zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_99.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} {s : ℝ}
  (hm : IsMinOn f Set.univ xm) (ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s)
  (hs : s > 0) (k : ℕ)
  (heq :
    (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
      {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x})
  (hnek : Nonempty { x // x ∈ Finset.range (k + 1) })
  (h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)) ⦃i : ℕ⦄ (left : i ≥ 0) (hi₂ : i ≤ k)
  (this :
    ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
          2 * subgradient_method.a f x₀ i *
            (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
        ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2 =
      ‖subgradient_method.x f x₀ i - xm‖ ^ 2 +
          -(2 * subgradient_method.a f x₀ i *
              (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) +
        ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2)
  (hxm : f xm ≥ f (subgradient_method.x f x₀ i) + ⟪subgradient_method.g f x₀ i, xm - subgradient_method.x f x₀ i⟫_ℝ) :
  k < k + 1 := sorry

Tactic state before:
case h.left
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
s : ℝ
hm : IsMinOn f Set.univ xm
ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s
hs : s > 0
k : ℕ
heq :
  (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
    {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}
hnek : Nonempty { x // x ∈ Finset.range (k + 1) }
h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)
i : ℕ
left✝ : i ≥ 0
hi₂ : i ≤ k
this :
  ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
        2 * subgradient_method.a f x₀ i *
          (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
      ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2 =
    ‖subgradient_method.x f x₀ i - xm‖ ^ 2 +
        -(2 * subgradient_method.a f x₀ i *
            (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) +
      ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2
hxm : f xm ≥ f (subgradient_method.x f x₀ i) + ⟪subgradient_method.g f x₀ i, xm - subgradient_method.x f x₀ i⟫_ℝ
⊢ k < k + 1

Tactic:
apply (Nat.lt_succ_self k)

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_100.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} {s : ℝ}
  (hm : IsMinOn f Set.univ xm) (ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s)
  (hs : s > 0) (k : ℕ)
  (heq :
    (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
      {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x})
  (hnek : Nonempty { x // x ∈ Finset.range (k + 1) })
  (h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)) ⦃i : ℕ⦄ (left : i ≥ 0) (hi₂ : i ≤ k)
  (this :
    ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
          2 * subgradient_method.a f x₀ i *
            (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
        ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2 =
      ‖subgradient_method.x f x₀ i - xm‖ ^ 2 +
          -(2 * subgradient_method.a f x₀ i *
              (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) +
        ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2)
  (hxm : f xm ≥ f (subgradient_method.x f x₀ i) + ⟪subgradient_method.g f x₀ i, xm - subgradient_method.x f x₀ i⟫_ℝ) :
  f (subgradient_method.x f x₀ i) = f (subgradient_method.x f x₀ i) := sorry

Tactic state before:
case h.right
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
s : ℝ
hm : IsMinOn f Set.univ xm
ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s
hs : s > 0
k : ℕ
heq :
  (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
    {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}
hnek : Nonempty { x // x ∈ Finset.range (k + 1) }
h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)
i : ℕ
left✝ : i ≥ 0
hi₂ : i ≤ k
this :
  ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
        2 * subgradient_method.a f x₀ i *
          (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
      ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2 =
    ‖subgradient_method.x f x₀ i - xm‖ ^ 2 +
        -(2 * subgradient_method.a f x₀ i *
            (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) +
      ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2
hxm : f xm ≥ f (subgradient_method.x f x₀ i) + ⟪subgradient_method.g f x₀ i, xm - subgradient_method.x f x₀ i⟫_ℝ
⊢ f (subgradient_method.x f x₀ i) = f (subgradient_method.x f x₀ i)

Tactic:
simp

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_102.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} {s : ℝ}
  (hm : IsMinOn f Set.univ xm) (ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s)
  (hs : s > 0) (k : ℕ)
  (heq :
    (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
      {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x})
  (hnek : Nonempty { x // x ∈ Finset.range (k + 1) })
  (h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)) ⦃i : ℕ⦄ (left : i ≥ 0) (hi₂ : i ≤ k)
  (this_1 :
    ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
          2 * subgradient_method.a f x₀ i *
            (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
        ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2 =
      ‖subgradient_method.x f x₀ i - xm‖ ^ 2 +
          -(2 * subgradient_method.a f x₀ i *
              (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) +
        ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2)
  (hxm : f xm ≥ f (subgradient_method.x f x₀ i) + ⟪subgradient_method.g f x₀ i, xm - subgradient_method.x f x₀ i⟫_ℝ)
  (this : f (subgradient_method.x f x₀ i) ∈ Set.range fun x => f (subgradient_method.x f x₀ ↑x)) :
  BddBelow (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
s : ℝ
hm : IsMinOn f Set.univ xm
ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s
hs : s > 0
k : ℕ
heq :
  (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
    {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}
hnek : Nonempty { x // x ∈ Finset.range (k + 1) }
h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)
i : ℕ
left✝ : i ≥ 0
hi₂ : i ≤ k
this✝ :
  ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
        2 * subgradient_method.a f x₀ i *
          (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
      ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2 =
    ‖subgradient_method.x f x₀ i - xm‖ ^ 2 +
        -(2 * subgradient_method.a f x₀ i *
            (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) +
      ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2
hxm : f xm ≥ f (subgradient_method.x f x₀ i) + ⟪subgradient_method.g f x₀ i, xm - subgradient_method.x f x₀ i⟫_ℝ
this : f (subgradient_method.x f x₀ i) ∈ Set.range fun x => f (subgradient_method.x f x₀ ↑x)
⊢ BddBelow (Set.range fun x => f (subgradient_method.x f x₀ ↑x))

Tactic:
apply Finite.bddBelow_range

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_105.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} {s : ℝ}
  (hm : IsMinOn f Set.univ xm) (ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s)
  (hs : s > 0) (k : ℕ)
  (heq :
    (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
      {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x})
  (hnek : Nonempty { x // x ∈ Finset.range (k + 1) })
  (h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)) ⦃i : ℕ⦄ (left : i ≥ 0) (hi₂ : i ≤ k)
  (this :
    ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
          2 * subgradient_method.a f x₀ i *
            (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
        ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2 =
      ‖subgradient_method.x f x₀ i - xm‖ ^ 2 +
          -(2 * subgradient_method.a f x₀ i *
              (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) +
        ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2)
  (hxm : f xm ≥ f (subgradient_method.x f x₀ i) + ⟪subgradient_method.g f x₀ i, xm - subgradient_method.x f x₀ i⟫_ℝ) :
  f (subgradient_method.x f x₀ i) + ⟪subgradient_method.g f x₀ i, xm - subgradient_method.x f x₀ i⟫_ℝ ≤ f xm := sorry

Tactic state before:
case a
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
s : ℝ
hm : IsMinOn f Set.univ xm
ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s
hs : s > 0
k : ℕ
heq :
  (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
    {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}
hnek : Nonempty { x // x ∈ Finset.range (k + 1) }
h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)
i : ℕ
left✝ : i ≥ 0
hi₂ : i ≤ k
this :
  ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
        2 * subgradient_method.a f x₀ i *
          (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
      ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2 =
    ‖subgradient_method.x f x₀ i - xm‖ ^ 2 +
        -(2 * subgradient_method.a f x₀ i *
            (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) +
      ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2
hxm : f xm ≥ f (subgradient_method.x f x₀ i) + ⟪subgradient_method.g f x₀ i, xm - subgradient_method.x f x₀ i⟫_ℝ
⊢ f (subgradient_method.x f x₀ i) + ⟪subgradient_method.g f x₀ i, xm - subgradient_method.x f x₀ i⟫_ℝ ≤ f xm

Tactic:
apply hxm

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_115.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} {s : ℝ}
  (hm : IsMinOn f Set.univ xm) (ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s)
  (hs : s > 0) (k : ℕ)
  (heq :
    (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
      {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x})
  (hnek : Nonempty { x // x ∈ Finset.range (k + 1) })
  (h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀))
  (h₁ :
    ∀ ⦃i : ℕ⦄,
      i ≥ 0 ∧ i ≤ k →
        ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 ≤
          ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
              2 * subgradient_method.a f x₀ i *
                (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
            ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2)
  (h₁' :
    ∀ ⦃i : ℕ⦄,
      i ≥ 0 ∧ i ≤ k →
        subgradient_method.a f x₀ i *
            (2 * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) ≤
          ‖subgradient_method.x f x₀ i - xm‖ ^ 2 - ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 + s ^ 2)
  (i : ℕ) (hi : i ∈ Finset.range (k + 1)) : i ≥ 0 := sorry

Tactic state before:
case h.a.left
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
s : ℝ
hm : IsMinOn f Set.univ xm
ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s
hs : s > 0
k : ℕ
heq :
  (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
    {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}
hnek : Nonempty { x // x ∈ Finset.range (k + 1) }
h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)
h₁ :
  ∀ ⦃i : ℕ⦄,
    i ≥ 0 ∧ i ≤ k →
      ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 ≤
        ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
            2 * subgradient_method.a f x₀ i *
              (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
          ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2
h₁' :
  ∀ ⦃i : ℕ⦄,
    i ≥ 0 ∧ i ≤ k →
      subgradient_method.a f x₀ i *
          (2 * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) ≤
        ‖subgradient_method.x f x₀ i - xm‖ ^ 2 - ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 + s ^ 2
i : ℕ
hi : i ∈ Finset.range (k + 1)
⊢ i ≥ 0

Tactic:
simp

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_116.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} {s : ℝ}
  (hm : IsMinOn f Set.univ xm) (ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s)
  (hs : s > 0) (k : ℕ)
  (heq :
    (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
      {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x})
  (hnek : Nonempty { x // x ∈ Finset.range (k + 1) })
  (h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀))
  (h₁ :
    ∀ ⦃i : ℕ⦄,
      i ≥ 0 ∧ i ≤ k →
        ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 ≤
          ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
              2 * subgradient_method.a f x₀ i *
                (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
            ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2)
  (h₁' :
    ∀ ⦃i : ℕ⦄,
      i ≥ 0 ∧ i ≤ k →
        subgradient_method.a f x₀ i *
            (2 * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) ≤
          ‖subgradient_method.x f x₀ i - xm‖ ^ 2 - ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 + s ^ 2)
  (i : ℕ) (hi : i ∈ Finset.range (k + 1)) (this : i < k + 1) : i ≤ k := sorry

Tactic state before:
case h.a.right
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
s : ℝ
hm : IsMinOn f Set.univ xm
ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s
hs : s > 0
k : ℕ
heq :
  (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
    {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}
hnek : Nonempty { x // x ∈ Finset.range (k + 1) }
h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)
h₁ :
  ∀ ⦃i : ℕ⦄,
    i ≥ 0 ∧ i ≤ k →
      ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 ≤
        ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
            2 * subgradient_method.a f x₀ i *
              (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
          ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2
h₁' :
  ∀ ⦃i : ℕ⦄,
    i ≥ 0 ∧ i ≤ k →
      subgradient_method.a f x₀ i *
          (2 * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) ≤
        ‖subgradient_method.x f x₀ i - xm‖ ^ 2 - ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 + s ^ 2
i : ℕ
hi : i ∈ Finset.range (k + 1)
this : i < k + 1
⊢ i ≤ k

Tactic:
apply (Nat.lt_add_one_iff).mp this

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_118.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} {s : ℝ}
  (hm : IsMinOn f Set.univ xm) (ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s)
  (hs : s > 0) (k : ℕ)
  (heq :
    (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
      {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x})
  (hnek : Nonempty { x // x ∈ Finset.range (k + 1) })
  (h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀))
  (h₁ :
    ∀ ⦃i : ℕ⦄,
      i ≥ 0 ∧ i ≤ k →
        ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 ≤
          ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
              2 * subgradient_method.a f x₀ i *
                (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
            ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2)
  (h₁' :
    ∀ ⦃i : ℕ⦄,
      i ≥ 0 ∧ i ≤ k →
        subgradient_method.a f x₀ i *
            (2 * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) ≤
          ‖subgradient_method.x f x₀ i - xm‖ ^ 2 - ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 + s ^ 2)
  (h₂ :
    (2 * ∑ i ∈ Finset.range (k + 1), subgradient_method.a f x₀ i) *
        (sInf {x | ∃ i < k + 1, f (subgradient_method.x f x₀ i) = x} - f xm) ≤
      ‖x₀ - xm‖ ^ 2 - ‖subgradient_method.x f x₀ (k + 1) - xm‖ ^ 2 + (↑k + 1) * s ^ 2) :
  s > 0 := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
s : ℝ
hm : IsMinOn f Set.univ xm
ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s
hs : s > 0
k : ℕ
heq :
  (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
    {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}
hnek : Nonempty { x // x ∈ Finset.range (k + 1) }
h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)
h₁ :
  ∀ ⦃i : ℕ⦄,
    i ≥ 0 ∧ i ≤ k →
      ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 ≤
        ‖subgradient_method.x f x₀ i - xm‖ ^ 2 -
            2 * subgradient_method.a f x₀ i *
              (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) +
          ‖subgradient_method.a f x₀ i‖ ^ 2 * ‖subgradient_method.g f x₀ i‖ ^ 2
h₁' :
  ∀ ⦃i : ℕ⦄,
    i ≥ 0 ∧ i ≤ k →
      subgradient_method.a f x₀ i *
          (2 * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm)) ≤
        ‖subgradient_method.x f x₀ i - xm‖ ^ 2 - ‖subgradient_method.x f x₀ (i + 1) - xm‖ ^ 2 + s ^ 2
h₂ :
  (2 * ∑ i ∈ Finset.range (k + 1), subgradient_method.a f x₀ i) *
      (sInf {x | ∃ i < k + 1, f (subgradient_method.x f x₀ i) = x} - f xm) ≤
    ‖x₀ - xm‖ ^ 2 - ‖subgradient_method.x f x₀ (k + 1) - xm‖ ^ 2 + (↑k + 1) * s ^ 2
⊢ s > 0

Tactic:
apply hs

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_149.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} (hm : IsMinOn f Set.univ xm)
  (ha' : Tendsto (subgradient_method.a f x₀) atTop (𝓝 0))
  (ha'' : Tendsto (fun k => (Finset.range (k + 1)).sum (subgradient_method.a f x₀)) atTop atTop)
  (h_original_goal :
    Tendsto (fun k => sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) atTop (𝓝 0))
  (s : Set ℝ) :
  s ∈ 𝓝 0 → ∃ a, ∀ (b : ℕ), a ≤ b → sInf {x | ∃ i < b + 1, f (subgradient_method.x f x₀ i) = x} - f xm ∈ s := sorry

Tactic state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
hm : IsMinOn f Set.univ xm
ha' : Tendsto (subgradient_method.a f x₀) atTop (𝓝 0)
ha'' : Tendsto (fun k => (Finset.range (k + 1)).sum (subgradient_method.a f x₀)) atTop atTop
⊢ Tendsto (fun k => sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) atTop (𝓝 0)

Tactic:
simp [tendsto_def]

Tactic state after:
[E : Type u_1
 inst✝¹ : NormedAddCommGroup E
 inst✝ : InnerProductSpace ℝ E
 f : E → ℝ
 xm x₀ : E
 alg : subgradient_method f x₀
 hm : IsMinOn f Set.univ xm
 ha' : Tendsto (subgradient_method.a f x₀) atTop (𝓝 0)
 ha'' : Tendsto (fun k => (Finset.range (k + 1)).sum (subgradient_method.a f x₀)) atTop atTop
 ⊢ ∀ s ∈ 𝓝 0, ∃ a, ∀ (b : ℕ), a ≤ b → sInf {x | ∃ i < b + 1, f (subgradient_method.x f x₀ i) = x} - f xm ∈ s]
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_150.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} (hm : IsMinOn f Set.univ xm)
  (ha'' :
    ∀ (s : Set ℝ) (x : ℝ),
      (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s)
  (s_1 : Set ℝ)
  (ha' :
    ∀ (s x : Set ℝ),
      x ⊆ s →
        (∀ 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)
  (t : Set ℝ) (hs₁ : t ⊆ s_1) (hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t) (hs₀ : 0 ∈ t) (ε : ℝ) (εpos : 0 < ε)
  (hs₂₀ : Metric.ball 0 ε ⊆ t) :
  let A := ‖x₀ - xm‖ ^ 2 / ε + 1;
  let s := {x | x > ‖x₀ - xm‖ ^ 2 / ε};
  ∀ (b : ℝ), A ≤ b → b ∈ s → ‖x₀ - xm‖ ^ 2 / ε < b := sorry

Tactic state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
hm : IsMinOn f Set.univ xm
ha'' :
  ∀ (s : Set ℝ) (x : ℝ),
    (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s
s✝ : Set ℝ
ha' :
  ∀ (s x : Set ℝ),
    x ⊆ s →
      (∀ 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
t : Set ℝ
hs₁ : t ⊆ s✝
hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t
hs₀ : 0 ∈ t
ε : ℝ
εpos : 0 < ε
hs₂₀ : Metric.ball 0 ε ⊆ t
A : ℝ := ‖x₀ - xm‖ ^ 2 / ε + 1
s : Set ℝ := {x | x > ‖x₀ - xm‖ ^ 2 / ε}
b : ℝ
hb : A ≤ b
⊢ b ∈ s

Tactic:
simp [s]

Tactic state after:
[E : Type u_1
 inst✝¹ : NormedAddCommGroup E
 inst✝ : InnerProductSpace ℝ E
 f : E → ℝ
 xm x₀ : E
 alg : subgradient_method f x₀
 hm : IsMinOn f Set.univ xm
 ha'' :
   ∀ (s : Set ℝ) (x : ℝ),
     (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s
 s✝ : Set ℝ
 ha' :
   ∀ (s x : Set ℝ),
     x ⊆ s →
       (∀ 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
 t : Set ℝ
 hs₁ : t ⊆ s✝
 hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t
 hs₀ : 0 ∈ t
 ε : ℝ
 εpos : 0 < ε
 hs₂₀ : Metric.ball 0 ε ⊆ t
 A : ℝ := ‖x₀ - xm‖ ^ 2 / ε + 1
 s : Set ℝ := {x | x > ‖x₀ - xm‖ ^ 2 / ε}
 b : ℝ
 hb : A ≤ b
 ⊢ ‖x₀ - xm‖ ^ 2 / ε < b]