| { |
| "summary": { |
| "total_with_gt": 239, |
| "matched": 31, |
| "accuracy": 0.1297071129707113 |
| }, |
| "details": [ |
| { |
| "id": 0, |
| "match": false, |
| "ground_truth": "apply HasDerivAt.sub", |
| "model_output": "apply HasDerivAt.const_sub" |
| }, |
| { |
| "id": 1, |
| "match": false, |
| "ground_truth": "have : (fun (t : ℝ) => c⁻¹ * t) = (fun t => t * c⁻¹) := by ext t; ring", |
| "model_output": "apply HasDerivAt.const_mul" |
| }, |
| { |
| "id": 2, |
| "match": false, |
| "ground_truth": "rw [← EReal.coe_sub]", |
| "model_output": "norm_cast at *" |
| }, |
| { |
| "id": 3, |
| "match": false, |
| "ground_truth": "ext x", |
| "model_output": "apply convex_indicator_empty" |
| }, |
| { |
| "id": 4, |
| "match": true, |
| "ground_truth": "exact ah", |
| "model_output": "exact ah" |
| }, |
| { |
| "id": 5, |
| "match": false, |
| "ground_truth": "lift a to ℝ using ⟨ha1, ha2⟩\n lift b to ℝ using ⟨hb1, hb2⟩\n rw [← EReal.coe_sub]\n apply bot_lt_iff_ne_bot.2\n let c := a - b\n suffices Real.toEReal c ≠ ⊥ by exact this\n simp", |
| "model_output": "apply gt_iff_ne_bot.mpr" |
| }, |
| { |
| "id": 6, |
| "match": false, |
| "ground_truth": "lift a to ℝ using ⟨ha1, ha2⟩", |
| "model_output": "simp" |
| }, |
| { |
| "id": 7, |
| "match": false, |
| "ground_truth": "apply hasDerivAt_mul_const c⁻¹", |
| "model_output": "apply HasDerivAt.mul_const" |
| }, |
| { |
| "id": 8, |
| "match": false, |
| "ground_truth": "by\n use Aφ (A := A) (x := x) H\n exact ⟨Aφ_mono (A := A) (x := x) H, α_subseq_converge (Aφ_sub H)⟩", |
| "model_output": "have H' := H" |
| }, |
| { |
| "id": 9, |
| "match": false, |
| "ground_truth": "· rw [ha1];simp", |
| "model_output": "aesop" |
| }, |
| { |
| "id": 10, |
| "match": false, |
| "ground_truth": "let g : EReal ≃o EReal := ⟨g', by simp [g'];apply add_le_add_right a⟩", |
| "model_output": "apply @iSup₂_add_const _ _ _ _ _ _ _" |
| }, |
| { |
| "id": 11, |
| "match": false, |
| "ground_truth": "ext i j", |
| "model_output": "have h2 := h 2" |
| }, |
| { |
| "id": 12, |
| "match": false, |
| "ground_truth": "apply glt", |
| "model_output": "aesop" |
| }, |
| { |
| "id": 13, |
| "match": false, |
| "ground_truth": "apply add_le_add_right a", |
| "model_output": "intro a b" |
| }, |
| { |
| "id": 14, |
| "match": false, |
| "ground_truth": "left", |
| "model_output": "exact Or.inl hdomsf" |
| }, |
| { |
| "id": 15, |
| "match": false, |
| "ground_truth": "ext j", |
| "model_output": "apply congr_arg" |
| }, |
| { |
| "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 hs" |
| }, |
| { |
| "id": 18, |
| "match": false, |
| "ground_truth": "constructor", |
| "model_output": "aesop" |
| }, |
| { |
| "id": 19, |
| "match": false, |
| "ground_truth": "by_cases hA : A = 0", |
| "model_output": "intro x" |
| }, |
| { |
| "id": 20, |
| "match": false, |
| "ground_truth": "by norm_num", |
| "model_output": "have := hf.1" |
| }, |
| { |
| "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" |
| }, |
| { |
| "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 hasGradientAt_inner" |
| }, |
| { |
| "id": 24, |
| "match": false, |
| "ground_truth": "refine Eq.symm (CancelDenoms.inv_subst (hA j) ?_)\n refine prod_erase_mul univ (fun «x» ↦ A «x» «x») ?_\n exact mem_univ j", |
| "model_output": "rw [← Finset.prod_erase_mul _ _ (Finset.mem_univ j)]" |
| }, |
| { |
| "id": 25, |
| "match": true, |
| "ground_truth": "intro x y hy", |
| "model_output": "intro x y h" |
| }, |
| { |
| "id": 26, |
| "match": false, |
| "ground_truth": "simp [LowerSemicontinuousHull]", |
| "model_output": "apply Set.ext" |
| }, |
| { |
| "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": "apply nh" |
| }, |
| { |
| "id": 29, |
| "match": false, |
| "ground_truth": "by_cases hx2 : f x = ⊥", |
| "model_output": "rw [← h] at hx1" |
| }, |
| { |
| "id": 30, |
| "match": false, |
| "ground_truth": "rw [@mem_iInter]", |
| "model_output": "simp only [intrinsicClosure, mem_iInter, mem_setOf_eq] at hx ⊢" |
| }, |
| { |
| "id": 31, |
| "match": false, |
| "ground_truth": "rcases h with ⟨δ, dpos, h⟩", |
| "model_output": "obtain ⟨δ, δpos, hδ⟩ := h" |
| }, |
| { |
| "id": 32, |
| "match": false, |
| "ground_truth": "exact half_lt_self dpos", |
| "model_output": "linarith" |
| }, |
| { |
| "id": 33, |
| "match": false, |
| "ground_truth": "apply pin", |
| "model_output": "exact pin" |
| }, |
| { |
| "id": 34, |
| "match": false, |
| "ground_truth": "by_cases hf_pinf : f = ⊤", |
| "model_output": "apply convexOn_convex_conjugate" |
| }, |
| { |
| "id": 35, |
| "match": false, |
| "ground_truth": "by\n apply intrinsicInterior_subset.antisymm\n nth_rw 1 [intrinsicInterior]\n rw [intrinsicInterior, image_subset_iff]\n rw [affineSpan_intrinsicInterior hsc]\n rw [Function.Injective.preimage_image Subtype.val_injective]\n simp [intrinsicInterior]", |
| "model_output": "apply le_antisymm" |
| }, |
| { |
| "id": 36, |
| "match": false, |
| "ground_truth": "rw [← sub_add, norm_add_sq_real, real_inner_comm, inner_sub_right]", |
| "model_output": "rw [pow_two]" |
| }, |
| { |
| "id": 37, |
| "match": false, |
| "ground_truth": "rw [EReal.coe_le_coe_iff] at hccleq", |
| "model_output": "norm_cast at hccleq" |
| }, |
| { |
| "id": 38, |
| "match": false, |
| "ground_truth": "intro z", |
| "model_output": "apply intrinsicInterior_mono" |
| }, |
| { |
| "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": "exact ⟨fun i => ⟨v i, hv.1 i⟩, fun i => rfl⟩" |
| }, |
| { |
| "id": 40, |
| "match": true, |
| "ground_truth": "rw [h2]", |
| "model_output": "rw [h2]" |
| }, |
| { |
| "id": 41, |
| "match": false, |
| "ground_truth": "apply banach_steinhaus", |
| "model_output": "have := hy 0" |
| }, |
| { |
| "id": 42, |
| "match": false, |
| "ground_truth": "intro h\n apply HasFDeriv_Convergence\n exact h", |
| "model_output": "intro hf ε ε_pos\nobtain ⟨δ, δ_pos, hδ⟩ := Metric.nhds_basis_closedBall.mem_iff.1 (hf.def ε ε_pos)" |
| }, |
| { |
| "id": 43, |
| "match": false, |
| "ground_truth": "· apply Convergence_HasFDeriv", |
| "model_output": "intro h" |
| }, |
| { |
| "id": 44, |
| "match": false, |
| "ground_truth": "by\n rw [←intrinsicInterior_affineSubspace_eq_self, ←nonempty_iff_ne_empty] at hn\n rw [←iIntersection_closure_eq_intrinsicInterior_closure' M.convex hC hn]\n rw [closure_affineSubspace_eq_self]", |
| "model_output": "have hM := M.isClosed.closure_eq" |
| }, |
| { |
| "id": 45, |
| "match": false, |
| "ground_truth": "apply alg.ha 0", |
| "model_output": "rw [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": "obtain ⟨y, ⟨n, hn, rfl⟩, hy⟩ := this" |
| }, |
| { |
| "id": 48, |
| "match": false, |
| "ground_truth": "have h2 := tendsto_add (A:=A) (B:=B)", |
| "model_output": "have h2 := tendsto_lim_inf B" |
| }, |
| { |
| "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": "split_ifs" |
| }, |
| { |
| "id": 51, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "simp at *" |
| }, |
| { |
| "id": 52, |
| "match": false, |
| "ground_truth": "by rw [h y ys, hf]; exact hg y ys", |
| "model_output": "rw [← h x (by simp [SubderivWithinAt, *])]" |
| }, |
| { |
| "id": 53, |
| "match": false, |
| "ground_truth": "exact in_of_le_sinf_of_b_top hx", |
| "model_output": "apply subset_closure" |
| }, |
| { |
| "id": 54, |
| "match": true, |
| "ground_truth": "intro hmin", |
| "model_output": "intro h" |
| }, |
| { |
| "id": 55, |
| "match": false, |
| "ground_truth": "simp [hx2, hw.2]", |
| "model_output": "simp only [hx2, Prod.tendsto_iff, tendsto_const_nhds_iff, and_self_iff, imp_true_iff,\n tendsto_nhds_top_iff_nat, true_and_iff]" |
| }, |
| { |
| "id": 56, |
| "match": false, |
| "ground_truth": "rw [smul_sub, smul_sub, add_comm_sub, sub_sub, ← add_smul, add_comm b a]", |
| "model_output": "rw [← sub_add, ← smul_add, ← sub_add, smul_sub, smul_sub]" |
| }, |
| { |
| "id": 57, |
| "match": false, |
| "ground_truth": "simp [ConvexCone.addPoint] at hx", |
| "model_output": "cases' hx with c hc y hy" |
| }, |
| { |
| "id": 58, |
| "match": false, |
| "ground_truth": "intro x", |
| "model_output": "intro x hx y hy" |
| }, |
| { |
| "id": 59, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "exact mem_univ _" |
| }, |
| { |
| "id": 60, |
| "match": false, |
| "ground_truth": "by\n simp only [mem_setOf_eq]\n intro _ _ _ _ hp1 hp2 hp3\n rw [vsub_eq_sub, vadd_eq_add, map_add, map_smul, map_sub,\n hp1, hp2, hp3, sub_self, smul_zero, zero_add]", |
| "model_output": "intro c p₁ p₂ p₃ hp₁ hp₂ hp₃" |
| }, |
| { |
| "id": 61, |
| "match": false, |
| "ground_truth": "linarith", |
| "model_output": "aesop" |
| }, |
| { |
| "id": 62, |
| "match": false, |
| "ground_truth": "simp [h]", |
| "model_output": "simp only [LinearIsometryEquiv.apply_symm_apply]" |
| }, |
| { |
| "id": 63, |
| "match": false, |
| "ground_truth": "simp [g]", |
| "model_output": "have := h.2" |
| }, |
| { |
| "id": 64, |
| "match": false, |
| "ground_truth": "rw [Prod.tendsto_iff] at xntend", |
| "model_output": "apply le_of_forall_lt'" |
| }, |
| { |
| "id": 65, |
| "match": false, |
| "ground_truth": "by_cases ha1 : a = 0", |
| "model_output": "simp [ConvexOn, h1]" |
| }, |
| { |
| "id": 66, |
| "match": false, |
| "ground_truth": "by_cases hb1 : b = 0", |
| "model_output": "apply le_trans (hf.2 _ _ ha hb hab)" |
| }, |
| { |
| "id": 67, |
| "match": true, |
| "ground_truth": "congr", |
| "model_output": "congr" |
| }, |
| { |
| "id": 68, |
| "match": false, |
| "ground_truth": "intro z", |
| "model_output": "intro y hy" |
| }, |
| { |
| "id": 69, |
| "match": false, |
| "ground_truth": "exact Batteries.compareOfLessAndEq_eq_lt.mp rfl", |
| "model_output": "norm_num" |
| }, |
| { |
| "id": 70, |
| "match": false, |
| "ground_truth": "intro y", |
| "model_output": "intro x'" |
| }, |
| { |
| "id": 71, |
| "match": false, |
| "ground_truth": "have hb : b = 1 := by linarith", |
| "model_output": "simp" |
| }, |
| { |
| "id": 72, |
| "match": false, |
| "ground_truth": "intro x y r", |
| "model_output": "aesop" |
| }, |
| { |
| "id": 73, |
| "match": false, |
| "ground_truth": "rw [hab]", |
| "model_output": "rw [hab, gradient_add, gradient_const, gradient_linear, transpose_single, one_mul]" |
| }, |
| { |
| "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 sub_le_sub_left" |
| }, |
| { |
| "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": "exact h x y r" |
| }, |
| { |
| "id": 77, |
| "match": false, |
| "ground_truth": "have h1 : M -ᵥ M ⊆ Submodule.comap (LinearMap.inl R α β) v := by\n rintro p ⟨c, hc, d, hd, rfl⟩\n exact Submodule.subset_span ⟨⟨c, n0⟩, ⟨hc, hn0⟩, ⟨d, n0⟩, ⟨hd, hn0⟩, by simp⟩", |
| "model_output": "apply Submodule.span_induction" |
| }, |
| { |
| "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": "apply exists_norm_eq_one_and_forall_le" |
| }, |
| { |
| "id": 79, |
| "match": false, |
| "ground_truth": "linarith", |
| "model_output": "apply lt_of_lt_of_le hy2 hx₂.le" |
| }, |
| { |
| "id": 80, |
| "match": false, |
| "ground_truth": "apply DifferentiableAt.hasGradientAt", |
| "model_output": "apply HasGradientAt.comp z.1" |
| }, |
| { |
| "id": 81, |
| "match": false, |
| "ground_truth": "rcases lbdf with ⟨L, Lbound⟩", |
| "model_output": "apply And.intro" |
| }, |
| { |
| "id": 82, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "aesop" |
| }, |
| { |
| "id": 83, |
| "match": true, |
| "ground_truth": "norm_num", |
| "model_output": "norm_num" |
| }, |
| { |
| "id": 84, |
| "match": false, |
| "ground_truth": "intro x hx", |
| "model_output": "intro r hr" |
| }, |
| { |
| "id": 85, |
| "match": false, |
| "ground_truth": "intro x _", |
| "model_output": "intro t ht\nexact h1 t" |
| }, |
| { |
| "id": 86, |
| "match": false, |
| "ground_truth": "rw [mem_setOf]", |
| "model_output": "apply Iff.intro" |
| }, |
| { |
| "id": 87, |
| "match": false, |
| "ground_truth": "have inequ₂'' : lim_inf (const_real_liminf (f y)) = f y := by\n apply liminf_const_eq\n rw[← inequ₂''];\n exact inequ₂' y nonempty lscf x_converge g_converge", |
| "model_output": "apply le_liminf_add_real_liminf" |
| }, |
| { |
| "id": 88, |
| "match": false, |
| "ground_truth": "refine norm_ne_zero_iff.mpr ?_", |
| "model_output": "apply norm_ne_zero_iff.mpr" |
| }, |
| { |
| "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": "apply hT.surjective₂" |
| }, |
| { |
| "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": "have := hx.1" |
| }, |
| { |
| "id": 91, |
| "match": false, |
| "ground_truth": "apply Finset.sum_eq_zero", |
| "model_output": "rw [Finset.sum_eq_zero]" |
| }, |
| { |
| "id": 92, |
| "match": false, |
| "ground_truth": "mul_inv_cancel₀ (by linarith),", |
| "model_output": "rw [mul_inv_cancel tpos.ne']" |
| }, |
| { |
| "id": 93, |
| "match": false, |
| "ground_truth": "by rw [one_smul]; rfl", |
| "model_output": "simp [one_smul]" |
| }, |
| { |
| "id": 94, |
| "match": false, |
| "ground_truth": "intro b", |
| "model_output": "apply h0.mp at h" |
| }, |
| { |
| "id": 95, |
| "match": false, |
| "ground_truth": "by linarith", |
| "model_output": "nlinarith" |
| }, |
| { |
| "id": 96, |
| "match": false, |
| "ground_truth": "rw [smul_sub, smul_sub, add_comm_sub, sub_sub, ← add_smul, add_comm b a]", |
| "model_output": "rw [smul_sub, smul_sub]" |
| }, |
| { |
| "id": 97, |
| "match": false, |
| "ground_truth": "· rw [intrinsicInterior_prod_eq_prod_intrinsicInterior]; exact mk_mem_prod (by simp) hx", |
| "model_output": "apply hx.2" |
| }, |
| { |
| "id": 98, |
| "match": false, |
| "ground_truth": "rw [alg.γeq k, alg.teq k]", |
| "model_output": "apply div_le_div <;> norm_cast" |
| }, |
| { |
| "id": 99, |
| "match": true, |
| "ground_truth": "intro i _", |
| "model_output": "intro i _" |
| }, |
| { |
| "id": 100, |
| "match": false, |
| "ground_truth": "rcases vt with ⟨c, d, ⟨a, ha⟩, ⟨vt1, vt2⟩⟩", |
| "model_output": "obtain ⟨c, d, ⟨⟨k, hk⟩, ctop, dtop⟩⟩ := vt" |
| }, |
| { |
| "id": 101, |
| "match": false, |
| "ground_truth": "exact hu.2.2 (mem_inter hu.2.1 hx)", |
| "model_output": "apply mem_of_mem_of_subset hx 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": "rw [add_comm, add_assoc]" |
| }, |
| { |
| "id": 103, |
| "match": false, |
| "ground_truth": "intro x", |
| "model_output": "intro v" |
| }, |
| { |
| "id": 104, |
| "match": false, |
| "ground_truth": "apply LinearMap.congr_fun\n ((LinearEquiv.symm_apply_eq (Module.dualProdDualEquivDual ℝ E ℝ)).1 (by rfl)) a", |
| "model_output": "apply Eq.symm" |
| }, |
| { |
| "id": 105, |
| "match": false, |
| "ground_truth": "rw[Metric.isOpen_iff] at isopen", |
| "model_output": "have := hf.isBoundedOn_inter_closedBall x_pos.1 (by positivity)" |
| }, |
| { |
| "id": 106, |
| "match": false, |
| "ground_truth": "rw [deriv_of_const_mul_func (f u - f x)]", |
| "model_output": "apply le_trans (mul_le_mul_of_nonneg_left ?_ (ENNReal.ofReal_nonneg 1))" |
| }, |
| { |
| "id": 107, |
| "match": false, |
| "ground_truth": "rcases exists_seq_tendsto_sInf hs hs' with ⟨fx, _, cfx, fxs⟩", |
| "model_output": "apply LowerSemicontinuous.exists_isMinOn_of_isCompact hf h2 hs'" |
| }, |
| { |
| "id": 108, |
| "match": false, |
| "ground_truth": "induction n with", |
| "model_output": "apply admm.kkt.converges" |
| }, |
| { |
| "id": 109, |
| "match": false, |
| "ground_truth": "simp at ht", |
| "model_output": "aesop" |
| }, |
| { |
| "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" |
| }, |
| { |
| "id": 111, |
| "match": false, |
| "ground_truth": "rw [subdiff_gradient g g' hg gradcon]", |
| "model_output": "apply (hg x).hasFDerivAt.mem_subdifferential" |
| }, |
| { |
| "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_pos (half_pos δpos)] at hd" |
| }, |
| { |
| "id": 113, |
| "match": false, |
| "ground_truth": "simp [phi_add] at this", |
| "model_output": "rw [phi_add] at this" |
| }, |
| { |
| "id": 114, |
| "match": false, |
| "ground_truth": "have h_eq : {z | f z ≤ α} = f ⁻¹' Iic α ∩ univ := by\n ext z\n simp [Set.mem_setOf_eq, Set.mem_preimage, Set.mem_Iic]", |
| "model_output": "apply isClosed_Iic.preimage h.continuous" |
| }, |
| { |
| "id": 115, |
| "match": false, |
| "ground_truth": "lift t to ℝ using ⟨LT.lt.ne_top fx_le_top, LT.lt.ne_bot fx_ge_bot⟩", |
| "model_output": "rw [← ht]" |
| }, |
| { |
| "id": 116, |
| "match": false, |
| "ground_truth": "rcases zin with ⟨u,hux,hufx,hv⟩", |
| "model_output": "obtain ⟨u, u_lim, fu_lim, v, v_lim⟩ := zin" |
| }, |
| { |
| "id": 117, |
| "match": false, |
| "ground_truth": "by linarith", |
| "model_output": "exact h.1" |
| }, |
| { |
| "id": 118, |
| "match": false, |
| "ground_truth": "apply convexHull_min (union_subset h1 h2) cone_conv_s_t.convex", |
| "model_output": "apply convexHull_min ?_ ?_" |
| }, |
| { |
| "id": 119, |
| "match": false, |
| "ground_truth": "have hΦ : ∃ r_Φ, range Φ ⊆ Metric.ball 0 r_Φ := by apply Φ_isBounded'", |
| "model_output": "apply admm.admm_kkt.2.2" |
| }, |
| { |
| "id": 120, |
| "match": false, |
| "ground_truth": "· simp", |
| "model_output": "exact Set.mem_singleton (g' x)" |
| }, |
| { |
| "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_norm_add_of_tendsto_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": "apply le_of_lt" |
| }, |
| { |
| "id": 123, |
| "match": false, |
| "ground_truth": "by_cases u₁_eq_0 : u₁ = 0", |
| "model_output": "rw [hu_eq, inner_add_left, inner_add_right, this, zero_add, hu₁]" |
| }, |
| { |
| "id": 124, |
| "match": false, |
| "ground_truth": "apply mul_pos", |
| "model_output": "linarith" |
| }, |
| { |
| "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" |
| }, |
| { |
| "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 inner_nonneg_of_nonneg_of_nonneg" |
| }, |
| { |
| "id": 128, |
| "match": false, |
| "ground_truth": "simp [grad_fun_comp,grad_comp] at h", |
| "model_output": "aesop" |
| }, |
| { |
| "id": 129, |
| "match": true, |
| "ground_truth": "exact hu₂", |
| "model_output": "exact hu₂" |
| }, |
| { |
| "id": 130, |
| "match": false, |
| "ground_truth": "apply Finset.card_eq_zero.2", |
| "model_output": "apply Set.card_eq_zero.mpr" |
| }, |
| { |
| "id": 131, |
| "match": false, |
| "ground_truth": "exact False.elim (by simp at this)", |
| "model_output": "linarith" |
| }, |
| { |
| "id": 132, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "nlinarith" |
| }, |
| { |
| "id": 133, |
| "match": false, |
| "ground_truth": "rw [← add_assoc, ← add_assoc]", |
| "model_output": "abel" |
| }, |
| { |
| "id": 134, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "rw [Metric.mem_ball, dist_eq_norm, sub_zero]" |
| }, |
| { |
| "id": 135, |
| "match": false, |
| "ground_truth": "intro y yin z zin a b anneg bnneg absum1", |
| "model_output": "intro x _ y _ a b ha hb hab\nhave h1 := hfun (by simp) (by simp) ha hb hab" |
| }, |
| { |
| "id": 136, |
| "match": false, |
| "ground_truth": "gcongr", |
| "model_output": "apply mul_lt_mul_of_pos_left _ 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 h1 := hmin (z 0)" |
| }, |
| { |
| "id": 138, |
| "match": false, |
| "ground_truth": "rfl", |
| "model_output": "have := h_A₁e₁_A₂e₂" |
| }, |
| { |
| "id": 139, |
| "match": false, |
| "ground_truth": "rw [mul_one] at htp", |
| "model_output": "linarith" |
| }, |
| { |
| "id": 140, |
| "match": true, |
| "ground_truth": "apply Continuous.tendsto", |
| "model_output": "apply Continuous.tendsto" |
| }, |
| { |
| "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_of_monotone" |
| }, |
| { |
| "id": 142, |
| "match": false, |
| "ground_truth": "apply A₁e₁_converge_zero", |
| "model_output": "apply tendsto_norm_zero.comp" |
| }, |
| { |
| "id": 143, |
| "match": false, |
| "ground_truth": "intro j", |
| "model_output": "apply hA_col_sum.eq_of_le" |
| }, |
| { |
| "id": 144, |
| "match": false, |
| "ground_truth": "rw[this]", |
| "model_output": "aesop" |
| }, |
| { |
| "id": 145, |
| "match": false, |
| "ground_truth": "· simp; linarith [μpos]", |
| "model_output": "apply inv_nonneg.2" |
| }, |
| { |
| "id": 146, |
| "match": false, |
| "ground_truth": "ext x", |
| "model_output": "simp [sub_eq_add_neg, add_comm, add_left_comm, add_assoc]" |
| }, |
| { |
| "id": 147, |
| "match": false, |
| "ground_truth": "have xBall_in_U : xBall ⊆ U := by\n apply l1Ball_sub_convexHull r_pos hn", |
| "model_output": "apply And.intro" |
| }, |
| { |
| "id": 148, |
| "match": false, |
| "ground_truth": "rw [← zero_pow]", |
| "model_output": "apply tendsto_pow_atTop_nhds_zero_of_le_one <;> norm_num" |
| }, |
| { |
| "id": 149, |
| "match": true, |
| "ground_truth": "apply mul_nonneg", |
| "model_output": "apply mul_nonneg" |
| }, |
| { |
| "id": 150, |
| "match": true, |
| "ground_truth": "by simp at h; simpa [h]", |
| "model_output": "simp at h" |
| }, |
| { |
| "id": 151, |
| "match": false, |
| "ground_truth": "rw [← mul_neg]", |
| "model_output": "rw [inner_sub_left, neg_eq_neg_one_mul, mul_neg, mul_comm]" |
| }, |
| { |
| "id": 152, |
| "match": true, |
| "ground_truth": "exact h", |
| "model_output": "exact h" |
| }, |
| { |
| "id": 153, |
| "match": false, |
| "ground_truth": "by rw [z]; simp; left; rw [z]; simp", |
| "model_output": "rw [← sub_eq_sub_left_iff, Prod.mk.eta]" |
| }, |
| { |
| "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_neg" |
| }, |
| { |
| "id": 155, |
| "match": false, |
| "ground_truth": "rcases this with ⟨z_, _, α, ⟨monoa, conv⟩⟩", |
| "model_output": "obtain ⟨z', z'cls, α, αmon, αlim⟩ := this" |
| }, |
| { |
| "id": 156, |
| "match": true, |
| "ground_truth": "intro n x", |
| "model_output": "intro n x" |
| }, |
| { |
| "id": 157, |
| "match": false, |
| "ground_truth": "simp [coef.mem_support_toFun]; rfl", |
| "model_output": "simp [Finsupp.mem_support_iff, ne_eq]" |
| }, |
| { |
| "id": 158, |
| "match": false, |
| "ground_truth": "simp only [Set.mem_union, Set.mem_iUnion, Set.mem_singleton_iff]", |
| "model_output": "cases' i with i" |
| }, |
| { |
| "id": 159, |
| "match": false, |
| "ground_truth": "have h2 : ‖x + xx - xx‖ ≤ ‖x + xx‖ + ‖xx‖ := by apply norm_sub_le", |
| "model_output": "apply norm_add_le" |
| }, |
| { |
| "id": 160, |
| "match": true, |
| "ground_truth": "apply eq4 u v", |
| "model_output": "apply eq4" |
| }, |
| { |
| "id": 161, |
| "match": false, |
| "ground_truth": "have : IsClosed ((MM x).carrier ∩ (f.Epi univ)) := by\n rw [ht]; exact IsClosed.prod isClosed_singleton isClosed_Ici", |
| "model_output": "have h14 : (MM x).carrier ∩ intrinsicInterior (Epi f univ) = {x} ×ˢ {μ | (f x).toReal ≤ μ} := by\n rw [intrinsicInterior_epi_eq] at hi'\n rw [ht] at hi'\n exact hi'" |
| }, |
| { |
| "id": 162, |
| "match": false, |
| "ground_truth": "simp [z]", |
| "model_output": "apply hs.segment_subset xin yin" |
| }, |
| { |
| "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.comp" |
| }, |
| { |
| "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": "apply lt_of_le_of_lt (hAC_le x j)" |
| }, |
| { |
| "id": 165, |
| "match": false, |
| "ground_truth": "dsimp [ey_diff, t_Ax_Ax_n1, t_A_diff_x', A2_e2_n1, A2_e2_n]\n linarith [Φ_isdescending_inequ5 n]", |
| "model_output": "apply le_of_sub_nonneg" |
| }, |
| { |
| "id": 166, |
| "match": true, |
| "ground_truth": "intro n", |
| "model_output": "intro n" |
| }, |
| { |
| "id": 167, |
| "match": false, |
| "ground_truth": "· apply ContinuousOn.comp (g := f₁) (f := fun x ↦ x + x₀) (t := univ) hcon\n · apply ContinuousOn.add continuousOn_id continuousOn_const\n · simp", |
| "model_output": "apply hcon.comp (continuous_add_right _)" |
| }, |
| { |
| "id": 168, |
| "match": true, |
| "ground_truth": "simp only [one_div, mul_inv_rev, add_sub_cancel_right]", |
| "model_output": "simp" |
| }, |
| { |
| "id": 169, |
| "match": false, |
| "ground_truth": "suffices 0 ≤ (a1 • tx + b1 • ty) by exact this", |
| "model_output": "intro i" |
| }, |
| { |
| "id": 170, |
| "match": false, |
| "ground_truth": "by\n rw [sub_right_comm]", |
| "model_output": "rw [norm_sub_rev]" |
| }, |
| { |
| "id": 171, |
| "match": false, |
| "ground_truth": "intro i hi", |
| "model_output": "intro x hx" |
| }, |
| { |
| "id": 172, |
| "match": false, |
| "ground_truth": "apply le_trans (Nat.le_add_left ax N)", |
| "model_output": "apply le_of_lt" |
| }, |
| { |
| "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": false, |
| "ground_truth": "exact trivial", |
| "model_output": "repeat' trivial" |
| }, |
| { |
| "id": 176, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "apply sq_nonneg" |
| }, |
| { |
| "id": 177, |
| "match": false, |
| "ground_truth": "intro _ epos", |
| "model_output": "intro ε εpos" |
| }, |
| { |
| "id": 178, |
| "match": true, |
| "ground_truth": "intro n", |
| "model_output": "intro n" |
| }, |
| { |
| "id": 179, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "aesop" |
| }, |
| { |
| "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) <;> aesop" |
| }, |
| { |
| "id": 182, |
| "match": false, |
| "ground_truth": "lift ⨆ u, Real.toEReal (inner ℝ u x) - f u to ℝ with c", |
| "model_output": "linarith" |
| }, |
| { |
| "id": 183, |
| "match": false, |
| "ground_truth": "simp at vt1", |
| "model_output": "use 1" |
| }, |
| { |
| "id": 184, |
| "match": false, |
| "ground_truth": "rw [pow_two]", |
| "model_output": "rw [← mul_assoc, mul_comm, ← div_eq_mul_inv, div_self, mul_one]" |
| }, |
| { |
| "id": 185, |
| "match": false, |
| "ground_truth": "rcases exist_open_sub_finite_cover with ⟨t, ht1, ht2, ht3⟩", |
| "model_output": "obtain ⟨b', _, b'finite, _⟩ := exist_open_sub_finite_cover" |
| }, |
| { |
| "id": 186, |
| "match": false, |
| "ground_truth": "intro b _", |
| "model_output": "aesop" |
| }, |
| { |
| "id": 187, |
| "match": true, |
| "ground_truth": "ring", |
| "model_output": "ring" |
| }, |
| { |
| "id": 188, |
| "match": true, |
| "ground_truth": "exact hcf", |
| "model_output": "exact hcf" |
| }, |
| { |
| "id": 189, |
| "match": false, |
| "ground_truth": "ring_nf", |
| "model_output": "simp [add_smul, sub_smul, smul_add, smul_sub, smul_smul, mul_comm, mul_assoc, mul_left_comm]" |
| }, |
| { |
| "id": 190, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "aesop" |
| }, |
| { |
| "id": 191, |
| "match": false, |
| "ground_truth": "simp [FeasSet, FeasPoint] at xf", |
| "model_output": "rw [hab]" |
| }, |
| { |
| "id": 192, |
| "match": false, |
| "ground_truth": "· simp [hx]", |
| "model_output": "exact Or.inr (Or.inr (smul_zero _))" |
| }, |
| { |
| "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 := eq3" |
| }, |
| { |
| "id": 194, |
| "match": false, |
| "ground_truth": "by ring", |
| "model_output": "simp" |
| }, |
| { |
| "id": 195, |
| "match": false, |
| "ground_truth": "by\n rw[sign_mul_abs]\n field_simp [bi_pos i]", |
| "model_output": "rw [abs_eq_self.mpr (norm_nonneg _)]" |
| }, |
| { |
| "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": "simp [Φ]" |
| }, |
| { |
| "id": 197, |
| "match": false, |
| "ground_truth": "· simp [V₂]", |
| "model_output": "simp [fromRows, submatrix]" |
| }, |
| { |
| "id": 198, |
| "match": false, |
| "ground_truth": "intro s", |
| "model_output": "intro s g _ sum_zero i" |
| }, |
| { |
| "id": 199, |
| "match": false, |
| "ground_truth": "rw[zero_add]", |
| "model_output": "simp" |
| }, |
| { |
| "id": 200, |
| "match": false, |
| "ground_truth": "linarith", |
| "model_output": "exact Nat.le_trans kleaw this" |
| }, |
| { |
| "id": 201, |
| "match": false, |
| "ground_truth": "repeat apply mul_nonneg", |
| "model_output": "apply mul_nonneg <;> norm_num" |
| }, |
| { |
| "id": 202, |
| "match": false, |
| "ground_truth": "exact Eq.symm (Nat.add_sub_of_le kge)", |
| "model_output": "rw [add_sub_cancel]" |
| }, |
| { |
| "id": 203, |
| "match": false, |
| "ground_truth": "rw [Filter.tendsto_congr' this]", |
| "model_output": "rw [this]" |
| }, |
| { |
| "id": 204, |
| "match": false, |
| "ground_truth": "intro c1", |
| "model_output": "intro c' c'pos" |
| }, |
| { |
| "id": 205, |
| "match": false, |
| "ground_truth": "exact min_le_right δ₀ δ₂", |
| "model_output": "apply min_le_right" |
| }, |
| { |
| "id": 206, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "congr" |
| }, |
| { |
| "id": 207, |
| "match": true, |
| "ground_truth": "simp", |
| "model_output": "simp" |
| }, |
| { |
| "id": 208, |
| "match": false, |
| "ground_truth": "rw[← mul_div]", |
| "model_output": "ring" |
| }, |
| { |
| "id": 209, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "apply Nat.zero_le" |
| }, |
| { |
| "id": 210, |
| "match": false, |
| "ground_truth": "intro c", |
| "model_output": "aesop" |
| }, |
| { |
| "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, two_mul, ← add_assoc, ← add_mul, ← inner_add_right, ← sub_eq_add_neg, ← sub_eq_add_neg,\n sub_add_sub_cancel, sub_eq_add_neg, sub_eq_add_neg, sub_eq_add_neg, sub_eq_add_neg]" |
| }, |
| { |
| "id": 213, |
| "match": false, |
| "ground_truth": "intro v", |
| "model_output": "simp" |
| }, |
| { |
| "id": 214, |
| "match": false, |
| "ground_truth": "by exact this", |
| "model_output": "rw [← this]" |
| }, |
| { |
| "id": 215, |
| "match": false, |
| "ground_truth": "apply lt_min δ₀pos (lt_min δ₄pos δ₅pos)", |
| "model_output": "aesop" |
| }, |
| { |
| "id": 216, |
| "match": false, |
| "ground_truth": "apply εpos", |
| "model_output": "all_goals positivity" |
| }, |
| { |
| "id": 217, |
| "match": true, |
| "ground_truth": "linarith [eq1]", |
| "model_output": "linarith" |
| }, |
| { |
| "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" |
| }, |
| { |
| "id": 219, |
| "match": false, |
| "ground_truth": "linarith [a1leab]", |
| "model_output": "apply Nat.succ_le_of_lt" |
| }, |
| { |
| "id": 220, |
| "match": false, |
| "ground_truth": "rw [←sub_add,norm_add_sq_real,←sub_add,norm_add_sq_real] at ieq", |
| "model_output": "apply le_trans ieq" |
| }, |
| { |
| "id": 221, |
| "match": false, |
| "ground_truth": "simp at din", |
| "model_output": "apply And.intro <;> linarith [Real.inter_Ioo.1 din]" |
| }, |
| { |
| "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" |
| }, |
| { |
| "id": 223, |
| "match": false, |
| "ground_truth": "by\n rw [WithLp.prod_norm_eq_of_L2]; simp\n exact (Real.le_sqrt (norm_nonneg _) (Left.add_nonneg (sq_nonneg _) (sq_nonneg _))).mpr\n (le_add_of_nonneg_right (sq_nonneg _))", |
| "model_output": "apply norm_fst_le_norm" |
| }, |
| { |
| "id": 224, |
| "match": false, |
| "ground_truth": "intro a t hat IH w hw_pos' hw_sum' hmem'", |
| "model_output": "intro a s has h_ind w hw₀ hw₁ hw₂" |
| }, |
| { |
| "id": 225, |
| "match": false, |
| "ground_truth": "by\n apply add_lt_add\n · apply ha₁ b hba₁\n · apply ha₂ b hba₂", |
| "model_output": "apply_rules [add_lt_add_iff_right, ha₁ b hba₁, ha₂ b hba₂]" |
| }, |
| { |
| "id": 226, |
| "match": false, |
| "ground_truth": "exact inv_sub_inv l' h₁", |
| "model_output": "field_simp" |
| }, |
| { |
| "id": 227, |
| "match": false, |
| "ground_truth": "apply hpos'", |
| "model_output": "norm_num" |
| }, |
| { |
| "id": 228, |
| "match": false, |
| "ground_truth": "apply Set.add_subset_add_right", |
| "model_output": "apply add_subset_add Subset.rfl" |
| }, |
| { |
| "id": 229, |
| "match": true, |
| "ground_truth": "rw [y1, y2]", |
| "model_output": "rw [y1, y2]" |
| }, |
| { |
| "id": 230, |
| "match": false, |
| "ground_truth": "linarith [hγ]", |
| "model_output": "apply zero_lt_one.trans hγ" |
| }, |
| { |
| "id": 231, |
| "match": true, |
| "ground_truth": "apply norm_sub_le", |
| "model_output": "apply norm_sub_le" |
| }, |
| { |
| "id": 232, |
| "match": false, |
| "ground_truth": "have h1 := hk (fun q ↦ alg.x (α q) - alg.x (α q - 1)) diflte1 (ε / (γ * l) / 3) (by positivity)", |
| "model_output": "apply (diflte2 ε epos).mono" |
| }, |
| { |
| "id": 233, |
| "match": true, |
| "ground_truth": "intro n", |
| "model_output": "intro n" |
| }, |
| { |
| "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": "field_simp [l0.ne']" |
| }, |
| { |
| "id": 236, |
| "match": false, |
| "ground_truth": "refine LICQ_injM z m Z Ax ?_ mlen ?_ eq2 eq1 ⟨heq1, heq2⟩", |
| "model_output": "have : z ∈ NullSpace Ax" |
| }, |
| { |
| "id": 237, |
| "match": false, |
| "ground_truth": "apply le_trans", |
| "model_output": "apply le_trans Nge" |
| }, |
| { |
| "id": 238, |
| "match": false, |
| "ground_truth": "apply mul_eq_mul_left_iff.mpr", |
| "model_output": "rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (by positivity)]" |
| } |
| ] |
| } |
