zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_187.{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) :
  let A_e_sum := (OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1));
  (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F - ADMM.ρ E₁ E₂ F + ADMM.ρ E₁ E₂ F + -(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)) • A_e_sum =
    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
A_e_sum : F := (OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))
A_x_sum : F := -(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1))
ey' : F := ey (n + 1)
Ae1 : F := (OptProblem.A₁ E₂) (e₁ (n + 1))
Ae2 : F := (OptProblem.A₂ E₁) (e₂ (n + 1))
⊢ (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F - ADMM.ρ E₁ E₂ F + ADMM.ρ E₁ E₂ F + -(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)) • A_e_sum = 0

Tactic:
rw [sub_eq_add_neg, add_assoc (τ * ρ) (-ρ) ρ, add_comm (-ρ) ρ, add_neg_cancel, add_zero, add_neg_cancel,
  zero_smul]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_188.{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) :
  let A_e_sum := (OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1));
  let A_x_sum := -(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1));
  let ey' := ey (n + 1);
  let Ae1 := (OptProblem.A₁ E₂) (e₁ (n + 1));
  -ey' + (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F - ADMM.ρ E₁ E₂ F) • A_e_sum =
      -ey' - (-(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) + ADMM.ρ E₁ E₂ F) • A_e_sum →
    ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_ℝ +
        ⟪-ey' + (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F - ADMM.ρ E₁ E₂ F) • A_e_sum, A_e_sum⟫_ℝ =
      ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_ℝ +
        ⟪-ey' - (-(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) + ADMM.ρ E₁ E₂ F) • A_e_sum, A_e_sum⟫_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
n : ℕ
A_e_sum : F := (OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))
A_x_sum : F := -(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1))
ey' : F := ey (n + 1)
Ae1 : F := (OptProblem.A₁ E₂) (e₁ (n + 1))
Ae2 : F := (OptProblem.A₂ E₁) (e₂ (n + 1))
sub :
  -ey' + (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F - ADMM.ρ E₁ E₂ F) • A_e_sum =
    -ey' - (-(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) + ADMM.ρ E₁ E₂ F) • A_e_sum
⊢ ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_ℝ + ⟪-ey' + (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F - ADMM.ρ E₁ E₂ F) • A_e_sum, A_e_sum⟫_ℝ =
    ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_ℝ +
      ⟪-ey' - (-(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) + ADMM.ρ E₁ E₂ F) • A_e_sum, A_e_sum⟫_ℝ

Tactic:
rw [sub]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_189.{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) :
  let A_e_sum := (OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1));
  let A_x_sum := -(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1));
  let ey' := ey (n + 1);
  let Ae1 := (OptProblem.A₁ E₂) (e₁ (n + 1));
  let Ae2 := (OptProblem.A₂ E₁) (e₂ (n + 1));
  ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum, Ae1 + Ae2⟫_ℝ + ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_ℝ =
    ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum, Ae1⟫_ℝ +
        ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum, Ae2⟫_ℝ +
      ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_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
n : ℕ
A_e_sum : F := (OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))
A_x_sum : F := -(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1))
ey' : F := ey (n + 1)
Ae1 : F := (OptProblem.A₁ E₂) (e₁ (n + 1))
Ae2 : F := (OptProblem.A₂ E₁) (e₂ (n + 1))
⊢ ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum, Ae1 + Ae2⟫_ℝ + ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_ℝ =
    ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum, Ae1⟫_ℝ +
        ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • A_e_sum, Ae2⟫_ℝ +
      ADMM.ρ E₁ E₂ F * ⟪A_x_sum, Ae1⟫_ℝ

Tactic:
rw [inner_add_right]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_192.{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) :
  ADMM.ρ E₁ E₂ F • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n)) =
    -1 • ADMM.ρ E₁ E₂ F • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 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
n : ℕ
A_e_sum : F := (OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))
A_x_sum : F := -(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1))
ey' : F := ey (n + 1)
Ae1 : F := (OptProblem.A₁ E₂) (e₁ (n + 1))
Ae2 : F := (OptProblem.A₂ E₁) (e₂ (n + 1))
⊢ ADMM.ρ E₁ E₂ F • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n)) =
    -1 • ADMM.ρ E₁ E₂ F • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)))

Tactic:
rw [smul_comm, neg_one_smul, neg_sub]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_197.{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) :
  0 ≤
    ⟪-ey (n + 1) -
          ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
            ((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) -
        ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)),
      (OptProblem.A₁ E₂) (e₁ (n + 1))⟫_ℝ := sorry

Tactic state before:
case hb
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
A_e_sum : F := (OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))
A_x_sum : F := -(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1))
ey' : F := ey (n + 1)
Ae1 : F := (OptProblem.A₁ E₂) (e₁ (n + 1))
Ae2 : F := (OptProblem.A₂ E₁) (e₂ (n + 1))
⊢ 0 ≤
    ⟪-ey (n + 1) -
          ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
            ((OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) -
        ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)),
      (OptProblem.A₁ E₂) (e₁ (n + 1))⟫_ℝ

Tactic:
apply expended_u_gt_zero

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_198.{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.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
      ADMM.ρ E₁ E₂ F *
        ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ) :
  -(ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ) =
    ADMM.ρ E₁ E₂ F *
      ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_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
n : ℕ
⊢ -ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
    ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ

Tactic:
rw [neg_mul (ρ) (inner (A₂ (x₂ (n + 1) - x₂ n)) (A₂ (e₂ (n + 1))))]

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.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ) =
     ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_199.{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.ρ E₁ E₂ F *
          ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ) =
      ADMM.ρ E₁ E₂ F *
        ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ) :
  ADMM.ρ E₁ E₂ F * -⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
    ADMM.ρ E₁ E₂ F *
      ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_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
n : ℕ
⊢ -(ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ) =
    ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ

Tactic:
rw [← mul_neg]

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.ρ E₁ E₂ F * -⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
     ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_200.{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.ρ E₁ E₂ F * -⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
      ADMM.ρ E₁ E₂ F *
        ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ) :
  ADMM.ρ E₁ E₂ F * ⟪-(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
    ADMM.ρ E₁ E₂ F *
      ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_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
n : ℕ
⊢ ADMM.ρ E₁ E₂ F * -⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
    ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ

Tactic:
rw [← inner_neg_left (A₂ (x₂ (n + 1) - x₂ n)) (A₂ (e₂ (n + 1)))]

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.ρ E₁ E₂ F * ⟪-(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
     ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

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} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (n : ℕ)
  (h_original_goal :
    ADMM.ρ E₁ E₂ F * ⟪-(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
      ADMM.ρ E₁ E₂ F *
        ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ) :
  ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (-(ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
    ADMM.ρ E₁ E₂ F *
      ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_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
n : ℕ
⊢ ADMM.ρ E₁ E₂ F * ⟪-(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
    ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ

Tactic:
rw [← map_neg A₂ (x₂ (n + 1) - x₂ n)]

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.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (-(ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
     ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

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} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (n : ℕ) :
  ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (-(ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
    ADMM.ρ E₁ E₂ F *
      ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_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
n : ℕ
⊢ ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (-(ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
    ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ

Tactic:
rw [neg_sub (x₂ (n + 1)) (x₂ n)]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

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} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (n : ℕ)
  (h :
    (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂ =
      (OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))) :
  (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂ -
      (OptProblem.A₂ E₁) (e₂ (n + 1)) =
    (OptProblem.A₁ E₂) (e₁ (n + 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
n : ℕ
h :
  (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂ =
    (OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))
⊢ (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂ -
      (OptProblem.A₂ E₁) (e₂ (n + 1)) =
    (OptProblem.A₁ E₂) (e₁ (n + 1))

Tactic:
simp [h]

Tactic state after: