zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_201.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm admm_kkt]
  (this :
    Tendsto (fun n => (ADMM.x₁ E₂ F (φ n), ADMM.x₂ E₁ F (φ n), ADMM.y E₁ E₂ (φ n))) atTop (𝓝 x₁'' ×ˢ 𝓝 x₂'' ×ˢ 𝓝 y'')) :
  Tendsto (fun n => (ADMM.x₁ E₂ F (φ n), ADMM.x₂ E₁ F (φ n), ADMM.y E₁ E₂ (φ n))) atTop
    (𝓝 x₁'' ×ˢ 𝓝 x₂'' ×ˢ 𝓝 y'') := sorry

Tactic state before:
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm admm_kkt
this : Tendsto (fun n => (ADMM.x₁ E₂ F (φ n), ADMM.x₂ E₁ F (φ n), ADMM.y E₁ E₂ (φ n))) atTop (𝓝 x₁'' ×ˢ 𝓝 x₂'' ×ˢ 𝓝 y'')
⊢ Tendsto (fun n => (ADMM.x₁ E₂ F (φ n), ADMM.x₂ E₁ F (φ n), ADMM.y E₁ E₂ (φ n))) atTop (𝓝 x₁'' ×ˢ 𝓝 x₂'' ×ˢ 𝓝 y'')

Tactic:
exact this

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_202.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm admm_kkt] (n : ℕ)
  (this : φ (n + 1) > 0) : ↑(φ₁ n) = φ (n + 1) := sorry

Tactic state before:
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝¹⁰ : NormedAddCommGroup E₁
inst✝⁹ : InnerProductSpace ℝ E₁
inst✝⁸ : FiniteDimensional ℝ E₁
inst✝⁷ : NormedAddCommGroup E₂
inst✝⁶ : InnerProductSpace ℝ E₂
inst✝⁵ : FiniteDimensional ℝ E₂
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝¹ inst✝ : Setting E₁ E₂ F admm admm_kkt
n : ℕ
this : φ (n + 1) > 0
⊢ ↑(φ₁ n) = φ (n + 1)

Tactic:
exact Nat.sub_one_add_one_eq_of_pos this

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_203.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm admm_kkt]
  (this : (fun n => ADMM.x₁ E₂ F ↑(φ₁ n)) = fun n => ADMM.x₁ E₂ F (φ (n + 1))) :
  Tendsto (fun n => ADMM.x₁ E₂ F (φ n)) atTop (𝓝 x₁'') := sorry

Tactic state before:
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm admm_kkt
this : (fun n => ADMM.x₁ E₂ F ↑(φ₁ n)) = fun n => ADMM.x₁ E₂ F (φ (n + 1))
⊢ Tendsto (fun n => ADMM.x₁ E₂ F (φ n)) atTop (𝓝 x₁'')

Tactic:
apply x₁_subseq_converge

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_204.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm admm_kkt]
  (this : (fun n => ADMM.x₂ E₁ F ↑(φ₁ n)) = fun n => ADMM.x₂ E₁ F (φ (n + 1))) :
  Tendsto (fun n => ADMM.x₂ E₁ F (φ n)) atTop (𝓝 x₂'') := sorry

Tactic state before:
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm admm_kkt
this : (fun n => ADMM.x₂ E₁ F ↑(φ₁ n)) = fun n => ADMM.x₂ E₁ F (φ (n + 1))
⊢ Tendsto (fun n => ADMM.x₂ E₁ F (φ n)) atTop (𝓝 x₂'')

Tactic:
apply x₂_subseq_converge

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_205.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (r : ℝ)
  (hr : ∀ (n : ℕ), ‖Φ n‖ < r) (n i : ℕ) :
  i ∈ Finset.range n → Φ i.succ - Φ (i + 1).succ = Φ i.succ - Φ (i.succ + 1) := sorry

Tactic state before:
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
r : ℝ
hr : ∀ (n : ℕ), ‖Φ n‖ < r
n : ℕ
φ₀ : ℕ → ℝ := fun i => Φ i.succ
⊢ ∀ i ∈ Finset.range n, Φ i.succ - Φ (i + 1).succ = Φ i.succ - Φ (i.succ + 1)

Tactic:
simp only [Finset.mem_range, Nat.succ_eq_add_one, implies_true]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_206.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (r : ℝ)
  (hr : ∀ (n : ℕ), ‖Φ n‖ < r) (n : ℕ) :
  let φ₀ := fun i => Φ i.succ;
  (∀ i ∈ Finset.range n, φ₀ i - φ₀ (i + 1) = Φ i.succ - Φ (i.succ + 1)) →
    Finset.range n = Finset.range n → φ₀ 0 - φ₀ n ≤ r + r → Φ (Nat.succ 0) - Φ n.succ ≤ r + r := sorry

Tactic state before:
case h
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
r : ℝ
hr : ∀ (n : ℕ), ‖Φ n‖ < r
n : ℕ
φ₀ : ℕ → ℝ := fun i => Φ i.succ
this : ∀ i ∈ Finset.range n, φ₀ i - φ₀ (i + 1) = Φ i.succ - Φ (i.succ + 1)
h : Finset.range n = Finset.range n
⊢ φ₀ 0 - φ₀ n ≤ r + r

Tactic:
simp only [φ₀]

Tactic state after:
[case h
 E₁ : Type u_1
 E₂ : Type u_2
 F : Type u_3
 inst✝⁹ : NormedAddCommGroup E₁
 inst✝⁸ : InnerProductSpace ℝ E₁
 inst✝⁷ : FiniteDimensional ℝ E₁
 inst✝⁶ : NormedAddCommGroup E₂
 inst✝⁵ : InnerProductSpace ℝ E₂
 inst✝⁴ : FiniteDimensional ℝ E₂
 inst✝³ : NormedAddCommGroup F
 inst✝² : InnerProductSpace ℝ F
 inst✝¹ : FiniteDimensional ℝ F
 admm : ADMM E₁ E₂ F
 admm_kkt : Existance_of_kkt admm
 inst✝ : Setting E₁ E₂ F admm admm_kkt
 r : ℝ
 hr : ∀ (n : ℕ), ‖Φ n‖ < r
 n : ℕ
 φ₀ : ℕ → ℝ := fun i => Φ i.succ
 this : ∀ i ∈ Finset.range n, φ₀ i - φ₀ (i + 1) = Φ i.succ - Φ (i.succ + 1)
 h : Finset.range n = Finset.range n
 ⊢ Φ (Nat.succ 0) - Φ n.succ ≤ r + r]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_207.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (r : ℝ)
  (hr : ∀ (n : ℕ), ‖Φ n‖ < r) (n : ℕ) :
  let φ₀ := fun i => Φ i.succ;
  (∀ i ∈ Finset.range n, φ₀ i - φ₀ (i + 1) = Φ i.succ - Φ (i.succ + 1)) →
    Finset.range n = Finset.range n → Φ (Nat.succ 0) - Φ n.succ ≤ |Φ (Nat.succ 0) - Φ n.succ| := sorry

Tactic state before:
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
r : ℝ
hr : ∀ (n : ℕ), ‖Φ n‖ < r
n : ℕ
φ₀ : ℕ → ℝ := fun i => Φ i.succ
this : ∀ i ∈ Finset.range n, φ₀ i - φ₀ (i + 1) = Φ i.succ - Φ (i.succ + 1)
h : Finset.range n = Finset.range n
⊢ Φ (Nat.succ 0) - Φ n.succ ≤ |Φ (Nat.succ 0) - Φ n.succ|

Tactic:
apply le_abs_self

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_208.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (r : ℝ)
  (hr : ∀ (n : ℕ), ‖Φ n‖ < r) (n : ℕ) :
  let φ₀ := fun i => Φ i.succ;
  (∀ i ∈ Finset.range n, φ₀ i - φ₀ (i + 1) = Φ i.succ - Φ (i.succ + 1)) →
    Finset.range n = Finset.range n → |Φ (Nat.succ 0) - Φ n.succ| ≤ |Φ (Nat.succ 0)| + |Φ n.succ| := sorry

Tactic state before:
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
r : ℝ
hr : ∀ (n : ℕ), ‖Φ n‖ < r
n : ℕ
φ₀ : ℕ → ℝ := fun i => Φ i.succ
this : ∀ i ∈ Finset.range n, φ₀ i - φ₀ (i + 1) = Φ i.succ - Φ (i.succ + 1)
h : Finset.range n = Finset.range n
⊢ |Φ (Nat.succ 0) - Φ n.succ| ≤ |Φ (Nat.succ 0)| + |Φ n.succ|

Tactic:
apply abs_sub

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_209.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (r : ℝ)
  (hr : ∀ (n : ℕ), ‖Φ n‖ < r) (n : ℕ) :
  let φ₀ := fun i => Φ i.succ;
  (∀ i ∈ Finset.range n, φ₀ i - φ₀ (i + 1) = Φ i.succ - Φ (i.succ + 1)) →
    Finset.range n = Finset.range n →
      |Φ (Nat.succ 0)| + |Φ n.succ| ≤ r + r → |Φ (Nat.succ 0)| + |Φ n.succ| < r + r := sorry

Tactic state before:
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
r : ℝ
hr : ∀ (n : ℕ), ‖Φ n‖ < r
n : ℕ
φ₀ : ℕ → ℝ := fun i => Φ i.succ
this : ∀ i ∈ Finset.range n, φ₀ i - φ₀ (i + 1) = Φ i.succ - Φ (i.succ + 1)
h : Finset.range n = Finset.range n
⊢ |Φ (Nat.succ 0)| + |Φ n.succ| ≤ r + r

Tactic:
apply le_of_lt

Tactic state after:
[case hab
 E₁ : Type u_1
 E₂ : Type u_2
 F : Type u_3
 inst✝⁹ : NormedAddCommGroup E₁
 inst✝⁸ : InnerProductSpace ℝ E₁
 inst✝⁷ : FiniteDimensional ℝ E₁
 inst✝⁶ : NormedAddCommGroup E₂
 inst✝⁵ : InnerProductSpace ℝ E₂
 inst✝⁴ : FiniteDimensional ℝ E₂
 inst✝³ : NormedAddCommGroup F
 inst✝² : InnerProductSpace ℝ F
 inst✝¹ : FiniteDimensional ℝ F
 admm : ADMM E₁ E₂ F
 admm_kkt : Existance_of_kkt admm
 inst✝ : Setting E₁ E₂ F admm admm_kkt
 r : ℝ
 hr : ∀ (n : ℕ), ‖Φ n‖ < r
 n : ℕ
 φ₀ : ℕ → ℝ := fun i => Φ i.succ
 this : ∀ i ∈ Finset.range n, φ₀ i - φ₀ (i + 1) = Φ i.succ - Φ (i.succ + 1)
 h : Finset.range n = Finset.range n
 ⊢ |Φ (Nat.succ 0)| + |Φ n.succ| < r + r]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_210.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (r : ℝ)
  (hr : ∀ (n : ℕ), ‖Φ n‖ < r) (n : ℕ) :
  let φ₀ := fun i => Φ i.succ;
  (∀ i ∈ Finset.range n, φ₀ i - φ₀ (i + 1) = Φ i.succ - Φ (i.succ + 1)) →
    Finset.range n = Finset.range n → |Φ (Nat.succ 0)| + |Φ n.succ| < r + r := sorry

Tactic state before:
case hab
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
r : ℝ
hr : ∀ (n : ℕ), ‖Φ n‖ < r
n : ℕ
φ₀ : ℕ → ℝ := fun i => Φ i.succ
this : ∀ i ∈ Finset.range n, φ₀ i - φ₀ (i + 1) = Φ i.succ - Φ (i.succ + 1)
h : Finset.range n = Finset.range n
⊢ |Φ (Nat.succ 0)| + |Φ n.succ| < r + r

Tactic:
apply add_lt_add (hr (Nat.succ 0)) (hr (Nat.succ n))

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_211.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (c : ℝ)
  (hn : ∀ (n : ℕ), ∑ i ∈ Finset.range n, (Φ i.succ - Φ (i.succ + 1)) ≤ c)
  (h_original_goal : Summable fun n => Φ (n + 1) - Φ (n + 1 + 1)) (n : ℕ) : 0 ≤ Φ (n + 1) - Φ (n + 1 + 1) := sorry

Tactic state before:
case intro
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
c : ℝ
hn : ∀ (n : ℕ), ∑ i ∈ Finset.range n, (Φ i.succ - Φ (i.succ + 1)) ≤ c
⊢ Summable fun n => Φ (n + 1) - Φ (n + 1 + 1)

Tactic:
apply summable_of_sum_range_le (c := c)

Tactic state after:
[case intro.hf
 E₁ : Type u_1
 E₂ : Type u_2
 F : Type u_3
 inst✝⁹ : NormedAddCommGroup E₁
 inst✝⁸ : InnerProductSpace ℝ E₁
 inst✝⁷ : FiniteDimensional ℝ E₁
 inst✝⁶ : NormedAddCommGroup E₂
 inst✝⁵ : InnerProductSpace ℝ E₂
 inst✝⁴ : FiniteDimensional ℝ E₂
 inst✝³ : NormedAddCommGroup F
 inst✝² : InnerProductSpace ℝ F
 inst✝¹ : FiniteDimensional ℝ F
 admm : ADMM E₁ E₂ F
 admm_kkt : Existance_of_kkt admm
 inst✝ : Setting E₁ E₂ F admm admm_kkt
 c : ℝ
 hn : ∀ (n : ℕ), ∑ i ∈ Finset.range n, (Φ i.succ - Φ (i.succ + 1)) ≤ c
 ⊢ ∀ (n : ℕ), 0 ≤ Φ (n + 1) - Φ (n + 1 + 1),
 case intro.h
 E₁ : Type u_1
 E₂ : Type u_2
 F : Type u_3
 inst✝⁹ : NormedAddCommGroup E₁
 inst✝⁸ : InnerProductSpace ℝ E₁
 inst✝⁷ : FiniteDimensional ℝ E₁
 inst✝⁶ : NormedAddCommGroup E₂
 inst✝⁵ : InnerProductSpace ℝ E₂
 inst✝⁴ : FiniteDimensional ℝ E₂
 inst✝³ : NormedAddCommGroup F
 inst✝² : InnerProductSpace ℝ F
 inst✝¹ : FiniteDimensional ℝ F
 admm : ADMM E₁ E₂ F
 admm_kkt : Existance_of_kkt admm
 inst✝ : Setting E₁ E₂ F admm admm_kkt
 c : ℝ
 hn : ∀ (n : ℕ), ∑ i ∈ Finset.range n, (Φ i.succ - Φ (i.succ + 1)) ≤ c
 ⊢ ∀ (n : ℕ), ∑ i ∈ Finset.range n, (Φ (i + 1) - Φ (i + 1 + 1)) ≤ c]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_212.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (c : ℝ)
  (hn : ∀ (n : ℕ), ∑ i ∈ Finset.range n, (Φ i.succ - Φ (i.succ + 1)) ≤ c)
  (h_original_goal : Summable fun n => Φ (n + 1) - Φ (n + 1 + 1)) (n : ℕ) :
  ∑ i ∈ Finset.range n, (Φ (i + 1) - Φ (i + 1 + 1)) ≤ c := sorry

Tactic state before:
case intro
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
c : ℝ
hn : ∀ (n : ℕ), ∑ i ∈ Finset.range n, (Φ i.succ - Φ (i.succ + 1)) ≤ c
⊢ Summable fun n => Φ (n + 1) - Φ (n + 1 + 1)

Tactic:
apply summable_of_sum_range_le (c := c)

Tactic state after:
[case intro.hf
 E₁ : Type u_1
 E₂ : Type u_2
 F : Type u_3
 inst✝⁹ : NormedAddCommGroup E₁
 inst✝⁸ : InnerProductSpace ℝ E₁
 inst✝⁷ : FiniteDimensional ℝ E₁
 inst✝⁶ : NormedAddCommGroup E₂
 inst✝⁵ : InnerProductSpace ℝ E₂
 inst✝⁴ : FiniteDimensional ℝ E₂
 inst✝³ : NormedAddCommGroup F
 inst✝² : InnerProductSpace ℝ F
 inst✝¹ : FiniteDimensional ℝ F
 admm : ADMM E₁ E₂ F
 admm_kkt : Existance_of_kkt admm
 inst✝ : Setting E₁ E₂ F admm admm_kkt
 c : ℝ
 hn : ∀ (n : ℕ), ∑ i ∈ Finset.range n, (Φ i.succ - Φ (i.succ + 1)) ≤ c
 ⊢ ∀ (n : ℕ), 0 ≤ Φ (n + 1) - Φ (n + 1 + 1),
 case intro.h
 E₁ : Type u_1
 E₂ : Type u_2
 F : Type u_3
 inst✝⁹ : NormedAddCommGroup E₁
 inst✝⁸ : InnerProductSpace ℝ E₁
 inst✝⁷ : FiniteDimensional ℝ E₁
 inst✝⁶ : NormedAddCommGroup E₂
 inst✝⁵ : InnerProductSpace ℝ E₂
 inst✝⁴ : FiniteDimensional ℝ E₂
 inst✝³ : NormedAddCommGroup F
 inst✝² : InnerProductSpace ℝ F
 inst✝¹ : FiniteDimensional ℝ F
 admm : ADMM E₁ E₂ F
 admm_kkt : Existance_of_kkt admm
 inst✝ : Setting E₁ E₂ F admm admm_kkt
 c : ℝ
 hn : ∀ (n : ℕ), ∑ i ∈ Finset.range n, (Φ i.succ - Φ (i.succ + 1)) ≤ c
 ⊢ ∀ (n : ℕ), ∑ i ∈ Finset.range n, (Φ (i + 1) - Φ (i + 1 + 1)) ≤ c]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_213.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (c : ℝ)
  (hn : ∀ (n : ℕ), ∑ i ∈ Finset.range n, (Φ i.succ - Φ (i.succ + 1)) ≤ c) (n : ℕ) :
  ∑ i ∈ Finset.range n, (Φ (i + 1) - Φ (i + 1 + 1)) ≤ c := sorry

Tactic state before:
case intro.h
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
c : ℝ
hn : ∀ (n : ℕ), ∑ i ∈ Finset.range n, (Φ i.succ - Φ (i.succ + 1)) ≤ c
⊢ ∀ (n : ℕ), ∑ i ∈ Finset.range n, (Φ (i + 1) - Φ (i + 1 + 1)) ≤ c

Tactic:
exact hn

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_214.{u_4} {β : Type u_4} {f g : β → ℝ} (hg : ∀ (n : β), 0 ≤ g n)
  (hgf : ∀ (n : β), g n ≤ f n) (hf : Summable f) (n : β) : 0 ≤ f n := sorry

Tactic state before:
β : Type u_4
f g : β → ℝ
hg : ∀ (n : β), 0 ≤ g n
hgf : ∀ (n : β), g n ≤ f n
hf : Summable f
n : β
⊢ 0 ≤ f n

Tactic:
apply le_trans (hg n) (hgf n)

Tactic state after: