zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

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] (n : ℕ) :
  let succ_n := (n + 1).toPNat';
  ⟪u succ_n + (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) admm_kkt.y, ADMM.x₁ E₂ F ↑succ_n - admm_kkt.x₁⟫_ℝ ≥
    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
Ae1 : F := (OptProblem.A₁ E₂) (e₁ (n + 1))
e' : E₁ := e₁ (n + 1)
block : F :=
  -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))
u' : E₁ :=
  -(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂))
      (ADMM.y E₁ E₂ (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)))
x_diff : E₁ := ADMM.x₁ E₂ F (n + 1) - admm_kkt.x₁
succ_n : ℕ+ := (n + 1).toPNat'
⊢ ⟪u succ_n + (ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) admm_kkt.y, ADMM.x₁ E₂ F ↑succ_n - admm_kkt.x₁⟫_ℝ ≥ 0

Tactic:
apply subgradientAt_mono_u

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

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] (n : ℕ) :
  let succ_n := (n + 1).toPNat';
  let ey' := ey ↑succ_n;
  let e₁' := e₁ ↑succ_n;
  let e₂' := e₂ ↑succ_n;
  ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂'),
      (OptProblem.A₂ E₁) e₂'⟫_ℝ =
    ⟪(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
        (-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')),
      e₂'⟫_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
succ_n : ℕ+ := (n + 1).toPNat'
ey' : F := ey ↑succ_n
e₁' : E₁ := e₁ ↑succ_n
e₂' : E₂ := e₂ ↑succ_n
y_k_1 : F := ADMM.y E₁ E₂ ↑succ_n
v_k_1 : E₂ := v ↑succ_n
x_diff : E₂ := ADMM.x₂ E₁ F ↑succ_n - admm_kkt.x₂
⊢ ⟪-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂'),
      (OptProblem.A₂ E₁) e₂'⟫_ℝ =
    ⟪(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
        (-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')),
      e₂'⟫_ℝ

Tactic:
rw [ContinuousLinearMap.adjoint_inner_left]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

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] (n : ℕ) :
  let succ_n := (n + 1).toPNat';
  let ey' := ey ↑succ_n;
  let e₁' := e₁ ↑succ_n;
  let e₂' := e₂ ↑succ_n;
  ⟪(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
        (-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')),
      e₂'⟫_ℝ =
    ⟪-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
          (ey' + ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')),
      e₂'⟫_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
succ_n : ℕ+ := (n + 1).toPNat'
ey' : F := ey ↑succ_n
e₁' : E₁ := e₁ ↑succ_n
e₂' : E₂ := e₂ ↑succ_n
y_k_1 : F := ADMM.y E₁ E₂ ↑succ_n
v_k_1 : E₂ := v ↑succ_n
x_diff : E₂ := ADMM.x₂ E₁ F ↑succ_n - admm_kkt.x₂
⊢ ⟪(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
        (-ey' - ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')),
      e₂'⟫_ℝ =
    ⟪-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
          (ey' + ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')),
      e₂'⟫_ℝ

Tactic:
rw [sub_eq_add_neg, ← neg_add, A₂†.map_neg]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

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] (n : ℕ) :
  let succ_n := (n + 1).toPNat';
  let ey' := ey ↑succ_n;
  let y_k_1 := ADMM.y E₁ E₂ ↑succ_n;
  ey' = y_k_1 - admm_kkt.y → ey ↑succ_n = ADMM.y E₁ E₂ ↑succ_n - admm_kkt.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
inst✝ : Setting E₁ E₂ F admm admm_kkt
n : ℕ
succ_n : ℕ+ := (n + 1).toPNat'
ey' : F := ey ↑succ_n
e₁' : E₁ := e₁ ↑succ_n
e₂' : E₂ := e₂ ↑succ_n
y_k_1 : F := ADMM.y E₁ E₂ ↑succ_n
v_k_1 : E₂ := v ↑succ_n
x_diff : E₂ := ADMM.x₂ E₁ F ↑succ_n - admm_kkt.x₂
⊢ ey' = y_k_1 - admm_kkt.y

Tactic:
simp [ey', y_k_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
 succ_n : ℕ+ := (n + 1).toPNat'
 ey' : F := ey ↑succ_n
 e₁' : E₁ := e₁ ↑succ_n
 e₂' : E₂ := e₂ ↑succ_n
 y_k_1 : F := ADMM.y E₁ E₂ ↑succ_n
 v_k_1 : E₂ := v ↑succ_n
 x_diff : E₂ := ADMM.x₂ E₁ F ↑succ_n - admm_kkt.x₂
 ⊢ ey ↑succ_n = ADMM.y E₁ E₂ ↑succ_n - admm_kkt.y]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

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] (n : ℕ) :
  let succ_n := (n + 1).toPNat';
  let ey' := ey ↑succ_n;
  let e₁' := e₁ ↑succ_n;
  let e₂' := e₂ ↑succ_n;
  let y_k_1 := ADMM.y E₁ E₂ ↑succ_n;
  ey' = y_k_1 - admm_kkt.y →
    ⟪-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
            (ey' + ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')),
        e₂'⟫_ℝ =
      ⟪-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
            (y_k_1 - admm_kkt.y +
              ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')),
        e₂'⟫_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
succ_n : ℕ+ := (n + 1).toPNat'
ey' : F := ey ↑succ_n
e₁' : E₁ := e₁ ↑succ_n
e₂' : E₂ := e₂ ↑succ_n
y_k_1 : F := ADMM.y E₁ E₂ ↑succ_n
v_k_1 : E₂ := v ↑succ_n
x_diff : E₂ := ADMM.x₂ E₁ F ↑succ_n - admm_kkt.x₂
sub : ey' = y_k_1 - admm_kkt.y
⊢ ⟪-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
          (ey' + ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')),
      e₂'⟫_ℝ =
    ⟪-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
          (y_k_1 - admm_kkt.y +
            ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')),
      e₂'⟫_ℝ

Tactic:
rw [sub]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

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] (n : ℕ) :
  let succ_n := (n + 1).toPNat';
  let e₁' := e₁ ↑succ_n;
  let e₂' := e₂ ↑succ_n;
  let y_k_1 := ADMM.y E₁ E₂ ↑succ_n;
  ⟪-((ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) (-admm_kkt.y) +
          (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
            (y_k_1 + ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂'))),
      e₂'⟫_ℝ =
    ⟪-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
            (y_k_1 + ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')) +
        (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) admm_kkt.y,
      e₂'⟫_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
succ_n : ℕ+ := (n + 1).toPNat'
ey' : F := ey ↑succ_n
e₁' : E₁ := e₁ ↑succ_n
e₂' : E₂ := e₂ ↑succ_n
y_k_1 : F := ADMM.y E₁ E₂ ↑succ_n
v_k_1 : E₂ := v ↑succ_n
x_diff : E₂ := ADMM.x₂ E₁ F ↑succ_n - admm_kkt.x₂
⊢ ⟪-((ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) (-admm_kkt.y) +
          (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
            (y_k_1 + ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂'))),
      e₂'⟫_ℝ =
    ⟪-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
            (y_k_1 + ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) e₁' + (OptProblem.A₂ E₁) e₂')) +
        (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) admm_kkt.y,
      e₂'⟫_ℝ

Tactic:
simp

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

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} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (n : ℕ) :
  let succ_n := (n + 1).toPNat';
  let v_k_1 := v ↑succ_n;
  let x_diff := ADMM.x₂ E₁ F ↑succ_n - admm_kkt.x₂;
  ⟪v_k_1 + (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) admm_kkt.y, x_diff⟫_ℝ ≥ 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
succ_n : ℕ+ := (n + 1).toPNat'
ey' : F := ey ↑succ_n
e₁' : E₁ := e₁ ↑succ_n
e₂' : E₂ := e₂ ↑succ_n
y_k_1 : F := ADMM.y E₁ E₂ ↑succ_n
v_k_1 : E₂ := v ↑succ_n
x_diff : E₂ := ADMM.x₂ E₁ F ↑succ_n - admm_kkt.x₂
⊢ ⟪v_k_1 + (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) admm_kkt.y, x_diff⟫_ℝ ≥ 0

Tactic:
apply subgradientAt_mono_v

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

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] (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', -A_e_sum⟫_ℝ + ⟪-((1 - ADMM.τ 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', A_e_sum⟫_ℝ + ⟪-((1 - ADMM.τ 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 := 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', -A_e_sum⟫_ℝ + ⟪-((1 - ADMM.τ 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', A_e_sum⟫_ℝ + ⟪-((1 - ADMM.τ 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

Tactic:
rw [inner_neg_right (𝕜 := ℝ), inner_neg_left (𝕜 := ℝ), inner_neg_left (𝕜 := ℝ)]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_177.{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 ey' := ey (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 →
    -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) =
      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))
⊢ -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

Tactic:
rw [← sub_eq_zero, sub_eq_add_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
 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' + (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) =
     0]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_178.{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 ey' := ey (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) =
      0 →
    -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)) =
      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))
⊢ -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) =
    0

Tactic:
rw [sub_eq_add_neg (G := F) (-ey') ((-(τ * ρ) + ρ) • A_e_sum)]

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
 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' + (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)) =
     0]