zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_453.{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} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)]
  (h_original_goal : Tendsto (fun n => ‖ey n‖) atTop (𝓝 0)) : Tendsto (fun n => ‖ey n‖) atTop (𝓝 √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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
⊢ Tendsto (fun n => ‖ey n‖) atTop (𝓝 0)

Tactic:
rw [← Real.sqrt_zero]

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
 fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
 fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
 inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
 ⊢ Tendsto (fun n => ‖ey n‖) atTop (𝓝 √0)]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_454.{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} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)]
  (n : ℕ) : ‖ey n‖ = √(‖ey n‖ ^ 2) := 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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
n : ℕ
⊢ ‖ey n‖ = √(‖ey n‖ ^ 2)

Tactic:
simp [Real.sqrt_sq]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_455.{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} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)]
  (this : (fun n => ‖ey n‖) = fun n => √(‖ey n‖ ^ 2)) (h_original_goal : Tendsto (fun n => ‖ey n‖) atTop (𝓝 √0)) :
  Tendsto (fun n => √(‖ey n‖ ^ 2)) atTop (𝓝 √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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
this : (fun n => ‖ey n‖) = fun n => √(‖ey n‖ ^ 2)
⊢ Tendsto (fun n => ‖ey n‖) atTop (𝓝 √0)

Tactic:
rw [this]

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
 fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
 fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
 inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
 this : (fun n => ‖ey n‖) = fun n => √(‖ey n‖ ^ 2)
 ⊢ Tendsto (fun n => √(‖ey n‖ ^ 2)) atTop (𝓝 √0)]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_456.{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} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)]
  (this : (fun n => ‖ey n‖) = fun n => √(‖ey n‖ ^ 2)) : Tendsto (fun n => √(‖ey n‖ ^ 2)) atTop (𝓝 √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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
this : (fun n => ‖ey n‖) = fun n => √(‖ey n‖ ^ 2)
⊢ Tendsto (fun n => √(‖ey n‖ ^ 2)) atTop (𝓝 √0)

Tactic:
apply Filter.Tendsto.sqrt (ey_sqnrm_converge_zero fullrank₁ fullrank₂)

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_457.{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} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)]
  (h_original_goal : Tendsto ey atTop (𝓝 0)) : Tendsto (fun x => ‖ey x‖) atTop (𝓝 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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
⊢ Tendsto ey atTop (𝓝 0)

Tactic:
apply tendsto_zero_iff_norm_tendsto_zero.2

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
 fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
 fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
 inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
 ⊢ Tendsto (fun x => ‖ey x‖) atTop (𝓝 0)]
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_458.{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} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)] :
  Tendsto (fun x => ‖ey x‖) atTop (𝓝 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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
⊢ Tendsto (fun x => ‖ey x‖) atTop (𝓝 0)

Tactic:
apply ey_nrm_converge_zero

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_459.{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} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)]
  (this : e₁ = fun n => ADMM.x₁ E₂ F n - x₁'') (h : Tendsto (ADMM.x₁ E₂ F) atTop (𝓝 x₁'')) :
  Tendsto (ADMM.x₁ E₂ F) atTop (𝓝 x₁'') := 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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
this : e₁ = fun n => ADMM.x₁ E₂ F n - x₁''
h : Tendsto (ADMM.x₁ E₂ F) atTop (𝓝 x₁'')
⊢ Tendsto (ADMM.x₁ E₂ F) atTop (𝓝 x₁'')

Tactic:
exact h

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_460.{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} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)]
  (this : e₂ = fun n => ADMM.x₂ E₁ F n - x₂'') (h : Tendsto (ADMM.x₂ E₁ F) atTop (𝓝 x₂'')) :
  Tendsto (ADMM.x₂ E₁ F) atTop (𝓝 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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
this : e₂ = fun n => ADMM.x₂ E₁ F n - x₂''
h : Tendsto (ADMM.x₂ E₁ F) atTop (𝓝 x₂'')
⊢ Tendsto (ADMM.x₂ E₁ F) atTop (𝓝 x₂'')

Tactic:
exact h

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_461.{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} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)]
  (this : ey = fun n => ADMM.y E₁ E₂ n - y'') (h : Tendsto (ADMM.y E₁ E₂) atTop (𝓝 y'')) :
  Tendsto (ADMM.y E₁ E₂) atTop (𝓝 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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
this : ey = fun n => ADMM.y E₁ E₂ n - y''
h : Tendsto (ADMM.y E₁ E₂) atTop (𝓝 y'')
⊢ Tendsto (ADMM.y E₁ E₂) atTop (𝓝 y'')

Tactic:
exact h

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter LinearMap ContinuousLinearMap InnerProduct Function

theorem extracted_formal_statement_0.{u_2, u_1} {X : Type u_1} {Y : Type u_2} [inst : NormedAddCommGroup X]
  [inst_1 : InnerProductSpace ℝ X] [inst_2 : NormedAddCommGroup Y] [inst_3 : InnerProductSpace ℝ Y] (A : X →L[ℝ] Y)
  [inst_4 : CompleteSpace X] [inst_5 : CompleteSpace Y] (x : X) (h : (ContinuousLinearMap.adjoint A) (A x) = 0) :
  ⟪A x, A x⟫_ℝ = ⟪x, (ContinuousLinearMap.adjoint A) (A x)⟫_ℝ := sorry

Tactic state before:
X : Type u_1
Y : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : InnerProductSpace ℝ X
inst✝³ : NormedAddCommGroup Y
inst✝² : InnerProductSpace ℝ Y
A : X →L[ℝ] Y
inst✝¹ : CompleteSpace X
inst✝ : CompleteSpace Y
x : X
h : (ContinuousLinearMap.adjoint A) (A x) = 0
⊢ ⟪A x, A x⟫_ℝ = ⟪x, (ContinuousLinearMap.adjoint A) (A x)⟫_ℝ

Tactic:
rw [ContinuousLinearMap.adjoint_inner_right]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter LinearMap ContinuousLinearMap InnerProduct Function

theorem extracted_formal_statement_1.{u_2, u_1} {X : Type u_1} {Y : Type u_2} [inst : NormedAddCommGroup X]
  [inst_1 : InnerProductSpace ℝ X] [inst_2 : NormedAddCommGroup Y] [inst_3 : InnerProductSpace ℝ Y] (A : X →L[ℝ] Y)
  [inst_4 : CompleteSpace X] [inst_5 : CompleteSpace Y] (x : X) (h : (ContinuousLinearMap.adjoint A) (A x) = 0) :
  ⟪x, (ContinuousLinearMap.adjoint A) (A x)⟫_ℝ = 0 := sorry

Tactic state before:
X : Type u_1
Y : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : InnerProductSpace ℝ X
inst✝³ : NormedAddCommGroup Y
inst✝² : InnerProductSpace ℝ Y
A : X →L[ℝ] Y
inst✝¹ : CompleteSpace X
inst✝ : CompleteSpace Y
x : X
h : (ContinuousLinearMap.adjoint A) (A x) = 0
⊢ ⟪x, (ContinuousLinearMap.adjoint A) (A x)⟫_ℝ = 0

Tactic:
rw [h, inner_zero_right]

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter LinearMap ContinuousLinearMap InnerProduct Function

theorem extracted_formal_statement_2.{u_2, u_1} {X : Type u_1} {Y : Type u_2} [inst : NormedAddCommGroup X]
  [inst_1 : InnerProductSpace ℝ X] [inst_2 : NormedAddCommGroup Y] [inst_3 : InnerProductSpace ℝ Y] (A : X →L[ℝ] Y)
  [inst_4 : CompleteSpace X] [inst_5 : CompleteSpace Y] (x : X) (h : (ContinuousLinearMap.adjoint A) (A x) = 0)
  (this : ⟪A x, A x⟫_ℝ = 0) : x ∈ LinearMap.ker A := sorry

Tactic state before:
case h.mpr
X : Type u_1
Y : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : InnerProductSpace ℝ X
inst✝³ : NormedAddCommGroup Y
inst✝² : InnerProductSpace ℝ Y
A : X →L[ℝ] Y
inst✝¹ : CompleteSpace X
inst✝ : CompleteSpace Y
x : X
h : (ContinuousLinearMap.adjoint A) (A x) = 0
this : ⟪A x, A x⟫_ℝ = 0
⊢ x ∈ LinearMap.ker A

Tactic:
apply inner_self_eq_zero.1 this

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter LinearMap ContinuousLinearMap InnerProduct Function

theorem extracted_formal_statement_3.{u_2, u_1} {X : Type u_1} {Y : Type u_2} [inst : NormedAddCommGroup X]
  [inst_1 : InnerProductSpace ℝ X] [inst_2 : NormedAddCommGroup Y] [inst_3 : InnerProductSpace ℝ Y] (A : X →L[ℝ] Y)
  [inst_4 : CompleteSpace X] [inst_5 : CompleteSpace Y] (fullrank : Injective ⇑A) : LinearMap.ker A = ⊥ := sorry

Tactic state before:
X : Type u_1
Y : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : InnerProductSpace ℝ X
inst✝³ : NormedAddCommGroup Y
inst✝² : InnerProductSpace ℝ Y
A : X →L[ℝ] Y
inst✝¹ : CompleteSpace X
inst✝ : CompleteSpace Y
fullrank : Injective ⇑A
⊢ LinearMap.ker A = ⊥

Tactic:
apply KerA_bot A fullrank

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter LinearMap ContinuousLinearMap InnerProduct Function

theorem extracted_formal_statement_4.{u_2, u_1} {X : Type u_1} {Y : Type u_2} [inst : NormedAddCommGroup X]
  [inst_1 : InnerProductSpace ℝ X] [inst_2 : NormedAddCommGroup Y] [inst_3 : InnerProductSpace ℝ Y] (A : X →L[ℝ] Y)
  [inst_4 : CompleteSpace X] [inst_5 : CompleteSpace Y] [inst_6 : FiniteDimensional ℝ X] (fullrank : Injective ⇑A)
  (H : Module.finrank ℝ X = Module.finrank ℝ X) : Surjective ⇑((ContinuousLinearMap.adjoint A).comp A) := sorry

Tactic state before:
X : Type u_1
Y : Type u_2
inst✝⁶ : NormedAddCommGroup X
inst✝⁵ : InnerProductSpace ℝ X
inst✝⁴ : NormedAddCommGroup Y
inst✝³ : InnerProductSpace ℝ Y
A : X →L[ℝ] Y
inst✝² : CompleteSpace X
inst✝¹ : CompleteSpace Y
inst✝ : FiniteDimensional ℝ X
fullrank : Injective ⇑A
H : Module.finrank ℝ X = Module.finrank ℝ X
⊢ Surjective ⇑((ContinuousLinearMap.adjoint A).comp A)

Tactic:
apply (injective_iff_surjective_of_finrank_eq_finrank H).1 (A'A_inj A fullrank)

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter LinearMap ContinuousLinearMap InnerProduct Function

theorem extracted_formal_statement_5.{u_2, u_1} {X : Type u_1} {Y : Type u_2} [inst : NormedAddCommGroup X]
  [inst_1 : InnerProductSpace ℝ X] [inst_2 : NormedAddCommGroup Y] [inst_3 : InnerProductSpace ℝ Y] (A : X →L[ℝ] Y)
  [inst_4 : CompleteSpace X] [inst_5 : CompleteSpace Y] [inst_6 : FiniteDimensional ℝ X] (fullrank : Injective ⇑A)
  (C : ℝ) (hc : C > 0) (hx : ∀ (y : X), ∃ x, ((ContinuousLinearMap.adjoint A).comp A) x = y ∧ ‖x‖ ≤ C * ‖y‖) (x x1 : X)
  (hx1 :
    ((ContinuousLinearMap.adjoint A).comp A) x = ((ContinuousLinearMap.adjoint A).comp A) x ∧
      ‖x‖ ≤ C * ‖((ContinuousLinearMap.adjoint A).comp A) x‖)
  (h : x = x1) : ‖x‖ ≤ C * ‖((ContinuousLinearMap.adjoint A).comp A) x‖ := sorry

Tactic state before:
case right.intro
X : Type u_1
Y : Type u_2
inst✝⁶ : NormedAddCommGroup X
inst✝⁵ : InnerProductSpace ℝ X
inst✝⁴ : NormedAddCommGroup Y
inst✝³ : InnerProductSpace ℝ Y
A : X →L[ℝ] Y
inst✝² : CompleteSpace X
inst✝¹ : CompleteSpace Y
inst✝ : FiniteDimensional ℝ X
fullrank : Injective ⇑A
C : ℝ
hc : C > 0
hx : ∀ (y : X), ∃ x, ((ContinuousLinearMap.adjoint A).comp A) x = y ∧ ‖x‖ ≤ C * ‖y‖
x x1 : X
hx1 :
  ((ContinuousLinearMap.adjoint A).comp A) x = ((ContinuousLinearMap.adjoint A).comp A) x ∧
    ‖x‖ ≤ C * ‖((ContinuousLinearMap.adjoint A).comp A) x‖
h : x = x1
⊢ ‖x‖ ≤ C * ‖((ContinuousLinearMap.adjoint A).comp A) x‖

Tactic:
exact hx1.2

Tactic state after:
import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter LinearMap ContinuousLinearMap InnerProduct Function

theorem extracted_formal_statement_6.{u_2, u_1} {X : Type u_1} {Y : Type u_2} [inst : NormedAddCommGroup X]
  [inst_1 : InnerProductSpace ℝ X] [inst_2 : NormedAddCommGroup Y] [inst_3 : InnerProductSpace ℝ Y] (A : X →L[ℝ] Y)
  [inst_4 : CompleteSpace X] [inst_5 : CompleteSpace Y] (this : IsBoundedLinearMap ℝ ⇑(ContinuousLinearMap.adjoint A))
  (M : ℝ) (h1 : 0 < M ∧ ∀ (x : Y), ‖(ContinuousLinearMap.adjoint A) x‖ ≤ M * ‖x‖) (x : X)
  (h2 : ‖(ContinuousLinearMap.adjoint A) (A x)‖ ≤ M * ‖A x‖) :
  ‖((ContinuousLinearMap.adjoint A).comp A) x‖ ≤ M * ‖A x‖ := sorry

Tactic state before:
case right
X : Type u_1
Y : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : InnerProductSpace ℝ X
inst✝³ : NormedAddCommGroup Y
inst✝² : InnerProductSpace ℝ Y
A : X →L[ℝ] Y
inst✝¹ : CompleteSpace X
inst✝ : CompleteSpace Y
this : IsBoundedLinearMap ℝ ⇑(ContinuousLinearMap.adjoint A)
M : ℝ
h1 : 0 < M ∧ ∀ (x : Y), ‖(ContinuousLinearMap.adjoint A) x‖ ≤ M * ‖x‖
x : X
h2 : ‖(ContinuousLinearMap.adjoint A) (A x)‖ ≤ M * ‖A x‖
⊢ ‖((ContinuousLinearMap.adjoint A).comp A) x‖ ≤ M * ‖A x‖

Tactic:
exact h2

Tactic state after: