zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_163.{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] (r1 : ℝ)
  (h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1) (r2 : ℝ)
  (h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2) :
  let r := r1 + r2;
  r = r1 + r2 → ∀ (n : ℕ), (⇑(OptProblem.A₂ E₁) ∘ e₂) n = (OptProblem.A₂ E₁) (e₂ n) := 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
r1 : ℝ
h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1
r2 : ℝ
h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2
r : ℝ := r1 + r2
hr : r = r1 + r2
n : ℕ
⊢ (⇑(OptProblem.A₂ E₁) ∘ e₂) n = (OptProblem.A₂ E₁) (e₂ n)

Tactic:
simp

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_164.{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] (r1 : ℝ)
  (h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1) (r2 : ℝ)
  (h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2) :
  let r := r1 + r2;
  r = r1 + r2 →
    ∀ (n : ℕ),
      (OptProblem.A₂ E₁) (e₂ n) ∈ range (⇑(OptProblem.A₂ E₁) ∘ e₂) → (OptProblem.A₂ E₁) (e₂ n) ∈ ball 0 r1 := 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
r1 : ℝ
h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1
r2 : ℝ
h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2
r : ℝ := r1 + r2
hr : r = r1 + r2
n : ℕ
haa : (OptProblem.A₂ E₁) (e₂ n) ∈ range (⇑(OptProblem.A₂ E₁) ∘ e₂)
⊢ (OptProblem.A₂ E₁) (e₂ n) ∈ ball 0 r1

Tactic:
apply h_A₂e₂ haa

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_165.{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] (r1 : ℝ)
  (h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1) (r2 : ℝ)
  (h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2) :
  let r := r1 + r2;
  r = r1 + r2 →
    ∀ (n : ℕ),
      (OptProblem.A₂ E₁) (e₂ n) ∈ range (⇑(OptProblem.A₂ E₁) ∘ e₂) →
        ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1 → ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1 := 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
r1 : ℝ
h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1
r2 : ℝ
h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2
r : ℝ := r1 + r2
hr : r = r1 + r2
n : ℕ
haa : (OptProblem.A₂ E₁) (e₂ n) ∈ range (⇑(OptProblem.A₂ E₁) ∘ e₂)
ha_in_ball : ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1
⊢ ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1

Tactic:
exact ha_in_ball

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_166.{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] (r1 : ℝ)
  (h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1) (r2 : ℝ)
  (h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2) :
  let r := r1 + r2;
  r = r1 + r2 →
    ∀ (n : ℕ),
      ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1 →
        (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) n =
          (OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n) := 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
r1 : ℝ
h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1
r2 : ℝ
h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2
r : ℝ := r1 + r2
hr : r = r1 + r2
n : ℕ
ha : ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1
⊢ (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) n = (OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)

Tactic:
simp

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_167.{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] (r1 : ℝ)
  (h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1) (r2 : ℝ)
  (h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2) :
  let r := r1 + r2;
  r = r1 + r2 →
    ∀ (n : ℕ),
      ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1 →
        (OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n) ∈
            range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) →
          (OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n) ∈ ball 0 r2 := 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
r1 : ℝ
h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1
r2 : ℝ
h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2
r : ℝ := r1 + r2
hr : r = r1 + r2
n : ℕ
ha : ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1
hbb :
  (OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n) ∈ range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂)
⊢ (OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n) ∈ ball 0 r2

Tactic:
apply h_A₁e₁_A₂e₂ hbb

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_168.{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] (r1 : ℝ)
  (h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1) (r2 : ℝ)
  (h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2) :
  let r := r1 + r2;
  r = r1 + r2 →
    ∀ (n : ℕ),
      ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1 →
        (OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n) ∈
            range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) →
          ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ < r2 →
            ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ < r2 := 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
r1 : ℝ
h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1
r2 : ℝ
h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2
r : ℝ := r1 + r2
hr : r = r1 + r2
n : ℕ
ha : ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1
hbb :
  (OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n) ∈ range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂)
hb_in_ball : ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ < r2
⊢ ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ < r2

Tactic:
exact hb_in_ball

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_169.{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] (r1 : ℝ)
  (h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1) (r2 : ℝ)
  (h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2) :
  let r := r1 + r2;
  r = r1 + r2 →
    ∀ (n : ℕ),
      ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1 + r2 →
        ‖(OptProblem.A₁ E₂) (e₁ n)‖ ≤
          ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖ := 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
r1 : ℝ
h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1
r2 : ℝ
h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2
r : ℝ := r1 + r2
hr : r = r1 + r2
n : ℕ
h : ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1 + r2
⊢ ‖(OptProblem.A₁ E₂) (e₁ n)‖ ≤ ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖

Tactic:
apply norm_le_add_norm_add

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_170.{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] (r1 : ℝ)
  (h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1) (r2 : ℝ)
  (h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2) :
  let r := r1 + r2;
  r = r1 + r2 →
    ∀ (n : ℕ),
      ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1 + r2 →
        ‖(OptProblem.A₁ E₂) (e₁ n)‖ ≤
            ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖ →
          ‖(OptProblem.A₁ E₂) (e₁ n) - 0‖ < r → ‖(OptProblem.A₁ E₂) (e₁ n) - 0‖ < 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
r1 : ℝ
h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1
r2 : ℝ
h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2
r : ℝ := r1 + r2
hr : r = r1 + r2
n : ℕ
h : ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1 + r2
h_ineq :
  ‖(OptProblem.A₁ E₂) (e₁ n)‖ ≤ ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖
h_norm : ‖(OptProblem.A₁ E₂) (e₁ n) - 0‖ < r
⊢ ‖(OptProblem.A₁ E₂) (e₁ n) - 0‖ < r

Tactic:
exact h_norm

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_171.{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] (r1 : ℝ)
  (h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1) (r2 : ℝ)
  (h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2) :
  let r := r1 + r2;
  r = r1 + r2 →
    ∀ (n : ℕ),
      ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1 + r2 →
        ‖(OptProblem.A₁ E₂) (e₁ n)‖ ≤
            ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖ →
          ‖(OptProblem.A₁ E₂) (e₁ n)‖ < r →
            (OptProblem.A₁ E₂) (e₁ n) ∈ ball 0 r → (⇑(OptProblem.A₁ E₂) ∘ e₁) n ∈ ball 0 r := sorry

Tactic state before:
case h.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
r1 : ℝ
h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1
r2 : ℝ
h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2
r : ℝ := r1 + r2
hr : r = r1 + r2
n : ℕ
h : ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1 + r2
h_ineq :
  ‖(OptProblem.A₁ E₂) (e₁ n)‖ ≤ ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖
h_norm : ‖(OptProblem.A₁ E₂) (e₁ n)‖ < r
h_dist : (OptProblem.A₁ E₂) (e₁ n) ∈ ball 0 r
⊢ (⇑(OptProblem.A₁ E₂) ∘ e₁) n ∈ ball 0 r

Tactic:
apply h_dist

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_172.{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)) (C : ℝ)
  (C_pos : C > 0) (hh : ∀ (x : E₁), ‖x‖ ≤ C * ‖(OptProblem.A₁ E₂) x‖) : C > 0 := sorry

Tactic state before:
case h.left
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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
C : ℝ
C_pos : C > 0
hh : ∀ (x : E₁), ‖x‖ ≤ C * ‖(OptProblem.A₁ E₂) x‖
⊢ C > 0

Tactic:
exact C_pos

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_173.{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]
  (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)) (C : ℝ) (C_pos : C > 0)
  (hh : ∀ (x : E₁), ‖x‖ ≤ C * ‖(OptProblem.A₁ E₂) x‖) (n : ℕ) : ‖e₁ n‖ ≤ C * ‖(OptProblem.A₁ E₂) (e₁ n)‖ := sorry

Tactic state before:
case h.right
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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
C : ℝ
C_pos : C > 0
hh : ∀ (x : E₁), ‖x‖ ≤ C * ‖(OptProblem.A₁ E₂) x‖
n : ℕ
⊢ ‖e₁ n‖ ≤ C * ‖(OptProblem.A₁ E₂) (e₁ n)‖

Tactic:
exact hh (e₁ n)

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_174.{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} (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) (C : ℝ)
  (C_pos : C > 0) (hh : ∀ (x : E₂), ‖x‖ ≤ C * ‖(OptProblem.A₂ E₁) x‖) : C > 0 := sorry

Tactic state before:
case h.left
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
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
C : ℝ
C_pos : C > 0
hh : ∀ (x : E₂), ‖x‖ ≤ C * ‖(OptProblem.A₂ E₁) x‖
⊢ C > 0

Tactic:
exact C_pos

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_175.{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]
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) (C : ℝ) (C_pos : C > 0)
  (hh : ∀ (x : E₂), ‖x‖ ≤ C * ‖(OptProblem.A₂ E₁) x‖) (n : ℕ) : ‖e₂ n‖ ≤ C * ‖(OptProblem.A₂ E₁) (e₂ n)‖ := sorry

Tactic state before:
case h.right
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
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
C : ℝ
C_pos : C > 0
hh : ∀ (x : E₂), ‖x‖ ≤ C * ‖(OptProblem.A₂ E₁) x‖
n : ℕ
⊢ ‖e₂ n‖ ≤ C * ‖(OptProblem.A₂ E₁) (e₂ n)‖

Tactic:
exact hh (e₂ n)

Tactic state after: