f ⁻¹' Set.Iic y), ∀ a ∈ (f ⁻¹' Set.Iic y), f x ≤ f a := by\n rcases (sInf_eq_inf hs hs').1 with ⟨x, hx⟩\n use x; constructor\n · exact hx\n intro a ha; exact le_csInf hs' ⟨a, ha, rfl⟩"
},
{
"id": 108,
"match": false,
"ground_truth": "induction n with",
"model_output": "have : Φ n ≤ C := by\n calc\n _ = ∑ i ∈ range n, Φ_i n + ∑ i ∈ range n, Ψ_i n := by rfl\n _ ≤ ∑ i ∈ range n, (φ i + ψ i) + ∑ i ∈ range n, (φ i + ψ i) := by\n apply add_le_add\n · apply finite_sum_inequ₁\n · apply finite_sum_inequ₂\n _ = 2 * ∑ i ∈ range n, (φ i + ψ i) :="
},
{
"id": 109,
"match": false,
"ground_truth": "simp at ht",
"model_output": "simp at ht [Icc]"
},
{
"id": 110,
"match": true,
"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; simp [c1];\n by_cases ht : i ∈ σ\n · simp [ht];\n specialize cpos i ht;\n exact cpos\n simp [ht]"
},
{
"id": 111,
"match": true,
"ground_truth": "rw [subdiff_gradient g g' hg gradcon]",
"model_output": "rw [subdiff_gradient g g' hg gradcon]"
},
{
"id": 112,
"match": false,
"ground_truth": "rw [div_mul_cancel₀ _ hne, abs_of_nonneg (by positivity), mul_comm] at hd",
"model_output": "rw [div_mul_cancel₀, abs_of_nonneg (by positivity)] at hd"
},
{
"id": 113,
"match": false,
"ground_truth": "simp [phi_add] at this",
"model_output": "simp [phi_add] at this [hz]"
},
{
"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 [Set.preimage] at hu_eq [USER]\n rw [← hu_eq] [ASSISTANT]\n apply hu_closed [USER]\n simp [Set.preimage] [ASSISTANT]"
},
{
"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_le_coe_iff] at fx_le_top ⊢ [fx_ge_bot]"
},
{
"id": 116,
"match": false,
"ground_truth": "rcases zin with ⟨u,hux,hufx,hv⟩",
"model_output": "rcases zin with ⟨u, utend, funtend, ⟨v, hv⟩⟩"
},
{
"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"
},
{
"id": 119,
"match": false,
"ground_truth": "have hΦ : ∃ r_Φ, range Φ ⊆ Metric.ball 0 r_Φ := by apply Φ_isBounded'",
"model_output": "rcases Φ_isBounded' with ⟨M , h⟩\n use M; intro x; simp\n rw [range] at h; rcases h x x.2 with hx\n simp at hx; rw [← dist_eq_norm]\n exact hx"
},
{
"id": 120,
"match": true,
"ground_truth": "· simp",
"model_output": "· simp"
},
{
"id": 121,
"match": false,
"ground_truth": "have : Tendsto (fun n => ‖A₁ (e₁ n) + A₂ (e₂ n)‖ ^ 2) atTop (𝓝 0) := by\n apply Summable.tendsto_atTop_zero Summable₁\n apply square_converge_zero₁ this",
"model_output": "have : Tendsto (fun n => √((‖A₁ (e₁ n) + A₂ (e₂ n)‖)^2)) atTop (𝓝 √0) := by\n apply Tendsto.sqrt (admm.E₁_A₁e₁_A₂e₂_subseq_converge_zero)"
},
{
"id": 122,
"match": false,
"ground_truth": "rw[← EReal.coe_toReal (x := f x) hfx (LT.lt.ne_bot (hsf.uninfinity x hx))]",
"model_output": "by_cases hfx2 : f x = ⊥"
},
{
"id": 123,
"match": false,
"ground_truth": "by_cases u₁_eq_0 : u₁ = 0",
"model_output": "have : (coef i : 𝕜) = 0 := by\n apply inj\n simp [coef]\n rw [← hu₁, Finsupp.sum_apply, Finsupp.sum_eq_single i (fun x a ↦ False.elim (a (Finset.mem_of_mem_insert_of_ne (Finset.mem_range.mp x) (Finset.mem_range.mp i).2)))]\n simp [Tu_eq_0, this]"
},
{
"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": 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"
},
{
"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": "by\n simp; apply sq_nonneg b; linarith [sq_nonneg (b - 1)]"
},
{
"id": 133,
"match": false,
"ground_truth": "rw [← add_assoc, ← add_assoc]",
"model_output": "by simp [add_assoc, add_comm, add_left_comm]"
},
{
"id": 134,
"match": false,
"ground_truth": "simp",
"model_output": "simp [Metric.ball, dist_eq_norm]"
},
{
"id": 135,
"match": false,
"ground_truth": "intro y yin z zin a b anneg bnneg absum1",
"model_output": "intro y _ a b ha hb hab [h]"
},
{
"id": 136,
"match": false,
"ground_truth": "gcongr",
"model_output": "by gcongr"
},
{
"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 : ℕ), z n ∈ FeasSet := by\n intro n\n have : z n ∈ Metric.ball loc ε ∩ FeasSet := by\n simp [z]; apply εball; simp\n exact this.2"
},
{
"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 ⊢ [ceq0]"
},
{
"id": 140,
"match": false,
"ground_truth": "apply Continuous.tendsto",
"model_output": "apply Continuous.tendsto\n apply ContinuousLinearMap.continuous"
},
{
"id": 141,
"match": true,
"ground_truth": "by\n have h₁: Tendsto A₂ (𝓝 0) (𝓝 (A₂ 0)) := by\n apply Continuous.tendsto; apply ContinuousLinearMap.continuous\n simp at h₁; apply Filter.tendsto_iff_seq_tendsto.1 h₁; apply e₂_subseq_converge_zero",
"model_output": "have 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"
},
{
"id": 142,
"match": false,
"ground_truth": "apply A₁e₁_converge_zero",
"model_output": "apply tendsto_zero_iff_norm_tendsto_zero.1; 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 row_sum_le : ∑ i, A i j ≤ 1 := hA_col_sum j\n have row_sum_ge : ∑ i, A i j ≥ 1 := by\n -- A is DoublySubStochastic\n calc\n ∑ i, A i j ≥ ∑ i, (∑ j, A i j) := Finset.single_le_sum (fun i _ => hA_row_sum i) j (Finset.mem_univ j)"
},
{
"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": "simp only [c]"
},
{
"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 inter_mem_nhdsWithin _ (t_isopen.mem_nhds x_in_xBall)\n apply mem_nhdsWithin_of_mem_nhds\n apply isOpen_interior.mem_nhds\n exact nonempty_of_mem hx"
},
{
"id": 148,
"match": false,
"ground_truth": "rw [← zero_pow]",
"model_output": "apply squeeze_zero_norm\n · apply Summable.hasSum\n · apply square_nonneg"
},
{
"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_nonneg; simp"
},
{
"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_eq_iff_eq_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": "by\n rw [z]; simp; left; rw [z]; simp"
},
{
"id": 154,
"match": false,
"ground_truth": "· simp\n apply neg_le.mpr\n · have pos: - 0 ≤ ε * ‖y - x‖:=by\n simp\n rw [mul_comm]\n apply (mul_nonneg_iff_left_nonneg_of_pos εpos).mpr\n simp\n apply pos",
"model_output": "· apply neg_le.mpr\n apply mul_nonneg\n · linarith [εpos]\n apply norm_nonneg"
},
{
"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": "intro n x\napply inner_gradient\n[SYSTEM]\nrepeat apply ADMM_iter_process₁'_eq3_2'"
},
{
"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": 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‖ := norm_sub_le (x + xx) xx"
},
{
"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 h1 : (MM x).carrier ∩ (f.Epi univ) = (MM x).carrier ∩ ((f.closure univ).Epi univ) := by\n rw [epi_closure_epi_eq_epi_closure' hc (by simp [hp]) (by simp [MM])]"
},
{
"id": 162,
"match": true,
"ground_truth": "simp [z]",
"model_output": "simp [z]"
},
{
"id": 163,
"match": true,
"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 Tendsto.sub_const\n apply Tendsto.add\n · apply Satisfying_equational_constraint1\n · apply Satisfying_equational_constraint2"
},
{
"id": 164,
"match": true,
"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 := by\n 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": "by\n dsimp [ey_diff, t_Ax_Ax_n1, t_A_diff_x', A2_e2_n1, A2_e2_n]\n linarith [Φ_isdescending_inequ5 n]"
},
{
"id": 166,
"match": false,
"ground_truth": "intro n",
"model_output": "intro n\nrw [SubderivAt, Submodule.mem_orthogonal]\n apply SubderivAt_wrt\n apply ADMM_iter_process₁'_eq3_1'"
},
{
"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_add, smul_sub, smul_smul]"
},
{
"id": 169,
"match": false,
"ground_truth": "suffices 0 ≤ (a1 • tx + b1 • ty) by exact this",
"model_output": "intro i [hi]"
},
{
"id": 170,
"match": true,
"ground_truth": "by\n rw [sub_right_comm]",
"model_output": "rw [sub_right_comm]"
},
{
"id": 171,
"match": true,
"ground_truth": "intro i hi",
"model_output": "intro i 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) [hax (ax + n) (by simp)]"
},
{
"id": 173,
"match": true,
"ground_truth": "rw [hdomain]; trivial",
"model_output": "rw [hdomain]"
},
{
"id": 174,
"match": false,
"ground_truth": "rfl",
"model_output": "simp only [Function.comp_apply]"
},
{
"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 _ epos [assumption]"
},
{
"id": 178,
"match": true,
"ground_truth": "intro n",
"model_output": "intro n"
},
{
"id": 179,
"match": false,
"ground_truth": "simp",
"model_output": "simp [mul_pow, mul_assoc, mul_comm, mul_left_comm]"
},
{
"id": 180,
"match": false,
"ground_truth": "· simp",
"model_output": "simp [tendsto_const_nhds_iff]"
},
{
"id": 181,
"match": false,
"ground_truth": "simp at hu",
"model_output": "apply h_Ω1 u (hO1 hu.1) <;> simp at hu <;> linarith [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 EReal.coe_le_coe_iff.mpr [add_le_add supa supb]"
},
{
"id": 183,
"match": false,
"ground_truth": "simp at vt1",
"model_output": "use 2 * (‖v‖ + ε) / ε"
},
{
"id": 184,
"match": false,
"ground_truth": "rw [pow_two]",
"model_output": "by\n rw [pow_two]; apply mul_le_mul_of_nonneg_right\n rw [mul_le_mul_iff_right₀ alg.tpos]; exact alg.step; simp; linarith [alg.hL]"
},
{
"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⟩"
},
{
"id": 186,
"match": false,
"ground_truth": "intro b _",
"model_output": "intro b _; simp [hG]; simp; apply div_nonneg; apply mul_nonneg; apply le_of_lt εpos; apply div_nonneg; apply le_of_lt; apply div_pos; apply εpos; apply zero_lt_two\n apply zero_le; apply Finset.sum_nonneg; apply alg.ha; apply sq_nonneg"
},
{
"id": 187,
"match": false,
"ground_truth": "ring",
"model_output": "by ring_nf"
},
{
"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 eq4 : 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 [mul_pos apos bpos]"
},
{
"id": 194,
"match": false,
"ground_truth": "by ring",
"model_output": "by ring_nf"
},
{
"id": 195,
"match": true,
"ground_truth": "by\n rw[sign_mul_abs]\n field_simp [bi_pos i]",
"model_output": "rw[sign_mul_abs]"
},
{
"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 eq0 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 [this, kleaw]"
},
{
"id": 201,
"match": false,
"ground_truth": "repeat apply mul_nonneg",
"model_output": "· apply mul_nonneg\n · apply mul_nonneg\n · apply inv_nonneg.mpr\n linarith [admm.hrho]\n · apply pow_nonneg; simp\n · linarith [norm_nonneg (ey n)]"
},
{
"id": 202,
"match": false,
"ground_truth": "exact Eq.symm (Nat.add_sub_of_le kge)",
"model_output": "by\n exact Eq.symm (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 c1 hc1 [atTop]"
},
{
"id": 205,
"match": true,
"ground_truth": "exact min_le_right δ₀ δ₂",
"model_output": "exact min_le_right δ₀ δ₂"
},
{
"id": 206,
"match": false,
"ground_truth": "simp",
"model_output": "by\n simp; left; rw [mul_pow, pow_two ‖alg.a‖, 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": "by 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'''] [hφciv] [hφcv] [hφcv'] [hφcv''] [hφcv'''] [hφcviv] [hφcvv] [hφcv'] [hφcv''] [hφcv'''] [hφcviv] [hφcvv] [hφcv'] [hφcv''] [hφcv'''] [hφcviv] [hφcvv] [hφcv']"
},
{
"id": 211,
"match": false,
"ground_truth": "intro i",
"model_output": "intro i hi"
},
{
"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": false,
"ground_truth": "apply εpos",
"model_output": "· linarith [εpos]"
},
{
"id": 217,
"match": false,
"ground_truth": "linarith [eq1]",
"model_output": "linarith [(alg.γbound k).1]"
},
{
"id": 218,
"match": true,
"ground_truth": "· apply Finset.sum_pos\n · intro i _; apply (alg.ha i)\n · simp",
"model_output": "· apply Finset.sum_pos\n · intro i _; apply (alg.ha i)\n · simp"
},
{
"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] 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": "have : (alg.x (α q - 1), alg.y (α q - 1)) ∈ Metric.closedBall (0 : WithLp 2 (E × F)) C1 := by\n simp; apply mem_closedBall_iff_norm.mpr\n rw [WithLp.prod_norm_eq_of_L2]\n apply le_trans (inin (alg.x (α q - 1), alg.y (α q - 1)) (mem_image_of_mem alg.z trivial))\n apply sq]"
},
{
"id": 223,
"match": true,
"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": "rw [WithLp.prod_norm_eq_of_L2]"
},
{
"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' hmem'"
},
{
"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 [sum_subset (fun i ↦ intrinsicInterior_subset) (fun i ↦ hC i)]"
},
{
"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": 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 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 apply And.intro\n · specialize allh' (n+1) (Nn n)\n apply allh'.left\n · apply tendsto_const_nhds\n apply tendsto_atTop_nhds.mpr\n intro s\n specialize vnkt (n+1)\n apply vnkt\n apply Nat.succ_pos"
},
{
"id": 234,
"match": false,
"ground_truth": "by\n apply zp5",
"model_output": "exact zp5"
},
{
"id": 235,
"match": true,
"ground_truth": "by\n apply Simplifying₂\n apply h₁\n apply norm_nonneg",
"model_output": "apply Simplifying₂"
},
{
"id": 236,
"match": false,
"ground_truth": "refine LICQ_injM z m Z Ax ?_ mlen ?_ eq2 eq1 ⟨heq1, heq2⟩",
"model_output": "rw [← Matrix.mulVec_eq_mul, ← Matrix.mulVec_eq_mul] at heq1 heq2"
},
{
"id": 237,
"match": false,
"ground_truth": "apply le_trans",
"model_output": "apply le_trans Nge [by apply add_le_add_right; simp]"
},
{
"id": 238,
"match": false,
"ground_truth": "apply mul_eq_mul_left_iff.mpr",
"model_output": "by\n rw [mul_comm, mul_assoc,mul_comm _ ‖_‖]\n apply mul_eq_mul_left_iff.mpr\n left\n refine\n {\n norm_num,\n eq_true (Eq.symm (norm_smul _ _))\n }"
}
]
} |