[
  {
    "conversations": [
      {
        "from": "human",
        "value": "a : ℝ\nb : ℝ\nc : EReal\nha : ¬a = 0\nha' : ¬a < 0\n⊢ ↑a * (↑b - c) = ↑a * ↑b - ↑a * c\n:::"
      },
      {
        "from": "gpt",
        "value": "have ha : a > 0 := by\n    simp at ha'\n    exact lt_of_le_of_ne ha' fun a_1 ↦ ha <| id (Eq.symm a_1)"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "a : ℝ\nha : a ≥ 0\nha1 : ¬a = 0\n⊢ a • ⊤ ≠ ⊥\n:::"
      },
      {
        "from": "gpt",
        "value": "refine bot_lt_iff_ne_bot.mp ?_"
      }
    ],
    "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\nx : α × EReal\nhx : x.1 ∈ s ∧ x.2 = ⊤\n⊢ x ∉ botline s\n:::"
      },
      {
        "from": "gpt",
        "value": "simp [botline]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "a : ℝ\nha : a ≥ 0\nb : EReal\nhb : b < ⊤\n⊢ ↑a * b < ⊤\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n  by_cases ha1 : a = 0\n  · rw [ha1]; simp\n  rwa [mul_lt_top_iff_lt_top (positivity)]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "A : real_liminf\nB : real_liminf\ninequ : ∀ (n : ℕ), A.x n ≥ B.x n\nk : ℕ\n⊢ real_liminf.α k ≥ real_liminf.α k\n:::"
      },
      {
        "from": "gpt",
        "value": "have h := α_inequ' (A:=A) (B:=B)"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : TopologicalSpace E\ninst✝2 : AddCommMonoid E\n⊢ ProperFunction univ ⊤ → LowerSemicontinuousHull univ ⊤ = ⊤\n:::"
      },
      {
        "from": "gpt",
        "value": "intro 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 → EReal\nm : ℝ\nhm : m ≥ 0\nhf : LowerSemicontinuous f\n⊢ LowerSemicontinuous (m • f)\n:::"
      },
      {
        "from": "gpt",
        "value": "refine lowerSemicontinuous_iff_le_liminf.mpr ?_"
      }
    ],
    "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\nhp : p.domain = ∅\n⊢ sup_value ≤ ⊤\n:::"
      },
      {
        "from": "gpt",
        "value": "simp only [le_top]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "β : Type u_4\nf : β → ℝ\ng : β → ℝ\nhg : ∀ (n : β), 0 ≤ g n\nhgf : ∀ (n : β), g n ≤ f n\nhf : Summable f\n⊢ ∀ (n : β), 0 ≤ f n\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✝ : TopologicalSpace E\ns : Set E\nf : E → EReal\nx : E\nhx : x ∈ s\nhsf : ¬ProperFunction s f\nhf : ∀ x ∈ s, f x ≠ ⊥\n⊢ f x = ⊤\n:::"
      },
      {
        "from": "gpt",
        "value": "apply top_of_ne_bot_of_ne_proper hsf hf hx"
      }
    ],
    "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⊢ IsClosed {p | p.1 ∈ s ∧ f p.1 ≤ p.2} → IsClosed s ∧ IsClosed {p | p.1 ∈ s ∧ f p.1 ≤ ↑p.2}\n:::"
      },
      {
        "from": "gpt",
        "value": "intro h"
      }
    ],
    "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\n⊢ ‖σ x‖ = ∑ i, ‖((finBasis ℝ α).repr x) i‖ * ‖(finBasis ℝ α) i‖\n:::"
      },
      {
        "from": "gpt",
        "value": "PiLp.norm_eq_of_nat 1 (norm_num)"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "𝕜 : Type u_1\nV : Type u_2\ninst✝ : Ring 𝕜\ninst✝2 : AddCommGroup V\ninst✝3 : Module 𝕜 V\ninst✝4 : TopologicalSpace V\ns : Set V\n⊢ intrinsicInterior 𝕜 s ⊆ intrinsicInterior 𝕜 (intrinsicClosure 𝕜 s)\n:::"
      },
      {
        "from": "gpt",
        "value": "simp [intrinsicInterior]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "d : EReal\nhd1 : d ≠ ⊤\nhd2 : d ≠ ⊥\na : ℝ\nha1 : ↑a ≠ ⊤\nha2 : ↑a ≠ ⊥\nb : ℝ\nhb1 : ↑b ≠ ⊤\nhb2 : ↑b ≠ ⊥\nc : ℝ\nhc1 : ↑c ≠ ⊤\nhc2 : ↑c ≠ ⊥\nh : ↑a ≥ ↑b + ↑c - d\n⊢ d ≥ ↑b + ↑c - ↑a\n:::"
      },
      {
        "from": "gpt",
        "value": "lift d to ℝ using ⟨hd1, hd2⟩"
      }
    ],
    "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\nt : Set EReal\n⊢ (∃ t_1, IsOpen t_1 ∧ Subtype.val ⁻¹' t_1 = Subtype.val ⁻¹' (f ⁻¹' t)) ↔ ∃ u, IsOpen u ∧ f ⁻¹' t ∩ s = u ∩ s\n:::"
      },
      {
        "from": "gpt",
        "value": "simp only [Subtype.preimage_coe_eq_preimage_coe_iff]"
      }
    ],
    "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\n⊢ ∀ (x y : EuclideanSpace ℝ (Fin ↑n)), ‖p.f' x - p.f' y‖ ≤ ↑p.L * ‖x - y‖\n:::"
      },
      {
        "from": "gpt",
        "value": "intro x"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "n : ℕ\nτ : Finset ℕ\nσ : Finset ℕ\ntrivial : ¬(τ ∪ σ).Nonempty\n⊢ ∀ (a b : ℕ → EuclideanSpace ℝ (Fin n)),\n  ∃ μ c, {z | ∃ lam mu, (∀ (i : { x // x ∈ σ }), 0 ≤ mu i) ∧ z = ∑ i, lam i • a ↑i + ∑ i, mu i • b ↑i} = cone μ c\n:::"
      },
      {
        "from": "gpt",
        "value": "· exact general_polyhedra_is_polyhedra_empty τ σ 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\nf : E → ℝ\nx : E\nc : ℝ\nhc : 0 < c\ng : E\ngg : E\nhgg : gg ∈ SubderivAt f x\neq : (fun x => c • x) gg = g\ny : E\n⊢ c * f x + ⟪c • gg, y - x⟫_ℝ = c * f x + ⟪g, y - x⟫_ℝ\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [← eq]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "𝕜 : Type u_1\nV : Type u_2\ninst✝ : Ring 𝕜\ninst✝2 : AddCommGroup V\ninst✝3 : Module 𝕜 V\ninst✝4 : TopologicalSpace V\ninst✝5 : ContinuousSub V\ninst✝6 : ContinuousAdd V\ns : Set V\nhs : s.Nonempty\n⊢ s ⊆ range (affSpanCoerce_pre' 𝕜 hs)\n:::"
      },
      {
        "from": "gpt",
        "value": "exact sub_range 𝕜 hs"
      }
    ],
    "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\n⊢ ContinuousAt φ y\n:::"
      },
      {
        "from": "gpt",
        "value": "have cont_φ := ContDiffOn.continuousOn h₃"
      }
    ],
    "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\ninst✝2 : NormedSpace ℝ E\nhf : Convex ℝ (Epi f s)\nx : E\nhx : x ∈ dom s f\ny : E\nhy : y ∈ dom s f\na : ℝ\nb : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\ng1 : ℝ\ng2 : ℝ\n⊢ a • x + b • y ∈ dom s f\n:::"
      },
      {
        "from": "gpt",
        "value": "have"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "case.1:\na : ℝ\nb : ℝ\nc : ENNReal\nhgeq : a ≥ b\napos : a > 0\nhc : c ≠ 0\nhctop : c = ⊤\n⊢ ENNReal.ofReal a * c ≥ ENNReal.ofReal b * c\n\ncase.2:\na : ℝ\nb : ℝ\nc : ENNReal\nhgeq : a ≥ b\napos : a > 0\nhc : c ≠ 0\nhctop : ¬c = ⊤\n⊢ ENNReal.ofReal a * c ≥ ENNReal.ofReal b * c\n:::"
      },
      {
        "from": "gpt",
        "value": "· rw [hctop]\n    have ha : (ENNReal.ofReal a) * ⊤ = ⊤ := by\n      refine ENNReal.mul_top ?h\n      simpa\n    rw [ha]\n    simp"
      }
    ],
    "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 ≠ 0\n:::"
      },
      {
        "from": "gpt",
        "value": "linarith"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\ninst✝ : SeminormedAddCommGroup E\nf : E → ℝ\nx : E\ns : Set E\nhc : ContinuousOn f (interior s)\nhx : x ∈ interior s\nt✝ : ℝ\na✝ : t✝ > f x\nh : IsOpen {p | p.1 ∈ interior s ∧ f p.1 < p.2}\nh' : {p | p.1 ∈ interior s ∧ f p.1 < p.2} ⊆ {p | p.1 ∈ s ∧ f p.1 ≤ p.2}\n⊢ f (x, t✝).1 < (x, t✝).2\n:::"
      },
      {
        "from": "gpt",
        "value": "dsimp"
      }
    ],
    "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\nhfun : ConvexOn ℝ univ f\nu : E\ng : E → ℝ\nhg : ConvexOn ℝ univ g\nhcg : ContinuousOn g univ\nz : E\n⊢ HasSubgradientAt (f + g) z u ↔ HasSubgradientAt (g + f) z u\n:::"
      },
      {
        "from": "gpt",
        "value": "constructor"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "X : Type u_1\ninst✝ : SeminormedAddCommGroup X\ninst✝2 : NormedSpace ℝ X\nf : X → ℝ\ninst✝3 : T0Space X\ns : Set X\nhs : ∀ x ∈ s, ∃ ε > 0, ball x ε ⊆ s\nf_upperbounded : BddAbove (f '' s)\nf_convex : ConvexOn ℝ s f\nx : X\nhx : x ∈ s\nr : ℝ\nhr : r > 0 ∧ ball x r ⊆ s\nthis : ball x (r / 2) ∈ 𝓝[s] x\n⊢ ball x (r - r / 2) ∈ 𝓝[s] x\n:::"
      },
      {
        "from": "gpt",
        "value": "have eq_r : r - r / 2 = r / 2 := linarith"
      }
    ],
    "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\nhx : x ∈ s\na : ℝ\nhatop : ¬↑a = ⊤\nhabot : ¬↑a = ⊥\nb : ℝ\nhbtop : ¬↑b = ⊤\nhab : ↑a ≤ ↑b\nxn : ℕ → E × EReal\nhxn : ∀ (n : ℕ), xn n ∈ {p | p.1 ∈ s ∧ f p.1 ≤ p.2}\nhlim : Tendsto xn atTop (𝓝 (x, ↑a))\n⊢ (x, ↑b) ∈ closure {p | p.1 ∈ s ∧ f p.1 ≤ p.2}\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [mem_closure_iff_seq_limit]"
      }
    ],
    "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\nh : ∀ x ∈ s, HasGradientAt f (f' x) x\nh₁ : Convex ℝ s\nh₂ : ∀ x ∈ s, ∀ y ∈ s, f x + ⟪f' x, y - x⟫_ℝ ≤ f y\nx : E\nh : x ∈ s → HasGradientAt f (f' x) x\n⊢ x ∈ s → HasFDerivAt f (?f' x) x\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [hasGradientAt_iff_hasFDerivAt] at 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\nf : E → ℝ\nx : E\nhconv : ConvexOn ℝ univ f\nv : E\n⊢ v ∈ f_subdifferential f x →\n  ∃ u,\n    Tendsto u atTop (𝓝 x) ∧\n      Tendsto (fun n => f (u n)) atTop (𝓝 (f x)) ∧\n        ∃ v_1, ∀ (n : ℕ), v_1 n ∈ f_subdifferential f (u n) ∧ Tendsto v_1 atTop (𝓝 v)\n:::"
      },
      {
        "from": "gpt",
        "value": "intro h"
      }
    ],
    "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⁻¹ * ⟪x - u, y - u⟫_ℝ ≤ f y\n⊢ (t • f) y ≥ (t • f) u + ⟪x - u, y - u⟫_ℝ\n:::"
      },
      {
        "from": "gpt",
        "value": "field_simp at cond"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "E : Type u_1\nxm : E\nf : E → ℝ\ng : ℕ → E\nmono : ∀ (k : ℕ), f (g (k + 1)) ≤ f (g k)\nj : ℕ\na✝ : f (g (j + 1)) - f xm ≤ (∑ k ∈ Finset.range (j + 1), (f (g (k + 1)) - f xm)) / (↑j + 1)\n⊢ (∑ k ∈ Finset.range (j.succ + 1), f (g (k + 1))) / (↑j.succ + 1) - f xm =\n  (∑ k ∈ Finset.range (j.succ + 1), f (g (k + 1)) - f xm * (↑j.succ + 1)) / (↑j.succ + 1)\n:::"
      },
      {
        "from": "gpt",
        "value": "field_simp"
      }
    ],
    "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\nhx : x ∈ s\na : ℝ\nhatop : ¬↑a = ⊤\nhabot : ¬↑a = ⊥\nb : ℝ\nhbtop : ¬↑b = ⊤\nhab : ↑a ≤ ↑b\nxn : ℕ → E × EReal\nhxn : ∀ (n : ℕ), xn n ∈ {p | p.1 ∈ s ∧ f p.1 ≤ p.2}\nhlim : Tendsto xn atTop (𝓝 (x, ↑a))\nxn' : ℕ → E × EReal\nhable : b - a ≥ 0\n⊢ (∀ (n : ℕ), xn' n ∈ {p | p.1 ∈ s ∧ f p.1 ≤ p.2}) ∧ Tendsto xn' atTop (𝓝 (x, ↑b))\n:::"
      },
      {
        "from": "gpt",
        "value": "constructor"
      }
    ],
    "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\nhfun : ConvexOn ℝ s f\nh : ∀ x ∈ s, HasGradientAt f (f' x) x\ng : E → StrongDual ℝ E\nh' : ∀ x ∈ s, HasFDerivAt f (g x) x\nequiv : ∀ (x y : E), ⟪f' x - f' y, x - y⟫_ℝ = (g x - g y) (x - y)\n⊢ ∀ x ∈ s, ∀ y ∈ s, ⟪f' x - f' y, x - y⟫_ℝ ≥ 0\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\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⊢ f (a • y + b • z) ≤ a * f y + b * f z\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [← smul_eq_mul, ← smul_eq_mul]"
      }
    ],
    "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\nhxold : x ∈ _root_.closure {(x, y) | x ∈ s ∧ f x ≤ y}\nw : ℕ → E × EReal\nhw : ∀ (n : ℕ), (w n).1 ∈ s ∧ f (w n).1 ≤ (w n).2\nh : Tendsto w atTop (𝓝 x)\nhx : x.1 ∈ _root_.closure s → ¬x.2 = ⊥\n⊢ x ∈ Prod.map id Real.toEReal '' _root_.closure (Epi f s) ∪ topline (_root_.closure s)\n:::"
      },
      {
        "from": "gpt",
        "value": "have"
      }
    ],
    "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\nxn : ℕ → E\nun : ℕ → E\nx : E\nu : E\nhx : ∀ (n : ℕ), (xn n, un n) ∈ subdifferential_Graph f\nhconv : Tendsto (fun n => (xn n, un n)) atTop (𝓝 (x, u))\nhf : Tendsto (fun n => f (xn n)) atTop (𝓝 (f x))\nn : ℕ\nthis : match (xn n, un n) with\n| (x, u) => (x, f x, u) ∈ closure (limsubdifferential_Graph f)\n⊢ (xn n, f (xn n), un n) ∈ closure (limsubdifferential_Graph f)\n:::"
      },
      {
        "from": "gpt",
        "value": "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\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\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\n⊢ False\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [IsLocalMinOn, IsMinFilter, eventually_iff_exists_mem] at hl"
      }
    ],
    "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 : ∀ (x₁ : E), HasGradientAt f (f' x₁) x₁\ngradcon : Continuous f'\nz : E\nx : E\n⊢ x ∈\n    {v₀ |\n      ∃ u,\n        Tendsto u atTop (𝓝 z) ∧\n          Tendsto (fun n => f (u n)) atTop (𝓝 (f z)) ∧\n            ∃ v, ∀ (n : ℕ), v n ∈ f_subdifferential f (u n) ∧ Tendsto v atTop (𝓝 v₀)} ↔\n  x ∈ {f' z}\n:::"
      },
      {
        "from": "gpt",
        "value": "constructor"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "case.1:\nα : 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\n⊢ Convex ℝ univ\n\ncase.2:\nα : 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\n⊢ ∀ ⦃x : α⦄,\n  x ∈ univ → ∀ ⦃y : α⦄, y ∈ univ → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y\n:::"
      },
      {
        "from": "gpt",
        "value": "· apply convex_univ"
      }
    ],
    "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\nhxold : x ∈ _root_.closure {(x, y) | x ∈ s ∧ f x ≤ y}\nw : ℕ → E × EReal\nhw : ∀ (n : ℕ), (w n).1 ∈ s ∧ f (w n).1 ≤ (w n).2\nh : Tendsto w atTop (𝓝 x)\nhx : x.1 ∈ _root_.closure s → ¬x.2 = ⊥\nhx1 : x.1 ∈ _root_.closure s\nhx2 : x.2 = ⊤\n⊢ x ∈ Prod.map id Real.toEReal '' _root_.closure (Epi f s) ∪ topline (_root_.closure s)\n:::"
      },
      {
        "from": "gpt",
        "value": "right\n    simpa [topline] using ⟨hx1, hx2⟩"
      }
    ],
    "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\nhabot : a = ⊥\nxn : ℕ → E × EReal\nhxn : ∀ (n : ℕ), (xn n).1 ∈ s ∧ f (xn n).1 ≤ (xn n).2\nhlim : Tendsto (fun n => (xn n).1) atTop (𝓝 x) ∧ Tendsto (fun n => (xn n).2) atTop (𝓝 a)\nb : ℝ\nhab : a ≤ ↑b\nhab : a < ↑b\nN : ℕ\nxn' : ℕ → E × EReal\nthis : ∀ (x : ℝ), ∃ a, ∀ (b : ℕ), a ≤ b → (xn b).2 < ↑x\nhbtop : True\nhn : ∃ a, ∀ (b_1 : ℕ), a ≤ b_1 → (xn b_1).2 < ↑b\nhN : ∀ (b_1 : ℕ), N ≤ b_1 → (xn b_1).2 < ↑b\nn : ℕ\n⊢ (xn (n + N)).1 ∈ s\n:::"
      },
      {
        "from": "gpt",
        "value": "apply (hxn (n + N)).1"
      }
    ],
    "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‖\nb : EReal\nh' : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝[≠] x, differential_fun x f u y > -ε\nhnb : ¬b = ⊥\nheq : ↑b.toReal = b\nhb : b.toReal < 0\nthis : b < b / ↑2\n⊢ b < b / 2\n:::"
      },
      {
        "from": "gpt",
        "value": "exact this"
      }
    ],
    "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\nscon : Convex ℝ S\ncn : ∃ lam mu, (∀ (i : { x // x ∈ σ }), 0 ≤ mu i) ∧ c = ∑ x, lam x • a ↑x + ∑ x, mu x • b ↑x\n⊢ False\n:::"
      },
      {
        "from": "gpt",
        "value": "rcases cn with ⟨lam, mu, ⟨cn1, cn2⟩⟩"
      }
    ],
    "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\nminpoint : ∀ (i : Fin ↑n), xm i = sign (x i) * max (|x i| - t * μ) 0\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\ni : Fin ↑n\n⊢ ((μ⁻¹ * t⁻¹) • (x - xm)) i * (y - xm) i = μ⁻¹ * t⁻¹ * (x i - xm i) * (y i - xm i)\n:::"
      },
      {
        "from": "gpt",
        "value": "simp [mul_comm, mul_left_comm]"
      }
    ],
    "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\nh1 : ∀ r ∈ Icc 0 1, HasDerivAt g (g' r) r\ne1 : f x✝ = g 0\ne2 : f y = g 1\ne3 : ⟪f' x✝, y - x✝⟫_ℝ = g' 0\nmono' : ∀ t ∈ Ioo 0 1, g' t ≥ g' 0\nx : ℝ\nhx : x ∈ Ioo 0 1\n⊢ x ∈ Icc 0 1\n:::"
      },
      {
        "from": "gpt",
        "value": "simp at hx"
      }
    ],
    "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\n⊢ posTangentConeAt FeasSet x = linearized_feasible_directions x\n:::"
      },
      {
        "from": "gpt",
        "value": "apply Set.Subset.antisymm"
      }
    ],
    "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 : 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 ≠ ⊥\nx : E\nhfx : f x = ⊤\n⊢ f x = ⊤\n\ncase.2:\nE : 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 ≠ ⊥\nx : E\nhfx : f x = ⊥\n⊢ f x = ⊤\n:::"
      },
      {
        "from": "gpt",
        "value": "· exact hfx"
      }
    ],
    "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\nhs : _root_.IsClosed s\ntfae_1_iff_2 : LowerSemicontinuousOn f s ↔ _root_.IsClosed {p | p.1 ∈ s ∧ f p.1 ≤ p.2}\n⊢ _root_.IsClosed {p | p.1 ∈ s ∧ f p.1 ≤ p.2} → _root_.IsClosed {p | p.1 ∈ s ∧ f p.1 ≤ ↑p.2}\n\ncase.2:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ns : Set E\nf : E → EReal\nhs : _root_.IsClosed s\ntfae_1_iff_2 : LowerSemicontinuousOn f s ↔ _root_.IsClosed {p | p.1 ∈ s ∧ f p.1 ≤ p.2}\n⊢ _root_.IsClosed {p | p.1 ∈ s ∧ f p.1 ≤ ↑p.2} → _root_.IsClosed {p | p.1 ∈ s ∧ f p.1 ≤ p.2}\n:::"
      },
      {
        "from": "gpt",
        "value": "· exact fun a ↦ Real_epi_closed_of_EReal_epi_closed a"
      }
    ],
    "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\ns : Set E\nf : E → ℝ\nm : ℝ\nf' : E → E\ninst✝3 : CompleteSpace E\nhsc : StrongConvexOn s m f\nhf : ∀ x ∈ s, HasGradientAt f (f' x) x\nx : E\nxs : x ∈ s\ny : E\nys : y ∈ s\ncvx : ConvexOn ℝ s fun x => f x - m / 2 * ‖x‖ ^ 2\ng : E → E\ngrd : HasGradientAt (fun x => f x - m / 2 * ‖x‖ ^ 2) (g x) x\nthis : g x = f' x - m • x\nthis : m = m / 2 * 2\nfoc : f x + inner ℝ (f' x) (y - x) + m / 2 * ‖y - x‖ ^ 2 ≤ f y\nthis : x - y = -(y - x)\n⊢ 0 ≤ f y - (f x + inner ℝ (f' x) (y - x) + m / 2 * ‖y - x‖ ^ 2)\n:::"
      },
      {
        "from": "gpt",
        "value": "linarith"
      }
    ],
    "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\nxn : ℕ → E × EReal\nx : E\ny : EReal\nhxy : ∀ (n : ℕ), (xn n).1 ∈ s ∧ f (xn n).1 ≤ (xn n).2\nhy : True\nh : ∀ ⦃x : ℕ → E × ℝ⦄ (a : E) (b : ℝ),\n  (∀ (n : ℕ), (x n).1 ∈ s ∧ f (x n).1 ≤ ↑(x n).2) → Tendsto x atTop (𝓝 (a, b)) → a ∈ s ∧ f a ≤ ↑b\nhlim : Tendsto (fun n => (xn n).1) atTop (𝓝 x) ∧ Tendsto (fun n => (xn n).2) atTop (𝓝 ⊥)\nhfx : f x ≠ ⊥\nhxnin : ∀ (n : ℕ), (xn n).1 ∈ s\n⊢ ∀ᶠ (x_1 : ℕ) in atTop, (xn x_1).2 ≤ ↑((f x).toReal - 1) → f (xn x_1).1 ≤ ↑(f x).toReal - 1\n:::"
      },
      {
        "from": "gpt",
        "value": "apply Filter.Eventually.of_forall"
      }
    ],
    "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\nn : ℕ+\nh : ADMM.τ E₁ E₂ F ≥ 0\nh₁ : √(ADMM.τ E₁ E₂ F) ^ 2 = ADMM.τ E₁ E₂ F\nh₂ : (ADMM.τ E₁ E₂ F)⁻¹ = (√(ADMM.τ E₁ E₂ F))⁻¹ ^ 2\nS1 : F\nS2 : F\ns1 : ℝ\nh1 : s1 ^ 2 * ‖S1‖ ^ 2 = ‖s1 • S1‖ ^ 2\nh2 : s1⁻¹ ^ 2 * ‖S2‖ ^ 2 = ‖s1⁻¹ • S2‖ ^ 2\n⊢ s1 = √(ADMM.τ E₁ E₂ F)\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\nhsr : ∀ b ∈ S₂, c ≤ f b\na : E\nb : ℝ\nhin : (0, 0) ∈ S₂\nhc0 : c ≤ 0\nhin1 : ∀ t > 0, (0, t) ∈ S₁\nt : ℝ\nht : t > 0\nhin1 : (0, t) ∈ S₁\nhsl : f (0, t) < c\nhab : f (0, t) = ⟪a, (0, t).1⟫_ℝ + b * (0, t).2\n⊢ b * t < c\n:::"
      },
      {
        "from": "gpt",
        "value": "simp at hab"
      }
    ],
    "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\ng : E → ℝ\nf : E → E →L[ℝ] ℝ\nf' : E → E →L[ℝ] E →L[ℝ] ℝ\ns : Set E\nhs : Convex ℝ s\nhg : ∀ x ∈ s, HasFDerivAt g (f x) x\nhf : ∀ x ∈ s, HasFDerivAt f (f' x) x\nx : E\ny : E\nxs : x ∈ interior s\nys : y ∈ interior s\nhxy : x ≠ y\nh : ℝ → E\nhx : x ∈ s\nhy : y ∈ s\nh_deriv : ∀ (t : ℝ), HasDerivAt h (y - x) t\nt₀ : ℝ\nht₀ : t₀ ∈ Set.Icc 0 1\ng1 : E → ℝ\nhg : HasFDerivAt (fun z => (f z) (y - x)) ((f' (x + t₀ • (y - x))) (y - x)) (x + t₀ • (y - x))\nthis : g1 ∘ h = fun t => (f (x + t • (y - x))) (y - x)\n⊢ HasDerivAt h (y - x) t₀\n:::"
      },
      {
        "from": "gpt",
        "value": "exact h_deriv t₀"
      }
    ],
    "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\ny' : E\nhy' : f y' < f x + inner ℝ g (y' - x)\nconvset : Convex ℝ univ\nxin : x ∈ univ\nyin' : y' ∈ univ\nconvfun : ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y') ≤ a • f x + b • f y'\npos : 1 / 2 * (f x + inner ℝ g (y' - x) - f y') / ‖y' - x‖ > 0\nT : Set E\ntin : T ⊆ {x_1 | inner ℝ g (x_1 - x) ≤ f x_1 - f x + 2⁻¹ * (f x + inner ℝ g (y' - x) - f y') / ‖y' - x‖ * ‖x_1 - x‖}\nxint : x ∈ T\nS : Set E\nST : Set E\nδ : ℝ\nposδ : δ > 0\nhδ : Metric.ball x δ ⊆ T\nδ' : ℝ\nδ'' : ℝ\n⊢ 0 < δ' / ‖y' - x‖\n:::"
      },
      {
        "from": "gpt",
        "value": "apply div_pos"
      }
    ],
    "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\nx : E\nxf : x ∈ FeasSet\ndiffable : ∀ i ∈ τ, DifferentiableAt ℝ (p.equality_constraints i) x\ndiffable₂ : ∀ i ∈ σ, DifferentiableAt ℝ (p.inequality_constraints i) x\nv : E\nhv : v ∈ posTangentConeAt FeasSet x\nj : ℕ\nhj : j ∈ σ\njs : j ∈ σ\nineq : p.inequality_constraints j x = 0\ni : ℕ\nis : i ∈ σ\ninezero : p.inequality_constraints i x = 0\n⊢ ∀ y ∈ FeasSet, p.inequality_constraints i x ≤ p.inequality_constraints i y\n:::"
      },
      {
        "from": "gpt",
        "value": "intro y yf\n          rw [FeasSet] at yf xf\n          rw [inezero]\n          apply yf.2.2 i is"
      }
    ],
    "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\nf : E → ℝ\ng : F → ℝ\nx : E\ny : F\nu : E\nv : F\nhf : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖\nhg : ∀ ε > 0, ∀ᶠ (y_1 : F) in 𝓝 y, g y_1 - g y - inner ℝ v (y_1 - y) ≥ -ε * ‖y_1 - y‖\nε : ℝ\nεpos : ε > 0\nε2pos : 0 < ε / 2\nz : WithLp 2 (E × F)\nhfz : f z.1 - f x - inner ℝ u (z.1 - x) ≥ -(ε / 2) * ‖z.1 - x‖\nhyz : g z.2 - g y - inner ℝ v (z.2 - y) ≥ -(ε / 2) * ‖z.2 - y‖\nz' : WithLp 2 (E × F)\n⊢ inner ℝ u (z.1 - x) + inner ℝ v (z.2 - y) ≤ f z.1 + g z.2 - (f x + g y) + ε * ‖z - (x, y)‖\n:::"
      },
      {
        "from": "gpt",
        "value": "have h1 : ‖z.1 - x‖ ≤ ‖z - z'‖ := fst_norm_le_prod_L2 (z - 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\ninst✝10 : Setting E₁ E₂ F admm admm_kkt\nh1 : 0 < ADMM.τ E₁ E₂ F\nright✝ : ADMM.τ E₁ E₂ F < (1 + √5) / 2\nh0 : 1 + ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F ^ 2 = -(ADMM.τ E₁ E₂ F - 1 / 2) ^ 2 + 5 / 4\nh2 : 5 / 4 = |√5 / 2| ^ 2\n⊢ |ADMM.τ E₁ E₂ F - 2⁻¹| < |(|√5 / 2|)|\n:::"
      },
      {
        "from": "gpt",
        "value": "simp only [abs_abs]"
      }
    ],
    "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\ninequ₁ : ‖a‖ ≤ γ * ↑l * ‖z (k + 1) - z k‖\n⊢ ‖(0, alg.y (k + 1) - alg.y k)‖ = ‖(z (k + 1)).2 - (z k).2‖\n:::"
      },
      {
        "from": "gpt",
        "value": "repeat rw [z]; simp; apply norm_prod_left_zero"
      }
    ],
    "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 f h f' x0\nxm : E\nminφ : IsMinOn (f + h) univ xm\nh1 : ∀ (k : ℕ),\n  Nesterov_first.y f h f' x0 k - Nesterov_first.x f h f' x0 (k + 1) -\n      Nesterov_first.t f h f' x0 k • f' (Nesterov_first.y f h f' x0 k) ∈\n    SubderivAt (Nesterov_first.t f h f' x0 k • h) (Nesterov_first.x f h f' x0 (k + 1))\n⊢ ∀ (k : ℕ),\n  f (Nesterov_first.x f h f' x0 (k + 1)) + h (Nesterov_first.x f h f' x0 (k + 1)) - f xm - h xm ≤\n    Nesterov_first.γ f h f' x0 k ^ 2 / (2 * Nesterov_first.t f h f' x0 k) * ‖x0 - xm‖ ^ 2\n:::"
      },
      {
        "from": "gpt",
        "value": "have"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  },
  {
    "conversations": [
      {
        "from": "human",
        "value": "X : Type u_1\ninst✝ : SeminormedAddCommGroup X\ninst✝2 : NormedSpace ℝ X\nx₀ : X\nr : ℝ\nf : X → ℝ\ninst✝3 : T0Space X\nhf : Convex ℝ (ball x₀ r) ∧\n  ∀ ⦃x : X⦄,\n    x ∈ ball x₀ r →\n      ∀ ⦃y : X⦄, y ∈ ball x₀ r → ∀ ⦃a b : ℝ⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y\nε : ℝ\nhε : 0 < ε ∧ ε < r\nM : ℝ\nhM : f '' ball x₀ r ⊆ ball 0 M\nK : ℝ\nK_pos : K ≥ 0\nx : X\nhx : x ∈ ball x₀ (r - ε)\ny : X\nhy : y ∈ ball x₀ (r - ε)\nuy : X\nvx : X\nh : uy ≠ vx\nhu : uy ∈ ball x₀ (r - ε)\nhv : vx ∈ ball x₀ (r - ε)\nsub : ball x₀ (r - ε) ⊆ ball x₀ r\nvx_pos : vx ∈ ball x₀ r\nuy_pos : uy ∈ ball x₀ r\nz : X\nhnot : dist uy vx ≤ 0\nthis : dist uy vx = 0\n⊢ False\n:::"
      },
      {
        "from": "gpt",
        "value": "have : Inseparable uy vx := Metric.inseparable_iff.mpr 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\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))\n⊢ Nonempty ↑(prox_set f x) ∧ IsCompact (prox_set f x)\n:::"
      },
      {
        "from": "gpt",
        "value": "have"
      }
    ],
    "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\nthis : Tendsto (fun n => ‖(OptProblem.A₂ E₁) (e₂ n)‖) atTop (𝓝 0)\nh : C * 0 = 0\n⊢ Tendsto (fun n => C * ‖(OptProblem.A₂ E₁) (e₂ n)‖) atTop (𝓝 (C * 0))\n:::"
      },
      {
        "from": "gpt",
        "value": "apply Tendsto.const_mul C this"
      }
    ],
    "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\nscon : Convex ℝ S\ncn : c ∉ S\nf : StrongDual ℝ (EuclideanSpace ℝ (Fin n))\nu : ℝ\nsep1 : f c < u\nd : EuclideanSpace ℝ (Fin n)\nfeq : ∀ (x : EuclideanSpace ℝ (Fin n)), f x = ⟪d, x⟫_ℝ\nuleq : u < 0\nhc : ⟪c, d⟫_ℝ < 0\nhb : ∀ (i : { x // x ∈ σ }), ⟪b ↑i, d⟫_ℝ ≥ 0\ni : { x // x ∈ τ }\nthis : ∀ (t : ℝ), t • a ↑i ∈ S\nt : ℝ\na✝ : t < 0\nsep2 : u < t * ⟪a ↑i, d⟫_ℝ\n⊢ t * ⟪a ↑i, d⟫_ℝ > u\n:::"
      },
      {
        "from": "gpt",
        "value": "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 : NormedSpace ℝ E\nf : E → ℝ\nf' : E → E →L[ℝ] ℝ\ns : Set E\nh₁ : Convex ℝ s\nh₂ : ∀ x ∈ s, ∀ y ∈ s, f x + (f' x) (y - x) ≤ f y\nx : E\nxs : x ∈ s\ny : E\nys : y ∈ s\na : ℝ\nb : ℝ\nanonneg : 0 ≤ a\nbnonneg : 0 ≤ b\nsumab : a + b = 1\nx' : E\nx'rfl : x' = a • x + b • y\nx'rfl_comm : x' = b • y + a • x\nsumba : b + a = 1\nh : x' ∈ s → HasFDerivAt f (f' x') x'\nx1s : x' ∈ s\nH1 : f x' + (f' x') (x - x') ≤ f x\nH2 : f x' + (f' x') (y - x') ≤ f y\nlnmp : ∀ (c : ℝ), (f' x') (c • (y - x)) = c * (f' x') (y - x)\nl1 : a • (f x' + (f' x') (x - x')) + b • (f x' + (f' x') (y - x')) =\n  (a + b) • f x' + a • (f' x') (x - x') + b • (f' x') (y - x')\n⊢ b * (a * (f' x') (y - x)) = b * a * (f' x') (y - x)\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [mul_assoc]"
      }
    ],
    "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\nbd : Bornology.IsBounded (z '' univ)\nε : ℝ\nepos : 0 < ε\nh : ∀ (x : ℕ),\n  ∃ x_1,\n    x ≤ x_1 ∧\n      (-ε < (EMetric.infEdist (z x_1) (limit_set z)).toReal → ε ≤ (EMetric.infEdist (z x_1) (limit_set z)).toReal)\nW : ℕ → ℕ\nmonoW : StrictMono W\nhcs : IsSeqCompact (closure (z ∘ W '' univ))\neven : ∀ (n : ℕ), (z ∘ W) n ∈ closure (z ∘ W '' univ)\n⊢ ∃ z_ ∈ closure (z ∘ W '' univ), ∃ α, StrictMono α ∧ Tendsto (fun n => (z ∘ W) (α n)) atTop (𝓝 z_)\n:::"
      },
      {
        "from": "gpt",
        "value": "apply hcs.subseq_of_frequently_in (Filter.Frequently.of_forall even)"
      }
    ],
    "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))\n⊢ map (fun n => z (α n)) atTop = map ((fun n => z n) ∘ α) atTop\n:::"
      },
      {
        "from": "gpt",
        "value": "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\nn : ℕ+\nA₂_x_diff : F\nr_n : F\nr_n_1 : F\nυ_diff : F\ny_diff : F\nx_diff : E₂\nA₂T : F →L[ℝ] E₂\n⊢ (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n_1⟫_ℝ + ⟪A₂_x_diff, y_diff⟫_ℝ =\n  (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n⟫_ℝ -\n        (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n⟫_ℝ +\n      (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ⟪A₂_x_diff, r_n_1⟫_ℝ +\n    ⟪A₂_x_diff, y_diff⟫_ℝ\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n            rw [sub_self ((1 - τ) * ρ * (inner ℝ A₂_x_diff r_n)), zero_add]"
      }
    ],
    "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\nn : ℕ\n⊢ 0 < ↑n + 1\n:::"
      },
      {
        "from": "gpt",
        "value": "linarith"
      }
    ],
    "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+\nh₄'' : ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =\n  ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖\nh₄' : ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ =\n  ‖(OptProblem.A₂ E₁) (e₂ ↑n)‖\n⊢ |ADMM.τ E₁ E₂ F| * |ADMM.ρ E₁ E₂ F| = ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F\n:::"
      },
      {
        "from": "gpt",
        "value": "have h1: τ ≥ 0 := rcases admm.htau with ⟨h₁, _⟩; apply le_of_lt 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\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 : ∀ (b : ℝ), ∀ᶠ (a : ℕ) in atTop, b ≤ c a\nvt2 : ∀ s ∈ 𝓝 v, ∃ a, ∀ b ≥ a, c b • d b ∈ s\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\n⊢ 0 < 1\n:::"
      },
      {
        "from": "gpt",
        "value": "norm_num"
      }
    ],
    "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  (fun x =>\n      Augmented_Lagrangian_Function E₁ E₂ F admm.toOptProblem (ADMM.ρ E₁ E₂ F)\n        (x, ADMM.x₂ E₁ F n.natPred, ADMM.y E₁ E₂ n.natPred)) =\n    fun x =>\n    OptProblem.f₁ E₂ F x + OptProblem.f₂ E₁ F (ADMM.x₂ E₁ F n.natPred) +\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      ADMM.ρ E₁ E₂ F / 2 * ‖(OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂‖ ^ 2\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n   intro n\n   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\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\nreduction : ∀ (k : ℕ),\n  ‖Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1) - xm‖ ^ 2 ≤\n    (1 -\n        Gradient_Descent_fix_stepsize.a f f' x₀ *\n          (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))) *\n      ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2\nthis : 0 < m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)\n⊢ (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) ^ 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) =\n  m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [pow_two]; 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\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\nx₁ : ℝ\nhx₁ : |x₁| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)\n⊢ 0 < (ε / (2 * ↑(subgradient_method.G f x₀) ^ 2) - |x₁|) / 2\n:::"
      },
      {
        "from": "gpt",
        "value": "apply div_pos\n            · simp[hx₁]\n            · 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\nvn : ℕ → E\nv : E\nvnin : ∀ (n : ℕ), vn n ∈ subdifferential f x\nvntov : Tendsto vn atTop (𝓝 v)\nxk : ℕ → E\nuk : ℕ → E\nu : E\nn : ℕ\nnpos : 0 < n\nxkt : Tendsto xk atTop (𝓝 x)\nfxkt : Tendsto (fun k => f (xk k)) atTop (𝓝 (f x))\nukt : Tendsto uk atTop (𝓝 u)\nak : ℕ → E\na : E\nakt : ∀ (U : Set E), a ∈ U → IsOpen U → ∃ N, ∀ (n : ℕ), N ≤ n → ak n ∈ U\n⊢ IsOpen (Metric.ball a (1 / (2 * ↑n)))\n\ncase.2:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nvn : ℕ → E\nv : E\nvnin : ∀ (n : ℕ), vn n ∈ subdifferential f x\nvntov : Tendsto vn atTop (𝓝 v)\nxk : ℕ → E\nuk : ℕ → E\nu : E\nn : ℕ\nnpos : 0 < n\nxkt : Tendsto xk atTop (𝓝 x)\nfxkt : Tendsto (fun k => f (xk k)) atTop (𝓝 (f x))\nukt : Tendsto uk atTop (𝓝 u)\nak : ℕ → E\na : E\nakt : ∃ N, ∀ (n_1 : ℕ), N ≤ n_1 → ak n_1 ∈ Metric.ball a (1 / (2 * ↑n))\n⊢ ∃ N, ∀ (k : ℕ), N ≤ k → ‖ak k - a‖ < 1 / (2 * ↑n)\n:::"
      },
      {
        "from": "gpt",
        "value": "exact Metric.isOpen_ball"
      }
    ],
    "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\nzf : E\nzg : E\nfgadd : z = zf + zg\nuf : ℕ → E\nufx : Tendsto uf atTop (𝓝 x)\nuffx : Tendsto (fun n => f (uf n)) atTop (𝓝 (f x))\nvf : ℕ → E\nvflim : ∀ (n : ℕ), vf n ∈ f_subdifferential f (uf n) ∧ Tendsto vf atTop (𝓝 zf)\nug : ℕ → E\nugx : Tendsto ug atTop (𝓝 x)\nuggx : Tendsto (fun n => g (ug n)) atTop (𝓝 (g x))\nvg : ℕ → E\nvglim : ∀ (n : ℕ), vg n ∈ f_subdifferential g (ug n) ∧ Tendsto vg atTop (𝓝 zg)\nn : ℕ\nlimgrad : ∀ (An : ℕ → E) (x : E), Tendsto An atTop (𝓝 x) → Tendsto (fun n => g' (An n)) atTop (𝓝 (g' x))\nvgin : vg n ∈ {g' (ug n)}\nvgtends : Tendsto (fun x => g' (ug x)) atTop (𝓝 zg)\nfuntrans : ∀ (n : ℕ), vg n = g' (ug n)\n⊢ Tendsto (fun x => g' (uf x)) atTop (𝓝 zg)\n:::"
      },
      {
        "from": "gpt",
        "value": "have"
      }
    ],
    "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\nx₁ : ℝ\nhx₁ : |x₁| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)\nx₂ : ℝ\nhx₂ : dist x₂ x₁ < (ε / (2 * ↑(subgradient_method.G f x₀) ^ 2) - |x₁|) / 2\n⊢ (ε / (2 * ↑(subgradient_method.G f x₀) ^ 2) + |x₁|) / 2 <\n  (ε / (2 * ↑(subgradient_method.G f x₀) ^ 2) + ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)) / 2\n:::"
      },
      {
        "from": "gpt",
        "value": "apply (mul_lt_mul_iff_of_pos_left zero_lt_two).mp\n                rw [mul_div_cancel₀, mul_div_cancel₀]\n                simp [hx₁]; simp; 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\nh₁ : ConcaveOn ℝ (Ico 0 η_min) φ_sum\nh₃ : φ_sum 0 = 0\n⊢ ContDiffOn ℝ 1 φ_sum (Ioo 0 η_min)\n:::"
      },
      {
        "from": "gpt",
        "value": "have : φ_sum = (fun c => ∑ x ∈ ht2.toFinset, φ x c) := ext c; simp [φ_sum]"
      }
    ],
    "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✝)\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\nj : ℕ\nhj : j ∈ σ\nhj1 : j ∈ active_set x\nhj' : j ∈ active_set x ∩ σ\na : EuclideanSpace ℝ (Fin n✝)\nc : ℝ\nhab : p.inequality_constraints j = fun y => ⟪a, y⟫_ℝ + c\nhg : gradient (p.inequality_constraints j) x = a\nxf : (x ∈ p.domain ∧ ∀ i ∈ τ, p.equality_constraints i x = 0) ∧ x ∈ p.domain ∧ ∀ j ∈ σ, 0 ≤ p.inequality_constraints j x\n⊢ 0 ≤ p.inequality_constraints j (x + z n)\n:::"
      },
      {
        "from": "gpt",
        "value": "have"
      }
    ],
    "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\nh1 : (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 =>\n  ADMM.ρ E₁ E₂ F / 2 * ‖(OptProblem.A₁ E₂) x + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)‖ ^ 2\nh2 : (fun x => ADMM.ρ E₁ E₂ F / 2 * ‖(OptProblem.A₁ E₂) x + c‖ ^ 2) = fun x =>\n  ADMM.ρ E₁ E₂ F / 2 * (⟪(OptProblem.A₁ E₂) x, (OptProblem.A₁ E₂) x⟫_ℝ + 2 * ⟪c, (OptProblem.A₁ E₂) x⟫_ℝ + ⟪c, c⟫_ℝ)\n⊢ HasGradientAt (fun x => ADMM.ρ E₁ E₂ F / 2 * ‖(OptProblem.A₁ E₂) x + c‖ ^ 2)\n  (ADMM.ρ E₁ E₂ F • (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + c))\n  (ADMM.x₁ E₂ F ↑n)\n:::"
      },
      {
        "from": "gpt",
        "value": "rw[h2]"
      }
    ],
    "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 hc1"
      }
    ],
    "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\nε : ℝ\nepos : ε > 0\nn1 : ℕ\nleft✝ : True\nieq1 : ∀ (x : ℕ), n1 ≤ x → ‖z (α x) - z_‖ < ε / 2\nb : ℕ\nn1leb : n1 ≤ b\n⊢ ‖z (α b) - z_‖ < ε / 2\n:::"
      },
      {
        "from": "gpt",
        "value": "exact ieq1 b n1leb"
      }
    ],
    "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+\n⊢ ‖Φ ↑k‖ < r_Φ\n:::"
      },
      {
        "from": "gpt",
        "value": "have"
      }
    ],
    "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✝)\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\nj : ℕ\nhj : j ∈ σ\nhj' : j ∈ active_set x ∩ σ\na : EuclideanSpace ℝ (Fin n✝)\nc : ℝ\nhab : p.inequality_constraints j = fun y => ⟪a, y⟫_ℝ + c\nhg : gradient (p.inequality_constraints j) x = a\nthis : ⟪a, z n⟫_ℝ ≥ 0\nleft✝ : x ∈ p.domain\nright✝ : ∀ i ∈ τ, p.equality_constraints i x = 0\nleft✝2 : x ∈ p.domain\nh2 : ∀ j ∈ σ, 0 ≤ p.inequality_constraints j x\nthis : j ∉ τ\nhj1 : j ∈ σ ∧ p.inequality_constraints j x = 0\n⊢ (fun y => ⟪a, y⟫_ℝ + c) x ≤ (fun y => ⟪a, y⟫_ℝ + c) (x + z n)\n:::"
      },
      {
        "from": "gpt",
        "value": "simp only [add_le_add_iff_right, ge_iff_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\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\n⊢ h (Nesterov_second.x f h f' x0 ↑k) ≤\n  (1 - Nesterov_second.γ f h f' x0 ↑k) * h (Nesterov_second.x f h f' x0 (↑k - 1)) +\n    Nesterov_second.γ f h f' x0 ↑k * h (Nesterov_second.y f h f' x0 ↑k)\n:::"
      },
      {
        "from": "gpt",
        "value": "have"
      }
    ],
    "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\ndiflte1 : ∀ ε > 0, ∀ᶠ (q : ℕ) in atTop, ‖alg.x (α q) - alg.x (α q - 1)‖ < ε\nε : ℝ\nepos : ε > 0\nn1 : ℕ\nleft✝ : True\nieq1 : ∀ (x : ℕ), n1 ≤ x → ‖z (α x) - z_‖ < ε / 2\nb : ℕ\nn1leb : n1 ≤ b\n⊢ ‖z_ - z (α b)‖ < ε / 2\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [norm_sub_rev]; exact ieq1 b n1leb"
      }
    ],
    "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⊢ 0 ≤ ‖f x - f x' + inner ℝ grad (x' - x)‖\n:::"
      },
      {
        "from": "gpt",
        "value": "apply norm_nonneg"
      }
    ],
    "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\nvn : ℕ → E\nv : E\nvnin : ∀ (n : ℕ), vn n ∈ subdifferential f x\nvntov : Tendsto vn atTop (𝓝 v)\nxk : ℕ → E\nuk : ℕ → E\nu : E\nn : ℕ\nnpos : 0 < n\nxkt : Tendsto xk atTop (𝓝 x)\nfxkt : Tendsto (fun k => f (xk k)) atTop (𝓝 (f x))\nukt : Tendsto uk atTop (𝓝 u)\nak : ℕ → E\na : E\nakt : ∀ (U : Set E), a ∈ U → IsOpen U → ∃ N, ∀ (n : ℕ), N ≤ n → ak n ∈ U\n⊢ n ≠ 0\n\ncase.2:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nvn : ℕ → E\nv : E\nvnin : ∀ (n : ℕ), vn n ∈ subdifferential f x\nvntov : Tendsto vn atTop (𝓝 v)\nxk : ℕ → E\nuk : ℕ → E\nu : E\nn : ℕ\nnpos : 0 < n\nxkt : Tendsto xk atTop (𝓝 x)\nfxkt : Tendsto (fun k => f (xk k)) atTop (𝓝 (f x))\nukt : Tendsto uk atTop (𝓝 u)\nak : ℕ → E\na : E\nakt : ∀ (U : Set E), a ∈ U → IsOpen U → ∃ N, ∀ (n : ℕ), N ≤ n → ak n ∈ U\n⊢ IsOpen (Metric.ball a (1 / (2 * ↑n)))\n\ncase.3:\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ninst✝2 : InnerProductSpace ℝ E\nf : E → ℝ\nx : E\nvn : ℕ → E\nv : E\nvnin : ∀ (n : ℕ), vn n ∈ subdifferential f x\nvntov : Tendsto vn atTop (𝓝 v)\nxk : ℕ → E\nuk : ℕ → E\nu : E\nn : ℕ\nnpos : 0 < n\nxkt : Tendsto xk atTop (𝓝 x)\nfxkt : Tendsto (fun k => f (xk k)) atTop (𝓝 (f x))\nukt : Tendsto uk atTop (𝓝 u)\nak : ℕ → E\na : E\nakt : ∃ N, ∀ (n_1 : ℕ), N ≤ n_1 → ak n_1 ∈ Metric.ball a (1 / (2 * ↑n))\n⊢ ∃ N, ∀ (k : ℕ), N ≤ k → ‖ak k - a‖ < 1 / (2 * ↑n)\n:::"
      },
      {
        "from": "gpt",
        "value": "apply Nat.ne_zero_iff_zero_lt.mpr npos"
      }
    ],
    "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⊢ ‖grad‖ ≤ ‖grad‖ + 1\n:::"
      },
      {
        "from": "gpt",
        "value": "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\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\nh₂ : (2 * ∑ i ∈ Finset.range (k + 1), subgradient_method.a f x₀ i) *\n    (sInf {x | ∃ i < k + 1, f (subgradient_method.x f x₀ i) = x} - f xm) ≤\n  ‖x₀ - xm‖ ^ 2 - ‖subgradient_method.x f x₀ (k + 1) - xm‖ ^ 2 + (↑k + 1) * s ^ 2\nhG : ↑(subgradient_method.G f x₀) > 0\ninq₁ : ∑ x ∈ Finset.range (k + 1), subgradient_method.a f x₀ x ≥ (↑k + 1) * (s / ↑(subgradient_method.G f x₀))\n⊢ 0 < ↑k + 1\n:::"
      },
      {
        "from": "gpt",
        "value": "apply add_pos_of_nonneg_of_pos (Nat.cast_nonneg k) zero_lt_one"
      }
    ],
    "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\nreduction : ∀ (k : ℕ),\n  ‖Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1) - xm‖ ^ 2 ≤\n    (1 -\n        Gradient_Descent_fix_stepsize.a f f' x₀ *\n          (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))) *\n      ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2\neq : 0 ≤\n  1 -\n    Gradient_Descent_fix_stepsize.a f f' x₀ *\n      (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))\nq : ℕ\nIH1 : ‖Gradient_Descent_fix_stepsize.x f f' x₀ q - xm‖ ^ 2 ≤\n  (1 -\n        Gradient_Descent_fix_stepsize.a f f' x₀ *\n          (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))) ^\n      q *\n    ‖x₀ - xm‖ ^ 2\n⊢ ‖Gradient_Descent_fix_stepsize.x f f' x₀ (q + 1) - xm‖ ^ 2 ≤\n  (1 -\n      Gradient_Descent_fix_stepsize.a f f' x₀ *\n        (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))) *\n    ‖Gradient_Descent_fix_stepsize.x f f' x₀ q - xm‖ ^ 2\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n      apply reduction"
      }
    ],
    "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)))\nm1 : ℕ\nie1 : ∀ (b : ℕ),\n  m1 ≤ b →\n    |inner ℝ (alg.x (α b) - alg.x (α b - 1)) ((1 / (γ * ↑l)) • grad_fst H (alg.y (α b - 1)) (alg.x (α b - 1)))| ≤\n      ε / (γ * ↑l) / 3\nm2 : ℕ\nie2 : ∀ (b : ℕ),\n  m2 ≤ b →\n    |inner ℝ (z_.1 - alg.x (α b - 1)) ((1 / (γ * ↑l)) • grad_fst H (alg.y (α b - 1)) (alg.x (α b - 1)))| ≤\n      ε / (γ * ↑l) / 3\nm3 : ℕ\nie3 : ∀ (b : ℕ), m3 ≤ b → ‖z_.1 - alg.x (α b - 1)‖ ^ 2 / 2 < ε / (γ * ↑l) / 3\nq : ℕ\nmleq : 1 + max (max m1 m2) m3 ≤ q\n⊢ f (z (α q)).1 < f z_.1 + ε\n:::"
      },
      {
        "from": "gpt",
        "value": "have m1le : m1 ≤ 1 + max (max m1 m2) m3:= by\n      linarith [(le_max_left m1 m2).trans (le_max_left _ m3)]"
      }
    ],
    "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)))\nh1 : ∀ᶠ (q : ℕ) in atTop,\n  |inner ℝ ((fun q => alg.x (α q) - alg.x (α q - 1)) q)\n        (alg.c (α q - 1) • grad_fst H (alg.y (α q - 1)) (alg.x (α q - 1)))| ≤\n    ε / (γ * ↑l) / 3\nh2 : ∀ᶠ (q : ℕ) in atTop,\n  |inner ℝ ((fun q => z_.1 - alg.x (α q - 1)) q) (alg.c (α q - 1) • grad_fst H (alg.y (α q - 1)) (alg.x (α q - 1)))| ≤\n    ε / (γ * ↑l) / 3\nx : ℕ\nassx : ‖z_.1 - alg.x (α x - 1)‖ < √(2 * (ε / (γ * ↑l) / 3))\nthis : ‖z_.1 - alg.x (α x - 1)‖ ^ 2 < 2 * (ε / (γ * ↑l) / 3)\n⊢ 2 * (ε / (γ * ↑l) / 3) / 2 = ε / (γ * ↑l) / 3\n:::"
      },
      {
        "from": "gpt",
        "value": "by\n          apply mul_div_cancel_left₀\n          linarith"
      }
    ],
    "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 : 1 < ADMM.τ E₁ E₂ F\nn : ℕ+\nx_diff : E₂\nr_n : F\nr_n_1 : F\ne_sum : F\ne_sum_1 : F\na₁ : ℝ\na₂ : ℝ\nb₁ : ℝ\nb₂ : ℝ\nc₁ : ℝ\nc₂ : ℝ\nd₁ : ℝ\nd₂ : ℝ\nM_inequ : 2 * M (n + 1) ≤\n  (ADMM.τ E₁ E₂ F ^ 2 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x_diff‖ ^ 2 +\n    (1 - 1 / ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖r_n‖ ^ 2\nH : 1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) * ‖ey ↑n‖ ^ 2 - 1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) * ‖ey (↑n + 1)‖ ^ 2 -\n              (2 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖e_sum_1‖ ^ 2 +\n            (ADMM.τ E₁ E₂ F ^ 2 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x_diff‖ ^ 2 +\n          (1 - 1 / ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖e_sum‖ ^ 2 -\n        ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x_diff‖ ^ 2 -\n      ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ ^ 2 +\n    ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) (e₂ ↑n)‖ ^ 2 ≥\n  0\nrw_rn_1 : r_n_1 = e_sum_1\nrw_rn : r_n = e_sum\n⊢ (1 - 1 / ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖e_sum_1‖ ^ 2 +\n    (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖e_sum_1‖ ^ 2 =\n  (1 - 1 / ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖e_sum_1‖ ^ 2 +\n    (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F * ‖r_n_1‖ ^ 2\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [rw_rn_1]"
      }
    ],
    "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_\nL1 : ∀ η > 0, ∃ l1, ∀ k ≥ l1, ψ z_ < ψ (z k) ∧ ψ (z k) < ψ z_ + η\nL2 : ∀ ε > 0, ∃ l2, ∀ k > l2, (EMetric.infEdist (z k) (limit_set z)).toReal < ε\nwklpt : ∀ z0 ∈ limit_set z, KL_point ψ z0\ncons : is_constant_on ψ (limit_set z)\nε : ℝ\nleft✝2 : ε ∈ Ioi 0\nη : ℝ\nleft✝3 : η ∈ Ioi 0\nφ : ℝ → ℝ\nhφ : φ ∈ desingularizing_function η\nLL : ℕ\nieq : ∀ n > LL,\n  (ψ z_ < ψ (z n) ∧ ψ (z n) < ψ z_ + η) ∧\n    ENNReal.ofReal (deriv φ (ψ (z n) - ψ z_)) * EMetric.infEdist 0 (subdifferential ψ (z n)) ≥ 1\na : ℕ → ℝ\nb : ℕ → ℝ\nc : ℕ → ℝ\nd : ℕ → ℝ\nsum : ℕ → ℝ\nhlin : ∀ (n : ℕ), ψ (z (n + LL + 1)) - ψ z_ ∈ Ioo 0 η\nρ : ℝ\nρpos : ρ > 0\nhsgub : ∀ (k : ℕ), ∃ dΨ ∈ f_subdifferential ψ (z (k + 1)), ‖dΨ‖ ≤ ρ * ‖z (k + 1) - z k‖\nC : ℝ\nhnnegC : 0 ≤ C\nhposa : ∀ (n : ℕ), 0 < a n\nn : ℕ\nhpc : d n * b n ≤ a n - a (n + 1)\n⊢ 2 * c (n + 1) ≤ c n + C * (a n - a (n + 1))\n:::"
      },
      {
        "from": "gpt",
        "value": "have"
      }
    ],
    "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)\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 }\nRzgrad : HasStrictFDerivAt Rz Mx x\nRxeq0 : Rz x = 0\nMxinj : Function.Injective ⇑Mx\nMxsurj : Function.Surjective ⇑Mx\nhvh1 : ∀ i ∈ τ, ⟪gradient (p.equality_constraints i) x, v⟫_ℝ = 0\nhvh2 : ∀ j ∈ σ ∩ active_set x, ⟪gradient (p.inequality_constraints j) x, v⟫_ℝ ≥ 0\nN : ℕ\nd : ℕ → EuclideanSpace ℝ (Fin n)\nderiv : Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖Rz x' - Rz x - Mx (x' - x)‖) (𝓝 x) (𝓝 0)\nderiv : Tendsto ((fun x' => ‖x' - x‖⁻¹ * ‖Rz x' - Rz x - Mx (x' - x)‖) ∘ d) atTop (𝓝 0)\ndtend : ∀ (ε : ℝ), 0 < ε → ∃ a, ∀ (b : ℕ), a ≤ b → ‖d b - x‖ < ε\nε : ℝ\nεpos : ε > 0\ninactive : ∀ i ∈ σ \\ active_set x, ∀ z ∈ Metric.ball x ε, 0 < p.inequality_constraints i z\nN' : ℕ\ndtendx : ∀ (b : ℕ), N' ≤ b → ‖d b - x‖ < ε\nnn : ℕ\nhnn : max N N' ≤ nn\nhv1 : c (d nn) = (↑nn)⁻¹ • Ax *ᵥ v\nhv2 : Zᵀ *ᵥ (d nn - x) = (↑nn)⁻¹ • (toEuclideanLin Zᵀ) v\ni : { x_1 // x_1 ∈ active_set x }\n⊢ ((fun i => gradc x i) *ᵥ v) i = gradc x i ⬝ᵥ v ∨ nn = 0\n:::"
      },
      {
        "from": "gpt",
        "value": "left"
      }
    ],
    "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)\nh3 : ∀ (k : ℕ+), Φ ↑k - Φ (↑k + 1) < 2 * r_Φ\nh4 : ∀ (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    2 * r_Φ\nh5 : ∀ (k : ℕ+),\n  ‖(OptProblem.A₁ E₂) (e₁ (↑k + 1)) + (OptProblem.A₂ E₁) (e₂ (↑k + 1))‖ <\n    √(2 * r_Φ / (min 1 (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F))\nh5' : ∀ (n : ℕ),\n  1 < n →\n    ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ <\n      √(2 * r_Φ / (min 1 (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F))\nh_n0 : ‖(OptProblem.A₁ E₂) (e₁ 0) + (OptProblem.A₂ E₁) (e₂ 0)‖ < ‖(OptProblem.A₁ E₂) (e₁ 0) + (OptProblem.A₂ E₁) (e₂ 0)‖ + 1\nh_n1 : ‖(OptProblem.A₁ E₂) (e₁ 1) + (OptProblem.A₂ E₁) (e₂ 1)‖ < ‖(OptProblem.A₁ E₂) (e₁ 1) + (OptProblem.A₂ E₁) (e₂ 1)‖ + 1\nr : ℝ\nhr' : r =\n  max\n    (max (√(2 * r_Φ / (min 1 (1 + 1 / ADMM.τ E₁ E₂ F - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F)))\n      (‖(OptProblem.A₁ E₂) (e₁ 0) + (OptProblem.A₂ E₁) (e₂ 0)‖ + 1))\n    (‖(OptProblem.A₁ E₂) (e₁ 1) + (OptProblem.A₂ E₁) (e₂ 1)‖ + 1)\nn : ℕ\nh_n : ∀ (n : ℕ), ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ < r\nh_n' : ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ < r\n⊢ dist ((OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)) 0 < r\n:::"
      },
      {
        "from": "gpt",
        "value": "rw[← sub_zero (A₁ (e₁ n) + A₂ (e₂ n))] at h_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\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\nfieq2 : ∀ (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 * ‖Gt x‖ ^ 2\nfieq3 : ∀ (x z : E), f x + inner ℝ (f' x) (z - x) ≤ f z\nhieq1 : ∀ (x z : E),\n  h (x - proximal_gradient_method.t f h f' x₀ • Gt x) +\n      inner ℝ (Gt x - f' x) (z - x + proximal_gradient_method.t f h f' x₀ • Gt x) ≤\n    h z\nhieq2 : ∀ (x z : E),\n  h (x - proximal_gradient_method.t f h f' x₀ • Gt x) ≤\n    h z - inner ℝ (Gt x - f' x) (z - x + proximal_gradient_method.t f h f' x₀ • Gt x)\nunivieq : ∀ (x z : E),\n  φ (x - proximal_gradient_method.t f h f' x₀ • Gt x) ≤\n    φ z + inner ℝ (Gt x) (x - z) - proximal_gradient_method.t f h f' x₀ / 2 * ‖Gt x‖ ^ 2\niter : ∀ (i : ℕ),\n  proximal_gradient_method.x f h f' x₀ (i + 1) =\n    proximal_gradient_method.x f h f' x₀ i -\n      proximal_gradient_method.t f h f' x₀ • Gt (proximal_gradient_method.x f h f' x₀ i)\ni : ℕ\nφieq1 : φ\n      (proximal_gradient_method.x f h f' x₀ i -\n        proximal_gradient_method.t f h f' x₀ • Gt (proximal_gradient_method.x f h f' x₀ i)) -\n    φ (proximal_gradient_method.xm f h f' x₀) ≤\n  1 / (2 * proximal_gradient_method.t f h f' x₀) *\n    (‖proximal_gradient_method.x f h f' x₀ i - proximal_gradient_method.xm f h f' x₀‖ ^ 2 -\n      ‖proximal_gradient_method.x f h f' x₀ i -\n              proximal_gradient_method.t f h f' x₀ • Gt (proximal_gradient_method.x f h f' x₀ i) -\n            proximal_gradient_method.xm f h f' x₀‖ ^\n        2)\n⊢ φ (proximal_gradient_method.x f h f' x₀ i) +\n      inner ℝ (Gt (proximal_gradient_method.x f h f' x₀ i))\n        (proximal_gradient_method.x f h f' x₀ i - proximal_gradient_method.x f h f' x₀ i) -\n    proximal_gradient_method.t f h f' x₀ / 2 * ‖Gt (proximal_gradient_method.x f h f' x₀ i)‖ ^ 2 ≤\n  φ (proximal_gradient_method.x f h f' x₀ i)\n:::"
      },
      {
        "from": "gpt",
        "value": "simp; apply mul_nonneg; linarith [alg.tpos]; apply sq_nonneg"
      }
    ],
    "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 f h f' x0\nxm : E\nminφ : IsMinOn (f + h) univ xm\nh1 : ∀ (k : ℕ),\n  Nesterov_first.y f h f' x0 k - Nesterov_first.x f h f' x0 (k + 1) -\n      Nesterov_first.t f h f' x0 k • f' (Nesterov_first.y f h f' x0 k) ∈\n    SubderivAt (Nesterov_first.t f h f' x0 k • h) (Nesterov_first.x f h f' x0 (k + 1))\nhieq1 : ∀ (z : E) (k : ℕ),\n  Nesterov_first.t f h f' x0 k * h (Nesterov_first.x f h f' x0 (k + 1)) +\n      inner ℝ\n        (Nesterov_first.y f h f' x0 k - Nesterov_first.x f h f' x0 (k + 1) -\n          Nesterov_first.t f h f' x0 k • f' (Nesterov_first.y f h f' x0 k))\n        (z - Nesterov_first.x f h f' x0 (k + 1)) ≤\n    Nesterov_first.t f h f' x0 k * h z\nhieq2 : ∀ (z : E) (k : ℕ),\n  h (Nesterov_first.x f h f' x0 (k + 1)) ≤\n    h z +\n      inner ℝ\n        (f' (Nesterov_first.y f h f' x0 k) +\n          (1 / Nesterov_first.t f h f' x0 k) • (Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k))\n        (z - Nesterov_first.x f h f' x0 (k + 1))\nfieq1 : ∀ (k : ℕ) (x y : E), f x ≤ f y + inner ℝ (f' y) (x - y) + 1 / (2 * Nesterov_first.t f h f' x0 k) * ‖x - y‖ ^ 2\nφ : E → ℝ\nφieq2 : ∀ (z : E) (k : ℕ),\n  φ (Nesterov_first.x f h f' x0 (k + 1)) ≤\n    φ z +\n        1 / Nesterov_first.t f h f' x0 k *\n          inner ℝ (Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k)\n            (z - Nesterov_first.x f h f' x0 (k + 1)) +\n      1 / (2 * Nesterov_first.t f h f' x0 k) * ‖Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k‖ ^ 2\nk : ℕ\nieq1 : φ (Nesterov_first.x f h f' x0 (k + 1)) - φ (Nesterov_first.x f h f' x0 k) ≤\n  (1 / Nesterov_first.t f h f' x0 k) •\n      inner ℝ (Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k)\n        (Nesterov_first.x f h f' x0 k - Nesterov_first.x f h f' x0 (k + 1)) +\n    1 / (2 * Nesterov_first.t f h f' x0 k) * ‖Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k‖ ^ 2\nieq2 : φ (Nesterov_first.x f h f' x0 (k + 1)) - φ xm ≤\n  (1 / Nesterov_first.t f h f' x0 k) •\n      inner ℝ (Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k)\n        (xm - Nesterov_first.x f h f' x0 (k + 1)) +\n    1 / (2 * Nesterov_first.t f h f' x0 k) * ‖Nesterov_first.x f h f' x0 (k + 1) - Nesterov_first.y f h f' x0 k‖ ^ 2\n⊢ φ (Nesterov_first.x f h f' x0 (k + 1)) - (1 * φ xm - (1 - Nesterov_first.γ f h f' x0 k) * φ xm) -\n    (1 - Nesterov_first.γ f h f' x0 k) * φ (Nesterov_first.x f h f' x0 k) =\n  φ (Nesterov_first.x f h f' x0 (k + 1)) -\n    (Nesterov_first.γ f h f' x0 k * φ xm + (1 - Nesterov_first.γ f h f' x0 k) * φ (Nesterov_first.x f h f' x0 k))\n:::"
      },
      {
        "from": "gpt",
        "value": "rw [← sub_mul, sub_sub_cancel, sub_sub]"
      }
    ],
    "system": "You are a helpful Lean assistant that outputs tactic needed."
  }
]