| { |
| "summary": { |
| "total_with_gt": 273, |
| "matched": 55, |
| "accuracy": 0.20146520146520147 |
| }, |
| "details": [ |
| { |
| "id": 0, |
| "match": false, |
| "ground_truth": "· exact mul_le_mul_of_nonneg_left hyu ha", |
| "model_output": "· apply mul_le_mul_of_nonneg_left hyu ha" |
| }, |
| { |
| "id": 1, |
| "match": false, |
| "ground_truth": "exact mul_le_mul_of_nonneg_left hyu ha", |
| "model_output": "apply mul_le_mul_of_nonneg_left hyu (le_of_lt ha)" |
| }, |
| { |
| "id": 2, |
| "match": false, |
| "ground_truth": "· exact mul_le_mul_of_nonneg_left hyv hb", |
| "model_output": "apply mul_le_mul_of_nonneg_left hyv hb" |
| }, |
| { |
| "id": 3, |
| "match": false, |
| "ground_truth": "exact mul_le_mul_of_nonneg_left hyv hb", |
| "model_output": "apply mul_le_mul_of_nonneg_left hyv hb" |
| }, |
| { |
| "id": 4, |
| "match": false, |
| "ground_truth": "rw [lipschitzWith_iff_norm_sub_le] at *", |
| "model_output": "apply lipschitz_of_norm_gradient_le\n rw [lipschitzWith_iff_norm_sub_le]\n intro z₁ z₂\n rw [← Prod.fst_sub, ← Prod.snd_sub, ← Prod.fst_sub, ← Prod.snd_sub]\n apply LipschitzWith.prodMk (lipschitzWith_iff_norm_grad_le.mpr hl) hl\n apply lipschitz_of_norm_gradient_le\n rw [lipschitzWith_iff_norm_grad_le]\n intro z₁ z₂\n apply LipschitzWith.prodMk (lipschitzWith_iff_norm" |
| }, |
| { |
| "id": 5, |
| "match": false, |
| "ground_truth": "by rw [grad_snd_eq h, grad_snd_eq h]", |
| "model_output": "rw [grad_eq_block_grad, grad_eq_block_grad]" |
| }, |
| { |
| "id": 6, |
| "match": false, |
| "ground_truth": "rw [grad_snd_eq h, grad_snd_eq h]", |
| "model_output": "rw [grad_eq_block_grad, grad_eq_block_grad]" |
| }, |
| { |
| "id": 7, |
| "match": true, |
| "ground_truth": "apply prox_well_define f hc lbdf x", |
| "model_output": "apply prox_well_define f hc lbdf x" |
| }, |
| { |
| "id": 8, |
| "match": false, |
| "ground_truth": "refine ⟨x₁, hx₁, x₂, hx₂, hx, ?_⟩", |
| "model_output": "refine ⟨x₁, ⟨hx₁, ⟨x₂, ⟨hx₂, ⟨hx, ?_⟩⟩⟩⟩⟩⟩\n rw [map_add, norm_add_sq_real, ← mul_two, ← sq, ← mul_two, ← sq]\n simp [hker hx₂]" |
| }, |
| { |
| "id": 9, |
| "match": false, |
| "ground_truth": "exact LinearMap.mem_ker.mp (hker hx₂)", |
| "model_output": "apply hker hx₂" |
| }, |
| { |
| "id": 10, |
| "match": false, |
| "ground_truth": "simp [Orthonormal, Pairwise] at *", |
| "model_output": "apply Orthonormal.mk\n · intro i hi j hj hij\n simp [restrict_apply, hi, hj, hij]\n exact hv.1 i j hij\n · intro i hi\n simp [restrict_apply, hi]\n exact hv.2 i hi" |
| }, |
| { |
| "id": 11, |
| "match": false, |
| "ground_truth": "by_cases hsf : ProperFunction s f", |
| "model_output": "apply closure_isClosed" |
| }, |
| { |
| "id": 12, |
| "match": false, |
| "ground_truth": "· exact closure_is_closed_of_proper f hs", |
| "model_output": "· apply closure_is_closed\n exact hs" |
| }, |
| { |
| "id": 13, |
| "match": false, |
| "ground_truth": "exact closure_is_closed_of_proper f hs", |
| "model_output": "apply closure_isClosed_of_closed\n exact hs" |
| }, |
| { |
| "id": 14, |
| "match": true, |
| "ground_truth": "apply IsClosed.prod hs isClosed_univ", |
| "model_output": "apply IsClosed.prod hs isClosed_univ" |
| }, |
| { |
| "id": 15, |
| "match": false, |
| "ground_truth": "simp [w, l, g, AffineMap.decomp']", |
| "model_output": "exact i.hx x _hx" |
| }, |
| { |
| "id": 16, |
| "match": true, |
| "ground_truth": "rw [φ₁_equ fullrank₁ fullrank₂ n]", |
| "model_output": "rw [φ₁_equ fullrank₁ fullrank₂ n]" |
| }, |
| { |
| "id": 17, |
| "match": false, |
| "ground_truth": "exact EReal.coe_le_coe_iff.mp h", |
| "model_output": "exact EReal.coe_le_coe_iff.1 h" |
| }, |
| { |
| "id": 18, |
| "match": false, |
| "ground_truth": "rw [← hT.span_range_w_eq_range]", |
| "model_output": "have h := hT.finrank_range_add_finrank_ker\n have : finrank 𝕜 (LinearMap.ker T) = 0 := by\n rw [← ker_eq_bot_of_range_eq_top hT.range_eq_top]\n exact finrank_bot 𝕜 V\n simp [this] at h\n exact h" |
| }, |
| { |
| "id": 19, |
| "match": false, |
| "ground_truth": "apply iff_iff_implies_and_implies.mpr", |
| "model_output": "rw [← closure_convex_support_eq hscl hscv, ← closure_convex_support_eq htcl htcv, h]" |
| }, |
| { |
| "id": 20, |
| "match": false, |
| "ground_truth": "apply convex_conjugate_ge_bot_univ'", |
| "model_output": "apply convex_conjugate_ge_bot_univ s.convex_indicator x" |
| }, |
| { |
| "id": 21, |
| "match": false, |
| "ground_truth": "exact HasGradientWithinAt.const_smul h c", |
| "model_output": "exact HasGradientAtFilter.const_smul h c" |
| }, |
| { |
| "id": 22, |
| "match": true, |
| "ground_truth": "exact HasGradientAtFilter.const_add hf c", |
| "model_output": "exact HasGradientAtFilter.const_add hf c" |
| }, |
| { |
| "id": 23, |
| "match": true, |
| "ground_truth": "exact HasGradientAtFilter.const_add hf c", |
| "model_output": "exact HasGradientAtFilter.const_add hf c" |
| }, |
| { |
| "id": 24, |
| "match": true, |
| "ground_truth": "exact HasGradientAtFilter.const_sub hf c", |
| "model_output": "exact HasGradientAtFilter.const_sub hf c" |
| }, |
| { |
| "id": 25, |
| "match": false, |
| "ground_truth": "apply convex_on_p_top\n ext x; exact h x", |
| "model_output": "apply convex_on_n_inf" |
| }, |
| { |
| "id": 26, |
| "match": false, |
| "ground_truth": "apply monotone_gradient_convex'", |
| "model_output": "apply LinearProgram_convex" |
| }, |
| { |
| "id": 27, |
| "match": false, |
| "ground_truth": "exact (fun x _ => affine_sq_gradient x)", |
| "model_output": "intro x _" |
| }, |
| { |
| "id": 28, |
| "match": false, |
| "ground_truth": "have h_zero_eq := (hS 0).mp h_zero_mem", |
| "model_output": "have h1 : (0 : E) ∈ S := h_zero_mem\n have h2 := hS (0 : E)\n simp at h2\n exact h2.1" |
| }, |
| { |
| "id": 29, |
| "match": false, |
| "ground_truth": "have : (0 : 𝕜) < (1 : 𝕜) := one_pos", |
| "model_output": "apply (neq_iff_lt_or_gt).mp neq" |
| }, |
| { |
| "id": 30, |
| "match": false, |
| "ground_truth": "rw [inner_add_left, real_inner_smul_left, real_inner_smul_left]", |
| "model_output": "rw [inner_add_left, ← sub_sub, sub_le_iff_le_add]" |
| }, |
| { |
| "id": 31, |
| "match": false, |
| "ground_truth": "rw [← one_mul (-ε * ‖y - x‖), ← hab, add_mul]", |
| "model_output": "apply add_nonneg (mul_nonneg ha (neg_mul_le_iff_mul_le_neg.mpr hyu))\n (mul_nonneg hb (neg_mul_le_iff_mul_le_neg.mpr hyv))" |
| }, |
| { |
| "id": 32, |
| "match": false, |
| "ground_truth": "calc\n _ = ‖(gradient H z).2 - (gradient H z').2‖ := by rw [grad_snd_eq h, grad_snd_eq h]\n _ = ‖(gradient H z - gradient H z').2‖ := rfl\n _ ≤ ‖(gradient H z - gradient H z')‖ := snd_norm_le_prod_L2 _\n _ ≤ _ := hl z z'", |
| "model_output": "have h1 := hl z z'" |
| }, |
| { |
| "id": 33, |
| "match": false, |
| "ground_truth": "by\n intro k\n specialize ck k; rw [ck]\n apply div_pos\n · norm_num\n apply mul_pos\n · linarith [hγ]\n · apply alg.lpos", |
| "model_output": "intro k\n rw [ck k]\n apply div_pos\n · linarith\n · apply mul_pos\n · linarith\n · linarith [alg.lpos]" |
| }, |
| { |
| "id": 34, |
| "match": false, |
| "ground_truth": "intro k\n specialize ck k; rw [ck]\n apply div_pos\n · norm_num\n apply mul_pos\n · linarith [hγ]\n · apply alg.lpos", |
| "model_output": "intro k\n rw [ck k]\n apply div_pos\n · linarith\n · apply mul_pos\n · linarith\n · linarith [alg.lpos]" |
| }, |
| { |
| "id": 35, |
| "match": true, |
| "ground_truth": "by\n apply prox_well_define f hc lbdf x", |
| "model_output": "apply prox_well_define f hc lbdf x" |
| }, |
| { |
| "id": 36, |
| "match": false, |
| "ground_truth": "by\n unfold colSubmodule\n rw [finrank_span_eq_card]\n simp [h]", |
| "model_output": "apply finrank_colSubmodule_eq_card" |
| }, |
| { |
| "id": 37, |
| "match": false, |
| "ground_truth": "unfold colSubmodule\n rw [finrank_span_eq_card]\n simp [h]", |
| "model_output": "apply finrank_colSubmodule_eq_card" |
| }, |
| { |
| "id": 38, |
| "match": false, |
| "ground_truth": "let ⟨x₁, hx₁, x₂, hx₂, hx⟩ := exists_orthogonal_decompose s x\n refine ⟨x₁, hx₁, x₂, hx₂, hx, ?_⟩\n rw [hx, LinearMap.map_add]\n suffices T x₂ = 0 from by\n rw [this, add_zero, norm_sq_eq_re_inner (𝕜:=𝕜)]\n exact LinearMap.mem_ker.mp (hker hx₂)", |
| "model_output": "have : x = (s.orthogonalProjection x) + (x - (s.orthogonalProjection x)) := by simp" |
| }, |
| { |
| "id": 39, |
| "match": false, |
| "ground_truth": "suffices T x₂ = 0 from by\n rw [this, add_zero, norm_sq_eq_re_inner (𝕜:=𝕜)]", |
| "model_output": "rw [mem_orthogonal'] at hx₂\n[ASSISTANT]\nhave : T x₂ = 0 := hker hx₂\n[ASSISTANT]\nrw [this, add_zero]" |
| }, |
| { |
| "id": 40, |
| "match": false, |
| "ground_truth": "by\n rw [this, add_zero, norm_sq_eq_re_inner (𝕜:=𝕜)]", |
| "model_output": "rw [this, add_zero]" |
| }, |
| { |
| "id": 41, |
| "match": false, |
| "ground_truth": "rw [this, add_zero, norm_sq_eq_re_inner (𝕜:=𝕜)]", |
| "model_output": "rw [this, add_zero]" |
| }, |
| { |
| "id": 42, |
| "match": false, |
| "ground_truth": "by\n simp [Orthonormal, Pairwise] at *\n aesop", |
| "model_output": "apply Orthonormal.mk\n · intro i hi j hj hij\n simp [restrict_apply, hi, hj, hij]\n exact hv.1 i j hij\n · intro i hi\n simp [restrict_apply, hi]\n exact hv.2 i hi" |
| }, |
| { |
| "id": 43, |
| "match": false, |
| "ground_truth": "simp [Orthonormal, Pairwise] at *\n aesop", |
| "model_output": "apply Orthonormal.mk\n · intro i hi j hj hij\n simp [restrict_apply, hi, hj, hij]\n exact hv.1 i j hij\n · intro i hi\n simp [restrict_apply, hi]\n exact hv.2 i hi" |
| }, |
| { |
| "id": 44, |
| "match": false, |
| "ground_truth": "by\n let s := Metric.ball x ε\n have h : ContDiffAt ℝ 1 f x := by\n apply ContDiffOn.contDiffAt hf\n rw [mem_nhds_iff]; use s\n exact ⟨Eq.subset rfl, ⟨Metric.isOpen_ball, Metric.mem_ball_self he⟩⟩\n exact gradient_continuous_of_contdiffat f h", |
| "model_output": "apply HasGradientAt.continuousAt\n apply HasGradientAt_of_HasFDerivAt\n apply HasFDerivAt_of_C1\n apply ContDiffOn.contDiffWithinAt\n apply hf\n apply he\n apply ball_mem_nhds x\n apply mem_ball_self he" |
| }, |
| { |
| "id": 45, |
| "match": false, |
| "ground_truth": "let s := Metric.ball x ε\n have h : ContDiffAt ℝ 1 f x := by\n apply ContDiffOn.contDiffAt hf\n rw [mem_nhds_iff]; use s\n exact ⟨Eq.subset rfl, ⟨Metric.isOpen_ball, Metric.mem_ball_self he⟩⟩\n exact gradient_continuous_of_contdiffat f h", |
| "model_output": "apply HasGradientAt.continuousAt\n apply HasGradientAt_of_HasFDerivAt\n apply HasFDerivAt_of_C1\n apply ContDiffOn.contDiffWithinAt\n apply hf\n apply he\n apply ball_mem_nhds x\n apply mem_ball_self he" |
| }, |
| { |
| "id": 46, |
| "match": false, |
| "ground_truth": "have h : ContDiffAt ℝ 1 f x := by\n apply ContDiffOn.contDiffAt hf\n rw [mem_nhds_iff]; use s\n exact ⟨Eq.subset rfl, ⟨Metric.isOpen_ball, Metric.mem_ball_self he⟩⟩", |
| "model_output": "apply HasGradientAt.continuousAt\n apply HasGradientAt.continuousAt\n apply ContDiffOn.continuousOn\n apply hf\n apply he\n apply s" |
| }, |
| { |
| "id": 47, |
| "match": false, |
| "ground_truth": "exact ⟨Eq.subset rfl, ⟨Metric.isOpen_ball, Metric.mem_ball_self he⟩⟩", |
| "model_output": "constructor" |
| }, |
| { |
| "id": 48, |
| "match": false, |
| "ground_truth": "exact gradient_continuous_of_contdiffat f h", |
| "model_output": "apply HasGradientAt.continuousAt h" |
| }, |
| { |
| "id": 49, |
| "match": false, |
| "ground_truth": "by\n simp [Function.closure]\n intro h; exfalso\n obtain h1 := h.2; simp at h1", |
| "model_output": "apply closure_univ_eq_top" |
| }, |
| { |
| "id": 50, |
| "match": false, |
| "ground_truth": "simp [Function.closure]\n intro h; exfalso\n obtain h1 := h.2; simp at h1", |
| "model_output": "apply closure_univ_eq_top" |
| }, |
| { |
| "id": 51, |
| "match": false, |
| "ground_truth": "· simp [hf]; simp [Function.IsClosed]\n have h1 : Epi ⊥ s = s ×ˢ (univ : Set ℝ) := by\n ext ⟨x, _⟩\n simp [Function.Epi]\n rw [h1]\n apply IsClosed.prod hs isClosed_univ", |
| "model_output": "· rw [if_pos hf]\n exact isClosed_univ_iff.mpr trivial" |
| }, |
| { |
| "id": 52, |
| "match": true, |
| "ground_truth": "simp [hf]; simp [Function.IsClosed]\n have h1 : Epi ⊥ s = s ×ˢ (univ : Set ℝ) := by\n ext ⟨x, _⟩\n simp [Function.Epi]\n rw [h1]\n apply IsClosed.prod hs isClosed_univ", |
| "model_output": "simp [hf]" |
| }, |
| { |
| "id": 53, |
| "match": false, |
| "ground_truth": "have h1 : Epi ⊥ s = s ×ˢ (univ : Set ℝ) := by\n ext ⟨x, _⟩\n simp [Function.Epi]", |
| "model_output": "apply isClosed_epi_bot_of_isClosed hs" |
| }, |
| { |
| "id": 54, |
| "match": false, |
| "ground_truth": "have h' : Epi ⊤ s = ∅ := by\n ext ⟨x, _⟩; simp [Function.Epi]", |
| "model_output": "apply isClosed_epi_of_closed_of_exist_top hs (not_proper_of_bot_not_in_image f hs hf)" |
| }, |
| { |
| "id": 55, |
| "match": false, |
| "ground_truth": "by\n constructor\n · intro h z\n unfold HasSubgradientAt at h\n obtain hz := h z\n simp; linarith\n intro h z\n unfold Banach_HasSubgradientAt at h\n obtain hz := h z; simp at hz\n linarith", |
| "model_output": "constructor\n · intro h z\n unfold Banach_HasSubgradientAt HasSubgradientAt\n specialize h z\n rw[← inner_apply_apply (toDual ℝ E g) z]\n exact le_of_eq (congrArg EReal.toEReal h)\n · apply Banach_HasSubgradientAt_to_HasSubgradientAt" |
| }, |
| { |
| "id": 56, |
| "match": false, |
| "ground_truth": "constructor\n · intro h z\n unfold HasSubgradientAt at h\n obtain hz := h z\n simp; linarith\n intro h z\n unfold Banach_HasSubgradientAt at h\n obtain hz := h z; simp at hz\n linarith", |
| "model_output": "constructor\n · intro h z\n unfold Banach_HasSubgradientAt HasSubgradientAt\n specialize h z\n rw[← inner_apply_apply (toDual ℝ E g) z]\n exact le_of_eq (congrArg EReal.toEReal h)\n · apply Banach_HasSubgradientAt_to_HasSubgradientAt" |
| }, |
| { |
| "id": 57, |
| "match": false, |
| "ground_truth": "· intro h z\n unfold HasSubgradientAt at h\n obtain hz := h z\n simp; linarith", |
| "model_output": "· intro h z\n unfold Banach_HasSubgradientAt\n rw[hasSubgradientAt_iff_isLittleO] at h\n apply Asymptotics.isLittleO_iff.mpr\n intro ε εpos\n specialize h (ε / 2) (half_pos εpos)\n rcases h with ⟨δ, δpos, hδ⟩\n use δ\n constructor\n · linarith\n · intro y hy\n rw[← sub_add, sub_self, sub_zero] at hy\n rw[← sub_add, sub_self, sub_zero]\n apply" |
| }, |
| { |
| "id": 58, |
| "match": false, |
| "ground_truth": "intro h z\n unfold HasSubgradientAt at h\n obtain hz := h z\n simp; linarith", |
| "model_output": "intro h z\n rw [hasSubgradientAt_iff_le] at h\n apply h z\n simp\n rw [← ContinuousLinearMap.map_sub]; simp\n apply real_inner_le_norm" |
| }, |
| { |
| "id": 59, |
| "match": false, |
| "ground_truth": "· intro x hx\n simpa [w, l, g, AffineMap.decomp', ← EReal.coe_sub, ← EReal.coe_add] using i.hlw x hx", |
| "model_output": "· exact i.hg" |
| }, |
| { |
| "id": 60, |
| "match": false, |
| "ground_truth": "intro x hx\n simpa [w, l, g, AffineMap.decomp', ← EReal.coe_sub, ← EReal.coe_add] using i.hlw x hx", |
| "model_output": "intro x xs\n rw[← EReal.coe_add, ← inner_add_left, ← inner_add_left, ← inner_add_left]\n apply le_trans\n · apply i.1\n · apply i.2 x xs" |
| }, |
| { |
| "id": 61, |
| "match": false, |
| "ground_truth": "apply Filter.tendsto_iff_seq_tendsto.1 Seq_converge_zero₃", |
| "model_output": "apply tendsto_zero_iff_norm_tendsto_zero.2\n apply tendsto_zero_iff_norm_tendsto_zero.2\n apply A₂_subseq_converge_zero" |
| }, |
| { |
| "id": 62, |
| "match": false, |
| "ground_truth": "have : (fun n => (Int.toNat (φ₁ n))) = (fun n => (φ (n+1))) := by\n ext n; rw [φ₁_equ fullrank₁ fullrank₂ n]; simp", |
| "model_output": "apply strictMono_nat_of_lt_succ\n intro n\n have : (φ₁ (n + 1)) > (φ₁ n) := by\n calc\n (φ₁ (n + 1)) ≥ n + 1 := StrictMono.id_le (hphi_StrictMono fullrank₁ fullrank₂) (n + 1)\n _ > n := by linarith\n apply this\n apply Nat.cast_lt.mpr\n apply this\n apply nplusone_le_phi" |
| }, |
| { |
| "id": 63, |
| "match": false, |
| "ground_truth": "suffices 0 ≤ (c - a - b).toEReal by exact this", |
| "model_output": "rw [← EReal.coe_add, ← EReal.coe_sub] at h" |
| }, |
| { |
| "id": 64, |
| "match": false, |
| "ground_truth": "have : a + b ≤ c := by exact EReal.coe_le_coe_iff.mp h", |
| "model_output": "apply EReal.coe_nonneg.mpr" |
| }, |
| { |
| "id": 65, |
| "match": false, |
| "ground_truth": "refine EReal.coe_nonneg.mpr ?intro.intro.a", |
| "model_output": "apply EReal.coe_nonneg.mpr" |
| }, |
| { |
| "id": 66, |
| "match": false, |
| "ground_truth": "by\n constructor\n · exact hs\n intro x hx y hy a b ha hb hab\n specialize hfun hx hy ha hb hab\n dsimp\n have : m / 2 * a * b * ‖x - y‖ ^ 2 = a * b * (m / 2 * ‖x - y‖ ^ 2) := by ring_nf\n simp at this;\n rw [← this]; exact hfun", |
| "model_output": "apply StrongConvexOn_of_ConvexOn hs\n intro x xs y ys a b ha hb hab\n specialize hfun xs ys ha hb hab\n rw [← sub_le_sub_iff, sub_add, sub_add_cancel, sub_le_sub_iff] at hfun\n linarith [hfun]" |
| }, |
| { |
| "id": 67, |
| "match": false, |
| "ground_truth": "constructor\n · exact hs\n intro x hx y hy a b ha hb hab\n specialize hfun hx hy ha hb hab\n dsimp\n have : m / 2 * a * b * ‖x - y‖ ^ 2 = a * b * (m / 2 * ‖x - y‖ ^ 2) := by ring_nf\n simp at this;\n rw [← this]; exact hfun", |
| "model_output": "apply StrongConvexOn_of_ConvexOn hs\n intro x xs y ys a b ha hb hab\n specialize hfun xs ys ha hb hab\n rw [← sub_le_sub_iff, sub_add, sub_add_cancel, sub_le_sub_iff] at hfun\n linarith [hfun]" |
| }, |
| { |
| "id": 68, |
| "match": false, |
| "ground_truth": "have : m / 2 * a * b * ‖x - y‖ ^ 2 = a * b * (m / 2 * ‖x - y‖ ^ 2) := by ring_nf", |
| "model_output": "linarith [hfun]" |
| }, |
| { |
| "id": 69, |
| "match": false, |
| "ground_truth": "by\n rw [← hT.span_range_w_eq_range]\n simp [finrank_span_eq_card (b:=hT.w) hT.w_orthonormal.linearIndependent]", |
| "model_output": "have h := hT.finrank_range_add_finrank_ker\n have : finrank 𝕜 (LinearMap.ker T) = 0 := by\n rw [← ker_eq_bot_of_range_eq_top hT.range_eq_top]\n exact finrank_bot 𝕜 V\n simp [this] at h\n exact h" |
| }, |
| { |
| "id": 70, |
| "match": false, |
| "ground_truth": "rw [← hT.span_range_w_eq_range]\n simp [finrank_span_eq_card (b:=hT.w) hT.w_orthonormal.linearIndependent]", |
| "model_output": "have h := hT.finrank_range_add_finrank_ker\n have : finrank 𝕜 (LinearMap.ker T) = 0 := by\n rw [← ker_eq_bot_of_range_eq_top hT.range_eq_top]\n exact finrank_bot 𝕜 V\n simp [this] at h\n exact h" |
| }, |
| { |
| "id": 71, |
| "match": false, |
| "ground_truth": "simp [finrank_span_eq_card (b:=hT.w) hT.w_orthonormal.linearIndependent]", |
| "model_output": "rw [finrank_span_eq_card]" |
| }, |
| { |
| "id": 72, |
| "match": false, |
| "ground_truth": "· exact fun a ↦ set_inclusion_convex_support_eq s t htcv htcl a h", |
| "model_output": "· intro zin\n have : s.convex_support z = 0 := by\n rw [← h z]; exact convex_support_of_mem zin\n have : t.convex_support z = 0 := by rw [← h]; exact this\n apply convex_support_eq_zero htcl hscv htcv; exact this\n apply convex_conjugate_of_zero_support\n apply convex_convex_support htcl hscv\n exact zin" |
| }, |
| { |
| "id": 73, |
| "match": false, |
| "ground_truth": "exact fun a ↦ set_inclusion_convex_support_eq s t htcv htcl a h", |
| "model_output": "intro hzs\n have h1 : s.convex_support z = ∅ := by\n rw [convex_support_eq_empty_iff]; exact hzs\n have h2 := congrFun h hzs\n rw [h1] at h2\n exact (Set.convex_indicator_empty t).convex_support_eq_closure h2 htcl\n apply subset_closure\n trivial" |
| }, |
| { |
| "id": 74, |
| "match": false, |
| "ground_truth": "exact fun a ↦ set_inclusion_convex_support_eq t s hscv hscl a (fun x => (h x).symm)", |
| "model_output": "intro zt\n rw [← h z]\n apply convex_support_subset hscl htcl zt" |
| }, |
| { |
| "id": 75, |
| "match": true, |
| "ground_truth": "by\n intro x _\n simp [convex_support]\n obtain := ConvexIndicator_is_proper_on_univ hs\n apply convex_conjugate_ge_bot_univ'", |
| "model_output": "intro x _" |
| }, |
| { |
| "id": 76, |
| "match": true, |
| "ground_truth": "intro x _\n simp [convex_support]\n obtain := ConvexIndicator_is_proper_on_univ hs\n apply convex_conjugate_ge_bot_univ'", |
| "model_output": "intro x _" |
| }, |
| { |
| "id": 77, |
| "match": false, |
| "ground_truth": "have convex_conjugate_at_zero :\n convex_conjugate univ (convex_indicator s) 0 ≤ 0 := by\n refine (conjugate_le_zero_iff s.convex_indicator).mpr ?_\n simp\n intro y\n simp [convex_indicator]\n split_ifs with hy <;> simp", |
| "model_output": "rcases hs with ⟨x, hx⟩\n use 0\n simp [convex_conjugate]\n rw[Set.mem_setOf_eq]\n exact ⟨hx, by simp⟩" |
| }, |
| { |
| "id": 78, |
| "match": false, |
| "ground_truth": "refine (conjugate_le_zero_iff s.convex_indicator).mpr ?_", |
| "model_output": "apply convex_conjugate_le_zero\n · simp\n · apply hs.convex_indicator_nonempty\n · simp" |
| }, |
| { |
| "id": 79, |
| "match": false, |
| "ground_truth": "exact lt_of_le_of_lt (convex_conjugate_at_zero) (zero_lt_top)", |
| "model_output": "apply convex_conjugate_lt_top_of_nonempty_convex hs convex_indicator_convex\n apply convex_univ\n apply convex_indicator_convex_convex_conjugate_nonempty hs convex_univ\n apply convex_univ" |
| }, |
| { |
| "id": 80, |
| "match": false, |
| "ground_truth": "by\n exact HasGradientWithinAt.const_smul h c", |
| "model_output": "exact HasGradientAtFilter.const_smul h c" |
| }, |
| { |
| "id": 81, |
| "match": true, |
| "ground_truth": "by\n exact HasGradientAtFilter.const_add hf c", |
| "model_output": "exact HasGradientAtFilter.const_add hf c" |
| }, |
| { |
| "id": 82, |
| "match": true, |
| "ground_truth": "by\n exact HasGradientAtFilter.const_add hf c", |
| "model_output": "exact HasGradientAtFilter.const_add hf c" |
| }, |
| { |
| "id": 83, |
| "match": false, |
| "ground_truth": "by\n simp only [sub_eq_add_neg, gradient_add_const]", |
| "model_output": "rw [gradient_sub_const c]" |
| }, |
| { |
| "id": 84, |
| "match": false, |
| "ground_truth": "simp only [sub_eq_add_neg, gradient_add_const]", |
| "model_output": "rw [gradient_sub_const c]" |
| }, |
| { |
| "id": 85, |
| "match": true, |
| "ground_truth": "by\n exact HasGradientAtFilter.const_sub hf c", |
| "model_output": "exact HasGradientAtFilter.const_sub hf c" |
| }, |
| { |
| "id": 86, |
| "match": false, |
| "ground_truth": "by\n obtain h1 := h.2; simp at h1\n rcases h1 with ⟨x, hx⟩; use x; simp [hx]", |
| "model_output": "simp [dom]" |
| }, |
| { |
| "id": 87, |
| "match": false, |
| "ground_truth": "obtain h1 := h.2; simp at h1\n rcases h1 with ⟨x, hx⟩; use x; simp [hx]", |
| "model_output": "simp [dom]" |
| }, |
| { |
| "id": 88, |
| "match": false, |
| "ground_truth": "by\n apply convex_on_p_top\n ext x; exact h x", |
| "model_output": "apply convex_on_n_inf" |
| }, |
| { |
| "id": 89, |
| "match": false, |
| "ground_truth": "by\n rw [sub_eq_add_neg u v, mulVec_add, mulVec_neg, sub_eq_add_neg]", |
| "model_output": "rw [Matrix.mulVec_sub]" |
| }, |
| { |
| "id": 90, |
| "match": false, |
| "ground_truth": "rw [sub_eq_add_neg u v, mulVec_add, mulVec_neg, sub_eq_add_neg]", |
| "model_output": "rw [Matrix.mulVec_sub]" |
| }, |
| { |
| "id": 91, |
| "match": false, |
| "ground_truth": "· exact (fun x _ => affine_sq_gradient x)", |
| "model_output": "apply linear_gradient" |
| }, |
| { |
| "id": 92, |
| "match": true, |
| "ground_truth": "· intro x _ y _\n rw [Matrix.mulVec_sub, Matrix.mulVec_sub, ← sub_add, sub_add_eq_add_sub, sub_add_cancel,\n ← Matrix.mulVec_sub, real_inner_eq_dot]\n rw [← dot_mul_eq_transpose_mul_dot,← Matrix.mulVec_sub, ← norm2eq_dot]\n apply sq_nonneg", |
| "model_output": "intro x _ y _" |
| }, |
| { |
| "id": 93, |
| "match": true, |
| "ground_truth": "intro x _ y _\n rw [Matrix.mulVec_sub, Matrix.mulVec_sub, ← sub_add, sub_add_eq_add_sub, sub_add_cancel,\n ← Matrix.mulVec_sub, real_inner_eq_dot]\n rw [← dot_mul_eq_transpose_mul_dot,← Matrix.mulVec_sub, ← norm2eq_dot]\n apply sq_nonneg", |
| "model_output": "intro x _ y _" |
| }, |
| { |
| "id": 94, |
| "match": false, |
| "ground_truth": "rw [Matrix.mulVec_sub, Matrix.mulVec_sub, ← sub_add, sub_add_eq_add_sub, sub_add_cancel,\n ← Matrix.mulVec_sub, real_inner_eq_dot]", |
| "model_output": "rw [← sub_add, sub_mulVec, sub_mulVec, sub_add, sub_add, sub_add, sub_add, sub_add]" |
| }, |
| { |
| "id": 95, |
| "match": false, |
| "ground_truth": "rw [← dot_mul_eq_transpose_mul_dot,← Matrix.mulVec_sub, ← norm2eq_dot]", |
| "model_output": "rw [← Matrix.mulVec_sub, ← sub_mulVec, ← sub_mulVec, ← sub_mulVec, ← sub_mulVec]" |
| }, |
| { |
| "id": 96, |
| "match": false, |
| "ground_truth": "have h_zero_mem : (0 : E) ∈ S := S.zero_mem", |
| "model_output": "have h1 : ∀ i, (hS (B (Fin.natAdd n i))).1 = (hS (B (Fin.natAdd n i))).2 := by\n intro i\n simp [hS]" |
| }, |
| { |
| "id": 97, |
| "match": false, |
| "ground_truth": "have h_repr_zero : B.repr 0 = 0 := LinearEquiv.map_zero B.repr", |
| "model_output": "simp at h_zero_eq" |
| }, |
| { |
| "id": 98, |
| "match": false, |
| "ground_truth": "simp only [map_zero, Finsupp.coe_zero, Pi.zero_apply] at h_zero_eq", |
| "model_output": "simp [h_repr_zero] at h_zero_eq" |
| }, |
| { |
| "id": 99, |
| "match": false, |
| "ground_truth": "simp only [Matrix.mulVec_zero, add_zero] at h_zero_eq", |
| "model_output": "simp [Matrix.mulVec_zero] at h_zero_eq" |
| }, |
| { |
| "id": 100, |
| "match": false, |
| "ground_truth": "by\n unfold sup_value\n unfold FeasSet FeasPoint\n simp [hp]", |
| "model_output": "unfold sup_value\n simp [hp]" |
| }, |
| { |
| "id": 101, |
| "match": false, |
| "ground_truth": "unfold sup_value\n unfold FeasSet FeasPoint\n simp [hp]", |
| "model_output": "unfold sup_value\n simp [hp]" |
| }, |
| { |
| "id": 102, |
| "match": true, |
| "ground_truth": "by\n intro b; rw [polyhedra_iff_cone b]\n exact closed_conic σ b", |
| "model_output": "intro b" |
| }, |
| { |
| "id": 103, |
| "match": true, |
| "ground_truth": "intro b; rw [polyhedra_iff_cone b]\n exact closed_conic σ b", |
| "model_output": "intro b" |
| }, |
| { |
| "id": 104, |
| "match": true, |
| "ground_truth": "by\n rw [h]; apply (local_Minimum_TangentCone' loc hl hf)", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 105, |
| "match": true, |
| "ground_truth": "rw [h]; apply (local_Minimum_TangentCone' loc hl hf)", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 106, |
| "match": false, |
| "ground_truth": "apply (local_Minimum_TangentCone' loc hl hf)", |
| "model_output": "rw [← h]" |
| }, |
| { |
| "id": 107, |
| "match": false, |
| "ground_truth": "by rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp", |
| "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply, ← ContinuousLinearMap.adjoint_comp_apply]" |
| }, |
| { |
| "id": 108, |
| "match": false, |
| "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp", |
| "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply, ← ContinuousLinearMap.adjoint_comp_apply]" |
| }, |
| { |
| "id": 109, |
| "match": false, |
| "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]", |
| "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply, ← ContinuousLinearMap.adjoint_comp_apply]" |
| }, |
| { |
| "id": 110, |
| "match": false, |
| "ground_truth": "by rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp", |
| "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply, ← ContinuousLinearMap.adjoint_comp_apply]" |
| }, |
| { |
| "id": 111, |
| "match": false, |
| "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp", |
| "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply, ← ContinuousLinearMap.adjoint_comp_apply]" |
| }, |
| { |
| "id": 112, |
| "match": false, |
| "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]", |
| "model_output": "rw [← ContinuousLinearMap.adjoint_comp_apply, ← ContinuousLinearMap.adjoint_comp_apply]" |
| }, |
| { |
| "id": 113, |
| "match": false, |
| "ground_truth": "by rw [← trace_transpose, ← trace_transpose_mul, transpose_mul]", |
| "model_output": "rw [Matrix.mul_comm, Matrix.trace_mul_comm]" |
| }, |
| { |
| "id": 114, |
| "match": false, |
| "ground_truth": "rw [← trace_transpose, ← trace_transpose_mul, transpose_mul]", |
| "model_output": "rw [Matrix.mul_comm, Matrix.trace_mul_comm]" |
| }, |
| { |
| "id": 115, |
| "match": false, |
| "ground_truth": "have : ConvexOn ℝ X fun _ ↦ (0: EReal) := by\n dsimp [ConvexOn]\n constructor\n exact hs\n intro x _ y _ a b _ _ _\n simp", |
| "model_output": "apply convex_indicator_convex hs" |
| }, |
| { |
| "id": 116, |
| "match": false, |
| "ground_truth": "have eq : EqOn (fun _ ↦ (0: EReal)) (Set.convex_indicator X) X := by\n dsimp[EqOn]\n intro x hx\n dsimp [Set.convex_indicator]\n rw [if_pos hx]", |
| "model_output": "apply ConvexOn.subset this (fun x a ↦ a)" |
| }, |
| { |
| "id": 117, |
| "match": false, |
| "ground_truth": "by\n apply Subset.antisymm\n · exact iIntersection_closure_sub_closure_iIntersection'' h hinter\n exact closure_iIntersection_sub_iIntersection_closure'", |
| "model_output": "apply Subset.antisymm\n · exact iIntersection_intrinsicClosure_sub_intrinsicClosure_iIntersection\n exact intrinsicClosure_iIntersection_sub_iIntersection_intrinsicClosure h hinter" |
| }, |
| { |
| "id": 118, |
| "match": false, |
| "ground_truth": "apply Subset.antisymm\n · exact iIntersection_closure_sub_closure_iIntersection'' h hinter\n exact closure_iIntersection_sub_iIntersection_closure'", |
| "model_output": "apply Subset.antisymm\n · exact iIntersection_intrinsicClosure_sub_intrinsicClosure_iIntersection\n exact intrinsicClosure_iIntersection_sub_iIntersection_intrinsicClosure h hinter" |
| }, |
| { |
| "id": 119, |
| "match": false, |
| "ground_truth": "· exact iIntersection_closure_sub_closure_iIntersection'' h hinter", |
| "model_output": "· apply iIntersection_intrinsicClosure_sub_intrinsicClosure_iIntersection\n · apply h\n · exact hinter" |
| }, |
| { |
| "id": 120, |
| "match": false, |
| "ground_truth": "exact iIntersection_closure_sub_closure_iIntersection'' h hinter", |
| "model_output": "apply iIntersection_intrinsicClosure_sub_intrinsicClosure_iIntersection" |
| }, |
| { |
| "id": 121, |
| "match": false, |
| "ground_truth": "exact closure_iIntersection_sub_iIntersection_closure'", |
| "model_output": "apply intrinsicClosure_minimal\n · intro x hx\n rw[mem_iInter] at hx\n exact fun i ↦ intrinsicClosure_mono (hx i)\n · exact convex_iInter h\n · exact hinter" |
| }, |
| { |
| "id": 122, |
| "match": false, |
| "ground_truth": "have eqone : a + (1 - a) = 1 := add_sub_cancel a 1", |
| "model_output": "have : a * (f x - f y) < 0 := mul_neg_of_pos_of_neg lta (sub_neg_of_lt hy (hx trivial))" |
| }, |
| { |
| "id": 123, |
| "match": false, |
| "ground_truth": "have lta' : 0 < 1 - a := sub_pos_of_lt alt", |
| "model_output": "have : f (a • x + (1 - a) • y) < f x := by\n apply hf' (hx trivial) (hy trivial) neq lta (sub_pos_of_lt alt) eqone\n apply lt_of_le_of_lt (le_of_lt lta) (le_of_lt alt)" |
| }, |
| { |
| "id": 124, |
| "match": false, |
| "ground_truth": "have h : f (a • x + (1 - a) • y) < f y := by\n apply Eq.trans_gt (max_eq_right (hx trivial))\n apply hf' neq lta lta' eqone", |
| "model_output": "have : a + (1 - a) > 0 := by linarith" |
| }, |
| { |
| "id": 125, |
| "match": false, |
| "ground_truth": "apply Eq.trans_gt (max_eq_right (hx trivial))", |
| "model_output": "apply hf' (x := x) (y := y) neq lta lta' eqone" |
| }, |
| { |
| "id": 126, |
| "match": false, |
| "ground_truth": "intro ε εpos", |
| "model_output": "intro ε εpos\n filter_upwards [hu ε εpos, hv ε εpos] with y huy hvy\n rw [eventually_and]\n rw [eventually_nhds_iff_ball] at huy hvy\n intro y hy\n specialize huy y hy\n specialize hvy y hy\n rw [inner_add_left, ← sub_sub, sub_le_sub_iff, ← sub_sub, sub_le_sub_iff] at huy hvy\n rw [← sub_sub, sub_le_sub_iff, ← sub_sub, sub_le_sub_iff]\n rw [← sub_sub," |
| }, |
| { |
| "id": 127, |
| "match": false, |
| "ground_truth": "rw [ge_iff_le] at *", |
| "model_output": "apply add_le_add (mul_le_mul_of_nonneg_left hyu (le_of_lt ha))\n (mul_le_mul_of_nonneg_left hyv (le_of_lt hb))" |
| }, |
| { |
| "id": 128, |
| "match": false, |
| "ground_truth": "apply add_le_add", |
| "model_output": "apply add_le_add (mul_le_mul_of_nonneg_left hyu ha) (mul_le_mul_of_nonneg_left hyv hb)" |
| }, |
| { |
| "id": 129, |
| "match": false, |
| "ground_truth": "apply div_pos", |
| "model_output": "apply div_pos zero_lt_one (mul_pos _ alg.lpos)\n[· linarith]" |
| }, |
| { |
| "id": 130, |
| "match": false, |
| "ground_truth": "apply mul_pos", |
| "model_output": "apply mul_pos _ alg.lpos\n[apply hγ]" |
| }, |
| { |
| "id": 131, |
| "match": true, |
| "ground_truth": "· linarith [hγ]", |
| "model_output": "· linarith [hγ]" |
| }, |
| { |
| "id": 132, |
| "match": true, |
| "ground_truth": "linarith [hγ]", |
| "model_output": "linarith [hγ]" |
| }, |
| { |
| "id": 133, |
| "match": true, |
| "ground_truth": "· apply alg.lpos", |
| "model_output": "apply alg.lpos" |
| }, |
| { |
| "id": 134, |
| "match": true, |
| "ground_truth": "apply alg.lpos", |
| "model_output": "apply alg.lpos" |
| }, |
| { |
| "id": 135, |
| "match": false, |
| "ground_truth": "rw [mem_nhds_iff]", |
| "model_output": "apply Metric.ball_mem_nhds\n exact he" |
| }, |
| { |
| "id": 136, |
| "match": false, |
| "ground_truth": "simp [Function.Epi]", |
| "model_output": "simp [Epi]" |
| }, |
| { |
| "id": 137, |
| "match": true, |
| "ground_truth": "simp [Function.Epi]", |
| "model_output": "simp [Function.Epi]" |
| }, |
| { |
| "id": 138, |
| "match": true, |
| "ground_truth": "exact isClosed_empty", |
| "model_output": "exact isClosed_empty" |
| }, |
| { |
| "id": 139, |
| "match": false, |
| "ground_truth": "constructor", |
| "model_output": "constructor\n · intro h z\n unfold Banach_HasSubgradientAt HasSubgradientAt\n specialize h z\n rw[← inner_apply_apply (toDual ℝ E g) z]\n exact le_of_eq (congrArg EReal.toEReal h)\n · apply Banach_HasSubgradientAt_to_HasSubgradientAt" |
| }, |
| { |
| "id": 140, |
| "match": true, |
| "ground_truth": "by\n ext g; rw [h]", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 141, |
| "match": true, |
| "ground_truth": "ext g; rw [h]", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 142, |
| "match": true, |
| "ground_truth": "constructor", |
| "model_output": "constructor" |
| }, |
| { |
| "id": 143, |
| "match": false, |
| "ground_truth": "intro x _hx", |
| "model_output": "intro x _ [H]\n rw [← H]\n rfl" |
| }, |
| { |
| "id": 144, |
| "match": false, |
| "ground_truth": "simp at this", |
| "model_output": "rw [← this]" |
| }, |
| { |
| "id": 145, |
| "match": false, |
| "ground_truth": "· simp [StrictMono]", |
| "model_output": "apply strictMono_add_const" |
| }, |
| { |
| "id": 146, |
| "match": false, |
| "ground_truth": "simp [StrictMono]", |
| "model_output": "apply strictMono_add_const" |
| }, |
| { |
| "id": 147, |
| "match": true, |
| "ground_truth": "by_cases hc2 : c = ⊤", |
| "model_output": "by_cases hc2 : c = ⊤" |
| }, |
| { |
| "id": 148, |
| "match": false, |
| "ground_truth": "· rw [hc2]; simp", |
| "model_output": "· rw [hc2, EReal.top_sub_coe, EReal.top_sub_coe]" |
| }, |
| { |
| "id": 149, |
| "match": false, |
| "ground_truth": "rw [hc2]; simp", |
| "model_output": "rw [hc2, EReal.top_sub_coe, EReal.coe_sub_top]" |
| }, |
| { |
| "id": 150, |
| "match": false, |
| "ground_truth": "by exact this", |
| "model_output": "exact EReal.coe_le_coe_iff.2 this" |
| }, |
| { |
| "id": 151, |
| "match": false, |
| "ground_truth": "constructor", |
| "model_output": "apply StrongConvexOn_of_ConvexOn hs\n intro x xs y ys a b ha hb hab\n specialize hfun xs ys ha hb hab\n rw [← sub_le_sub_iff, sub_add, sub_add_cancel, sub_le_sub_iff] at hfun\n linarith [hfun]" |
| }, |
| { |
| "id": 152, |
| "match": false, |
| "ground_truth": "rw [← this]", |
| "model_output": "rw [this] at hfun\n exact hfun" |
| }, |
| { |
| "id": 153, |
| "match": false, |
| "ground_truth": "constructor", |
| "model_output": "constructor\n · intro x\n rw[x]\n · intro x\n rw[x]" |
| }, |
| { |
| "id": 154, |
| "match": false, |
| "ground_truth": "simp [funext_iff]", |
| "model_output": "intro h" |
| }, |
| { |
| "id": 155, |
| "match": true, |
| "ground_truth": "constructor", |
| "model_output": "constructor" |
| }, |
| { |
| "id": 156, |
| "match": false, |
| "ground_truth": "· apply convex_univ", |
| "model_output": "apply convex_univ [f'']" |
| }, |
| { |
| "id": 157, |
| "match": true, |
| "ground_truth": "apply convex_univ", |
| "model_output": "apply convex_univ" |
| }, |
| { |
| "id": 158, |
| "match": true, |
| "ground_truth": "intro x _ y _", |
| "model_output": "intro x _ y _" |
| }, |
| { |
| "id": 159, |
| "match": true, |
| "ground_truth": "apply sq_nonneg", |
| "model_output": "apply sq_nonneg" |
| }, |
| { |
| "id": 160, |
| "match": true, |
| "ground_truth": "exact h_zero_eq.symm", |
| "model_output": "exact h_zero_eq.symm" |
| }, |
| { |
| "id": 161, |
| "match": false, |
| "ground_truth": "dsimp [ConvexOn]", |
| "model_output": "apply convexOn_const hs 0" |
| }, |
| { |
| "id": 162, |
| "match": false, |
| "ground_truth": "constructor", |
| "model_output": "constructor <;> simp [Convex] <;> tauto" |
| }, |
| { |
| "id": 163, |
| "match": false, |
| "ground_truth": "dsimp[EqOn]", |
| "model_output": "intro x hx\n rw[← convex_indicator_eq_zero hs] at hx\n rw[← hx]\n simp only [Pi.zero_apply, zero_eq_iff]" |
| }, |
| { |
| "id": 164, |
| "match": false, |
| "ground_truth": "rw [if_pos hx]", |
| "model_output": "rw[if_pos hx]" |
| }, |
| { |
| "id": 165, |
| "match": false, |
| "ground_truth": "by_contra neq", |
| "model_output": "by_contra hxy" |
| }, |
| { |
| "id": 166, |
| "match": false, |
| "ground_truth": "intro u hu v hv a b ha hb hab", |
| "model_output": "apply convex_set_of_norm_le_one" |
| }, |
| { |
| "id": 167, |
| "match": true, |
| "ground_truth": "rw [has_f_subdiff_iff] at *", |
| "model_output": "rw [has_f_subdiff_iff] at *" |
| }, |
| { |
| "id": 168, |
| "match": true, |
| "ground_truth": "rw [finrank_span_eq_card]", |
| "model_output": "rw [finrank_span_eq_card]" |
| }, |
| { |
| "id": 169, |
| "match": false, |
| "ground_truth": "rw [hx, LinearMap.map_add]", |
| "model_output": "rw [hx, map_add, norm_add_sq_real]" |
| }, |
| { |
| "id": 170, |
| "match": false, |
| "ground_truth": "apply ContDiffOn.contDiffAt hf", |
| "model_output": "apply ContDiffOn.contDiffAt hf (Metric.ball x ε) (Metric.mem_ball_self he)" |
| }, |
| { |
| "id": 171, |
| "match": false, |
| "ground_truth": "simp [Function.closure]", |
| "model_output": "apply closure_univ_eq_top" |
| }, |
| { |
| "id": 172, |
| "match": false, |
| "ground_truth": "simp [Function.closure, hsf]", |
| "model_output": "simp [Function.IsClosed, hsf]" |
| }, |
| { |
| "id": 173, |
| "match": false, |
| "ground_truth": "by_cases hf : ∃ x ∈ s, f x = ⊥", |
| "model_output": "by_cases h : ∃ x ∈ s, f x = ⊥\n · simp [h]\n push_neg at hsf\n simp [h]\n exact isClosed_univ" |
| }, |
| { |
| "id": 174, |
| "match": false, |
| "ground_truth": "simp [Function.IsClosed]", |
| "model_output": "apply isClosed_of_subset_closure\n · exact fun x hx ↦ by simp\n rcases hf with ⟨x, hx, hfx⟩\n exact hfx ▸ hx" |
| }, |
| { |
| "id": 175, |
| "match": false, |
| "ground_truth": "simp [Function.IsClosed]", |
| "model_output": "simp [Function.IsClosed, hs]" |
| }, |
| { |
| "id": 176, |
| "match": false, |
| "ground_truth": "apply StrictMono.tendsto_atTop", |
| "model_output": "apply StrictMono.tendsto_atTop\n apply strictMono_nat_of_lt_succ\n intro n\n have : φ₁ (n + 1) > φ₁ n := by\n apply StrictMono.strictMonoOn\n apply hphi_StrictMono fullrank₁ fullrank₂\n apply lt_add_one\n apply this\n apply Nat.cast_lt.mpr\n apply this\n apply nlinarith\n apply Nat.le_add_left\n apply le_add_of_le_add_left\n apply le_add_of_le_add_right\n apply le_add_of_le_add_left\n apply le_add" |
| }, |
| { |
| "id": 177, |
| "match": false, |
| "ground_truth": "apply StrictMono.comp", |
| "model_output": "apply StrictMono.comp\n · apply hphi_StrictMono\n · apply StrictMono.add\n · apply strictMono_id\n · apply strictMono_nat_of_lt_succ\n intro n\n simp only [Nat.one_add, Nat.add_one, Nat.lt_succ_self]" |
| }, |
| { |
| "id": 178, |
| "match": true, |
| "ground_truth": "· apply hphi_StrictMono", |
| "model_output": "· apply hphi_StrictMono" |
| }, |
| { |
| "id": 179, |
| "match": true, |
| "ground_truth": "apply hphi_StrictMono", |
| "model_output": "apply hphi_StrictMono" |
| }, |
| { |
| "id": 180, |
| "match": false, |
| "ground_truth": "lift a to ℝ using ⟨ha2, ha1⟩", |
| "model_output": "have : a < ⊤ := LT.lt.ne_top ha2" |
| }, |
| { |
| "id": 181, |
| "match": false, |
| "ground_truth": "lift c to ℝ using ⟨hc2, hc1⟩", |
| "model_output": "lift c to ℝ using ⟨hc2, LT.lt.ne_bot hc1⟩\n rw [← EReal.coe_add, ← EReal.coe_sub, EReal.coe_le_coe_iff]\n rw [← EReal.coe_sub, EReal.coe_sub_le_coe_sub_iff]\n exact h" |
| }, |
| { |
| "id": 182, |
| "match": false, |
| "ground_truth": "intro x hx y hy a b ha hb hab", |
| "model_output": "intro x xs y ys a b ha hb hab\n specialize hfun xs ys a ha b hb hab\n simp at hfun\n exact hfun" |
| }, |
| { |
| "id": 183, |
| "match": false, |
| "ground_truth": "· exact fun h => (by rw [h])", |
| "model_output": "· intro eq; rw [eq]" |
| }, |
| { |
| "id": 184, |
| "match": false, |
| "ground_truth": "exact fun h => (by rw [h])", |
| "model_output": "intro hst\n rw [hst]" |
| }, |
| { |
| "id": 185, |
| "match": false, |
| "ground_truth": "simp [convex_support]", |
| "model_output": "apply convex_support_nonempty hs x trivial" |
| }, |
| { |
| "id": 186, |
| "match": false, |
| "ground_truth": "simp [convex_support]", |
| "model_output": "rcases hs with ⟨x, hx⟩\n use x, trivial\n rw [convex_support]\n simp\n exact hx.2" |
| }, |
| { |
| "id": 187, |
| "match": false, |
| "ground_truth": "simp [convex_indicator]", |
| "model_output": "by_cases hy : y ∈ s\n · rw [convex_indicator_of_mem hy]; simp\n simp [convex_indicator, hy]" |
| }, |
| { |
| "id": 188, |
| "match": false, |
| "ground_truth": "split_ifs with hy <;> simp", |
| "model_output": "by_cases h : y ∈ s\n · simp [h]\n simp [h]" |
| }, |
| { |
| "id": 189, |
| "match": false, |
| "ground_truth": "rcases h1 with ⟨x, hx⟩", |
| "model_output": "rcases h1 with ⟨x, hx⟩\n use x\n simp [dom]\n exact hx.1" |
| }, |
| { |
| "id": 190, |
| "match": false, |
| "ground_truth": "apply convex_on_p_top", |
| "model_output": "apply convex_on_n_inf" |
| }, |
| { |
| "id": 191, |
| "match": false, |
| "ground_truth": "rw [polyhedra_iff_cone b]", |
| "model_output": "apply isClosed_setOf_sum" |
| }, |
| { |
| "id": 192, |
| "match": false, |
| "ground_truth": "exact closed_conic σ b", |
| "model_output": "apply cone_is_closed" |
| }, |
| { |
| "id": 193, |
| "match": false, |
| "ground_truth": "intro x _ y _ a b _ _ _", |
| "model_output": "simp" |
| }, |
| { |
| "id": 194, |
| "match": false, |
| "ground_truth": "dsimp [Set.convex_indicator]", |
| "model_output": "rw[convex_indicator_of_mem hx]" |
| }, |
| { |
| "id": 195, |
| "match": false, |
| "ground_truth": "apply ConvexOn.congr this eq", |
| "model_output": "apply EqOn.convexOn eq this" |
| }, |
| { |
| "id": 196, |
| "match": false, |
| "ground_truth": "apply Subset.antisymm", |
| "model_output": "apply Subset.antisymm\n · exact iIntersection_intrinsicClosure_sub_intrinsicClosure_iIntersection\n exact intrinsicClosure_iIntersection_sub_iIntersection_intrinsicClosure h hinter" |
| }, |
| { |
| "id": 197, |
| "match": false, |
| "ground_truth": "apply hf' neq lta lta' eqone", |
| "model_output": "apply lt_max_iff.2" |
| }, |
| { |
| "id": 198, |
| "match": false, |
| "ground_truth": "simp only [isMinOn_iff] at hy", |
| "model_output": "have : f (a • x + (1 - a) • y) < f x := by\n apply lt_of_lt_of_le h (hy (mem_univ (a • x + (1 - a) • y)))" |
| }, |
| { |
| "id": 199, |
| "match": false, |
| "ground_truth": "apply not_le_of_gt h hy", |
| "model_output": "linarith [h, hy]" |
| }, |
| { |
| "id": 200, |
| "match": false, |
| "ground_truth": "intro u", |
| "model_output": "apply convex_set_of_norm_le_one" |
| }, |
| { |
| "id": 201, |
| "match": false, |
| "ground_truth": "intro ε", |
| "model_output": "intro ε εpos\n filter_upwards [hu ε εpos, hv ε εpos] with y huy hvy\n rw [eventually_and]\n rw [eventually_nhds_iff_ball] at huy hvy\n intro y hy\n specialize huy y hy\n specialize hvy y hy\n rw [inner_add_left, ← sub_sub, sub_le_sub_iff, ← sub_sub, sub_le_sub_iff] at huy hvy\n rw [← sub_sub, sub_le_sub_iff, ← sub_sub, sub_le_sub_iff]\n rw [← sub_sub," |
| }, |
| { |
| "id": 202, |
| "match": false, |
| "ground_truth": "intro z z'", |
| "model_output": "intro x y\n have h1 := hl x y\n rw [← Prod.fst_sub, ← Prod.snd_sub, ← Prod.fst_sub, ← Prod.snd_sub] at h1\n apply le_trans (fst_norm_le_prod_L2 _) at h1\n apply le_trans (snd_norm_le_prod_L2 _) at h1\n apply le_trans (fst_norm_le_prod_L2 _) at h1\n apply le_trans (snd_norm_le_prod_L2 _) at h1\n apply le_trans (fst_norm_le_prod_L2 _) at h1\n apply le_trans" |
| }, |
| { |
| "id": 203, |
| "match": false, |
| "ground_truth": "intro z", |
| "model_output": "intro x y\n have h1 := hl x y\n rw [← Prod.fst_sub, ← Prod.snd_sub, ← Prod.fst_sub, ← Prod.snd_sub] at h1\n apply le_trans (fst_norm_le_prod_L2 _) at h1\n apply le_trans (snd_norm_le_prod_L2 _) at h1\n apply le_trans (fst_norm_le_prod_L2 _) at h1\n apply le_trans (snd_norm_le_prod_L2 _) at h1\n apply le_trans (fst_norm_le_prod_L2 _) at h1\n apply le_trans" |
| }, |
| { |
| "id": 204, |
| "match": false, |
| "ground_truth": "intro k", |
| "model_output": "intro k\n rw [ck k]\n apply div_pos\n · linarith\n · apply mul_pos\n · linarith\n · linarith [alg.lpos]" |
| }, |
| { |
| "id": 205, |
| "match": true, |
| "ground_truth": "rw [ck]", |
| "model_output": "rw [ck]" |
| }, |
| { |
| "id": 206, |
| "match": false, |
| "ground_truth": "· norm_num", |
| "model_output": "· linarith\n apply mul_pos\n · linarith\n · simp; linarith [alg.lpos]" |
| }, |
| { |
| "id": 207, |
| "match": false, |
| "ground_truth": "norm_num", |
| "model_output": "linarith" |
| }, |
| { |
| "id": 208, |
| "match": false, |
| "ground_truth": "simp [h]", |
| "model_output": "apply linearIndependent_iff.mpr\n intro x a\n simp [colVec, Matrix.mulVec, dotProduct]\n rw [← Matrix.mulVec, ← Matrix.mulVec, h] at a\n exact a.1" |
| }, |
| { |
| "id": 209, |
| "match": false, |
| "ground_truth": "aesop", |
| "model_output": "constructor <;> intro a ha b hb hab <;> exact hv.1 a (ha hb) <;> exact hv.2 a b hab" |
| }, |
| { |
| "id": 210, |
| "match": false, |
| "ground_truth": "intro h", |
| "model_output": "intro h\n exact univ_lowersemicontinoushull_top_eq_top h" |
| }, |
| { |
| "id": 211, |
| "match": false, |
| "ground_truth": "exfalso", |
| "model_output": "apply le_antisymm_iff.mpr\n constructor\n · exact lowerSemicontinuousHull_le_top\n exact fun a ↦ by\n rw [← closure_eq_top_iff, ← closure_epi_eq_top_iff_lowerSemicontinuousHull_eq_top]\n apply closure_epi_eq_top_of_properFunction h" |
| }, |
| { |
| "id": 212, |
| "match": true, |
| "ground_truth": "simp at h1", |
| "model_output": "simp at h1" |
| }, |
| { |
| "id": 213, |
| "match": true, |
| "ground_truth": "simp [hf]", |
| "model_output": "simp [hf]" |
| }, |
| { |
| "id": 214, |
| "match": false, |
| "ground_truth": "ext ⟨x, _⟩", |
| "model_output": "simp [Function.Epi]" |
| }, |
| { |
| "id": 215, |
| "match": true, |
| "ground_truth": "rw [h1]", |
| "model_output": "rw [h1]" |
| }, |
| { |
| "id": 216, |
| "match": true, |
| "ground_truth": "simp [hf]", |
| "model_output": "simp [hf]" |
| }, |
| { |
| "id": 217, |
| "match": false, |
| "ground_truth": "ext ⟨x, _⟩", |
| "model_output": "simp [ProperFunction] at hsf" |
| }, |
| { |
| "id": 218, |
| "match": true, |
| "ground_truth": "rw [h']", |
| "model_output": "rw [h']" |
| }, |
| { |
| "id": 219, |
| "match": false, |
| "ground_truth": "intro h z", |
| "model_output": "intro h z\n rw [hasSubgradientAt_iff_le] at h\n apply h z\n simp\n rw [← ContinuousLinearMap.map_sub]; simp\n apply real_inner_le_norm" |
| }, |
| { |
| "id": 220, |
| "match": false, |
| "ground_truth": "intro h", |
| "model_output": "intro h z\n rw [hasSubgradientAt_iff_le] at h\n apply h z\n simp\n rw [← ContinuousLinearMap.map_sub]; simp\n apply real_inner_le_norm" |
| }, |
| { |
| "id": 221, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "simp [toDual_apply]" |
| }, |
| { |
| "id": 222, |
| "match": false, |
| "ground_truth": "linarith", |
| "model_output": "linarith [hz]" |
| }, |
| { |
| "id": 223, |
| "match": false, |
| "ground_truth": "intro h z", |
| "model_output": "intro h z hz\n rw [← mem_gradientSet_iff_hasSubgradientAt, ← Banach_gradientSet_iff] at h\n rcases h with ⟨_, h⟩\n exact h z hz" |
| }, |
| { |
| "id": 224, |
| "match": false, |
| "ground_truth": "intro h", |
| "model_output": "intro h z hz\n rw [← mem_gradientSet_iff_hasSubgradientAt, ← Banach_gradientSet_iff] at h\n rcases h with ⟨_, h⟩\n exact h z hz" |
| }, |
| { |
| "id": 225, |
| "match": false, |
| "ground_truth": "simp at hz", |
| "model_output": "simp [toDual_apply] at hz" |
| }, |
| { |
| "id": 226, |
| "match": false, |
| "ground_truth": "linarith", |
| "model_output": "linarith [hz]" |
| }, |
| { |
| "id": 227, |
| "match": false, |
| "ground_truth": "ext g", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 228, |
| "match": true, |
| "ground_truth": "rw [h]", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 229, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "apply And.intro" |
| }, |
| { |
| "id": 230, |
| "match": false, |
| "ground_truth": "intro x hx", |
| "model_output": "intro x xs\n rw[← EReal.coe_add, ← inner_add_left, ← inner_add_left, ← inner_add_left]\n apply le_trans\n · apply i.1\n · apply i.2 x xs" |
| }, |
| { |
| "id": 231, |
| "match": false, |
| "ground_truth": "intro x", |
| "model_output": "intro x xs\n rw[← EReal.coe_add, ← inner_add_left, ← inner_add_left, ← inner_add_left]\n apply le_trans\n · apply i.1\n · apply i.2 x xs" |
| }, |
| { |
| "id": 232, |
| "match": false, |
| "ground_truth": "intro x", |
| "model_output": "intro x _ [H]\n rw [← H]\n rfl" |
| }, |
| { |
| "id": 233, |
| "match": false, |
| "ground_truth": "ext n", |
| "model_output": "ext n\n congr\n apply Nat.sub_one_add_one_eq_of_pos (φ_is_positive (n + 1))" |
| }, |
| { |
| "id": 234, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "simp [φ]" |
| }, |
| { |
| "id": 235, |
| "match": true, |
| "ground_truth": "rw [this]", |
| "model_output": "rw [this]" |
| }, |
| { |
| "id": 236, |
| "match": false, |
| "ground_truth": "rw [hc2]", |
| "model_output": "rw [hc2, EReal.top_sub_coe, EReal.coe_sub_top]" |
| }, |
| { |
| "id": 237, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "simp [← EReal.coe_add] at *" |
| }, |
| { |
| "id": 238, |
| "match": false, |
| "ground_truth": "exact this", |
| "model_output": "exact EReal.coe_le_coe_iff.2 this" |
| }, |
| { |
| "id": 239, |
| "match": false, |
| "ground_truth": "linarith", |
| "model_output": "linarith [this]" |
| }, |
| { |
| "id": 240, |
| "match": false, |
| "ground_truth": "· exact hs", |
| "model_output": "apply hs" |
| }, |
| { |
| "id": 241, |
| "match": false, |
| "ground_truth": "exact hs", |
| "model_output": "apply hs" |
| }, |
| { |
| "id": 242, |
| "match": false, |
| "ground_truth": "intro x", |
| "model_output": "intro x xs y ys a b ha hb hab\n specialize hfun xs ys a ha b hb hab\n simp at hfun\n exact hfun" |
| }, |
| { |
| "id": 243, |
| "match": false, |
| "ground_truth": "dsimp", |
| "model_output": "simp [← sub_nonneg] at hfun ⊢ [← mul_nonneg] at hab ⊢ [← ha] at hab ⊢ [← hb] at hab ⊢\n rw [← hab, sub_mul, sub_le_sub_iff_left, mul_comm, mul_comm (a * b)]\n apply hfun" |
| }, |
| { |
| "id": 244, |
| "match": false, |
| "ground_truth": "ring_nf", |
| "model_output": "linarith" |
| }, |
| { |
| "id": 245, |
| "match": false, |
| "ground_truth": "exact hfun", |
| "model_output": "apply hfun" |
| }, |
| { |
| "id": 246, |
| "match": true, |
| "ground_truth": "by rw [h]", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 247, |
| "match": true, |
| "ground_truth": "rw [h]", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 248, |
| "match": false, |
| "ground_truth": "intro h", |
| "model_output": "intro h\n have h1 := fun x a ↦ (convex_support_eq_closure h x a).1\n have h2 := fun x a ↦ (convex_support_eq_closure h x a).2\n apply Subset.antisymm\n · intro x hx\n rw [← closure_convex_support_eq hscl hscv, ← closure_convex_support_eq htcl htcv, h]\n exact h1 x hx\n · intro x hx\n rw [← closure_convex_support_eq hscl hscv, ← closure_convex_support_eq htcl htcv" |
| }, |
| { |
| "id": 249, |
| "match": false, |
| "ground_truth": "ext z", |
| "model_output": "rw [← closure_convex_support_eq s hscl, ← closure_convex_support_eq t htcl, h]" |
| }, |
| { |
| "id": 250, |
| "match": true, |
| "ground_truth": "intro x _", |
| "model_output": "intro x _" |
| }, |
| { |
| "id": 251, |
| "match": false, |
| "ground_truth": "intro x", |
| "model_output": "intro x _" |
| }, |
| { |
| "id": 252, |
| "match": false, |
| "ground_truth": "right", |
| "model_output": "right\n rcases hs with ⟨x, hx⟩\n use x, trivial\n rw [convex_support]\n simp\n exact Ne.lt_top' (id (Ne.symm hx))" |
| }, |
| { |
| "id": 253, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "intro y\n simp\n exact convex_indicator_ge_zero hs y" |
| }, |
| { |
| "id": 254, |
| "match": false, |
| "ground_truth": "intro y", |
| "model_output": "intro y\n by_cases hy : y ∈ s\n · rw [convex_indicator_of_mem hy]; simp\n · simp [convex_indicator, hy]" |
| }, |
| { |
| "id": 255, |
| "match": true, |
| "ground_truth": "simp", |
| "model_output": "simp" |
| }, |
| { |
| "id": 256, |
| "match": true, |
| "ground_truth": "simp at h1", |
| "model_output": "simp at h1" |
| }, |
| { |
| "id": 257, |
| "match": false, |
| "ground_truth": "simp [hx]", |
| "model_output": "simp [dom]" |
| }, |
| { |
| "id": 258, |
| "match": false, |
| "ground_truth": "ext x", |
| "model_output": "ext x\n exact Eq.symm (h x)" |
| }, |
| { |
| "id": 259, |
| "match": true, |
| "ground_truth": "exact h x", |
| "model_output": "exact h x" |
| }, |
| { |
| "id": 260, |
| "match": false, |
| "ground_truth": "intro x", |
| "model_output": "intro x _ y _" |
| }, |
| { |
| "id": 261, |
| "match": true, |
| "ground_truth": "simp [hp]", |
| "model_output": "simp [hp]" |
| }, |
| { |
| "id": 262, |
| "match": true, |
| "ground_truth": "intro b", |
| "model_output": "intro b" |
| }, |
| { |
| "id": 263, |
| "match": false, |
| "ground_truth": "by rfl", |
| "model_output": "constructor\n · intro h\n exact (IsLinear_iff'.1 h)\n intro ⟨a, b, h⟩\n exact (IsLinear_iff'.2 (fun x y ↦ by simp [h]; exact AddMonoidHomClass.map_add _ _ _))" |
| }, |
| { |
| "id": 264, |
| "match": false, |
| "ground_truth": "rfl", |
| "model_output": "constructor\n · intro h\n exact (IsLinear_iff'.1 h)\n intro ⟨a, b, h⟩\n exact (IsLinear_iff'.2 (fun x y ↦ by simp [h]; exact AddMonoidHomClass.map_add _ _ _))" |
| }, |
| { |
| "id": 265, |
| "match": true, |
| "ground_truth": "rw [h]", |
| "model_output": "rw [h]" |
| }, |
| { |
| "id": 266, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "simp [ContinuousLinearMap.adjoint]" |
| }, |
| { |
| "id": 267, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "simp [ContinuousLinearMap.adjoint]" |
| }, |
| { |
| "id": 268, |
| "match": false, |
| "ground_truth": "exact hs", |
| "model_output": "· apply hs" |
| }, |
| { |
| "id": 269, |
| "match": false, |
| "ground_truth": "intro x", |
| "model_output": "simp" |
| }, |
| { |
| "id": 270, |
| "match": false, |
| "ground_truth": "simp", |
| "model_output": "simp [smul_zero, add_zero]" |
| }, |
| { |
| "id": 271, |
| "match": false, |
| "ground_truth": "intro x hx", |
| "model_output": "intro x hx\n rw[convex_indicator_of_mem hx]" |
| }, |
| { |
| "id": 272, |
| "match": false, |
| "ground_truth": "intro x", |
| "model_output": "intro x hx\n rw[convex_indicator_of_mem hx]" |
| } |
| ] |
| } |
