zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_0.{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} : Convex ℝ univ := sorry

Tactic state before:
case left
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
n : ℕ+
⊢ Convex ℝ univ

Tactic:
apply convex_univ

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_1.{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+) ⦃xx : E₁⦄ (a : xx ∈ univ) ⦃yy : E₁⦄
  (a_1 : yy ∈ univ) ⦃aa bb : ℝ⦄ (a_2 : 0 ≤ aa) (a_3 : 0 ≤ bb) (abh : aa + bb = 1)
  (h_original_goal :
    aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) -
        OptProblem.b E₁ E₂ =
      aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
        (aa + bb) • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)) :
  aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) -
      OptProblem.b E₁ E₂ =
    aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
      1 • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - 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+
xx : E₁
a✝³ : xx ∈ univ
yy : E₁
a✝² : yy ∈ univ
aa bb : ℝ
a✝¹ : 0 ≤ aa
a✝ : 0 ≤ bb
abh : aa + bb = 1
⊢ aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) -
      OptProblem.b E₁ E₂ =
    aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
      (aa + bb) • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)

Tactic:
rw [abh]

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+
 xx : E₁
 a✝³ : xx ∈ univ
 yy : E₁
 a✝² : yy ∈ univ
 aa bb : ℝ
 a✝¹ : 0 ≤ aa
 a✝ : 0 ≤ bb
 abh : aa + bb = 1
 ⊢ aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) -
       OptProblem.b E₁ E₂ =
     aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
       1 • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_2.{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+) ⦃xx : E₁⦄ (a : xx ∈ univ) ⦃yy : E₁⦄
  (a_1 : yy ∈ univ) ⦃aa bb : ℝ⦄ (a_2 : 0 ≤ aa) (a_3 : 0 ≤ bb) (abh : aa + bb = 1)
  (h_original_goal :
    aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) -
        OptProblem.b E₁ E₂ =
      aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
        1 • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)) :
  aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) -
      OptProblem.b E₁ E₂ =
    aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
      ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - 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+
xx : E₁
a✝³ : xx ∈ univ
yy : E₁
a✝² : yy ∈ univ
aa bb : ℝ
a✝¹ : 0 ≤ aa
a✝ : 0 ≤ bb
abh : aa + bb = 1
⊢ aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) -
      OptProblem.b E₁ E₂ =
    aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
      1 • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)

Tactic:
rw [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+
 xx : E₁
 a✝³ : xx ∈ univ
 yy : E₁
 a✝² : yy ∈ univ
 aa bb : ℝ
 a✝¹ : 0 ≤ aa
 a✝ : 0 ≤ bb
 abh : aa + bb = 1
 ⊢ aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) -
       OptProblem.b E₁ E₂ =
     aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
       ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_3.{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+) ⦃xx : E₁⦄ (a : xx ∈ univ) ⦃yy : E₁⦄
  (a_1 : yy ∈ univ) ⦃aa bb : ℝ⦄ (a_2 : 0 ≤ aa) (a_3 : 0 ≤ bb) (abh : aa + bb = 1) :
  aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) -
      OptProblem.b E₁ E₂ =
    aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
      ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - 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+
xx : E₁
a✝³ : xx ∈ univ
yy : E₁
a✝² : yy ∈ univ
aa bb : ℝ
a✝¹ : 0 ≤ aa
a✝ : 0 ≤ bb
abh : aa + bb = 1
⊢ aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) -
      OptProblem.b E₁ E₂ =
    aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
      ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)

Tactic:
rw [add_sub]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_5.{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+) ⦃xx : E₁⦄ (a : xx ∈ univ) ⦃yy : E₁⦄
  (a_1 : yy ∈ univ) ⦃aa bb : ℝ⦄ (a_2 : 0 ≤ aa) (a_3 : 0 ≤ bb) (abh : aa + bb = 1) :
  aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
      (aa • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) +
        bb • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)) =
    aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
        aa • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) +
      bb • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - 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+
xx : E₁
a✝³ : xx ∈ univ
yy : E₁
a✝² : yy ∈ univ
aa bb : ℝ
a✝¹ : 0 ≤ aa
a✝ : 0 ≤ bb
abh : aa + bb = 1
⊢ aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
      (aa • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) +
        bb • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)) =
    aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
        aa • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) +
      bb • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)

Tactic:
rw [add_assoc (aa • A₁ (xx) + bb • A₁ (yy))]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_8.{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+) ⦃xx : E₁⦄ (a : xx ∈ univ) ⦃yy : E₁⦄
  (a_1 : yy ∈ univ) ⦃aa bb : ℝ⦄ (a_2 : 0 ≤ aa) (a_3 : 0 ≤ bb) (abh : aa + bb = 1) :
  aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
        aa • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) +
      bb • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) =
    aa • (OptProblem.A₁ E₂) xx + aa • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) +
      (bb • (OptProblem.A₁ E₂) yy + bb • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - 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+
xx : E₁
a✝³ : xx ∈ univ
yy : E₁
a✝² : yy ∈ univ
aa bb : ℝ
a✝¹ : 0 ≤ aa
a✝ : 0 ≤ bb
abh : aa + bb = 1
⊢ aa • (OptProblem.A₁ E₂) xx + bb • (OptProblem.A₁ E₂) yy +
        aa • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) +
      bb • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) =
    aa • (OptProblem.A₁ E₂) xx + aa • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂) +
      (bb • (OptProblem.A₁ E₂) yy + bb • ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))

Tactic:
rw [← add_assoc, add_assoc (aa • A₁ (xx)), add_comm (bb • A₁ (yy)), ← add_assoc]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_9.{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+) ⦃xx : E₁⦄ (a : xx ∈ univ) ⦃yy : E₁⦄
  (a_1 : yy ∈ univ) ⦃aa bb : ℝ⦄ (a_2 : 0 ≤ aa) (a_3 : 0 ≤ bb) (abh : aa + bb = 1)
  (this :
    (OptProblem.A₁ E₂) (aa • xx + bb • yy) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂ =
      aa • ((OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)) +
        bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))) :
  ⟪ADMM.y E₁ E₂ n.natPred,
      (OptProblem.A₁ E₂) (aa • xx + bb • yy) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ =
    ⟪ADMM.y E₁ E₂ n.natPred,
      aa • ((OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)) +
        bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ 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+
xx : E₁
a✝³ : xx ∈ univ
yy : E₁
a✝² : yy ∈ univ
aa bb : ℝ
a✝¹ : 0 ≤ aa
a✝ : 0 ≤ bb
abh : aa + bb = 1
this :
  (OptProblem.A₁ E₂) (aa • xx + bb • yy) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂ =
    aa • ((OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)) +
      bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))
⊢ ⟪ADMM.y E₁ E₂ n.natPred,
      (OptProblem.A₁ E₂) (aa • xx + bb • yy) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ =
    ⟪ADMM.y E₁ E₂ n.natPred,
      aa • ((OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)) +
        bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))⟫_ℝ

Tactic:
rw [this]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter InnerProduct

open scoped Pointwise

theorem extracted_formal_statement_10.{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+) ⦃xx : E₁⦄ (a : xx ∈ univ) ⦃yy : E₁⦄
  (a_1 : yy ∈ univ) ⦃aa bb : ℝ⦄ (a_2 : 0 ≤ aa) (a_3 : 0 ≤ bb) (abh : aa + bb = 1)
  (this :
    (OptProblem.A₁ E₂) (aa • xx + bb • yy) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂ =
      aa • ((OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)) +
        bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))) :
  ⟪ADMM.y E₁ E₂ n.natPred,
      aa • ((OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)) +
        bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))⟫_ℝ =
    ⟪ADMM.y E₁ E₂ n.natPred,
        aa • ((OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))⟫_ℝ +
      ⟪ADMM.y E₁ E₂ n.natPred,
        bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ 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+
xx : E₁
a✝³ : xx ∈ univ
yy : E₁
a✝² : yy ∈ univ
aa bb : ℝ
a✝¹ : 0 ≤ aa
a✝ : 0 ≤ bb
abh : aa + bb = 1
this :
  (OptProblem.A₁ E₂) (aa • xx + bb • yy) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂ =
    aa • ((OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)) +
      bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))
⊢ ⟪ADMM.y E₁ E₂ n.natPred,
      aa • ((OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)) +
        bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))⟫_ℝ =
    ⟪ADMM.y E₁ E₂ n.natPred,
        aa • ((OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))⟫_ℝ +
      ⟪ADMM.y E₁ E₂ n.natPred,
        bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))⟫_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_12.{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+) ⦃xx : E₁⦄ (a : xx ∈ univ) ⦃yy : E₁⦄
  (a_1 : yy ∈ univ) ⦃aa bb : ℝ⦄ (a_2 : 0 ≤ aa) (a_3 : 0 ≤ bb) (abh : aa + bb = 1)
  (this :
    (OptProblem.A₁ E₂) (aa • xx + bb • yy) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂ =
      aa • ((OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)) +
        bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))) :
  aa *
        ⟪ADMM.y E₁ E₂ n.natPred,
          (OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)⟫_ℝ +
      ⟪ADMM.y E₁ E₂ n.natPred,
        bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))⟫_ℝ =
    aa *
        ⟪ADMM.y E₁ E₂ n.natPred,
          (OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)⟫_ℝ +
      bb *
        ⟪ADMM.y E₁ E₂ n.natPred,
          (OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ 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+
xx : E₁
a✝³ : xx ∈ univ
yy : E₁
a✝² : yy ∈ univ
aa bb : ℝ
a✝¹ : 0 ≤ aa
a✝ : 0 ≤ bb
abh : aa + bb = 1
this :
  (OptProblem.A₁ E₂) (aa • xx + bb • yy) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂ =
    aa • ((OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)) +
      bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))
⊢ aa *
        ⟪ADMM.y E₁ E₂ n.natPred,
          (OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)⟫_ℝ +
      ⟪ADMM.y E₁ E₂ n.natPred,
        bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))⟫_ℝ =
    aa *
        ⟪ADMM.y E₁ E₂ n.natPred,
          (OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)⟫_ℝ +
      bb *
        ⟪ADMM.y E₁ E₂ n.natPred,
          (OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)⟫_ℝ

Tactic:
rw [inner_smul_right]

Tactic state after: