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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
M : Type u_1
inst✝² : Category.{v_1, u_1} M
inst✝¹ : MonoidalCategory M
F : M ⥤ C ⥤ C
n : M
X : C
inst✝ : F.Monoidal
⊢ (F.obj n).map ((ε F).app X) = (F.map (λ_ n).inv).app X ≫ (δ F (𝟙_ M) n).app X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
β : Type u_1
X Y : GradedObject β C
f g : X ⟶ Y
h : ∀ (x : β), f x = g x
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
β : Type u_1
X Y : GradedObject β C
f : X ⟶ Y
hf : ∀ (i : β), IsIso (f i)
⊢ IsIso f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
β : Type u_1
X Y : GradedObject β C
f : X ⟶ Y
inst✝ : IsIso f
i : β
⊢ IsIso (f i)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u_1
J : Type u_4
inst✝ : Category.{v_1, u_1} C
X Y : GradedObject J C
e : X ≅ Y
j : J
⊢ e.hom j ≫ e.inv j = 𝟙 (X j)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u_1
J : Type u_4
inst✝ : Category.{v_1, u_1} C
X Y : GradedObject J C
e : X ≅ Y
j : J
⊢ e.inv j ≫ e.hom j = 𝟙 (Y j)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.End

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
M : Type u_1
inst✝² : Category.{v_1, u_1} M
inst✝¹ : MonoidalCategory M
F : M ⥤ C ⥤ C
n : M
X : C
inst✝ : F.Monoidal
⊢ (F.obj n).map ((η F).app X) = (μ F (𝟙_ M) n).app X ≫ (F.map (λ_ n).hom).app X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u_1
D : Type u_2
J : Type u_4
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Category.{v_2, u_2} D
X Y : GradedObject J C
e : X ≅ Y
F : C ⥤ D
j : J
⊢ F.map (e.hom j) ≫ F.map (e.inv j) = 𝟙 (F.obj (X j))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.End

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
M : Type u_1
inst✝² : Category.{v_1, u_1} M
inst✝¹ : MonoidalCategory M
F : M ⥤ C ⥤ C
n : M
X : C
inst✝ : F.Monoidal
⊢ (ε F).app ((F.obj n).obj X) = (F.map (ρ_ n).inv).app X ≫ (δ F n (𝟙_ M)).app X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u_1
D : Type u_2
J : Type u_4
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Category.{v_2, u_2} D
X Y : GradedObject J C
e : X ≅ Y
F : C ⥤ D
j : J
⊢ F.map (e.inv j) ≫ F.map (e.hom j) = 𝟙 (F.obj (Y j))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : S
⊢ 1 ⊗ₜ[R] a - a ⊗ₜ[R] 1 ∈ ideal R S

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
J : Type u_4
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Category.{v_3, u_3} E
X Y✝ : GradedObject J C
e : X ≅ Y✝
F : C ⥤ D ⥤ E
j : J
Y : D
⊢ (F.map (e.hom j)).app Y ≫ (F.map (e.inv j)).app Y = 𝟙 ((F.obj (X j)).obj Y)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.End

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
M : Type u_1
inst✝² : Category.{v_1, u_1} M
inst✝¹ : MonoidalCategory M
F : M ⥤ C ⥤ C
n : M
X : C
inst✝ : F.Monoidal
⊢ (η F).app ((F.obj n).obj X) = (μ F n (𝟙_ M)).app X ≫ (F.map (ρ_ n).hom).app X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
J : Type u_4
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Category.{v_3, u_3} E
X Y✝ : GradedObject J C
e : X ≅ Y✝
F : C ⥤ D ⥤ E
j : J
Y : D
⊢ (F.map (e.inv j)).app Y ≫ (F.map (e.hom j)).app Y = 𝟙 ((F.obj (Y✝ j)).obj Y)

Tactic:
rw [← NatTrans.comp_app, ← F.map_comp, inv_hom_id_eval,
    Functor.map_id, NatTrans.id_app]

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
J : Type u_4
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Category.{v_3, u_3} E
X Y✝ : GradedObject J C
e : X ≅ Y✝
F : C ⥤ D ⥤ E
j : J
Y : D
⊢ (F.map (e.inv j)).app Y ≫ (F.map (e.hom j)).app Y = 𝟙 ((F.obj (Y✝ j)).obj Y)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
I : Type u_1
x x' : GradedObject I C
h : x = x'
i : I
⊢ eqToHom h i = eqToHom ⋯

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
β γ : Type w
f g : β → γ
h : f = g
X : GradedObject γ C
b : β
⊢ (comap C f).obj X b = (comap C g).obj X b

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
β γ : Type w
f g : β → γ
h : f = g
X : GradedObject γ C
b : β
⊢ (comap C g).obj X b = (comap C f).obj X b

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
β γ : Type w
f g : β → γ
h : f = g
⊢ comapEq C ⋯ = (comapEq C h).symm

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
β γ : Type w
f g h : β → γ
k : f = g
l : g = h
⊢ comapEq C ⋯ = comapEq C k ≪≫ comapEq C l

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
β : Type w
X Y : β → C
h : X = Y
b : β
⊢ X b = Y b

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.End

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
M : Type u_1
inst✝² : Category.{v_1, u_1} M
inst✝¹ : MonoidalCategory M
F : M ⥤ C ⥤ C
m₁ m₂ m₃ : M
X : C
inst✝ : F.LaxMonoidal
⊢ (F.obj m₃).map ((μ F m₁ m₂).app X) ≫ (μ F (m₁ ⊗ m₂) m₃).app X ≫ (F.map (α_ m₁ m₂ m₃).hom).app X =
    (μ F m₂ m₃).app ((F.obj m₁).obj X) ≫ (μ F m₁ (m₂ ⊗ m₃)).app X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
β : Type w
X Y : β → C
h : X = Y
b : β
⊢ eqToHom h b = eqToHom ⋯

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
β γ : Type w
e : β ≃ γ
⊢ (fun i ↦ i) = ⇑e.symm ∘ ⇑e

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
β γ : Type w
e : β ≃ γ
⊢ ⇑e ∘ ⇑e.symm = fun i ↦ i

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
β : Type u_1
inst✝ : AddCommGroup β
s : β
⊢ (fun b ↦ b + 0 • s) = id

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
β : Type u_1
inst✝ : AddCommGroup β
s : β
m n : ℤ
⊢ (fun b ↦ b + (m + n) • s) = (fun b ↦ b + m • s) ∘ fun b ↦ b + n • s

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
β : Type w
⊢ HasZeroObject (GradedObject β C)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.End

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
M : Type u_1
inst✝² : Category.{v_1, u_1} M
inst✝¹ : MonoidalCategory M
F : M ⥤ C ⥤ C
m₁ m₂ m₃ : M
X : C
inst✝ : F.Monoidal
⊢ (F.obj m₃).map ((μ F m₁ m₂).app X) =
    (μ F m₂ m₃).app ((F.obj m₁).obj X) ≫
      (μ F m₁ (m₂ ⊗ m₃)).app X ≫ (F.map (α_ m₁ m₂ m₃).inv).app X ≫ (δ F (m₁ ⊗ m₂) m₃).app X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: β : Type
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasCoproducts C
inst✝ : HasZeroMorphisms C
X Y : GradedObject β C
f g : X ⟶ Y
i : β
w :
  (Discrete.natTrans fun X ↦ (fun i ↦ f i) X.as).app { as := i } ≫
      colimit.ι (Discrete.functor fun i ↦ Y i) { as := i } =
    (Discrete.natTrans fun X ↦ (fun i ↦ g i) X.as).app { as := i } ≫
      colimit.ι (Discrete.functor fun i ↦ Y i) { as := i }
⊢ f i ≫ colimit.ι (Discrete.functor Y) { as := i } = g i ≫ colimit.ι (Discrete.functor Y) { as := i }

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: β : Type
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasCoproducts C
inst✝ : HasZeroMorphisms C
X Y : GradedObject β C
f g : X ⟶ Y
w : (total β C).map f = (total β C).map g
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: I : Type u_1
J : Type u_2
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
X Y : GradedObject I C
e : X ≅ Y
p : I → J
inst✝ : X.HasMap p
j : J
⊢ HasCoproduct (Y.mapObjFun p j)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: I : Type u_1
J : Type u_2
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
X : GradedObject I C
p : I → J
inst✝ : X.HasMap p
A : C
j : J
φ : (i : I) → p i = j → (X i ⟶ A)
i : I
hi : p i = j
⊢ X.ιMapObj p i j hi ≫ X.descMapObj p φ = φ i hi

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: I : Type u_1
J : Type u_2
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
X : GradedObject I C
p : I → J
j : J
inst✝ : X.HasMap p
c : X.CofanMapObjFun p j
hc : IsColimit c
i : I
hi : p i = j
⊢ Cofan.inj c ⟨i, hi⟩ ≫ (iso hc).hom = X.ιMapObj p i j hi

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: I : Type u_1
J : Type u_2
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
X : GradedObject I C
p : I → J
j : J
inst✝ : X.HasMap p
c : X.CofanMapObjFun p j
hc : IsColimit c
i : I
hi : p i = j
⊢ X.ιMapObj p i j hi ≫ (iso hc).inv = Cofan.inj c ⟨i, hi⟩

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: I : Type u_1
J : Type u_2
C : Type u_4
inst✝² : Category.{v_1, u_4} C
X Y : GradedObject I C
φ : X ⟶ Y
p : I → J
inst✝¹ : X.HasMap p
inst✝ : Y.HasMap p
i : I
j : J
hij : p i = j
⊢ X.ιMapObj p i j hij ≫ mapMap φ p j = φ i ≫ Y.ιMapObj p i j hij

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: I : Type u_1
J : Type u_2
C : Type u_4
inst✝² : Category.{v_1, u_4} C
X Y : GradedObject I C
p : I → J
inst✝¹ : X.HasMap p
inst✝ : Y.HasMap p
φ₁ φ₂ : X ⟶ Y
h : φ₁ = φ₂
⊢ mapMap φ₁ p = mapMap φ₂ p

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: I : Type u_1
J : Type u_2
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
X : GradedObject I C
p : I → J
inst✝ : X.HasMap p
⊢ mapMap (𝟙 X) p = 𝟙 (X.mapObj p)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: I : Type u_1
J : Type u_2
C : Type u_4
inst✝³ : Category.{v_1, u_4} C
X Y Z : GradedObject I C
φ : X ⟶ Y
ψ : Y ⟶ Z
p : I → J
inst✝² : X.HasMap p
inst✝¹ : Y.HasMap p
inst✝ : Z.HasMap p
⊢ mapMap (φ ≫ ψ) p = mapMap φ p ≫ mapMap ψ p

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: I : Type u_1
J : Type u_2
K : Type u_3
C : Type u_4
inst✝² : Category.{v_1, u_4} C
X Y Z : GradedObject I C
φ : X ⟶ Y
e : X ≅ Y
ψ : Y ⟶ Z
p : I → J
j : J
inst✝¹ : X.HasMap p
inst✝ : Y.HasMap p
q : J → K
r : I → K
hpqr : ∀ (i : I), q (p i) = r i
k : K
c : (j : J) → q j = k → X.CofanMapObjFun p j
hc : (j : J) → (hj : q j = k) → IsColimit (c j hj)
c' : Cofan fun j ↦ (c ↑j ⋯).pt
hc' : IsColimit c'
i : I
hi : r i = k
⊢ q (p i) = k

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.End

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
M : Type u_1
inst✝² : Category.{v_1, u_1} M
inst✝¹ : MonoidalCategory M
F : M ⥤ C ⥤ C
m₁ m₂ m₃ : M
X : C
inst✝ : F.Monoidal
⊢ (F.obj m₃).map ((δ F m₁ m₂).app X) =
    (μ F (m₁ ⊗ m₂) m₃).app X ≫
      (F.map (α_ m₁ m₂ m₃).hom).app X ≫ (δ F m₁ (m₂ ⊗ m₃)).app X ≫ (δ F m₂ m₃).app ((F.obj m₁).obj X)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: I : Type u_1
J : Type u_2
K : Type u_3
C : Type u_4
inst✝² : Category.{v_1, u_4} C
X Y Z : GradedObject I C
φ : X ⟶ Y
e : X ≅ Y
ψ : Y ⟶ Z
p : I → J
j : J
inst✝¹ : X.HasMap p
inst✝ : Y.HasMap p
q : J → K
r : I → K
hpqr : ∀ (i : I), q (p i) = r i
k : K
c : (j : J) → q j = k → X.CofanMapObjFun p j
hc : (j : J) → (hj : q j = k) → IsColimit (c j hj)
c' : Cofan fun j ↦ (c ↑j ⋯).pt
hc' : IsColimit c'
i : I
hi : r i = k
⊢ p i ∈ q ⁻¹' {k}

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: I : Type u_1
J : Type u_2
K : Type u_3
C : Type u_4
inst✝² : Category.{v_1, u_4} C
X Y Z : GradedObject I C
φ : X ⟶ Y
e : X ≅ Y
ψ : Y ⟶ Z
p : I → J
j✝ : J
inst✝¹ : X.HasMap p
inst✝ : Y.HasMap p
q : J → K
r : I → K
hpqr : ∀ (i : I), q (p i) = r i
k : K
c : (j : J) → q j = k → X.CofanMapObjFun p j
hc : (j : J) → (hj : q j = k) → IsColimit (c j hj)
c' : Cofan fun j ↦ (c ↑j ⋯).pt
hc' : IsColimit c'
s : Cofan (X.mapObjFun r k)
x✝¹ : ↑(q ⁻¹' {k})
j : J
hj : q j = k
x✝ : ↑(p ⁻¹' {j})
i : I
hi : p i = j
⊢ i ∈ r ⁻¹' {k}

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: I : Type u_1
J : Type u_2
K : Type u_3
C : Type u_4
inst✝² : Category.{v_1, u_4} C
X Y Z : GradedObject I C
φ : X ⟶ Y
e : X ≅ Y
ψ : Y ⟶ Z
p : I → J
j : J
inst✝¹ : X.HasMap p
inst✝ : Y.HasMap p
q : J → K
r : I → K
hpqr : ∀ (i : I), q (p i) = r i
k : K
c : (j : J) → q j = k → X.CofanMapObjFun p j
hc : (j : J) → (hj : q j = k) → IsColimit (c j hj)
c' : Cofan fun j ↦ (c ↑j ⋯).pt
hc' : IsColimit c'
s : Cofan (X.mapObjFun r k)
x✝ : ↑(r ⁻¹' {k})
i : I
hi : r i = k
⊢ (Cofan.inj (X.cofanMapObjComp p q r hpqr k c c') ⟨i, hi⟩ ≫
      Cofan.IsColimit.desc hc' fun x ↦
        match x with
        | ⟨j, hj⟩ =>
          Cofan.IsColimit.desc (hc j hj) fun x ↦
            match x with
            | ⟨i, hi⟩ => s.inj ⟨i, ⋯⟩) =
    s.inj ⟨i, hi⟩

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.End

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
M : Type u_1
inst✝² : Category.{v_1, u_1} M
inst✝¹ : MonoidalCategory M
F : M ⥤ C ⥤ C
m : M
X Y : C
f : X ⟶ (F.obj m).obj Y
inst✝ : F.Monoidal
⊢ (F.obj (𝟙_ M)).map f ≫ (μ F m (𝟙_ M)).app Y = (η F).app X ≫ f ≫ (F.map (ρ_ m).inv).app Y

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.End

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
M : Type u_1
inst✝² : Category.{v_1, u_1} M
inst✝¹ : MonoidalCategory M
F : M ⥤ C ⥤ C
m₁ m₂ : M
X : C
inst✝ : F.Monoidal
⊢ (μ F (𝟙_ M) m₂).app ((F.obj m₁).obj X) ≫
      (μ F m₁ (𝟙_ M ⊗ m₂)).app X ≫ (F.map (α_ m₁ (𝟙_ M) m₂).inv).app X ≫ (δ F (m₁ ⊗ 𝟙_ M) m₂).app X =
    (μ F (𝟙_ M) m₂).app ((F.obj m₁).obj X) ≫
      (F.map (λ_ m₂).hom).app ((F.obj m₁).obj X) ≫ (F.obj m₂).map ((F.map (ρ_ m₁).inv).app X)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.End

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
M : Type u_1
inst✝² : Category.{v_1, u_1} M
inst✝¹ : MonoidalCategory M
F : M ⥤ C ⥤ C
m n : M
h₁ : m ⊗ n ≅ 𝟙_ M
h₂ : n ⊗ m ≅ 𝟙_ M
H : h₁.hom ▷ m ≫ (λ_ m).hom = (α_ m n m).hom ≫ m ◁ h₂.hom ≫ (ρ_ m).hom
inst✝ : F.Monoidal
X : C
⊢ (F.obj m).map ((unitOfTensorIsoUnit F m n h₁).symm.hom.app X) ≫
      (unitOfTensorIsoUnit F n m h₂).hom.app ((F.obj m).obj X) =
    𝟙 ((F.obj m).obj X)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: I : Type u_1
J : Type u_2
K : Type u_3
C : Type u_4
inst✝² : Category.{v_1, u_4} C
X Y Z : GradedObject I C
φ : X ⟶ Y
e : X ≅ Y
ψ : Y ⟶ Z
p : I → J
j : J
inst✝¹ : X.HasMap p
inst✝ : Y.HasMap p
q : J → K
r : I → K
hpqr : ∀ (i : I), q (p i) = r i
k : K
c : (j : J) → q j = k → X.CofanMapObjFun p j
hc : (j : J) → (hj : q j = k) → IsColimit (c j hj)
c' : Cofan fun j ↦ (c ↑j ⋯).pt
hc' : IsColimit c'
s : Cofan (X.mapObjFun r k)
m : (X.cofanMapObjComp p q r hpqr k c c').pt ⟶ s.pt
hm : ∀ (j : ↑(r ⁻¹' {k})), Cofan.inj (X.cofanMapObjComp p q r hpqr k c c') j ≫ m = s.inj j
⊢ m =
    Cofan.IsColimit.desc hc' fun x ↦
      match x with
      | ⟨j, hj⟩ =>
        Cofan.IsColimit.desc (hc j hj) fun x ↦
          match x with
          | ⟨i, hi⟩ => s.inj ⟨i, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: I : Type u_1
J : Type u_2
C : Type u_4
inst✝⁴ : Category.{v_1, u_4} C
X Y : GradedObject I C
φ : X ⟶ Y
p : I → J
inst✝³ : X.HasMap p
inst✝² : Y.HasMap p
inst✝¹ : HasZeroMorphisms C
inst✝ : DecidableEq J
i : I
j : J
⊢ X.ιMapObjOrZero p i j ≫ mapMap φ p j = φ i ≫ Y.ιMapObjOrZero p i j

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Single

Goal state before:
pp: V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
inst✝¹ : HasZeroObject V
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
X✝ Y✝ : V
f : X✝ ⟶ Y✝
i : ι
h : i = j
⊢ { X := fun i ↦ if i = j then X✝ else 0, d := fun x x_1 ↦ 0, shape := ⋯, d_comp_d' := ⋯ }.X i = X✝

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Single

Goal state before:
pp: V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
inst✝¹ : HasZeroObject V
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
X✝ Y✝ : V
f : X✝ ⟶ Y✝
i : ι
h : i = j
⊢ Y✝ = { X := fun i ↦ if i = j then Y✝ else 0, d := fun x x_1 ↦ 0, shape := ⋯, d_comp_d' := ⋯ }.X i

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Single

Goal state before:
pp: V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
inst✝¹ : HasZeroObject V
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : V
⊢ { f := fun i ↦ if h : i = j then eqToHom ⋯ ≫ 𝟙 A ≫ eqToHom ⋯ else 0, comm' := ⋯ } =
    𝟙 { X := fun i ↦ if i = j then A else 0, d := fun x x_1 ↦ 0, shape := ⋯, d_comp_d' := ⋯ }

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Shift.Basic

Goal state before:
pp: C : Type u
A : Type u_1
inst✝¹ : Category.{v, u} C
inst✝ : AddMonoid A
F : A → C ⥤ C
zero : F 0 ≅ 𝟭 C
add : (n m : A) → F (n + m) ≅ F n ⋙ F m
m₁ m₂ m₃ : A
X : C
⊢ (F (m₁ + m₂ + m₃)).obj X = (F (m₁ + (m₂ + m₃))).obj X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Shift.Basic

Goal state before:
pp: C : Type u
A : Type u_1
inst✝¹ : Category.{v, u} C
inst✝ : AddMonoid A
F : A → C ⥤ C
zero : F 0 ≅ 𝟭 C
add : (n m : A) → F (n + m) ≅ F n ⋙ F m
assoc_hom_app :
  ∀ (m₁ m₂ m₃ : A) (X : C),
    (add (m₁ + m₂) m₃).hom.app X ≫ (F m₃).map ((add m₁ m₂).hom.app X) =
      eqToHom ⋯ ≫ (add m₁ (m₂ + m₃)).hom.app X ≫ (add m₂ m₃).hom.app ((F m₁).obj X)
n : A
X : C
⊢ (F (0 + n)).obj X = (F n).obj ((𝟭 C).obj X)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Single

Goal state before:
pp: V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
inst✝¹ : HasZeroObject V
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
X✝ Y✝ Z✝ : V
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ { f := fun i ↦ if h : i = j then eqToHom ⋯ ≫ (f ≫ g) ≫ eqToHom ⋯ else 0, comm' := ⋯ } =
    { f := fun i ↦ if h : i = j then eqToHom ⋯ ≫ f ≫ eqToHom ⋯ else 0, comm' := ⋯ } ≫
      { f := fun i ↦ if h : i = j then eqToHom ⋯ ≫ g ≫ eqToHom ⋯ else 0, comm' := ⋯ }

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Shift.Basic

Goal state before:
pp: C : Type u
A : Type u_1
inst✝¹ : Category.{v, u} C
inst✝ : AddMonoid A
F : A → C ⥤ C
zero : F 0 ≅ 𝟭 C
add : (n m : A) → F (n + m) ≅ F n ⋙ F m
assoc_hom_app :
  ∀ (m₁ m₂ m₃ : A) (X : C),
    (add (m₁ + m₂) m₃).hom.app X ≫ (F m₃).map ((add m₁ m₂).hom.app X) =
      eqToHom ⋯ ≫ (add m₁ (m₂ + m₃)).hom.app X ≫ (add m₂ m₃).hom.app ((F m₁).obj X)
zero_add_hom_app : ∀ (n : A) (X : C), (add 0 n).hom.app X = eqToHom ⋯ ≫ (F n).map (zero.inv.app X)
n : A
X : C
⊢ (F (n + 0)).obj X = (𝟭 C).obj ((F n).obj X)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Shift.Basic

Goal state before:
pp: C : Type u
A : Type u_1
inst✝¹ : Category.{v, u} C
inst✝ : AddMonoid A
h : ShiftMkCore C A
m₁ m₂ m₃ : A
X : C
⊢ (h.F (m₁ + (m₂ + m₃))).obj X = (h.F (m₁ + m₂ + m₃)).obj X

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Single

Goal state before:
pp: V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
inst✝¹ : HasZeroObject V
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : V
i : ι
hi : i ≠ j
⊢ IsZero (((single V c j).obj A).X i)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Single

Goal state before:
pp: V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
inst✝¹ : HasZeroObject V
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : V
i : ι
hi : i = j
⊢ ((single V c j).obj A).X i = A

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Single

Goal state before:
pp: V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
inst✝¹ : HasZeroObject V
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A B : V
f : A ⟶ B
⊢ ((single V c j).map f).f j = (singleObjXSelf c j A).hom ≫ f ≫ (singleObjXSelf c j B).inv

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Single

Goal state before:
pp: V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
inst✝¹ : HasZeroObject V
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A B : V
f : A ⟶ B
⊢ (single V c j ⋙ eval V c j).map f ≫ (singleObjXSelf c j B).hom = (singleObjXSelf c j A).hom ≫ (𝟭 V).map f

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomologicalComplex

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C : HomologicalComplex V c
i j k : ι
⊢ C.d i j ≫ C.d j k = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Single

Goal state before:
pp: V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
inst✝¹ : HasZeroObject V
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
K : HomologicalComplex V c
j : ι
A : V
f g : (single V c j).obj A ⟶ K
hfg : f.f j = g.f j
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Shift.Basic

Goal state before:
pp: C : Type u
A : Type u_1
inst✝¹ : Category.{v, u} C
inst✝ : AddMonoid A
h : ShiftMkCore C A
m₁ m₂ m₃ : A
X : C
⊢ (h.F m₃).map ((h.add m₁ m₂).inv.app X) ≫ (h.add (m₁ + m₂) m₃).inv.app X =
    (h.add m₂ m₃).inv.app ((h.F m₁).obj X) ≫ (h.add m₁ (m₂ + m₃)).inv.app X ≫ eqToHom ⋯

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
D : Derivation R S M
x y : S ⊗[R] S
⊢ D.tensorProductTo (x * y) =
    (TensorProduct.lmul' R) x • D.tensorProductTo y + (TensorProduct.lmul' R) y • D.tensorProductTo x

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Shift.Basic

Goal state before:
pp: C : Type u
A : Type u_1
inst✝¹ : Category.{v, u} C
inst✝ : AddMonoid A
h : ShiftMkCore C A
n : A
X : C
⊢ (h.F n).obj ((𝟭 C).obj X) = (h.F (0 + n)).obj X

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Single

Goal state before:
pp: V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
inst✝¹ : HasZeroObject V
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
K : HomologicalComplex V c
j : ι
A : V
f g : K ⟶ (single V c j).obj A
hfg : f.f j = g.f j
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S ⊗[R] S
hx : (TensorProduct.lmul' R) x = 0
⊢ x - (TensorProduct.lmul' R) x ⊗ₜ[R] 1 = x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Single

Goal state before:
pp: V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
inst✝¹ : HasZeroObject V
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A B : V
f g : A ⟶ B
w : (single V c j).map f = (single V c j).map g
⊢ f = g

Tactic:
by

Goal state after: