mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X X' Y : C
inst✝ : ExactPairing X' Y
i : X ≅ X'
⊢ Y ◁ η_ X' Y ⊗≫ ε_ X' Y ▷ Y = (ρ_ Y).hom ≫ (λ_ Y).inv

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X X' Y : C
inst✝ : ExactPairing X' Y
i : X ≅ X'
⊢ Y ◁ (η_ X' Y ≫ i.inv ▷ Y) ≫ (α_ Y X Y).inv ≫ (Y ◁ i.hom ≫ ε_ X' Y) ▷ Y = (ρ_ Y).hom ≫ (λ_ Y).inv

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.FiniteDimensional.Lemmas

Goal state before:
pp: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
m : ℕ
hm : finrank K V ≤ m
k : ℕ
h_k_le : k ≤ finrank K V
hk : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ k.succ)
⊢ LinearMap.ker (f ^ (k + (finrank K V - k))) = LinearMap.ker (f ^ finrank K V)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.FiniteDimensional.Lemmas

Goal state before:
pp: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
m : ℕ
hm : finrank K V ≤ m
⊢ LinearMap.ker (f ^ m) = LinearMap.ker (f ^ finrank K V)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X X' Y : C
inst✝ : ExactPairing X' Y
i : X ≅ X'
⊢ (η_ X' Y ≫ i.inv ▷ Y) ▷ X ≫ (α_ X Y X).hom ≫ X ◁ (Y ◁ i.hom ≫ ε_ X' Y) =
    η_ X' Y ▷ X ⊗≫ (i.inv ▷ (Y ⊗ X) ≫ X ◁ Y ◁ i.hom) ⊗≫ X ◁ ε_ X' Y

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.FiniteDimensional.Lemmas

Goal state before:
pp: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
m : ℕ
⊢ LinearMap.ker (f ^ m) ≤ LinearMap.ker (f ^ finrank K V)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X X' Y : C
inst✝ : ExactPairing X' Y
i : X ≅ X'
⊢ η_ X' Y ▷ X ⊗≫ (i.inv ▷ (Y ⊗ X) ≫ X ◁ Y ◁ i.hom) ⊗≫ X ◁ ε_ X' Y =
    𝟙 (𝟙_ C ⊗ X) ⊗≫ (η_ X' Y ▷ X ≫ (X' ⊗ Y) ◁ i.hom) ⊗≫ (i.inv ▷ (Y ⊗ X') ≫ X ◁ ε_ X' Y) ⊗≫ 𝟙 (X ⊗ 𝟙_ C)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.FiniteDimensional.Lemmas

Goal state before:
pp: K : Type u
V : Type v
W : Type v'
inst✝⁶ : DivisionRing K
inst✝⁵ : AddCommGroup W
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
inst✝² : Module K W
f : V →ₗ[K] W
p : Submodule K W
inst✝¹ : FiniteDimensional K ↥p
inst✝ : FiniteDimensional K ↥f.ker
⊢ FiniteDimensional K ↥(comap f p)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.FiniteDimensional.Lemmas

Goal state before:
pp: K : Type u
V : Type v
W : Type v'
inst✝⁶ : DivisionRing K
inst✝⁵ : AddCommGroup W
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
inst✝² : Module K W
f : V →ₗ[K] W
p : Submodule K V
inst✝¹ : FiniteDimensional K (V ⧸ p)
inst✝ : FiniteDimensional K (W ⧸ f.range)
⊢ FiniteDimensional K (W ⧸ map f p)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X X' Y : C
inst✝ : ExactPairing X' Y
i : X ≅ X'
⊢ 𝟙 (𝟙_ C ⊗ X) ⊗≫ (η_ X' Y ▷ X ≫ (X' ⊗ Y) ◁ i.hom) ⊗≫ (i.inv ▷ (Y ⊗ X') ≫ X ◁ ε_ X' Y) ⊗≫ 𝟙 (X ⊗ 𝟙_ C) =
    𝟙 (𝟙_ C ⊗ X) ⊗≫ i.hom ⊗≫ (η_ X' Y ▷ X' ⊗≫ X' ◁ ε_ X' Y) ⊗≫ i.inv ⊗≫ 𝟙 (X ⊗ 𝟙_ C)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X X' Y : C
inst✝ : ExactPairing X' Y
i : X ≅ X'
⊢ 𝟙 (𝟙_ C ⊗ X) ⊗≫ i.hom ⊗≫ (η_ X' Y ▷ X' ⊗≫ X' ◁ ε_ X' Y) ⊗≫ i.inv ⊗≫ 𝟙 (X ⊗ 𝟙_ C) =
    𝟙 (𝟙_ C ⊗ X) ⊗≫ (i.hom ≫ i.inv) ⊗≫ 𝟙 (X ⊗ 𝟙_ C)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X X' Y : C
inst✝ : ExactPairing X' Y
i : X ≅ X'
⊢ 𝟙 (𝟙_ C ⊗ X) ⊗≫ (i.hom ≫ i.inv) ⊗≫ 𝟙 (X ⊗ 𝟙_ C) = (λ_ X).hom ≫ (ρ_ X).inv

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y Y' : C
inst✝ : ExactPairing X Y'
i : Y ≅ Y'
⊢ Y ◁ (η_ X Y' ≫ X ◁ i.inv) ≫ (α_ Y X Y).inv ≫ (i.hom ▷ X ≫ ε_ X Y') ▷ Y =
    Y ◁ η_ X Y' ⊗≫ (Y ◁ X ◁ i.inv ≫ i.hom ▷ (X ⊗ Y)) ⊗≫ ε_ X Y' ▷ Y

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y Y' : C
inst✝ : ExactPairing X Y'
i : Y ≅ Y'
⊢ Y ◁ η_ X Y' ⊗≫ (Y ◁ X ◁ i.inv ≫ i.hom ▷ (X ⊗ Y)) ⊗≫ ε_ X Y' ▷ Y =
    𝟙 (Y ⊗ 𝟙_ C) ⊗≫ (Y ◁ η_ X Y' ≫ i.hom ▷ (X ⊗ Y')) ⊗≫ ((Y' ⊗ X) ◁ i.inv ≫ ε_ X Y' ▷ Y) ⊗≫ 𝟙 (𝟙_ C ⊗ Y)

Tactic:
whisker_exchange

Goal state after:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y Y' : C
inst✝ : ExactPairing X Y'
i : Y ≅ Y'
⊢ Y ◁ η_ X Y' ⊗≫ (i.hom ▷ (X ⊗ Y') ≫ Y' ◁ X ◁ i.inv) ⊗≫ ε_ X Y' ▷ Y =
    𝟙 (Y ⊗ 𝟙_ C) ⊗≫ (Y ◁ η_ X Y' ≫ i.hom ▷ (X ⊗ Y')) ⊗≫ ((Y' ⊗ X) ◁ i.inv ≫ ε_ X Y' ▷ Y) ⊗≫ 𝟙 (𝟙_ C ⊗ Y)
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y Y' : C
inst✝ : ExactPairing X Y'
i : Y ≅ Y'
⊢ Y ◁ η_ X Y' ⊗≫ (Y ◁ X ◁ i.inv ≫ i.hom ▷ (X ⊗ Y)) ⊗≫ ε_ X Y' ▷ Y =
    𝟙 (Y ⊗ 𝟙_ C) ⊗≫ (Y ◁ η_ X Y' ≫ i.hom ▷ (X ⊗ Y')) ⊗≫ ((Y' ⊗ X) ◁ i.inv ≫ ε_ X Y' ▷ Y) ⊗≫ 𝟙 (𝟙_ C ⊗ Y)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y Y' : C
inst✝ : ExactPairing X Y'
i : Y ≅ Y'
⊢ 𝟙 (Y ⊗ 𝟙_ C) ⊗≫ (i.hom ▷ 𝟙_ C ≫ Y' ◁ η_ X Y') ⊗≫ ((Y' ⊗ X) ◁ i.inv ≫ ε_ X Y' ▷ Y) ⊗≫ 𝟙 (𝟙_ C ⊗ Y) =
    𝟙 (Y ⊗ 𝟙_ C) ⊗≫ i.hom ⊗≫ (Y' ◁ η_ X Y' ⊗≫ ε_ X Y' ▷ Y') ⊗≫ i.inv ⊗≫ 𝟙 (𝟙_ C ⊗ Y)

Tactic:
whisker_exchange

Goal state after:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y Y' : C
inst✝ : ExactPairing X Y'
i : Y ≅ Y'
⊢ 𝟙 (Y ⊗ 𝟙_ C) ⊗≫ (i.hom ▷ 𝟙_ C ≫ Y' ◁ η_ X Y') ⊗≫ (ε_ X Y' ▷ Y' ≫ 𝟙_ C ◁ i.inv) ⊗≫ 𝟙 (𝟙_ C ⊗ Y) =
    𝟙 (Y ⊗ 𝟙_ C) ⊗≫ i.hom ⊗≫ (Y' ◁ η_ X Y' ⊗≫ ε_ X Y' ▷ Y') ⊗≫ i.inv ⊗≫ 𝟙 (𝟙_ C ⊗ Y)
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y Y' : C
inst✝ : ExactPairing X Y'
i : Y ≅ Y'
⊢ 𝟙 (Y ⊗ 𝟙_ C) ⊗≫ (Y ◁ η_ X Y' ≫ i.hom ▷ (X ⊗ Y')) ⊗≫ ((Y' ⊗ X) ◁ i.inv ≫ ε_ X Y' ▷ Y) ⊗≫ 𝟙 (𝟙_ C ⊗ Y) =
    𝟙 (Y ⊗ 𝟙_ C) ⊗≫ i.hom ⊗≫ (Y' ◁ η_ X Y' ⊗≫ ε_ X Y' ▷ Y') ⊗≫ i.inv ⊗≫ 𝟙 (𝟙_ C ⊗ Y)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y Y' : C
inst✝ : ExactPairing X Y'
i : Y ≅ Y'
⊢ 𝟙 (Y ⊗ 𝟙_ C) ⊗≫ i.hom ⊗≫ (Y' ◁ η_ X Y' ⊗≫ ε_ X Y' ▷ Y') ⊗≫ i.inv ⊗≫ 𝟙 (𝟙_ C ⊗ Y) =
    𝟙 (Y ⊗ 𝟙_ C) ⊗≫ (i.hom ≫ i.inv) ⊗≫ 𝟙 (𝟙_ C ⊗ Y)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y Y' : C
inst✝ : ExactPairing X Y'
i : Y ≅ Y'
⊢ 𝟙 (Y ⊗ 𝟙_ C) ⊗≫ (i.hom ≫ i.inv) ⊗≫ 𝟙 (𝟙_ C ⊗ Y) = (ρ_ Y).hom ≫ (λ_ Y).inv

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y Y' : C
inst✝ : ExactPairing X Y'
i : Y ≅ Y'
⊢ (η_ X Y' ≫ X ◁ i.inv) ▷ X ≫ (α_ X Y X).hom ≫ X ◁ (i.hom ▷ X ≫ ε_ X Y') =
    η_ X Y' ▷ X ⊗≫ X ◁ (i.inv ≫ i.hom) ▷ X ≫ X ◁ ε_ X Y'

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y Y' : C
inst✝ : ExactPairing X Y'
i : Y ≅ Y'
⊢ η_ X Y' ▷ X ⊗≫ X ◁ (i.inv ≫ i.hom) ▷ X ≫ X ◁ ε_ X Y' = η_ X Y' ▷ X ⊗≫ X ◁ ε_ X Y'

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y Y' : C
inst✝ : ExactPairing X Y'
i : Y ≅ Y'
⊢ η_ X Y' ▷ X ⊗≫ X ◁ ε_ X Y' = (λ_ X).hom ≫ (ρ_ X).inv

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y Y' : C
inst✝ : ExactPairing X Y'
i : Y ≅ Y'
⊢ (η_ X Y' ≫ X ◁ i.inv) ▷ X ≫ (α_ X Y X).hom ≫ X ◁ (i.hom ▷ X ≫ ε_ X Y') = (λ_ X).hom ≫ (ρ_ X).inv

Tactic:
calc
      _ = η_ X Y' ▷ X ⊗≫ X ◁ (i.inv ≫ i.hom) ▷ X ≫ X ◁ ε_ X Y' := by
        monoidal
      _ = η_ X Y' ▷ X ⊗≫ X ◁ ε_ X Y' := by
        rw [Iso.inv_hom_id]; monoidal
      _ = _ := by
        rw [evaluation_coevaluation'']
        simp

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y Y' : C
inst✝ : ExactPairing X Y'
i : Y ≅ Y'
⊢ (η_ X Y' ≫ X ◁ i.inv) ▷ X ≫ (α_ X Y X).hom ≫ X ◁ (i.hom ▷ X ≫ ε_ X Y') = (λ_ X).hom ≫ (ρ_ X).inv

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
X Y₁ Y₂ : C
p₁ : ExactPairing X Y₁
p₂ : ExactPairing X Y₂
⊢ 𝟙 Xᘁ ≫ 𝟙 Xᘁ = 𝟙 Y₁

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
X Y₁ Y₂ : C
p₁ : ExactPairing X Y₁
p₂ : ExactPairing X Y₂
⊢ 𝟙 Xᘁ ≫ 𝟙 Xᘁ = 𝟙 Y₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
X₁ X₂ Y : C
p₁ : ExactPairing X₁ Y
p₂ : ExactPairing X₂ Y
⊢ ((ᘁ𝟙 Y) ≫ ᘁ𝟙 Y) = 𝟙 X₁

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
X₁ X₂ Y : C
p₁ : ExactPairing X₁ Y
p₂ : ExactPairing X₂ Y
⊢ ((ᘁ𝟙 Y) ≫ ᘁ𝟙 Y) = 𝟙 X₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
X Y : C
p : ExactPairing X Y
⊢ rightDualIso p p = Iso.refl Y

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
X Y : C
p : ExactPairing X Y
⊢ leftDualIso p p = Iso.refl X

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Coevaluation

Goal state before:
pp: K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
⊢ (coevaluation K V) 1 =
    let bV := Basis.ofVectorSpace K V;
    ∑ i, bV i ⊗ₜ[K] bV.coord i

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring R₁
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : Module R₁ M₁
inst✝⁴ : CommSemiring R₂
inst✝³ : AddCommMonoid M₂
inst✝² : Module R₂ M₂
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I₁ : R₁ →+* R
I₂ : R₂ →+* R
B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M
x : M₂
⊢ B.IsOrtho 0 x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
R₁ : Type u_2
M : Type u_5
M₁ : Type u_6
inst✝⁵ : CommSemiring R
inst✝⁴ : CommSemiring R₁
inst✝³ : AddCommMonoid M₁
inst✝² : Module R₁ M₁
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I₁ I₁' : R₁ →+* R
B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M
x y : M₁
⊢ B.IsOrtho x y ↔ B.flip.IsOrtho y x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Coevaluation

Goal state before:
pp: K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
⊢ LinearMap.rTensor (Dual K V) (contractLeft K V) ∘ₗ
      ↑(TensorProduct.assoc K (Dual K V) V (Dual K V)).symm ∘ₗ LinearMap.lTensor (Dual K V) (coevaluation K V) =
    ↑(TensorProduct.lid K (Dual K V)).symm ∘ₗ ↑(TensorProduct.rid K (Dual K V))

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
R₁ : Type u_2
M : Type u_5
M₁ : Type u_6
n : Type u_19
inst✝⁵ : CommSemiring R
inst✝⁴ : CommSemiring R₁
inst✝³ : AddCommMonoid M₁
inst✝² : Module R₁ M₁
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I₁ I₁' : R₁ →+* R
B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M
v : n → M₁
⊢ B.IsOrthoᵢ v ↔ B.flip.IsOrthoᵢ v

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Coevaluation

Goal state before:
pp: K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
⊢ LinearMap.lTensor V (contractLeft K V) ∘ₗ
      ↑(TensorProduct.assoc K V (Dual K V) V) ∘ₗ LinearMap.rTensor V (coevaluation K V) =
    ↑(TensorProduct.rid K V).symm ∘ₗ ↑(TensorProduct.lid K V)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.FreeModule.Finite.Matrix

Goal state before:
pp: R : Type u
S : Type u'
M : Type v
N : Type w
inst✝¹² : Ring R
inst✝¹¹ : Ring S
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : Free R M
inst✝⁷ : Module.Finite R M
inst✝⁶ : AddCommGroup N
inst✝⁵ : Module R N
inst✝⁴ : Module S N
inst✝³ : SMulCommClass R S N
inst✝² : StrongRankCondition R
inst✝¹ : StrongRankCondition S
inst✝ : Free S N
⊢ Module.rank S (M →ₗ[R] N) = lift.{w, v} (Module.rank R M) * lift.{v, w} (Module.rank S N)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.FreeModule.Finite.Matrix

Goal state before:
pp: R : Type u
S : Type u'
M : Type v
N : Type w
inst✝¹² : Ring R
inst✝¹¹ : Ring S
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : Free R M
inst✝⁷ : Module.Finite R M
inst✝⁶ : AddCommGroup N
inst✝⁵ : Module R N
inst✝⁴ : Module S N
inst✝³ : SMulCommClass R S N
inst✝² : StrongRankCondition R
inst✝¹ : StrongRankCondition S
inst✝ : Free S N
⊢ finrank S (M →ₗ[R] N) = finrank R M * finrank S N

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.FreeModule.Finite.Matrix

Goal state before:
pp: R : Type u
S : Type u'
M : Type v
inst✝⁹ : Ring R
inst✝⁸ : Ring S
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : Free R M
inst✝⁴ : Module.Finite R M
inst✝³ : StrongRankCondition R
inst✝² : StrongRankCondition S
inst✝¹ : Module R S
inst✝ : SMulCommClass R S S
⊢ Module.rank S (M →ₗ[R] S) = lift.{u', v} (Module.rank R M)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
inst✝⁸ : Field K
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module K V
inst✝⁵ : Field K₁
inst✝⁴ : AddCommGroup V₁
inst✝³ : Module K₁ V₁
inst✝² : Field K₂
inst✝¹ : AddCommGroup V₂
inst✝ : Module K₂ V₂
I₁ : K₁ →+* K
I₂ : K₂ →+* K
B : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] V
x : V₁
y : V₂
a : K₁
ha : a ≠ 0
⊢ B.IsOrtho x y ↔ B.IsOrtho (a • x) y

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : AddCommMonoid P
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R P
f : N →ₗ[R] P
⊢ dualTensorHom R M P ∘ₗ lTensor (Dual R M) f = compRight R f ∘ₗ dualTensorHom R M N

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
inst✝⁸ : Field K
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module K V
inst✝⁵ : Field K₁
inst✝⁴ : AddCommGroup V₁
inst✝³ : Module K₁ V₁
inst✝² : Field K₂
inst✝¹ : AddCommGroup V₂
inst✝ : Module K₂ V₂
I₁ : K₁ →+* K
I₂ : K₂ →+* K
B : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] V
x : V₁
y : V₂
a : K₂
ha : a ≠ 0
⊢ B.IsOrtho x y ↔ B.IsOrtho x (a • y)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.FreeModule.Finite.Matrix

Goal state before:
pp: R : Type u
S : Type u'
M : Type v
inst✝⁹ : Ring R
inst✝⁸ : Ring S
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : Free R M
inst✝⁴ : Module.Finite R M
inst✝³ : StrongRankCondition R
inst✝² : StrongRankCondition S
inst✝¹ : Module R S
inst✝ : SMulCommClass R S S
⊢ finrank S (M →ₗ[R] S) = finrank R M

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : AddCommMonoid P
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R P
f : M →ₗ[R] N
⊢ dualTensorHom R M P ∘ₗ rTensor P f.dualMap = lcomp R P f ∘ₗ dualTensorHom R N P

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.FreeModule.Finite.Matrix

Goal state before:
pp: K : Type u_1
M : Type u_2
L : Type v
inst✝⁷ : CommRing K
inst✝⁶ : Ring M
inst✝⁵ : Algebra K M
inst✝⁴ : Module.Free K M
inst✝³ : Module.Finite K M
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
⊢ #(M →ₐ[K] L) ≤ lift.{v, u_2} (Module.rank K M)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: R : Type u_2
M : Type u_3
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f : Dual R M
m : M
⊢ Dual.transpose ((dualTensorHom R M M) (f ⊗ₜ[R] m)) =
    (dualTensorHom R (Dual R M) (Dual R M)) ((Dual.eval R M) m ⊗ₜ[R] f)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.FreeModule.Finite.Matrix

Goal state before:
pp: case e'_4
K : Type u_1
M : Type u_2
L : Type v
inst✝⁷ : CommRing K
inst✝⁶ : Ring M
inst✝⁵ : Algebra K M
inst✝⁴ : Module.Free K M
inst✝³ : Module.Finite K M
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
⊢ finrank K M = toNat (lift.{v, u_2} (Module.rank K M))

Tactic:
toNat_lift,

Goal state after:
case e'_4
K : Type u_1
M : Type u_2
L : Type v
inst✝⁷ : CommRing K
inst✝⁶ : Ring M
inst✝⁵ : Algebra K M
inst✝⁴ : Module.Free K M
inst✝³ : Module.Finite K M
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
⊢ finrank K M = toNat (Module.rank K M)
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: K : Type u_13
K₁ : Type u_14
V : Type u_16
V₁ : Type u_17
n : Type u_19
inst✝⁵ : Field K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
inst✝² : Field K₁
inst✝¹ : AddCommGroup V₁
inst✝ : Module K₁ V₁
I₁ I₁' : K₁ →+* K
B : V₁ →ₛₗ[I₁] V₁ →ₛₗ[I₁'] V
v : n → V₁
hv₁ : B.IsOrthoᵢ v
hv₂ : ∀ (i : n), (B (v i)) (v i) ≠ 0
s : Finset n
w : n → K₁
hs : ∑ i ∈ s, w i • v i = 0
i : n
hi : i ∈ s
⊢ (B (∑ i ∈ s, w i • v i)) (v i) = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.FreeModule.Finite.Matrix

Goal state before:
pp: K : Type u_1
M : Type u_2
L : Type v
inst✝⁷ : CommRing K
inst✝⁶ : Ring M
inst✝⁵ : Algebra K M
inst✝⁴ : Module.Free K M
inst✝³ : Module.Finite K M
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
⊢ Nat.card (M →ₐ[K] L) ≤ finrank K M

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.FGModuleCat.Basic

Goal state before:
pp: R : Type u
inst✝ : Ring R
⊢ ModuleCat.isFG R (ModuleCat.of R PUnit.{v + 1})

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.FGModuleCat.Basic

Goal state before:
pp: R : Type u
inst✝⁶ : Ring R
V W : Type v
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : Module.Finite R V
inst✝² : AddCommGroup W
inst✝¹ : Module R W
inst✝ : Module.Finite R W
e : V ≃ₗ[R] W
⊢ ConcreteCategory.ofHom ↑e ≫ ConcreteCategory.ofHom ↑e.symm = 𝟙 (of R V)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.FGModuleCat.Basic

Goal state before:
pp: R : Type u
inst✝⁶ : Ring R
V W : Type v
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : Module.Finite R V
inst✝² : AddCommGroup W
inst✝¹ : Module R W
inst✝ : Module.Finite R W
e : V ≃ₗ[R] W
⊢ ConcreteCategory.ofHom ↑e.symm ≫ ConcreteCategory.ofHom ↑e = 𝟙 (of R W)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.FGModuleCat.Basic

Goal state before:
pp: K : Type u
inst✝ : Field K
V : FGModuleCat K
⊢ FGModuleCatDual K V ◁ FGModuleCatCoevaluation K V ≫
      (α_ (FGModuleCatDual K V) V (FGModuleCatDual K V)).inv ≫ FGModuleCatEvaluation K V ▷ FGModuleCatDual K V =
    (ρ_ (FGModuleCatDual K V)).hom ≫ (λ_ (FGModuleCatDual K V)).inv

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.FGModuleCat.Basic

Goal state before:
pp: K : Type u
inst✝ : Field K
V : FGModuleCat K
⊢ FGModuleCatCoevaluation K V ▷ V ≫ (α_ V (FGModuleCatDual K V) V).hom ≫ V ◁ FGModuleCatEvaluation K V =
    (λ_ V).hom ≫ (ρ_ V).inv

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Products

Goal state before:
pp: R : Type u
inst✝ : Ring R
ι : Type v
Z : ι → ModuleCat R
s : Cone (Discrete.functor Z)
m : s.pt ⟶ (productCone Z).pt
w : ∀ (j : Discrete ι), m ≫ (productCone Z).π.app j = s.π.app j
⊢ m = ↟(LinearMap.pi fun j ↦ Hom.hom (s.π.app { as := j }))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Products

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
ι : Type v
Z : ι → ModuleCat R
inst✝ : DecidableEq ι
⊢ ∀ (s : Cocone (Discrete.functor Z)) (j : Discrete ι),
    (coproductCocone Z).ι.app j ≫ ↟(toModule R ι ↑s.1 fun i ↦ Hom.hom (s.ι.app { as := i })) = s.ι.app j

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Products

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
ι : Type v
Z : ι → ModuleCat R
inst✝ : DecidableEq ι
⊢ ∀ (s : Cocone (Discrete.functor Z)) (m : (coproductCocone Z).pt ⟶ s.pt),
    (∀ (j : Discrete ι), (coproductCocone Z).ι.app j ≫ m = s.ι.app j) →
      m = ↟(toModule R ι ↑s.1 fun i ↦ Hom.hom (s.ι.app { as := i }))

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
Q : Type u_6
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid N
inst✝⁵ : AddCommMonoid P
inst✝⁴ : AddCommMonoid Q
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module R P
inst✝ : Module R Q
f : Dual R M
p : P
⊢ ((dualTensorHom R M P) (f ⊗ₜ[R] p)).prodMap 0 =
    (dualTensorHom R (M × N) (P × Q)) ((f ∘ₗ fst R M N) ⊗ₜ[R] (inl R P Q) p)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: K : Type u_13
K₁ : Type u_14
V : Type u_16
V₁ : Type u_17
n : Type u_19
inst✝⁵ : Field K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
inst✝² : Field K₁
inst✝¹ : AddCommGroup V₁
inst✝ : Module K₁ V₁
I₁ I₁' : K₁ →+* K
B : V₁ →ₛₗ[I₁] V₁ →ₛₗ[I₁'] V
v : n → V₁
hv₁ : B.IsOrthoᵢ v
hv₂ : ∀ (i : n), (B (v i)) (v i) ≠ 0
s : Finset n
w : n → K₁
hs : ∑ i ∈ s, w i • v i = 0
i : n
hi : i ∈ s
this : (B (∑ i ∈ s, w i • v i)) (v i) = 0
⊢ ∑ j ∈ s, I₁ (w j) • (B (v j)) (v i) = I₁ (w i) • (B (v i)) (v i)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
Q : Type u_6
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid N
inst✝⁵ : AddCommMonoid P
inst✝⁴ : AddCommMonoid Q
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module R P
inst✝ : Module R Q
g : Dual R N
q : Q
⊢ prodMap 0 ((dualTensorHom R N Q) (g ⊗ₜ[R] q)) =
    (dualTensorHom R (M × N) (P × Q)) ((g ∘ₗ snd R M N) ⊗ₜ[R] (inr R P Q) q)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
Q : Type u_6
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid N
inst✝⁵ : AddCommMonoid P
inst✝⁴ : AddCommMonoid Q
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module R P
inst✝ : Module R Q
f : Dual R M
p : P
g : Dual R N
q : Q
⊢ TensorProduct.map ((dualTensorHom R M P) (f ⊗ₜ[R] p)) ((dualTensorHom R N Q) (g ⊗ₜ[R] q)) =
    (dualTensorHom R (M ⊗[R] N) (P ⊗[R] Q)) ((dualDistrib R M N) (f ⊗ₜ[R] g) ⊗ₜ[R] (p ⊗ₜ[R] q))

Tactic:
by

Goal state after: