zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_261.{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] (n : ℕ)
  (h_original_goal :
    (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) • (1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)) • (ey (n + 1) - ey n) =
      ey (n + 1) - ey n) :
  (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) • ((ADMM.ρ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F)⁻¹) • (ey (n + 1) - ey n) =
    ey (n + 1) - ey 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
n : ℕ
⊢ (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) • (1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)) • (ey (n + 1) - ey n) = ey (n + 1) - ey n

Tactic:
simp only [one_div, mul_inv_rev]

Tactic state after:
[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
 n : ℕ
 ⊢ (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) • ((ADMM.ρ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F)⁻¹) • (ey (n + 1) - ey n) =
     ey (n + 1) - ey n]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_262.{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] (n : ℕ)
  (h_original_goal :
    (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) • ((ADMM.ρ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F)⁻¹) • (ey (n + 1) - ey n) =
      ey (n + 1) - ey n) :
  (ADMM.τ E₁ E₂ F * (ADMM.ρ E₁ E₂ F * ((ADMM.ρ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F)⁻¹))) • (ey (n + 1) - ey n) =
    ey (n + 1) - ey 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
n : ℕ
⊢ (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) • ((ADMM.ρ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F)⁻¹) • (ey (n + 1) - ey n) =
    ey (n + 1) - ey n

Tactic:
rw [smul_smul, mul_assoc]

Tactic state after:
[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
 n : ℕ
 ⊢ (ADMM.τ E₁ E₂ F * (ADMM.ρ E₁ E₂ F * ((ADMM.ρ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F)⁻¹))) • (ey (n + 1) - ey n) =
     ey (n + 1) - ey n]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_263.{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] (n : ℕ)
  (h_original_goal :
    (ADMM.τ E₁ E₂ F * (ADMM.ρ E₁ E₂ F * (ADMM.ρ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F)⁻¹)) • (ey (n + 1) - ey n) =
      ey (n + 1) - ey n) :
  ADMM.τ E₁ E₂ F ≠ 0 := 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
n : ℕ
⊢ (ADMM.τ E₁ E₂ F * (ADMM.ρ E₁ E₂ F * (ADMM.ρ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F)⁻¹)) • (ey (n + 1) - ey n) =
    ey (n + 1) - ey n

Tactic:
rw [mul_inv_cancel₀, one_mul, mul_inv_cancel₀, one_smul]

Tactic state after:
[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
 n : ℕ
 ⊢ ADMM.τ E₁ E₂ F ≠ 0,
 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
 n : ℕ
 ⊢ ADMM.ρ E₁ E₂ F ≠ 0]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_264.{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] (n : ℕ)
  (h_original_goal :
    (ADMM.τ E₁ E₂ F * (ADMM.ρ E₁ E₂ F * (ADMM.ρ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F)⁻¹)) • (ey (n + 1) - ey n) =
      ey (n + 1) - ey n) :
  ADMM.ρ E₁ E₂ F ≠ 0 := 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
n : ℕ
⊢ (ADMM.τ E₁ E₂ F * (ADMM.ρ E₁ E₂ F * (ADMM.ρ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F)⁻¹)) • (ey (n + 1) - ey n) =
    ey (n + 1) - ey n

Tactic:
rw [mul_inv_cancel₀, one_mul, mul_inv_cancel₀, one_smul]

Tactic state after:
[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
 n : ℕ
 ⊢ ADMM.τ E₁ E₂ F ≠ 0,
 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
 n : ℕ
 ⊢ ADMM.ρ E₁ E₂ F ≠ 0]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_265.{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} : ADMM.ρ E₁ E₂ F ≠ 0 := 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
n : ℕ
⊢ ADMM.ρ E₁ E₂ F ≠ 0

Tactic:
apply ne_of_gt admm.hrho

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_266.{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} (n : ℕ+) :
  ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
    ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.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
inst✝ : Setting E₁ E₂ F admm admm_kkt
n : ℕ+
⊢ ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
    ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖

Tactic:
simp only [neg_sub]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_267.{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] (n : ℕ+)
  (h₄'' :
    ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
      ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖)
  (h_original_goal :
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ =
      ‖(OptProblem.A₂ E₁) (e₂ ↑n)‖) :
  ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) -
        (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - admm_kkt.x₂)‖ =
    ‖(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
n : ℕ+
h₄'' :
  ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
    ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖
⊢ ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ =
    ‖(OptProblem.A₂ E₁) (e₂ ↑n)‖

Tactic:
rw [e₂]

Tactic state after:
[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
 n : ℕ+
 h₄'' :
   ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
     ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖
 ⊢ ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) -
         (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - admm_kkt.x₂)‖ =
     ‖(OptProblem.A₂ E₁) (e₂ ↑n)‖]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_268.{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} (n : ℕ+)
  (h₄'' :
    ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
      ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖)
  (h_original_goal :
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) -
          (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - admm_kkt.x₂)‖ =
      ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - admm_kkt.x₂)‖) :
  ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.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
inst✝ : Setting E₁ E₂ F admm admm_kkt
n : ℕ+
h₄'' :
  ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
    ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖
⊢ ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) -
        (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - admm_kkt.x₂)‖ =
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n - admm_kkt.x₂)‖

Tactic:
simp only [map_sub, sub_sub_sub_cancel_left]

Tactic state after:
[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
 n : ℕ+
 h₄'' :
   ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
     ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖
 ⊢ ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
     ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂‖]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_269.{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} (n : ℕ+)
  (h₄'' :
    ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
      ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖) :
  ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.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
inst✝ : Setting E₁ E₂ F admm admm_kkt
n : ℕ+
h₄'' :
  ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
    ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖
⊢ ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂‖

Tactic:
rw [h₄'', norm_neg]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_272.{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] (n : ℕ+)
  (h₄'' :
    ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
      ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖)
  (h₄' :
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ =
      ‖(OptProblem.A₂ E₁) (e₂ ↑n)‖)
  (h₁ : 0 < ADMM.τ E₁ E₂ F) (right : ADMM.τ E₁ E₂ F < (1 + √5) / 2) : ADMM.τ E₁ E₂ F ≥ 0 := 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
n : ℕ+
h₄'' :
  ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
    ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖
h₄' :
  ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ =
    ‖(OptProblem.A₂ E₁) (e₂ ↑n)‖
h₁ : 0 < ADMM.τ E₁ E₂ F
right✝ : ADMM.τ E₁ E₂ F < (1 + √5) / 2
⊢ ADMM.τ E₁ E₂ F ≥ 0

Tactic:
apply le_of_lt h₁

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_273.{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] (n : ℕ+)
  (h₄'' :
    ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
      ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖)
  (h₄' :
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ =
      ‖(OptProblem.A₂ E₁) (e₂ ↑n)‖)
  (h1 : ADMM.τ E₁ E₂ F ≥ 0) : ADMM.ρ E₁ E₂ F ≥ 0 := 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
n : ℕ+
h₄'' :
  ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
    ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖
h₄' :
  ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ =
    ‖(OptProblem.A₂ E₁) (e₂ ↑n)‖
h1 : ADMM.τ E₁ E₂ F ≥ 0
⊢ ADMM.ρ E₁ E₂ F ≥ 0

Tactic:
apply le_of_lt admm.hrho

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_274.{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] (n : ℕ+)
  (h₄'' :
    ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
      ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖)
  (h₄' :
    ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ =
      ‖(OptProblem.A₂ E₁) (e₂ ↑n)‖)
  (h1 : ADMM.τ E₁ E₂ F ≥ 0) (h2 : ADMM.ρ E₁ E₂ F ≥ 0) : |ADMM.τ E₁ E₂ F| = ADMM.τ E₁ E₂ F := 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
n : ℕ+
h₄'' :
  ‖(OptProblem.A₂ E₁) admm_kkt.x₂ - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n)‖ =
    ‖-((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) admm_kkt.x₂)‖
h₄' :
  ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑n + 1) - ADMM.x₂ E₁ F ↑n) - (OptProblem.A₂ E₁) (e₂ (↑n + 1))‖ =
    ‖(OptProblem.A₂ E₁) (e₂ ↑n)‖
h1 : ADMM.τ E₁ E₂ F ≥ 0
h2 : ADMM.ρ E₁ E₂ F ≥ 0
⊢ |ADMM.τ E₁ E₂ F| = ADMM.τ E₁ E₂ F

Tactic:
apply abs_eq_self.mpr h1

Tactic state after: