zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_14.{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+) (x : E₂)
  (h_original_goal :
    ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) x - OptProblem.b E₁ E₂⟫_ℝ =
      ((fun x => ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) x⟫_ℝ) + fun x =>
          ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ)
        x) :
  ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) x + ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂)⟫_ℝ =
    ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) x⟫_ℝ +
      ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ := sorry

Tactic state before:
case h
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
n : ℕ+
x✝ : E₂
c : F := ADMM.y E₁ E₂ n.natPred
c₁ : ℝ := ⟪c, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ
x : E₂
⊢ ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) x - OptProblem.b E₁ E₂⟫_ℝ =
    ((fun x => ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) x⟫_ℝ) + fun x =>
        ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ)
      x

Tactic:
simp only [Pi.add_apply]
    rw [add_comm]
    rw [← add_sub (A₂ x) (A₁ (x₁ n)) b]

Tactic state after:
[case h
 E₁ : Type u_1
 E₂ : Type u_2
 F : Type u_3
 inst✝⁹ : NormedAddCommGroup E₁
 inst✝⁸ : InnerProductSpace ℝ E₁
 inst✝⁷ : FiniteDimensional ℝ E₁
 inst✝⁶ : NormedAddCommGroup E₂
 inst✝⁵ : InnerProductSpace ℝ E₂
 inst✝⁴ : FiniteDimensional ℝ E₂
 inst✝³ : NormedAddCommGroup F
 inst✝² : InnerProductSpace ℝ F
 inst✝¹ : FiniteDimensional ℝ F
 admm : ADMM E₁ E₂ F
 admm_kkt : Existance_of_kkt admm
 inst✝ : Setting E₁ E₂ F admm admm_kkt
 n : ℕ+
 x✝ : E₂
 c : F := ADMM.y E₁ E₂ n.natPred
 c₁ : ℝ := ⟪c, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ
 x : E₂
 ⊢ ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) x + ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂)⟫_ℝ =
     ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) x⟫_ℝ +
       ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_15.{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+) (x : E₂) :
  let c := ADMM.y E₁ E₂ n.natPred;
  let c₁ := ⟪c, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ;
  ((fun x =>
        ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) x - OptProblem.b E₁ E₂⟫_ℝ) =
      (fun x => ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) x⟫_ℝ) + fun x =>
        ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ) →
    HasFDerivAt (fun x => ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ + c₁)
      ((toDual ℝ E₂) ((ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) c)) x := 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+
x : E₂
c : F := ADMM.y E₁ E₂ n.natPred
c₁ : ℝ := ⟪c, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ
this :
  (fun x =>
      ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) x - OptProblem.b E₁ E₂⟫_ℝ) =
    (fun x => ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) x⟫_ℝ) + fun x =>
      ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ
⊢ HasFDerivAt (fun x => ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ + c₁)
    ((toDual ℝ E₂) ((ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) c)) x

Tactic:
apply HasFDerivAt.add_const c₁ _
    show HasGradientAt ((fun x => ⟪c, (A₂ x)⟫)) (A₂† c) x
    apply inner_gradient

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_19.{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+)
  (this : ↑n - 1 + 1 = ↑n) :
  let vn := v ↑n;
  vn = v ↑n →
    let yn := ADMM.y E₁ E₂ ↑n;
    let yn' := ADMM.y E₁ E₂ (↑n - 1);
    yn = ADMM.y E₁ E₂ ↑n →
      yn' = ADMM.y E₁ E₂ (↑n - 1) →
        let x₁n := ADMM.x₁ E₂ F ↑n;
        let x₂n := ADMM.x₂ E₁ F ↑n;
        x₁n = ADMM.x₁ E₂ F ↑n →
          x₂n = ADMM.x₂ E₁ F ↑n →
            yn' =
                yn -
                  (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) •
                    ((OptProblem.A₁ E₂) x₁n + (OptProblem.A₂ E₁) x₂n - OptProblem.b E₁ E₂) →
              yn -
                  (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F + (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
                    ((OptProblem.A₁ E₂) x₁n + (OptProblem.A₂ E₁) x₂n - OptProblem.b E₁ E₂) =
                yn - ADMM.ρ E₁ E₂ F • ((OptProblem.A₁ E₂) x₁n + (OptProblem.A₂ E₁) x₂n - OptProblem.b E₁ E₂) := 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+
this : ↑n - 1 + 1 = ↑n
vn : E₂ := v ↑n
h₁ : vn = v ↑n
yn : F := ADMM.y E₁ E₂ ↑n
yn' : F := ADMM.y E₁ E₂ (↑n - 1)
h₂ : yn = ADMM.y E₁ E₂ ↑n
h₃ : yn' = ADMM.y E₁ E₂ (↑n - 1)
x₁n : E₁ := ADMM.x₁ E₂ F ↑n
x₂n : E₂ := ADMM.x₂ E₁ F ↑n
h₄ : x₁n = ADMM.x₁ E₂ F ↑n
h₅ : x₂n = ADMM.x₂ E₁ F ↑n
aux :
  yn' = yn - (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) x₁n + (OptProblem.A₂ E₁) x₂n - OptProblem.b E₁ E₂)
⊢ yn -
      (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F + (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
        ((OptProblem.A₁ E₂) x₁n + (OptProblem.A₂ E₁) x₂n - OptProblem.b E₁ E₂) =
    yn - ADMM.ρ E₁ E₂ F • ((OptProblem.A₁ E₂) x₁n + (OptProblem.A₂ E₁) x₂n - OptProblem.b E₁ E₂)

Tactic:
simp
    rw [sub_mul, add_sub]
    simp

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_20.{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)
  (h_original_goal :
    (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂ =
      (1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)) • (ADMM.y E₁ E₂ (n + 1) - ADMM.y E₁ E₂ n)) :
  (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂ =
    ((ADMM.ρ E₁ E₂ F)⁻¹ * ((ADMM.τ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F))) •
      ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) -
        OptProblem.b E₁ E₂) := 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.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂ =
    (1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)) • (ADMM.y E₁ E₂ (n + 1) - ADMM.y E₁ E₂ n)

Tactic:
rw [admm.itey n, add_comm (y n)]
    simp only [one_div, mul_inv_rev, add_sub_cancel_right]
    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
 ⊢ (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂ =
     ((ADMM.ρ E₁ E₂ F)⁻¹ * ((ADMM.τ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F))) •
       ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - OptProblem.b E₁ E₂)]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_21.{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+) :
  ⟪u n - -(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) admm_kkt.y, ADMM.x₁ E₂ F ↑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+
⊢ ⟪u n - -(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) admm_kkt.y, ADMM.x₁ E₂ F ↑n - admm_kkt.x₁⟫_ℝ ≥ 0

Tactic:
apply subgradientAt_mono
    apply u_inthesubgradient
    exact admm_kkt.h.subgrad₁

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_22.{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+) :
  ⟪v ↑n - -(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) admm_kkt.y, ADMM.x₂ E₁ F ↑n - admm_kkt.x₂⟫_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+
⊢ ⟪v ↑n - -(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) admm_kkt.y, ADMM.x₂ E₁ F ↑n - admm_kkt.x₂⟫_ℝ ≥ 0

Tactic:
apply subgradientAt_mono
    apply v_inthesubgradient
    exact admm_kkt.h.subgrad₂

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_28.{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 +
        -1 • (-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]
    rw [sub_eq_add_neg (G := F) (-ey') ((-(τ * ρ) + ρ) • A_e_sum)]
    rw [← neg_one_smul (R := ℝ) (-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 +
       -1 • (-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_29.{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 +
        -1 • (-ey' + -((-(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) + ADMM.ρ E₁ E₂ F) • A_e_sum)) =
      0 →
    (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F - ADMM.ρ E₁ E₂ F) • A_e_sum +
        ((-(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) + ADMM.ρ E₁ E₂ F) • A_e_sum + ey' + -ey') =
      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 +
      -1 • (-ey' + -((-(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) + ADMM.ρ E₁ E₂ F) • A_e_sum)) =
    0

Tactic:
rw [smul_add (-1) (-ey') (-((-(τ * ρ) + ρ) • A_e_sum))]
    rw [neg_smul_neg, neg_smul_neg, one_smul, one_smul]
    rw [add_assoc, add_comm, add_assoc, add_comm 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))
 ⊢ (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F - ADMM.ρ E₁ E₂ F) • A_e_sum +
       ((-(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) + ADMM.ρ E₁ E₂ F) • A_e_sum + ey' + -ey') =
     0]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_30.{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);
  (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F - ADMM.ρ E₁ E₂ F) • A_e_sum +
        ((-(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) + ADMM.ρ E₁ E₂ F) • A_e_sum + ey' + -ey') =
      0 →
    (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F - ADMM.ρ E₁ E₂ F + (-(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))
⊢ (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F - ADMM.ρ E₁ E₂ F) • A_e_sum +
      ((-(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) + ADMM.ρ E₁ E₂ F) • A_e_sum + ey' + -ey') =
    0

Tactic:
rw [add_assoc]
    rw [add_neg_cancel, add_zero]
    rw [← add_smul (τ * ρ - ρ) (-(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))
 ⊢ (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F - ADMM.ρ E₁ E₂ F + (-(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) + ADMM.ρ E₁ E₂ F)) •
       A_e_sum =
     0]