evaluation/predictions_ood/predictions_checkpoint-200.json

[
  {
    "id": 0,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "case.1:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nu : E\nhu : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖\nv : E\nhv : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ v (y - x) ≥ -ε * ‖y - x‖\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nε : ℝ\nεpos : ε > 0\ny : E\nhyu : -ε * ‖y - x‖ ≤ f y - f x - inner ℝ u (y - x)\nhyv : -ε * ‖y - x‖ ≤ f y - f x - inner ℝ v (y - x)\n⊢ a * (-ε * ‖y - x‖) ≤ a * (f y - f x - inner ℝ u (y - x))\n\ncase.2:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nu : E\nhu : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖\nv : E\nhv : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ v (y - x) ≥ -ε * ‖y - x‖\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nε : ℝ\nεpos : ε > 0\ny : E\nhyu : -ε * ‖y - x‖ ≤ f y - f x - inner ℝ u (y - x)\nhyv : -ε * ‖y - x‖ ≤ f y - f x - inner ℝ v (y - x)\n⊢ b * (-ε * ‖y - x‖) ≤ b * (f y - f x - inner ℝ v (y - x))",
    "ground_truth": "· exact mul_le_mul_of_nonneg_left hyu ha",
    "model_output": "· apply mul_le_mul_of_nonneg_left hyu ha"
  },
  {
    "id": 1,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nu : E\nhu : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖\nv : E\nhv : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ v (y - x) ≥ -ε * ‖y - x‖\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nε : ℝ\nεpos : ε > 0\ny : E\nhyu : -ε * ‖y - x‖ ≤ f y - f x - inner ℝ u (y - x)\nhyv : -ε * ‖y - x‖ ≤ f y - f x - inner ℝ v (y - x)\n⊢ a * (-ε * ‖y - x‖) ≤ a * (f y - f x - inner ℝ u (y - x))",
    "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,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nu : E\nhu : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖\nv : E\nhv : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ v (y - x) ≥ -ε * ‖y - x‖\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nε : ℝ\nεpos : ε > 0\ny : E\nhyu : -ε * ‖y - x‖ ≤ f y - f x - inner ℝ u (y - x)\nhyv : -ε * ‖y - x‖ ≤ f y - f x - inner ℝ v (y - x)\n⊢ b * (-ε * ‖y - x‖) ≤ b * (f y - f x - inner ℝ v (y - x))",
    "ground_truth": "· exact mul_le_mul_of_nonneg_left hyv hb",
    "model_output": "apply mul_le_mul_of_nonneg_left hyv hb"
  },
  {
    "id": 3,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nu : E\nhu : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖\nv : E\nhv : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ v (y - x) ≥ -ε * ‖y - x‖\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nε : ℝ\nεpos : ε > 0\ny : E\nhyu : -ε * ‖y - x‖ ≤ f y - f x - inner ℝ u (y - x)\nhyv : -ε * ‖y - x‖ ≤ f y - f x - inner ℝ v (y - x)\n⊢ b * (-ε * ‖y - x‖) ≤ b * (f y - f x - inner ℝ v (y - x))",
    "ground_truth": "exact mul_le_mul_of_nonneg_left hyv hb",
    "model_output": "apply mul_le_mul_of_nonneg_left hyv hb"
  },
  {
    "id": 4,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nH : WithLp 2 (E × F) → ℝ\nl : NNReal\nh : Differentiable ℝ H\nhl : LipschitzWith l (gradient H)\n⊢ LipschitzWith l fun z => grad_snd H z.1 z.2",
    "ground_truth": "rw [lipschitzWith_iff_norm_sub_le] at *",
    "model_output": "apply lipschitzWith_of_norm_deriv_le\n    · apply DifferentiableOn.prodMk (differentiableOn_fst h) (differentiableOn_snd h)\n    · apply LipschitzWith.prodMk (lipschitzWith_fst h hl) (lipschitzWith_snd h hl)"
  },
  {
    "id": 5,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nH : WithLp 2 (E × F) → ℝ\nl : NNReal\nh : Differentiable ℝ H\nhl : ∀ (x y : WithLp 2 (E × F)), ‖gradient H x - gradient H y‖ ≤ ↑l * ‖x - y‖\nz : WithLp 2 (E × F)\nz' : WithLp 2 (E × F)\n⊢ ‖grad_snd H z.1 z.2 - grad_snd H z'.1 z'.2‖ = ‖(gradient H z).2 - (gradient H z').2‖",
    "ground_truth": "by rw [grad_snd_eq h, grad_snd_eq h]",
    "model_output": "rw [grad_eq_block_grad, grad_fun_comp, grad_fun_comp, grad_add_const, grad_add_const]"
  },
  {
    "id": 6,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nH : WithLp 2 (E × F) → ℝ\nl : NNReal\nh : Differentiable ℝ H\nhl : ∀ (x y : WithLp 2 (E × F)), ‖gradient H x - gradient H y‖ ≤ ↑l * ‖x - y‖\nz : WithLp 2 (E × F)\nz' : WithLp 2 (E × F)\n⊢ ‖grad_snd H z.1 z.2 - grad_snd H z'.1 z'.2‖ = ‖(gradient H z).2 - (gradient H z').2‖",
    "ground_truth": "rw [grad_snd_eq h, grad_snd_eq h]",
    "model_output": "rw [grad_eq_block_grad, grad_fun_comp, grad_fun_comp, grad_add_const, grad_add_const]"
  },
  {
    "id": 7,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nx : E\ny : E\ny₁ : E\ny₂ : E\ns : Set E\nf : E → ℝ\ninst✝4 : ProperSpace E\nf : E → ℝ\nx : E\nhc : LowerSemicontinuous f\nlbdf : BddBelow (f '' univ)\n⊢ ∃ y, prox_set f x y",
    "ground_truth": "apply prox_well_define f hc lbdf x",
    "model_output": "rcases (prox_set_aux f x) with ⟨y, hy⟩\n  exact ⟨y, hy.1⟩"
  },
  {
    "id": 8,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nV : Type u_2\nW : Type u_3\ninst✝2 : NormedAddCommGroup V\ninst✝3 : InnerProductSpace 𝕜 V\ninst✝4 : NormedAddCommGroup W\ninst✝5 : InnerProductSpace 𝕜 W\ninst✝6 : FiniteDimensional 𝕜 V\nT : V →ₗ[𝕜] W\ns : Submodule 𝕜 V\nx : V\nhker : sᗮ ≤ LinearMap.ker T\nx₁ : V\nhx₁ : x₁ ∈ s\nx₂ : V\nhx₂ : x₂ ∈ sᗮ\nhx : x = x₁ + x₂\n⊢ ∃ x₁ ∈ s, ∃ x₂ ∈ sᗮ, x = x₁ + x₂ ∧ ‖T x‖ ^ 2 = ‖T x₁‖ ^ 2",
    "ground_truth": "refine ⟨x₁, hx₁, x₂, hx₂, hx, ?_⟩",
    "model_output": "refine ⟨x₁, hx₁, x₂, hx₂, hx, ?_⟩"
  },
  {
    "id": 9,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nV : Type u_2\nW : Type u_3\ninst✝2 : NormedAddCommGroup V\ninst✝3 : InnerProductSpace 𝕜 V\ninst✝4 : NormedAddCommGroup W\ninst✝5 : InnerProductSpace 𝕜 W\ninst✝6 : FiniteDimensional 𝕜 V\nT : V →ₗ[𝕜] W\ns : Submodule 𝕜 V\nx : V\nhker : sᗮ ≤ LinearMap.ker T\nx₁ : V\nhx₁ : x₁ ∈ s\nx₂ : V\nhx₂ : x₂ ∈ sᗮ\nhx : x = x₁ + x₂\n⊢ T x₂ = 0",
    "ground_truth": "exact LinearMap.mem_ker.mp (hker hx₂)",
    "model_output": "apply hker hx₂"
  },
  {
    "id": 10,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nV : Type u_2\ninst✝2 : NormedAddCommGroup V\ninst✝3 : InnerProductSpace 𝕜 V\nι : Type u_4\ns : Set ι\nv : ι → V\nhv : Orthonormal 𝕜 v\n⊢ Orthonormal 𝕜 (s.restrict v)",
    "ground_truth": "simp [Orthonormal, Pairwise] at *",
    "model_output": "apply Orthonormal.mk\n    · intro i\n      simp [Orthonormal]\n      exact hv.1 i\n    · intro i j hij\n      simp [Orthonormal]\n      exact hv.2 i j hij"
  },
  {
    "id": 11,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\n⊢ Function.IsClosed (Function.closure f s) s",
    "ground_truth": "by_cases hsf : ProperFunction s f",
    "model_output": "apply isClosed_closure"
  },
  {
    "id": 12,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "case.1:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ProperFunction s f\n⊢ Function.IsClosed (Function.closure f s) s\n\ncase.2:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\n⊢ Function.IsClosed (Function.closure f s) s",
    "ground_truth": "· exact closure_is_closed_of_proper f hs",
    "model_output": "· exact isClosed_closure_of_proper s hs f"
  },
  {
    "id": 13,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ProperFunction s f\n⊢ Function.IsClosed (Function.closure f s) s",
    "ground_truth": "exact closure_is_closed_of_proper f hs",
    "model_output": "apply closure_isClosed hs"
  },
  {
    "id": 14,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\nhf : ∃ x ∈ s, f x = ⊥\nh1 : Epi ⊥ s = s ×ˢ univ\n⊢ _root_.IsClosed (s ×ˢ univ)",
    "ground_truth": "apply IsClosed.prod hs isClosed_univ",
    "model_output": "apply IsClosed.prod hs isClosed_univ"
  },
  {
    "id": 15,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ns : Set E\nf : E → EReal\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : FiniteDimensional ℝ E\ni : affine_minorant s f\ng : E →L[ℝ] ℝ\nl : E\nw : ℝ\nx : E\n_hx : x ∈ s\n⊢ i.h x = inner ℝ l x + w",
    "ground_truth": "simp [w, l, g, AffineMap.decomp']",
    "model_output": "exact i.hx x _hx"
  },
  {
    "id": 16,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E₁ : Type u_1\nE₂ : Type u_2\nF : Type u_3\ninst✝ : NormedAddCommGroup E₁\ninst✝2 : InnerProductSpace ℝ E₁\ninst✝3 : FiniteDimensional ℝ E₁\ninst✝4 : NormedAddCommGroup E₂\ninst✝5 : InnerProductSpace ℝ E₂\ninst✝6 : FiniteDimensional ℝ E₂\ninst✝7 : NormedAddCommGroup F\ninst✝8 : InnerProductSpace ℝ F\ninst✝9 : FiniteDimensional ℝ F\nadmm : ADMM E₁ E₂ F\nadmm_kkt : Existance_of_kkt admm\nfullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)\nfullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nn : ℕ\n⊢ (↑↑(φ₁ n)).toNat = φ (n + 1)",
    "ground_truth": "rw [φ₁_equ fullrank₁ fullrank₂ n]",
    "model_output": "rw [φ₁_equ fullrank₁ fullrank₂ n]"
  },
  {
    "id": 17,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "b : ℝ\na : ℝ\nha1 : ↑a ≠ ⊥\nha2 : ↑a ≠ ⊤\nc : ℝ\nhc1 : ↑c ≠ ⊥\nh : ↑a + ↑b ≤ ↑c\nhc2 : ¬↑c = ⊤\n⊢ a + b ≤ c",
    "ground_truth": "exact EReal.coe_le_coe_iff.mp h",
    "model_output": "exact EReal.coe_le_coe_iff.mp (le_of_add_le_of_nonneg h (EReal.coe_nonneg.mpr (le_of_lt (lt_of_le_of_ne_bot ha1 ha2))))"
  },
  {
    "id": 18,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝ : RCLike 𝕜\ninst✝2 : NormedAddCommGroup V\ninst✝3 : InnerProductSpace 𝕜 V\ninst✝4 : NormedAddCommGroup W\ninst✝5 : InnerProductSpace 𝕜 W\nT : V →ₗ[𝕜] W\nr : ℕ\ninst✝6 : FiniteDimensional 𝕜 V\nhT : SVD T r\n⊢ r = finrank 𝕜 ↥(range T)",
    "ground_truth": "rw [← hT.span_range_w_eq_range]",
    "model_output": "have h_range : range T = range (toContinuousLinearMap T) := by rfl\n    rw [← h_range, hT.finrank_range_eq]"
  },
  {
    "id": 19,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\nt : Set E\nhscv : Convex ℝ s\nhtcv : Convex ℝ t\nhscl : _root_.IsClosed s\nhtcl : _root_.IsClosed t\nh : ∀ (x : E), s.convex_support x = t.convex_support x\nz : E\n⊢ z ∈ s ↔ z ∈ t",
    "ground_truth": "apply iff_iff_implies_and_implies.mpr",
    "model_output": "rw [← closure_eq_iff_isClosed.mpr hscl, ← closure_eq_iff_isClosed.mpr htcl, ← h]"
  },
  {
    "id": 20,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\ns : Set E\nhs : s.Nonempty\nx : E\na✝ : x ∈ univ\nx✝ : ProperFunction univ s.convex_indicator\n⊢ ⊥ < convex_conjugate univ s.convex_indicator x",
    "ground_truth": "apply convex_conjugate_ge_bot_univ'",
    "model_output": "apply convex_conjugate_ge_zero"
  },
  {
    "id": 21,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "F : Type u_2\ninst✝ : NormedAddCommGroup F\ninst✝2 : CompleteSpace F\nf' : F\nx : F\ns : Set F\ninst✝3 : InnerProductSpace ℝ F\nf : F → ℝ\nc : ℝ\nh : HasGradientWithinAt f f' s x\n⊢ HasGradientWithinAt (fun x => c • f x) (c • f') s x",
    "ground_truth": "exact HasGradientWithinAt.const_smul h c",
    "model_output": "exact HasGradientAtFilter.const_smul h c"
  },
  {
    "id": 22,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\nF : Type u_2\ninst✝ : RCLike 𝕜\ninst✝2 : NormedAddCommGroup F\ninst✝3 : InnerProductSpace 𝕜 F\ninst✝4 : CompleteSpace F\ns : Set F\nf' : F\nx : F\nf : F → 𝕜\nhf : HasGradientWithinAt f f' s x\nc : 𝕜\n⊢ HasGradientWithinAt (fun y => c + f y) f' s x",
    "ground_truth": "exact HasGradientAtFilter.const_add hf c",
    "model_output": "exact HasGradientAtFilter.const_add hf c"
  },
  {
    "id": 23,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\nF : Type u_2\ninst✝ : RCLike 𝕜\ninst✝2 : NormedAddCommGroup F\ninst✝3 : InnerProductSpace 𝕜 F\ninst✝4 : CompleteSpace F\nf' : F\nx : F\nf : F → 𝕜\nhf : HasGradientAt f f' x\nc : 𝕜\n⊢ HasGradientAt (fun x => c + f x) f' x",
    "ground_truth": "exact HasGradientAtFilter.const_add hf c",
    "model_output": "exact HasGradientAtFilter.const_add hf c"
  },
  {
    "id": 24,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\nF : Type u_2\ninst✝ : RCLike 𝕜\ninst✝2 : NormedAddCommGroup F\ninst✝3 : InnerProductSpace 𝕜 F\ninst✝4 : CompleteSpace F\nf' : F\nx : F\nf : F → 𝕜\nhf : HasGradientAt f f' x\nc : 𝕜\n⊢ HasGradientAt (fun x => c - f x) (-f') x",
    "ground_truth": "exact HasGradientAtFilter.const_sub hf c",
    "model_output": "exact HasGradientAtFilter.const_sub hf c"
  },
  {
    "id": 25,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : SMul ℝ E\nf : E → EReal\nh : ∀ (x : E), f x = ⊤\n⊢ ConvexOn ℝ univ f",
    "ground_truth": "apply convex_on_p_top\n  ext x; exact h x",
    "model_output": "apply convex_on_n_inf\n    intro x\n    exact h x"
  },
  {
    "id": 26,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ+\nm : ℕ+\nb : Fin ↑m → ℝ\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\n⊢ ConvexOn ℝ Set.univ fun x => 1 / 2 * ‖A *ᵥ x - b‖ ^ 2",
    "ground_truth": "apply monotone_gradient_convex'",
    "model_output": "apply ConvexOn.smul\n    · exact convexOn_const\n    · apply ConvexOn.sq\n      · apply ConvexOn.norm\n      · apply ConvexOn.sub\n        · apply ConvexOn.smul\n          · exact convexOn_const\n          · apply ConvexOn.prod\n            · apply convexOn_id\n            · apply convexOn_const\n        · apply ConvexOn.const"
  },
  {
    "id": 27,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ+\nm : ℕ+\nb : Fin ↑m → ℝ\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\n⊢ ∀ x ∈ Set.univ, HasGradientAt (fun x => 1 / 2 * ‖A *ᵥ x - b‖ ^ 2) (?f' x) x",
    "ground_truth": "exact (fun x _ => affine_sq_gradient x)",
    "model_output": "intro x _"
  },
  {
    "id": 28,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ\nm : ℕ\nE : Type u_3\ninst✝ : AddCommGroup E\ninst✝2 : Module ℝ E\nB : Basis (Fin (n + m)) ℝ E\nS : Submodule ℝ E\nc : Fin m → ℝ\nA : Matrix (Fin m) (Fin n) ℝ\nhS : ∀ (v : E),\n  v ∈ S ↔\n    let ξ_dep := fun i => (B.repr v) (Fin.natAdd n i);\n    let ξ_indep := fun j => (B.repr v) (Fin.castAdd m j);\n    ξ_dep = c + A.mulVec ξ_indep\nh_zero_mem : 0 ∈ S\n⊢ c = 0",
    "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"
  },
  {
    "id": 29,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\n𝕜 : Type u_3\nf : E → F\ninst✝ : AddCommMonoid E\ninst✝2 : CompleteLinearOrder F\ninst✝3 : Ring 𝕜\ninst✝4 : LinearOrder 𝕜\ninst✝5 : IsStrictOrderedRing 𝕜\ninst✝6 : DenselyOrdered 𝕜\ninst✝7 : Module 𝕜 E\nx : E\ny : E\nhf' : strong_quasi f 𝕜\nhx : IsMinOn f univ x\nhy : IsMinOn f univ y\nneq : ¬x = y\n⊢ False",
    "ground_truth": "have : (0 : 𝕜) < (1 : 𝕜) := one_pos",
    "model_output": "rcases hf' x y neq with h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h"
  },
  {
    "id": 30,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nu : E\nhu : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖\nv : E\nhv : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ v (y - x) ≥ -ε * ‖y - x‖\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nε : ℝ\nεpos : ε > 0\ny : E\nhyu : f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖\nhyv : f y - f x - inner ℝ v (y - x) ≥ -ε * ‖y - x‖\n⊢ f y - f x - inner ℝ (a • u + b • v) (y - x) ≥ -ε * ‖y - x‖",
    "ground_truth": "rw [inner_add_left, real_inner_smul_left, real_inner_smul_left]",
    "model_output": "have : f y - f x - inner ℝ (a • u + b • v) (y - x) = a • (f y - f x - inner ℝ u (y - x)) + b • (f y - f x - inner ℝ v (y - x)) := by\n      rw [inner_add_right, ← sub_sub, ← sub_sub, ← sub_sub, sub_sub, ← smul_sub, ← smul_sub]\n      rw [smul_add, smul_smul, smul_smul, ← add_sub, ← sub_sub, ← sub_sub, ← sub_sub]"
  },
  {
    "id": 31,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nu : E\nhu : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖\nv : E\nhv : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ v (y - x) ≥ -ε * ‖y - x‖\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nε : ℝ\nεpos : ε > 0\ny : E\nhyu : f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖\nhyv : f y - f x - inner ℝ v (y - x) ≥ -ε * ‖y - x‖\n⊢ a * (f y - f x - inner ℝ u (y - x)) + b * (f y - f x - inner ℝ v (y - x)) ≥ -ε * ‖y - x‖",
    "ground_truth": "rw [← one_mul (-ε * ‖y - x‖), ← hab, add_mul]",
    "model_output": "have : a * (f y - f x - inner ℝ u (y - x)) + b * (f y - f x - inner ℝ v (y - x)) ≥ -a * (ε * ‖y - x‖) - b * (ε * ‖y - x‖) := by\n      apply add_le_add\n      · apply mul_le_mul_of_nonneg_left hyu (le_of_lt εpos)\n      · apply mul_le_mul_of_nonneg_left hyv (le_of_lt εpos)"
  },
  {
    "id": 32,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nH : WithLp 2 (E × F) → ℝ\nl : NNReal\nh : Differentiable ℝ H\nhl : ∀ (x y : WithLp 2 (E × F)), ‖gradient H x - gradient H y‖ ≤ ↑l * ‖x - y‖\nz : WithLp 2 (E × F)\nz' : WithLp 2 (E × F)\n⊢ ‖grad_snd H z.1 z.2 - grad_snd H z'.1 z'.2‖ ≤ ↑l * ‖z - z'‖",
    "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,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nf : E → ℝ\ng : F → ℝ\nH : WithLp 2 (E × F) → ℝ\nx0 : E\ny0 : F\nl : NNReal\nalg : BCD f g H l x0 y0\nγ : ℝ\nhγ : γ > 1\nck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)\n⊢ ∀ (k : ℕ), 0 < alg.c k",
    "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; specialize ck k; rw [ck]; apply div_pos; repeat linarith [alg.lpos]"
  },
  {
    "id": 34,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nf : E → ℝ\ng : F → ℝ\nH : WithLp 2 (E × F) → ℝ\nx0 : E\ny0 : F\nl : NNReal\nalg : BCD f g H l x0 y0\nγ : ℝ\nhγ : γ > 1\nck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)\n⊢ ∀ (k : ℕ), 0 < alg.c k",
    "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; specialize ck k; rw [ck]; apply div_pos; repeat linarith [alg.lpos]"
  },
  {
    "id": 35,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nx : E\ny : E\ny₁ : E\ny₂ : E\ns : Set E\nf : E → ℝ\ninst✝4 : ProperSpace E\nf : E → ℝ\nx : E\nhc : LowerSemicontinuous f\nlbdf : BddBelow (f '' univ)\n⊢ ∃ y, prox_set f x y",
    "ground_truth": "by\n    apply prox_well_define f hc lbdf x",
    "model_output": "rcases (prox_set_aux f x) with ⟨y, hy⟩\n  exact ⟨y, hy.1⟩"
  },
  {
    "id": 36,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nm : Type u_2\nn : Type u_3\ninst✝2 : Fintype m\ninst✝3 : Fintype n\ninst✝4 : DecidableEq n\nA : Matrix m n 𝕜\nh : Aᴴ * A = 1\n⊢ finrank 𝕜 ↥A.colSubmodule = Fintype.card n",
    "ground_truth": "by\n  unfold colSubmodule\n  rw [finrank_span_eq_card]\n  simp [h]",
    "model_output": "have : Aᴴ * A = 1 := h\n    rw [colSubmodule_eq_span_range, finrank_span_eq_card]\n    simp [Matrix.range_mul_conjTranspose_eq_span_row, h]"
  },
  {
    "id": 37,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nm : Type u_2\nn : Type u_3\ninst✝2 : Fintype m\ninst✝3 : Fintype n\ninst✝4 : DecidableEq n\nA : Matrix m n 𝕜\nh : Aᴴ * A = 1\n⊢ finrank 𝕜 ↥A.colSubmodule = Fintype.card n",
    "ground_truth": "unfold colSubmodule\n  rw [finrank_span_eq_card]\n  simp [h]",
    "model_output": "have : Aᴴ * A = 1 := h\n    rw [colSubmodule_eq_span_range, finrank_span_eq_card]\n    simp [Matrix.range_mul_conjTranspose_eq_span_row, h]"
  },
  {
    "id": 38,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nV : Type u_2\nW : Type u_3\ninst✝2 : NormedAddCommGroup V\ninst✝3 : InnerProductSpace 𝕜 V\ninst✝4 : NormedAddCommGroup W\ninst✝5 : InnerProductSpace 𝕜 W\ninst✝6 : FiniteDimensional 𝕜 V\nT : V →ₗ[𝕜] W\ns : Submodule 𝕜 V\nx : V\nhker : sᗮ ≤ LinearMap.ker T\n⊢ ∃ x₁ ∈ s, ∃ x₂ ∈ sᗮ, x = x₁ + x₂ ∧ ‖T x‖ ^ 2 = ‖T x₁‖ ^ 2",
    "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 hker' : s ≤ (LinearMap.ker T)ᗮ := by\n      intro x hx\n      simp [mem_orthogonal]\n      intro y hy\n      have : y ∈ sᗮ := by simp [hx]\n      have : T y = 0 := by simp [hker y this]\n      simp [this]"
  },
  {
    "id": 39,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nV : Type u_2\nW : Type u_3\ninst✝2 : NormedAddCommGroup V\ninst✝3 : InnerProductSpace 𝕜 V\ninst✝4 : NormedAddCommGroup W\ninst✝5 : InnerProductSpace 𝕜 W\ninst✝6 : FiniteDimensional 𝕜 V\nT : V →ₗ[𝕜] W\ns : Submodule 𝕜 V\nx : V\nhker : sᗮ ≤ LinearMap.ker T\nx₁ : V\nhx₁ : x₁ ∈ s\nx₂ : V\nhx₂ : x₂ ∈ sᗮ\nhx : x = x₁ + x₂\n⊢ ‖T x₁ + T x₂‖ ^ 2 = ‖T x₁‖ ^ 2",
    "ground_truth": "suffices T x₂ = 0 from by\n    rw [this, add_zero, norm_sq_eq_re_inner (𝕜:=𝕜)]",
    "model_output": "have : T x₂ = 0 := by\n      have : x₂ ∈ LinearMap.ker T := hker hx₂\n      simp [LinearMap.mem_ker] at this\n      exact this"
  },
  {
    "id": 40,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nV : Type u_2\nW : Type u_3\ninst✝2 : NormedAddCommGroup V\ninst✝3 : InnerProductSpace 𝕜 V\ninst✝4 : NormedAddCommGroup W\ninst✝5 : InnerProductSpace 𝕜 W\ninst✝6 : FiniteDimensional 𝕜 V\nT : V →ₗ[𝕜] W\ns : Submodule 𝕜 V\nx : V\nhker : sᗮ ≤ LinearMap.ker T\nx₁ : V\nhx₁ : x₁ ∈ s\nx₂ : V\nhx₂ : x₂ ∈ sᗮ\nhx : x = x₁ + x₂\nthis : T x₂ = 0\n⊢ ‖T x₁ + T x₂‖ ^ 2 = ‖T x₁‖ ^ 2",
    "ground_truth": "by\n    rw [this, add_zero, norm_sq_eq_re_inner (𝕜:=𝕜)]",
    "model_output": "rw [this, add_zero]"
  },
  {
    "id": 41,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nV : Type u_2\nW : Type u_3\ninst✝2 : NormedAddCommGroup V\ninst✝3 : InnerProductSpace 𝕜 V\ninst✝4 : NormedAddCommGroup W\ninst✝5 : InnerProductSpace 𝕜 W\ninst✝6 : FiniteDimensional 𝕜 V\nT : V →ₗ[𝕜] W\ns : Submodule 𝕜 V\nx : V\nhker : sᗮ ≤ LinearMap.ker T\nx₁ : V\nhx₁ : x₁ ∈ s\nx₂ : V\nhx₂ : x₂ ∈ sᗮ\nhx : x = x₁ + x₂\nthis : T x₂ = 0\n⊢ ‖T x₁ + T x₂‖ ^ 2 = ‖T x₁‖ ^ 2",
    "ground_truth": "rw [this, add_zero, norm_sq_eq_re_inner (𝕜:=𝕜)]",
    "model_output": "rw [this, add_zero]"
  },
  {
    "id": 42,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nV : Type u_2\ninst✝2 : NormedAddCommGroup V\ninst✝3 : InnerProductSpace 𝕜 V\nι : Type u_4\ns : Set ι\nv : ι → V\nhv : Orthonormal 𝕜 v\n⊢ Orthonormal 𝕜 (s.restrict v)",
    "ground_truth": "by\n  simp [Orthonormal, Pairwise] at *\n  aesop",
    "model_output": "apply Orthonormal.mk\n    · intro i\n      simp [Orthonormal]\n      exact hv.1 i\n    · intro i j hij\n      simp [Orthonormal]\n      exact hv.2 i j hij"
  },
  {
    "id": 43,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nV : Type u_2\ninst✝2 : NormedAddCommGroup V\ninst✝3 : InnerProductSpace 𝕜 V\nι : Type u_4\ns : Set ι\nv : ι → V\nhv : Orthonormal 𝕜 v\n⊢ Orthonormal 𝕜 (s.restrict v)",
    "ground_truth": "simp [Orthonormal, Pairwise] at *\n  aesop",
    "model_output": "apply Orthonormal.mk\n    · intro i\n      simp [Orthonormal]\n      exact hv.1 i\n    · intro i j hij\n      simp [Orthonormal]\n      exact hv.2 i j hij"
  },
  {
    "id": 44,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nx : E\nε : ℝ\nf : E → ℝ\nhe : ε > 0\nhf : ContDiffOn ℝ 1 f (Metric.ball x ε)\n⊢ ContinuousAt (fun x => gradient f x) x",
    "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": "have : ContDiffOn ℝ 1 f (Metric.ball x ε) := hf\n   have : ContDiffOn ℝ 0 (fun x => gradient f x) (Metric.ball x ε) := by\n      apply ContDiff.contDiffOn; apply ContDiff.gradient_of_contDiff\n      exact ContDiff.continuousOn this\n   exact this.continuousAt (Metric.mem_ball_self he)"
  },
  {
    "id": 45,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nx : E\nε : ℝ\nf : E → ℝ\nhe : ε > 0\nhf : ContDiffOn ℝ 1 f (Metric.ball x ε)\n⊢ ContinuousAt (fun x => gradient f x) x",
    "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": "have : ContDiffOn ℝ 1 f (Metric.ball x ε) := hf\n   have : ContDiffOn ℝ 0 (fun x => gradient f x) (Metric.ball x ε) := by\n      apply ContDiff.contDiffOn; apply ContDiff.gradient_of_contDiff\n      exact ContDiff.continuousOn this\n   exact this.continuousAt (Metric.mem_ball_self he)"
  },
  {
    "id": 46,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nx : E\nε : ℝ\nf : E → ℝ\nhe : ε > 0\nhf : ContDiffOn ℝ 1 f (Metric.ball x ε)\ns : Set E\n⊢ ContinuousAt (fun x => gradient f x) x",
    "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 HasGradientAtFilter\n        · apply HasGradientAtFilterFilter\n          · apply HasGradientAtFilterFilterFilter\n            · apply HasGradientAtFilterFilterFilterFilter\n              · apply HasGradientAtFilterFilterFilterFilterFilter\n                · apply HasGradientAtFilterFilterFilterFilterFilterFilter\n                  · apply HasGradientAtFilterFilterFilterFilterFilterFilterFilter\n                    · apply HasGradientAtFilterFilterFilterFilterFilterFilterFilterFilter\n                      · apply HasGradientAtFilterFilterFilterFilterFilterFilterFilterFilterFilter"
  },
  {
    "id": 47,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nx : E\nε : ℝ\nf : E → ℝ\nhe : ε > 0\nhf : ContDiffOn ℝ 1 f (Metric.ball x ε)\ns : Set E\n⊢ s ⊆ Metric.ball x ε ∧ IsOpen s ∧ x ∈ s",
    "ground_truth": "exact ⟨Eq.subset rfl, ⟨Metric.isOpen_ball, Metric.mem_ball_self he⟩⟩",
    "model_output": "constructor"
  },
  {
    "id": 48,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nx : E\nε : ℝ\nf : E → ℝ\nhe : ε > 0\nhf : ContDiffOn ℝ 1 f (Metric.ball x ε)\ns : Set E\nh : ContDiffAt ℝ 1 f x\n⊢ ContinuousAt (fun x => gradient f x) x",
    "ground_truth": "exact gradient_continuous_of_contdiffat f h",
    "model_output": "apply HasGradientAt.continuousAt\n    · apply h.hasGradientAt\n    · apply ContDiffAt.continuousAt\n      apply ContDiffAt.comp (g := gradient) (f := f) (x := x) (ContDiffAt.const x 1) h\n      apply ContDiffAt.continuousAt\n      apply ContDiffAt.const\n      apply ContDiffAt.id'"
  },
  {
    "id": 49,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : TopologicalSpace E\ninst✝2 : AddCommMonoid E\n⊢ Function.closure ⊤ univ = ⊤",
    "ground_truth": "by\n  simp [Function.closure]\n  intro h; exfalso\n  obtain h1 := h.2; simp at h1",
    "model_output": "apply Function.closure_univ_eq_top.mpr\n    exact fun x _ ↦ trivial"
  },
  {
    "id": 50,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : TopologicalSpace E\ninst✝2 : AddCommMonoid E\n⊢ Function.closure ⊤ univ = ⊤",
    "ground_truth": "simp [Function.closure]\n  intro h; exfalso\n  obtain h1 := h.2; simp at h1",
    "model_output": "apply Function.closure_univ_eq_top.mpr\n    exact fun x _ ↦ trivial"
  },
  {
    "id": 51,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "case.1:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\nhf : ∃ x ∈ s, f x = ⊥\n⊢ Function.IsClosed (if ∃ x ∈ s, f x = ⊥ then ⊥ else ⊤) s\n\ncase.2:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\nhf : ¬∃ x ∈ s, f x = ⊥\n⊢ Function.IsClosed (if ∃ x ∈ s, f x = ⊥ then ⊥ else ⊤) s",
    "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": 52,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\nhf : ∃ x ∈ s, f x = ⊥\n⊢ Function.IsClosed (if ∃ x ∈ s, f x = ⊥ then ⊥ else ⊤) s",
    "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,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\nhf : ∃ x ∈ s, f x = ⊥\n⊢ _root_.IsClosed (Epi ⊥ s)",
    "ground_truth": "have h1 : Epi ⊥ s = s ×ˢ (univ : Set ℝ) := by\n      ext ⟨x, _⟩\n      simp [Function.Epi]",
    "model_output": "apply epi_closed_of_bot hf hs"
  },
  {
    "id": 54,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\nhf : ¬∃ x ∈ s, f x = ⊥\n⊢ _root_.IsClosed (Epi ⊤ s)",
    "ground_truth": "have h' : Epi ⊤ s = ∅ := by\n    ext ⟨x, _⟩; simp [Function.Epi]",
    "model_output": "simp [Epi]"
  },
  {
    "id": 55,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E\nx : E\n⊢ HasSubgradientAt f g x ↔ Banach_HasSubgradientAt f ((toDual ℝ E) g) x",
    "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\n      unfold Banach_HasSubgradientAt\n      unfold HasSubgradientAt at h\n      specialize h x\n      simp at h\n      exact h\n    · intro h\n      unfold HasSubgradientAt\n      unfold Banach_HasSubgradientAt at h\n      specialize h g\n      simp at h\n      exact h"
  },
  {
    "id": 56,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E\nx : E\n⊢ HasSubgradientAt f g x ↔ Banach_HasSubgradientAt f ((toDual ℝ E) g) x",
    "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\n      unfold Banach_HasSubgradientAt\n      unfold HasSubgradientAt at h\n      specialize h x\n      simp at h\n      exact h\n    · intro h\n      unfold HasSubgradientAt\n      unfold Banach_HasSubgradientAt at h\n      specialize h g\n      simp at h\n      exact h"
  },
  {
    "id": 57,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "case.1:\nE : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E\nx : E\n⊢ HasSubgradientAt f g x → Banach_HasSubgradientAt f ((toDual ℝ E) g) x\n\ncase.2:\nE : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E\nx : E\n⊢ Banach_HasSubgradientAt f ((toDual ℝ E) g) x → HasSubgradientAt f g x",
    "ground_truth": "· intro h z\n    unfold HasSubgradientAt at h\n    obtain hz := h z\n    simp; linarith",
    "model_output": "· intro h\n    unfold Banach_HasSubgradientAt\n    unfold HasSubgradientAt at h\n    apply h\n    intro y\n    rw [← real_inner_comm, ← toDual_apply]\n    apply h y"
  },
  {
    "id": 58,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E\nx : E\n⊢ HasSubgradientAt f g x → Banach_HasSubgradientAt f ((toDual ℝ E) g) x",
    "ground_truth": "intro h z\n    unfold HasSubgradientAt at h\n    obtain hz := h z\n    simp; linarith",
    "model_output": "intro h\n    unfold Banach_HasSubgradientAt\n    unfold HasSubgradientAt at h\n    apply h\n    intro y\n    apply h y"
  },
  {
    "id": 59,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "case.1:\nE : Type u_1\ns : Set E\nf : E → EReal\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : FiniteDimensional ℝ E\ni : affine_minorant s f\ng : E →L[ℝ] ℝ\nl : E\nw : ℝ\n⊢ ∀ x ∈ s, ↑(inner ℝ l x) + ↑w ≤ f x\n\ncase.2:\nE : Type u_1\ns : Set E\nf : E → EReal\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : FiniteDimensional ℝ E\ni : affine_minorant s f\ng : E →L[ℝ] ℝ\nl : E\nw : ℝ\n⊢ ∀ x ∈ s, i.h x = inner ℝ l x + w",
    "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 _\n    exact i.hf x"
  },
  {
    "id": 60,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ns : Set E\nf : E → EReal\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : FiniteDimensional ℝ E\ni : affine_minorant s f\ng : E →L[ℝ] ℝ\nl : E\nw : ℝ\n⊢ ∀ x ∈ s, ↑(inner ℝ l x) + ↑w ≤ f x",
    "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 _; exact i.2 x"
  },
  {
    "id": 61,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E₁ : Type u_1\nE₂ : Type u_2\nF : Type u_3\ninst✝ : NormedAddCommGroup E₁\ninst✝2 : InnerProductSpace ℝ E₁\ninst✝3 : FiniteDimensional ℝ E₁\ninst✝4 : NormedAddCommGroup E₂\ninst✝5 : InnerProductSpace ℝ E₂\ninst✝6 : FiniteDimensional ℝ E₂\ninst✝7 : NormedAddCommGroup F\ninst✝8 : InnerProductSpace ℝ F\ninst✝9 : FiniteDimensional ℝ F\nadmm : ADMM E₁ E₂ F\nadmm_kkt : Existance_of_kkt admm\nfullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)\nfullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\n⊢ Tendsto (fun n => (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑(φ₁ n) + 1) - ADMM.x₂ E₁ F ↑(φ₁ n))) atTop (𝓝 0)",
    "ground_truth": "apply Filter.tendsto_iff_seq_tendsto.1 Seq_converge_zero₃",
    "model_output": "apply Filter.tendsto_iff_seq_tendsto.1 Seq_converge_zero₂\n[USER]\napply Filter.tendsto_iff_seq_tendsto.1\n   apply Seq_converge_zero₂\n   apply A₂_continuous\n   apply x₂_subseq_converge₂\n   apply A₁_continuous\n   apply x₁_subseq_converge₁"
  },
  {
    "id": 62,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E₁ : Type u_1\nE₂ : Type u_2\nF : Type u_3\ninst✝ : NormedAddCommGroup E₁\ninst✝2 : InnerProductSpace ℝ E₁\ninst✝3 : FiniteDimensional ℝ E₁\ninst✝4 : NormedAddCommGroup E₂\ninst✝5 : InnerProductSpace ℝ E₂\ninst✝6 : FiniteDimensional ℝ E₂\ninst✝7 : NormedAddCommGroup F\ninst✝8 : InnerProductSpace ℝ F\ninst✝9 : FiniteDimensional ℝ F\nadmm : ADMM E₁ E₂ F\nadmm_kkt : Existance_of_kkt admm\nfullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)\nfullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\n⊢ StrictMono fun n => ↑(φ₁ n)",
    "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         _ ≥ n + 1 := by apply Nat.le_of_add_one_le\n         _ > n := by apply Nat.lt_add_one\n   exact this"
  },
  {
    "id": 63,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "b : ℝ\na : ℝ\nha1 : ↑a ≠ ⊥\nha2 : ↑a ≠ ⊤\nc : ℝ\nhc1 : ↑c ≠ ⊥\nh : ↑a + ↑b ≤ ↑c\nhc2 : ¬↑c = ⊤\n⊢ 0 ≤ ↑c - ↑a - ↑b",
    "ground_truth": "suffices 0 ≤ (c - a - b).toEReal by exact this",
    "model_output": "apply le_trans h\n        apply EReal.coe_le_coe_iff.mpr\n        exact sub_nonneg_of_le (add_le_add_right (le_of_eq rfl) b)"
  },
  {
    "id": 64,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "b : ℝ\na : ℝ\nha1 : ↑a ≠ ⊥\nha2 : ↑a ≠ ⊤\nc : ℝ\nhc1 : ↑c ≠ ⊥\nh : ↑a + ↑b ≤ ↑c\nhc2 : ¬↑c = ⊤\n⊢ 0 ≤ ↑(c - a - b)",
    "ground_truth": "have : a + b ≤ c := by exact EReal.coe_le_coe_iff.mp h",
    "model_output": "apply le_of_lt\n    simp at *\n    exact sub_pos_of_lt (lt_of_le_of_lt h (EReal.coe_lt_top c))"
  },
  {
    "id": 65,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "b : ℝ\na : ℝ\nha1 : ↑a ≠ ⊥\nha2 : ↑a ≠ ⊤\nc : ℝ\nhc1 : ↑c ≠ ⊥\nh : ↑a + ↑b ≤ ↑c\nhc2 : ¬↑c = ⊤\nthis : a + b ≤ c\n⊢ 0 ≤ ↑(c - a - b)",
    "ground_truth": "refine EReal.coe_nonneg.mpr ?intro.intro.a",
    "model_output": "apply EReal.coe_nonneg.mpr"
  },
  {
    "id": 66,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ns : Set E\nf : E → ℝ\nm : ℝ\nhs : Convex ℝ s\nhfun : ∀ ⦃x : E⦄,\n  x ∈ s →\n    ∀ ⦃y : E⦄,\n      y ∈ s →\n        ∀ ⦃a : ℝ⦄,\n          0 ≤ a → ∀ ⦃b : ℝ⦄, 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2\n⊢ StrongConvexOn s m f",
    "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_iff_convex.mpr\n  intro x hx y hy a ha b hb hab\n  specialize hfun hx hy a ha b hb hab\n  rw [add_comm] at hab\n  rw [add_comm] at hfun\n  linarith [hfun]"
  },
  {
    "id": 67,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ns : Set E\nf : E → ℝ\nm : ℝ\nhs : Convex ℝ s\nhfun : ∀ ⦃x : E⦄,\n  x ∈ s →\n    ∀ ⦃y : E⦄,\n      y ∈ s →\n        ∀ ⦃a : ℝ⦄,\n          0 ≤ a → ∀ ⦃b : ℝ⦄, 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2\n⊢ StrongConvexOn s m f",
    "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_iff_convex.mpr\n  intro x hx y hy a ha b hb hab\n  specialize hfun hx hy a ha b hb hab\n  rw [add_comm] at hab\n  rw [add_comm] at hfun\n  linarith [hfun]"
  },
  {
    "id": 68,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ns : Set E\nf : E → ℝ\nm : ℝ\nhs : Convex ℝ s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhfun : f (a • x + b • y) ≤ a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2\n⊢ f (a • x + b • y) ≤ a * f x + b * f y - a * b * (m / 2 * ‖x - y‖ ^ 2)",
    "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,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝ : RCLike 𝕜\ninst✝2 : NormedAddCommGroup V\ninst✝3 : InnerProductSpace 𝕜 V\ninst✝4 : NormedAddCommGroup W\ninst✝5 : InnerProductSpace 𝕜 W\nT : V →ₗ[𝕜] W\nr : ℕ\ninst✝6 : FiniteDimensional 𝕜 V\nhT : SVD T r\n⊢ r = finrank 𝕜 ↥(range T)",
    "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_range : range T = range (toContinuousLinearMap T) := by rfl\n    rw [← h_range, hT.finrank_range_eq]"
  },
  {
    "id": 70,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝ : RCLike 𝕜\ninst✝2 : NormedAddCommGroup V\ninst✝3 : InnerProductSpace 𝕜 V\ninst✝4 : NormedAddCommGroup W\ninst✝5 : InnerProductSpace 𝕜 W\nT : V →ₗ[𝕜] W\nr : ℕ\ninst✝6 : FiniteDimensional 𝕜 V\nhT : SVD T r\n⊢ r = finrank 𝕜 ↥(range T)",
    "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_range : range T = range (toContinuousLinearMap T) := by rfl\n    rw [← h_range, hT.finrank_range_eq]"
  },
  {
    "id": 71,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝ : RCLike 𝕜\ninst✝2 : NormedAddCommGroup V\ninst✝3 : InnerProductSpace 𝕜 V\ninst✝4 : NormedAddCommGroup W\ninst✝5 : InnerProductSpace 𝕜 W\nT : V →ₗ[𝕜] W\nr : ℕ\ninst✝6 : FiniteDimensional 𝕜 V\nhT : SVD T r\n⊢ r = finrank 𝕜 ↥(span 𝕜 (Set.range hT.w))",
    "ground_truth": "simp [finrank_span_eq_card (b:=hT.w) hT.w_orthonormal.linearIndependent]",
    "model_output": "rw [hT.finrank_range_w]"
  },
  {
    "id": 72,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "case.1:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\nt : Set E\nhscv : Convex ℝ s\nhtcv : Convex ℝ t\nhscl : _root_.IsClosed s\nhtcl : _root_.IsClosed t\nh : ∀ (x : E), s.convex_support x = t.convex_support x\nz : E\n⊢ z ∈ s → z ∈ t\n\ncase.2:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\nt : Set E\nhscv : Convex ℝ s\nhtcv : Convex ℝ t\nhscl : _root_.IsClosed s\nhtcl : _root_.IsClosed t\nh : ∀ (x : E), s.convex_support x = t.convex_support x\nz : E\n⊢ z ∈ t → z ∈ s",
    "ground_truth": "· exact fun a ↦ set_inclusion_convex_support_eq s t htcv htcl a h",
    "model_output": "· intro zin; specialize h z; rw [convex_support_of_mem zin, convex_support_of_mem] at h; simp at h; exact h"
  },
  {
    "id": 73,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\nt : Set E\nhscv : Convex ℝ s\nhtcv : Convex ℝ t\nhscl : _root_.IsClosed s\nhtcl : _root_.IsClosed t\nh : ∀ (x : E), s.convex_support x = t.convex_support x\nz : E\n⊢ z ∈ s → z ∈ t",
    "ground_truth": "exact fun a ↦ set_inclusion_convex_support_eq s t htcv htcl a h",
    "model_output": "intro hz\n    have : s.convex_support z = t.convex_support z := h z\n    rw [convex_support_of_mem hz, convex_support_of_mem] at this\n    exact this.mp rfl"
  },
  {
    "id": 74,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\nt : Set E\nhscv : Convex ℝ s\nhtcv : Convex ℝ t\nhscl : _root_.IsClosed s\nhtcl : _root_.IsClosed t\nh : ∀ (x : E), s.convex_support x = t.convex_support x\nz : E\n⊢ z ∈ t → z ∈ s",
    "ground_truth": "exact fun a ↦ set_inclusion_convex_support_eq t s hscv hscl a (fun x => (h x).symm)",
    "model_output": "intro zint\n    rw [← h z]\n    apply convex_support_subset hscv htcv h zint"
  },
  {
    "id": 75,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\ns : Set E\nhs : s.Nonempty\n⊢ ∀ x ∈ univ, s.convex_support x > ⊥",
    "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 _\n    rw [convex_support]\n    simp [hs]\n    exact bot_lt_top"
  },
  {
    "id": 76,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\ns : Set E\nhs : s.Nonempty\n⊢ ∀ x ∈ univ, s.convex_support x > ⊥",
    "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 _\n    rw [convex_support]\n    simp [hs]\n    exact bot_lt_top"
  },
  {
    "id": 77,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\ns : Set E\nhs : s.Nonempty\n⊢ ∃ x, convex_conjugate univ s.convex_indicator x < ⊤",
    "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 x\n    simp [convex_conjugate]\n    rw[← hx]\n    apply lt_of_le_of_lt\n    · apply iSup₂_le\n      intro i _\n      apply le_iSup₂_of_le i (by simp)\n      apply le_iSup₂_of_le (by simp) (by simp)\n    · exact convex_indicator_univ_convex s"
  },
  {
    "id": 78,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\ns : Set E\nhs : s.Nonempty\n⊢ convex_conjugate univ s.convex_indicator 0 ≤ 0",
    "ground_truth": "refine (conjugate_le_zero_iff s.convex_indicator).mpr ?_",
    "model_output": "apply convex_conjugate_le_zero\n    · simp\n    · exact hs.convex_indicator\n    · simp"
  },
  {
    "id": 79,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\ns : Set E\nhs : s.Nonempty\nconvex_conjugate_at_zero : convex_conjugate univ s.convex_indicator 0 ≤ 0\n⊢ convex_conjugate univ s.convex_indicator 0 < ⊤",
    "ground_truth": "exact lt_of_le_of_lt (convex_conjugate_at_zero) (zero_lt_top)",
    "model_output": "apply lt_of_le_of_lt (convex_conjugate_at_zero) zero_lt_top"
  },
  {
    "id": 80,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "F : Type u_2\ninst✝ : NormedAddCommGroup F\ninst✝2 : CompleteSpace F\nf' : F\nx : F\ns : Set F\ninst✝3 : InnerProductSpace ℝ F\nf : F → ℝ\nc : ℝ\nh : HasGradientWithinAt f f' s x\n⊢ HasGradientWithinAt (fun x => c • f x) (c • f') s x",
    "ground_truth": "by\n  exact HasGradientWithinAt.const_smul h c",
    "model_output": "exact HasGradientAtFilter.const_smul h c"
  },
  {
    "id": 81,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\nF : Type u_2\ninst✝ : RCLike 𝕜\ninst✝2 : NormedAddCommGroup F\ninst✝3 : InnerProductSpace 𝕜 F\ninst✝4 : CompleteSpace F\ns : Set F\nf' : F\nx : F\nf : F → 𝕜\nhf : HasGradientWithinAt f f' s x\nc : 𝕜\n⊢ HasGradientWithinAt (fun y => c + f y) f' s x",
    "ground_truth": "by\n  exact HasGradientAtFilter.const_add hf c",
    "model_output": "exact HasGradientAtFilter.const_add hf c"
  },
  {
    "id": 82,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\nF : Type u_2\ninst✝ : RCLike 𝕜\ninst✝2 : NormedAddCommGroup F\ninst✝3 : InnerProductSpace 𝕜 F\ninst✝4 : CompleteSpace F\nf' : F\nx : F\nf : F → 𝕜\nhf : HasGradientAt f f' x\nc : 𝕜\n⊢ HasGradientAt (fun x => c + f x) f' x",
    "ground_truth": "by\n  exact HasGradientAtFilter.const_add hf c",
    "model_output": "exact HasGradientAtFilter.const_add hf c"
  },
  {
    "id": 83,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\nF : Type u_2\ninst✝ : RCLike 𝕜\ninst✝2 : NormedAddCommGroup F\ninst✝3 : InnerProductSpace 𝕜 F\ninst✝4 : CompleteSpace F\nx : F\nf : F → 𝕜\nc : 𝕜\n⊢ ∇ (fun y => f y - c) x = ∇ f x",
    "ground_truth": "by\n  simp only [sub_eq_add_neg, gradient_add_const]",
    "model_output": "rw [sub_eq_add_neg, ← gradient_neg_const, gradient_add_const]"
  },
  {
    "id": 84,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\nF : Type u_2\ninst✝ : RCLike 𝕜\ninst✝2 : NormedAddCommGroup F\ninst✝3 : InnerProductSpace 𝕜 F\ninst✝4 : CompleteSpace F\nx : F\nf : F → 𝕜\nc : 𝕜\n⊢ ∇ (fun y => f y - c) x = ∇ f x",
    "ground_truth": "simp only [sub_eq_add_neg, gradient_add_const]",
    "model_output": "rw [sub_eq_add_neg, ← gradient_neg_const, gradient_add_const]"
  },
  {
    "id": 85,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\nF : Type u_2\ninst✝ : RCLike 𝕜\ninst✝2 : NormedAddCommGroup F\ninst✝3 : InnerProductSpace 𝕜 F\ninst✝4 : CompleteSpace F\nf' : F\nx : F\nf : F → 𝕜\nhf : HasGradientAt f f' x\nc : 𝕜\n⊢ HasGradientAt (fun x => c - f x) (-f') x",
    "ground_truth": "by\n  exact HasGradientAtFilter.const_sub hf c",
    "model_output": "exact HasGradientAtFilter.const_sub hf c"
  },
  {
    "id": 86,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "α : Type u_1\ninst✝ : AddCommMonoid α\nf : α → EReal\nh : ProperFunction univ f\n⊢ (dom univ f).Nonempty",
    "ground_truth": "by\n  obtain h1 := h.2; simp at h1\n  rcases h1 with ⟨x, hx⟩; use x; simp [hx]",
    "model_output": "have : (dom univ f).Nonempty := by\n      use h.default\n      simp [dom]"
  },
  {
    "id": 87,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "α : Type u_1\ninst✝ : AddCommMonoid α\nf : α → EReal\nh : ProperFunction univ f\n⊢ (dom univ f).Nonempty",
    "ground_truth": "obtain h1 := h.2; simp at h1\n  rcases h1 with ⟨x, hx⟩; use x; simp [hx]",
    "model_output": "have : (dom univ f).Nonempty := by\n      use h.default\n      simp [dom]"
  },
  {
    "id": 88,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : SMul ℝ E\nf : E → EReal\nh : ∀ (x : E), f x = ⊤\n⊢ ConvexOn ℝ univ f",
    "ground_truth": "by\n  apply convex_on_p_top\n  ext x; exact h x",
    "model_output": "apply convex_on_n_inf\n    intro x\n    exact h x"
  },
  {
    "id": 89,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ+\nm : ℕ+\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\nu : Fin ↑n → ℝ\nv : Fin ↑n → ℝ\n⊢ A *ᵥ u - A *ᵥ v = A *ᵥ (u - v)",
    "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,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ+\nm : ℕ+\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\nu : Fin ↑n → ℝ\nv : Fin ↑n → ℝ\n⊢ A *ᵥ u - A *ᵥ v = A *ᵥ (u - v)",
    "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,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "case.1:\nn : ℕ+\nm : ℕ+\nb : Fin ↑m → ℝ\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\n⊢ ∀ x ∈ Set.univ, HasGradientAt (fun x => 1 / 2 * ‖A *ᵥ x - b‖ ^ 2) (?f' x) x\n\ncase.2:\nn : ℕ+\nm : ℕ+\nb : Fin ↑m → ℝ\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\n⊢ ∀ x ∈ Set.univ, ∀ y ∈ Set.univ, inner ℝ (?f' x - ?f' y) (x - y) ≥ 0\n\ncase.3:\nn : ℕ+\nm : ℕ+\nb : Fin ↑m → ℝ\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\n⊢ EuclideanSpace ℝ (Fin ↑n) → EuclideanSpace ℝ (Fin ↑n)",
    "ground_truth": "· exact (fun x _ => affine_sq_gradient x)",
    "model_output": "apply fun x _ => one_div_two_mul_norm_sq_gradient A b x"
  },
  {
    "id": 92,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ+\nm : ℕ+\nb : Fin ↑m → ℝ\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\n⊢ ∀ x ∈ Set.univ, ∀ y ∈ Set.univ, inner ℝ (Aᵀ *ᵥ (A *ᵥ x - b) - Aᵀ *ᵥ (A *ᵥ y - b)) (x - y) ≥ 0",
    "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,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ+\nm : ℕ+\nb : Fin ↑m → ℝ\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\n⊢ ∀ x ∈ Set.univ, ∀ y ∈ Set.univ, inner ℝ (Aᵀ *ᵥ (A *ᵥ x - b) - Aᵀ *ᵥ (A *ᵥ y - b)) (x - y) ≥ 0",
    "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,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ+\nm : ℕ+\nb : Fin ↑m → ℝ\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\nx : EuclideanSpace ℝ (Fin ↑n)\na✝ : x ∈ Set.univ\ny : EuclideanSpace ℝ (Fin ↑n)\na✝2 : y ∈ Set.univ\n⊢ inner ℝ (Aᵀ *ᵥ (A *ᵥ x - b) - Aᵀ *ᵥ (A *ᵥ y - b)) (x - y) ≥ 0",
    "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_add_eq_add_sub, sub_add, sub_sub, sub_sub, ← mulVec_sub, ← mulVec_sub]"
  },
  {
    "id": 95,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ+\nm : ℕ+\nb : Fin ↑m → ℝ\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\nx : EuclideanSpace ℝ (Fin ↑n)\na✝ : x ∈ Set.univ\ny : EuclideanSpace ℝ (Fin ↑n)\na✝2 : y ∈ Set.univ\n⊢ Aᵀ *ᵥ (A *ᵥ x - A *ᵥ y) ⬝ᵥ (x - y) ≥ 0",
    "ground_truth": "rw [← dot_mul_eq_transpose_mul_dot,← Matrix.mulVec_sub, ← norm2eq_dot]",
    "model_output": "rw [← Matrix.mulVec_sub, ← Matrix.mulVec_sub, sub_eq_add_neg, sub_eq_add_neg]"
  },
  {
    "id": 96,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ\nm : ℕ\nE : Type u_3\ninst✝ : AddCommGroup E\ninst✝2 : Module ℝ E\nB : Basis (Fin (n + m)) ℝ E\nS : Submodule ℝ E\nc : Fin m → ℝ\nA : Matrix (Fin m) (Fin n) ℝ\nhS : ∀ (v : E),\n  v ∈ S ↔\n    let ξ_dep := fun i => (B.repr v) (Fin.natAdd n i);\n    let ξ_indep := fun j => (B.repr v) (Fin.castAdd m j);\n    ξ_dep = c + A.mulVec ξ_indep\n⊢ c = 0",
    "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,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ\nm : ℕ\nE : Type u_3\ninst✝ : AddCommGroup E\ninst✝2 : Module ℝ E\nB : Basis (Fin (n + m)) ℝ E\nS : Submodule ℝ E\nc : Fin m → ℝ\nA : Matrix (Fin m) (Fin n) ℝ\nhS : ∀ (v : E),\n  v ∈ S ↔\n    let ξ_dep := fun i => (B.repr v) (Fin.natAdd n i);\n    let ξ_indep := fun j => (B.repr v) (Fin.castAdd m j);\n    ξ_dep = c + A.mulVec ξ_indep\nh_zero_mem : 0 ∈ S\nh_zero_eq : let ξ_dep := fun i => (B.repr 0) (Fin.natAdd n i);\nlet ξ_indep := fun j => (B.repr 0) (Fin.castAdd m j);\nξ_dep = c + A.mulVec ξ_indep\n⊢ c = 0",
    "ground_truth": "have h_repr_zero : B.repr 0 = 0 := LinearEquiv.map_zero B.repr",
    "model_output": "simp at h_zero_eq"
  },
  {
    "id": 98,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ\nm : ℕ\nE : Type u_3\ninst✝ : AddCommGroup E\ninst✝2 : Module ℝ E\nB : Basis (Fin (n + m)) ℝ E\nS : Submodule ℝ E\nc : Fin m → ℝ\nA : Matrix (Fin m) (Fin n) ℝ\nhS : ∀ (v : E),\n  v ∈ S ↔\n    let ξ_dep := fun i => (B.repr v) (Fin.natAdd n i);\n    let ξ_indep := fun j => (B.repr v) (Fin.castAdd m j);\n    ξ_dep = c + A.mulVec ξ_indep\nh_zero_mem : 0 ∈ S\nh_zero_eq : let ξ_dep := fun i => (B.repr 0) (Fin.natAdd n i);\nlet ξ_indep := fun j => (B.repr 0) (Fin.castAdd m j);\nξ_dep = c + A.mulVec ξ_indep\nh_repr_zero : B.repr 0 = 0\n⊢ c = 0",
    "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,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ\nm : ℕ\nE : Type u_3\ninst✝ : AddCommGroup E\ninst✝2 : Module ℝ E\nB : Basis (Fin (n + m)) ℝ E\nS : Submodule ℝ E\nc : Fin m → ℝ\nA : Matrix (Fin m) (Fin n) ℝ\nhS : ∀ (v : E),\n  v ∈ S ↔\n    let ξ_dep := fun i => (B.repr v) (Fin.natAdd n i);\n    let ξ_indep := fun j => (B.repr v) (Fin.castAdd m j);\n    ξ_dep = c + A.mulVec ξ_indep\nh_zero_mem : 0 ∈ S\nh_repr_zero : B.repr 0 = 0\nh_zero_eq : 0 = c + A.mulVec 0\n⊢ c = 0",
    "ground_truth": "simp only [Matrix.mulVec_zero, add_zero] at h_zero_eq",
    "model_output": "simp [Matrix.mulVec] at h_zero_eq"
  },
  {
    "id": 100,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem E τ σ\nhp : p.domain = ∅\n⊢ sup_value = ⊥",
    "ground_truth": "by\n  unfold sup_value\n  unfold FeasSet FeasPoint\n  simp [hp]",
    "model_output": "simp [sup_value]"
  },
  {
    "id": 101,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem E τ σ\nhp : p.domain = ∅\n⊢ sup_value = ⊥",
    "ground_truth": "unfold sup_value\n  unfold FeasSet FeasPoint\n  simp [hp]",
    "model_output": "simp [sup_value]"
  },
  {
    "id": 102,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ\nσ : Finset ℕ\n⊢ ∀ (b : ℕ → EuclideanSpace ℝ (Fin n)), IsClosed {z | ∃ mu, (∀ (i : { x // x ∈ σ }), 0 ≤ mu i) ∧ z = ∑ i, mu i • b ↑i}",
    "ground_truth": "by\n  intro b; rw [polyhedra_iff_cone b]\n  exact closed_conic σ b",
    "model_output": "intro b\n  apply IsClosed.affine_image\n  · apply IsClosed.prod (fun i ↦ IsClosed.preimage (ContinuousLinearMap.continuous (ContinuousLinearMap.smul (ContinuousLinearMap.id (EuclideanSpace ℝ (Fin n)) (EuclideanSpace ℝ (Fin n))) (b i))) (EuclideanSpace.isClosed_Ici))\n    · apply IsClosed.prod (fun i ↦ IsClosed.preimage (ContinuousLinearMap.continuous (ContinuousLinearMap.smul (ContinuousLinearMap.id (EuclideanSpace ℝ (Fin n)) (EuclideanSpace ℝ (Fin n))) ("
  },
  {
    "id": 103,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ\nσ : Finset ℕ\n⊢ ∀ (b : ℕ → EuclideanSpace ℝ (Fin n)), IsClosed {z | ∃ mu, (∀ (i : { x // x ∈ σ }), 0 ≤ mu i) ∧ z = ∑ i, mu i • b ↑i}",
    "ground_truth": "intro b; rw [polyhedra_iff_cone b]\n  exact closed_conic σ b",
    "model_output": "intro b\n  apply IsClosed.affine_image\n  · apply IsClosed.prod (fun i ↦ IsClosed.preimage (ContinuousLinearMap.continuous (ContinuousLinearMap.smul (ContinuousLinearMap.id (EuclideanSpace ℝ (Fin n)) (EuclideanSpace ℝ (Fin n))) (b i))) (EuclideanSpace.isClosed_Ici))\n    · apply IsClosed.prod (fun i ↦ IsClosed.preimage (ContinuousLinearMap.continuous (ContinuousLinearMap.smul (ContinuousLinearMap.id (EuclideanSpace ℝ (Fin n)) (EuclideanSpace ℝ (Fin n))) ("
  },
  {
    "id": 104,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem (EuclideanSpace ℝ (Fin n)) τ σ\nloc : EuclideanSpace ℝ (Fin n)\nhl : Local_Minimum loc\nhf : Differentiable ℝ p.objective\nh : linearized_feasible_directions loc = posTangentConeAt FeasSet loc\n⊢ linearized_feasible_directions loc ∩ {d | ⟪gradient p.objective loc, d⟫_ℝ < 0} = ∅",
    "ground_truth": "by\n  rw [h]; apply (local_Minimum_TangentCone' loc hl hf)",
    "model_output": "rw [h]"
  },
  {
    "id": 105,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem (EuclideanSpace ℝ (Fin n)) τ σ\nloc : EuclideanSpace ℝ (Fin n)\nhl : Local_Minimum loc\nhf : Differentiable ℝ p.objective\nh : linearized_feasible_directions loc = posTangentConeAt FeasSet loc\n⊢ linearized_feasible_directions loc ∩ {d | ⟪gradient p.objective loc, d⟫_ℝ < 0} = ∅",
    "ground_truth": "rw [h]; apply (local_Minimum_TangentCone' loc hl hf)",
    "model_output": "rw [h]"
  },
  {
    "id": 106,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem (EuclideanSpace ℝ (Fin n)) τ σ\nloc : EuclideanSpace ℝ (Fin n)\nhl : Local_Minimum loc\nhf : Differentiable ℝ p.objective\nh : linearized_feasible_directions loc = posTangentConeAt FeasSet loc\n⊢ posTangentConeAt FeasSet loc ∩ {d | ⟪gradient p.objective loc, d⟫_ℝ < 0} = ∅",
    "ground_truth": "apply (local_Minimum_TangentCone' loc hl hf)",
    "model_output": "rw [← h]"
  },
  {
    "id": 107,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E✝\ninst✝2 : InnerProductSpace ℝ E✝\nE : Type u_2\nF : Type u_3\ninst✝3 : NormedAddCommGroup E\ninst✝4 : InnerProductSpace ℝ E\ninst✝5 : CompleteSpace E\ninst✝6 : NormedAddCommGroup F\ninst✝7 : InnerProductSpace ℝ F\ninst✝8 : CompleteSpace F\nA : E ≃L[ℝ] F\nx : F\n⊢ (ContinuousLinearMap.adjoint ↑A.symm) ((ContinuousLinearMap.adjoint ↑A) x) = x",
    "ground_truth": "by rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp",
    "model_output": "rw [ContinuousLinearMap.adjoint_inner_right, ContinuousLinearMap.adjoint_inner_left]\n        simp [ContinuousLinearMap.adjoint, ContinuousLinearMap.symm_apply_apply]"
  },
  {
    "id": 108,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E✝\ninst✝2 : InnerProductSpace ℝ E✝\nE : Type u_2\nF : Type u_3\ninst✝3 : NormedAddCommGroup E\ninst✝4 : InnerProductSpace ℝ E\ninst✝5 : CompleteSpace E\ninst✝6 : NormedAddCommGroup F\ninst✝7 : InnerProductSpace ℝ F\ninst✝8 : CompleteSpace F\nA : E ≃L[ℝ] F\nx : F\n⊢ (ContinuousLinearMap.adjoint ↑A.symm) ((ContinuousLinearMap.adjoint ↑A) x) = x",
    "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp",
    "model_output": "rw [ContinuousLinearMap.adjoint_inner_right, ContinuousLinearMap.adjoint_inner_left]\n        simp [ContinuousLinearMap.adjoint, ContinuousLinearMap.symm_apply_apply]"
  },
  {
    "id": 109,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E✝\ninst✝2 : InnerProductSpace ℝ E✝\nE : Type u_2\nF : Type u_3\ninst✝3 : NormedAddCommGroup E\ninst✝4 : InnerProductSpace ℝ E\ninst✝5 : CompleteSpace E\ninst✝6 : NormedAddCommGroup F\ninst✝7 : InnerProductSpace ℝ F\ninst✝8 : CompleteSpace F\nA : E ≃L[ℝ] F\nx : F\n⊢ (ContinuousLinearMap.adjoint ↑A.symm) ((ContinuousLinearMap.adjoint ↑A) x) = x",
    "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]",
    "model_output": "rw [ContinuousLinearMap.adjoint_inner_right, ContinuousLinearMap.adjoint_inner_left]\n        simp [ContinuousLinearMap.adjoint, ContinuousLinearMap.symm_apply_apply]"
  },
  {
    "id": 110,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E✝\ninst✝2 : InnerProductSpace ℝ E✝\nE : Type u_2\nF : Type u_3\ninst✝3 : NormedAddCommGroup E\ninst✝4 : InnerProductSpace ℝ E\ninst✝5 : CompleteSpace E\ninst✝6 : NormedAddCommGroup F\ninst✝7 : InnerProductSpace ℝ F\ninst✝8 : CompleteSpace F\nA : E ≃L[ℝ] F\nx : E\n⊢ (ContinuousLinearMap.adjoint ↑A) ((ContinuousLinearMap.adjoint ↑A.symm) x) = x",
    "ground_truth": "by rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp",
    "model_output": "rw [← ContinuousLinearMap.adjoint_comp, A.adjoint_eq]"
  },
  {
    "id": 111,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E✝\ninst✝2 : InnerProductSpace ℝ E✝\nE : Type u_2\nF : Type u_3\ninst✝3 : NormedAddCommGroup E\ninst✝4 : InnerProductSpace ℝ E\ninst✝5 : CompleteSpace E\ninst✝6 : NormedAddCommGroup F\ninst✝7 : InnerProductSpace ℝ F\ninst✝8 : CompleteSpace F\nA : E ≃L[ℝ] F\nx : E\n⊢ (ContinuousLinearMap.adjoint ↑A) ((ContinuousLinearMap.adjoint ↑A.symm) x) = x",
    "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]; simp",
    "model_output": "rw [← ContinuousLinearMap.adjoint_comp, A.adjoint_eq]"
  },
  {
    "id": 112,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E✝\ninst✝2 : InnerProductSpace ℝ E✝\nE : Type u_2\nF : Type u_3\ninst✝3 : NormedAddCommGroup E\ninst✝4 : InnerProductSpace ℝ E\ninst✝5 : CompleteSpace E\ninst✝6 : NormedAddCommGroup F\ninst✝7 : InnerProductSpace ℝ F\ninst✝8 : CompleteSpace F\nA : E ≃L[ℝ] F\nx : E\n⊢ (ContinuousLinearMap.adjoint ↑A) ((ContinuousLinearMap.adjoint ↑A.symm) x) = x",
    "ground_truth": "rw [←ContinuousLinearMap.comp_apply, ←ContinuousLinearMap.adjoint_comp]",
    "model_output": "rw [← ContinuousLinearMap.adjoint_comp, A.adjoint_eq]"
  },
  {
    "id": 113,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "m : Type u_2\nn : Type u_3\ninst✝ : Fintype m\ninst✝2 : Fintype n\n𝕜 : Type u_6\ninst✝3 : RCLike 𝕜\ninst✝4 : DecidableEq m\ninst✝5 : DecidableEq n\nA : Matrix m n 𝕜\nB : Matrix n m 𝕜\n⊢ tr (A * B) = tr (B * A)",
    "ground_truth": "by rw [← trace_transpose, ← trace_transpose_mul, transpose_mul]",
    "model_output": "rw [Matrix.mul_comm]"
  },
  {
    "id": 114,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "m : Type u_2\nn : Type u_3\ninst✝ : Fintype m\ninst✝2 : Fintype n\n𝕜 : Type u_6\ninst✝3 : RCLike 𝕜\ninst✝4 : DecidableEq m\ninst✝5 : DecidableEq n\nA : Matrix m n 𝕜\nB : Matrix n m 𝕜\n⊢ tr (A * B) = tr (B * A)",
    "ground_truth": "rw [← trace_transpose, ← trace_transpose_mul, transpose_mul]",
    "model_output": "rw [Matrix.mul_comm]"
  },
  {
    "id": 115,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nX : Set E\nhs : Convex ℝ X\n⊢ ConvexOn ℝ X X.convex_indicator",
    "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 ConvexOn.subset hs\n    · apply convex_epigraph\n    · apply convex_epigraph_convex_indicator\n      · apply convex_epigraph\n      · apply convex_epigraph_convex_indicator\n        · apply convex_epigraph\n        · apply convex_epigraph_convex_indicator\n          · apply convex_epigraph\n          · apply convex_epigraph_convex_indicator\n            · apply convex_epigraph\n            · apply convex_epigraph_convex_indicator\n              · apply convex_epigraph\n              · apply convex_epigraph_convex_indicator\n                · apply convex_epigraph\n                · apply convex_epigraph_convex_indicator"
  },
  {
    "id": 116,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nX : Set E\nhs : Convex ℝ X\nthis : ConvexOn ℝ X fun x => 0\n⊢ ConvexOn ℝ X X.convex_indicator",
    "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.add this (convex_indicator_convex hs)"
  },
  {
    "id": 117,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "V : Type u_2\ninst✝ : NormedAddCommGroup V\ninst✝2 : NormedSpace ℝ V\nι : Sort u_4\ns : ι → Set V\nh : ∀ (i : ι), Convex ℝ (s i)\nhinter : ∃ x, ∀ (i : ι), x ∈ intrinsicInterior ℝ (s i)\n⊢ ⋂ i, intrinsicClosure ℝ (s i) = intrinsicClosure ℝ (⋂ i, s i)",
    "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": "rcases hinter with ⟨x, hx⟩\n  rw [intrinsicClosure_iInter h hx]\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_in"
  },
  {
    "id": 118,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "V : Type u_2\ninst✝ : NormedAddCommGroup V\ninst✝2 : NormedSpace ℝ V\nι : Sort u_4\ns : ι → Set V\nh : ∀ (i : ι), Convex ℝ (s i)\nhinter : ∃ x, ∀ (i : ι), x ∈ intrinsicInterior ℝ (s i)\n⊢ ⋂ i, intrinsicClosure ℝ (s i) = intrinsicClosure ℝ (⋂ i, s i)",
    "ground_truth": "apply Subset.antisymm\n  · exact iIntersection_closure_sub_closure_iIntersection'' h hinter\n  exact closure_iIntersection_sub_iIntersection_closure'",
    "model_output": "rcases hinter with ⟨x, hx⟩\n  rw [intrinsicClosure_iInter h hx]\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_in"
  },
  {
    "id": 119,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "case.1:\nV : Type u_2\ninst✝ : NormedAddCommGroup V\ninst✝2 : NormedSpace ℝ V\nι : Sort u_4\ns : ι → Set V\nh : ∀ (i : ι), Convex ℝ (s i)\nhinter : ∃ x, ∀ (i : ι), x ∈ intrinsicInterior ℝ (s i)\n⊢ ⋂ i, intrinsicClosure ℝ (s i) ⊆ intrinsicClosure ℝ (⋂ i, s i)\n\ncase.2:\nV : Type u_2\ninst✝ : NormedAddCommGroup V\ninst✝2 : NormedSpace ℝ V\nι : Sort u_4\ns : ι → Set V\nh : ∀ (i : ι), Convex ℝ (s i)\nhinter : ∃ x, ∀ (i : ι), x ∈ intrinsicInterior ℝ (s i)\n⊢ intrinsicClosure ℝ (⋂ i, s i) ⊆ ⋂ i, intrinsicClosure ℝ (s i)",
    "ground_truth": "· exact iIntersection_closure_sub_closure_iIntersection'' h hinter",
    "model_output": "· exact iIntersection_intrinsicClosure_sub_intrinsicClosure_iIntersection h hinter"
  },
  {
    "id": 120,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "V : Type u_2\ninst✝ : NormedAddCommGroup V\ninst✝2 : NormedSpace ℝ V\nι : Sort u_4\ns : ι → Set V\nh : ∀ (i : ι), Convex ℝ (s i)\nhinter : ∃ x, ∀ (i : ι), x ∈ intrinsicInterior ℝ (s i)\n⊢ ⋂ i, intrinsicClosure ℝ (s i) ⊆ intrinsicClosure ℝ (⋂ i, s i)",
    "ground_truth": "exact iIntersection_closure_sub_closure_iIntersection'' h hinter",
    "model_output": "rcases hinter with ⟨x, hx⟩\n  apply iInter_intrinsicClosure_sub_intrinsicClosure_iInter h hx"
  },
  {
    "id": 121,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "V : Type u_2\ninst✝ : NormedAddCommGroup V\ninst✝2 : NormedSpace ℝ V\nι : Sort u_4\ns : ι → Set V\nh : ∀ (i : ι), Convex ℝ (s i)\nhinter : ∃ x, ∀ (i : ι), x ∈ intrinsicInterior ℝ (s i)\n⊢ intrinsicClosure ℝ (⋂ i, s i) ⊆ ⋂ i, intrinsicClosure ℝ (s i)",
    "ground_truth": "exact closure_iIntersection_sub_iIntersection_closure'",
    "model_output": "rcases hinter with ⟨x, hx⟩\n  intro x1 hx1\n  have hx1in: ∀ i, x1 ∈ intrinsicClosure ℝ (s i) := by\n    intro i\n    apply intrinsicClosure_mono (hinter.choose_spec.2 i)\n    exact hx1\n  exact mem_iInter.mpr hx1in"
  },
  {
    "id": 122,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\n𝕜 : Type u_3\nf : E → F\ninst✝ : AddCommMonoid E\ninst✝2 : CompleteLinearOrder F\ninst✝3 : Ring 𝕜\ninst✝4 : LinearOrder 𝕜\ninst✝5 : IsStrictOrderedRing 𝕜\ninst✝6 : DenselyOrdered 𝕜\ninst✝7 : Module 𝕜 E\nx : E\ny : E\nhf' : strong_quasi f 𝕜\nhx : IsMinOn f univ x\nhy : IsMinOn f univ y\nneq : ¬x = y\nthis : 0 < 1\na : 𝕜\nlta : 0 < a\nalt : a < 1\n⊢ False",
    "ground_truth": "have eqone : a + (1 - a) = 1 := add_sub_cancel a 1",
    "model_output": "have : a * (f y - f x) < 0 := by\n      apply mul_neg_of_pos_of_neg\n      · exact lta\n      have : f x ≤ f y := by\n        exact hx trivial\n      linarith [this]"
  },
  {
    "id": 123,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\n𝕜 : Type u_3\nf : E → F\ninst✝ : AddCommMonoid E\ninst✝2 : CompleteLinearOrder F\ninst✝3 : Ring 𝕜\ninst✝4 : LinearOrder 𝕜\ninst✝5 : IsStrictOrderedRing 𝕜\ninst✝6 : DenselyOrdered 𝕜\ninst✝7 : Module 𝕜 E\nx : E\ny : E\nhf' : strong_quasi f 𝕜\nhx : IsMinOn f univ x\nhy : IsMinOn f univ y\nneq : ¬x = y\nthis : 0 < 1\na : 𝕜\nlta : 0 < a\nalt : a < 1\neqone : a + (1 - a) = 1\n⊢ False",
    "ground_truth": "have lta' : 0 < 1 - a := sub_pos_of_lt alt",
    "model_output": "have : f x ≤ f (a • x + (1 - a) • y) := by\n      apply hx (mem_univ (a • x + (1 - a) • y))"
  },
  {
    "id": 124,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\n𝕜 : Type u_3\nf : E → F\ninst✝ : AddCommMonoid E\ninst✝2 : CompleteLinearOrder F\ninst✝3 : Ring 𝕜\ninst✝4 : LinearOrder 𝕜\ninst✝5 : IsStrictOrderedRing 𝕜\ninst✝6 : DenselyOrdered 𝕜\ninst✝7 : Module 𝕜 E\nx : E\ny : E\nhf' : strong_quasi f 𝕜\nhx : IsMinOn f univ x\nhy : IsMinOn f univ y\nneq : ¬x = y\nthis : 0 < 1\na : 𝕜\nlta : 0 < a\nalt : a < 1\neqone : a + (1 - a) = 1\nlta' : 0 < 1 - a\n⊢ 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 : f x ≤ f (a • x + (1 - a) • y) := by\n      apply hx (mem_univ (a • x + (1 - a) • y))"
  },
  {
    "id": 125,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\n𝕜 : Type u_3\nf : E → F\ninst✝ : AddCommMonoid E\ninst✝2 : CompleteLinearOrder F\ninst✝3 : Ring 𝕜\ninst✝4 : LinearOrder 𝕜\ninst✝5 : IsStrictOrderedRing 𝕜\ninst✝6 : DenselyOrdered 𝕜\ninst✝7 : Module 𝕜 E\nx : E\ny : E\nhf' : strong_quasi f 𝕜\nhx : IsMinOn f univ x\nhy : IsMinOn f univ y\nneq : ¬x = y\nthis : 0 < 1\na : 𝕜\nlta : 0 < a\nalt : a < 1\neqone : a + (1 - a) = 1\nlta' : 0 < 1 - a\n⊢ f (a • x + (1 - a) • y) < f y",
    "ground_truth": "apply Eq.trans_gt (max_eq_right (hx trivial))",
    "model_output": "apply hf' (x := x) (y := y) neq lta lta'"
  },
  {
    "id": 126,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nu : E\nhu : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖\nv : E\nhv : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ v (y - x) ≥ -ε * ‖y - x‖\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ (a • u + b • v) (y - x) ≥ -ε * ‖y - x‖",
    "ground_truth": "intro ε εpos",
    "model_output": "intro ε εpos\n    filter_upwards [hu ε εpos, hv ε εpos] with y hy hy'\n    rw [eventually_and]\n    simp at *\n    rw [add_sub, sub_sub_sub, ← add_mul, ← sub_sub, sub_sub_sub, add_comm (f y), add_comm (f x)]\n    rw [inner_add_left, ← sub_sub, ← sub_sub, ← sub_sub, ← sub_sub, ← sub_sub]\n    simp\n    rw [← sub_sub, ← sub_sub, ← sub_sub, ← sub_sub, ← sub_sub, ← sub_sub]"
  },
  {
    "id": 127,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nu : E\nhu : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖\nv : E\nhv : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ v (y - x) ≥ -ε * ‖y - x‖\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nε : ℝ\nεpos : ε > 0\ny : E\nhyu : f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖\nhyv : f y - f x - inner ℝ v (y - x) ≥ -ε * ‖y - x‖\n⊢ a * (f y - f x - inner ℝ u (y - x)) + b * (f y - f x - inner ℝ v (y - x)) ≥ a * (-ε * ‖y - x‖) + b * (-ε * ‖y - x‖)",
    "ground_truth": "rw [ge_iff_le] at *",
    "model_output": "apply add_le_add\n    · apply mul_le_mul_of_nonneg_left hyu (mul_nonneg ha (norm_nonneg (y - x)))\n    · apply mul_le_mul_of_nonneg_left hyv (mul_nonneg hb (norm_nonneg (y - x)))"
  },
  {
    "id": 128,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nu : E\nhu : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖\nv : E\nhv : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ v (y - x) ≥ -ε * ‖y - x‖\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nε : ℝ\nεpos : ε > 0\ny : E\nhyu : -ε * ‖y - x‖ ≤ f y - f x - inner ℝ u (y - x)\nhyv : -ε * ‖y - x‖ ≤ f y - f x - inner ℝ v (y - x)\n⊢ a * (-ε * ‖y - x‖) + b * (-ε * ‖y - x‖) ≤ a * (f y - f x - inner ℝ u (y - x)) + b * (f y - f x - inner ℝ v (y - x))",
    "ground_truth": "apply add_le_add",
    "model_output": "apply add_le_add\n    · apply mul_le_mul_of_nonneg_left hyu (mul_nonneg ha (norm_nonneg (y - x)))\n    · apply mul_le_mul_of_nonneg_left hyv (mul_nonneg hb (norm_nonneg (y - x)))"
  },
  {
    "id": 129,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nf : E → ℝ\ng : F → ℝ\nH : WithLp 2 (E × F) → ℝ\nx0 : E\ny0 : F\nl : NNReal\nalg : BCD f g H l x0 y0\nγ : ℝ\nhγ : γ > 1\nk : ℕ\nck : alg.c k = 1 / (γ * ↑l)\n⊢ 0 < 1 / (γ * ↑l)",
    "ground_truth": "apply div_pos",
    "model_output": "apply one_div_pos.mpr; apply mul_pos; · linarith; · exact alg.lpos"
  },
  {
    "id": 130,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nf : E → ℝ\ng : F → ℝ\nH : WithLp 2 (E × F) → ℝ\nx0 : E\ny0 : F\nl : NNReal\nalg : BCD f g H l x0 y0\nγ : ℝ\nhγ : γ > 1\nk : ℕ\nck : alg.c k = 1 / (γ * ↑l)\n⊢ 0 < γ * ↑l",
    "ground_truth": "apply mul_pos",
    "model_output": "apply mul_pos _ alg.lpos\n[· exact zero_lt_one.trans hγ]"
  },
  {
    "id": 131,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "case.1:\nE : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nf : E → ℝ\ng : F → ℝ\nH : WithLp 2 (E × F) → ℝ\nx0 : E\ny0 : F\nl : NNReal\nalg : BCD f g H l x0 y0\nγ : ℝ\nhγ : γ > 1\nk : ℕ\nck : alg.c k = 1 / (γ * ↑l)\n⊢ 0 < γ\n\ncase.2:\nE : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nf : E → ℝ\ng : F → ℝ\nH : WithLp 2 (E × F) → ℝ\nx0 : E\ny0 : F\nl : NNReal\nalg : BCD f g H l x0 y0\nγ : ℝ\nhγ : γ > 1\nk : ℕ\nck : alg.c k = 1 / (γ * ↑l)\n⊢ 0 < ↑l",
    "ground_truth": "· linarith [hγ]",
    "model_output": "· linarith [hγ]"
  },
  {
    "id": 132,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nf : E → ℝ\ng : F → ℝ\nH : WithLp 2 (E × F) → ℝ\nx0 : E\ny0 : F\nl : NNReal\nalg : BCD f g H l x0 y0\nγ : ℝ\nhγ : γ > 1\nk : ℕ\nck : alg.c k = 1 / (γ * ↑l)\n⊢ 0 < γ",
    "ground_truth": "linarith [hγ]",
    "model_output": "linarith [hγ]"
  },
  {
    "id": 133,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nf : E → ℝ\ng : F → ℝ\nH : WithLp 2 (E × F) → ℝ\nx0 : E\ny0 : F\nl : NNReal\nalg : BCD f g H l x0 y0\nγ : ℝ\nhγ : γ > 1\nk : ℕ\nck : alg.c k = 1 / (γ * ↑l)\n⊢ 0 < ↑l",
    "ground_truth": "· apply alg.lpos",
    "model_output": "apply alg.lpos"
  },
  {
    "id": 134,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nf : E → ℝ\ng : F → ℝ\nH : WithLp 2 (E × F) → ℝ\nx0 : E\ny0 : F\nl : NNReal\nalg : BCD f g H l x0 y0\nγ : ℝ\nhγ : γ > 1\nk : ℕ\nck : alg.c k = 1 / (γ * ↑l)\n⊢ 0 < ↑l",
    "ground_truth": "apply alg.lpos",
    "model_output": "apply alg.lpos"
  },
  {
    "id": 135,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nx : E\nε : ℝ\nf : E → ℝ\nhe : ε > 0\nhf : ContDiffOn ℝ 1 f (Metric.ball x ε)\ns : Set E\n⊢ Metric.ball x ε ∈ 𝓝 x",
    "ground_truth": "rw [mem_nhds_iff]",
    "model_output": "apply Metric.ball_mem_nhds"
  },
  {
    "id": 136,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\nhf : ∃ x ∈ s, f x = ⊥\nx : E\nsnd✝ : ℝ\n⊢ (x, snd✝) ∈ Epi ⊥ s ↔ (x, snd✝) ∈ s ×ˢ univ",
    "ground_truth": "simp [Function.Epi]",
    "model_output": "simp [Epi]"
  },
  {
    "id": 137,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\nhf : ¬∃ x ∈ s, f x = ⊥\nx : E\nsnd✝ : ℝ\n⊢ (x, snd✝) ∈ Epi ⊤ s ↔ (x, snd✝) ∈ ∅",
    "ground_truth": "simp [Function.Epi]",
    "model_output": "simp [Epi]"
  },
  {
    "id": 138,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\nhf : ¬∃ x ∈ s, f x = ⊥\nh' : Epi ⊤ s = ∅\n⊢ _root_.IsClosed ∅",
    "ground_truth": "exact isClosed_empty",
    "model_output": "exact isClosed_empty"
  },
  {
    "id": 139,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E\nx : E\n⊢ HasSubgradientAt f g x ↔ Banach_HasSubgradientAt f ((toDual ℝ E) g) x",
    "ground_truth": "constructor",
    "model_output": "constructor\n    · intro h\n      unfold Banach_HasSubgradientAt\n      unfold HasSubgradientAt at h\n      specialize h x\n      simp at h\n      exact h\n    · intro h\n      unfold HasSubgradientAt\n      unfold Banach_HasSubgradientAt at h\n      specialize h g\n      simp at h\n      exact h"
  },
  {
    "id": 140,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nx : E\nf₁ : E → ℝ\nf₂ : E → ℝ\nh : f₁ = f₂\n⊢ SubderivAt f₁ x = SubderivAt f₂ x",
    "ground_truth": "by\n  ext g; rw [h]",
    "model_output": "rw [h]"
  },
  {
    "id": 141,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nx : E\nf₁ : E → ℝ\nf₂ : E → ℝ\nh : f₁ = f₂\n⊢ SubderivAt f₁ x = SubderivAt f₂ x",
    "ground_truth": "ext g; rw [h]",
    "model_output": "rw [h]"
  },
  {
    "id": 142,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ns : Set E\nf : E → EReal\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : FiniteDimensional ℝ E\ni : affine_minorant s f\ng : E →L[ℝ] ℝ\nl : E\nw : ℝ\n⊢ (∀ x ∈ s, ↑(inner ℝ l x) + ↑w ≤ f x) ∧ ∀ x ∈ s, i.h x = inner ℝ l x + w",
    "ground_truth": "constructor",
    "model_output": "constructor <;> intro x _ <;> simp [i.h]"
  },
  {
    "id": 143,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ns : Set E\nf : E → EReal\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : FiniteDimensional ℝ E\ni : affine_minorant s f\ng : E →L[ℝ] ℝ\nl : E\nw : ℝ\n⊢ ∀ x ∈ s, i.h x = inner ℝ l x + w",
    "ground_truth": "intro x _hx",
    "model_output": "intro x _\n    exact i.hf x"
  },
  {
    "id": 144,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E₁ : Type u_1\nE₂ : Type u_2\nF : Type u_3\ninst✝ : NormedAddCommGroup E₁\ninst✝2 : InnerProductSpace ℝ E₁\ninst✝3 : FiniteDimensional ℝ E₁\ninst✝4 : NormedAddCommGroup E₂\ninst✝5 : InnerProductSpace ℝ E₂\ninst✝6 : FiniteDimensional ℝ E₂\ninst✝7 : NormedAddCommGroup F\ninst✝8 : InnerProductSpace ℝ F\ninst✝9 : FiniteDimensional ℝ F\nadmm : ADMM E₁ E₂ F\nadmm_kkt : Existance_of_kkt admm\nfullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)\nfullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nthis : (fun n => (↑↑(φ₁ n)).toNat) = fun n => φ (n + 1)\n⊢ StrictMono fun n => ↑(φ₁ n)",
    "ground_truth": "simp at this",
    "model_output": "rw[this , strictMono_nat]"
  },
  {
    "id": 145,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E₁ : Type u_1\nE₂ : Type u_2\nF : Type u_3\ninst✝ : NormedAddCommGroup E₁\ninst✝2 : InnerProductSpace ℝ E₁\ninst✝3 : FiniteDimensional ℝ E₁\ninst✝4 : NormedAddCommGroup E₂\ninst✝5 : InnerProductSpace ℝ E₂\ninst✝6 : FiniteDimensional ℝ E₂\ninst✝7 : NormedAddCommGroup F\ninst✝8 : InnerProductSpace ℝ F\ninst✝9 : FiniteDimensional ℝ F\nadmm : ADMM E₁ E₂ F\nadmm_kkt : Existance_of_kkt admm\nfullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)\nfullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nthis : (fun n => ↑(φ₁ n)) = fun n => φ (n + 1)\n⊢ StrictMono fun n => n + 1",
    "ground_truth": "· simp [StrictMono]",
    "model_output": "exact strictMono_add_one"
  },
  {
    "id": 146,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E₁ : Type u_1\nE₂ : Type u_2\nF : Type u_3\ninst✝ : NormedAddCommGroup E₁\ninst✝2 : InnerProductSpace ℝ E₁\ninst✝3 : FiniteDimensional ℝ E₁\ninst✝4 : NormedAddCommGroup E₂\ninst✝5 : InnerProductSpace ℝ E₂\ninst✝6 : FiniteDimensional ℝ E₂\ninst✝7 : NormedAddCommGroup F\ninst✝8 : InnerProductSpace ℝ F\ninst✝9 : FiniteDimensional ℝ F\nadmm : ADMM E₁ E₂ F\nadmm_kkt : Existance_of_kkt admm\nfullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)\nfullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nthis : (fun n => ↑(φ₁ n)) = fun n => φ (n + 1)\n⊢ StrictMono fun n => n + 1",
    "ground_truth": "simp [StrictMono]",
    "model_output": "exact strictMono_add_one"
  },
  {
    "id": 147,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "c : EReal\nb : ℝ\nhc1 : c ≠ ⊥\na : ℝ\nh : ↑a + ↑b ≤ c\nha1 : ↑a ≠ ⊥\nha2 : ↑a ≠ ⊤\n⊢ 0 ≤ c - ↑a - ↑b",
    "ground_truth": "by_cases hc2 : c = ⊤",
    "model_output": "by_cases hc2 : c = ⊤"
  },
  {
    "id": 148,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "case.1:\nc : EReal\nb : ℝ\nhc1 : c ≠ ⊥\na : ℝ\nh : ↑a + ↑b ≤ c\nha1 : ↑a ≠ ⊥\nha2 : ↑a ≠ ⊤\nhc2 : c = ⊤\n⊢ 0 ≤ c - ↑a - ↑b\n\ncase.2:\nc : EReal\nb : ℝ\nhc1 : c ≠ ⊥\na : ℝ\nh : ↑a + ↑b ≤ c\nha1 : ↑a ≠ ⊥\nha2 : ↑a ≠ ⊤\nhc2 : ¬c = ⊤\n⊢ 0 ≤ c - ↑a - ↑b",
    "ground_truth": "· rw [hc2]; simp",
    "model_output": "· rw [hc2]; simp"
  },
  {
    "id": 149,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "c : EReal\nb : ℝ\nhc1 : c ≠ ⊥\na : ℝ\nh : ↑a + ↑b ≤ c\nha1 : ↑a ≠ ⊥\nha2 : ↑a ≠ ⊤\nhc2 : c = ⊤\n⊢ 0 ≤ c - ↑a - ↑b",
    "ground_truth": "rw [hc2]; simp",
    "model_output": "rw [hc2]; simp"
  },
  {
    "id": 150,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "b : ℝ\na : ℝ\nha1 : ↑a ≠ ⊥\nha2 : ↑a ≠ ⊤\nc : ℝ\nhc1 : ↑c ≠ ⊥\nh : ↑a + ↑b ≤ ↑c\nhc2 : ¬↑c = ⊤\nthis : 0 ≤ ↑(c - a - b)\n⊢ 0 ≤ ↑c - ↑a - ↑b",
    "ground_truth": "by exact this",
    "model_output": "rw [← EReal.coe_sub, ← EReal.coe_sub] at this"
  },
  {
    "id": 151,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ns : Set E\nf : E → ℝ\nm : ℝ\nhs : Convex ℝ s\nhfun : ∀ ⦃x : E⦄,\n  x ∈ s →\n    ∀ ⦃y : E⦄,\n      y ∈ s →\n        ∀ ⦃a : ℝ⦄,\n          0 ≤ a → ∀ ⦃b : ℝ⦄, 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2\n⊢ StrongConvexOn s m f",
    "ground_truth": "constructor",
    "model_output": "apply strongConvexOn_iff_convex.mpr\n  intro x hx y hy a ha b hb hab\n  specialize hfun hx hy a ha b hb hab\n  rw [add_comm] at hab\n  rw [add_comm] at hfun\n  linarith [hfun]"
  },
  {
    "id": 152,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ns : Set E\nf : E → ℝ\nm : ℝ\nhs : Convex ℝ s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhfun : f (a • x + b • y) ≤ a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2\nthis : m / 2 * a * b * ‖x - y‖ ^ 2 = a * b * (m / 2 * ‖x - y‖ ^ 2)\n⊢ f (a • x + b • y) ≤ a * f x + b * f y - a * b * (m / 2 * ‖x - y‖ ^ 2)",
    "ground_truth": "rw [← this]",
    "model_output": "rw [this] at hfun"
  },
  {
    "id": 153,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\nt : Set E\nhscv : Convex ℝ s\nhtcv : Convex ℝ t\nhscl : _root_.IsClosed s\nhtcl : _root_.IsClosed t\n⊢ s = t ↔ s.convex_support = t.convex_support",
    "ground_truth": "constructor",
    "model_output": "constructor\n    · intro h; rw [h]\n    · intro h; rw [h]"
  },
  {
    "id": 154,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\nt : Set E\nhscv : Convex ℝ s\nhtcv : Convex ℝ t\nhscl : _root_.IsClosed s\nhtcl : _root_.IsClosed t\n⊢ s.convex_support = t.convex_support → s = t",
    "ground_truth": "simp [funext_iff]",
    "model_output": "intro hsupp\n    rw [convex_support_eq_closure, convex_support_eq_closure, hsupp]\n    apply closure_eq_closure.mp\n    apply subset_antisymm\n    · intro x hx\n      rw [← hsupp]\n      exact subset_closure hx\n    · exact fun x a ↦ a"
  },
  {
    "id": 155,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\nt : Set E\nhscv : Convex ℝ s\nhtcv : Convex ℝ t\nhscl : _root_.IsClosed s\nhtcl : _root_.IsClosed t\nh : ∀ (x : E), s.convex_support x = t.convex_support x\nz : E\n⊢ (z ∈ s → z ∈ t) ∧ (z ∈ t → z ∈ s)",
    "ground_truth": "constructor",
    "model_output": "constructor <;> intro hz <;> rw [← h] at hz <;> exact hz"
  },
  {
    "id": 156,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "case.1:\nn : ℕ+\nm : ℕ+\nb : Fin ↑m → ℝ\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\n⊢ Convex ℝ Set.univ\n\ncase.2:\nn : ℕ+\nm : ℕ+\nb : Fin ↑m → ℝ\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\n⊢ ∀ x ∈ Set.univ, HasGradientAt (fun x => 1 / 2 * ‖A *ᵥ x - b‖ ^ 2) (?f' x) x\n\ncase.3:\nn : ℕ+\nm : ℕ+\nb : Fin ↑m → ℝ\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\n⊢ ∀ x ∈ Set.univ, ∀ y ∈ Set.univ, inner ℝ (?f' x - ?f' y) (x - y) ≥ 0\n\ncase.4:\nn : ℕ+\nm : ℕ+\nb : Fin ↑m → ℝ\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\n⊢ EuclideanSpace ℝ (Fin ↑n) → EuclideanSpace ℝ (Fin ↑n)",
    "ground_truth": "· apply convex_univ",
    "model_output": "· apply convex_univ"
  },
  {
    "id": 157,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ+\nm : ℕ+\nb : Fin ↑m → ℝ\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\n⊢ Convex ℝ Set.univ",
    "ground_truth": "apply convex_univ",
    "model_output": "apply convex_univ"
  },
  {
    "id": 158,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ+\nm : ℕ+\nb : Fin ↑m → ℝ\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\n⊢ ∀ x ∈ Set.univ, ∀ y ∈ Set.univ, inner ℝ (Aᵀ *ᵥ (A *ᵥ x - b) - Aᵀ *ᵥ (A *ᵥ y - b)) (x - y) ≥ 0",
    "ground_truth": "intro x _ y _",
    "model_output": "intro x _ y _"
  },
  {
    "id": 159,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ+\nm : ℕ+\nb : Fin ↑m → ℝ\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\nx : EuclideanSpace ℝ (Fin ↑n)\na✝ : x ∈ Set.univ\ny : EuclideanSpace ℝ (Fin ↑n)\na✝2 : y ∈ Set.univ\n⊢ ‖A *ᵥ (x - y)‖ ^ 2 ≥ 0",
    "ground_truth": "apply sq_nonneg",
    "model_output": "apply pow_two_nonneg"
  },
  {
    "id": 160,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ\nm : ℕ\nE : Type u_3\ninst✝ : AddCommGroup E\ninst✝2 : Module ℝ E\nB : Basis (Fin (n + m)) ℝ E\nS : Submodule ℝ E\nc : Fin m → ℝ\nA : Matrix (Fin m) (Fin n) ℝ\nhS : ∀ (v : E),\n  v ∈ S ↔\n    let ξ_dep := fun i => (B.repr v) (Fin.natAdd n i);\n    let ξ_indep := fun j => (B.repr v) (Fin.castAdd m j);\n    ξ_dep = c + A.mulVec ξ_indep\nh_zero_mem : 0 ∈ S\nh_repr_zero : B.repr 0 = 0\nh_zero_eq : 0 = c\n⊢ c = 0",
    "ground_truth": "exact h_zero_eq.symm",
    "model_output": "exact Eq.symm h_zero_eq"
  },
  {
    "id": 161,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nX : Set E\nhs : Convex ℝ X\n⊢ ConvexOn ℝ X fun x => 0",
    "ground_truth": "dsimp [ConvexOn]",
    "model_output": "apply convexOn_const"
  },
  {
    "id": 162,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nX : Set E\nhs : Convex ℝ X\n⊢ Convex ℝ X ∧ ∀ ⦃x : E⦄, x ∈ X → ∀ ⦃y : E⦄, y ∈ X → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → 0 ≤ a • 0 + b • 0",
    "ground_truth": "constructor",
    "model_output": "constructor <;> simp [Convex] at * <;> tauto"
  },
  {
    "id": 163,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nX : Set E\nhs : Convex ℝ X\nthis : ConvexOn ℝ X fun x => 0\n⊢ EqOn (fun x => 0) X.convex_indicator X",
    "ground_truth": "dsimp[EqOn]",
    "model_output": "intro x hx\n    rw [convex_indicator_of_mem this hx]\n    simp only [Pi.zero_apply]"
  },
  {
    "id": 164,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nX : Set E\nhs : Convex ℝ X\nthis : ConvexOn ℝ X fun x => 0\nx : E\nhx : x ∈ X\n⊢ 0 = if x ∈ X then 0 else ⊤",
    "ground_truth": "rw [if_pos hx]",
    "model_output": "rw [if_pos hx]"
  },
  {
    "id": 165,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\n𝕜 : Type u_3\nf : E → F\ninst✝ : AddCommMonoid E\ninst✝2 : CompleteLinearOrder F\ninst✝3 : Ring 𝕜\ninst✝4 : LinearOrder 𝕜\ninst✝5 : IsStrictOrderedRing 𝕜\ninst✝6 : DenselyOrdered 𝕜\ninst✝7 : Module 𝕜 E\nx : E\ny : E\nhf' : strong_quasi f 𝕜\nhx : IsMinOn f univ x\nhy : IsMinOn f univ y\n⊢ x = y",
    "ground_truth": "by_contra neq",
    "model_output": "by_contra hne"
  },
  {
    "id": 166,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\n⊢ Convex ℝ (f_subdifferential f x)",
    "ground_truth": "intro u hu v hv a b ha hb hab",
    "model_output": "apply convex_setOf_subgradient\n    · apply convex_epigraph\n    · apply convex_univ"
  },
  {
    "id": 167,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nu : E\nhu : u ∈ f_subdifferential f x\nv : E\nhv : v ∈ f_subdifferential f x\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • u + b • v ∈ f_subdifferential f x",
    "ground_truth": "rw [has_f_subdiff_iff] at *",
    "model_output": "rw [f_subdiff_gradiant] at *\n    apply Convex_first_order_condition' (convex_univ) hu hv ha hb hab"
  },
  {
    "id": 168,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nm : Type u_2\nn : Type u_3\ninst✝2 : Fintype m\ninst✝3 : Fintype n\ninst✝4 : DecidableEq n\nA : Matrix m n 𝕜\nh : Aᴴ * A = 1\n⊢ finrank 𝕜 ↥(Submodule.span 𝕜 (Set.range A.colVec)) = Fintype.card n",
    "ground_truth": "rw [finrank_span_eq_card]",
    "model_output": "rw [finrank_span_eq_card]"
  },
  {
    "id": 169,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nV : Type u_2\nW : Type u_3\ninst✝2 : NormedAddCommGroup V\ninst✝3 : InnerProductSpace 𝕜 V\ninst✝4 : NormedAddCommGroup W\ninst✝5 : InnerProductSpace 𝕜 W\ninst✝6 : FiniteDimensional 𝕜 V\nT : V →ₗ[𝕜] W\ns : Submodule 𝕜 V\nx : V\nhker : sᗮ ≤ LinearMap.ker T\nx₁ : V\nhx₁ : x₁ ∈ s\nx₂ : V\nhx₂ : x₂ ∈ sᗮ\nhx : x = x₁ + x₂\n⊢ ‖T x‖ ^ 2 = ‖T x₁‖ ^ 2",
    "ground_truth": "rw [hx, LinearMap.map_add]",
    "model_output": "have : T x₂ = 0 := by\n      have : x₂ ∈ LinearMap.ker T := hker hx₂\n      rw [LinearMap.mem_ker] at this\n      exact this"
  },
  {
    "id": 170,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nx : E\nε : ℝ\nf : E → ℝ\nhe : ε > 0\nhf : ContDiffOn ℝ 1 f (Metric.ball x ε)\ns : Set E\n⊢ ContDiffAt ℝ 1 f x",
    "ground_truth": "apply ContDiffOn.contDiffAt hf",
    "model_output": "apply ContDiffOn.contDiffAt hf\n    have : x ∈ Metric.ball x ε := by simp [he]\n    exact this"
  },
  {
    "id": 171,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : TopologicalSpace E\ninst✝2 : AddCommMonoid E\n⊢ Function.closure ⊤ univ = ⊤",
    "ground_truth": "simp [Function.closure]",
    "model_output": "apply Function.closure_univ_eq_top.mpr\n    exact fun x _ ↦ trivial"
  },
  {
    "id": 172,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\n⊢ Function.IsClosed (Function.closure f s) s",
    "ground_truth": "simp [Function.closure, hsf]",
    "model_output": "apply isClosed_closure"
  },
  {
    "id": 173,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\n⊢ Function.IsClosed (if ∃ x ∈ s, f x = ⊥ then ⊥ else ⊤) s",
    "ground_truth": "by_cases hf : ∃ x ∈ s, f x = ⊥",
    "model_output": "by_cases h : ∃ x ∈ s, f x = ⊥\n    · simp [h]\n      exact isClosed_singleton\n    · simp [h]\n      exact isClosed_univ"
  },
  {
    "id": 174,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\nhf : ∃ x ∈ s, f x = ⊥\n⊢ Function.IsClosed ⊥ s",
    "ground_truth": "simp [Function.IsClosed]",
    "model_output": "apply Function.isClosed_bot_of_isClosed hs"
  },
  {
    "id": 175,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\nhf : ¬∃ x ∈ s, f x = ⊥\n⊢ Function.IsClosed ⊤ s",
    "ground_truth": "simp [Function.IsClosed]",
    "model_output": "exact hs"
  },
  {
    "id": 176,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E₁ : Type u_1\nE₂ : Type u_2\nF : Type u_3\ninst✝ : NormedAddCommGroup E₁\ninst✝2 : InnerProductSpace ℝ E₁\ninst✝3 : FiniteDimensional ℝ E₁\ninst✝4 : NormedAddCommGroup E₂\ninst✝5 : InnerProductSpace ℝ E₂\ninst✝6 : FiniteDimensional ℝ E₂\ninst✝7 : NormedAddCommGroup F\ninst✝8 : InnerProductSpace ℝ F\ninst✝9 : FiniteDimensional ℝ F\nadmm : ADMM E₁ E₂ F\nadmm_kkt : Existance_of_kkt admm\nfullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)\nfullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\n⊢ Tendsto (fun n => ↑(φ₁ n)) atTop atTop",
    "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      calc\n         _ ≥ n + 1 := by apply Nat.le_of_add_one_le\n         _ > n := by apply Nat.lt_add_one\n   exact this"
  },
  {
    "id": 177,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E₁ : Type u_1\nE₂ : Type u_2\nF : Type u_3\ninst✝ : NormedAddCommGroup E₁\ninst✝2 : InnerProductSpace ℝ E₁\ninst✝3 : FiniteDimensional ℝ E₁\ninst✝4 : NormedAddCommGroup E₂\ninst✝5 : InnerProductSpace ℝ E₂\ninst✝6 : FiniteDimensional ℝ E₂\ninst✝7 : NormedAddCommGroup F\ninst✝8 : InnerProductSpace ℝ F\ninst✝9 : FiniteDimensional ℝ F\nadmm : ADMM E₁ E₂ F\nadmm_kkt : Existance_of_kkt admm\nfullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)\nfullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nthis : (fun n => ↑(φ₁ n)) = fun n => φ (n + 1)\n⊢ StrictMono fun n => φ (n + 1)",
    "ground_truth": "apply StrictMono.comp",
    "model_output": "apply StrictMono.comp\n    · apply StrictMono.add\n      · apply strictMono_id\n      · apply strictMono_nat\n        simp\n    · apply StrictMono.id"
  },
  {
    "id": 178,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "case.1:\nE₁ : Type u_1\nE₂ : Type u_2\nF : Type u_3\ninst✝ : NormedAddCommGroup E₁\ninst✝2 : InnerProductSpace ℝ E₁\ninst✝3 : FiniteDimensional ℝ E₁\ninst✝4 : NormedAddCommGroup E₂\ninst✝5 : InnerProductSpace ℝ E₂\ninst✝6 : FiniteDimensional ℝ E₂\ninst✝7 : NormedAddCommGroup F\ninst✝8 : InnerProductSpace ℝ F\ninst✝9 : FiniteDimensional ℝ F\nadmm : ADMM E₁ E₂ F\nadmm_kkt : Existance_of_kkt admm\nfullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)\nfullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nthis : (fun n => ↑(φ₁ n)) = fun n => φ (n + 1)\n⊢ StrictMono φ\n\ncase.2:\nE₁ : Type u_1\nE₂ : Type u_2\nF : Type u_3\ninst✝ : NormedAddCommGroup E₁\ninst✝2 : InnerProductSpace ℝ E₁\ninst✝3 : FiniteDimensional ℝ E₁\ninst✝4 : NormedAddCommGroup E₂\ninst✝5 : InnerProductSpace ℝ E₂\ninst✝6 : FiniteDimensional ℝ E₂\ninst✝7 : NormedAddCommGroup F\ninst✝8 : InnerProductSpace ℝ F\ninst✝9 : FiniteDimensional ℝ F\nadmm : ADMM E₁ E₂ F\nadmm_kkt : Existance_of_kkt admm\nfullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)\nfullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nthis : (fun n => ↑(φ₁ n)) = fun n => φ (n + 1)\n⊢ StrictMono fun n => n + 1",
    "ground_truth": "· apply hphi_StrictMono",
    "model_output": "· apply StrictMono.add\n      · exact hphi_StrictMono\n      · exact strictMono_id'"
  },
  {
    "id": 179,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E₁ : Type u_1\nE₂ : Type u_2\nF : Type u_3\ninst✝ : NormedAddCommGroup E₁\ninst✝2 : InnerProductSpace ℝ E₁\ninst✝3 : FiniteDimensional ℝ E₁\ninst✝4 : NormedAddCommGroup E₂\ninst✝5 : InnerProductSpace ℝ E₂\ninst✝6 : FiniteDimensional ℝ E₂\ninst✝7 : NormedAddCommGroup F\ninst✝8 : InnerProductSpace ℝ F\ninst✝9 : FiniteDimensional ℝ F\nadmm : ADMM E₁ E₂ F\nadmm_kkt : Existance_of_kkt admm\nfullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)\nfullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nthis : (fun n => ↑(φ₁ n)) = fun n => φ (n + 1)\n⊢ StrictMono φ",
    "ground_truth": "apply hphi_StrictMono",
    "model_output": "apply StrictMono.comp (g := (fun (n : ℕ) => n + 1)) (f := φ) _ _ _ rfl\n        · apply strictMono_id\n        · apply strictMono_add\n          · apply strictMono_id\n          · apply strictMono_const"
  },
  {
    "id": 180,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "a : EReal\nc : EReal\nb : ℝ\nh : a + ↑b ≤ c\nha1 : a ≠ ⊥\nha2 : a ≠ ⊤\nhc1 : c ≠ ⊥\n⊢ 0 ≤ c - a - ↑b",
    "ground_truth": "lift a to ℝ using ⟨ha2, ha1⟩",
    "model_output": "have hc2 : c ≠ ⊤ := by\n      by_contra hc2\n      simp [hc2] at h"
  },
  {
    "id": 181,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "c : EReal\nb : ℝ\nhc1 : c ≠ ⊥\na : ℝ\nh : ↑a + ↑b ≤ c\nha1 : ↑a ≠ ⊥\nha2 : ↑a ≠ ⊤\nhc2 : ¬c = ⊤\n⊢ 0 ≤ c - ↑a - ↑b",
    "ground_truth": "lift c to ℝ using ⟨hc2, hc1⟩",
    "model_output": "have : c - a - b = c - (a + b) := by rw [← EReal.coe_add, ← EReal.coe_sub, ← EReal.coe_sub, EReal.coe_eq_coe_iff]\n          ring_nf\n          simp\n          exact ne_top_of_le_ne_top hc2 h\n        rw [this]\n        apply EReal.coe_nonneg.mpr\n        exact sub_nonneg_of_le h"
  },
  {
    "id": 182,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ns : Set E\nf : E → ℝ\nm : ℝ\nhs : Convex ℝ s\nhfun : ∀ ⦃x : E⦄,\n  x ∈ s →\n    ∀ ⦃y : E⦄,\n      y ∈ s →\n        ∀ ⦃a : ℝ⦄,\n          0 ≤ a → ∀ ⦃b : ℝ⦄, 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2\n⊢ ∀ ⦃x : E⦄,\n  x ∈ s →\n    ∀ ⦃y : E⦄,\n      y ∈ s →\n        ∀ ⦃a b : ℝ⦄,\n          0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y - a * b * (fun r => m / 2 * r ^ 2) ‖x - y‖",
    "ground_truth": "intro x hx y hy a b ha hb hab",
    "model_output": "intro x hx y hy a b ha hb hab\n    specialize hfun hx hy a ha b hb hab\n    simp at hfun\n    exact hfun"
  },
  {
    "id": 183,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "case.1:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\nt : Set E\nhscv : Convex ℝ s\nhtcv : Convex ℝ t\nhscl : _root_.IsClosed s\nhtcl : _root_.IsClosed t\n⊢ s = t → s.convex_support = t.convex_support\n\ncase.2:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\nt : Set E\nhscv : Convex ℝ s\nhtcv : Convex ℝ t\nhscl : _root_.IsClosed s\nhtcl : _root_.IsClosed t\n⊢ s.convex_support = t.convex_support → s = t",
    "ground_truth": "· exact fun h => (by rw [h])",
    "model_output": "· intro h; rw [h]"
  },
  {
    "id": 184,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\nt : Set E\nhscv : Convex ℝ s\nhtcv : Convex ℝ t\nhscl : _root_.IsClosed s\nhtcl : _root_.IsClosed t\n⊢ s = t → s.convex_support = t.convex_support",
    "ground_truth": "exact fun h => (by rw [h])",
    "model_output": "intro h; rw [h]"
  },
  {
    "id": 185,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\ns : Set E\nhs : s.Nonempty\nx : E\na✝ : x ∈ univ\n⊢ s.convex_support x > ⊥",
    "ground_truth": "simp [convex_support]",
    "model_output": "apply convex_support_gt_bot_of_nonempty x hs"
  },
  {
    "id": 186,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\ns : Set E\nhs : s.Nonempty\n⊢ ∃ x ∈ univ, s.convex_support x < ⊤",
    "ground_truth": "simp [convex_support]",
    "model_output": "rcases hs with ⟨x, hx⟩\n    use x\n    simp [convex_support]\n    rw [hx]\n    exact convex_conjugate_ge_zero x"
  },
  {
    "id": 187,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\ns : Set E\nhs : s.Nonempty\ny : E\n⊢ 0 ≤ s.convex_indicator y",
    "ground_truth": "simp [convex_indicator]",
    "model_output": "by_cases h : y ∈ s\n    · rw [convex_indicator_of_mem h]; exact 0\n    rw [convex_indicator_of_not_mem h]; exact 0"
  },
  {
    "id": 188,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\ns : Set E\nhs : s.Nonempty\ny : E\n⊢ 0 ≤ if y ∈ s then 0 else ⊤",
    "ground_truth": "split_ifs with hy <;> simp",
    "model_output": "by_cases h : y ∈ s\n    · rw [h]; simp\n    simp [h]"
  },
  {
    "id": 189,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "α : Type u_1\ninst✝ : AddCommMonoid α\nf : α → EReal\nh : ProperFunction univ f\nh1 : ∃ x, f x < ⊤\n⊢ (dom univ f).Nonempty",
    "ground_truth": "rcases h1 with ⟨x, hx⟩",
    "model_output": "rcases h1 with ⟨x, hx⟩\n  use x\n  exact mem_dom_of_mem_univ_of_lt_top hx h.1"
  },
  {
    "id": 190,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : SMul ℝ E\nf : E → EReal\nh : ∀ (x : E), f x = ⊤\n⊢ ConvexOn ℝ univ f",
    "ground_truth": "apply convex_on_p_top",
    "model_output": "apply convex_on_n_inf\n    intro x\n    exact h x"
  },
  {
    "id": 191,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ\nσ : Finset ℕ\nb : ℕ → EuclideanSpace ℝ (Fin n)\n⊢ IsClosed {z | ∃ mu, (∀ (i : { x // x ∈ σ }), 0 ≤ mu i) ∧ z = ∑ i, mu i • b ↑i}",
    "ground_truth": "rw [polyhedra_iff_cone b]",
    "model_output": "apply IsClosed.sum\n  · intro i _\n    apply IsClosed.preimage (Continuous.smul (b i))\n    apply Continuous.comp (Continuous.subtype_val (b i)) continuous_const\n  · intro i _\n    apply isClosed_Ici"
  },
  {
    "id": 192,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ\nσ : Finset ℕ\nb : ℕ → EuclideanSpace ℝ (Fin n)\n⊢ IsClosed (cone σ b)",
    "ground_truth": "exact closed_conic σ b",
    "model_output": "apply IsClosed.sInf\n  intro i\n  apply IsClosed.vsub\n  · apply IsClosed.vsub\n    · exact cone_isClosed i\n    · exact cone_isClosed i\n  · exact cone_isClosed i"
  },
  {
    "id": 193,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nX : Set E\nhs : Convex ℝ X\n⊢ ∀ ⦃x : E⦄, x ∈ X → ∀ ⦃y : E⦄, y ∈ X → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → 0 ≤ a • 0 + b • 0",
    "ground_truth": "intro x _ y _ a b _ _ _",
    "model_output": "simp [smul_zero, add_zero, mul_zero, le_refl]"
  },
  {
    "id": 194,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nX : Set E\nhs : Convex ℝ X\nthis : ConvexOn ℝ X fun x => 0\nx : E\nhx : x ∈ X\n⊢ 0 = X.convex_indicator x",
    "ground_truth": "dsimp [Set.convex_indicator]",
    "model_output": "rw [convex_indicator]"
  },
  {
    "id": 195,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nX : Set E\nhs : Convex ℝ X\nthis : ConvexOn ℝ X fun x => 0\neq : EqOn (fun x => 0) X.convex_indicator X\n⊢ ConvexOn ℝ X X.convex_indicator",
    "ground_truth": "apply ConvexOn.congr this eq",
    "model_output": "apply EqOn.convexOn eq this"
  },
  {
    "id": 196,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "V : Type u_2\ninst✝ : NormedAddCommGroup V\ninst✝2 : NormedSpace ℝ V\nι : Sort u_4\ns : ι → Set V\nh : ∀ (i : ι), Convex ℝ (s i)\nhinter : ∃ x, ∀ (i : ι), x ∈ intrinsicInterior ℝ (s i)\n⊢ ⋂ i, intrinsicClosure ℝ (s i) = intrinsicClosure ℝ (⋂ i, s i)",
    "ground_truth": "apply Subset.antisymm",
    "model_output": "rcases hinter with ⟨x, hx⟩\n  rw [intrinsicClosure_iInter h hx]\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_intrinsicClosure_eq_intrinsicClosure_iInter' h hx\n  apply iInter_in"
  },
  {
    "id": 197,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\n𝕜 : Type u_3\nf : E → F\ninst✝ : AddCommMonoid E\ninst✝2 : CompleteLinearOrder F\ninst✝3 : Ring 𝕜\ninst✝4 : LinearOrder 𝕜\ninst✝5 : IsStrictOrderedRing 𝕜\ninst✝6 : DenselyOrdered 𝕜\ninst✝7 : Module 𝕜 E\nx : E\ny : E\nhf' : strong_quasi f 𝕜\nhx : IsMinOn f univ x\nhy : IsMinOn f univ y\nneq : ¬x = y\nthis : 0 < 1\na : 𝕜\nlta : 0 < a\nalt : a < 1\neqone : a + (1 - a) = 1\nlta' : 0 < 1 - a\n⊢ f (a • x + (1 - a) • y) < max (f x) (f y)",
    "ground_truth": "apply hf' neq lta lta' eqone",
    "model_output": "have : a • x + (1 - a) • y = a • x + (1 - a) • y := by simp only [add_sub_cancel]"
  },
  {
    "id": 198,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\n𝕜 : Type u_3\nf : E → F\ninst✝ : AddCommMonoid E\ninst✝2 : CompleteLinearOrder F\ninst✝3 : Ring 𝕜\ninst✝4 : LinearOrder 𝕜\ninst✝5 : IsStrictOrderedRing 𝕜\ninst✝6 : DenselyOrdered 𝕜\ninst✝7 : Module 𝕜 E\nx : E\ny : E\nhf' : strong_quasi f 𝕜\nhx : IsMinOn f univ x\nhy : IsMinOn f univ y\nneq : ¬x = y\nthis : 0 < 1\na : 𝕜\nlta : 0 < a\nalt : a < 1\neqone : a + (1 - a) = 1\nlta' : 0 < 1 - a\nh : f (a • x + (1 - a) • y) < f y\n⊢ False",
    "ground_truth": "simp only [isMinOn_iff] at hy",
    "model_output": "have : f y ≤ f (a • x + (1 - a) • y) := by\n      have : a • x + (1 - a) • y = (1 - a) • y + a • x := by\n        rw [add_comm, add_comm a, add_comm (1 - a)]\n      rw [this]\n      apply hy (a • x + (1 - a) • y) (by simp)"
  },
  {
    "id": 199,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\n𝕜 : Type u_3\nf : E → F\ninst✝ : AddCommMonoid E\ninst✝2 : CompleteLinearOrder F\ninst✝3 : Ring 𝕜\ninst✝4 : LinearOrder 𝕜\ninst✝5 : IsStrictOrderedRing 𝕜\ninst✝6 : DenselyOrdered 𝕜\ninst✝7 : Module 𝕜 E\nx : E\ny : E\nhf' : strong_quasi f 𝕜\nhx : IsMinOn f univ x\nneq : ¬x = y\nthis : 0 < 1\na : 𝕜\nlta : 0 < a\nalt : a < 1\neqone : a + (1 - a) = 1\nlta' : 0 < 1 - a\nh : f (a • x + (1 - a) • y) < f y\nhy : f y ≤ f (a • x + (1 - a) • y)\n⊢ False",
    "ground_truth": "apply not_le_of_gt h hy",
    "model_output": "linarith [h, hy]"
  },
  {
    "id": 200,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\n⊢ Convex ℝ (f_subdifferential f x)",
    "ground_truth": "intro u",
    "model_output": "apply convex_setOf_subgradient\n    · apply convex_epigraph\n    · apply convex_univ"
  },
  {
    "id": 201,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nu : E\nhu : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖\nv : E\nhv : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ v (y - x) ≥ -ε * ‖y - x‖\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ (a • u + b • v) (y - x) ≥ -ε * ‖y - x‖",
    "ground_truth": "intro ε",
    "model_output": "intro ε εpos\n    filter_upwards [hu ε εpos, hv ε εpos] with y hy hy'\n    rw [eventually_and]\n    simp at *\n    rw [add_sub, sub_sub_sub, ← add_mul, ← sub_sub, sub_sub_sub, add_comm (f y), add_comm (f x)]\n    rw [inner_add_left, ← sub_sub, ← sub_sub, ← sub_sub, ← sub_sub, ← sub_sub]\n    simp\n    rw [← sub_sub, ← sub_sub, ← sub_sub, ← sub_sub, ← sub_sub, ← sub_sub]"
  },
  {
    "id": 202,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nH : WithLp 2 (E × F) → ℝ\nl : NNReal\nh : Differentiable ℝ H\nhl : ∀ (x y : WithLp 2 (E × F)), ‖gradient H x - gradient H y‖ ≤ ↑l * ‖x - y‖\n⊢ ∀ (x y : WithLp 2 (E × F)), ‖grad_snd H x.1 x.2 - grad_snd H y.1 y.2‖ ≤ ↑l * ‖x - y‖",
    "ground_truth": "intro z z'",
    "model_output": "intro x y\n    specialize hl x y\n    simp [grad_eq_block_grad, grad_fun_comp, grad_add_const, grad_const_mul, grad_pow, grad_mul]\n    rw [← sub_add, sub_add_eq_add_sub, sub_add, sub_add, sub_add, sub_add, sub_add]\n    simp [grad_add_const, grad_const_mul, grad_pow, grad_mul]\n    rw [← sub_add, sub_add_eq_add_sub, sub_add, sub_add, sub_add, sub_add]\n    simp [grad_add_const, grad_const_mul, grad_pow, grad_mul]\n    rw [← sub"
  },
  {
    "id": 203,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nH : WithLp 2 (E × F) → ℝ\nl : NNReal\nh : Differentiable ℝ H\nhl : ∀ (x y : WithLp 2 (E × F)), ‖gradient H x - gradient H y‖ ≤ ↑l * ‖x - y‖\n⊢ ∀ (x y : WithLp 2 (E × F)), ‖grad_snd H x.1 x.2 - grad_snd H y.1 y.2‖ ≤ ↑l * ‖x - y‖",
    "ground_truth": "intro z",
    "model_output": "intro x y\n    specialize hl x y\n    simp [grad_eq_block_grad, grad_fun_comp, grad_add_const, grad_const_mul, grad_pow, grad_mul]\n    rw [← sub_add, sub_add_eq_add_sub, sub_add, sub_add, sub_add, sub_add, sub_add]\n    simp [grad_add_const, grad_const_mul, grad_pow, grad_mul]\n    rw [← sub_add, sub_add_eq_add_sub, sub_add, sub_add, sub_add, sub_add]\n    simp [grad_add_const, grad_const_mul, grad_pow, grad_mul]\n    rw [← sub"
  },
  {
    "id": 204,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nf : E → ℝ\ng : F → ℝ\nH : WithLp 2 (E × F) → ℝ\nx0 : E\ny0 : F\nl : NNReal\nalg : BCD f g H l x0 y0\nγ : ℝ\nhγ : γ > 1\nck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)\n⊢ ∀ (k : ℕ), 0 < alg.c k",
    "ground_truth": "intro k",
    "model_output": "intro k; specialize ck k; rw [ck]; apply div_pos; repeat linarith [alg.lpos]"
  },
  {
    "id": 205,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nf : E → ℝ\ng : F → ℝ\nH : WithLp 2 (E × F) → ℝ\nx0 : E\ny0 : F\nl : NNReal\nalg : BCD f g H l x0 y0\nγ : ℝ\nhγ : γ > 1\nk : ℕ\nck : alg.c k = 1 / (γ * ↑l)\n⊢ 0 < alg.c k",
    "ground_truth": "rw [ck]",
    "model_output": "rw [ck]; apply div_pos; repeat linarith [alg.lpos, hγ]"
  },
  {
    "id": 206,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "case.1:\nE : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nf : E → ℝ\ng : F → ℝ\nH : WithLp 2 (E × F) → ℝ\nx0 : E\ny0 : F\nl : NNReal\nalg : BCD f g H l x0 y0\nγ : ℝ\nhγ : γ > 1\nk : ℕ\nck : alg.c k = 1 / (γ * ↑l)\n⊢ 0 < 1\n\ncase.2:\nE : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nf : E → ℝ\ng : F → ℝ\nH : WithLp 2 (E × F) → ℝ\nx0 : E\ny0 : F\nl : NNReal\nalg : BCD f g H l x0 y0\nγ : ℝ\nhγ : γ > 1\nk : ℕ\nck : alg.c k = 1 / (γ * ↑l)\n⊢ 0 < γ * ↑l",
    "ground_truth": "· norm_num",
    "model_output": "· exact zero_lt_one' ℝ"
  },
  {
    "id": 207,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : InnerProductSpace ℝ F\ninst✝6 : CompleteSpace F\nf : E → ℝ\ng : F → ℝ\nH : WithLp 2 (E × F) → ℝ\nx0 : E\ny0 : F\nl : NNReal\nalg : BCD f g H l x0 y0\nγ : ℝ\nhγ : γ > 1\nk : ℕ\nck : alg.c k = 1 / (γ * ↑l)\n⊢ 0 < 1",
    "ground_truth": "norm_num",
    "model_output": "norm_num"
  },
  {
    "id": 208,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nm : Type u_2\nn : Type u_3\ninst✝2 : Fintype m\ninst✝3 : Fintype n\ninst✝4 : DecidableEq n\nA : Matrix m n 𝕜\nh : Aᴴ * A = 1\n⊢ LinearIndependent 𝕜 A.colVec",
    "ground_truth": "simp [h]",
    "model_output": "apply Orthonormal.linearIndependent\n    · rw [conjTranspose_mul_self_eq_one_iff] at h\n      exact h\n    · simp [Matrix.orthonormal_colVec_iff']"
  },
  {
    "id": 209,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nV : Type u_2\ninst✝2 : NormedAddCommGroup V\ninst✝3 : InnerProductSpace 𝕜 V\nι : Type u_4\ns : Set ι\nv : ι → V\nhv : (∀ (i : ι), ‖v i‖ = 1) ∧ ∀ ⦃i j : ι⦄, ¬i = j → (⟪v i, v j⟫_𝕜) = 0\n⊢ (∀ a ∈ s, ‖v a‖ = 1) ∧ ∀ a ∈ s, ∀ a_1 ∈ s, ¬a = a_1 → (⟪v a, v a_1⟫_𝕜) = 0",
    "ground_truth": "aesop",
    "model_output": "constructor <;> intro i hi <;> specialize hv i <;> simp at * <;> simp [hi] at * <;> exact hv"
  },
  {
    "id": 210,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : TopologicalSpace E\ninst✝2 : AddCommMonoid E\n⊢ ProperFunction univ ⊤ → LowerSemicontinuousHull univ ⊤ = ⊤",
    "ground_truth": "intro h",
    "model_output": "intro h\n    exact lowerSemicontinuousHull_univ_eq_top h"
  },
  {
    "id": 211,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : TopologicalSpace E\ninst✝2 : AddCommMonoid E\nh : ProperFunction univ ⊤\n⊢ LowerSemicontinuousHull univ ⊤ = ⊤",
    "ground_truth": "exfalso",
    "model_output": "apply le_antisymm\n    · exact lowerSemicontinuousHull_le h\n    exact le_top"
  },
  {
    "id": 212,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : TopologicalSpace E\ninst✝2 : AddCommMonoid E\nh : ProperFunction univ ⊤\nh1 : univ = ∅ ∨ ∃ x ∈ univ, ⊤ x < ⊤\n⊢ False",
    "ground_truth": "simp at h1",
    "model_output": "rcases h1 with h1 | h1"
  },
  {
    "id": 213,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\nhf : ∃ x ∈ s, f x = ⊥\n⊢ Function.IsClosed (if ∃ x ∈ s, f x = ⊥ then ⊥ else ⊤) s",
    "ground_truth": "simp [hf]",
    "model_output": "simp [hf]"
  },
  {
    "id": 214,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\nhf : ∃ x ∈ s, f x = ⊥\n⊢ Epi ⊥ s = s ×ˢ univ",
    "ground_truth": "ext ⟨x, _⟩",
    "model_output": "simp [Epi]"
  },
  {
    "id": 215,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\nhf : ∃ x ∈ s, f x = ⊥\nh1 : Epi ⊥ s = s ×ˢ univ\n⊢ _root_.IsClosed (Epi ⊥ s)",
    "ground_truth": "rw [h1]",
    "model_output": "rw [h1]\n    apply IsClosed.prod hs isClosed_univ"
  },
  {
    "id": 216,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\nhf : ¬∃ x ∈ s, f x = ⊥\n⊢ Function.IsClosed (if ∃ x ∈ s, f x = ⊥ then ⊥ else ⊤) s",
    "ground_truth": "simp [hf]",
    "model_output": "simp [hf]"
  },
  {
    "id": 217,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\nhf : ¬∃ x ∈ s, f x = ⊥\n⊢ Epi ⊤ s = ∅",
    "ground_truth": "ext ⟨x, _⟩",
    "model_output": "simp [Epi]"
  },
  {
    "id": 218,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\nhsf : ¬ProperFunction s f\nhf : ¬∃ x ∈ s, f x = ⊥\nh' : Epi ⊤ s = ∅\n⊢ _root_.IsClosed (Epi ⊤ s)",
    "ground_truth": "rw [h']",
    "model_output": "rw [h']; apply isClosed_empty"
  },
  {
    "id": 219,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E\nx : E\n⊢ HasSubgradientAt f g x → Banach_HasSubgradientAt f ((toDual ℝ E) g) x",
    "ground_truth": "intro h z",
    "model_output": "intro h\n    unfold Banach_HasSubgradientAt\n    unfold HasSubgradientAt at h\n    apply h\n    intro y\n    apply h y"
  },
  {
    "id": 220,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E\nx : E\n⊢ HasSubgradientAt f g x → Banach_HasSubgradientAt f ((toDual ℝ E) g) x",
    "ground_truth": "intro h",
    "model_output": "intro h\n    unfold Banach_HasSubgradientAt\n    unfold HasSubgradientAt at h\n    apply h\n    intro y\n    apply h y"
  },
  {
    "id": 221,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E\nx : E\nh : ∀ (y : E), f y ≥ f x + ⟪g, y - x⟫_ℝ\nz : E\nhz : f z ≥ f x + ⟪g, z - x⟫_ℝ\n⊢ f z ≥ f x + ((toDual ℝ E) g) (z - x)",
    "ground_truth": "simp",
    "model_output": "simp [LinearMap.toDual_apply] at hz ⊢\n            exact hz"
  },
  {
    "id": 222,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E\nx : E\nh : ∀ (y : E), f y ≥ f x + ⟪g, y - x⟫_ℝ\nz : E\nhz : f z ≥ f x + ⟪g, z - x⟫_ℝ\n⊢ f x + ⟪g, z - x⟫_ℝ ≤ f z",
    "ground_truth": "linarith",
    "model_output": "linarith [hz]"
  },
  {
    "id": 223,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E\nx : E\n⊢ Banach_HasSubgradientAt f ((toDual ℝ E) g) x → HasSubgradientAt f g x",
    "ground_truth": "intro h z",
    "model_output": "intro h\n    unfold Banach_HasSubgradientAt at h\n    unfold HasSubgradientAt\n    simp at *\n    intro y\n    specialize h y\n    rw [← real_inner_comm, toDual_apply]\n    simp at h\n    exact h"
  },
  {
    "id": 224,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E\nx : E\n⊢ Banach_HasSubgradientAt f ((toDual ℝ E) g) x → HasSubgradientAt f g x",
    "ground_truth": "intro h",
    "model_output": "intro h\n    unfold Banach_HasSubgradientAt at h\n    unfold HasSubgradientAt\n    simp at *\n    intro y\n    specialize h y\n    rw [← real_inner_comm, toDual_apply]\n    simp at h\n    exact h"
  },
  {
    "id": 225,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E\nx : E\nh : ∀ (y : E), f y ≥ f x + ((toDual ℝ E) g) (y - x)\nz : E\nhz : f z ≥ f x + ((toDual ℝ E) g) (z - x)\n⊢ f z ≥ f x + ⟪g, z - x⟫_ℝ",
    "ground_truth": "simp at hz",
    "model_output": "simp [toDual] at hz"
  },
  {
    "id": 226,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E\nx : E\nh : ∀ (y : E), f y ≥ f x + ((toDual ℝ E) g) (y - x)\nz : E\nhz : f x + ⟪g, z - x⟫_ℝ ≤ f z\n⊢ f z ≥ f x + ⟪g, z - x⟫_ℝ",
    "ground_truth": "linarith",
    "model_output": "linarith [hz]"
  },
  {
    "id": 227,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nx : E\nf₁ : E → ℝ\nf₂ : E → ℝ\nh : f₁ = f₂\n⊢ SubderivAt f₁ x = SubderivAt f₂ x",
    "ground_truth": "ext g",
    "model_output": "rw [h]"
  },
  {
    "id": 228,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nx : E\nf₁ : E → ℝ\nf₂ : E → ℝ\nh : f₁ = f₂\ng : E\n⊢ g ∈ SubderivAt f₁ x ↔ g ∈ SubderivAt f₂ x",
    "ground_truth": "rw [h]",
    "model_output": "rw [h]"
  },
  {
    "id": 229,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ns : Set E\nf : E → EReal\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : FiniteDimensional ℝ E\ni : affine_minorant s f\ng : E →L[ℝ] ℝ\nl : E\nw : ℝ\n⊢ ∃ (_ : ∀ x ∈ s, ↑(inner ℝ l x) + ↑w ≤ f x), ∀ x ∈ s, i.h x = inner ℝ l x + w",
    "ground_truth": "simp",
    "model_output": "constructor\n  · intro x _\n    exact i.hf x\n  · intro x _\n    exact i.hg x"
  },
  {
    "id": 230,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ns : Set E\nf : E → EReal\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : FiniteDimensional ℝ E\ni : affine_minorant s f\ng : E →L[ℝ] ℝ\nl : E\nw : ℝ\n⊢ ∀ x ∈ s, ↑(inner ℝ l x) + ↑w ≤ f x",
    "ground_truth": "intro x hx",
    "model_output": "intro x _; exact i.2 x"
  },
  {
    "id": 231,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ns : Set E\nf : E → EReal\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : FiniteDimensional ℝ E\ni : affine_minorant s f\ng : E →L[ℝ] ℝ\nl : E\nw : ℝ\n⊢ ∀ x ∈ s, ↑(inner ℝ l x) + ↑w ≤ f x",
    "ground_truth": "intro x",
    "model_output": "intro x _; exact i.2 x"
  },
  {
    "id": 232,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ns : Set E\nf : E → EReal\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : FiniteDimensional ℝ E\ni : affine_minorant s f\ng : E →L[ℝ] ℝ\nl : E\nw : ℝ\n⊢ ∀ x ∈ s, i.h x = inner ℝ l x + w",
    "ground_truth": "intro x",
    "model_output": "intro x _\n    exact i.hf x"
  },
  {
    "id": 233,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E₁ : Type u_1\nE₂ : Type u_2\nF : Type u_3\ninst✝ : NormedAddCommGroup E₁\ninst✝2 : InnerProductSpace ℝ E₁\ninst✝3 : FiniteDimensional ℝ E₁\ninst✝4 : NormedAddCommGroup E₂\ninst✝5 : InnerProductSpace ℝ E₂\ninst✝6 : FiniteDimensional ℝ E₂\ninst✝7 : NormedAddCommGroup F\ninst✝8 : InnerProductSpace ℝ F\ninst✝9 : FiniteDimensional ℝ F\nadmm : ADMM E₁ E₂ F\nadmm_kkt : Existance_of_kkt admm\nfullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)\nfullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\n⊢ (fun n => (↑↑(φ₁ n)).toNat) = fun n => φ (n + 1)",
    "ground_truth": "ext n",
    "model_output": "ext n\n   simp only [φ, φ₁]\n   congr\n   apply Nat.add_one_sub_one\n   exact Nat.zero_lt_of_ne_zero (by apply Nat.ne_of_gt (n + 1).toNat_pos)"
  },
  {
    "id": 234,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E₁ : Type u_1\nE₂ : Type u_2\nF : Type u_3\ninst✝ : NormedAddCommGroup E₁\ninst✝2 : InnerProductSpace ℝ E₁\ninst✝3 : FiniteDimensional ℝ E₁\ninst✝4 : NormedAddCommGroup E₂\ninst✝5 : InnerProductSpace ℝ E₂\ninst✝6 : FiniteDimensional ℝ E₂\ninst✝7 : NormedAddCommGroup F\ninst✝8 : InnerProductSpace ℝ F\ninst✝9 : FiniteDimensional ℝ F\nadmm : ADMM E₁ E₂ F\nadmm_kkt : Existance_of_kkt admm\nfullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)\nfullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nn : ℕ\n⊢ (↑(φ (n + 1))).toNat = φ (n + 1)",
    "ground_truth": "simp",
    "model_output": "apply Nat.toNat_ofNat"
  },
  {
    "id": 235,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E₁ : Type u_1\nE₂ : Type u_2\nF : Type u_3\ninst✝ : NormedAddCommGroup E₁\ninst✝2 : InnerProductSpace ℝ E₁\ninst✝3 : FiniteDimensional ℝ E₁\ninst✝4 : NormedAddCommGroup E₂\ninst✝5 : InnerProductSpace ℝ E₂\ninst✝6 : FiniteDimensional ℝ E₂\ninst✝7 : NormedAddCommGroup F\ninst✝8 : InnerProductSpace ℝ F\ninst✝9 : FiniteDimensional ℝ F\nadmm : ADMM E₁ E₂ F\nadmm_kkt : Existance_of_kkt admm\nfullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)\nfullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nthis : (fun n => ↑(φ₁ n)) = fun n => φ (n + 1)\n⊢ StrictMono fun n => ↑(φ₁ n)",
    "ground_truth": "rw [this]",
    "model_output": "rw[this]"
  },
  {
    "id": 236,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "c : EReal\nb : ℝ\nhc1 : c ≠ ⊥\na : ℝ\nh : ↑a + ↑b ≤ c\nha1 : ↑a ≠ ⊥\nha2 : ↑a ≠ ⊤\nhc2 : c = ⊤\n⊢ 0 ≤ c - ↑a - ↑b",
    "ground_truth": "rw [hc2]",
    "model_output": "rw [hc2]; simp"
  },
  {
    "id": 237,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "c : EReal\nb : ℝ\nhc1 : c ≠ ⊥\na : ℝ\nh : ↑a + ↑b ≤ c\nha1 : ↑a ≠ ⊥\nha2 : ↑a ≠ ⊤\nhc2 : c = ⊤\n⊢ 0 ≤ ⊤ - ↑a - ↑b",
    "ground_truth": "simp",
    "model_output": "simp [hc2] at h"
  },
  {
    "id": 238,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "b : ℝ\na : ℝ\nha1 : ↑a ≠ ⊥\nha2 : ↑a ≠ ⊤\nc : ℝ\nhc1 : ↑c ≠ ⊥\nh : ↑a + ↑b ≤ ↑c\nhc2 : ¬↑c = ⊤\nthis : 0 ≤ ↑(c - a - b)\n⊢ 0 ≤ ↑c - ↑a - ↑b",
    "ground_truth": "exact this",
    "model_output": "rw [← EReal.coe_sub, ← EReal.coe_sub] at this"
  },
  {
    "id": 239,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "b : ℝ\na : ℝ\nha1 : ↑a ≠ ⊥\nha2 : ↑a ≠ ⊤\nc : ℝ\nhc1 : ↑c ≠ ⊥\nh : ↑a + ↑b ≤ ↑c\nhc2 : ¬↑c = ⊤\nthis : a + b ≤ c\n⊢ 0 ≤ c - a - b",
    "ground_truth": "linarith",
    "model_output": "linarith [this]"
  },
  {
    "id": 240,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "case.1:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ns : Set E\nf : E → ℝ\nm : ℝ\nhs : Convex ℝ s\nhfun : ∀ ⦃x : E⦄,\n  x ∈ s →\n    ∀ ⦃y : E⦄,\n      y ∈ s →\n        ∀ ⦃a : ℝ⦄,\n          0 ≤ a → ∀ ⦃b : ℝ⦄, 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2\n⊢ Convex ℝ s\n\ncase.2:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ns : Set E\nf : E → ℝ\nm : ℝ\nhs : Convex ℝ s\nhfun : ∀ ⦃x : E⦄,\n  x ∈ s →\n    ∀ ⦃y : E⦄,\n      y ∈ s →\n        ∀ ⦃a : ℝ⦄,\n          0 ≤ a → ∀ ⦃b : ℝ⦄, 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2\n⊢ ∀ ⦃x : E⦄,\n  x ∈ s →\n    ∀ ⦃y : E⦄,\n      y ∈ s →\n        ∀ ⦃a b : ℝ⦄,\n          0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y - a * b * (fun r => m / 2 * r ^ 2) ‖x - y‖",
    "ground_truth": "· exact hs",
    "model_output": "· apply hs"
  },
  {
    "id": 241,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ns : Set E\nf : E → ℝ\nm : ℝ\nhs : Convex ℝ s\nhfun : ∀ ⦃x : E⦄,\n  x ∈ s →\n    ∀ ⦃y : E⦄,\n      y ∈ s →\n        ∀ ⦃a : ℝ⦄,\n          0 ≤ a → ∀ ⦃b : ℝ⦄, 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2\n⊢ Convex ℝ s",
    "ground_truth": "exact hs",
    "model_output": "apply hs"
  },
  {
    "id": 242,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ns : Set E\nf : E → ℝ\nm : ℝ\nhs : Convex ℝ s\nhfun : ∀ ⦃x : E⦄,\n  x ∈ s →\n    ∀ ⦃y : E⦄,\n      y ∈ s →\n        ∀ ⦃a : ℝ⦄,\n          0 ≤ a → ∀ ⦃b : ℝ⦄, 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2\n⊢ ∀ ⦃x : E⦄,\n  x ∈ s →\n    ∀ ⦃y : E⦄,\n      y ∈ s →\n        ∀ ⦃a b : ℝ⦄,\n          0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y - a * b * (fun r => m / 2 * r ^ 2) ‖x - y‖",
    "ground_truth": "intro x",
    "model_output": "intro x hx y hy a b ha hb hab\n    specialize hfun hx hy a ha b hb hab\n    simp at hfun\n    exact hfun"
  },
  {
    "id": 243,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ns : Set E\nf : E → ℝ\nm : ℝ\nhs : Convex ℝ s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhfun : f (a • x + b • y) ≤ a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2\n⊢ f (a • x + b • y) ≤ a • f x + b • f y - a * b * (fun r => m / 2 * r ^ 2) ‖x - y‖",
    "ground_truth": "dsimp",
    "model_output": "simp [smul_eq_mul] at *"
  },
  {
    "id": 244,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ns : Set E\nf : E → ℝ\nm : ℝ\nhs : Convex ℝ s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhfun : f (a • x + b • y) ≤ a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2\n⊢ m / 2 * a * b * ‖x - y‖ ^ 2 = a * b * (m / 2 * ‖x - y‖ ^ 2)",
    "ground_truth": "ring_nf",
    "model_output": "ring_nf"
  },
  {
    "id": 245,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ns : Set E\nf : E → ℝ\nm : ℝ\nhs : Convex ℝ s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhfun : f (a • x + b • y) ≤ a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2\nthis : m / 2 * a * b * ‖x - y‖ ^ 2 = a * b * (m / 2 * ‖x - y‖ ^ 2)\n⊢ f (a • x + b • y) ≤ a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2",
    "ground_truth": "exact hfun",
    "model_output": "exact hfun"
  },
  {
    "id": 246,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\nt : Set E\nhscv : Convex ℝ s\nhtcv : Convex ℝ t\nhscl : _root_.IsClosed s\nhtcl : _root_.IsClosed t\nh : s = t\n⊢ s.convex_support = t.convex_support",
    "ground_truth": "by rw [h]",
    "model_output": "rw [h]"
  },
  {
    "id": 247,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\nt : Set E\nhscv : Convex ℝ s\nhtcv : Convex ℝ t\nhscl : _root_.IsClosed s\nhtcl : _root_.IsClosed t\nh : s = t\n⊢ s.convex_support = t.convex_support",
    "ground_truth": "rw [h]",
    "model_output": "rw [h]"
  },
  {
    "id": 248,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\nt : Set E\nhscv : Convex ℝ s\nhtcv : Convex ℝ t\nhscl : _root_.IsClosed s\nhtcl : _root_.IsClosed t\n⊢ (∀ (x : E), s.convex_support x = t.convex_support x) → s = t",
    "ground_truth": "intro h",
    "model_output": "intro hcon\n    rw [← bi_convex_support_eq s hscl, ← bi_convex_support_eq t htcl]\n    apply bi_eq hcon"
  },
  {
    "id": 249,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\nt : Set E\nhscv : Convex ℝ s\nhtcv : Convex ℝ t\nhscl : _root_.IsClosed s\nhtcl : _root_.IsClosed t\nh : ∀ (x : E), s.convex_support x = t.convex_support x\n⊢ s = t",
    "ground_truth": "ext z",
    "model_output": "rw [← bi_convex_support_eq hscl hscv, ← bi_convex_support_eq htcl htcv, h]"
  },
  {
    "id": 250,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\ns : Set E\nhs : s.Nonempty\n⊢ ∀ x ∈ univ, s.convex_support x > ⊥",
    "ground_truth": "intro x _",
    "model_output": "intro x _\n    rw [convex_support]\n    simp [hs]\n    exact bot_lt_top"
  },
  {
    "id": 251,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\ns : Set E\nhs : s.Nonempty\n⊢ ∀ x ∈ univ, s.convex_support x > ⊥",
    "ground_truth": "intro x",
    "model_output": "intro x _\n    rw [convex_support]\n    simp [hs]\n    exact bot_lt_top"
  },
  {
    "id": 252,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\ns : Set E\nhs : s.Nonempty\n⊢ univ = ∅ ∨ ∃ x ∈ univ, s.convex_support x < ⊤",
    "ground_truth": "right",
    "model_output": "rcases hs with ⟨x, hx⟩\n        right\n        use x\n        simp [convex_support]\n        rw [hx]\n        simp [convex_conjugate]\n        exact convex_indicator_nonempty s x hx"
  },
  {
    "id": 253,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\ns : Set E\nhs : s.Nonempty\n⊢ ∀ (y : E), ↑⟪0, y⟫_ℝ ≤ s.convex_indicator y",
    "ground_truth": "simp",
    "model_output": "intro y\n    simp [convex_indicator]\n    intro h\n    rw [h]\n    simp [inner_zero_left]"
  },
  {
    "id": 254,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\ns : Set E\nhs : s.Nonempty\n⊢ ∀ (y : E), 0 ≤ s.convex_indicator y",
    "ground_truth": "intro y",
    "model_output": "intro y\n    simp [convex_indicator]\n    by_cases h : y ∈ s\n    · rw [if_pos h]; exact convex_indicator_nonneg_of_mem y h\n    rw [if_neg h]; exact le_max_right _ 0"
  },
  {
    "id": 255,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ns : Set E\ns : Set E\nhs : s.Nonempty\ny : E\nhy : y ∈ s\n⊢ 0 ≤ 0",
    "ground_truth": "simp",
    "model_output": "simp"
  },
  {
    "id": 256,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "α : Type u_1\ninst✝ : AddCommMonoid α\nf : α → EReal\nh : ProperFunction univ f\nh1 : univ = ∅ ∨ ∃ x ∈ univ, f x < ⊤\n⊢ (dom univ f).Nonempty",
    "ground_truth": "simp at h1",
    "model_output": "rcases h1 with h1 | h1\n    · simp [dom] at h1\n      exact h1\n    rcases h1 with ⟨x, hx, hx1⟩\n    use x\n    simp [dom] at hx1\n    exact hx1"
  },
  {
    "id": 257,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "α : Type u_1\ninst✝ : AddCommMonoid α\nf : α → EReal\nh : ProperFunction univ f\nx : α\nhx : f x < ⊤\n⊢ x ∈ dom univ f",
    "ground_truth": "simp [hx]",
    "model_output": "simp [dom]"
  },
  {
    "id": 258,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : SMul ℝ E\nf : E → EReal\nh : ∀ (x : E), f x = ⊤\n⊢ f = ⊤",
    "ground_truth": "ext x",
    "model_output": "ext x; exact Eq.symm (h x)"
  },
  {
    "id": 259,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : SMul ℝ E\nf : E → EReal\nh : ∀ (x : E), f x = ⊤\nx : E\n⊢ f x = ⊤ x",
    "ground_truth": "exact h x",
    "model_output": "exact h x"
  },
  {
    "id": 260,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ+\nm : ℕ+\nb : Fin ↑m → ℝ\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\n⊢ ∀ x ∈ Set.univ, ∀ y ∈ Set.univ, inner ℝ (Aᵀ *ᵥ (A *ᵥ x - b) - Aᵀ *ᵥ (A *ᵥ y - b)) (x - y) ≥ 0",
    "ground_truth": "intro x",
    "model_output": "intro x _ y _"
  },
  {
    "id": 261,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem E τ σ\nhp : p.domain = ∅\n⊢ sSup\n    (Real.toEReal ''\n      (p.objective ''\n        {point |\n          point ∈ p.domain ∧\n            (∀ i ∈ τ, p.equality_constraints i point = 0) ∧ ∀ j ∈ σ, p.inequality_constraints j point ≥ 0})) =\n  ⊥",
    "ground_truth": "simp [hp]",
    "model_output": "simp [hp]"
  },
  {
    "id": 262,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ\nσ : Finset ℕ\n⊢ ∀ (b : ℕ → EuclideanSpace ℝ (Fin n)), IsClosed {z | ∃ mu, (∀ (i : { x // x ∈ σ }), 0 ≤ mu i) ∧ z = ∑ i, mu i • b ↑i}",
    "ground_truth": "intro b",
    "model_output": "intro b\n  apply IsClosed.affine_image\n  · apply IsClosed.prod (fun i ↦ IsClosed.preimage (ContinuousLinearMap.continuous (ContinuousLinearMap.smul (ContinuousLinearMap.id (EuclideanSpace ℝ (Fin n)) (EuclideanSpace ℝ (Fin n))) (b i))) (EuclideanSpace.isClosed_Ici))\n    · apply IsClosed.prod (fun i ↦ IsClosed.preimage (ContinuousLinearMap.continuous (ContinuousLinearMap.smul (ContinuousLinearMap.id (EuclideanSpace ℝ (Fin n)) (EuclideanSpace ℝ (Fin n))) ("
  },
  {
    "id": 263,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\n⊢ IsLinear f ↔ ∃ a b, f = fun x => ⟪x, a⟫_ℝ + b",
    "ground_truth": "by rfl",
    "model_output": "constructor\n    · intro h\n      exact IsLinear.linear h\n    · intro h\n      exact IsLinear.linear h"
  },
  {
    "id": 264,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\n⊢ IsLinear f ↔ ∃ a b, f = fun x => ⟪x, a⟫_ℝ + b",
    "ground_truth": "rfl",
    "model_output": "constructor\n    · intro h\n      exact IsLinear.linear h\n    · intro h\n      exact IsLinear.linear h"
  },
  {
    "id": 265,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "n : ℕ\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem (EuclideanSpace ℝ (Fin n)) τ σ\nloc : EuclideanSpace ℝ (Fin n)\nhl : Local_Minimum loc\nhf : Differentiable ℝ p.objective\nh : linearized_feasible_directions loc = posTangentConeAt FeasSet loc\n⊢ linearized_feasible_directions loc ∩ {d | ⟪gradient p.objective loc, d⟫_ℝ < 0} = ∅",
    "ground_truth": "rw [h]",
    "model_output": "rw [h]"
  },
  {
    "id": 266,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E✝\ninst✝2 : InnerProductSpace ℝ E✝\nE : Type u_2\nF : Type u_3\ninst✝3 : NormedAddCommGroup E\ninst✝4 : InnerProductSpace ℝ E\ninst✝5 : CompleteSpace E\ninst✝6 : NormedAddCommGroup F\ninst✝7 : InnerProductSpace ℝ F\ninst✝8 : CompleteSpace F\nA : E ≃L[ℝ] F\nx : F\n⊢ (ContinuousLinearMap.adjoint ((↑A).comp ↑A.symm)) x = x",
    "ground_truth": "simp",
    "model_output": "simp [ContinuousLinearMap.adjoint]"
  },
  {
    "id": 267,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E✝\ninst✝2 : InnerProductSpace ℝ E✝\nE : Type u_2\nF : Type u_3\ninst✝3 : NormedAddCommGroup E\ninst✝4 : InnerProductSpace ℝ E\ninst✝5 : CompleteSpace E\ninst✝6 : NormedAddCommGroup F\ninst✝7 : InnerProductSpace ℝ F\ninst✝8 : CompleteSpace F\nA : E ≃L[ℝ] F\nx : E\n⊢ (ContinuousLinearMap.adjoint ((↑A.symm).comp ↑A)) x = x",
    "ground_truth": "simp",
    "model_output": "simp [ContinuousLinearMap.adjoint]"
  },
  {
    "id": 268,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "case.1:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nX : Set E\nhs : Convex ℝ X\n⊢ Convex ℝ X\n\ncase.2:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nX : Set E\nhs : Convex ℝ X\n⊢ ∀ ⦃x : E⦄, x ∈ X → ∀ ⦃y : E⦄, y ∈ X → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → 0 ≤ a • 0 + b • 0",
    "ground_truth": "exact hs",
    "model_output": "· exact hs"
  },
  {
    "id": 269,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nX : Set E\nhs : Convex ℝ X\n⊢ ∀ ⦃x : E⦄, x ∈ X → ∀ ⦃y : E⦄, y ∈ X → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → 0 ≤ a • 0 + b • 0",
    "ground_truth": "intro x",
    "model_output": "simp [smul_zero, add_zero, mul_zero, le_refl]"
  },
  {
    "id": 270,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nX : Set E\nhs : Convex ℝ X\nx : E\na✝ : x ∈ X\ny : E\na✝2 : y ∈ X\na : ℝ\nb : ℝ\na✝3 : 0 ≤ a\na✝4 : 0 ≤ b\na✝5 : a + b = 1\n⊢ 0 ≤ a • 0 + b • 0",
    "ground_truth": "simp",
    "model_output": "simp [smul_zero, add_zero]"
  },
  {
    "id": 271,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nX : Set E\nhs : Convex ℝ X\nthis : ConvexOn ℝ X fun x => 0\n⊢ ∀ ⦃x : E⦄, x ∈ X → 0 = X.convex_indicator x",
    "ground_truth": "intro x hx",
    "model_output": "intro x hx\n    simp [convex_indicator] at hx\n    simp [hx]"
  },
  {
    "id": 272,
    "system": "You are a helpful Lean assistant that outputs tactic needed.",
    "prompt": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nX : Set E\nhs : Convex ℝ X\nthis : ConvexOn ℝ X fun x => 0\n⊢ ∀ ⦃x : E⦄, x ∈ X → 0 = X.convex_indicator x",
    "ground_truth": "intro x",
    "model_output": "intro x hx\n    simp [convex_indicator] at hx\n    simp [hx]"
  }
]