evaluation/valid_in.json

[
  {
    "conversations": [
      {
        "from": "human",
        "value": "t : ℝ\n⊢ HasDerivAt (fun t => 1 - t) (0 - 1) t\n:::"
      },
      {
        "from": "gpt",
        "value": "apply HasDerivAt.sub"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "c : ℝ\nx : ℝ\n⊢ HasDerivAt (fun t => c⁻¹ * t) c⁻¹ x\n:::"
      },
      {
        "from": "gpt",
        "value": "have : (fun (t : ℝ) => c⁻¹ * t) = (fun t => t * c⁻¹) := by ext t; ring"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "a : ℝ\nb : ℝ\nh₁ : ¬↑b = ⊤\nh₂ : ¬↑b = ⊥\n⊢ ↑a - ↑b ≤ 0 ↔ ↑a ≤ ↑b\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [← EReal.coe_sub]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ns : Set E\nh : s = ∅\n⊢ ∅.convex_indicator = ⊤\n:::"
      },
      {
        "from": "gpt",
        "value": "ext x"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "s : Set EReal\nthis : ∀ a ∈ -s, sInf (-s) ≤ a\na : EReal\nah : a ∈ s\n⊢ a ∈ s\n:::"
      },
      {
        "from": "gpt",
        "value": "exact ah"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "a : EReal\nb : EReal\nha1 : a ≠ ⊤\nha2 : a ≠ ⊥\nhb1 : b ≠ ⊤\nhb2 : b ≠ ⊥\n⊢ a - b > ⊥\n:::"
      },
      {
        "from": "gpt",
        "value": "lift a to ℝ using ⟨ha1, ha2⟩\n  lift b to ℝ using ⟨hb1, hb2⟩\n  rw [← EReal.coe_sub]\n  apply bot_lt_iff_ne_bot.2\n  let c := a - b\n  suffices Real.toEReal c ≠ ⊥ by exact this\n  simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "a : EReal\nb : EReal\nha1 : a ≠ ⊤\nha2 : a ≠ ⊥\nhb1 : b ≠ ⊤\nhb2 : b ≠ ⊥\n⊢ a ≥ a + (b - b)\n:::"
      },
      {
        "from": "gpt",
        "value": "lift a to ℝ using ⟨ha1, ha2⟩"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "c : ℝ\nx : ℝ\nthis : (fun t => c⁻¹ * t) = fun t => t * c⁻¹\n⊢ HasDerivAt (fun t => t * c⁻¹) c⁻¹ x\n:::"
      },
      {
        "from": "gpt",
        "value": "apply hasDerivAt_mul_const c⁻¹"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "A : real_liminf\nx : ℝ\nH : lim_inf A = x\n⊢ ∃ φ, StrictMono φ ∧ Tendsto (A.x ∘ φ) atTop (𝓝 x)\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n  use Aφ (A := A) (x := x) H\n  exact ⟨Aφ_mono (A := A) (x := x) H, α_subseq_converge (Aφ_sub H)⟩"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "case.1:\na : ℝ\nha : a ≥ 0\nha1 : a = 0\n⊢ a • ⊤ ≠ ⊥\n\ncase.2:\na : ℝ\nha : a ≥ 0\nha1 : ¬a = 0\n⊢ a • ⊤ ≠ ⊥\n:::"
      },
      {
        "from": "gpt",
        "value": "· rw [ha1];simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "α : Type u_1\na : ℝ\nf : α → EReal\ns : Set α\ng' : EReal ≃ EReal\n⊢ (⨆ m ∈ s, f m) + ↑a = ⨆ m ∈ s, f m + ↑a\n:::"
      },
      {
        "from": "gpt",
        "value": "let g : EReal ≃o EReal := ⟨g', by simp [g'];apply add_le_add_right a⟩"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nτ : Finset ℕ\nm : ℕ\nf : ℕ → EuclideanSpace ℝ (Fin n)\nh : ℕ → ℕ\ng : ℕ → EuclideanSpace ℝ (Fin n)\n⊢ f = g ∘ h\n:::"
      },
      {
        "from": "gpt",
        "value": "ext i j"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "x : ℝ\nh : ¬x = 0\ng : ℝ\nhx : x > 0\ngne : g ∉ {x.sign}\nglt : ¬g < 1\nthis : x + 1 > 0\nhg : ⟪g, 1⟫_ℝ ≤ 1\n⊢ False\n:::"
      },
      {
        "from": "gpt",
        "value": "apply glt"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "α : Type u_1\na : ℝ\nf : α → EReal\ns : Set α\ng' : EReal ≃ EReal\n⊢ ∀ {a_1 b : EReal}, a_1 + ↑a ≤ b + ↑a ↔ a_1 ≤ b\n:::"
      },
      {
        "from": "gpt",
        "value": "apply add_le_add_right a"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "α : Type u_1\ns : Set α\nf : α → EReal\nhsf : ProperFunction s f\nhdomsf : dom s f = ∅\n⊢ dom s f = ∅ ∨ ∃ x ∈ dom s f, f x < ⊤\n:::"
      },
      {
        "from": "gpt",
        "value": "left"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\ns : Type u_1\nM : Matrix s (Fin n) ℝ\ny : EuclideanSpace ℝ (Fin n)\n⊢ M *ᵥ y = WithLp.toLp 2 (M *ᵥ WithLp.ofLp y)\n:::"
      },
      {
        "from": "gpt",
        "value": "ext j"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nA : Matrix (Fin n) (Fin n) ℝ\nhdiag : ∀ (i : Fin n), A i i > 0\n⊢ ∀ (i j : Fin n), (negsqrtdiagPart A * sqrtdiagPart A) i j = 1 i j\n:::"
      },
      {
        "from": "gpt",
        "value": "intro i"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "case.1:\nE : Type u_1\ns : Set E\n⊢ s.convex_indicator = ⊤ → s = ∅\n\ncase.2:\nE : Type u_1\ns : Set E\n⊢ s = ∅ → s.convex_indicator = ⊤\n:::"
      },
      {
        "from": "gpt",
        "value": "· intro h; by_contra! hs\n    obtain ⟨x, xs⟩:= Set.nonempty_def.1 hs\n    have h1: s.convex_indicator x = 0 := by\n      simp [convex_indicator]; exact xs\n    have h2 := congrFun h x; simp [h1] at h2"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ns : Set E\nf : E → EReal\ng : E → EReal\n⊢ (∀ x ∈ s, f x ≤ g x) ↔ {p | p.1 ∈ s ∧ g p.1 ≤ p.2} ⊆ {p | p.1 ∈ s ∧ f p.1 ≤ p.2}\n:::"
      },
      {
        "from": "gpt",
        "value": "constructor"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ+\nm : ℕ+\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\n⊢ ∀ (x : EuclideanSpace ℝ (Fin ↑n)), HasGradientAt (fun x => A *ᵥ x ⬝ᵥ A *ᵥ x) (2 • Aᵀ *ᵥ A *ᵥ x) x\n:::"
      },
      {
        "from": "gpt",
        "value": "by_cases hA : A = 0"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : AddCommGroup E\ninst✝2 : Module ℝ E\nf : E → EReal\nhf : ConvexOn ℝ univ f\nx : E\nhfx : f x = ⊥\ny : E\na : ℝ\nb : ℝ\n⊢ a + a = 1\n:::"
      },
      {
        "from": "gpt",
        "value": "by norm_num"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx₀ : E\nhx : IsLocalMin f x₀\n⊢ 0 ∈ subdifferential f x₀\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n  obtain hcon := subdifferential_subset f x₀\n  apply hcon; exact first_order_optimality_condition f x₀ hx"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\nz1 : ℝ\nf : E → EReal\ny : E\nthis : {z | (y, z) ∈ {p | p.1 ∈ dom univ f ∧ f p.1 ≤ ↑p.2}} = {z | f y ≤ ↑z}\nh : z1 ∈ {z | f y < ↑z}\n⊢ ↑z1 > f y\n:::"
      },
      {
        "from": "gpt",
        "value": "exact h"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nx : E\na : E\n⊢ HasGradientAt (fun x => ⟪x, a⟫_ℝ) a x\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n  have : (fun x ↦ (inner (ℝ) x a : ℝ)) = fun x ↦ (inner (ℝ) a x : ℝ) := by\n    ext y; exact real_inner_comm a y\n  rw [this]; apply gradient_of_inner_const"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nA : Matrix (Fin n) (Fin n) ℝ\nhA : ∀ (i : Fin n), A i i ≠ 0\ni : Fin n\nj : Fin n\nh : i = j\n⊢ ∏ i ∈ univ.erase j, A i i = (∏ i, A i i) * (A j j)⁻¹\n:::"
      },
      {
        "from": "gpt",
        "value": "refine Eq.symm (CancelDenoms.inv_subst (hA j) ?_)\n      refine prod_erase_mul univ (fun «x» ↦ A «x» «x») ?_\n      exact mem_univ j"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : TopologicalSpace E\nC : Set E\nhc : IsClosed C\n⊢ ∀ (x : E), ∀ y < if x ∈ C then 0 else ⊤, ∀ᶠ (x' : E) in 𝓝 x, y < if x' ∈ C then 0 else ⊤\n:::"
      },
      {
        "from": "gpt",
        "value": "intro x y hy"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\n⊢ {p | p.1 ∈ s ∧ LowerSemicontinuousHull s f p.1 ≤ p.2} = ⋂ i, {p | p.1 ∈ s ∧ i.h p.1 ≤ p.2}\n:::"
      },
      {
        "from": "gpt",
        "value": "simp [LowerSemicontinuousHull]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\nz1 : ℝ\nf : E → EReal\ny : E\nh : z1 ∈ intrinsicInterior ℝ {z | (y, z) ∈ {p | p.1 ∈ dom univ f ∧ f p.1 ≤ ↑p.2}}\nz : ℝ\nh : f y ≤ ↑z\nhfy : ⊤ ≤ f y\n⊢ False\n:::"
      },
      {
        "from": "gpt",
        "value": "simp at hfy"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nnh : ∀ (f : Fin (n + 1) → EReal), (∀ (i : Fin (n + 1)), f i ≠ ⊥) → ∑ i, f i.succ ≠ ⊥\nf : Fin (n + 1 + 1) → EReal\np : ∀ (i : Fin (n + 1 + 1)), f i ≠ ⊥\n⊢ ∑ i, f i.succ ≠ ⊥\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [Fin.sum_univ_succ]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → EReal\nh : convex_conjugate univ f = ⊥\nx : E\nh : ↑⟪x, x⟫_ℝ - f x = ⊥\nhx1 : f x ≠ ⊤\n⊢ False\n:::"
      },
      {
        "from": "gpt",
        "value": "by_cases hx2 : f x = ⊥"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "V : Type u_2\ninst✝ : NormedAddCommGroup V\ninst✝2 : NormedSpace ℝ V\nι : Sort u_4\ns : ι → Set V\nx : V\nhx : x ∈ intrinsicClosure ℝ (⋂ i, s i)\n⊢ x ∈ ⋂ i, intrinsicClosure ℝ (s i)\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [@mem_iInter]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : E → ℝ\nx : E\nε : ℝ\nepos : ε > 0\nA : Set ℝ\nh : ∃ ε > 0, Metric.ball x ε ⊆ f ⁻¹' A\n⊢ ∃ δ > 0, ∀ (x' : E), ‖x - x'‖ ≤ δ → ‖f x' - f x‖ ≤ ε\n:::"
      },
      {
        "from": "gpt",
        "value": "rcases h with ⟨δ, dpos, h⟩"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : E → ℝ\nx : E\nε : ℝ\nepos : ε > 0\nA : Set ℝ\nδ : ℝ\ndpos : δ > 0\nh : Metric.ball x δ ⊆ f ⁻¹' A\nx' : E\nx1le : ‖x - x'‖ ≤ δ / 2\n⊢ δ / 2 < δ\n:::"
      },
      {
        "from": "gpt",
        "value": "exact half_lt_self dpos"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nu : E\nf : E → ℝ\np : E\npin : HasSubgradientWithinAt f p univ u\n⊢ p ∈ {g | HasSubgradientWithinAt f g univ u}\n:::"
      },
      {
        "from": "gpt",
        "value": "apply pin"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → EReal\ninst✝3 : FiniteDimensional ℝ E\nhf : ¬ProperFunction univ f\n⊢ ConvexOn ℝ univ (convex_conjugate univ f)\n:::"
      },
      {
        "from": "gpt",
        "value": "by_cases hf_pinf : f = ⊤"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "V : Type u_2\ninst✝ : NormedAddCommGroup V\ninst✝2 : NormedSpace ℝ V\ns : Set V\ninst✝3 : FiniteDimensional ℝ V\nhsc : Convex ℝ s\n⊢ intrinsicInterior ℝ (intrinsicInterior ℝ s) = intrinsicInterior ℝ s\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n  apply intrinsicInterior_subset.antisymm\n  nth_rw 1 [intrinsicInterior]\n  rw [intrinsicInterior, image_subset_iff]\n  rw [affineSpan_intrinsicInterior hsc]\n  rw [Function.Injective.preimage_image Subtype.val_injective]\n  simp [intrinsicInterior]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nx : E\na : E\nf : E → ℝ\nz : E\nv : E\n⊢ ‖v - (x - a)‖ ^ 2 / 2 = ‖v - x‖ ^ 2 / 2 + ⟪a, v⟫_ℝ + (‖a‖ ^ 2 / 2 - ⟪a, x⟫_ℝ)\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [← sub_add, norm_add_sq_real, real_inner_comm, inner_sub_right]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → EReal\na : ℝ\nl : E\nx : E\nfx : ℝ\nhccleq : ↑(inner ℝ x l - fx) ≤ ↑a\nhfx1 : ↑fx ≠ ⊤\nhfx2 : ↑fx ≠ ⊥\n⊢ ↑(inner ℝ l x - a) ≤ ↑fx\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [EReal.coe_le_coe_iff] at hccleq"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "V : Type u_2\ninst✝ : NormedAddCommGroup V\ninst✝2 : NormedSpace ℝ V\ns : Set V\nh : Convex ℝ s\nhn : (intrinsicInterior ℝ s).Nonempty\n⊢ intrinsicInterior ℝ (intrinsicClosure ℝ s) ⊆ intrinsicInterior ℝ s\n:::"
      },
      {
        "from": "gpt",
        "value": "intro z"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nV : Type u_2\ninst✝2 : NormedAddCommGroup V\ninst✝3 : InnerProductSpace 𝕜 V\nι : Type u_4\ninst✝4 : Fintype ι\nv : ι → V\nhv : Orthonormal 𝕜 v\n⊢ ∃ b, ∀ (i : ι), ↑(b i) = v i\n:::"
      },
      {
        "from": "gpt",
        "value": "have : FiniteDimensional 𝕜 (Submodule.span 𝕜 (Set.range v)) := by\n    apply FiniteDimensional.span_of_finite\n    simp [Set.finite_range]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nA : Matrix (Fin n) (Fin n) ℝ\nhdiag : ∀ (i : Fin n), A i i > 0\ni : Fin n\nj : Fin n\nh1 : (negsqrtdiagPart A * sqrtdiagPart A) i j = ∑ k, negsqrtdiagPart A i k * sqrtdiagPart A k j\nh2 : i = j\n⊢ IsUnit √(A i j)\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [h2]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "α : Type u_2\ninst✝ : NormedAddCommGroup α\ninst✝2 : InnerProductSpace ℝ α\ninst✝3 : CompleteSpace α\ns : Set α\nhy : ∀ (y : α), ⊥ < s.convex_support y ∧ s.convex_support y < ⊤\n⊢ ∃ C, ∀ (x : ↑s), ‖(innerSL ℝ) ↑x‖ ≤ C\n:::"
      },
      {
        "from": "gpt",
        "value": "apply banach_steinhaus"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nf : E → ℝ\nx : E\nf'x : E →L[ℝ] ℝ\n⊢ HasFDerivAt f f'x x → ∀ ε > 0, ∃ δ > 0, ∀ (x' : E), ‖x - x'‖ ≤ δ → ‖f x' - f x - f'x (x' - x)‖ ≤ ε * ‖x - x'‖\n:::"
      },
      {
        "from": "gpt",
        "value": "intro h\n    apply HasFDeriv_Convergence\n    exact h"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nf : E → ℝ\nx : E\nf'x : E →L[ℝ] ℝ\n⊢ (∀ ε > 0, ∃ δ > 0, ∀ (x' : E), ‖x - x'‖ ≤ δ → ‖f x' - f x - f'x (x' - x)‖ ≤ ε * ‖x - x'‖) → HasFDerivAt f f'x x\n:::"
      },
      {
        "from": "gpt",
        "value": "· apply Convergence_HasFDeriv"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "V : Type u_2\ninst✝ : NormedAddCommGroup V\ninst✝2 : NormedSpace ℝ V\ninst✝3 : FiniteDimensional ℝ V\nC : Set V\nhC : Convex ℝ C\nM : AffineSubspace ℝ V\nhn : ↑M ∩ intrinsicInterior ℝ C ≠ ∅\n⊢ closure (↑M ∩ C) = ↑M ∩ closure C\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n  rw [←intrinsicInterior_affineSubspace_eq_self, ←nonempty_iff_ne_empty] at hn\n  rw [←iIntersection_closure_eq_intrinsicInterior_closure' M.convex hC hn]\n  rw [closure_affineSubspace_eq_self]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nxm : E\nx₀ : E\nalg : subgradient_method f x₀\nt : ℝ\nha' : ∀ (n : ℕ), subgradient_method.a f x₀ n = t\nk : ℕ\n⊢ subgradient_method.a f x₀ 0 > 0\n:::"
      },
      {
        "from": "gpt",
        "value": "apply alg.ha 0"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nA : Matrix (Fin n) (Fin n) ℝ\nhdiag : ∀ (i : Fin n), A i i > 0\ni : Fin n\nj : Fin n\nh1 : (sqrtdiagPart A * sqrtdiagPart A) i j = ∑ k, sqrtdiagPart A i k * sqrtdiagPart A k j\nh2 : i = j\n⊢ √(A i j) * √(A j j) = A i i\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [h2]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "A : real_liminf\nx : ℝ\nε : ℝ\nk : ℕ\nhε : ε > 0\nh : x - ε < sInf {x | ∃ n, k ≤ n ∧ A.x n = x} ∧ sInf {x | ∃ n, k ≤ n ∧ A.x n = x} ≤ x\nthis : ∃ x_1 ∈ {y | ∃ n ≥ k, A.x n = y}, x_1 < x + ε\n⊢ ∃ n, k ≤ n ∧ x - ε < A.x n ∧ A.x n < x + ε\n:::"
      },
      {
        "from": "gpt",
        "value": "rcases this with ⟨xn, hxn1, hxn2⟩"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "A : real_liminf\nB : real_liminf\nhAa : BddAbove (range A.x)\nhBa : BddAbove (range B.x)\nh1 : Tendsto real_liminf.α atTop (𝓝 (lim_inf (add_real_liminf A B hAa hBa)))\n⊢ lim_inf (add_real_liminf A B hAa hBa) ≥ lim_inf A + lim_inf B\n:::"
      },
      {
        "from": "gpt",
        "value": "have h2 := tendsto_add (A:=A) (B:=B)"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : NormedAddCommGroup F\ninst✝4 : InnerProductSpace ℝ F\nx : E\ny : F\nz : WithLp 2 (E × F)\na : ℝ\nb : WithLp 2 (E × F)\n⊢ ‖a • b‖ ≤ ‖a‖ * ‖b‖\n:::"
      },
      {
        "from": "gpt",
        "value": "have : ‖a • b‖ ^ 2 ≤ (‖a‖ * ‖b‖) ^ 2 := by\n      rw [mul_pow, WithLp.prod_norm_sq_eq_of_L2, WithLp.prod_norm_sq_eq_of_L2]\n      simp only [WithLp.smul_fst, WithLp.smul_snd]\n      rw [norm_smul, norm_smul, mul_add, mul_pow, mul_pow]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nσ : Finset ℕ\nb : ℕ → EuclideanSpace ℝ (Fin n)\nz : EuclideanSpace ℝ (Fin n)\nc : { x // x ∈ σ } → ℝ\ncpos : ∀ (a : ℕ) (b : a ∈ σ), 0 ≤ c ⟨a, b⟩\nh : z = ∑ x ∈ σ.attach, c x • b ↑x\nc1 : ℕ → ℝ\ni : ℕ\n⊢ 0 ≤ if w : i ∈ σ then c ⟨i, w⟩ else 0\n:::"
      },
      {
        "from": "gpt",
        "value": "by_cases ht : i ∈ σ"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "α : Type u_1\ninst✝ : AddCommMonoid α\ninst✝2 : SMul ℝ α\nd : Set α\nhd : Convex ℝ d\nt : Fin 0 → α → EReal\nhst : ∀ (i : Fin 0), ProperFunction d (t i)\nhdti : ⋂ i, dom d (t i) ≠ ∅\nh : ∀ (i : Fin 0), ConvexOn ℝ d (t i)\n⊢ ConvexOn ℝ d (∑ i, t i)\n:::"
      },
      {
        "from": "gpt",
        "value": "simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nx : E\nf₁ : E → ℝ\nf₂ : E → ℝ\ns : Set E\nh : ∀ y ∈ s, f₁ y = f₂ y\nhf : f₁ x = f₂ x\ng : E\nhg : g ∈ SubderivWithinAt f₂ s x\ny : E\nys : y ∈ s\n⊢ f₁ y ≥ f₁ x + ⟪g, y - x⟫_ℝ\n:::"
      },
      {
        "from": "gpt",
        "value": "by rw [h y ys, hf]; exact hg y ys"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nx : E\na : EReal\nb : EReal\nhx : x ∈ s\nha : (x, a) ∈ closure {p | p.1 ∈ s ∧ f p.1 ≤ p.2}\nhab : a ≤ b\nhatop : ¬a = ⊤\nhbtop : b = ⊤\n⊢ (x, ⊤) ∈ closure {p | p.1 ∈ s ∧ f p.1 ≤ p.2}\n:::"
      },
      {
        "from": "gpt",
        "value": "exact in_of_le_sinf_of_b_top hx"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nxm : E\nf : E → ℝ\nf' : E → E\nhf : ∀ (x : E), HasGradientAt f (f' x) x\nhcon : ConvexOn ℝ univ f\nhfc : ContinuousOn f' univ\n⊢ IsMinOn f univ xm → f' xm = 0\n:::"
      },
      {
        "from": "gpt",
        "value": "intro hmin"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nx : E × EReal\nhx2 : x.2 = ⊤\nw : ℕ → E\nhw : (∀ (n : ℕ), w n ∈ s) ∧ Tendsto w atTop (𝓝 x.1)\n⊢ Tendsto (fun n => (w n, ⊤).1) atTop (𝓝 x.1) ∧ Tendsto (fun n => (w n, ⊤).2) atTop (𝓝 x.2)\n:::"
      },
      {
        "from": "gpt",
        "value": "simp [hx2, hw.2]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ns : Set E\nx : E\nconv : Convex ℝ s\ny : E\na✝ : y ∈ s\nz : E\na✝2 : z ∈ s\na : ℝ\nb : ℝ\nanneg : 0 ≤ a\nbnneg : 0 ≤ b\nabsum1 : a + b = 1\n⊢ a • y + b • z - x = a • (y - x) + b • (z - x)\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [smul_sub, smul_sub, add_comm_sub, sub_sub, ← add_smul, add_comm b a]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "𝕜 : Type u_4\nM : Type u_5\ninst✝ : Field 𝕜\ninst✝2 : LinearOrder 𝕜\ninst✝3 : IsStrictOrderedRing 𝕜\ninst✝4 : AddCommGroup M\ninst✝5 : Module 𝕜 M\ns : Set M\nhs : Convex 𝕜 s\nhne : s.Nonempty\nx : M\nhx : x ∈ (Convex.toCone s hs).addPoint\n⊢ ∃ c, 0 ≤ c ∧ ∃ y ∈ s, c • y = x\n:::"
      },
      {
        "from": "gpt",
        "value": "simp [ConvexCone.addPoint] at hx"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nf' : E → E\ns : Set E\nhs : Convex ℝ s\nhf : ∀ x ∈ s, HasGradientAt f (f' x) x\n⊢ ∀ x ∈ s, ∀ y ∈ s, ∃ c ∈ Ioo 0 1, ⟪f' (x + c • (y - x)), y - x⟫_ℝ = f y - f x\n:::"
      },
      {
        "from": "gpt",
        "value": "intro x"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "case.1:\nE : Type u_1\nz1 : ℝ\nf : E → EReal\ny : E\nh : ↑z1 > f y\nhfy : f y < ⊤\nhfy2 : f y = ⊥\nx : ℝ\nhx : ⊥ < ↑2 * ↑x\n⊢ x ∈ univ\n\ncase.2:\nE : Type u_1\nz1 : ℝ\nf : E → EReal\ny : E\nh : ↑z1 > f y\nhfy : f y < ⊤\nhfy2 : f y = ⊥\nx : ℝ\n⊢ x ∈ univ → x ∈ {x | ⊥ < ↑2 * ↑x}\n:::"
      },
      {
        "from": "gpt",
        "value": "simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedAddCommGroup F\ninst✝3 : Module ℝ F\ninst✝4 : Module ℝ E\nb : F\nA : E →ₗ[ℝ] F\n⊢ ∀ (c : ℝ) {p₁ p₂ p₃ : E},\n  p₁ ∈ {x | A x = b} → p₂ ∈ {x | A x = b} → p₃ ∈ {x | A x = b} → c • (p₁ -ᵥ p₂) +ᵥ p₃ ∈ {x | A x = b}\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n        simp only [mem_setOf_eq]\n        intro _ _ _ _  hp1 hp2 hp3\n        rw [vsub_eq_sub, vadd_eq_add, map_add, map_smul, map_sub,\n        hp1, hp2, hp3, sub_self, smul_zero, zero_add]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nx : E\nf : E → ℝ\nt : ℝ\nhfun : ConvexOn ℝ univ f\nht : 0 < t\nu : E\ng : E → ℝ\ntnonneg : 0 ≤ t\ngconv : ConvexOn ℝ univ g\ny : E\ncond : f u * t + ⟪x - u, y - u⟫_ℝ ≤ t * f y\n⊢ 0 < t\n:::"
      },
      {
        "from": "gpt",
        "value": "linarith"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nx : E\ns : Set E\nf' : E → E →L[ℝ] ℝ\nhx : x ∈ interior s\nhf : ConvexOn ℝ s f\nh : HasFDerivAt f (f' x) x\n⊢ HasFDerivAt f ((toDual ℝ E) ((toDual ℝ E).symm (f' x))) x\n:::"
      },
      {
        "from": "gpt",
        "value": "simp [h]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nxm : E\nf : E → ℝ\nf' : E → E\nhf : ∀ (x : E), HasGradientAt f (f' x) x\nmin : IsMinOn f univ xm\nhfc : ContinuousOn f' univ\nd : E\nh : DescentDirection d xm ⋯\ng : ℝ → ℝ\n⊢ g 0 = ⟪f' xm, d⟫_ℝ\n:::"
      },
      {
        "from": "gpt",
        "value": "simp [g]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : E → ℝ\nhc : LowerSemicontinuousOn f univ\nunderboundf : ∃ b, ∀ (x : E), b ≤ f x\nxn : ℕ → E × ℝ\np : E × ℝ\nxnin : ∀ (n : ℕ), xn n ∈ {p | f p.1 ≤ p.2}\nxntend : Tendsto xn atTop (𝓝 p)\nxncond : ∀ (n : ℕ), f (xn n).1 ≤ (xn n).2\n⊢ f p.1 ≤ p.2\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [Prod.tendsto_iff] at xntend"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "α : Type u_1\nf : α → EReal\ninst✝ : NormedAddCommGroup α\ninst✝2 : NormedSpace ℝ α\nhsf : ProperFunction univ f\nhf : ConvexOn ℝ (dom univ f) f\nx : α\na✝ : x ∈ univ\ny : α\na✝2 : y ∈ univ\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nh1 : f x = ⊤\n⊢ f (a • x + b • y) ≤ a • ⊤ + b • f y\n:::"
      },
      {
        "from": "gpt",
        "value": "by_cases ha1 : a = 0"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "α : Type u_1\nf : α → EReal\ninst✝ : NormedAddCommGroup α\ninst✝2 : NormedSpace ℝ α\nhsf : ProperFunction univ f\nhf : ConvexOn ℝ (dom univ f) f\nx : α\na✝ : x ∈ univ\ny : α\na✝2 : y ∈ univ\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nh1 : ¬f x = ⊤\nh2 : f y = ⊤\n⊢ f (a • x + b • y) ≤ a • f x + b • ⊤\n:::"
      },
      {
        "from": "gpt",
        "value": "by_cases hb1 : b = 0"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\nf : E → EReal\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nx : E\na✝ : x ∈ univ\nl : E\n⊢ sSup (range fun a => ⨆ (_ : ↑a ≥ convex_conjugate univ f l), ↑(inner ℝ l x - a)) =\n  sSup (range fun a => ⨆ (_ : ∀ (x : E), f x ≥ ↑(inner ℝ l x) - ↑a), ↑(inner ℝ l x - a))\n:::"
      },
      {
        "from": "gpt",
        "value": "congr"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nx : E\nf : E → ℝ\nf' : E → E →L[ℝ] ℝ\nu : Set E\nhu1 : u ∈ 𝓝 x\nhu2 : ContinuousOn f' u\nhu3 : ∀ x ∈ u, HasFDerivAt f (f' x) x\n⊢ EqOn (fun y => (toDual ℝ E).symm (f' y)) (fun y => gradient f y) u\n:::"
      },
      {
        "from": "gpt",
        "value": "intro z"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nf : E → EReal\nhf : ConvexOn ℝ univ f\nhfcl : LowerSemicontinuous f\nx : E\nhsf : ¬ProperFunction univ f\nthis : ∀ x ∈ univ, f x = ⊤ ∨ f x = ⊥\nhfex : ∀ (x : E), f x ≠ ⊥\nhtop : ∀ (x : E), f x = ⊤\nb : EReal\nhb : b < ⊤\nhb' : b = ⊥\n⊢ ⊥ < 1\n:::"
      },
      {
        "from": "gpt",
        "value": "exact Batteries.compareOfLessAndEq_eq_lt.mp rfl"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ+\nm : ℕ+\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\nb : Fin ↑m → ℝ\nμ : ℝ\nμpos : 0 < μ\nAne0 : A ≠ 0\nx₀ : EuclideanSpace ℝ (Fin ↑n)\np : LASSO A b μ μpos Ane0 x₀\nx : EuclideanSpace ℝ (Fin ↑n)\na✝ : x ∈ Set.univ\nε : ℝ\nεpos : ε > 0\n⊢ ∀ (x' : EuclideanSpace ℝ (Fin ↑n)), ‖x - x'‖ ≤ ε / ↑↑n / μ → ‖p.h x' - p.h x‖ ≤ ε\n:::"
      },
      {
        "from": "gpt",
        "value": "intro y"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → EReal\nhsf : ProperFunction univ f\nx : E\ny : E\na : ℝ\nb : ℝ\napos : 0 ≤ a\nbpos : 0 ≤ b\nabsum : a + b = 1\nha : a = 0\n⊢ convex_conjugate univ f (0 • x + b • y) ≤ ↑0 * convex_conjugate univ f x + ↑b * convex_conjugate univ f y\n:::"
      },
      {
        "from": "gpt",
        "value": "have hb : b = 1 := by linarith"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedAddCommGroup F\ninst✝3 : Module ℝ E\ninst✝4 : Module ℝ F\nT : E → F\nh : ¬∀ (x y : E) (r : ℝ), T ((1 - r) • x + r • y) = (1 - r) • T x + r • T y\nA : E →ᵃ[ℝ] F\nhEq : ⇑A = T\n⊢ ∀ (x y : E) (r : ℝ), T ((1 - r) • x + r • y) = (1 - r) • T x + r • T y\n:::"
      },
      {
        "from": "gpt",
        "value": "intro x y r"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem (EuclideanSpace ℝ (Fin n)) τ σ\nx : EuclideanSpace ℝ (Fin n)\nLx : LinearCQ x\ni : ℕ\nhi : i ∈ active_set x ∩ σ\na : EuclideanSpace ℝ (Fin n)\nb : ℝ\nhab : p.inequality_constraints i = fun y => ⟪a, y⟫_ℝ + b\n⊢ gradient (p.inequality_constraints i) x = a\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [hab]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "φ : ℝ → ℝ\nη : ℝ\nx : ℝ\ny : ℝ\nhx : x ∈ Ioo 0 η\nhy : y ∈ Ioo 0 η\nh1 : ConcaveOn ℝ (Ico 0 η) φ\nleft✝ : φ 0 = 0\nleft✝2 : ContDiffOn ℝ 1 φ (Ioo 0 η)\nleft✝3 : ContinuousAt φ 0\nh6 : ∀ x ∈ Ioo 0 η, deriv φ x > 0\nhdiff : DifferentiableAt ℝ φ x\nhx' : x ∈ Ico 0 η\nhy' : y ∈ Ico 0 η\nxgt : y < x\n⊢ φ x - φ y ≥ deriv φ x * (x - y)\n:::"
      },
      {
        "from": "gpt",
        "value": "obtain h := ConcaveOn.deriv_le_slope h1 hy' hx' xgt hdiff\n      rw [slope_def_field, le_div_iff₀] at h\n      repeat linarith"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nv : EuclideanSpace ℝ (Fin n)\nhv : v ≠ 0\ns : Set (EuclideanSpace ℝ (Fin n))\na : ℝ\napos : a > 0\nha : Metric.ball v a ⊆ s\nμ : ℝ\nvpos : 0 < ‖v‖\ndenompos : 0 < a + ‖v‖\neq : μ * ‖v‖ = (1 - μ) * a\nμle : 0 < 1 - μ\nμpos : 0 < μ\nr : ℝ\nz : EuclideanSpace ℝ (Fin n)\nzin : ‖if z = 0 then v else ‖z‖⁻¹ • (z - v)‖ < r\n⊢ z ∈ s\n:::"
      },
      {
        "from": "gpt",
        "value": "have ze : z ≠ 0 := by\n      by_contra hz; simp [hz] at zin; simp [r] at zin"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedAddCommGroup F\ninst✝3 : Module ℝ E\ninst✝4 : Module ℝ F\nT : E → F\nh : ∀ (x y : E) (r : ℝ), T ((1 - r) • x + r • y) = (1 - r) • T x + r • T y\nA : E →ᵃ[ℝ] F\nhA : ⇑A = T\nhEq : ⇑A = T\nx : E\ny : E\nr : ℝ\n⊢ T ((1 - r) • x + r • y) = (1 - r) • T x + r • T y\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [← hEq]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "R : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝ : Field R\ninst✝2 : AddCommGroup α\ninst✝3 : Module R α\ninst✝4 : AddCommGroup β\ninst✝5 : Module R β\nM : Set α\nN : Set β\nm : α\nn : β\nhm : m ∈ vectorSpan R M\nhn : n ∈ vectorSpan R N\nv : Submodule R (α × β)\nn0 : β\nhn0 : n0 ∈ N\nm0 : α\nhm0 : m0 ∈ M\n⊢ (m, n) ∈ vectorSpan R (M ×ˢ N)\n:::"
      },
      {
        "from": "gpt",
        "value": "have h1 : M -ᵥ M ⊆ Submodule.comap (LinearMap.inl R α β) v := by\n    rintro p ⟨c, hc, d, hd, rfl⟩\n    exact Submodule.subset_span ⟨⟨c, n0⟩, ⟨hc, hn0⟩, ⟨d, n0⟩, ⟨hd, hn0⟩, by simp⟩"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝6 : Nontrivial V\ninst✝7 : FiniteDimensional 𝕜 V\nT : V →ₗ[𝕜] W\nS : Set V\nh_compact : IsCompact S\n⊢ ∃ u, ‖u‖ = 1 ∧ ∀ (v : V), ‖v‖ = 1 → ‖T v‖ ≤ ‖T u‖\n:::"
      },
      {
        "from": "gpt",
        "value": "have h_nonempty : S.Nonempty := by\n    obtain ⟨a, ha⟩ := @NormedSpace.sphere_nonempty_rclike 𝕜 _ V _ _ _ 1 zero_le_one\n    use a"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "φ : ℝ → ℝ\nη : ℝ\nx : ℝ\nhx₁ : 0 < x\nhx₂ : x < η\nleft✝ : ConcaveOn ℝ (Ico 0 η) φ\nh₂ : φ 0 = 0\nh₃ : ContDiffOn ℝ 1 φ (Ioo 0 η)\nh₄ : ContinuousAt φ 0\nh₅ : ∀ (x : ℝ), 0 < x → x < η → 0 < deriv φ x\ny : ℝ\nhy : y ∈ Icc 0 x\nhy0 : y ≠ 0\ncont_φ : ContinuousOn φ (Ioo 0 η)\nisopen : IsOpen (Ioo 0 η)\nhyx : y ≠ x\nleft✝2 : 0 < y\nhy2 : y < x\n⊢ y < η\n:::"
      },
      {
        "from": "gpt",
        "value": "linarith"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\nh : Differentiable ℝ H\nz : WithLp 2 (E × F)\n⊢ HasGradientAt (fun x => H (x, z.2)) (grad_fst H z.2 z.1) z.1\n:::"
      },
      {
        "from": "gpt",
        "value": "apply DifferentiableAt.hasGradientAt"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : ProperSpace E\nf : E → ℝ\nhc : LowerSemicontinuous f\nlbdf : BddBelow (f '' univ)\nx : E\ng : E → ℝ\nepi : Set (E × ℝ)\nS : Set E\neq : S = g ⁻¹' Iic (g x)\nImS : Set ℝ\nneImS : ImS.Nonempty\n⊢ Nonempty ↑(prox_set f x) ∧ IsCompact (prox_set f x)\n:::"
      },
      {
        "from": "gpt",
        "value": "rcases lbdf with ⟨L, Lbound⟩"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ+\nm : ℕ+\nA : Matrix (Fin ↑m) (Fin ↑n) ℝ\nb : Fin ↑m → ℝ\nμ : ℝ\nμpos : 0 < μ\nAne0 : A ≠ 0\nx₀ : EuclideanSpace ℝ (Fin ↑n)\np : LASSO A b μ μpos Ane0 x₀\nx : EuclideanSpace ℝ (Fin ↑n)\na✝ : x ∈ Set.univ\nε : ℝ\nεpos : ε > 0\ny : EuclideanSpace ℝ (Fin ↑n)\nydist : ‖x - y‖ ≤ ε / ↑↑n / μ\nle : ∀ (i : Fin ↑n), ||y i| - |x i|| ≤ ε / ↑↑n / μ\n⊢ 0 < |μ|\n:::"
      },
      {
        "from": "gpt",
        "value": "simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nf' : E → E\nx : E\np : E\nhf : ∀ y ∈ Metric.closedBall x ‖p‖, HasGradientAt f (f' y) y\ng : ℝ → ℝ\ng' : ℝ → ℝ\nh1 : ∀ r ∈ Icc 0 1, HasDerivAt g (g' r) r\ne1 : f (x + p) = g 1\ne2 : f x = g 0\ne3 : ∀ (t : ℝ), ⟪f' (x + t • p), p⟫_ℝ = g' t\n⊢ 0 < 1\n:::"
      },
      {
        "from": "gpt",
        "value": "norm_num"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nf' : E → E\ns : Set E\nhs : Convex ℝ s\nhf : ∀ x ∈ s, HasGradientAt f (f' x) x\nx : E\nxs : x ∈ s\ny : E\nys : y ∈ s\ng : ℝ → ℝ\ng' : ℝ → ℝ\nh1 : ∀ r ∈ Icc 0 1, HasDerivAt g (g' r) r\ne1 : f y = g 1\ne2 : f x = g 0\n⊢ ∀ x ∈ Ioo 0 1, HasDerivAt g (g' x) x\n:::"
      },
      {
        "from": "gpt",
        "value": "intro x hx"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nf' : E → E\nhf : ∀ (x : E), HasGradientAt f (f' x) x\nx : E\np : E\ng : ℝ → ℝ\ng' : ℝ → ℝ\nh1 : ∀ (r : ℝ), HasDerivAt g (g' r) r\ne1 : f (x + p) = g 1\ne2 : f x = g 0\ne3 : ∀ (t : ℝ), ⟪f' (x + t • p), p⟫_ℝ = g' t\n⊢ ∀ x ∈ Icc 0 1, HasDerivAt g (g' x) x\n:::"
      },
      {
        "from": "gpt",
        "value": "intro x _"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nhconv : ConvexOn ℝ univ f\nv : E\n⊢ v ∈ f_subdifferential f x ↔\n  v ∈\n    {v₀ |\n      ∃ u,\n        Tendsto u atTop (𝓝 x) ∧\n          Tendsto (fun n => f (u n)) atTop (𝓝 (f x)) ∧\n            ∃ v, ∀ (n : ℕ), v n ∈ f_subdifferential f (u n) ∧ Tendsto v atTop (𝓝 v₀)}\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [mem_setOf]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx' : E\ng' : E\nx : ℕ → E\ng : ℕ → E\ny : E\nnonempty : ∀ (n : ℕ), g n ∈ SubderivAt f (x n)\nlscf : LowerSemicontinuous f\nx_converge : Tendsto x atTop (𝓝 x')\ng_converge : Tendsto g atTop (𝓝 g')\n⊢ f y ≥ lim_inf (add_real_liminf (_root_.fx lscf y nonempty x_converge g_converge) G ⋯ ⋯)\n:::"
      },
      {
        "from": "gpt",
        "value": "have inequ₂'' : lim_inf (const_real_liminf (f y)) =  f y := by\n    apply liminf_const_eq\n  rw[← inequ₂''];\n  exact inequ₂' y nonempty lscf x_converge g_converge"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "α : Type u_1\ninst✝ : NormedAddCommGroup α\ninst✝2 : InnerProductSpace ℝ α\ninst✝3 : FiniteDimensional ℝ α\nf : α → ℝ\ns : Set α\nhs_convex : Convex ℝ s\nhs_isopen : ∀ x ∈ s, ∃ ε > 0, ball x ε ⊆ s\nhf : ConvexOn ℝ s f\nx : α\nhx : x ∈ s\nr₀ : ℝ\nhr₀ : r₀ > 0 ∧ ball x r₀ ⊆ s\nr : ℝ\nr_pos : r > 0\nn : ℕ\nι : Type\nb : Module.Basis (Fin (Module.finrank ℝ α)) ℝ α\ni : ι\n⊢ ‖b i‖ ≠ 0\n:::"
      },
      {
        "from": "gpt",
        "value": "refine norm_ne_zero_iff.mpr ?_"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\nhT : SVD T r\ninst✝6 : FiniteDimensional 𝕜 V\nw : W\nx : V\nhx : T x = w\nx₁ : V\nx₂ : V\nhx₂ : x₂ ∈ (ker T)ᗮ\nxeq : x = x₁ + x₂\nhx₁ : T x₁ = 0\n⊢ ∃ c, ∑ i, c i • hT.w i = T x₂\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [← hT.span_range_v_eq_orthogonal_ker,\n      mem_span_range_iff_exists_fun] at hx₂"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "case.1:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nx : E × EReal\nhx : x.1 ∈ _root_.closure s ∧ x.2 = ⊤\n⊢ ∃ x_1, (∀ (n : ℕ), (x_1 n).1 ∈ s ∧ f (x_1 n).1 ≤ (x_1 n).2) ∧ Tendsto x_1 atTop (𝓝 x)\n\ncase.2:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nx : E × EReal\nhx : x.1 ∈ _root_.closure s ∧ x.2 = ⊤\n⊢ x ∉ botline (_root_.closure s)\n:::"
      },
      {
        "from": "gpt",
        "value": "· simp [mem_closure_iff_seq_limit] at hx\n    rcases hx with ⟨⟨w, hw⟩, hx2⟩\n    use (fun n => (w n, ⊤))\n    constructor\n    · intro n\n      simpa using hw.1 n\n    rw [Prod.tendsto_iff]\n    simp [hx2, hw.2]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem E τ σ\nlambda1 : { x // x ∈ τ } → ℝ\nlambda2 : { x // x ∈ σ } → ℝ\nmpos : lambda2 ≥ 0\na : E\nha : a ∈ FeasSet\nminL : EReal\nx : E\nhx : (x ∈ p.domain ∧ ∀ i ∈ τ, p.equality_constraints i x = 0) ∧ x ∈ p.domain ∧ ∀ j ∈ σ, 0 ≤ p.inequality_constraints j x\n⊢ ∑ i, lambda1 i * p.equality_constraints (↑i) x = 0\n:::"
      },
      {
        "from": "gpt",
        "value": "apply Finset.sum_eq_zero"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nx : E\nt : ℝ\ntpos : 0 < t\nf : E → ℝ\nz : E\ny : E\ncond : t ^ 2 * (t⁻¹ * f (t⁻¹ • z) + ‖t⁻¹ • z - t⁻¹ • x‖ ^ 2 / 2) ≤ t ^ 2 * (t⁻¹ * f (t⁻¹ • y) + ‖t⁻¹ • y - t⁻¹ • x‖ ^ 2 / 2)\ntsq : 0 < t ^ 2\n⊢ t * f (t⁻¹ • z) + ‖z - x‖ ^ 2 / 2 = t * (t * t⁻¹) * f (t⁻¹ • z) + t * t * (‖t⁻¹‖ ^ 2 * ‖z - x‖ ^ 2 / 2)\n:::"
      },
      {
        "from": "gpt",
        "value": "mul_inv_cancel₀ (by linarith),"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\nF : Type u_2\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedAddCommGroup F\ninst✝3 : Module ℝ E\ninst✝4 : Module ℝ F\nT : E → F\nh : ∀ (x y : E) (r : ℝ), T ((1 - r) • x + r • y) = (1 - r) • T x + r • T y\no : E\nx : E\ny : E\nh1 : T ((1 - 1 / 2) • (x +ᵥ o) + (1 / 2) • (y +ᵥ o)) = (1 - 1 / 2) • T (x +ᵥ o) + (1 / 2) • T (y +ᵥ o)\n⊢ (1 / 2) • (x + y) + 1 • o = (1 / 2) • (x + y) +ᵥ o\n:::"
      },
      {
        "from": "gpt",
        "value": "by rw [one_smul]; rfl"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nu : E\nh0 : (∀ ε > 0, ∀ᶠ (y : E) in 𝓝[≠] x, f y - f x - inner ℝ u (y - x) > -ε * ‖y - x‖) ↔\n  ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖\nsa : Set EReal\nh : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝[≠] x, f y - f x - inner ℝ u (y - x) > -ε * ‖y - x‖\n⊢ ∀ b ∈ upperBounds sa, 0 ≤ b\n:::"
      },
      {
        "from": "gpt",
        "value": "intro b"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_4\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\ninst✝3 : FiniteDimensional ℝ E\nC : Set E\nhC : Convex ℝ C\nhn : C.Nonempty\nK : ConvexCone ℝ (ℝ × E)\nhK : K = Convex.toCone {x | ∃ z ∈ C, (1, z) = x} ⋯\nCy : ℝ → Set E\nhCy : ∀ (y : ℝ), Cy y = {z | (y, z) ∈ K}\nhCyt : ∀ t > 0, Cy t = t • C\nt : ℝ\nw✝ : E\nc : ℝ\nhc : 0 < c\nleft✝ : c • w✝ ∈ C\nright✝ : 1 = c * t\n⊢ 0 < c * t\n:::"
      },
      {
        "from": "gpt",
        "value": "by linarith"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nconv : Convex ℝ univ\nhfun : ∀ ⦃x : E⦄,\n  x ∈ univ → ∀ ⦃y : E⦄, y ∈ univ → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y\ny : E\nyin : y ∈ univ\nz : E\nzin : z ∈ univ\na : ℝ\nb : ℝ\nanneg : 0 ≤ a\nbnneg : 0 ≤ b\nabsum1 : a + b = 1\n⊢ a • y + b • z - x = a • (y - x) + b • (z - x)\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [smul_sub, smul_sub, add_comm_sub, sub_sub, ← add_smul, add_comm b a]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_5\nF : Type u_4\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\ninst✝3 : FiniteDimensional ℝ E\ninst✝4 : NormedAddCommGroup F\ninst✝5 : NormedSpace ℝ F\ninst✝6 : FiniteDimensional ℝ F\nC : Set F\nA : E →L[ℝ] F\nhC : Convex ℝ C\nD : Set (E × F)\nM : Submodule ℝ (E × F)\nhAP : ⇑A ⁻¹' C = ⇑Proj '' (↑M ∩ D)\nx : E\nhx : x ∈ ⇑A ⁻¹' intrinsicInterior ℝ C\n⊢ (x, A x) ∈ intrinsicInterior ℝ D\n:::"
      },
      {
        "from": "gpt",
        "value": "· rw [intrinsicInterior_prod_eq_prod_intrinsicInterior]; exact mk_mem_prod (by simp) hx"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nh : E → ℝ\nf' : E → E\nx0 : E\nalg : Nesterov_first_fix_stepsize f h f' x0\nxm : E\nminφ : IsMinOn (f + h) univ xm\nk : ℕ\n⊢ Nesterov_first_fix_stepsize.γ f h f' x0 k ^ 2 / (2 * Nesterov_first_fix_stepsize.t f h f' x0 k) ≤\n  2 * ↑(Nesterov_first_fix_stepsize.l f h f' x0) / (↑k + 2) ^ 2\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [alg.γeq k, alg.teq k]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "τ : Finset ℕ\nσ : Finset ℕ\nn : ℕ\na : ℕ → EuclideanSpace ℝ (Fin n)\nb : ℕ → EuclideanSpace ℝ (Fin n)\nc : EuclideanSpace ℝ (Fin n)\nlam : { x // x ∈ τ } → ℝ\nmu : { x // x ∈ σ } → ℝ\nh1 : ∀ (i : { x // x ∈ σ }), 0 ≤ mu i\nh2 : c = ∑ i, lam i • a ↑i + ∑ i, mu i • b ↑i\nz : EuclideanSpace ℝ (Fin n)\nh31 : ∀ i ∈ τ, ⟪a i, z⟫_ℝ = 0\nh32 : ∀ i ∈ σ, ⟪b i, z⟫_ℝ ≥ 0\nh33 : ⟪c, z⟫_ℝ < 0\n⊢ ∀ i ∈ univ, 0 ≤ mu i * ⟪b ↑i, z⟫_ℝ\n:::"
      },
      {
        "from": "gpt",
        "value": "intro i _"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem E τ σ\nf : E → ℝ\nloc : E\nhl : IsLocalMinOn f FeasSet loc\nhf : DifferentiableAt ℝ f loc\nv : E\nvt : ∃ c d, (∃ a, ∀ (b : ℕ), a ≤ b → loc + d b ∈ FeasSet) ∧ Tendsto c atTop atTop ∧ Tendsto (fun n => c n • d n) atTop (𝓝 v)\n⊢ ⟪gradient f loc, v⟫_ℝ ≥ 0\n:::"
      },
      {
        "from": "gpt",
        "value": "rcases vt with ⟨c, d, ⟨a, ha⟩, ⟨vt1, vt2⟩⟩"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "α : Type u_1\ninst✝ : NormedAddCommGroup α\ninst✝2 : InnerProductSpace ℝ α\ninst✝3 : FiniteDimensional ℝ α\nf : α → ℝ\ns : Set α\nhs : IsOpen s\nh : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, Convex ℝ t ∧ IsOpen t ∧ BddAbove (f '' t)\nhf : ConvexOn ℝ s f\nx : α\nhx : x ∈ s\nt : Set α\nht : t ∈ 𝓝[s] x ∧ Convex ℝ t ∧ IsOpen t ∧ BddAbove (f '' t)\nt_isOpen : IsOpen t\nisopen : IsOpen (t ∩ s)\nu : Set α\nhu : IsOpen u ∧ x ∈ u ∧ u ∩ s ⊆ t\n⊢ x ∈ t\n:::"
      },
      {
        "from": "gpt",
        "value": "exact hu.2.2 (mem_inter hu.2.1 hx)"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : Type u_3\ninst✝ : Fintype n\ninst✝2 : DecidableEq n\nA : Matrix n n ℝ\nhT_pos : T A > 0\nhA_pos : ∀ (i j : n), 0 ≤ A i j\nhA_row_sum : ∀ (i : n), ∑ j, A i j ≤ 1\nhA_col_sum : ∀ (j : n), ∑ i, A i j ≤ 1\nj : n\nhj_col_lt : ∑ i, A i j < 1\nx : n\nhi_row : ∑ j, A x j < 1\nε : ℝ\nhε_def : ε = min (1 - ∑ j, A x j) (1 - ∑ i, A i j)\nhε_pos : 0 < ε\nC : Matrix n n ℝ\n⊢ A x j + ε + ∑ x_1 with ¬x_1 = j, A x x_1 = ε + ∑ x_1, A x x_1\n:::"
      },
      {
        "from": "gpt",
        "value": "have h1 :A x j + ε + ∑ x_1 with ¬x_1 = j, A x x_1 = ε + A x j + ∑ x_1\n              with ¬x_1 = j, A x x_1 := by ring_nf"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝6 : Nontrivial V\ninst✝7 : FiniteDimensional 𝕜 V\nT : V →ₗ[𝕜] W\nu : V\nhu : ‖u‖ = 1\nhT : ∀ (v : V), ‖v‖ = 1 → ‖T v‖ ≤ ‖T u‖\nhnorm : ∀ (v : V), ‖T v‖ = ‖(toContinuousLinearMap T) v‖\n⊢ ∀ (x : V), ‖(toContinuousLinearMap T) x‖ ≤ ‖T u‖ * ‖x‖\n:::"
      },
      {
        "from": "gpt",
        "value": "intro x"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : SeminormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nf : E → ℝ\nx : E\ns : Set E\nhf : ConvexOn ℝ s f\nhc : ContinuousOn f (interior s)\nhx : x ∈ interior s\nhepi_f₁ : Convex ℝ (interior (Epi f s))\ndisj : (x, f x) ∉ interior (Epi f s)\nφ : StrongDual ℝ (E × ℝ)\nhφ : ∀ b ∈ interior (Epi f s), φ (x, f x) < φ b\ng : Module.Dual ℝ E × Module.Dual ℝ ℝ\na : E × ℝ\n⊢ φ a = ((Module.dualProdDualEquivDual ℝ E ℝ) g) a\n:::"
      },
      {
        "from": "gpt",
        "value": "apply LinearMap.congr_fun\n      ((LinearEquiv.symm_apply_eq (Module.dualProdDualEquivDual ℝ E ℝ)).1 (by rfl)) a"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "α : Type u_1\ninst✝ : NormedAddCommGroup α\ninst✝2 : InnerProductSpace ℝ α\ninst✝3 : FiniteDimensional ℝ α\nf : α → ℝ\ns : Set α\nhs : IsOpen s\nh : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, Convex ℝ t ∧ IsOpen t ∧ BddAbove (f '' t)\nhf : ConvexOn ℝ s f\nx : α\nhx : x ∈ s\nt : Set α\nht : t ∈ 𝓝[s] x ∧ Convex ℝ t ∧ IsOpen t ∧ BddAbove (f '' t)\nt_isOpen : IsOpen t\nisopen : IsOpen (t ∩ s)\nx_pos : x ∈ t ∩ s\n⊢ ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t\n:::"
      },
      {
        "from": "gpt",
        "value": "rw[Metric.isOpen_iff] at isopen"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nh_noncrit : 0 ∉ subdifferential f x\nc : ℝ\nhc_pos : c > 0\nh : ∀ (u : E), ‖u - x‖ + ‖f u - f x‖ < c → EMetric.infEdist 0 (subdifferential f u) ≥ ENNReal.ofReal c\nφ : ℝ → ℝ\nu : E\nhu : u ∈ Metric.ball x (c / 2) ∩ {y | f x < f y ∧ f y < f x + c / 2}\n⊢ ENNReal.ofReal (deriv φ (f u - f x)) * EMetric.infEdist 0 (subdifferential f u) ≥ ENNReal.ofReal 1\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [deriv_of_const_mul_func (f u - f x)]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\nF : Type u_2\ninst✝ : CompleteLinearOrder F\nf : E → F\ninst✝2 : TopologicalSpace E\ninst✝3 : TopologicalSpace F\ninst✝4 : OrderTopology F\ninst✝5 : FirstCountableTopology E\ninst✝6 : FirstCountableTopology F\nhf : LowerSemicontinuous f\ny : F\nh1 : (f ⁻¹' Iic y).Nonempty\nh2 : IsCompact (f ⁻¹' Iic y)\nhs : {x | ∃ x_1 ∈ f ⁻¹' Iic y, f x_1 = x}.Nonempty\nhs' : BddBelow {x | ∃ x_1 ∈ f ⁻¹' Iic y, f x_1 = x}\n⊢ ∃ x, IsMinOn f univ x\n:::"
      },
      {
        "from": "gpt",
        "value": "rcases exists_seq_tendsto_sInf hs hs' with ⟨fx, _, cfx, fxs⟩"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nC : ℝ\nn : ℕ\n⊢ Φ n < C\n:::"
      },
      {
        "from": "gpt",
        "value": "induction n with"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nf' : E → E\ns : Set E\nh₁ : Convex ℝ s\nhf : ∀ x ∈ s, HasGradientAt f (f' x) x\nmono : ∀ x ∈ s, ∀ y ∈ s, ⟪f' x - f' y, x - y⟫_ℝ ≥ 0\nx : E\nxs : x ∈ s\ny : E\nys : y ∈ s\ng : ℝ → ℝ\ng' : ℝ → ℝ\nh : ℝ → E\nthis : g = f ∘ h\nt : ℝ\nht : t ∈ Icc 0 1\nthis : ⟪f' (x + t • (y - x)), y - x⟫_ℝ = ((toDual ℝ E) (f' (x + t • (y - x)))) (y - x)\n⊢ 0 ≤ t ∧ t ≤ 1\n:::"
      },
      {
        "from": "gpt",
        "value": "simp at ht"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "case.1:\nn : ℕ\nσ : Finset ℕ\nb : ℕ → EuclideanSpace ℝ (Fin n)\nz : EuclideanSpace ℝ (Fin n)\nc : { x // x ∈ σ } → ℝ\ncpos : ∀ (a : ℕ) (b : a ∈ σ), 0 ≤ c ⟨a, b⟩\nh : z = ∑ x ∈ σ.attach, c x • b ↑x\nc1 : ℕ → ℝ\n⊢ ∀ (i : ℕ), 0 ≤ c1 i\n\ncase.2:\nn : ℕ\nσ : Finset ℕ\nb : ℕ → EuclideanSpace ℝ (Fin n)\nz : EuclideanSpace ℝ (Fin n)\nc : { x // x ∈ σ } → ℝ\ncpos : ∀ (a : ℕ) (b : a ∈ σ), 0 ≤ c ⟨a, b⟩\nh : z = ∑ x ∈ σ.attach, c x • b ↑x\nc1 : ℕ → ℝ\n⊢ ∑ x ∈ σ, c1 x • b x = z\n:::"
      },
      {
        "from": "gpt",
        "value": "· intro i; simp [c1];\n      by_cases ht : i ∈ σ\n      · simp [ht];\n        specialize cpos i ht;\n        exact cpos\n      simp [ht]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E → ℝ\ng' : E → E\nx : E\nhg : ∀ (x₁ : E), HasGradientAt g (g' x₁) x₁\ngcon : Continuous g\ngradcon : Continuous g'\nz : E\nu : ℕ → E\nhux : Tendsto u atTop (𝓝 x)\nhufx : Tendsto (fun n => (f + g) (u n)) atTop (𝓝 ((f + g) x))\nv : ℕ → E\nvlim : ∀ (n : ℕ), v n ∈ f_subdifferential (f + g) (u n) ∧ Tendsto v atTop (𝓝 z)\n⊢ g' x ∈ subdifferential g x\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [subdiff_gradient g g' hg gradcon]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nu : E\nv : E\nhu : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖\nhv : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, (-f) y - (-f) x - inner ℝ v (y - x) ≥ -ε * ‖y - x‖\nheq : ¬u + v = 0\nε : ℝ\nεpos : ε > 0\nδ : ℝ\nδpos : δ > 0\nhne : ‖u + v‖ ≠ 0\nhb : x + (δ / 2 / ‖u + v‖) • (u + v) ∈ Metric.ball x δ\nhd : δ / 2 * ‖u + v‖ ≤ ε * (|δ / 2| / ‖u + v‖ * ‖u + v‖)\n⊢ ‖u + v‖ ≤ ε\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [div_mul_cancel₀ _ hne, abs_of_nonneg (by positivity), mul_comm] at hd"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\ns : Set E\nx : E × ℝ\nφ : Dual ℝ E × Dual ℝ ℝ\nf : E → EReal\nhsf : ProperFunction s f\nh : E × ℝ →L[ℝ] ℝ\nw : ℝ\nu : ℝ\nhua : ∀ a ∈ Function.Epi f s, h a < u\nhux : u < h x\nphi_add : ∀ (x : E × ℝ), h x = φ.1 x.1 + w * x.2\nhw : w < 0\nil : E →ₗ[ℝ] ℝ\ni1 : E →ᵃ[ℝ] ℝ\nz : E\nhz : z ∈ s\nhfx : ¬f z = ⊤\nthis : (z, (f z).toReal) ∈ Function.Epi f s\nthis : h (z, (f z).toReal) < u\n⊢ (f z).toReal * w ≤ -φ.1 z + u\n:::"
      },
      {
        "from": "gpt",
        "value": "simp [phi_add] at this"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : E → EReal\nx : E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : FiniteDimensional ℝ E\nhp : ProperFunction univ f\nh : ConvexOn ℝ univ f\nhx : x ∈ dom univ f\nhf : Function.IsClosed f univ\ny : E\nγ : ℝ → E\nφ : ℝ → EReal\nh1 : ConvexOn ℝ univ φ\nα : EReal\nh_lsc : ∀ (y : EReal), ∃ u, _root_.IsClosed u ∧ f ⁻¹' Iic y ∩ univ = u ∩ univ\nu : Set E\nhu_closed : _root_.IsClosed u\nhu_eq : f ⁻¹' Iic α ∩ univ = u ∩ univ\n⊢ _root_.IsClosed {z | f z ≤ α}\n:::"
      },
      {
        "from": "gpt",
        "value": "have h_eq : {z | f z ≤ α} = f ⁻¹' Iic α ∩ univ := by\n      ext z\n      simp [Set.mem_setOf_eq, Set.mem_preimage, Set.mem_Iic]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\ninst✝3 : FiniteDimensional ℝ E\nf : E → EReal\nhp : ProperFunction univ f\nhc : ConvexOn ℝ (dom univ f) f\nx : E\nhx : x ∈ intrinsicInterior ℝ (dom univ f)\nclf_leq_f : Function.closure f univ x ≤ f x\nhclf_top : ¬Function.closure f univ x = ⊤\nh13 : Convex ℝ (Epi f univ)\nx' : E\ny : ℝ\nt : EReal\nht : t = f x'\nhx : x' ∈ intrinsicInterior ℝ (dom univ f)\nfx_le_top : t < ⊤\nfx_ge_bot : t > ⊥\n⊢ t ≤ ↑y ↔ t.toReal ≤ y\n:::"
      },
      {
        "from": "gpt",
        "value": "lift t to ℝ using ⟨LT.lt.ne_top fx_le_top, LT.lt.ne_bot fx_ge_bot⟩"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E → ℝ\ng' : E → E\nx : E\nhg : ∀ (x₁ : E), HasGradientAt g (g' x₁) x₁\ngcon : Continuous g\ngradcon : Continuous g'\nz : E\nzin : ∃ u,\n  Tendsto u atTop (𝓝 x) ∧\n    Tendsto (fun n => (f + g) (u n)) atTop (𝓝 ((f + g) x)) ∧\n      ∃ v, ∀ (n : ℕ), v n ∈ f_subdifferential (f + g) (u n) ∧ Tendsto v atTop (𝓝 z)\n⊢ ∃ zf ∈ subdifferential f x, ∃ zg ∈ subdifferential g x, z = zf + zg\n:::"
      },
      {
        "from": "gpt",
        "value": "rcases zin with ⟨u,hux,hufx,hv⟩"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nh : 0 < ADMM.τ E₁ E₂ F ∧ ADMM.τ E₁ E₂ F ≤ 1\n⊢ 0 < ADMM.τ E₁ E₂ F\n:::"
      },
      {
        "from": "gpt",
        "value": "by linarith"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "𝕜 : Type u_4\nM : Type u_5\ninst✝ : Field 𝕜\ninst✝2 : LinearOrder 𝕜\ninst✝3 : IsStrictOrderedRing 𝕜\ninst✝4 : AddCommGroup M\ninst✝5 : Module 𝕜 M\ns : Set M\nt : Set M\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nconv_cone_s_cone_t : Set M\ncone_conv_s_t : ConvexCone 𝕜 M\nhs1 : s ⊆ ↑cone_conv_s_t\nht1 : t ⊆ ↑cone_conv_s_t\nh1 : ↑(toCone_Point s hs) ⊆ ↑cone_conv_s_t\nh2 : ↑(toCone_Point t ht) ⊆ ↑cone_conv_s_t\n⊢ (convexHull 𝕜) (↑(toCone_Point s hs) ∪ ↑(toCone_Point t ht)) ⊆ ↑(toCone_Point ((convexHull 𝕜) (s ∪ t)) ⋯)\n:::"
      },
      {
        "from": "gpt",
        "value": "apply convexHull_min (union_subset h1 h2) cone_conv_s_t.convex"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\n⊢ ∃ r, range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r\n:::"
      },
      {
        "from": "gpt",
        "value": "have hΦ : ∃ r_Φ, range Φ ⊆ Metric.ball 0 r_Φ := by apply Φ_isBounded'"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "case.1:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E → ℝ\ng' : E → E\nx : E\nhg : ∀ (x₁ : E), HasGradientAt g (g' x₁) x₁\nz : E\nzin : z ∈ f_subdifferential (f + g) x\n⊢ g' x ∈ {g' x}\n\ncase.2:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\ng : E → ℝ\ng' : E → E\nx : E\nhg : ∀ (x₁ : E), HasGradientAt g (g' x₁) x₁\nz : E\nzin : z ∈ f_subdifferential (f + g) x\n⊢ HasGradientAt g (g' x) x\n:::"
      },
      {
        "from": "gpt",
        "value": "· simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\n⊢ Tendsto (fun n => ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖) atTop (𝓝 0)\n:::"
      },
      {
        "from": "gpt",
        "value": "have : Tendsto (fun n => ‖A₁ (e₁ n) + A₂ (e₂ n)‖ ^ 2) atTop (𝓝 0) := by\n      apply Summable.tendsto_atTop_zero Summable₁\n   apply square_converge_zero₁ this"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\ns : Set E\nf : E → EReal\nhsf : ProperFunction s f\nhf : ConvexOn ℝ s f\nhfcl : Function.IsClosed f s\nhs : ¬s = ∅\nx : E\nhx : x✝ ∈ s\nhtop : f x✝ < ⊤\nl : StrongDual ℝ (E × ℝ)\nt : ℝ\nhl1 : (φ ↑l).1 (x✝, (f x✝).toReal - 1).1 + (φ ↑l).2 (x✝, (f x✝).toReal - 1).2 < t\nhl2 : ∀ b ∈ Function.Epi f s, t < l b\nthis : ∀ (x : E × ℝ), l x = ↑l x\nhl3 : 0 < u ↑l\nil : E →ₗ[ℝ] ℝ\ni1 : E →ᵃ[ℝ] ℝ\nx : E\nhx : x ∈ s\nhfx : ¬f x = ⊤\n⊢ ↑((-(φ ↑l).1 x + t) / u ↑l) ≤ f x\n:::"
      },
      {
        "from": "gpt",
        "value": "rw[← EReal.coe_toReal (x := f x) hfx (LT.lt.ne_bot (hsf.uninfinity x hx))]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\nhT : SVD T r\ninst✝6 : FiniteDimensional 𝕜 V\ni : Fin r\nu : V\nu₁ : V\nu₂ : V\nhu₂ : u₂ ∈ (span 𝕜 (Set.range hT.v))ᗮ\nhu_eq : u = u₁ + u₂\nthis : T u₂ = 0\nTu_eq_0 : T u₁ = 0\ninj : Function.Injective ⇑(Finsupp.linearCombination 𝕜 hT.w)\ncoef : Fin r →₀ 𝕜\nhu₁ : ∑ x ∈ coef.support, coef x • hT.v x = u₁\n⊢ ⟪u, hT.v i⟫_𝕜 = 0\n:::"
      },
      {
        "from": "gpt",
        "value": "by_cases u₁_eq_0 : u₁ = 0"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "case.1:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : ProperSpace E\nx₀ : E\nf : E → ℝ\nf' : E → E\nh : E → ℝ\nalg : proximal_gradient_method f h f' x₀\nk : ℕ+\nkpos : 0 < ↑↑k\n⊢ 0 < 2 * ↑↑k\n\ncase.2:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : ProperSpace E\nx₀ : E\nf : E → ℝ\nf' : E → E\nh : E → ℝ\nalg : proximal_gradient_method f h f' x₀\nk : ℕ+\nkpos : 0 < ↑↑k\n⊢ 0 < proximal_gradient_method.t f h f' x₀\n:::"
      },
      {
        "from": "gpt",
        "value": "apply mul_pos"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "α : Type u_1\ninst✝ : NormedAddCommGroup α\ninst✝2 : InnerProductSpace ℝ α\ninst✝3 : FiniteDimensional ℝ α\nf : α → ℝ\ns : Set α\nhs : IsOpen s\nh : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, Convex ℝ t ∧ IsOpen t ∧ BddAbove (f '' t)\nhf : ConvexOn ℝ s f\nx : α\nhx : x ∈ s\nt : Set α\nht : t ∈ 𝓝[s] x ∧ Convex ℝ t ∧ IsOpen t ∧ BddAbove (f '' t)\nt_isOpen : IsOpen t\nisopen : ∀ x ∈ t ∩ s, ∃ ε > 0, ball x ε ⊆ t ∩ s\nx_pos : x ∈ t ∩ s\nr : ℝ\nhr : r > 0 ∧ ball x r ⊆ t ∩ s\nthis : ball x (r / 2) ∈ 𝓝[s] x\neq_r : r - r / 2 = r / 2\n⊢ ball x (r / 2) ∈ 𝓝[s] x\n:::"
      },
      {
        "from": "gpt",
        "value": "exact this"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : Type u_3\ninst✝ : Fintype n\ninst✝2 : DecidableEq n\nA : Matrix n n ℝ\nhT_pos : T A > 0\nhA_pos : ∀ (i j : n), 0 ≤ A i j\nhA_row_sum : ∀ (i : n), ∑ j, A i j ≤ 1\nhA_col_sum : ∀ (j : n), ∑ i, A i j ≤ 1\ni : n\nhi_row : ∑ j, A i j < 1\nj : n\nhj_col_lt : ∑ i, A i j < 1\nε : ℝ\nhε_def : ε = min (1 - ∑ j, A i j) (1 - ∑ i, A i j)\nhε_pos : 0 < ε\nC : Matrix n n ℝ\nhC_DoublySubStochastic : DoublySubStochastic C\nhAC_le : ∀ (x y : n), A x y ≤ C x y\nh_L : ε = 1 - ∑ i, A i j\ncol_sum_C_eq : ∑ i, C i j = 1\n⊢ ∀ (y : n), y ≠ j → ∑ i, C i y = ∑ i, A i y\n:::"
      },
      {
        "from": "gpt",
        "value": "intro y"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nn : ℕ+\n⊢ 0 ≤ ⟪u n + (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) admm_kkt.y, ADMM.x₁ E₂ F ↑n - admm_kkt.x₁⟫_ℝ\n:::"
      },
      {
        "from": "gpt",
        "value": "calc\n      _= inner ℝ (u (n) - (- A₁† y')) (x₁ (n) - x₁') := by simp\n      _≥ (0 : ℝ) := by\n         apply subgradientAt_mono\n         · apply u_inthesubgradient\n         · exact admm_kkt.h.subgrad₁"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\nself : BCD f g H l x0 y0\nh : ∀ (x y : WithLp 2 (E × F)), ‖grad_fun_comp H x - grad_fun_comp H y‖ ≤ ↑l * ‖x - y‖\nx : E\ny1 : F\ny2 : F\nh : ‖grad_fun_comp H (x, y1) - grad_fun_comp H (x, y2)‖ ≤ ↑l * ‖(x, y1) - (x, y2)‖\n⊢ ‖grad_snd H x y1 - grad_snd H x y2‖ ≤ ↑l * ‖y1 - y2‖\n:::"
      },
      {
        "from": "gpt",
        "value": "simp [grad_fun_comp,grad_comp] at h"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\nhT : SVD T r\ninst✝6 : FiniteDimensional 𝕜 V\ni : Fin r\nu : V\nu₁ : V\nu₂ : V\nhu₂ : u₂ ∈ (span 𝕜 (Set.range hT.v))ᗮ\nhu_eq : u = u₁ + u₂\nthis : T u₂ = 0\nTu_eq_0 : T u₁ = 0\ninj : Function.Injective ⇑(Finsupp.linearCombination 𝕜 hT.w)\ncoef : Fin r →₀ 𝕜\nhu₁ : ∑ x ∈ coef.support, coef x • hT.v x = u₁\nu₁_eq_0 : u₁ = 0\n⊢ u₂ ∈ (span 𝕜 (Set.range hT.v))ᗮ\n:::"
      },
      {
        "from": "gpt",
        "value": "exact hu₂"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : Type u_3\ninst✝ : Fintype n\ninst✝2 : DecidableEq n\nA : Matrix n n ℝ\nh1 : A ∈ doublyStochastic ℝ n ↔ (∀ (i j : n), 0 ≤ A i j) ∧ (∀ (i : n), ∑ j, A i j = 1) ∧ ∀ (j : n), ∑ i, A i j = 1\nh2 : DoublySubStochastic A ↔ (∀ (i j : n), 0 ≤ A i j) ∧ (∀ (i : n), ∑ j, A i j ≤ 1) ∧ ∀ (j : n), ∑ i, A i j ≤ 1\nhA_pos : ∀ (i j : n), 0 ≤ A i j\nhA_row_sum : ∀ (i : n), ∑ j, A i j ≤ 1\nhA_col_sum : ∀ (j : n), ∑ i, A i j ≤ 1\nh_nonneg : ∀ (i j : n), 0 ≤ A i j\nh_row_eq : ∀ (i : n), ∑ j, A i j = 1\nh_col_eq : ∀ (j : n), ∑ i, A i j = 1\n⊢ {i | ∑ j, A i j < 1}.card = 0\n:::"
      },
      {
        "from": "gpt",
        "value": "apply Finset.card_eq_zero.2"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "𝕜 : Type u_1\nW : Type u_3\ninst✝ : RCLike 𝕜\ninst✝2 : NormedAddCommGroup W\ninst✝3 : InnerProductSpace 𝕜 W\nr : ℕ\nih : ∀ {V : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace 𝕜 V] [FiniteDimensional 𝕜 V] (T : V →ₗ[𝕜] W),\n  r = finrank 𝕜 ↥(range T) → Nonempty (SVD T r)\nV : Type u_2\ninst✝4 : NormedAddCommGroup V\ninst✝5 : InnerProductSpace 𝕜 V\ninst✝6 : FiniteDimensional 𝕜 V\nT : V →ₗ[𝕜] W\nhr : r + 1 = finrank 𝕜 ↥(range T)\nhV : ¬Nontrivial V\nthis : Subsingleton V\nthis : finrank 𝕜 ↥(range T) = 0\nthis : r + 1 = 0\n⊢ Nonempty (SVD T (r + 1))\n:::"
      },
      {
        "from": "gpt",
        "value": "exact False.elim (by simp at this)"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : ProperSpace E\nf : E → ℝ\nhc : LowerSemicontinuous f\nx : E\ng : E → ℝ\nepi : Set (E × ℝ)\nS : Set E\neq : S = g ⁻¹' Iic (g x)\nImS : Set ℝ\nneImS : ImS.Nonempty\nL : ℝ\nLbound : ∀ x ∈ f '' univ, L ≤ x\nboundg : ∀ (x : E), L ≤ g x\nhg : LowerSemicontinuous g\nS_bddbelow : BddBelow ImS\nepi_closed : IsClosed epi\ncloseds : IsClosed S\nB : Set E\nu : E\nuin : g u ≤ g x\nieq : ‖u - x‖ ^ 2 / 2 ≤ f x - L\na : ℝ\nb : ℝ\nh1 : a ^ 2 ≤ b * 2\nh2 : |a| = a\n⊢ 2 * b ≤ b ^ 2 + 1 + 2 * b\n:::"
      },
      {
        "from": "gpt",
        "value": "simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf₁ : E → ℝ\nf₂ : E → ℝ\nh₁ : ConvexOn ℝ univ f₁\nh₂ : ConvexOn ℝ univ f₂\nhcon : ContinuousOn f₁ univ\nx₀ : E\ng : E\nhg : HasSubgradientAt (f₁ + f₂) g x₀\nS₁ : Set (E × ℝ)\nS₂ : Set (E × ℝ)\ni : E × ℝ\nj : E × ℝ\na : ℝ\nb : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhi : a * (f₁ (i.1 + x₀) - f₁ x₀ - ⟪g, i.1⟫_ℝ) < a * i.2\nhj : b * (f₁ (j.1 + x₀) - f₁ x₀ - ⟪g, j.1⟫_ℝ) < b * j.2\nthis : x₀ = 1 • x₀\n⊢ a • i.1 + b • j.1 + (a • x₀ + b • x₀) = a • i.1 + a • x₀ + (b • j.1 + b • x₀)\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [← add_assoc, ← add_assoc]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem (EuclideanSpace ℝ (Fin n)) τ σ\nx : EuclideanSpace ℝ (Fin n)\nm : ℕ\nMx : EuclideanSpace ℝ (Fin n) →L[ℝ] EuclideanSpace ℝ { x_1 // x_1 ∈ active_set x } × (Fin (n - m) → ℝ)\nv : EuclideanSpace ℝ (Fin n)\nvne0 : v ≠ 0\nMxbij : (LinearMap.range Mx).topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c ⇑Mx\nc : NNReal\ns : Set (EuclideanSpace ℝ (Fin n))\na : ℝ\napos : a > 0\nha : Metric.ball 0 a ⊆ s\nhc : 0 ≠ c\nhc' : 0 < c\nz : EuclideanSpace ℝ (Fin n)\nzin : ‖Mx z‖ < a / ↑c\nantil : ‖z‖ ≤ ↑c * ‖Mx z‖\n⊢ z ∈ Metric.ball 0 a\n:::"
      },
      {
        "from": "gpt",
        "value": "simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nconv : Convex ℝ univ\nhfun : ∀ ⦃x : E⦄,\n  x ∈ univ → ∀ ⦃y : E⦄, y ∈ univ → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y\n⊢ ∀ ⦃x_1 : E⦄,\n  x_1 ∈ univ →\n    ∀ ⦃y : E⦄,\n      y ∈ univ →\n        ∀ ⦃a b : ℝ⦄,\n          0 ≤ a →\n            0 ≤ b →\n              a + b = 1 →\n                f (a • x_1 + b • y) + ‖a • x_1 + b • y - x‖ ^ 2 / 2 ≤\n                  a • (f x_1 + ‖x_1 - x‖ ^ 2 / 2) + b • (f y + ‖y - x‖ ^ 2 / 2) - a * b * (1 / 2 * ‖x_1 - y‖ ^ 2)\n:::"
      },
      {
        "from": "gpt",
        "value": "intro y yin z zin a b anneg bnneg absum1"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem (EuclideanSpace ℝ (Fin n)) τ σ\nx : EuclideanSpace ℝ (Fin n)\nm : ℕ\nMx : EuclideanSpace ℝ (Fin n) →L[ℝ] EuclideanSpace ℝ { x_1 // x_1 ∈ active_set x } × (Fin (n - m) → ℝ)\nv : EuclideanSpace ℝ (Fin n)\nvne0 : v ≠ 0\nMxbij : (LinearMap.range Mx).topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c ⇑Mx\nc : NNReal\ns : Set (EuclideanSpace ℝ (Fin n))\na : ℝ\napos : a > 0\nha : Metric.ball 0 a ⊆ s\nhc : 0 ≠ c\nhc' : 0 < c\nz : EuclideanSpace ℝ (Fin n)\nzin : ‖Mx z‖ < a / ↑c\nantil : ‖z‖ ≤ ↑c * ‖Mx z‖\n⊢ ↑c * ‖Mx z‖ < ↑c * (a / ↑c)\n:::"
      },
      {
        "from": "gpt",
        "value": "gcongr"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem E τ σ\nf : E → ℝ\nloc : E\nhf : DifferentiableAt ℝ f loc\nv : E\nc : ℕ → ℝ\nd : ℕ → E\na : ℕ\nha : ∀ (b : ℕ), a ≤ b → loc + d b ∈ FeasSet\nvt1 : Tendsto c atTop atTop\nvt2 : Tendsto (fun n => c n • d n) atTop (𝓝 v)\nproneg : ⟪gradient f loc, v⟫_ℝ < 0\ns : Set E\nhs : s ∈ 𝓝 loc ⊓ 𝓟 FeasSet\nhs2 : ∀ y ∈ s, f loc ≤ f y\nε : ℝ\nεpos : ε > 0\nεball : Metric.ball loc ε ∩ FeasSet ⊆ s\ns1 : Set E\nhmin : ∀ y ∈ s1, f loc ≤ f y\nz : ℕ → E\n⊢ False\n:::"
      },
      {
        "from": "gpt",
        "value": "have hzd : ∀ n, d n = z n - loc := fun _ => eq_sub_of_add_eq' rfl"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nr1 : ℝ\nh_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1\nr2 : ℝ\nh_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2\nr : ℝ\n⊢ r = r1 + r2\n:::"
      },
      {
        "from": "gpt",
        "value": "rfl"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf₁ : E → ℝ\nf₂ : E → ℝ\nh₁ : ConvexOn ℝ univ f₁\nh₂ : ConvexOn ℝ univ f₂\nhcon : ContinuousOn f₁ univ\nx₀ : E\ng : E\nhg : HasSubgradientAt (f₁ + f₂) g x₀\nS₁ : Set (E × ℝ)\nS₂ : Set (E × ℝ)\nhs1 : Convex ℝ S₁\nhs2 : Convex ℝ S₂\nhint : Disjoint S₁ S₂\nhso : IsOpen S₁\nf : StrongDual ℝ (E × ℝ)\nc : ℝ\nhsl : ∀ a ∈ S₁, f a < c\nhsr : ∀ b ∈ S₂, c ≤ f b\na : E\nb : ℝ\nhab : ∀ (p : E × ℝ), f p = ⟪a, p.1⟫_ℝ + b * p.2\nhin : (0, 0) ∈ S₂\nhc0 : c ≤ 0\nhin1 : ∀ t > 0, (0, t) ∈ S₁\nceq0 : c = 0\nhtp : b * 1 < 0\n⊢ b < 0\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [mul_one] at htp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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_kkt₁ fullrank₁ fullrank₂)\n⊢ Tendsto (⇑(OptProblem.A₂ E₁)) (𝓝 0) (𝓝 ((OptProblem.A₂ E₁) 0))\n:::"
      },
      {
        "from": "gpt",
        "value": "apply Continuous.tendsto"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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_kkt₁ fullrank₁ fullrank₂)\n⊢ Tendsto (fun n => (OptProblem.A₂ E₁) (e₂ (φ n))) atTop (𝓝 0)\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n   have h₁: Tendsto A₂ (𝓝 0) (𝓝 (A₂ 0)) := by\n      apply Continuous.tendsto; apply ContinuousLinearMap.continuous\n   simp at h₁; apply Filter.tendsto_iff_seq_tendsto.1 h₁; apply e₂_subseq_converge_zero"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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_kkt₁ fullrank₁ fullrank₂)\nC : ℝ\n⊢ Tendsto (fun n => ‖(OptProblem.A₁ E₂) (e₁ n)‖) atTop (𝓝 0)\n:::"
      },
      {
        "from": "gpt",
        "value": "apply A₁e₁_converge_zero"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : Type u_3\ninst✝ : Fintype n\ninst✝2 : DecidableEq n\nA : Matrix n n ℝ\nh1 : A ∈ doublyStochastic ℝ n ↔ (∀ (i j : n), 0 ≤ A i j) ∧ (∀ (i : n), ∑ j, A i j = 1) ∧ ∀ (j : n), ∑ i, A i j = 1\nh2 : DoublySubStochastic A ↔ (∀ (i j : n), 0 ≤ A i j) ∧ (∀ (i : n), ∑ j, A i j ≤ 1) ∧ ∀ (j : n), ∑ i, A i j ≤ 1\nhA_pos : ∀ (i j : n), 0 ≤ A i j\nhA_row_sum : ∀ (i : n), ∑ j, A i j ≤ 1\nhA_col_sum : ∀ (j : n), ∑ i, A i j ≤ 1\nh : {i | ∑ j, A i j < 1}.card + {j | ∑ i, A i j < 1}.card = 0\nh_row_card : {i | ∑ j, A i j < 1}.card = 0\nh_col_card : {j | ∑ i, A i j < 1}.card = 0\nh3 : ∀ (i : n), ∑ j, A i j = 1\n⊢ ∀ (j : n), ∑ i, A i j = 1\n:::"
      },
      {
        "from": "gpt",
        "value": "intro j"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "α : Type u_1\ninst✝ : NormedAddCommGroup α\ninst✝2 : InnerProductSpace ℝ α\ninst✝3 : FiniteDimensional ℝ α\nx : α\nr : ℝ\nhr : r > 0\nhn : finrank ℝ α ≠ 0\nx₀ : α\nhx₀ : ‖σ (x₀ - x)‖ < r\nsum_le_r : ∑ i, ‖(finBasis ℝ α).equivFun (x₀ - x) i‖ * ‖(finBasis ℝ α) i‖ / r ≤ 1\nn : ℕ\nι : Type\nι₀ : Type\nw₀ : Fin (finrank ℝ α) → ℝ\nrepr : ∑ i, w₀ i • (finBasis ℝ α) i = x₀ - x\nw₁ : ι → ℝ\nsum : ℝ\nsum_pos : 1 - sum ≥ 0\nw : ι₀ → ℝ\nlem_i : ∀ {i : ℕ}, i < n + 2 → ¬i = n + 1 → ¬i = n → i < n\nn_pos : n > 0\nfin0 : ι\nz : ι₀ → α\nhw₀ : ∀ (i : ι₀), 0 ≤ w i\nthis : n + 1 + 1 = n + 2\nthis : ∑ x, w₁ x = sum\n⊢ ∑ x, w₁ x + (1 - sum) / 2 + (1 - sum) / 2 = 1\n:::"
      },
      {
        "from": "gpt",
        "value": "rw[this]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ+\nt : ℝ\nμ : ℝ\nh : EuclideanSpace ℝ (Fin ↑n) → ℝ\nlasso : h = fun y => μ • ∑ i, ‖y i‖\nx : EuclideanSpace ℝ (Fin ↑n)\nxm : EuclideanSpace ℝ (Fin ↑n)\ntpos : 0 < t\nμpos : 0 < μ\ng : EuclideanSpace ℝ (Fin ↑n) → ℝ\ngeqth : g = t • h\ntμpos : 0 < t * μ\ny : EuclideanSpace ℝ (Fin ↑n)\ninv_eq : 1 / (t * μ) = μ⁻¹ * t⁻¹\nexpand_term : ∀ (i : Fin ↑n), ((μ⁻¹ * t⁻¹) • (x - xm)) i * (y - xm) i = μ⁻¹ * t⁻¹ * (x i - xm i) * (y i - xm i)\ni : Fin ↑n\na✝ : i ∈ Finset.univ\nabs_subg : SubderivAt abs (xm i) = if xm i = 0 then Set.Icc (-1) 1 else {sign (xm i)}\nhxm : xm i = 0\nminpoint : x i = 0 ∨ |x i| ≤ t * μ\naux : |x i| ≤ t * μ\n⊢ 0 ≤ μ⁻¹\n:::"
      },
      {
        "from": "gpt",
        "value": "· simp; linarith [μpos]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nn : ℕ+\nc : F\n⊢ (fun x =>\n    ADMM.ρ E₁ E₂ F / 2 *\n      ‖(OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂‖ ^ 2) =\n  fun x => ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₁ E₂) x - c‖ ^ 2 / 2\n:::"
      },
      {
        "from": "gpt",
        "value": "ext x"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "α : Type u_1\ninst✝ : NormedAddCommGroup α\ninst✝2 : InnerProductSpace ℝ α\ninst✝3 : FiniteDimensional ℝ α\nf : α → ℝ\ns : Set α\nhs_convex : Convex ℝ s\nhs_isopen : ∀ x ∈ s, ∃ ε > 0, ball x ε ⊆ s\nhf : ConvexOn ℝ s f\nx : α\nhx : x ∈ s\nr₀ : ℝ\nhr₀ : r₀ > 0 ∧ ball x r₀ ⊆ s\nr : ℝ\nr_pos : r > 0\nn : ℕ\nι : Type\nb : Module.Basis ι ℝ α\nbi_pos : ∀ (i : ι), ‖b i‖ ≠ 0\nhn : ¬n = 0\nn_pos : n > 0\nfin0 : ι\nu₀ : Set α\nu : Set α\nu₀_isFinite : u₀.Finite\nu_isFinite : u.Finite\nu_pos : u ⊆ s\nU : Set α\nt : Set α\nt_isopen : IsOpen t\nxBall : Set α\nxBall_isopen : IsOpen xBall\nx_in_xBall : x ∈ xBall\n⊢ t ∈ 𝓝[s] x ∧ Convex ℝ t ∧ IsOpen t ∧ BddAbove (f '' t)\n:::"
      },
      {
        "from": "gpt",
        "value": "have xBall_in_U : xBall ⊆ U := by\n    apply l1Ball_sub_convexHull r_pos hn"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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_kkt₁ fullrank₁ fullrank₂)\n⊢ Tendsto (fun n => ‖(OptProblem.A₁ E₂) (e₁ (φ n)) + (OptProblem.A₂ E₁) (e₂ (φ n))‖ ^ 2) atTop (𝓝 0)\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [← zero_pow]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nn : ℕ\nh1 : 0 < ADMM.τ E₁ E₂ F\nright✝ : ADMM.τ E₁ E₂ F < (1 + √5) / 2\nh2 : Ψ n ≥ 0\n⊢ max (1 - ADMM.τ E₁ E₂ F) (1 - 1 / ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F *\n    ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ ^ 2 ≥\n  0\n:::"
      },
      {
        "from": "gpt",
        "value": "apply mul_nonneg"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_4\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\ninst✝3 : FiniteDimensional ℝ E\nι : Type\ninst✝4 : Fintype ι\nC : ι → Set E\nhC : ∀ (i : ι), Convex ℝ (C i)\nhne : ∀ (i : ι), (C i).Nonempty\nih : ∀ {ι : Type} [inst : Fintype ι] (C : ι → Set E),\n  (∀ (i : ι), Convex ℝ (C i)) →\n    (∀ (i : ι), (C i).Nonempty) →\n      Fintype.card ι = 0 →\n        intrinsicInterior ℝ ((convexHull ℝ) (⋃ i, C i)) =\n          ⋃ s ∈ {t | (∀ (i : ι), t i > 0) ∧ ∑ i, t i = 1}, ∑ i, s i • intrinsicInterior ℝ (C i)\nhι : Fintype.card ι = 1\ni₀ : ι\nhi₀ : ∀ (y : ι), y = i₀\nx✝ : ι → ℝ\nl : ι → ℝ\nh : l ∈ {fun i => 1}\n⊢ l ∈ {t | (∀ (i : ι), t i > 0) ∧ ∑ i, t i = 1}\n:::"
      },
      {
        "from": "gpt",
        "value": "by simp at h; simpa [h]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nn : ℕ\n⊢ -(ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ) =\n  ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [← mul_neg]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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_kkt₁ fullrank₁ fullrank₂)\nthis : e₂ = fun n => ADMM.x₂ E₁ F n - x₂''\nh : Tendsto (ADMM.x₂ E₁ F) atTop (𝓝 x₂'')\n⊢ Tendsto (ADMM.x₂ E₁ F) atTop (𝓝 x₂'')\n:::"
      },
      {
        "from": "gpt",
        "value": "exact h"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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)\ndk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l)\nk : ℕ\nlip : ∀ (x y : WithLp 2 (E × F)), ‖gradient H x - gradient H y‖ ≤ ↑l * ‖x - y‖\ncpos' : (alg.c k)⁻¹ ≥ 0\ndpos' : (alg.d k)⁻¹ ≥ 0\nh1 : ‖(subdiff k).1‖ ≤ ↑l * (γ + 1) * ‖z (k + 1) - z k‖\na : F\n⊢ 1 / alg.d k * ‖alg.y (k + 1) - alg.y k‖ = 1 / alg.d k * ‖(z (k + 1) - z k).2‖\n:::"
      },
      {
        "from": "gpt",
        "value": "by rw [z]; simp; left; rw [z]; simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "case.1:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nhconv : ConvexOn ℝ univ f\ng : E\nh : ∀ (y : E), f y ≥ f x + inner ℝ g (y - x)\nε : ℝ\nεpos : ε > 0\ny : E\n⊢ -ε * ‖y - x‖ ≤ ?h.mpr.h.b\n\ncase.2:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nhconv : ConvexOn ℝ univ f\ng : E\nh : ∀ (y : E), f y ≥ f x + inner ℝ g (y - x)\nε : ℝ\nεpos : ε > 0\ny : E\n⊢ ?h.mpr.h.b ≤ f y - f x - inner ℝ g (y - x)\n\ncase.3:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nhconv : ConvexOn ℝ univ f\ng : E\nh : ∀ (y : E), f y ≥ f x + inner ℝ g (y - x)\nε : ℝ\nεpos : ε > 0\ny : E\n⊢ ℝ\n:::"
      },
      {
        "from": "gpt",
        "value": "· simp\n      apply neg_le.mpr\n      · have pos: - 0 ≤ ε * ‖y - x‖:=by\n          simp\n          rw [mul_comm]\n          apply (mul_nonneg_iff_left_nonneg_of_pos εpos).mpr\n          simp\n        apply pos"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝7 : ProperSpace E\ninst✝8 : ProperSpace F\nγ : ℝ\nhγ : γ > 1\nck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)\ndk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l)\nbd : Bornology.IsBounded (z '' univ)\nhψ : KL_function ψ\nlbdψ : BddBelow (ψ '' univ)\nthis : ∃ z_ ∈ closure (z '' univ), ∃ α, StrictMono α ∧ Tendsto (fun n => z (α n)) atTop (𝓝 z_)\n⊢ ∃ M, ∀ (n : ℕ), ∑ k ∈ Finset.range n, ‖z (k + 1) - z k‖ ≤ M\n:::"
      },
      {
        "from": "gpt",
        "value": "rcases this with ⟨z_, _, α, ⟨monoa, conv⟩⟩"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\n⊢ ∀ (n : ℕ+) (x : E₁),\n  HasGradientAt\n    (fun x =>\n      ⟪ADMM.y E₁ E₂ n.natPred,\n        (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ)\n    ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ n.natPred)) x\n:::"
      },
      {
        "from": "gpt",
        "value": "intro n x"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\nhT : SVD T r\ninst✝6 : FiniteDimensional 𝕜 V\ni : Fin r\nu : V\nu₁ : V\nu₂ : V\nhu₂ : u₂ ∈ (span 𝕜 (Set.range hT.v))ᗮ\nhu_eq : u = u₁ + u₂\nthis : T u₂ = 0\ninj : Function.Injective ⇑(Finsupp.linearCombination 𝕜 hT.w)\ncoef : Fin r →₀ 𝕜\nhu₁ : ∑ x ∈ coef.support, coef x • hT.v x = u₁\nu₁_eq_0 : ¬u₁ = 0\nthis : ∀ (x : Fin r), coef x • T (hT.v x) = (coef x * ↑↑(hT.σ x)) • hT.w x\nTu_eq_0 : ∑ x ∈ coef.support, (coef x * ↑↑(hT.σ x)) • hT.w x = 0\na : Fin r\n⊢ a ∈ coef.support ↔ coef a • ↑↑(hT.σ a) ≠ 0\n:::"
      },
      {
        "from": "gpt",
        "value": "simp [coef.mem_support_toFun]; rfl"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "α : Type u_1\ninst✝ : NormedAddCommGroup α\ninst✝2 : InnerProductSpace ℝ α\ninst✝3 : FiniteDimensional ℝ α\nx : α\nr : ℝ\nhr : r > 0\nhn : finrank ℝ α ≠ 0\nx₀ : α\nhx₀ : ‖σ (x₀ - x)‖ < r\nsum_le_r : ∑ i, ‖(finBasis ℝ α).equivFun (x₀ - x) i‖ * ‖(finBasis ℝ α) i‖ / r ≤ 1\nn : ℕ\nι : Type\nι₀ : Type\nw₀ : Fin (finrank ℝ α) → ℝ\nrepr : ∑ i, w₀ i • (finBasis ℝ α) i = x₀ - x\nw₁ : ι → ℝ\nsum : ℝ\nsum_pos : 1 - sum ≥ 0\nw : ι₀ → ℝ\nlem_i : ∀ {i : ℕ}, i < n + 2 → ¬i = n + 1 → ¬i = n → i < n\nn_pos : n > 0\nfin0 : ι\nz : ι₀ → α\nhw₀ : ∀ (i : ι₀), 0 ≤ w i\nhw₁ : ∑ i, w i = 1\ni : ℕ\nhi : i < n + 2\n⊢ z ⟨i, hi⟩ ∈ (⋃ i, {(r / ‖(finBasis ℝ α) i‖) • (finBasis ℝ α) i}) ∪ ⋃ i, {-(r / ‖(finBasis ℝ α) i‖) • (finBasis ℝ α) i}\n:::"
      },
      {
        "from": "gpt",
        "value": "simp only [Set.mem_union, Set.mem_iUnion, Set.mem_singleton_iff]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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_kkt₁ fullrank₁ fullrank₂)\nn : ℕ\nx : F\nxx : F\nh1 : ‖x‖ = ‖x + xx - xx‖\n⊢ ‖(OptProblem.A₁ E₂) (e₁ n)‖ ≤ ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖\n:::"
      },
      {
        "from": "gpt",
        "value": "have h2 : ‖x + xx - xx‖ ≤ ‖x + xx‖ + ‖xx‖ := by apply norm_sub_le"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nconv : Convex ℝ univ\nhfun : ∀ ⦃x : E⦄,\n  x ∈ univ → ∀ ⦃y : E⦄, y ∈ univ → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y\ny : E\nyin : y ∈ univ\nz : E\nzin : z ∈ univ\na : ℝ\nb : ℝ\nanneg : 0 ≤ a\nbnneg : 0 ≤ b\nabsum1 : a + b = 1\neq1 : a • y + b • z - x = a • (y - x) + b • (z - x)\nhab : a = 1 - b\neq2 : ∀ (u v : E), ‖a • u + b • v‖ ^ 2 = b * ‖v‖ ^ 2 + a * ‖u‖ ^ 2 - a * b * ‖u - v‖ ^ 2\neq3 : y - z = y - x - (z - x)\neq4 : ∀ (u v : E), ‖a • u + b • v‖ ^ 2 ≤ b * ‖v‖ ^ 2 + a * ‖u‖ ^ 2 - a * b * ‖u - v‖ ^ 2\nu : E\nv : E\n⊢ ‖a • u + b • v‖ ^ 2 ≤ b * ‖v‖ ^ 2 + a * ‖u‖ ^ 2 - a * b * ‖u - v‖ ^ 2\n:::"
      },
      {
        "from": "gpt",
        "value": "apply eq4 u v"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\ninst✝3 : FiniteDimensional ℝ E\nf : E → EReal\nhp : ProperFunction univ f\nhc : ConvexOn ℝ (dom univ f) f\nx : E\nhx : x ∈ intrinsicInterior ℝ (dom univ f)\nclf_leq_f : Function.closure f univ x ≤ f x\nhclf_top : ¬Function.closure f univ x = ⊤\nh13 : Convex ℝ (Epi f univ)\nge_f_iff : ∀ (x' : E) (y : ℝ) (t : EReal), t = f x' → x' ∈ intrinsicInterior ℝ (dom univ f) → (t ≤ ↑y ↔ t.toReal ≤ y)\nge_fx_iff : ∀ (x' : E) (y : ℝ), x' ∈ intrinsicInterior ℝ (dom univ f) → (f x' ≤ ↑y ↔ (f x').toReal ≤ y)\nhi' : (MM x).carrier ∩ intrinsicInterior ℝ (Epi f univ) ≠ ∅\nht : (MM x).carrier ∩ Epi f univ = {x} ×ˢ {μ | (f x).toReal ≤ μ}\n⊢ Function.closure f univ x = f x\n:::"
      },
      {
        "from": "gpt",
        "value": "have : IsClosed ((MM x).carrier ∩ (f.Epi univ)) := by\n    rw [ht]; exact IsClosed.prod isClosed_singleton isClosed_Ici"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nf' : E → E\ns : Set E\nhs : Convex ℝ s\nhf : ∀ x ∈ s, HasGradientAt f (f' x) x\nmono : ∀ x ∈ s, ∀ y ∈ s, x ≠ y → ⟪f' x - f' y, x - y⟫_ℝ > 0\nx : E\nxin : x ∈ s\ny : E\nyin : y ∈ s\nxney : x ≠ y\na : ℝ\nb : ℝ\napos : 0 < a\nbpos : 0 < b\nabsum1 : a + b = 1\nh₀ : a • f x + b • f y ≤ f (a • x + b • y)\nanneg : 0 ≤ a\nbnneg : 0 ≤ b\nmono' : ∀ x ∈ s, ∀ y ∈ s, ⟪f' x - f' y, x - y⟫_ℝ ≥ 0\nleft✝ : Convex ℝ s\nconvf : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y\neq : f (a • x + b • y) ≤ a • f x + b • f y\neq2 : f (a • x + b • y) = a • f x + b • f y\nz : E\n⊢ z ∈ s\n:::"
      },
      {
        "from": "gpt",
        "value": "simp [z]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\n  (fun n => ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (φ n)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (φ n)) - OptProblem.b E₁ E₂‖)\n  atTop (𝓝 ‖(OptProblem.A₁ E₂) x₁'' + (OptProblem.A₂ E₁) x₂'' - OptProblem.b E₁ E₂‖)\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n   apply Tendsto.norm\n   apply Tendsto.sub_const\n   apply Tendsto.add\n   · apply Satisfying_equational_constraint1\n   · apply Satisfying_equational_constraint2"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : Type u_3\ninst✝ : Fintype n\ninst✝2 : DecidableEq n\nA : Matrix n n ℝ\nhT_pos : T A > 0\nhA_pos : ∀ (i j : n), 0 ≤ A i j\nhA_row_sum : ∀ (i : n), ∑ j, A i j ≤ 1\nhA_col_sum : ∀ (j : n), ∑ i, A i j ≤ 1\ni : n\nhi_row : ∑ j, A i j < 1\nj : n\nhj_col_lt : ∑ i, A i j < 1\nε : ℝ\nhε_def : ε = min (1 - ∑ j, A i j) (1 - ∑ i, A i j)\nhε_pos : 0 < ε\nC : Matrix n n ℝ\nhC_DoublySubStochastic : DoublySubStochastic C\nhAC_le : ∀ (x y : n), A x y ≤ C x y\nh_L : ε = 1 - ∑ i, A i j\ncol_sum_C_eq : ∑ i, C i j = 1\ncol_sum_C_eq_1 : ∀ (y : n), y ≠ j → ∑ i, C i y = ∑ i, A i y\nh_col_card_lt : {j | ∑ i, C i j < 1}.card < {j | ∑ i, A i j < 1}.card\nx : n\nhx : x ∈ Finset.univ ∧ ∑ j, C x j < 1\nh_univ : x ∈ Finset.univ\nh_sum_lt : ∑ j, C x j < 1\nh : ¬x = i\n⊢ ∑ j, A x j < 1\n:::"
      },
      {
        "from": "gpt",
        "value": "· have h0 : ∑ j, C x j = ∑ j, A x j := by\n              dsimp [C]\n              simp [h]\n            rw [← h0]\n            exact h_sum_lt"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nhtau : 0 < ADMM.τ E₁ E₂ F ∧ ADMM.τ E₁ E₂ F ≤ 1\nn : ℕ+\ne_y_n : ℝ\ne_y_n1 : ℝ\nA2_e2_n : ℝ\nA2_e2_n1 : ℝ\nAe_Ae_n : ℝ\nAe_Ae_n1 : ℝ\nAx_Ax_n1 : ℝ\nt_Ax_Ax_n1 : ℝ\ntt_Ax_Ax_n : ℝ\nA_diff_x : ℝ\nA_diff_x' : ℝ\nt_A_diff_x : ℝ\nt_A_diff_x' : ℝ\ntt_A_diff_x' : ℝ\ney_diff : ℝ\n⊢ 0 ≤ ey_diff - t_Ax_Ax_n1 + 2 * M (n + 1) - t_A_diff_x' - A2_e2_n1 + A2_e2_n\n:::"
      },
      {
        "from": "gpt",
        "value": "dsimp [ey_diff, t_Ax_Ax_n1, t_A_diff_x', A2_e2_n1, A2_e2_n]\n            linarith [Φ_isdescending_inequ5 n]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\n⊢ ∀ (n : ℕ+),\n  -(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ (↑n - 1)) -\n      ADMM.ρ E₁ E₂ F •\n        (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂))\n          ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n - 1)) - OptProblem.b E₁ E₂) ∈\n    SubderivAt (OptProblem.f₁ E₂ F) (ADMM.x₁ E₂ F ↑n)\n:::"
      },
      {
        "from": "gpt",
        "value": "intro n"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "case.1:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf₁ : E → ℝ\nf₂ : E → ℝ\nh₁ : ConvexOn ℝ univ f₁\nh₂ : ConvexOn ℝ univ f₂\nhcon : ContinuousOn f₁ univ\nx₀ : E\ng : E\nhg : HasSubgradientAt (f₁ + f₂) g x₀\nS₁ : Set (E × ℝ)\nS₂ : Set (E × ℝ)\nhs1 : Convex ℝ S₁\nhs2 : Convex ℝ S₂\nhint : Disjoint S₁ S₂\n⊢ ContinuousOn (fun x => f₁ (x + x₀)) univ\n\ncase.2:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf₁ : E → ℝ\nf₂ : E → ℝ\nh₁ : ConvexOn ℝ univ f₁\nh₂ : ConvexOn ℝ univ f₂\nhcon : ContinuousOn f₁ univ\nx₀ : E\ng : E\nhg : HasSubgradientAt (f₁ + f₂) g x₀\nS₁ : Set (E × ℝ)\nS₂ : Set (E × ℝ)\nhs1 : Convex ℝ S₁\nhs2 : Convex ℝ S₂\nhint : Disjoint S₁ S₂\n⊢ ContinuousOn (fun x => f₁ x₀) univ\n:::"
      },
      {
        "from": "gpt",
        "value": "· apply ContinuousOn.comp (g := f₁) (f := fun x ↦ x + x₀) (t := univ) hcon\n        · apply ContinuousOn.add continuousOn_id continuousOn_const\n        · simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nn : ℕ\n⊢ (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂ =\n  (1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)) •\n    ((ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) •\n          ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂) +\n        ADMM.y E₁ E₂ n -\n      ADMM.y E₁ E₂ n)\n:::"
      },
      {
        "from": "gpt",
        "value": "simp only [one_div, mul_inv_rev, add_sub_cancel_right]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "τ : Finset ℕ\nσ : Finset ℕ\nn : ℕ\na : ℕ → EuclideanSpace ℝ (Fin n)\nb : ℕ → EuclideanSpace ℝ (Fin n)\nc : EuclideanSpace ℝ (Fin n)\nh : ¬∃ z, (∀ i ∈ τ, ⟪a i, z⟫_ℝ = 0) ∧ (∀ i ∈ σ, ⟪b i, z⟫_ℝ ≥ 0) ∧ ⟪c, z⟫_ℝ < 0\nh1 : ¬∃ lam mu, (∀ (i : { x // x ∈ σ }), 0 ≤ mu i) ∧ c = ∑ i, lam i • a ↑i + ∑ i, mu i • b ↑i\nS : Set (EuclideanSpace ℝ (Fin n))\nsc : IsClosed S\nx : EuclideanSpace ℝ (Fin n)\ny : EuclideanSpace ℝ (Fin n)\na1 : ℝ\nb1 : ℝ\nanng : 0 < a1\nbnng : 0 < b1\na✝ : a1 + b1 = 1\nsx : { x // x ∈ τ } → ℝ\ntx : { x // x ∈ σ } → ℝ\ntxcond : ∀ (i : { x // x ∈ σ }), 0 ≤ tx i\neqx : x = ∑ i, sx i • a ↑i + ∑ i, tx i • b ↑i\nsy : { x // x ∈ τ } → ℝ\nty : { x // x ∈ σ } → ℝ\ntycond : ∀ (i : { x // x ∈ σ }), 0 ≤ ty i\neqy : y = ∑ i, sy i • a ↑i + ∑ i, ty i • b ↑i\n⊢ ∀ (i : { x // x ∈ σ }), 0 ≤ (a1 • tx + b1 • ty) i\n:::"
      },
      {
        "from": "gpt",
        "value": "suffices 0 ≤ (a1 • tx + b1 • ty) by exact this"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nm : ℝ\nf' : E → E\nxm : E\nx₀ : E\nalg : Gradient_Descent_fix_stepsize f f' x₀\nhm : m > 0\nmin : IsMinOn f univ xm\nstep₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀))\nhsc : StrongConvexOn univ m f\nthis : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'\nthis : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0\nk : ℕ\n⊢ ‖Gradient_Descent_fix_stepsize.x f f' x₀ k -\n          Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) -\n        xm‖ ^\n    2 =\n  ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm -\n        Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^\n    2\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n        rw [sub_right_comm]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "case.1:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ns : Set E\nt : Set E\nhns : s.Nonempty\nhnt : t.Nonempty\nb : E\nc : ℝ\nhs : ∀ x ∈ s, inner ℝ b x ≥ c\nht : ∀ y ∈ t, inner ℝ b y ≤ c\nhs_neq : ¬s ⊆ {x | inner ℝ b x = c}\nright✝ : ¬t ⊆ {x | inner ℝ b x = c}\nht1 : ∀ y ∈ t, ↑(inner ℝ b y) ≤ ↑c\nhs1 : ∀ x ∈ s, ↑(inner ℝ b x) ≥ ↑c\n⊢ ∀ i ∈ s, ⨆ y ∈ t, ↑(inner ℝ b y) ≤ ↑(inner ℝ b i)\n\ncase.2:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ns : Set E\nt : Set E\nhns : s.Nonempty\nhnt : t.Nonempty\nb : E\nc : ℝ\nhs : ∀ x ∈ s, inner ℝ b x ≥ c\nht : ∀ y ∈ t, inner ℝ b y ≤ c\nhs_neq : ¬s ⊆ {x | inner ℝ b x = c}\nright✝ : ¬t ⊆ {x | inner ℝ b x = c}\nht1 : ∀ y ∈ t, ↑(inner ℝ b y) ≤ ↑c\nhs1 : ∀ x ∈ s, ↑(inner ℝ b x) ≥ ↑c\n⊢ ⨆ x ∈ s, ↑(inner ℝ b x) > ⨅ y ∈ t, ↑(inner ℝ b y)\n:::"
      },
      {
        "from": "gpt",
        "value": "intro i hi"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝7 : ProperSpace E\ninst✝8 : ProperSpace F\nγ : ℝ\nhγ : γ > 1\nck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)\ndk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l)\nbd : Bornology.IsBounded (z '' univ)\nlbdψ : BddBelow (ψ '' univ)\nhz : ∀ (n : ℕ), z n ∈ z '' univ\na : WithLp 2 (E × F)\nleft✝ : a ∈ closure (z '' univ)\nφ : ℕ → ℕ\nhmφ : StrictMono φ\ns : Set (WithLp 2 (E × F))\nU : Set ℕ\nax : ℕ\nhax : ∀ b ≥ ax, b ∈ U\nt : Set (WithLp 2 (E × F))\nt_s : t ⊆ s\nisopent : IsOpen t\na_t : a ∈ t\nN : ℕ\nhN : ∀ (n : ℕ), N ≤ n → (z ∘ φ) n ∈ t\nn : ℕ\n⊢ φ n ≥ ax\n:::"
      },
      {
        "from": "gpt",
        "value": "apply le_trans (Nat.le_add_left ax N)"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem (EuclideanSpace ℝ (Fin n✝)) τ σ\nx : EuclideanSpace ℝ (Fin n✝)\nxf : x ∈ FeasSet\nconte : ∀ i ∈ τ, ContDiffAt ℝ (↑1) (p.equality_constraints i) x\nconti : ∀ i ∈ σ, ContDiffAt ℝ (↑1) (p.inequality_constraints i) x\nLx : LinearCQ x\nhdomain : p.domain = univ\ndiffable : ∀ i ∈ τ, DifferentiableAt ℝ (p.equality_constraints i) x\ndiffable₂ : ∀ i ∈ σ, DifferentiableAt ℝ (p.inequality_constraints i) x\nv : EuclideanSpace ℝ (Fin n✝)\nt_ : ℝ\nht_ : t_ > 0\nht : ∀ t ∈ Icc 0 t_, ∀ i ∈ σ \\ active_set x, 0 < p.inequality_constraints i (x + t • v)\nhv1 : ∀ i ∈ τ, ⟪gradient (p.equality_constraints i) x, v⟫_ℝ = 0\nhv2 : ∀ j ∈ σ ∩ active_set x, ⟪gradient (p.inequality_constraints j) x, v⟫_ℝ ≥ 0\nz : ℕ → EuclideanSpace ℝ (Fin n✝)\nc : ℕ → ℝ\nn : ℕ\nhn : 0 ≤ n\n⊢ x + z n ∈ p.domain\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [hdomain]; trivial"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\nF : Type u_2\ninst✝ : CompleteLinearOrder F\nf : E → F\ninst✝2 : TopologicalSpace E\ninst✝3 : TopologicalSpace F\ninst✝4 : OrderTopology F\ninst✝5 : FirstCountableTopology E\ninst✝6 : FirstCountableTopology F\nhf : LowerSemicontinuous f\ny : F\nh1 : (f ⁻¹' Iic y).Nonempty\nh2 : IsCompact (f ⁻¹' Iic y)\nhs : {x | ∃ x_1 ∈ f ⁻¹' Iic y, f x_1 = x}.Nonempty\nhs' : BddBelow {x | ∃ x_1 ∈ f ⁻¹' Iic y, f x_1 = x}\nfx : ℕ → F\nleft✝ : Antitone fx\ncfx : Tendsto fx atTop (𝓝 (sInf {x | ∃ x_1 ∈ f ⁻¹' Iic y, f x_1 = x}))\nx : ℕ → E\nxsub : ∀ (n : ℕ), x n ∈ f ⁻¹' Iic y\nxeq : ∀ (n : ℕ), f (x n) = fx n\nx' : E\nxsub' : x' ∈ f ⁻¹' Iic y\nk : ℕ → ℕ\nmono : StrictMono k\ncxk : Tendsto (x ∘ k) atTop (𝓝 x')\ncfxk : Tendsto (f ∘ x ∘ k) atTop (𝓝 (sInf {x | ∃ x_1 ∈ f ⁻¹' Iic y, f x_1 = x}))\nn : ℕ\n⊢ f ((x ∘ k) n, (f ∘ x ∘ k) n).1 = ((x ∘ k) n, (f ∘ x ∘ k) n).2\n:::"
      },
      {
        "from": "gpt",
        "value": "rfl"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : Type u_3\ninst✝ : Fintype n\ninst✝2 : DecidableEq n\nA : Matrix n n ℝ\nhT_pos : T A > 0\nhA_pos : ∀ (i j : n), 0 ≤ A i j\nhA_row_sum : ∀ (i : n), ∑ j, A i j ≤ 1\nhA_col_sum : ∀ (j : n), ∑ i, A i j ≤ 1\ni : n\nhi_row_lt : ∑ j, A i j < 1\ny : n\nh_univ : y ∈ Finset.univ\nhj_col : ∑ i, A i y < 1\nε : ℝ\nhε_def : ε = min (1 - ∑ j, A i j) (1 - ∑ i, A i y)\nhε_pos : 0 < ε\nC : Matrix n n ℝ\nhC_DoublySubStochastic : DoublySubStochastic C\nhAC_le : ∀ (x y : n), A x y ≤ C x y\nh_H : ε = 1 - ∑ j, A i j\nrow_sum_C_eq : ∑ j, C i j = 1\nrow_sum_C_eq_1 : ∀ (x : n), x ≠ i → ∑ j, C x j = ∑ j, A x j\nh_row_card_lt : {i | ∑ j, C i j < 1}.card < {i | ∑ j, A i j < 1}.card\nhy : y ∈ Finset.univ ∧ ∑ i, C i y < 1\nh_sum_lt : ∑ i, C i y < 1\nh1 : A i y + ε + ∑ i_1 with ¬i_1 = i, A i_1 y = ε + A i y + ∑ i_1 with ¬i_1 = i, A i_1 y\nx✝ : n\na✝ : x✝ ∈ Finset.univ.erase i\n⊢ True\n:::"
      },
      {
        "from": "gpt",
        "value": "exact trivial"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nm : ℝ\nf' : E → E\nx : E\ny : E\nl : NNReal\nmp : m > 0\nhf : ∀ (x : E), HasGradientAt f (f' x) x\nh₂ : LipschitzWith l f'\nhl : ↑l > 0\ncov : Convex ℝ univ\nhsc : ∀ ⦃x : E⦄,\n  x ∈ univ →\n    ∀ ⦃y : E⦄,\n      y ∈ univ →\n        ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y - a * b * (m / 2 * ‖x - y‖ ^ 2)\nphi : E → ℝ\nconvphi : ConvexOn ℝ univ phi\ng : E → ℝ\ng' : E → E\nh : E → ℝ\ngderiv : ∀ (x : E), HasGradientAt g (g' x) x\nconvg : ConvexOn ℝ univ g\nconvh : ConvexOn ℝ univ h\ncoef : ¬0 < ↑l - m\nalpha : E\nbeta : E\neq1 : m * ‖beta‖ ^ 2 ≤ inner ℝ alpha beta\neq2 : 1 / ↑l * ‖alpha‖ ^ 2 ≤ inner ℝ alpha beta\nmlpos : 0 < m + ↑l\nu : E\nv : E\nh1 : m * ‖v‖ ^ 2 ≤ inner ℝ u v\nh2 : 1 / ↑l * ‖u‖ ^ 2 ≤ inner ℝ u v\n⊢ 0 ≤ ‖v‖ ^ 2\n:::"
      },
      {
        "from": "gpt",
        "value": "simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : ProperSpace E\nf : E → ℝ\nhc : LowerSemicontinuous f\nx : E\ng : E → ℝ\nepi : Set (E × ℝ)\nS : Set E\neq : S = g ⁻¹' Iic (g x)\nImS : Set ℝ\nneImS : ImS.Nonempty\nL : ℝ\nLbound : ∀ x ∈ f '' univ, L ≤ x\nboundg : ∀ (x : E), L ≤ g x\nhg : LowerSemicontinuous g\nS_bddbelow : BddBelow ImS\nepi_closed : IsClosed epi\ncloseds : IsClosed S\ncompacts : IsCompact S\nfx : ℕ → ℝ\nleft✝ : Antitone fx\ncfx : Tendsto fx atTop (𝓝 (sInf ImS))\nxn : ℕ → E\nxsub : ∀ (n : ℕ), xn n ∈ S\nxeq : ∀ (n : ℕ), g (xn n) = fx n\nx' : E\nleft✝2 : x' ∈ S\nk : ℕ → ℕ\nmono : StrictMono k\ncxk : Tendsto (xn ∘ k) atTop (𝓝 x')\ncfxk : Tendsto (g ∘ xn ∘ k) atTop (𝓝 (sInf ImS))\ninepi : (x', sInf ImS) ∈ epi\nminima_ieq : g x' ≤ sInf ImS\nw : E\nhw : w ∈ S\ngwin : g w ∈ ImS\n⊢ ∀ (ε : ℝ), 0 < ε → ∃ x ∈ ImS, x < g w + ε\n:::"
      },
      {
        "from": "gpt",
        "value": "intro _ epos"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : SeminormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\nf : E → ℝ\nx : E\ns : Set E\nhf : ConvexOn ℝ s f\nhc : ContinuousOn f (interior s)\nhx : x ∈ interior s\nhepi_f₁ : Convex ℝ (interior (Epi f s))\ndisj : (x, f x) ∉ interior (Epi f s)\nφ : StrongDual ℝ (E × ℝ)\nhφ : ∀ b ∈ interior (Epi f s), φ (x, f x) < φ b\ng : Module.Dual ℝ E × Module.Dual ℝ ℝ\nhg : ∀ (a : E × ℝ), φ a = g.1 a.1 + g.2 a.2\nu : ℝ\nhu : ∀ (a : ℝ), g.2 a = u * a\nhgu : ∀ a ∈ interior (Epi f s), g.1 x + u * f x < g.1 a.1 + u * a.2\nhu0 : u > 0\nh : Module.Dual ℝ E\nC : ℝ\nhC : ∀ (y : E), |g.1 y| ≤ C * ‖y‖\nhh : ∃ C, ∀ (x : E), ‖h x‖ ≤ C * ‖x‖\nh' : E →L[ℝ] ℝ\nkey1 : ∀ a ∈ interior (Epi f s), h' (a.1 - x) + f x < a.2\nkey2₀ : ∀ a ∈ Epi f s, a.1 ∈ interior s → h' (a.1 - x) + f x ≤ a.2\na : E × ℝ\nha : a ∈ Epi f s\na✝ : a.1 ∉ interior s\nan : ℕ → E × ℝ\n⊢ ∀ (n : ℕ), h' ((an n).1 - x) + f x ≤ (an n).2\n:::"
      },
      {
        "from": "gpt",
        "value": "intro n"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝6 : FiniteDimensional 𝕜 V\nT : V →ₗ[𝕜] W\ne : V\nhe : ‖e‖ = 1\nu : V\nhu : ⟪e, u⟫_𝕜 = 0\nσ : ℝ\nhT : σ = ‖T e‖\nhσ : σ = T.opNorm\nhσ_zero : σ ≠ 0\nhu_zero : ‖u‖ ≠ 0\nh_ineq : ∀ (a : 𝕜), ‖T (e + a • u)‖ ^ 2 ≤ σ ^ 2 * (1 + ‖a‖ ^ 2 * ‖u‖ ^ 2)\nh_norm_expansion : ∀ (a : 𝕜), ‖T (e + a • u)‖ ^ 2 ≥ σ ^ 2 + 2 * re (a * ⟪T e, T u⟫_𝕜)\nh_combined_ineq : ∀ (a : 𝕜), 2 * re (a * ⟪T e, T u⟫_𝕜) ≤ σ ^ 2 * ‖a‖ ^ 2 * ‖u‖ ^ 2\nh : ⟪T e, T u⟫_𝕜 ≠ 0\nε : ℝ\nhε : ε = 1 / (σ ^ 2 * ‖u‖ ^ 2)\nhε_pos : 0 < ε\na : 𝕜\nha : a = ⟪T u, T e⟫_𝕜 * ↑ε\nh_combined_ineq_a : 2 * re (a * ⟪T e, T u⟫_𝕜) ≤ σ ^ 2 * ‖a‖ ^ 2 * ‖u‖ ^ 2\n⊢ σ ^ 2 * ‖⟪T e, T u⟫_𝕜‖ ^ 2 * |1 / (σ ^ 2 * ‖u‖ ^ 2)| ^ 2 * ‖u‖ ^ 2 = 1 * ‖⟪T e, T u⟫_𝕜‖ ^ 2 * (1 / (σ ^ 2 * ‖u‖ ^ 2))\n:::"
      },
      {
        "from": "gpt",
        "value": "simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem (EuclideanSpace ℝ (Fin n)) τ σ\nx : EuclideanSpace ℝ (Fin n)\nm : ℕ\nv : EuclideanSpace ℝ (Fin n)\nMx : EuclideanSpace ℝ (Fin n) →L[ℝ] EuclideanSpace ℝ { x_1 // x_1 ∈ active_set x } × (Fin (n - m) → ℝ)\nRz : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ { x_1 // x_1 ∈ active_set x } × (Fin (n - m) → ℝ)\nRt : ℝ → EuclideanSpace ℝ { x_1 // x_1 ∈ active_set x } × (Fin (n - m) → ℝ)\nRteq : Rt = fun t => t • Mx v\nRxeq0 : Rz x = 0\nRzgrad : HasStrictFDerivAt Rz Mx x\nMxsurj : LinearMap.range Mx = ⊤\ng : EuclideanSpace ℝ { x_1 // x_1 ∈ active_set x } × (Fin (n - m) → ℝ) → ↥(LinearMap.ker Mx) → EuclideanSpace ℝ (Fin n)\nε : ℝ\nεpos : ε > 0\nnhdsin : Metric.ball (0, 0) ε ⊆ {x_1 | Rz (g x_1.1 x_1.2) = x_1.1}\nRtleε : ∀ ε > 0, ∃ N, ∀ m_1 ≥ N, ‖Rt (↑m_1)⁻¹‖ < ε\nN : ℕ\nRtle : ∀ m_1 ≥ N, ‖Rt (↑m_1)⁻¹‖ < ε\n⊢ Tendsto (fun t => 0) atTop (𝓝 0)\n:::"
      },
      {
        "from": "gpt",
        "value": "· simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nΩ : Set E\nh_compact : IsCompact Ω\nh_Ω2 : is_constant_on f Ω\nh_nonempty : Ω.Nonempty\nμ : ℝ\nconstant_value : ∀ x ∈ Ω, f x = μ\nx : E\nhx : x ∈ Ω\nh_Ω1 : ∀ x ∈ Ω,\n  ∃ η,\n    0 < η ∧\n      ∃ s ∈ 𝓝 x,\n        ∃ φ ∈ desingularizing_function η,\n          ∀ x_1 ∈ s,\n            f x < f x_1 →\n              f x_1 < f x + η → 1 ≤ ENNReal.ofReal (deriv φ (f x_1 - f x)) * EMetric.infEdist 0 (subdifferential f x_1)\nη : ℝ\nhη : 0 < η\ns : Set E\nhs : s ∈ 𝓝 x\nφ : ℝ → ℝ\nhφ : φ ∈ desingularizing_function η\nh_Ω1 : ∀ x_1 ∈ s,\n  f x < f x_1 →\n    f x_1 < f x + η → 1 ≤ ENNReal.ofReal (deriv φ (f x_1 - f x)) * EMetric.infEdist 0 (subdifferential f x_1)\nO : Set E\nhO1 : O ⊆ s\nhO2 : IsOpen O\nhO3 : x ∈ O\nu : E\nhu : u ∈ O ∩ {y | f x < f y ∧ f y < f x + η}\n⊢ ENNReal.ofReal (deriv φ (f u - f x)) * EMetric.infEdist 0 (subdifferential f u) ≥ 1\n:::"
      },
      {
        "from": "gpt",
        "value": "simp at hu"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → EReal\nhsf : ProperFunction univ f\nx : E\ny : E\na : ℝ\nb : ℝ\napos : 0 ≤ a\nbpos : 0 ≤ b\nabsum : a + b = 1\nha : a ≠ 0\nhb : b ≠ 0\nha' : a > 0\nhb' : b > 0\nh1 : ¬convex_conjugate univ f x = ⊤\nu : E\nhcgx2 : ⨆ u, ↑(inner ℝ u x) - f u > ⊥\nhcgx2 : ⨆ u, ↑(inner ℝ u x) - f u ≠ ⊥\nhcgy2 : ⨆ u, ↑(inner ℝ u y) - f u > ⊥\nhcgy2 : ⨆ u, ↑(inner ℝ u y) - f u ≠ ⊥\nhcgx1 : ⨆ u, ↑(inner ℝ u x) - f u ≠ ⊤\nfu : ℝ\nhfu2 : ↑fu ≠ ⊥\nhfu1 : ↑fu ≠ ⊤\nsup₁ : ↑(inner ℝ u x) - ↑fu ≤ ⨆ u, ↑(inner ℝ u x) - f u\nsup₂ : ↑(inner ℝ u y) - ↑fu ≤ ⨆ u, ↑(inner ℝ u y) - f u\nsupa : ↑a * (↑(inner ℝ u x) - ↑fu) ≤ ↑a * ⨆ u, ↑(inner ℝ u x) - f u\nsupb : ↑b * (↑(inner ℝ u y) - ↑fu) ≤ ↑b * ⨆ u, ↑(inner ℝ u y) - f u\nhcgy1 : ⨆ u, ↑(inner ℝ u y) - f u ≠ ⊤\n⊢ ↑a * (↑(inner ℝ u x) - ↑fu) + ↑b * (↑(inner ℝ u y) - ↑fu) ≤\n  (↑a * ⨆ u, ↑(inner ℝ u x) - f u) + ↑b * ⨆ u, ↑(inner ℝ u y) - f u\n:::"
      },
      {
        "from": "gpt",
        "value": "lift ⨆ u, Real.toEReal (inner ℝ u x) - f u to ℝ with c"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem E τ σ\nf : E → ℝ\nloc : E\nhf : DifferentiableAt ℝ f loc\nv : E\nc : ℕ → ℝ\nd : ℕ → E\na : ℕ\nha : ∀ (b : ℕ), a✝ ≤ b → loc + d b ∈ FeasSet\nproneg : ⟪v, gradient f loc⟫_ℝ < 0\ns : Set E\nhs : s ∈ 𝓝 loc ⊓ 𝓟 FeasSet\nhs2 : ∀ y ∈ s, f loc ≤ f y\nε : ℝ\nεpos : ε > 0\nεball : Metric.ball loc ε ∩ FeasSet ⊆ s\ns1 : Set E\nhmin : ∀ y ∈ s1, f loc ≤ f y\nz : ℕ → E\nhzd : ∀ (n : ℕ), d n = z n - loc\nhcp : ∀ᶠ (n : ℕ) in atTop, c n > 0\nhz3 : ∀ᶠ (n : ℕ) in atTop, 1 / c n > 0\nhzt : Tendsto z atTop (𝓝 loc)\nhz : (fun n => f (z n) - f loc - ⟪z n - loc, gradient f loc⟫_ℝ) =o[atTop] fun n => z n - loc\nthis : Metric.ball v 1 ∈ 𝓝 v\na : ℕ\nha : ∀ b ≥ a, c b • d b ∈ Metric.ball v 1\nvt1 : ∀ᶠ (a : ℕ) in atTop, 2 * (‖v‖ + ε) / ε ≤ c a\n⊢ ∃ c_1, ∀ᶠ (x : ℕ) in atTop, ‖z x - loc‖ ≤ c_1 * ‖1 / c x‖\n:::"
      },
      {
        "from": "gpt",
        "value": "simp at vt1"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\ninst✝4 : ProperSpace E\nx₀ : E\nf : E → ℝ\nf' : E → E\nh : E → ℝ\nalg : proximal_gradient_method f h f' x₀\nk : ℕ+\nth : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ\nth' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)\nGt : E → E\nφ : E → ℝ\nhG : ∀ (z : E), Gt z - f' z ∈ SubderivAt h (z - proximal_gradient_method.t f h f' x₀ • Gt z)\nfieq1 : ∀ (x : E),\n  f (x - proximal_gradient_method.t f h f' x₀ • Gt x) ≤\n    f x - proximal_gradient_method.t f h f' x₀ * inner ℝ (f' x) (Gt x) +\n      proximal_gradient_method.t f h f' x₀ ^ 2 * ↑(proximal_gradient_method.L f h f' x₀) / 2 * ‖Gt x‖ ^ 2\nx : E\n⊢ proximal_gradient_method.t f h f' x₀ ^ 2 * ↑(proximal_gradient_method.L f h f' x₀) ≤\n  proximal_gradient_method.t f h f' x₀ * (1 / ↑(proximal_gradient_method.L f h f' x₀)) *\n    ↑(proximal_gradient_method.L f h f' x₀)\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [pow_two]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nΩ : Set E\nh_compact : IsCompact Ω\nh_Ω1 : ∀ x ∈ Ω, KL_point f x\nh_Ω2 : is_constant_on f Ω\nh_nonempty : Ω.Nonempty\nμ : ℝ\nconstant_value : ∀ x ∈ Ω, f x = μ\nη : E → ℝ\nhη2 : ∀ x ∈ Ω, η x ∈ Ioi 0\nO : E → Set E\nh_Ox : ∀ x ∈ Ω, IsOpen (O x)\nh_Ox' : ∀ x ∈ Ω, x ∈ O x\nφ : E → ℝ → ℝ\nhφ : ∀ x ∈ Ω, φ x ∈ desingularizing_function (η x)\nh_exist : ∀ x ∈ Ω,\n  ∀ u ∈ O x ∩ {y | f x < f y ∧ f y < f x + η x},\n    ENNReal.ofReal (deriv (φ x) (f u - f x)) * EMetric.infEdist 0 (subdifferential f u) ≥ 1\nexist_open_cover : Ω ⊆ ⋃ x ∈ Ω, O x\nexist_open_sub_finite_cover : ∃ b' ⊆ Ω, b'.Finite ∧ Ω ⊆ ⋃ i ∈ b', O i\n⊢ ∃ ε ∈ Ioi 0,\n  ∃ η ∈ Ioi 0,\n    ∃ φ ∈ desingularizing_function η,\n      ∀ u ∈ Ω,\n        ∀ x ∈ {y | (EMetric.infEdist y Ω).toReal < ε} ∩ {y | f u < f y ∧ f y < f u + η},\n          ENNReal.ofReal (deriv φ (f x - f u)) * EMetric.infEdist 0 (subdifferential f x) ≥ 1\n:::"
      },
      {
        "from": "gpt",
        "value": "rcases exist_open_sub_finite_cover with ⟨t, ht1, ht2, ht3⟩"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nxm : E\nx₀ : E\nalg : subgradient_method f x₀\nhm : IsMinOn f Set.univ xm\nha'' : ∀ (s : Set ℝ) (x : ℝ),\n  (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s\ns : Set ℝ\nha' : ∀ (s x : Set ℝ),\n  x ⊆ s →\n    (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) → 0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s\nt : Set ℝ\nhs₁ : t ⊆ s\nhs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t\nhs₀ : 0 ∈ t\nε : ℝ\nεpos : 0 < ε\nhs₂₀ : Metric.ball 0 ε ⊆ t\nha₁ : ∃ a₁, ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2\nhG : subgradient_method.G f x₀ = 0\n⊢ ∀ (b : ℕ),\n  0 ≤ b →\n    (↑(subgradient_method.G f x₀) ^ 2 * ∑ i ∈ Finset.range (b + 1), subgradient_method.a f x₀ i ^ 2) /\n        (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) <\n      ε / 2\n:::"
      },
      {
        "from": "gpt",
        "value": "intro b _"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nn : ℕ+\nhtau : 0 < ADMM.τ E₁ E₂ F ∧ ADMM.τ E₁ E₂ F ≤ 1\n⊢ (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - ADMM.x₂ E₁ F (↑n + 1))‖ ^ 2 +\n    (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F *\n      ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂‖ ^ 2 =\n  (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F *\n    (‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - ADMM.x₂ E₁ F (↑n + 1))‖ ^ 2 +\n      ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂‖ ^ 2)\n:::"
      },
      {
        "from": "gpt",
        "value": "ring"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_2\ns : Set E\nι : Type u_1\ninst✝ : AddCommGroup E\ninst✝2 : Module ℝ E\ninst✝3 : DecidableEq ι\nf : E → EReal\np : ι → E\nhf : ConvexOn ℝ s f\na : ι\nt : Finset ι\nhat : a ∉ t\nhc : ∀ (w : ι → ℝ),\n  (∀ i ∈ t, 0 ≤ w i) → ∑ i ∈ t, w i = 1 → (∀ i ∈ t, p i ∈ s) → f (∑ i ∈ t, w i • p i) ≤ ∑ i ∈ t, w i • f (p i)\nw : ι → ℝ\nh₀ : ∀ i ∈ insert a t, 0 ≤ w i\nh₁ : ∑ i ∈ insert a t, w i = 1\nhmem : ∀ i ∈ insert a t, p i ∈ s\nhw : ∀ i ∈ t, w i ≥ 0\nha : ¬w a = 1\nhab0 : w a ≥ 0\nhab1 : 1 - w a > 0\nthis : 1 - w a ≤ 1\nw' : ι → ℝ\nh₀' : ∀ i ∈ t, 0 ≤ w' i\nh₁' : ∑ i ∈ t, w' i = 1\nhmem' : ∀ i ∈ t, p i ∈ s\nhc : f (∑ i ∈ t, w' i • p i) ≤ ∑ i ∈ t, w' i • f (p i)\nhtt : ∑ i ∈ t, w i • p i = (1 - w a) • ∑ i ∈ t, w' i • p i\nhp2 : ∑ i ∈ t, w' i • p i ∈ s\nhp3 : 1 - w a ≥ 0\nhcf : f (w a • p a + (1 - w a) • ∑ i ∈ t, w' i • p i) ≤ ↑(w a) * f (p a) + (1 - ↑(w a)) * f (∑ i ∈ t, w' i • p i)\n⊢ f (w a • p a + (1 - w a) • ∑ i ∈ t, w' i • p i) ≤ ↑(w a) * f (p a) + (1 - ↑(w a)) * f (∑ i ∈ t, w' i • p i)\n:::"
      },
      {
        "from": "gpt",
        "value": "exact hcf"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nf' : E → E\nxm : E\nx₀ : E\nγ : ℕ+ → ℝ\nalg : Nesterov f f' γ x₀\nhfun : ConvexOn ℝ univ f\nhg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)\nh2 : ∀ (k : ℕ+) (x' : E),\n  f (Nesterov.x f f' γ x₀ ↑k) - f x' ≤\n    ↑(Nesterov.l f f' γ x₀) *\n        inner ℝ (Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k) (x' - Nesterov.x f f' γ x₀ ↑k) +\n      ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.x f f' γ x₀ ↑k - Nesterov.y f f' γ x₀ k‖ ^ 2\nk : ℕ+\nthis : f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) =\n  γ k * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) +\n    (1 - γ k) * (f (Nesterov.x f f' γ x₀ ↑k) - f (Nesterov.x f f' γ x₀ (↑k - 1)))\nlzero : 0 < ↑↑k + 1\nhz : γ k ≥ 0\nhl : γ k ≤ 1\n⊢ γ k • xm + (1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) - (γ k + (1 - γ k)) • Nesterov.x f f' γ x₀ ↑k =\n  (1 - γ k) • Nesterov.x f f' γ x₀ (↑k - 1) + γ k • xm - Nesterov.x f f' γ x₀ ↑k\n:::"
      },
      {
        "from": "gpt",
        "value": "ring_nf"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "R : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝ : NontriviallyNormedField R\ninst✝2 : NormedAddCommGroup α\ninst✝3 : NormedSpace R α\ninst✝4 : NormedAddCommGroup β\ninst✝5 : NormedSpace R β\nM : Set α\nN : Set β\nneM : M.Nonempty\nneN : N.Nonempty\nphiM' : ↥(affineSpan R M) → α\nphiN' : ↥(affineSpan R N) → β\nphi' : ↥(affineSpan R (M ×ˢ N)) → α × β\nsubtype : Type (max 0 u_5 u_6)\nsubtypeM : Type (max 0 u_5)\nsubtypeN : Type (max 0 u_6)\naffSpan_iff : ∀ (x : α × β),\n  (fun x => x ∈ affineSpan R (M ×ˢ N)) x ↔\n    (fun x => x.1 ∈ affineSpan R M ∧ x.2 ∈ affineSpan R N) ((Homeomorph.refl (α × β)) x)\neqv1 : { x // (fun x => x ∈ affineSpan R (M ×ˢ N)) x } ≃ₜ { y // (fun x => x.1 ∈ affineSpan R M ∧ x.2 ∈ affineSpan R N) y }\neqv2 : { x // x.1 ∈ affineSpan R M ∧ x.2 ∈ affineSpan R N } ≃ₜ ↥(affineSpan R M) × ↥(affineSpan R N)\neqv : subtype ≃ₜ subtypeM × subtypeN\nx : { y // (fun x => x.1 ∈ affineSpan R M ∧ x.2 ∈ affineSpan R N) y }\n⊢ (fun x => x ∈ affineSpan R (M ×ˢ N)) ((Homeomorph.refl (α × β)).symm ↑x)\n:::"
      },
      {
        "from": "gpt",
        "value": "simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem (EuclideanSpace ℝ (Fin n✝)) τ σ\nx : EuclideanSpace ℝ (Fin n✝)\nxf : x ∈ FeasSet\nconte : ∀ i ∈ τ, ContDiffAt ℝ (↑1) (p.equality_constraints i) x\nconti : ∀ i ∈ σ, ContDiffAt ℝ (↑1) (p.inequality_constraints i) x\nLx : LinearCQ x\nhdomain : p.domain = univ\ndiffable : ∀ i ∈ τ, DifferentiableAt ℝ (p.equality_constraints i) x\ndiffable₂ : ∀ i ∈ σ, DifferentiableAt ℝ (p.inequality_constraints i) x\nv : EuclideanSpace ℝ (Fin n✝)\nt_ : ℝ\nht_ : t_ > 0\nht : ∀ t ∈ Icc 0 t_, ∀ i ∈ σ \\ active_set x, 0 < p.inequality_constraints i (x + t • v)\nhv1 : ∀ i ∈ τ, ⟪gradient (p.equality_constraints i) x, v⟫_ℝ = 0\nhv2 : ∀ j ∈ σ ∩ active_set x, ⟪gradient (p.inequality_constraints j) x, v⟫_ℝ ≥ 0\nz : ℕ → EuclideanSpace ℝ (Fin n✝)\nc : ℕ → ℝ\nn : ℕ\nhn : 0 ≤ n\ni : ℕ\nhi : i ∈ τ\na : EuclideanSpace ℝ (Fin n✝)\nc : ℝ\nhab : p.equality_constraints i = fun y => ⟪a, y⟫_ℝ + c\nhg : gradient (p.equality_constraints i) x = a\n⊢ p.equality_constraints i (x + z n) = 0\n:::"
      },
      {
        "from": "gpt",
        "value": "simp [FeasSet, FeasPoint] at xf"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "case.1:\n𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\nE : Type ?u.1591003\nF : Type ?u.1591108\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\ninst✝3 : NormedAddCommGroup F\ninst✝4 : NormedSpace ℝ F\ninst✝5 : FiniteDimensional ℝ E\ninst✝6 : FiniteDimensional ℝ F\n𝕜 : Type u_1\nM : Type u_4\ninst✝7 : Field 𝕜\ninst✝8 : LinearOrder 𝕜\ninst✝9 : IsStrictOrderedRing 𝕜\ninst✝10 : AddCommGroup M\ninst✝11 : Module 𝕜 M\ns : Set M\nhs : Convex 𝕜 s\na : 𝕜\nha : 0 < a\nx : M\nhx : x = 0\n⊢ (a = 0 ∨ x = 0) ∨ a • x ∈ toCone s hs\n\ncase.2:\n𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\nE : Type ?u.1591003\nF : Type ?u.1591108\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\ninst✝3 : NormedAddCommGroup F\ninst✝4 : NormedSpace ℝ F\ninst✝5 : FiniteDimensional ℝ E\ninst✝6 : FiniteDimensional ℝ F\n𝕜 : Type u_1\nM : Type u_4\ninst✝7 : Field 𝕜\ninst✝8 : LinearOrder 𝕜\ninst✝9 : IsStrictOrderedRing 𝕜\ninst✝10 : AddCommGroup M\ninst✝11 : Module 𝕜 M\ns : Set M\nhs : Convex 𝕜 s\na : 𝕜\nha : 0 < a\nx : M\nhx : x ∈ toCone s hs\n⊢ (a = 0 ∨ x = 0) ∨ a • x ∈ toCone s hs\n:::"
      },
      {
        "from": "gpt",
        "value": "· simp [hx]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nf' : E → E\ns : Set E\nhs : Convex ℝ s\nhf : ∀ x ∈ s, HasGradientAt f (f' x) x\nmono : ∀ x ∈ s, ∀ y ∈ s, x ≠ y → ⟪f' x - f' y, x - y⟫_ℝ > 0\nx : E\nxin : x ∈ s\ny : E\nyin : y ∈ s\nxney : x ≠ y\na : ℝ\nb : ℝ\napos : 0 < a\nbpos : 0 < b\nabsum1 : a + b = 1\nh₀ : a • f x + b • f y ≤ f (a • x + b • y)\nanneg : 0 ≤ a\nbnneg : 0 ≤ b\nmono' : ∀ x ∈ s, ∀ y ∈ s, ⟪f' x - f' y, x - y⟫_ℝ ≥ 0\nleft✝ : Convex ℝ s\nconvf : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y\neq : f (a • x + b • y) ≤ a • f x + b • f y\neq2 : f (a • x + b • y) = a • f x + b • f y\nz : E\nzin : z ∈ s\nc : ℝ\ncin : c ∈ Ioo 0 1\ne1 : ⟪f' (x + c • (z - x)), z - x⟫_ℝ = f z - f x\nd : ℝ\ndin : d ∈ Ioo 0 1\ne2 : ⟪f' (z + d • (y - z)), y - z⟫_ℝ = f y - f z\neq3 : a * b * ⟪f' (z + d • (y - z)) - f' (x + c • (z - x)), y - x⟫_ℝ = 0\nthis1 : b • (y - z) = a • b • (y - x)\nthis2 : a • (z - x) = a • b • (y - x)\n⊢ False\n:::"
      },
      {
        "from": "gpt",
        "value": "have eq0 : inner ℝ (f' (z + d • (y - z)) - f' (x + c • (z - x))) (y - x) = (0 : ℝ) := by\n    contrapose! eq3\n    rw [mul_ne_zero_iff]\n    constructor\n    · rw [mul_ne_zero_iff]; constructor <;> linarith\n    · exact eq3"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nf' : E → E\nxm : E\nx₀ : E\nγ : ℕ+ → ℝ\nalg : Nesterov f f' γ x₀\nhfun : ConvexOn ℝ univ f\nhg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)\nmin : IsMinOn f univ xm\nk : ℕ+\nh4 : f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤\n  ↑(Nesterov.l f f' γ x₀) * γ k ^ 2 / 2 *\n    (‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2)\ncon : (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2\nγk_sq_pos : γ k ^ 2 > 0\nh4' : (f (Nesterov.x f f' γ x₀ ↑k) - f xm) / γ k ^ 2 - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) / γ k ^ 2 ≤\n  ↑(Nesterov.l f f' γ x₀) / 2 * (‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2)\n⊢ 1 / γ k ^ 2 * (f (Nesterov.x f f' γ x₀ ↑k) - f xm) + ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2 =\n  (f (Nesterov.x f f' γ x₀ ↑k) - f xm) / γ k ^ 2 + ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2\n:::"
      },
      {
        "from": "gpt",
        "value": "by ring"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "α : Type u_1\ninst✝ : NormedAddCommGroup α\ninst✝2 : InnerProductSpace ℝ α\ninst✝3 : FiniteDimensional ℝ α\nx : α\nr : ℝ\nhr : r > 0\nhn : finrank ℝ α ≠ 0\nx₀ : α\nhx₀ : ‖σ (x₀ - x)‖ < r\nsum_le_r : ∑ i, ‖(finBasis ℝ α).equivFun (x₀ - x) i‖ * ‖(finBasis ℝ α) i‖ / r ≤ 1\nn : ℕ\nι : Type\nι₀ : Type\nw₀ : Fin (finrank ℝ α) → ℝ\nrepr : ∑ i, w₀ i • (finBasis ℝ α) i = x₀ - x\nw₁ : ι → ℝ\nsum : ℝ\nsum_pos : 1 - sum ≥ 0\nw : ι₀ → ℝ\nlem_i : ∀ {i : ℕ}, i < n + 2 → ¬i = n + 1 → ¬i = n → i < n\nn_pos : n > 0\nfin0 : ι\nz : ι₀ → α\nhw₀ : ∀ (i : ι₀), 0 ≤ w i\nhw₁ : ∑ i, w i = 1\nhz : ∀ (i : ι₀),\n  z i ∈ (⋃ i, {(r / ‖(finBasis ℝ α) i‖) • (finBasis ℝ α) i}) ∪ ⋃ i, {-(r / ‖(finBasis ℝ α) i‖) • (finBasis ℝ α) i}\nbi_pos : ∀ (i : ι), ‖(finBasis ℝ α) i‖ ≠ 0\nthis : ((1 - sum) / 2) • z (Fin.last n).castSucc + ((1 - sum) / 2) • z (Fin.last (n + 1)) = 0\ni : Fin n\nh₁ : ↑i ≠ n + 1\nh₂ : ↑i ≠ n\n⊢ ↑(SignType.sign ((finBasis ℝ α).equivFun (x₀ - x) i)) * |(finBasis ℝ α).equivFun (x₀ - x) i| *\n    (‖(finBasis ℝ α) i‖ / r * (r / ‖(finBasis ℝ α) i‖)) =\n  (finBasis ℝ α).equivFun (x₀ - x) i\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n          rw[sign_mul_abs]\n          field_simp [bi_pos i]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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_kkt₁ fullrank₁ fullrank₂)\nh_add : Tendsto\n  (fun n =>\n    1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) * ‖ey (φ n)‖ ^ 2 + ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (e₂ (φ n))‖ ^ 2)\n  atTop (𝓝 0)\nh_add' : Tendsto\n  (fun n =>\n    1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) * ‖ey (φ n)‖ ^ 2 + ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (e₂ (φ n))‖ ^ 2 +\n      max (1 - ADMM.τ E₁ E₂ F) (1 - 1 / ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F *\n        ‖(OptProblem.A₁ E₂) (e₁ (φ n)) + (OptProblem.A₂ E₁) (e₂ (φ n))‖ ^ 2)\n  atTop (𝓝 0)\n⊢ Tendsto (fun n => Φ (φ n)) atTop (𝓝 0)\n:::"
      },
      {
        "from": "gpt",
        "value": "have : (fun n => Φ (φ n)) = (fun n => (1 / (τ * ρ)) * ‖ey (φ n)‖^2 +\n       ρ * ‖A₂ (e₂ (φ n))‖^2 +\n       ((max (1-τ) (1-(1/τ)))*ρ) * ‖A₁ (e₁ (φ n)) + A₂ (e₂ (φ n))‖^2) := by\n      funext n; rw [Φ, Ψ]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "case.1:\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nι : Type u_4\nκ₁ : Type u_5\nκ₂ : Type u_6\ninst✝2 : Fintype ι\ninst✝3 : Fintype κ₁\ninst✝4 : Fintype κ₂\ninst✝5 : DecidableEq ι\ninst✝6 : DecidableEq κ₁\ninst✝7 : DecidableEq κ₂\ne : ι ≃ κ₁ ⊕ κ₂\nV₁ : Matrix κ₁ ι 𝕜\nh₁ : V₁ * V₁ᴴ = 1\nV₁' : Matrix κ₁ (κ₁ ⊕ κ₂) 𝕜\nh₁' : V₁' * V₁'ᴴ = 1\nV₂' : Matrix κ₂ (κ₁ ⊕ κ₂) 𝕜\nh₂' : V₁'.fromRows V₂' ∈ unitaryGroup (κ₁ ⊕ κ₂) 𝕜\nV₂ : Matrix κ₂ ι 𝕜\ni : κ₂\nj : κ₁\n⊢ V₁'.fromRows V₂' (Sum.inr i) (Sum.inl j) = (V₁.fromRows V₂).submatrix id (⇑e.symm) (Sum.inr i) (Sum.inl j)\n\ncase.2:\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nι : Type u_4\nκ₁ : Type u_5\nκ₂ : Type u_6\ninst✝2 : Fintype ι\ninst✝3 : Fintype κ₁\ninst✝4 : Fintype κ₂\ninst✝5 : DecidableEq ι\ninst✝6 : DecidableEq κ₁\ninst✝7 : DecidableEq κ₂\ne : ι ≃ κ₁ ⊕ κ₂\nV₁ : Matrix κ₁ ι 𝕜\nh₁ : V₁ * V₁ᴴ = 1\nV₁' : Matrix κ₁ (κ₁ ⊕ κ₂) 𝕜\nh₁' : V₁' * V₁'ᴴ = 1\nV₂' : Matrix κ₂ (κ₁ ⊕ κ₂) 𝕜\nh₂' : V₁'.fromRows V₂' ∈ unitaryGroup (κ₁ ⊕ κ₂) 𝕜\nV₂ : Matrix κ₂ ι 𝕜\ni : κ₂\nj : κ₂\n⊢ V₁'.fromRows V₂' (Sum.inr i) (Sum.inr j) = (V₁.fromRows V₂).submatrix id (⇑e.symm) (Sum.inr i) (Sum.inr j)\n:::"
      },
      {
        "from": "gpt",
        "value": "· simp [V₂]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem (EuclideanSpace ℝ (Fin n)) τ σ\nx : EuclideanSpace ℝ (Fin n)\nLIx : LinearIndependent ℝ fun i =>\n  if ↑i ∈ τ then gradient (p.equality_constraints ↑i) x else gradient (p.inequality_constraints ↑i) x\nm : ℕ\nmeq : m = (active_set x).card\nM : Matrix { x_1 // x_1 ∈ active_set x } (Fin n) ℝ\neq : M = fun i => if ↑i ∈ τ then gradient (p.equality_constraints ↑i) x else gradient (p.inequality_constraints ↑i) x\nmlen : m ≤ n\nfullrowrank : M.rank = Fintype.card { x_1 // x_1 ∈ active_set x }\nmap : (Fin n → ℝ) →ₗ[ℝ] { x_1 // x_1 ∈ active_set x } → ℝ\neq : m + Module.finrank ℝ ↥(LinearMap.ker (toLin' M)) = n\naux : Module.finrank ℝ ↥(LinearMap.range (toLin' M)) = m\nkernel : Submodule ℝ (Fin n → ℝ)\ndim_ker : Module.finrank ℝ ↥kernel = n - m\nbase : Module.Basis (Fin (n - m)) ℝ ↥kernel\nZ : Matrix (Fin (n - m)) (Fin n) ℝ\nbase_indep : LinearIndependent ℝ ⇑base\n⊢ ∀ (s : Finset (Fin (n - m))) (g : Fin (n - m) → ℝ),\n  (∀ i ∉ s, g i = 0) → ∑ i ∈ s, g i • row (fun i => ↑(base i)) i = 0 → ∀ (i : Fin (n - m)), g i = 0\n:::"
      },
      {
        "from": "gpt",
        "value": "intro s"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nn : ℕ+\n⊢ 0 + SubderivAt (OptProblem.f₂ E₁ F) (ADMM.x₂ E₁ F ↑n) +\n      {(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) (ADMM.y E₁ E₂ n.natPred)} +\n    {ADMM.ρ E₁ E₂ F •\n        (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))\n          ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂)} =\n  SubderivAt (OptProblem.f₂ E₁ F) (ADMM.x₂ E₁ F ↑n) +\n      {(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) (ADMM.y E₁ E₂ n.natPred)} +\n    {ADMM.ρ E₁ E₂ F •\n        (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))\n          ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂)}\n:::"
      },
      {
        "from": "gpt",
        "value": "rw[zero_add]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝7 : ProperSpace E\ninst✝8 : ProperSpace F\nγ : ℝ\nhγ : γ > 1\nck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)\ndk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l)\nbd : Bornology.IsBounded (z '' univ)\nhψ : KL_function ψ\nlbdψ : BddBelow (ψ '' univ)\nz_ : WithLp 2 (E × F)\nleft✝ : z_ ∈ closure (z '' univ)\nα : ℕ → ℕ\nmonoa : StrictMono α\nconv : Tendsto (fun n => z (α n)) atTop (𝓝 z_)\nρ1 : ℝ\nρ1pos : ρ1 > 0\nsuff_des : ρ1 = (γ - 1) * ↑l ∧ ∀ (k : ℕ), ρ1 / 2 * ‖z (k + 1) - z k‖ ^ 2 ≤ ψ (z k) - ψ (z (k + 1))\nz_in : z_ ∈ limit_set z\nfinal : ∀ (m : ℕ), ∃ n, m ≤ α n\npsiconv : Tendsto (fun n => ψ (z (α n))) atTop (𝓝 (ψ z_))\nmonopsi : ∀ (m n : ℕ), ψ (z (m + n)) ≤ ψ (z m)\nn : ℕ\neq : ψ (z (α n)) ≤ ψ z_\nN : ℕ\ni : ℕ\nige : i ≥ N\nk : ℕ\nkge : k ≥ N\ncon : ψ (z k) < ψ z_\nw : ℕ\nkleaw : k ≤ α w\nx : ℕ\nwlex : x ∈ Ici w\nthis : α w ≤ α x\n⊢ k ≤ α x\n:::"
      },
      {
        "from": "gpt",
        "value": "linarith"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nn : ℕ\nh1 : 0 < ADMM.τ E₁ E₂ F\nright✝ : ADMM.τ E₁ E₂ F < (1 + √5) / 2\n⊢ 0 ≤ (ADMM.ρ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F)⁻¹ * ‖ey n‖ ^ 2\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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nn : ℕ\nh1 : 0 < ADMM.τ E₁ E₂ F\nright✝ : ADMM.τ E₁ E₂ F < (1 + √5) / 2\n⊢ 0 ≤ ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (e₂ n)‖ ^ 2\n:::"
      },
      {
        "from": "gpt",
        "value": "repeat apply mul_nonneg"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝7 : ProperSpace E\ninst✝8 : ProperSpace F\nγ : ℝ\nhγ : γ > 1\nck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)\ndk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l)\nbd : Bornology.IsBounded (z '' univ)\nhψ : KL_function ψ\nlbdψ : BddBelow (ψ '' univ)\nz_ : WithLp 2 (E × F)\nleft✝ : z_ ∈ closure (z '' univ)\nα : ℕ → ℕ\nmonoa : StrictMono α\nconv : Tendsto (fun n => z (α n)) atTop (𝓝 z_)\nρ1 : ℝ\nρ1pos : ρ1 > 0\nsuff_des : ρ1 = (γ - 1) * ↑l ∧ ∀ (k : ℕ), ρ1 / 2 * ‖z (k + 1) - z k‖ ^ 2 ≤ ψ (z k) - ψ (z (k + 1))\nz_in : z_ ∈ limit_set z\nfinal : ∀ (m : ℕ), ∃ n, m ≤ α n\npsiconv : Tendsto (fun n => ψ (z (α n))) atTop (𝓝 (ψ z_))\nmonopsi : ∀ (m n : ℕ), ψ (z (m + n)) ≤ ψ (z m)\nh : ∀ (n : ℕ), ψ (z (α n)) > ψ z_\nη : ℝ\nepos : η > 0\nl1 : ℕ\nleft✝2 : True\nieq : ∀ x ∈ Ici l1, ψ (z (α x)) ∈ Ioo (ψ z_ - η) (ψ z_ + η)\nk : ℕ\nkge : k ≥ α l1\n⊢ k = α l1 + (k - α l1)\n:::"
      },
      {
        "from": "gpt",
        "value": "exact Eq.symm (Nat.add_sub_of_le kge)"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem E τ σ\nf : E → ℝ\nloc : E\nhf : DifferentiableAt ℝ f loc\nv : E\nc : ℕ → ℝ\nd : ℕ → E\na : ℕ\nha : ∀ (b : ℕ), a ≤ b → loc + d b ∈ FeasSet\nvt1 : Tendsto c atTop atTop\nvt2 : Tendsto (fun n => c n • d n) atTop (𝓝 v)\nproneg : ⟪v, gradient f loc⟫_ℝ < 0\ns : Set E\nhs : s ∈ 𝓝 loc ⊓ 𝓟 FeasSet\nhs2 : ∀ y ∈ s, f loc ≤ f y\nε : ℝ\nεpos : ε > 0\nεball : Metric.ball loc ε ∩ FeasSet ⊆ s\ns1 : Set E\nhmin : ∀ y ∈ s1, f loc ≤ f y\nz : ℕ → E\nhzd : ∀ (n : ℕ), d n = z n - loc\nhcp : ∀ᶠ (n : ℕ) in atTop, c n > 0\nhz3 : ∀ᶠ (n : ℕ) in atTop, 1 / c n > 0\nhzt : Tendsto z atTop (𝓝 loc)\nhz : (fun n => f (z n) - f loc - ⟪z n - loc, gradient f loc⟫_ℝ) =o[atTop] fun n => z n - loc\nt1 : (fun n => z n - loc) =O[atTop] fun n => 1 / c n\nt2 : (fun n => f (z n) - f loc - ⟪z n - loc, gradient f loc⟫_ℝ) =o[atTop] fun n => 1 / c n\nthis : (fun x => ‖z x - loc - (1 / c x) • v‖ / ‖1 / c x‖) =ᶠ[atTop] fun x => ‖c x • (z x - loc) - v‖\n⊢ Tendsto (fun x => ‖z x - loc - (1 / c x) • v‖ / ‖1 / c x‖) atTop (𝓝 0)\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [Filter.tendsto_congr' this]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem E τ σ\nf : E → ℝ\nloc : E\nhf : DifferentiableAt ℝ f loc\nv : E\nc : ℕ → ℝ\nd : ℕ → E\na : ℕ\nha : ∀ (b : ℕ), a ≤ b → loc + d b ∈ FeasSet\nvt1 : Tendsto c atTop atTop\nvt2 : Tendsto (fun n => c n • d n) atTop (𝓝 v)\nproneg : ⟪v, gradient f loc⟫_ℝ < 0\ns : Set E\nhs : s ∈ 𝓝 loc ⊓ 𝓟 FeasSet\nhs2 : ∀ y ∈ s, f loc ≤ f y\nε : ℝ\nεpos : ε > 0\nεball : Metric.ball loc ε ∩ FeasSet ⊆ s\ns1 : Set E\nhmin : ∀ y ∈ s1, f loc ≤ f y\nz : ℕ → E\nhzd : ∀ (n : ℕ), d n = z n - loc\nhcp : ∀ᶠ (n : ℕ) in atTop, c n > 0\nhz3 : ∀ᶠ (n : ℕ) in atTop, 1 / c n > 0\nhzt : Tendsto z atTop (𝓝 loc)\nhz : (fun n => f (z n) - f loc - ⟪z n - loc, gradient f loc⟫_ℝ) =o[atTop] fun n => z n - loc\nt1 : (fun n => z n - loc) =O[atTop] fun n => 1 / c n\nt2 : (fun n => f (z n) - f loc - ⟪z n - loc, gradient f loc⟫_ℝ) =o[atTop] fun n => 1 / c n\nt5 : (fun n => z n - loc - (1 / c n) • v) =o[atTop] fun n => 1 / c n\n⊢ ∀ ⦃c_1 : ℝ⦄, 0 < c_1 → ∀ᶠ (x : ℕ) in atTop, ‖⟪z x - loc - (1 / c x) • v, gradient f loc⟫_ℝ‖ ≤ c_1 * ‖1 / c x‖\n:::"
      },
      {
        "from": "gpt",
        "value": "intro c1"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nx : E\ngrad : E\nhf : ∀ ε > 0, ∃ δ > 0, ∀ (x' : E), ‖x - x'‖ ≤ δ → ‖f x' - f x - inner ℝ grad (x' - x)‖ ≤ ε * ‖x - x'‖\nh₁ : ¬f x = 0\nhf' : HasGradientAt f grad x\nε : ℝ\nεpos : ε > 0\nδ' : ℝ\nleft✝ : δ' > 0\nright✝ : ∀ (x' : E), ‖x - x'‖ ≤ δ' → ‖|f x'| - |f x|‖ ≤ ε\nε1pos : 0 < ε * ‖f x‖ * ‖f x‖ / 4\nδ₀ : ℝ\nδ₀pos : δ₀ > 0\nhδ₀ : ∀ (x' : E), ‖x - x'‖ ≤ δ₀ → ‖f x‖ / 2 ≤ ‖f x'‖\nδ'' : ℝ\nδ''pos : δ'' > 0\nhm2 : ∀ (x' : E), ‖x - x'‖ ≤ δ'' → ‖f x' - f x - inner ℝ grad (x' - x)‖ ≤ ε * ‖f x‖ * ‖f x‖ / 4 * ‖x - x'‖\nδ₂ : ℝ\nδ₂pos : δ₂ > 0\nhδ₂ : ∀ (x' : E), ‖x - x'‖ ≤ δ₂ → ‖f x - f x' + inner ℝ grad (x' - x)‖ ≤ ε * ‖f x‖ * ‖f x‖ / 4 * ‖x - x'‖\nε2pos : 0 < ε * ‖f x‖ * ‖f x‖ * ‖f x‖ / (4 * (‖grad‖ + 1))\nδ₃ : ℝ\nδ₃pos : δ₃ > 0\nhδ₃ : ∀ (x' : E), ‖x - x'‖ ≤ δ₃ → ‖f x' - f x‖ ≤ ε * ‖f x‖ * ‖f x‖ * ‖f x‖ / (4 * (‖grad‖ + 1))\nδ₄ : ℝ\nδ₄pos : δ₄ > 0\nhδ₄ : ∀ (x' : E),\n  ‖x - x'‖ ≤ δ₄ → ‖(f x' * inner ℝ grad (x' - x) - f x * inner ℝ grad (x' - x)) / (f x * f x * f x')‖ ≤ ε / 2 * ‖x' - x‖\nx' : E\nh' : ‖x - x'‖ ≤ min δ₀ δ₂\nhp₁ : ‖x - x'‖ ≤ δ₀\n⊢ min δ₀ δ₂ ≤ δ₂\n:::"
      },
      {
        "from": "gpt",
        "value": "exact min_le_right δ₀ δ₂"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nm : ℝ\nf' : E → E\nxm : E\nx₀ : E\nalg : Gradient_Descent_fix_stepsize f f' x₀\nhm : m > 0\nmin : IsMinOn f univ xm\nstep₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀))\nhsc : StrongConvexOn univ m f\nthis : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'\nthis : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0\nk : ℕ\n⊢ ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -\n      Gradient_Descent_fix_stepsize.a f f' x₀ *\n          inner ℝ (Gradient_Descent_fix_stepsize.x f f' x₀ k - xm) (f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)) *\n        2 +\n    (‖Gradient_Descent_fix_stepsize.a f f' x₀‖ * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖) ^ 2 =\n  ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 -\n      Gradient_Descent_fix_stepsize.a f f' x₀ *\n          inner ℝ (Gradient_Descent_fix_stepsize.x f f' x₀ k - xm) (f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)) *\n        2 +\n    Gradient_Descent_fix_stepsize.a f f' x₀ ^ 2 * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2\n:::"
      },
      {
        "from": "gpt",
        "value": "simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "𝕜 : Type u_1\nW : Type u_3\ninst✝ : RCLike 𝕜\ninst✝2 : NormedAddCommGroup W\ninst✝3 : InnerProductSpace 𝕜 W\nr : ℕ\nih : ∀ {V : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace 𝕜 V] [FiniteDimensional 𝕜 V] (T : V →ₗ[𝕜] W),\n  r = finrank 𝕜 ↥(range T) → Nonempty (SVD T r)\nV : Type u_2\ninst✝4 : NormedAddCommGroup V\ninst✝5 : InnerProductSpace 𝕜 V\ninst✝6 : FiniteDimensional 𝕜 V\nT : V →ₗ[𝕜] W\nhr : r + 1 = finrank 𝕜 ↥(range T)\nhV : Nontrivial V\ne₁ : V\nhe₁_unit : ‖e₁‖ = 1\nhT_e₁_max : T.opNorm = ‖T e₁‖\nhT_e₁_nonzero : T e₁ ≠ 0\nσ₁ : ℝ≥0\nw₁ : W\nV₂ : Submodule 𝕜 V\nT₂ : ↥V₂ →ₗ[𝕜] W\nhT_eq : ∀ (v' : Fin r → ↥V₂) (j : Fin r), T ↑(v' j) = T₂ (v' j)\nhrank_T₂ : finrank 𝕜 ↥(range T₂) = r\nσ' : Fin r → ℝ≥0\nhσ_pos' : ∀ (i : Fin r), σ' i > 0\nhσ_anti' : Antitone σ'\nv' : Fin r → ↥V₂\nw' : Fin r → W\nhv_ortho' : Orthonormal 𝕜 v'\nhw_ortho' : Orthonormal 𝕜 w'\nhTv' : ∀ (i : Fin r), T₂ (v' i) = ↑↑(σ' i) • w' i\nhker' : (span 𝕜\n      (Set.range\n        { σ := σ', σ_pos := hσ_pos', σ_antitone := hσ_anti', v := v', w := w', v_orthonormal := hv_ortho',\n            w_orthonormal := hw_ortho', map := hTv' }.v))ᗮ ≤\n  ker T₂\nh : 0 ≤ 0\n⊢ Fin.cons σ₁ σ' 0 ≤ Fin.cons σ₁ σ' 0\n:::"
      },
      {
        "from": "gpt",
        "value": "simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "α : Type u_1\ninst✝ : NormedAddCommGroup α\ninst✝2 : InnerProductSpace ℝ α\ninst✝3 : FiniteDimensional ℝ α\nx : α\nr : ℝ\nhr : r > 0\nhn : finrank ℝ α ≠ 0\nx₀ : α\nhx₀ : ‖σ (x₀ - x)‖ < r\nsum_le_r : ∑ i, ‖(finBasis ℝ α).equivFun (x₀ - x) i‖ * ‖(finBasis ℝ α) i‖ / r ≤ 1\nn : ℕ\nι : Type\nι₀ : Type\nw₀ : Fin (finrank ℝ α) → ℝ\nrepr : ∑ i, w₀ i • (finBasis ℝ α) i = x₀ - x\nw₁ : ι → ℝ\nsum : ℝ\nsum_pos : 1 - sum ≥ 0\nw : ι₀ → ℝ\nlem_i : ∀ {i : ℕ}, i < n + 2 → ¬i = n + 1 → ¬i = n → i < n\nn_pos : n > 0\nfin0 : ι\nz : ι₀ → α\nhw₀ : ∀ (i : ι₀), 0 ≤ w i\nhw₁ : ∑ i, w i = 1\nhz : ∀ (i : ι₀),\n  z i ∈ (⋃ i, {(r / ‖(finBasis ℝ α) i‖) • (finBasis ℝ α) i}) ∪ ⋃ i, {-(r / ‖(finBasis ℝ α) i‖) • (finBasis ℝ α) i}\nbi_pos : ∀ (i : ι), ‖(finBasis ℝ α) i‖ ≠ 0\nthis : ((1 - sum) / 2) • z (Fin.last n).castSucc + ((1 - sum) / 2) • z (Fin.last (n + 1)) = 0\ni : Fin n\nh₁ : ↑i ≠ n + 1\nh₂ : ↑i ≠ n\n⊢ |(finBasis ℝ α).equivFun (x₀ - x) i| * ‖(finBasis ℝ α) i‖ / r *\n    (↑(SignType.sign ((finBasis ℝ α).equivFun (x₀ - x) i)) * (r / ‖(finBasis ℝ α) i‖)) =\n  |(finBasis ℝ α).equivFun (x₀ - x) i| * (‖(finBasis ℝ α) i‖ / r) *\n      ↑(SignType.sign ((finBasis ℝ α).equivFun (x₀ - x) i)) *\n    (r / ‖(finBasis ℝ α) i‖)\n:::"
      },
      {
        "from": "gpt",
        "value": "rw[← mul_div]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nxm : E\nx₀ : E\nalg : subgradient_method f x₀\ns : ℝ\nhm : IsMinOn f Set.univ xm\nha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s\nhs : s > 0\nk : ℕ\nheq : (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =\n  {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}\nhnek : Nonempty { x // x ∈ Finset.range (k + 1) }\nh' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)\nh₁ : ∀ ⦃i : ℕ⦄,\n  i ≥ 0 ∧ i ≤ k →\n    ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 ≤\n      ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -\n          2 * subgradient_method.a f x₀ i *\n            (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +\n        ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2\nh₁' : ∀ ⦃i : ℕ⦄,\n  i ≥ 0 ∧ i ≤ k →\n    subgradient_method.a f x₀ i *\n        (2 * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) ≤\n      ‖subgradient_method.x f x₀ i - xm‖ ^ 2 - ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 + s ^ 2\ni : ℕ\nhi : i ∈ Finset.range (k + 1)\n⊢ i ≥ 0\n:::"
      },
      {
        "from": "gpt",
        "value": "simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nΩ : Set E\nh_compact : IsCompact Ω\nh_Ω1 : ∀ x ∈ Ω, KL_point f x\nh_Ω2 : is_constant_on f Ω\nh_nonempty : Ω.Nonempty\nμ : ℝ\nconstant_value : ∀ x ∈ Ω, f x = μ\nη : E → ℝ\nhη2 : ∀ x ∈ Ω, η x ∈ Ioi 0\nO : E → Set E\nh_Ox : ∀ x ∈ Ω, IsOpen (O x)\nh_Ox' : ∀ x ∈ Ω, x ∈ O x\nφ : E → ℝ → ℝ\nh_exist : ∀ x ∈ Ω,\n  ∀ u ∈ O x ∩ {y | f x < f y ∧ f y < f x + η x},\n    ENNReal.ofReal (deriv (φ x) (f u - f x)) * EMetric.infEdist 0 (subdifferential f u) ≥ 1\nexist_open_cover : Ω ⊆ ⋃ x ∈ Ω, O x\nt : Set E\nht1 : t ⊆ Ω\nht2 : t.Finite\nht3 : Ω ⊆ ⋃ i ∈ t, O i\nmem_t_in_Ω : ∀ x ∈ ht2.toFinset, x ∈ Ω\nt_nonempty : ht2.toFinset.Nonempty\nφ_sum : ℝ → ℝ\nη_min : ℝ\nηpos : η_min ∈ Ioi 0\nhmin : ∀ x ∈ t, η x ≥ η_min\nhφ : ∀ x ∈ Ω,\n  ConcaveOn ℝ (Ico 0 (η x)) (φ x) ∧\n    φ x 0 = 0 ∧\n      ContDiffOn ℝ 1 (φ x) (Ioo 0 (η x)) ∧ ContinuousAt (φ x) 0 ∧ ∀ (x_1 : ℝ), 0 < x_1 → x_1 < η x → 0 < deriv (φ x) x_1\nx : ℝ\nxpos : x ∈ Ico 0 η_min\ny : ℝ\nypos : y ∈ Ico 0 η_min\na : ℝ\nb : ℝ\napos : 0 ≤ a\nbpos : 0 ≤ b\nabsum : a + b = 1\nthis : ∀ (d x : ℝ), d * ∑ c ∈ ht2.toFinset, φ c x = ∑ c ∈ ht2.toFinset, d * φ c x\n⊢ ∀ c ∈ t, a * φ c x + b * φ c y ≤ φ c (a * x + b * y)\n:::"
      },
      {
        "from": "gpt",
        "value": "intro c"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem (EuclideanSpace ℝ (Fin n)) τ σ\nx : EuclideanSpace ℝ (Fin n)\nm : ℕ\nN : ℕ\nMx : EuclideanSpace ℝ (Fin n) →L[ℝ] EuclideanSpace ℝ { x_1 // x_1 ∈ active_set x } × (Fin (n - m) → ℝ)\nRz : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ { x_1 // x_1 ∈ active_set x } × (Fin (n - m) → ℝ)\nRt : ℝ → EuclideanSpace ℝ { x_1 // x_1 ∈ active_set x } × (Fin (n - m) → ℝ)\nd : ℕ → EuclideanSpace ℝ (Fin n)\nv : EuclideanSpace ℝ (Fin n)\nvne0 : v ≠ 0\nRteq : Rt = fun t => t • Mx v\nRxeq0 : Rz x = 0\nhfd : ∀ i ≥ N, Rz (d i) = Rt (1 / ↑i)\ndtend : ∀ (ε : ℝ), 0 < ε → ∃ a, ∀ (b : ℕ), a ≤ b → ‖d b - x‖ < ε\nMxbij : Function.Bijective ⇑Mx\nderiv : Tendsto ((fun x' => ‖x' - x‖⁻¹ * ‖Rz x' - Rz x - Mx (x' - x)‖) ∘ d) atTop (𝓝 0)\ndne : ∀ i ≥ N.succ, d i ≠ x\neq1 : (fun x' => ‖x' - x‖⁻¹ * ‖Rz x' - Rz x - Mx (x' - x)‖) ∘ d = fun i => ‖d i - x‖⁻¹ * ‖Rz (d i) - Rz x - Mx (d i - x)‖\neq2 : (fun i => ‖d i - x‖⁻¹ * ‖Rz (d i) - Rz x - Mx (d i - x)‖) = fun i =>\n  ‖d i - x‖⁻¹ * ‖Rz (d i) - Rt (1 / ↑i) - Rz x - Mx (d i - x - (1 / ↑i) • v)‖\n⊢ ∀ b ≥ N,\n  ‖d b - x‖⁻¹ * ‖Rz (d b) - Rt (1 / ↑b) - Rz x - Mx (d b - x - (1 / ↑b) • v)‖ =\n    ‖d b - x‖⁻¹ * ‖Mx (d b - x - (1 / ↑b) • v)‖\n:::"
      },
      {
        "from": "gpt",
        "value": "intro i"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nn : ℕ+\nhtau : 0 < ADMM.τ E₁ E₂ F ∧ ADMM.τ E₁ E₂ F ≤ 1\nthis : (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - ADMM.x₂ E₁ F (↑n + 1))‖ ^ 2 +\n    (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F *\n      ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂‖ ^ 2 =\n  (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F *\n    (‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - ADMM.x₂ E₁ F (↑n + 1))‖ ^ 2 +\n      ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂‖ ^ 2)\n⊢ 2 * M (n + 1) =\n  (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F *\n    (2 *\n      ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n),\n        (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ)\n:::"
      },
      {
        "from": "gpt",
        "value": "dsimp [M]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "𝕜 : Type u_1\nW : Type u_3\ninst✝ : RCLike 𝕜\ninst✝2 : NormedAddCommGroup W\ninst✝3 : InnerProductSpace 𝕜 W\nr : ℕ\nih : ∀ {V : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace 𝕜 V] [FiniteDimensional 𝕜 V] (T : V →ₗ[𝕜] W),\n  r = finrank 𝕜 ↥(range T) → Nonempty (SVD T r)\nV : Type u_2\ninst✝4 : NormedAddCommGroup V\ninst✝5 : InnerProductSpace 𝕜 V\ninst✝6 : FiniteDimensional 𝕜 V\nT : V →ₗ[𝕜] W\nhr : r + 1 = finrank 𝕜 ↥(range T)\nhV : Nontrivial V\ne₁ : V\nhe₁_unit : ‖e₁‖ = 1\nhT_e₁_max : T.opNorm = ‖T e₁‖\nhT_e₁_nonzero : T e₁ ≠ 0\nσ₁ : ℝ≥0\nw₁ : W\nV₂ : Submodule 𝕜 V\nT₂ : ↥V₂ →ₗ[𝕜] W\nhT_eq : ∀ (v' : Fin r → ↥V₂) (j : Fin r), T ↑(v' j) = T₂ (v' j)\nhrank_T₂ : finrank 𝕜 ↥(range T₂) = r\nσ' : Fin r → ℝ≥0\nhσ_pos' : ∀ (i : Fin r), σ' i > 0\nhσ_anti' : Antitone σ'\nv' : Fin r → ↥V₂\nw' : Fin r → W\nhv_ortho' : Orthonormal 𝕜 v'\nhw_ortho' : Orthonormal 𝕜 w'\nhTv' : ∀ (i : Fin r), T₂ (v' i) = ↑↑(σ' i) • w' i\nhker' : (span 𝕜\n      (Set.range\n        { σ := σ', σ_pos := hσ_pos', σ_antitone := hσ_anti', v := v', w := w', v_orthonormal := hv_ortho',\n            w_orthonormal := hw_ortho', map := hTv' }.v))ᗮ ≤\n  ker T₂\nj : Fin r\nh : 0 ≤ j.succ\nhnorm : ‖T ↑(v' j)‖ = ↑(σ' j)\n⊢ ∀ (v : V), ‖T v‖ = ‖(toContinuousLinearMap T) v‖\n:::"
      },
      {
        "from": "gpt",
        "value": "intro v"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nf' : E → E\ns : Set E\nhs : Convex ℝ s\nhf : ∀ x ∈ s, HasGradientAt f (f' x) x\nmono : ∀ x ∈ s, ∀ y ∈ s, x ≠ y → ⟪f' x - f' y, x - y⟫_ℝ > 0\nx : E\nxin : x ∈ s\ny : E\nyin : y ∈ s\nxney : x ≠ y\na : ℝ\nb : ℝ\napos : 0 < a\nbpos : 0 < b\nabsum1 : a + b = 1\nh₀ : a • f x + b • f y ≤ f (a • x + b • y)\nanneg : 0 ≤ a\nbnneg : 0 ≤ b\nmono' : ∀ x ∈ s, ∀ y ∈ s, ⟪f' x - f' y, x - y⟫_ℝ ≥ 0\nleft✝ : Convex ℝ s\nconvf : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y\neq : f (a • x + b • y) ≤ a • f x + b • f y\neq2 : f (a • x + b • y) = a • f x + b • f y\nz : E\nzin : z ∈ s\nc : ℝ\ncin : c ∈ Ioo 0 1\ne1 : ⟪f' (x + c • (z - x)), z - x⟫_ℝ = f z - f x\nd : ℝ\ndin : d ∈ Ioo 0 1\ne2 : ⟪f' (z + d • (y - z)), y - z⟫_ℝ = f y - f z\neq3 : a * b * ⟪f' (z + d • (y - z)) - f' (x + c • (z - x)), y - x⟫_ℝ = 0\nthis1 : b • (y - z) = a • b • (y - x)\nthis2 : a • (z - x) = a • b • (y - x)\neq0 : ⟪f' (z + d • (y - z)) - f' (x + c • (z - x)), y - x⟫_ℝ = 0\nzeq : z = x + b • (y - x)\nu : E\nv : E\nthis : z + d • (y - z) = x + (b + d) • (y - x) - d • b • (y - x)\n⊢ u = x + (b + d) • (y - x) - d • b • (y - x)\n:::"
      },
      {
        "from": "gpt",
        "value": "by exact this"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nx : E\ngrad : E\nhf : ∀ ε > 0, ∃ δ > 0, ∀ (x' : E), ‖x - x'‖ ≤ δ → ‖f x' - f x - inner ℝ grad (x' - x)‖ ≤ ε * ‖x - x'‖\nh₁ : ¬f x = 0\nhf' : HasGradientAt f grad x\nε : ℝ\nεpos : ε > 0\nδ' : ℝ\nleft✝ : δ' > 0\nright✝ : ∀ (x' : E), ‖x - x'‖ ≤ δ' → ‖|f x'| - |f x|‖ ≤ ε\nε1pos : 0 < ε * ‖f x‖ * ‖f x‖ / 4\nδ₀ : ℝ\nδ₀pos : δ₀ > 0\nhδ₀ : ∀ (x' : E), ‖x - x'‖ ≤ δ₀ → ‖f x‖ / 2 ≤ ‖f x'‖\nδ'' : ℝ\nδ''pos : δ'' > 0\nhm2 : ∀ (x' : E), ‖x - x'‖ ≤ δ'' → ‖f x' - f x - inner ℝ grad (x' - x)‖ ≤ ε * ‖f x‖ * ‖f x‖ / 4 * ‖x - x'‖\nδ₂ : ℝ\nδ₂pos : δ₂ > 0\nhδ₂ : ∀ (x' : E), ‖x - x'‖ ≤ δ₂ → ‖f x - f x' + inner ℝ grad (x' - x)‖ ≤ ε * ‖f x‖ * ‖f x‖ / 4 * ‖x - x'‖\nε2pos : 0 < ε * ‖f x‖ * ‖f x‖ * ‖f x‖ / (4 * (‖grad‖ + 1))\nδ₃ : ℝ\nδ₃pos : δ₃ > 0\nhδ₃ : ∀ (x' : E), ‖x - x'‖ ≤ δ₃ → ‖f x' - f x‖ ≤ ε * ‖f x‖ * ‖f x‖ * ‖f x‖ / (4 * (‖grad‖ + 1))\nδ₄ : ℝ\nδ₄pos : δ₄ > 0\nhδ₄ : ∀ (x' : E),\n  ‖x - x'‖ ≤ δ₄ → ‖(f x' * inner ℝ grad (x' - x) - f x * inner ℝ grad (x' - x)) / (f x * f x * f x')‖ ≤ ε / 2 * ‖x' - x‖\nδ₅ : ℝ\nδ₅pos : δ₅ > 0\nhδ₅ : ∀ (x' : E), ‖x - x'‖ ≤ δ₅ → ‖f x * (f x - f x' + inner ℝ grad (x' - x)) / (f x * f x * f x')‖ ≤ ε / 2 * ‖x' - x‖\n⊢ min δ₀ (min δ₄ δ₅) > 0\n:::"
      },
      {
        "from": "gpt",
        "value": "apply lt_min δ₀pos (lt_min δ₄pos δ₅pos)"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "case.1:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nx : E\ngrad : E\nhf : HasGradientAt f grad x\nh₁ : ¬f x = 0\nhf' : HasGradientAt f grad x\nε : ℝ\nεpos : ε > 0\nδ' : ℝ\nleft✝ : δ' > 0\nright✝ : ∀ (x' : E), ‖x - x'‖ ≤ δ' → ‖|f x'| - |f x|‖ ≤ ε\n⊢ 0 < ε\n\ncase.2:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nx : E\ngrad : E\nhf : HasGradientAt f grad x\nh₁ : ¬f x = 0\nhf' : HasGradientAt f grad x\nε : ℝ\nεpos : ε > 0\nδ' : ℝ\nleft✝ : δ' > 0\nright✝ : ∀ (x' : E), ‖x - x'‖ ≤ δ' → ‖|f x'| - |f x|‖ ≤ ε\n⊢ 0 < ‖f x‖\n\ncase.3:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nx : E\ngrad : E\nhf : HasGradientAt f grad x\nh₁ : ¬f x = 0\nhf' : HasGradientAt f grad x\nε : ℝ\nεpos : ε > 0\nδ' : ℝ\nleft✝ : δ' > 0\nright✝ : ∀ (x' : E), ‖x - x'‖ ≤ δ' → ‖|f x'| - |f x|‖ ≤ ε\n⊢ 0 < ‖f x‖\n\ncase.4:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nx : E\ngrad : E\nhf : HasGradientAt f grad x\nh₁ : ¬f x = 0\nhf' : HasGradientAt f grad x\nε : ℝ\nεpos : ε > 0\nδ' : ℝ\nleft✝ : δ' > 0\nright✝ : ∀ (x' : E), ‖x - x'‖ ≤ δ' → ‖|f x'| - |f x|‖ ≤ ε\n⊢ 0 < 4\n:::"
      },
      {
        "from": "gpt",
        "value": "apply εpos"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nx0 : E\nf : E → ℝ\nh : E → ℝ\nf' : E → E\nalg : Nesterov_second f h f' x0\nxm : E\nminφ : IsMinOn (f + h) univ xm\nφ : E → ℝ\nφdef : ∀ (z : E), φ z = f z + h z\nh1 : ∀ (k : ℕ+),\n  Nesterov_second.γ f h f' x0 ↑k • (Nesterov_second.y f h f' x0 (↑k - 1) - Nesterov_second.y f h f' x0 ↑k) -\n      Nesterov_second.t f h f' x0 ↑k • f' (Nesterov_second.z f h f' x0 k) ∈\n    SubderivAt (Nesterov_second.t f h f' x0 ↑k • h) (Nesterov_second.y f h f' x0 ↑k)\nhieq1 : ∀ (w : E) (k : ℕ+),\n  Nesterov_second.t f h f' x0 ↑k * h w ≥\n    Nesterov_second.t f h f' x0 ↑k * h (Nesterov_second.y f h f' x0 ↑k) +\n      inner ℝ\n        (Nesterov_second.γ f h f' x0 ↑k • (Nesterov_second.y f h f' x0 (↑k - 1) - Nesterov_second.y f h f' x0 ↑k) -\n          Nesterov_second.t f h f' x0 ↑k • f' (Nesterov_second.z f h f' x0 k))\n        (w - Nesterov_second.y f h f' x0 ↑k)\nk : ℕ+\nleft✝ : Convex ℝ univ\nfall : ∀ ⦃x : E⦄,\n  x ∈ univ → ∀ ⦃y : E⦄, y ∈ univ → ∀ ⦃a b : ℝ⦄, 0 < a → 0 < b → a + b = 1 → h (a • x + b • y) ≤ a • h x + b • h y\nmem1 : Nesterov_second.x f h f' x0 (↑k - 1) ∈ univ\nmem2 : Nesterov_second.y f h f' x0 ↑k ∈ univ\neq1 : 0 = 1 - Nesterov_second.γ f h f' x0 ↑k\n⊢ Nesterov_second.γ f h f' x0 ↑k = 1\n:::"
      },
      {
        "from": "gpt",
        "value": "linarith [eq1]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nxm : E\nx₀ : E\nalg : subgradient_method f x₀\nhm : IsMinOn f Set.univ xm\nha'' : ∀ (s : Set ℝ) (x : ℝ),\n  (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s\ns : Set ℝ\nha' : ∀ (s x : Set ℝ),\n  x ⊆ s →\n    (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) → 0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s\nt : Set ℝ\nhs₁ : t ⊆ s✝\nhs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t\nhs₀ : 0 ∈ t\nε : ℝ\nεpos : 0 < ε\nhs₂₀ : Metric.ball 0 ε ⊆ t\nha₁ : ∃ a₁, ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2\nhG : ¬subgradient_method.G f x₀ = 0\nhpos' : ↑(subgradient_method.G f x₀) ^ 2 > 0\ns : Set ℝ\nc₁ : ∀ x_1 ∈ s, x_1 ∈ s\nc₂ : ∀ x_1 ∈ s, ∃ ε, 0 < ε ∧ ∀ (x_2 : ℝ), dist x_2 x_1 < ε → x_2 ∈ s\nc₃ : 0 ∈ s\na₂ : ℕ\nha₂ : ∀ (b : ℕ), a₂ ≤ b → |subgradient_method.a f x₀ b| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)\nA : ℝ\ns₁ : Set ℝ\nthis : ∀ (b : ℝ), A ≤ b → b ∈ s₁\na₁ : ℕ\nhasA : ∀ (b : ℕ), a₁ ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s₁\nb : ℕ\nhb : max a₁ (a₂ + 1) ≤ b\nhba₁ : b ≥ a₁\nhba₂' : b ≥ a₂ + 1\nhba₂ : b ≥ a₂\n⊢ 0 < (Finset.range (b + 1)).sum (subgradient_method.a f x₀)\n:::"
      },
      {
        "from": "gpt",
        "value": "· apply Finset.sum_pos\n            · intro i _; apply (alg.ha i)\n            · simp"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝7 : ProperSpace E\ninst✝8 : ProperSpace F\nγ : ℝ\nhγ : γ > 1\nck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)\ndk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l)\nα : ℕ → ℕ\nz_ : WithLp 2 (E × F)\nmonoa : StrictMono α\nconv : Tendsto (fun n => z (α n)) atTop (𝓝 z_)\nbd : Bornology.IsBounded (z '' univ)\nlbdψ : BddBelow (ψ '' univ)\nlpos : ↑l > 0\nε : ℝ\nepos : 0 < ε✝\nlef : ∀ᶠ (x : ℕ) in atTop, f (z (α x)).1 > f z_.1 - ε✝\nieq : ∀ (q : ℕ),\n  1 ≤ α q →\n    alg.c (α q - 1) * f (alg.x (α q)) +\n        inner ℝ (alg.x (α q) - alg.x (α q - 1)) (alg.c (α q - 1) • grad_fst H (alg.y (α q - 1)) (alg.x (α q - 1))) ≤\n      alg.c (α q - 1) * f z_.1 + ‖z_.1 - alg.x (α q - 1)‖ ^ 2 / 2 +\n        inner ℝ (z_.1 - alg.x (α q - 1)) (alg.c (α q - 1) • grad_fst H (alg.y (α q - 1)) (alg.x (α q - 1)))\nC : ℝ\nhbd : ∀ (q : ℕ), ‖grad_fst H (alg.y (α q - 1)) (alg.x (α q - 1))‖ ≤ C\nε : ℝ\nepos : ε > 0\na : ℕ\nie : ∀ (b : ℕ), a ≤ b → ‖z (b + 1) - z b‖ < ε\nA : ℕ\niee : a + 1 ≤ α A\nb : ℕ\nAleb : A ≤ b\na1leab : a + 1 ≤ α b\nht : ‖z (α b - 1 + 1) - z (α b - 1)‖ < ε\n⊢ 1 ≤ α b\n:::"
      },
      {
        "from": "gpt",
        "value": "linarith [a1leab]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝7 : ProperSpace E\ninst✝8 : ProperSpace F\nγ : ℝ\nhγ : γ > 1\nck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)\ndk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l)\nα : ℕ → ℕ\nz_ : WithLp 2 (E × F)\nmonoa : StrictMono α\nconv : Tendsto (fun n => z (α n)) atTop (𝓝 z_)\nbd : Bornology.IsBounded (z '' univ)\nlbdψ : BddBelow (ψ '' univ)\nε : ℝ\nepos : 0 < ε\nlef : ∀ᶠ (x : ℕ) in atTop, g (z (α x)).2 > g z_.2 - ε\nq : ℕ\nhq : 1 ≤ α q\nieq : alg.d (α q - 1) * g (alg.y (α q - 1 + 1)) +\n    ‖alg.y (α q - 1 + 1) - (alg.y (α q - 1) - alg.d (α q - 1) • grad_snd H (alg.x (α q - 1 + 1)) (alg.y (α q - 1)))‖ ^\n        2 /\n      2 ≤\n  alg.d (α q - 1) * g z_.2 +\n    ‖z_.2 - (alg.y (α q - 1) - alg.d (α q - 1) • grad_snd H (alg.x (α q - 1 + 1)) (alg.y (α q - 1)))‖ ^ 2 / 2\n⊢ alg.d (α q - 1) * g (alg.y (α q)) +\n    inner ℝ (alg.y (α q) - alg.y (α q - 1)) (alg.d (α q - 1) • grad_snd H (alg.x (α q)) (alg.y (α q - 1))) ≤\n  alg.d (α q - 1) * g z_.2 + ‖z_.2 - alg.y (α q - 1)‖ ^ 2 / 2 +\n    inner ℝ (z_.2 - alg.y (α q - 1)) (alg.d (α q - 1) • grad_snd H (alg.x (α q)) (alg.y (α q - 1)))\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [←sub_add,norm_add_sq_real,←sub_add,norm_add_sq_real] at ieq"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nf' : E → E\ns : Set E\nhs : Convex ℝ s\nhf : ∀ x ∈ s, HasGradientAt f (f' x) x\nmono : ∀ x ∈ s, ∀ y ∈ s, x ≠ y → ⟪f' x - f' y, x - y⟫_ℝ > 0\nx : E\nxin : x ∈ s\ny : E\nyin : y ∈ s\nxney : x ≠ y\na : ℝ\nb : ℝ\napos : 0 < a\nbpos : 0 < b\nabsum1 : a + b = 1\nh₀ : a • f x + b • f y ≤ f (a • x + b • y)\nanneg : 0 ≤ a\nbnneg : 0 ≤ b\nmono' : ∀ x ∈ s, ∀ y ∈ s, ⟪f' x - f' y, x - y⟫_ℝ ≥ 0\nleft✝ : Convex ℝ s\nconvf : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y\neq : f (a • x + b • y) ≤ a • f x + b • f y\neq2 : f (a • x + b • y) = a • f x + b • f y\nz : E\nzin : z ∈ s\nc : ℝ\ncin : c ∈ Ioo 0 1\ne1 : ⟪f' (x + c • (z - x)), z - x⟫_ℝ = f z - f x\nd : ℝ\ndin : d ∈ Ioo 0 1\ne2 : ⟪f' (z + d • (y - z)), y - z⟫_ℝ = f y - f z\neq3 : a * b * ⟪f' (z + d • (y - z)) - f' (x + c • (z - x)), y - x⟫_ℝ = 0\nthis1 : b • (y - z) = a • b • (y - x)\nthis2 : a • (z - x) = a • b • (y - x)\neq0 : ⟪f' (z + d • (y - z)) - f' (x + c • (z - x)), y - x⟫_ℝ = 0\nzeq : z = x + b • (y - x)\nu : E\nv : E\nueq : u = x + (b + d) • (y - x) - d • b • (y - x)\nveq : v = x + c • b • (y - x)\nusubv : u - v = (b + d - d * b - c * b) • (y - x)\neeq0 : ⟪f' u - f' v, u - v⟫_ℝ = 0\ncoefne0 : b + d - d * b - c * b > 0\n⊢ 0 ≤ d ∧ d ≤ 1\n:::"
      },
      {
        "from": "gpt",
        "value": "simp at din"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝7 : ProperSpace E\ninst✝8 : ProperSpace F\nγ : ℝ\nhγ : γ > 1\nck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)\ndk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l)\nα : ℕ → ℕ\nz_ : WithLp 2 (E × F)\nmonoa : StrictMono α\nconv : Tendsto (fun n => z (α n)) atTop (𝓝 z_)\nbd : Bornology.IsBounded (z '' univ)\nlbdψ : BddBelow (ψ '' univ)\nlpos : ↑l > 0\nε : ℝ\nepos : 0 < ε\nlef : ∀ᶠ (x : ℕ) in atTop, f (z (α x)).1 > f z_.1 - ε\nieq : ∀ (q : ℕ),\n  1 ≤ α q →\n    alg.c (α q - 1) * f (alg.x (α q)) +\n        inner ℝ (alg.x (α q) - alg.x (α q - 1)) (alg.c (α q - 1) • grad_fst H (alg.y (α q - 1)) (alg.x (α q - 1))) ≤\n      alg.c (α q - 1) * f z_.1 + ‖z_.1 - alg.x (α q - 1)‖ ^ 2 / 2 +\n        inner ℝ (z_.1 - alg.x (α q - 1)) (alg.c (α q - 1) • grad_fst H (alg.y (α q - 1)) (alg.x (α q - 1)))\nC1 : ℝ\ninin : ∀ x ∈ z '' univ, ‖x‖ ≤ C1\ncon11H : ContinuousOn\n  (fun x =>\n    match x with\n    | (x, y) => grad_fst H y x)\n  (Metric.closedBall 0 C1)\nC : ℝ\nsq : ∀ x ∈ Metric.closedBall 0 C1,\n  ‖match x with\n      | (x, y) => grad_fst H y x‖ ≤\n    C\nq : ℕ\n⊢ ‖grad_fst H (alg.y (α q - 1)) (alg.x (α q - 1))‖ ≤ C\n:::"
      },
      {
        "from": "gpt",
        "value": "have : (alg.x (α q -1),alg.y (α q -1)) ∈ Metric.closedBall (0 : WithLp 2 (E × F)) C1 := by\n        apply mem_closedBall_iff_norm.mpr; simp\n        apply inin (alg.x (α q -1),alg.y (α q -1))\n        exact mem_image_of_mem alg.z trivial"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝7 : ProperSpace E\ninst✝8 : ProperSpace F\nγ : ℝ\nhγ : γ > 1\nck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)\ndk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l)\nα : ℕ → ℕ\nz_ : WithLp 2 (E × F)\nmonoa : StrictMono α\nconv : Tendsto (fun n => z (α n)) atTop (𝓝 z_)\nbd : Bornology.IsBounded (z '' univ)\nlbdψ : BddBelow (ψ '' univ)\nlpos : ↑l > 0\nε : ℝ\nepos : 0 < ε✝\nlef : ∀ᶠ (x : ℕ) in atTop, f (z (α x)).1 > f z_.1 - ε✝\nieq : ∀ (q : ℕ),\n  1 ≤ α q →\n    alg.c (α q - 1) * f (alg.x (α q)) +\n        inner ℝ (alg.x (α q) - alg.x (α q - 1)) (alg.c (α q - 1) • grad_fst H (alg.y (α q - 1)) (alg.x (α q - 1))) ≤\n      alg.c (α q - 1) * f z_.1 + ‖z_.1 - alg.x (α q - 1)‖ ^ 2 / 2 +\n        inner ℝ (z_.1 - alg.x (α q - 1)) (alg.c (α q - 1) • grad_fst H (alg.y (α q - 1)) (alg.x (α q - 1)))\nC : ℝ\nhbd : ∀ (q : ℕ), ‖grad_fst H (alg.y (α q - 1)) (alg.x (α q - 1))‖ ≤ C\nε : ℝ\nepos : ε > 0\na : ℕ\nie : ∀ (b : ℕ), a ≤ b → ‖z (b + 1) - z b‖ < ε\nA : ℕ\niee : a + 1 ≤ α A\nb : ℕ\nAleb : A ≤ b\na1leab : a + 1 ≤ α b\n⊢ ‖alg.x (α b) - alg.x (α b - 1)‖ ≤ ‖(alg.x (α b) - alg.x (α b - 1), alg.y (α b) - alg.y (α b - 1))‖\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n          rw [WithLp.prod_norm_eq_of_L2]; simp\n          exact (Real.le_sqrt (norm_nonneg _) (Left.add_nonneg (sq_nonneg _) (sq_nonneg _))).mpr\n            (le_add_of_nonneg_right (sq_nonneg _))"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "α : Type u_1\ninst✝ : NormedAddCommGroup α\ninst✝2 : InnerProductSpace ℝ α\ninst✝3 : FiniteDimensional ℝ α\nf : α → ℝ\ns : Set α\nhs_convex : Convex ℝ s✝\nhs_isopen : ∀ x ∈ s✝, ∃ ε > 0, ball x ε ⊆ s✝\nhf : ConvexOn ℝ s✝ f\nx : α\nhx : x ∈ s✝\nr₀ : ℝ\nhr₀ : r₀ > 0 ∧ ball x r₀ ⊆ s✝\nr : ℝ\nr_pos : r > 0\nn : ℕ\nι : Type\nb : Module.Basis ι ℝ α\nbi_pos : ∀ (i : ι), ‖b i‖ ≠ 0\nhn : ¬n = 0\nn_pos : n > 0\nfin0 : ι\nu₀ : Set α\nu : Set α\nu₀_isFinite : u₀.Finite\nu_isFinite : u.Finite\nu_pos : u ⊆ s✝\nU : Set α\nt : Set α\nt_isopen : IsOpen t\nxBall : Set α\nxBall_isopen : IsOpen xBall\nx_in_xBall : x ∈ xBall\nxBall_in_U : xBall ⊆ U\nx_in_t : x ∈ t\nu' : Finset α\nu_nonempty : u'.Nonempty\nfx : ℝ\nx₁ : α\nhx₁ : x₁ ∈ t ∧ f x₁ = fx\nhf : ConvexOn ℝ ((convexHull ℝ) ↑u') f\nt_pos : t ⊆ U\nsup_ub : ∀ y ∈ u', f y ≤ u'.sup' u_nonempty f\nw : α → ℝ\nhw_pos : ∀ y ∈ u', 0 ≤ w y\nhw_sum : ∑ y ∈ u', w y = 1\nhx₁_eq : u'.centerMass w id = x₁\nw_sum_pos : 0 < ∑ i ∈ u', w i\nthis : ∑ j ∈ u', w j = 1\ns : Finset α\n⊢ ∀ (a : α) (s : Finset α),\n  a ∉ s →\n    (∀ (w : α → ℝ),\n        (∀ i ∈ s, 0 ≤ w i) →\n          ∑ i ∈ s, w i = 1 → (∀ i ∈ s, i ∈ (convexHull ℝ) ↑u') → f (∑ i ∈ s, w i • i) ≤ ∑ i ∈ s, w i * f i) →\n      ∀ (w : α → ℝ),\n        (∀ i ∈ insert a s, 0 ≤ w i) →\n          ∑ i ∈ insert a s, w i = 1 →\n            (∀ i ∈ insert a s, i ∈ (convexHull ℝ) ↑u') → f (∑ i ∈ insert a s, w i • i) ≤ ∑ i ∈ insert a s, w i * f i\n:::"
      },
      {
        "from": "gpt",
        "value": "intro a t hat IH w hw_pos' hw_sum' hmem'"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nxm : E\nx₀ : E\nalg : subgradient_method f x₀\nhm : IsMinOn f Set.univ xm\nha'' : ∀ (s : Set ℝ) (x : ℝ),\n  (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s\ns : Set ℝ\nha' : ∀ (s x : Set ℝ),\n  x ⊆ s →\n    (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) → 0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s\nt : Set ℝ\nhs₁ : t ⊆ s\nhs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t\nhs₀ : 0 ∈ t\nε : ℝ\nεpos : 0 < ε\nhs₂₀ : Metric.ball 0 ε ⊆ t\na₁ : ℕ\nha₁ : ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2\na₂ : ℕ\nha₂ : ∀ (b : ℕ),\n  a₂ ≤ b →\n    (↑(subgradient_method.G f x₀) ^ 2 * ∑ i ∈ Finset.range (b + 1), subgradient_method.a f x₀ i ^ 2) /\n        (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) <\n      ε / 2\nb : ℕ\nhb : max a₁ a₂ ≤ b\nhba₁ : b ≥ a₁\nhba₂ : b ≥ a₂\nhne : sInf {x | ∃ i < b + 1, f (subgradient_method.x f x₀ i) = x} - f xm ≥ 0\nhpos : 2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀) > 0\n⊢ ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) +\n    (↑(subgradient_method.G f x₀) ^ 2 * ∑ i ∈ Finset.range (b + 1), subgradient_method.a f x₀ i ^ 2) /\n      (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) <\n  ε / 2 + ε / 2\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n        apply add_lt_add\n        · apply ha₁ b hba₁\n        · apply ha₂ b hba₂"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nx : E\ngrad : E\nhf : ∀ ε > 0, ∃ δ > 0, ∀ (x' : E), ‖x - x'‖ ≤ δ → ‖f x' - f x - inner ℝ grad (x' - x)‖ ≤ ε * ‖x - x'‖\nh₁ : ¬f x = 0\nhf' : HasGradientAt f grad x\nε : ℝ\nεpos : ε > 0\nδ' : ℝ\nleft✝ : δ' > 0\nright✝ : ∀ (x' : E), ‖x - x'‖ ≤ δ' → ‖|f x'| - |f x|‖ ≤ ε\nε1pos : 0 < ε * ‖f x‖ * ‖f x‖ / 4\nδ₀ : ℝ\nδ₀pos : δ₀ > 0\nhδ₀ : ∀ (x' : E), ‖x - x'‖ ≤ δ₀ → ‖f x‖ / 2 ≤ ‖f x'‖\nδ'' : ℝ\nδ''pos : δ'' > 0\nhm2 : ∀ (x' : E), ‖x - x'‖ ≤ δ'' → ‖f x' - f x - inner ℝ grad (x' - x)‖ ≤ ε * ‖f x‖ * ‖f x‖ / 4 * ‖x - x'‖\nδ₂ : ℝ\nδ₂pos : δ₂ > 0\nhδ₂ : ∀ (x' : E), ‖x - x'‖ ≤ δ₂ → ‖f x - f x' + inner ℝ grad (x' - x)‖ ≤ ε * ‖f x‖ * ‖f x‖ / 4 * ‖x - x'‖\nε2pos : 0 < ε * ‖f x‖ * ‖f x‖ * ‖f x‖ / (4 * (‖grad‖ + 1))\nδ₃ : ℝ\nδ₃pos : δ₃ > 0\nhδ₃ : ∀ (x' : E), ‖x - x'‖ ≤ δ₃ → ‖f x' - f x‖ ≤ ε * ‖f x‖ * ‖f x‖ * ‖f x‖ / (4 * (‖grad‖ + 1))\nδ₄ : ℝ\nδ₄pos : δ₄ > 0\nhδ₄ : ∀ (x' : E),\n  ‖x - x'‖ ≤ δ₄ → ‖(f x' * inner ℝ grad (x' - x) - f x * inner ℝ grad (x' - x)) / (f x * f x * f x')‖ ≤ ε / 2 * ‖x' - x‖\nδ₅ : ℝ\nδ₅pos : δ₅ > 0\nhδ₅ : ∀ (x' : E), ‖x - x'‖ ≤ δ₅ → ‖f x * (f x - f x' + inner ℝ grad (x' - x)) / (f x * f x * f x')‖ ≤ ε / 2 * ‖x' - x‖\nx' : E\nh' : ‖x - x'‖ ≤ min δ₀ (min δ₄ δ₅)\nhp₀ : ‖x - x'‖ ≤ δ₀\nhp₁ : ‖x - x'‖ ≤ δ₄\nhp₂ : ‖x - x'‖ ≤ δ₅\nl0 : 0 < ‖f x'‖\nl' : f x' ≠ 0\n⊢ (f x')⁻¹ - (f x)⁻¹ = (f x - f x') / (f x' * f x)\n:::"
      },
      {
        "from": "gpt",
        "value": "exact inv_sub_inv l' h₁"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nxm : E\nx₀ : E\nalg : subgradient_method f x₀\nhm : IsMinOn f Set.univ xm\nha'' : ∀ (s : Set ℝ) (x : ℝ),\n  (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s\ns : Set ℝ\nha' : ∀ (s x : Set ℝ),\n  x ⊆ s →\n    (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) → 0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s\nt : Set ℝ\nhs₁ : t ⊆ s✝\nhs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t\nhs₀ : 0 ∈ t\nε : ℝ\nεpos : 0 < ε\nhs₂₀ : Metric.ball 0 ε ⊆ t\nha₁ : ∃ a₁, ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2\nhG : ¬subgradient_method.G f x₀ = 0\nhpos' : ↑(subgradient_method.G f x₀) ^ 2 > 0\ns : Set ℝ\nc₁ : ∀ x_1 ∈ s, x_1 ∈ s\nc₂ : ∀ x_1 ∈ s, ∃ ε, 0 < ε ∧ ∀ (x_2 : ℝ), dist x_2 x_1 < ε → x_2 ∈ s\nc₃ : 0 ∈ s\na₂ : ℕ\nha₂ : ∀ (b : ℕ), a₂ ≤ b → |subgradient_method.a f x₀ b| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)\nA : ℝ\ns₁ : Set ℝ\nthis : ∀ (b : ℝ), A ≤ b → b ∈ s₁\na₁ : ℕ\nhasA : ∀ (b : ℕ), a₁ ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s₁\nb : ℕ\nhb : max a₁ (a₂ + 1) ≤ b\nhba₁ : b ≥ a₁\nhba₂' : b ≥ a₂ + 1\nhba₂ : b ≥ a₂\nhpos : 2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀) > 0\nhpos'' : (Finset.range (b + 1)).sum (subgradient_method.a f x₀) > 0\n⊢ 0 < ↑(subgradient_method.G f x₀) ^ 2\n:::"
      },
      {
        "from": "gpt",
        "value": "apply hpos'"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_4\ninst✝ : NormedAddCommGroup E\ninst✝2 : NormedSpace ℝ E\ninst✝3 : FiniteDimensional ℝ E\nι : Type\ninst✝4 : Fintype ι\nC : ι → Set E\nhC : ∀ (i : ι), Convex ℝ (C i)\nhne : ∀ (i : ι), (C i).Nonempty\nn : ℕ\nih : ∀ {ι : Type} [inst : Fintype ι] (C : ι → Set E),\n  (∀ (i : ι), Convex ℝ (C i)) →\n    (∀ (i : ι), (C i).Nonempty) →\n      Fintype.card ι = n + 1 →\n        intrinsicInterior ℝ ((convexHull ℝ) (⋃ i, C i)) =\n          ⋃ s ∈ {t | (∀ (i : ι), t i > 0) ∧ ∑ i, t i = 1}, ∑ i, s i • intrinsicInterior ℝ (C i)\nhι : Fintype.card ι = n + 1 + 1\ni₀ : ι\nι' : Type\nthis : Fintype ι'\nj : ι\nhj : j ≠ i₀\nh1 : intrinsicInterior ℝ ((convexHull ℝ) (⋃ i, (fun i => C ↑i) i)) =\n  ⋃ s ∈ {t | (∀ (i : ι'), t i > 0) ∧ ∑ i, t i = 1}, ∑ i, s i • intrinsicInterior ℝ ((fun i => C ↑i) i)\na : Set E\nhna : a.Nonempty\nh2 : intrinsicInterior ℝ ((convexHull ℝ) (a ∪ C i₀)) =\n  ⋃ s ∈ {(m, n) | m > 0 ∧ n > 0 ∧ m + n = 1}, s.1 • intrinsicInterior ℝ a + s.2 • intrinsicInterior ℝ (C i₀)\nx : E\nl : ι → ℝ\nhx : x ∈ ∑ i, l i • intrinsicInterior ℝ (C i)\nhl1 : ∀ (i : ι), 0 < l i\nhl2 : ∑ i, l i = 1\nhl0_pos : 0 < 1 - l i₀\nhl0_ne0 : 1 - l i₀ ≠ 0\nhx' : x ∈ ∑ i, l ↑i • intrinsicInterior ℝ (C ↑i) + l i₀ • intrinsicInterior ℝ (C i₀)\n⊢ ∑ i, l ↑i • intrinsicInterior ℝ (C ↑i) + l i₀ • intrinsicInterior ℝ (C i₀) ⊆\n  ((1 - l i₀) • ⋃ s, ⋃ (_ : (∀ (i : ι'), 0 < s i) ∧ ∑ i, s i = 1), ∑ x, s x • intrinsicInterior ℝ (C ↑x)) +\n    l i₀ • intrinsicInterior ℝ (C i₀)\n:::"
      },
      {
        "from": "gpt",
        "value": "apply Set.add_subset_add_right"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nf' : E → E\nxm : E\nx₀ : E\nγ : ℕ+ → ℝ\nalg : Nesterov f f' γ x₀\nhfun : ConvexOn ℝ univ f\nhg : ∀ (k : ℕ+), γ k = 2 / (↑↑k + 1)\nmin : IsMinOn f univ xm\ncon : ∀ (k : ℕ+), (1 - γ k) / γ k ^ 2 ≤ 1 / γ (k - 1) ^ 2\nh4 : ∀ (k : ℕ+),\n  f (Nesterov.x f f' γ x₀ ↑k) - f xm - (1 - γ k) * (f (Nesterov.x f f' γ x₀ (↑k - 1)) - f xm) ≤\n    ↑(Nesterov.l f f' γ x₀) * γ k ^ 2 / 2 *\n      (‖Nesterov.v f f' γ x₀ (↑k - 1) - xm‖ ^ 2 - ‖Nesterov.v f f' γ x₀ ↑k - xm‖ ^ 2)\nj : ℕ+\nIH : 1 / γ j ^ 2 * (f (Nesterov.x f f' γ x₀ ↑j) - f xm) + ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑j - xm‖ ^ 2 ≤\n  1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) + ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2\nh5 : 1 / γ (j + 1) ^ 2 * (f (Nesterov.x f f' γ x₀ ↑(j + 1)) - f xm) +\n    ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ ↑(j + 1) - xm‖ ^ 2 ≤\n  1 / γ (j + 1 - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑(j + 1) - 1)) - f xm) +\n    ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑(j + 1) - 1) - xm‖ ^ 2\ny1 : ↑(j + 1) - 1 = ↑j\ny2 : j + 1 - 1 = j\n⊢ 1 / γ (j + 1 - 1) ^ 2 * (f (Nesterov.x f f' γ x₀ (↑(j + 1) - 1)) - f xm) +\n    ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ (↑(j + 1) - 1) - xm‖ ^ 2 ≤\n  1 / γ 1 ^ 2 * (f (Nesterov.x f f' γ x₀ 1) - f xm) + ↑(Nesterov.l f f' γ x₀) / 2 * ‖Nesterov.v f f' γ x₀ 1 - xm‖ ^ 2\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [y1, y2]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝7 : ProperSpace E\ninst✝8 : ProperSpace F\nγ : ℝ\nhγ : γ > 1\nck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)\ndk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l)\nα : ℕ → ℕ\nz_ : WithLp 2 (E × F)\nmonoa : StrictMono α\nconv : Tendsto (fun n => z (α n)) atTop (𝓝 z_)\nbd : Bornology.IsBounded (z '' univ)\nlbdψ : BddBelow (ψ '' univ)\nε : ℝ\nepos : 0 < ε✝\nlef : ∀ᶠ (x : ℕ) in atTop, g (z (α x)).2 > g z_.2 - ε✝\nieq : ∀ (q : ℕ),\n  1 ≤ α q →\n    alg.d (α q - 1) * g (alg.y (α q)) +\n        inner ℝ (alg.y (α q) - alg.y (α q - 1)) (alg.d (α q - 1) • grad_snd H (alg.x (α q)) (alg.y (α q - 1))) ≤\n      alg.d (α q - 1) * g z_.2 + ‖z_.2 - alg.y (α q - 1)‖ ^ 2 / 2 +\n        inner ℝ (z_.2 - alg.y (α q - 1)) (alg.d (α q - 1) • grad_snd H (alg.x (α q)) (alg.y (α q - 1)))\nC : ℝ\nhbd : ∀ (q : ℕ), ‖grad_snd H (alg.x (α q)) (alg.y (α q - 1))‖ ≤ C\ndiflte1 : ∀ ε > 0, ∀ᶠ (q : ℕ) in atTop, ‖alg.y (α q) - alg.y (α q - 1)‖ < ε\ndiflte2 : ∀ ε > 0, ∀ᶠ (q : ℕ) in atTop, ‖z_.2 - alg.y (α q - 1)‖ < ε\nk : ℕ → F\nε : ℝ\nepos : ε > 0\ndefle : ∀ (ε : ℝ), 0 < ε → ∃ a, ∀ (b : ℕ), a ≤ b → ‖k b‖ < ε\nCpos : 0 < C\nthis : 0 < ε / (C / (γ * ↑l))\nnn : ℕ\nieq : ∀ (b : ℕ), nn ≤ b → ‖k b‖ < ε / (C / (γ * ↑l))\nb : ℕ\nnleb : nn ≤ b\n⊢ 0 < γ\n:::"
      },
      {
        "from": "gpt",
        "value": "linarith [hγ]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nr_Φ : ℝ\nhΦ : range Φ ⊆ ball 0 r_Φ\nh1 : ∀ (k : ℕ+),\n  min 1 (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F *\n      ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 ≤\n    min (ADMM.τ E₁ E₂ F) (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2) * ADMM.ρ E₁ E₂ F *\n        ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑k - ADMM.x₂ E₁ F (↑k + 1))‖ ^ 2 +\n      min 1 (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F *\n        ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2\nh2 : ∀ (k : ℕ+),\n  min (ADMM.τ E₁ E₂ F) (1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2) * ADMM.ρ E₁ E₂ F *\n        ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑k - ADMM.x₂ E₁ F (↑k + 1))‖ ^ 2 +\n      min 1 (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F *\n        ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ ^ 2 ≤\n    Φ ↑k - Φ (↑k + 1)\nk : ℕ+\nh3a : ‖Φ ↑k‖ < r_Φ\nh3b : ‖Φ (↑k + 1)‖ < r_Φ\nh3' : ‖Φ ↑k‖ + ‖Φ (↑k + 1)‖ < r_Φ + r_Φ\nh3'' : ‖Φ ↑k‖ + ‖Φ (↑k + 1)‖ < 2 * r_Φ\n⊢ ‖Φ ↑k - Φ (↑k + 1)‖ ≤ ‖Φ ↑k‖ + ‖Φ (↑k + 1)‖\n:::"
      },
      {
        "from": "gpt",
        "value": "apply norm_sub_le"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝7 : ProperSpace E\ninst✝8 : ProperSpace F\nγ : ℝ\nhγ : γ > 1\nck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)\ndk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l)\nα : ℕ → ℕ\nz_ : WithLp 2 (E × F)\nmonoa : StrictMono α\nconv : Tendsto (fun n => z (α n)) atTop (𝓝 z_)\nbd : Bornology.IsBounded (z '' univ)\nlbdψ : BddBelow (ψ '' univ)\nlpos : ↑l > 0\nε : ℝ\nepos : 0 < ε\nlef : ∀ᶠ (x : ℕ) in atTop, f (z (α x)).1 > f z_.1 - ε\nC : ℝ\nhbd : ∀ (q : ℕ), ‖grad_fst H (alg.y (α q - 1)) (alg.x (α q - 1))‖ ≤ C\ndiflte1 : ∀ ε > 0, ∀ᶠ (q : ℕ) in atTop, ‖alg.x (α q) - alg.x (α q - 1)‖ < ε\ndiflte2 : ∀ ε > 0, ∀ᶠ (q : ℕ) in atTop, ‖z_.1 - alg.x (α q - 1)‖ < ε\nhk : ∀ (k : ℕ → E),\n  (∀ ε > 0, ∀ᶠ (q : ℕ) in atTop, ‖k q‖ < ε) →\n    ∀ ε > 0, ∀ᶠ (q : ℕ) in atTop, |inner ℝ (k q) (alg.c (α q - 1) • grad_fst H (alg.y (α q - 1)) (alg.x (α q - 1)))| ≤ ε\nieq : ∀ (q : ℕ),\n  1 ≤ α q →\n    1 / (γ * ↑l) * f (alg.x (α q)) +\n        inner ℝ (alg.x (α q) - alg.x (α q - 1)) ((1 / (γ * ↑l)) • grad_fst H (alg.y (α q - 1)) (alg.x (α q - 1))) ≤\n      1 / (γ * ↑l) * f z_.1 + ‖z_.1 - alg.x (α q - 1)‖ ^ 2 / 2 +\n        inner ℝ (z_.1 - alg.x (α q - 1)) ((1 / (γ * ↑l)) • grad_fst H (alg.y (α q - 1)) (alg.x (α q - 1)))\n⊢ ∀ᶠ (x : ℕ) in atTop, f (z (α x)).1 < f z_.1 + ε\n:::"
      },
      {
        "from": "gpt",
        "value": "have h1 := hk (fun q ↦ alg.x (α q) - alg.x (α q - 1)) diflte1 (ε / (γ * l) / 3) (by positivity)"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nvn : ℕ → E\nv : E\nvnin : ∀ (n : ℕ),\n  vn n ∈\n    {v₀ |\n      ∃ u,\n        Tendsto u atTop (𝓝 x) ∧\n          Tendsto (fun n => f (u n)) atTop (𝓝 (f x)) ∧\n            ∃ v, ∀ (n : ℕ), v n ∈ f_subdifferential f (u n) ∧ Tendsto v atTop (𝓝 v₀)}\nvntov : Tendsto vn atTop (𝓝 v)\nh' : ∀ (xk uk : ℕ → E) (u : E) (n : ℕ),\n  0 < n →\n    Tendsto xk atTop (𝓝 x) →\n      Tendsto (fun k => f (xk k)) atTop (𝓝 (f x)) →\n        Tendsto uk atTop (𝓝 u) →\n          ∃ N, ‖uk N - u‖ < 1 / (2 * ↑n) ∧ ‖xk N - x‖ < 1 / (2 * ↑n) ∧ ‖f (xk N) - f x‖ < 1 / (2 * ↑n)\nxnk : ℕ → ℕ → E\nallh : ∀ (x_1 : ℕ),\n  Tendsto (xnk x_1) atTop (𝓝 x) ∧\n    Tendsto (fun k => f (xnk x_1 k)) atTop (𝓝 (f x)) ∧\n      ∃ v, ∀ (n_1 : ℕ), v n_1 ∈ f_subdifferential f (xnk x_1 n_1) ∧ Tendsto v atTop (𝓝 (vn x_1))\nxnkt : ∀ (n : ℕ), Tendsto (xnk n) atTop (𝓝 x)\nfxnkt : ∀ (n : ℕ), Tendsto (fun k => f (xnk n k)) atTop (𝓝 (f x))\nvnk : ℕ → ℕ → E\nallh' : ∀ (x n_1 : ℕ), vnk x n_1 ∈ f_subdifferential f (xnk x n_1) ∧ Tendsto (vnk x) atTop (𝓝 (vn x))\nvnkt : ∀ (n : ℕ), Tendsto (vnk n) atTop (𝓝 (vn n))\nnpos : ∀ (n : ℕ), 0 < n + 1\nNn : ℕ → ℕ\nhNn' : ∀ (x_1 : ℕ),\n  ‖vnk (x_1 + 1) (Nn x_1) - vn (x_1 + 1)‖ < 1 / (2 * ↑(x_1 + 1)) ∧\n    ‖xnk (x_1 + 1) (Nn x_1) - x‖ < 1 / (2 * ↑(x_1 + 1)) ∧ ‖f (xnk (x_1 + 1) (Nn x_1)) - f x‖ < 1 / (2 * ↑(x_1 + 1))\n⊢ ∀ (n : ℕ),\n  (fun n => vnk (n + 1) (Nn n)) n ∈ f_subdifferential f (xnk (n + 1) (Nn n)) ∧\n    Tendsto (fun n => vnk (n + 1) (Nn n)) atTop (𝓝 v)\n:::"
      },
      {
        "from": "gpt",
        "value": "intro n"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nx : E\ngrad : E\nhf : ∀ ε > 0, ∃ δ > 0, ∀ (x' : E), ‖x - x'‖ ≤ δ → ‖f x' - f x - inner ℝ grad (x' - x)‖ ≤ ε * ‖x - x'‖\nh₁ : ¬f x = 0\nhf' : HasGradientAt f grad x\nε : ℝ\nεpos : ε > 0\nδ' : ℝ\nleft✝ : δ' > 0\nright✝ : ∀ (x' : E), ‖x - x'‖ ≤ δ' → ‖|f x'| - |f x|‖ ≤ ε\nε1pos : 0 < ε * ‖f x‖ * ‖f x‖ / 4\nδ₀ : ℝ\nδ₀pos : δ₀ > 0\nhδ₀ : ∀ (x' : E), ‖x - x'‖ ≤ δ₀ → ‖f x‖ / 2 ≤ ‖f x'‖\nδ'' : ℝ\nδ''pos : δ'' > 0\nhm2 : ∀ (x' : E), ‖x - x'‖ ≤ δ'' → ‖f x' - f x - inner ℝ grad (x' - x)‖ ≤ ε * ‖f x‖ * ‖f x‖ / 4 * ‖x - x'‖\nδ₂ : ℝ\nδ₂pos : δ₂ > 0\nhδ₂ : ∀ (x' : E), ‖x - x'‖ ≤ δ₂ → ‖f x - f x' + inner ℝ grad (x' - x)‖ ≤ ε * ‖f x‖ * ‖f x‖ / 4 * ‖x - x'‖\nε2pos : 0 < ε * ‖f x‖ * ‖f x‖ * ‖f x‖ / (4 * (‖grad‖ + 1))\nδ₃ : ℝ\nδ₃pos : δ₃ > 0\nhδ₃ : ∀ (x' : E), ‖x - x'‖ ≤ δ₃ → ‖f x' - f x‖ ≤ ε * ‖f x‖ * ‖f x‖ * ‖f x‖ / (4 * (‖grad‖ + 1))\nδ₄ : ℝ\nδ₄pos : δ₄ > 0\nhδ₄ : ∀ (x' : E),\n  ‖x - x'‖ ≤ δ₄ → ‖(f x' * inner ℝ grad (x' - x) - f x * inner ℝ grad (x' - x)) / (f x * f x * f x')‖ ≤ ε / 2 * ‖x' - x‖\nx' : E\nh' : ‖x - x'‖ ≤ min δ₀ δ₂\nhp₁ : ‖x - x'‖ ≤ δ₀\nhp₂ : ‖x - x'‖ ≤ δ₂\nzp1 : ‖f x * (f x - f x' + inner ℝ grad (x' - x)) / (f x * f x * f x')‖ =\n  ‖(f x - f x' + inner ℝ grad (x' - x)) / (f x * f x')‖\nzp2 : ‖f x‖ * ‖f x‖ / 2 ≤ ‖f x * f x'‖\nzp3 : ‖1 / (f x * f x')‖ ≤ 2 / (‖f x‖ * ‖f x‖)\nzp5 : ε * ‖f x‖ * ‖f x‖ / 4 * ‖x - x'‖ * (2 / (‖f x‖ * ‖f x‖)) = ε / 2 * ‖x' - x‖\n⊢ ε * ‖f x‖ * ‖f x‖ / 4 * ‖x - x'‖ * (2 / (‖f x‖ * ‖f x‖)) = ε / 2 * ‖x' - x‖\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n        apply zp5"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\ninst✝3 : CompleteSpace E\nf : E → ℝ\nx : E\ngrad : E\nhf : ∀ ε > 0, ∃ δ > 0, ∀ (x' : E), ‖x - x'‖ ≤ δ → ‖f x' - f x - inner ℝ grad (x' - x)‖ ≤ ε * ‖x - x'‖\nh₁ : ¬f x = 0\nhf' : HasGradientAt f grad x\nε : ℝ\nεpos : ε > 0\nδ' : ℝ\nleft✝ : δ' > 0\nright✝ : ∀ (x' : E), ‖x - x'‖ ≤ δ' → ‖|f x'| - |f x|‖ ≤ ε\nε1pos : 0 < ε * ‖f x‖ * ‖f x‖ / 4\nδ₀ : ℝ\nδ₀pos : δ₀ > 0\nhδ₀ : ∀ (x' : E), ‖x - x'‖ ≤ δ₀ → ‖f x‖ / 2 ≤ ‖f x'‖\nδ'' : ℝ\nδ''pos : δ'' > 0\nhm2 : ∀ (x' : E), ‖x - x'‖ ≤ δ'' → ‖f x' - f x - inner ℝ grad (x' - x)‖ ≤ ε * ‖f x‖ * ‖f x‖ / 4 * ‖x - x'‖\nδ₂ : ℝ\nδ₂pos : δ₂ > 0\nhδ₂ : ∀ (x' : E), ‖x - x'‖ ≤ δ₂ → ‖f x - f x' + inner ℝ grad (x' - x)‖ ≤ ε * ‖f x‖ * ‖f x‖ / 4 * ‖x - x'‖\nε2pos : 0 < ε * ‖f x‖ * ‖f x‖ * ‖f x‖ / (4 * (‖grad‖ + 1))\nδ₃ : ℝ\nδ₃pos : δ₃ > 0\nhδ₃ : ∀ (x' : E), ‖x - x'‖ ≤ δ₃ → ‖f x' - f x‖ ≤ ε * ‖f x‖ * ‖f x‖ * ‖f x‖ / (4 * (‖grad‖ + 1))\nx' : E\nh' : ‖x - x'‖ ≤ min δ₀ δ₃\nhp₁ : ‖x - x'‖ ≤ δ₀\nhp₂ : ‖x - x'‖ ≤ δ₃\nl0 : 0 < ‖f x'‖\nl1 : ‖(f x' - f x) * inner ℝ grad (x' - x) / (f x * f x * f x')‖ =\n  ‖(f x' - f x) * inner ℝ grad (x' - x)‖ * ‖1 / (f x * f x * f x')‖\nl2 : ‖(f x' - f x) * inner ℝ grad (x' - x)‖ * ‖1 / (f x * f x * f x')‖ =\n  ‖f x' - f x‖ * ‖inner ℝ grad (x' - x)‖ * ‖1 / (f x * f x * f x')‖\nl3 : ‖f x' - f x‖ * ‖inner ℝ grad (x' - x)‖ ≤ ‖f x' - f x‖ * ‖grad‖ * ‖x' - x‖\nl5 : ‖f x' - f x‖ ≤ ε * ‖f x‖ * ‖f x‖ * ‖f x‖ / (4 * (‖grad‖ + 1))\n⊢ ‖2 / (f x * f x * f x)‖ * ‖x' - x‖ * (‖grad‖ + 1) * (ε * ‖f x‖ * ‖f x‖ * ‖f x‖ / (4 * (‖grad‖ + 1))) = ε / 2 * ‖x' - x‖\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n        apply Simplifying₂\n        apply h₁\n        apply norm_nonneg"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nτ : Finset ℕ\nσ : Finset ℕ\np : Constrained_OptimizationProblem (EuclideanSpace ℝ (Fin n)) τ σ\nx : EuclideanSpace ℝ (Fin n)\nxf : x ∈ FeasSet\nconte : ∀ i ∈ τ, ContDiffAt ℝ (↑1) (p.equality_constraints i) x\nconti : ∀ i ∈ σ, ContDiffAt ℝ (↑1) (p.inequality_constraints i) x\nLIx : LinearIndependent ℝ fun i =>\n  if ↑i ∈ τ then gradient (p.equality_constraints ↑i) x else gradient (p.inequality_constraints ↑i) x\nhdomain : p.domain = univ\nv : EuclideanSpace ℝ (Fin n)\nhv : v ∈ linearized_feasible_directions x\nveq0 : ¬v = 0\ngradc : EuclideanSpace ℝ (Fin n) → { x_1 // x_1 ∈ active_set x } → EuclideanSpace ℝ (Fin n)\nAx : Matrix { x_1 // x_1 ∈ active_set x } (Fin n) ℝ\nm : ℕ\nmlen : m ≤ n\nZ : Matrix (Fin n) (Fin (n - m)) ℝ\neq1 : Ax * Z = 0\neq2 : Z.rank = n - m\nMx : EuclideanSpace ℝ (Fin n) →L[ℝ] EuclideanSpace ℝ { x_1 // x_1 ∈ active_set x } × (Fin (n - m) → ℝ)\nc : EuclideanSpace ℝ (Fin n) → { x_1 // x_1 ∈ active_set x } → ℝ\nRz : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ { x_1 // x_1 ∈ active_set x } × (Fin (n - m) → ℝ)\nRt : ℝ → EuclideanSpace ℝ { x_1 // x_1 ∈ active_set x } × (Fin (n - m) → ℝ)\nA : EuclideanSpace ℝ (Fin n) → Matrix { x_1 // x_1 ∈ active_set x } (Fin n) ℝ\nJz : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin n) →L[ℝ] EuclideanSpace ℝ { x_1 // x_1 ∈ active_set x }\ncgrad_atx : Jz x = LinearMap.toContinuousLinearMap (toEuclideanLin Ax)\nRzgrad : HasStrictFDerivAt Rz Mx x\nRxeq0 : Rz x = 0\nz : EuclideanSpace ℝ (Fin n)\nMzeq0 : (toEuclideanLin Ax) z = 0 ∧ (toEuclideanLin Zᵀ) z = 0\nheq1 : Ax *ᵥ z = 0\nheq2 : Zᵀ *ᵥ z = 0\n⊢ z = 0\n:::"
      },
      {
        "from": "gpt",
        "value": "refine LICQ_injM z m Z Ax ?_ mlen ?_ eq2 eq1 ⟨heq1, heq2⟩"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nvn : ℕ → E\nv : E\nvnin : ∀ (n : ℕ),\n  vn n ∈\n    {v₀ |\n      ∃ u,\n        Tendsto u atTop (𝓝 x) ∧\n          Tendsto (fun n => f (u n)) atTop (𝓝 (f x)) ∧\n            ∃ v, ∀ (n : ℕ), v n ∈ f_subdifferential f (u n) ∧ Tendsto v atTop (𝓝 v₀)}\nvntov : Tendsto vn atTop (𝓝 v)\nh' : ∀ (xk uk : ℕ → E) (u : E) (n : ℕ),\n  0 < n →\n    Tendsto xk atTop (𝓝 x) →\n      Tendsto (fun k => f (xk k)) atTop (𝓝 (f x)) →\n        Tendsto uk atTop (𝓝 u) →\n          ∃ N, ‖uk N - u‖ < 1 / (2 * ↑n) ∧ ‖xk N - x‖ < 1 / (2 * ↑n) ∧ ‖f (xk N) - f x‖ < 1 / (2 * ↑n)\nxnk : ℕ → ℕ → E\nallh : ∀ (x_1 : ℕ),\n  Tendsto (xnk x_1) atTop (𝓝 x) ∧\n    Tendsto (fun k => f (xnk x_1 k)) atTop (𝓝 (f x)) ∧\n      ∃ v, ∀ (n_1 : ℕ), v n_1 ∈ f_subdifferential f (xnk x_1 n_1) ∧ Tendsto v atTop (𝓝 (vn x_1))\nxnkt : ∀ (n : ℕ), Tendsto (xnk n) atTop (𝓝 x)\nfxnkt : ∀ (n : ℕ), Tendsto (fun k => f (xnk n k)) atTop (𝓝 (f x))\nvnk : ℕ → ℕ → E\nallh' : ∀ (x n_1 : ℕ), vnk x n_1 ∈ f_subdifferential f (xnk x n_1) ∧ Tendsto (vnk x) atTop (𝓝 (vn x))\nvnkt : ∀ (n : ℕ), Tendsto (vnk n) atTop (𝓝 (vn n))\nnpos : ∀ (n : ℕ), 0 < n + 1\nNn : ℕ → ℕ\nhNn' : ∀ (x_1 : ℕ),\n  ‖vnk (x_1 + 1) (Nn x_1) - vn (x_1 + 1)‖ < 1 / (2 * ↑(x_1 + 1)) ∧\n    ‖xnk (x_1 + 1) (Nn x_1) - x‖ < 1 / (2 * ↑(x_1 + 1)) ∧ ‖f (xnk (x_1 + 1) (Nn x_1)) - f x‖ < 1 / (2 * ↑(x_1 + 1))\nu : Set ℝ\nfxinu : f x ∈ u\nε : ℝ\nεpos : ε > 0\nεin : Metric.ball (f x) ε ⊆ u\nε' : ℤ\nhε' : ↑ε' ≤ 1 / ε\nhε'' : ∀ (z : ℤ), ↑z ≤ 1 / ε → z ≤ ε'\nN : ℕ\nleft✝ : ↑N ≤ 1 / ε\nn : ℕ\nnge : N ≤ n\nNge : ε⁻¹ ≤ ↑N + 1\n⊢ ε⁻¹ ≤ ↑n + 1\n:::"
      },
      {
        "from": "gpt",
        "value": "apply le_trans"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "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\ninst✝7 : ProperSpace E\ninst✝8 : ProperSpace F\nγ : ℝ\nhγ : γ > 1\nck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)\ndk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l)\nα : ℕ → ℕ\nz_ : WithLp 2 (E × F)\nmonoa : StrictMono α\nconv : Tendsto (fun n => z (α n)) atTop (𝓝 z_)\nbd : Bornology.IsBounded (z '' univ)\nlbdψ : BddBelow (ψ '' univ)\nε : ℝ\nepos : 0 < ε✝\nlef : ∀ᶠ (x : ℕ) in atTop, g (z (α x)).2 > g z_.2 - ε✝\nieq : ∀ (q : ℕ),\n  1 ≤ α q →\n    alg.d (α q - 1) * g (alg.y (α q)) +\n        inner ℝ (alg.y (α q) - alg.y (α q - 1)) (alg.d (α q - 1) • grad_snd H (alg.x (α q)) (alg.y (α q - 1))) ≤\n      alg.d (α q - 1) * g z_.2 + ‖z_.2 - alg.y (α q - 1)‖ ^ 2 / 2 +\n        inner ℝ (z_.2 - alg.y (α q - 1)) (alg.d (α q - 1) • grad_snd H (alg.x (α q)) (alg.y (α q - 1)))\nC : ℝ\nhbd : ∀ (q : ℕ), ‖grad_snd H (alg.x (α q)) (alg.y (α q - 1))‖ ≤ C\ndiflte1 : ∀ ε > 0, ∀ᶠ (q : ℕ) in atTop, ‖alg.y (α q) - alg.y (α q - 1)‖ < ε\ndiflte2 : ∀ ε > 0, ∀ᶠ (q : ℕ) in atTop, ‖z_.2 - alg.y (α q - 1)‖ < ε\nk : ℕ → F\nε : ℝ\nepos : ε > 0\ndefle : ∀ (ε : ℝ), 0 < ε → ∃ a, ∀ (b : ℕ), a ≤ b → ‖k b‖ < ε\nCpos : 0 < C\nthis : 0 < ε / (C / (γ * ↑l))\nnn : ℕ\nieq : ∀ (b : ℕ), nn ≤ b → ‖k b‖ < ε / (C / (γ * ↑l))\nb : ℕ\nnleb : nn ≤ b\n⊢ ε / (C / (γ * ↑l)) * ‖(1 / (γ * ↑l)) • grad_snd H (alg.x (α b)) (alg.y (α b - 1))‖ =\n  ε / (C / (γ * ↑l)) * (1 / (γ * ↑l) * ‖grad_snd H (alg.x (α b)) (alg.y (α b - 1))‖)\n:::"
      },
      {
        "from": "gpt",
        "value": "apply mul_eq_mul_left_iff.mpr"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  }
]