mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Limits.Shapes.Kernels

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasCokernels C
f g : Arrow C
u : f ⟶ g
⊢ f.hom ≫ Arrow.Hom.right u ≫ cokernel.π g.hom = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Kernels

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasCokernels C
⊢ Arrow.leftToRight ≫ π C = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FunctorCategory

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
n : ℕ
α : F ⟶ G
X Y : C
f : X ⟶ Y
⊢ F.map f ≫ (n • α.app) Y = (n • α.app) X ≫ G.map f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FunctorCategory

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
n : ℤ
α : F ⟶ G
X Y : C
f : X ⟶ Y
⊢ F.map f ≫ (n • α.app) Y = (n • α.app) X ≫ G.map f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.FunctorCategory

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
ι : Type u_3
s : Finset ι
X : C
α : ι → (F ⟶ G)
⊢ (∑ i ∈ s, α i).app X = ∑ i ∈ s, (α i).app X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.AdditiveFunctor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F : C ⥤ D
inst✝ : F.Additive
G : C ⥤ D
e : F ≅ G
⊢ G.Additive

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.AdditiveFunctor

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Category.{v_3, u_3} E
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
inst✝⁴ : Preadditive E
F : C ⥤ D
inst✝³ : F.Additive
inst✝² : F.Full
inst✝¹ : F.EssSurj
G : D ⥤ E
inst✝ : (F ⋙ G).Additive
X Y : D
f g : X ⟶ Y
⊢ G.map (f + g) = G.map f + G.map g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
g : K → C
e : J ≃ K
w : (j : J) → g (e j) ≅ f j
inst✝¹ : HasBiproduct f
inst✝ : HasBiproduct g
⊢ (whiskerEquiv e w).hom = lift fun k ↦ π f (e.symm k) ≫ (w (e.symm k)).inv ≫ eqToHom ⋯

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
J : Type u_1
inst✝ : Fintype J
f : J → C
b : Bicone f
total : ∑ j, b.π j ≫ b.ι j = 𝟙 b.pt
s : Cone (Discrete.functor f)
j : Discrete J
⊢ (∑ j, s.π.app { as := j } ≫ b.ι j) ≫ b.toCone.π.app j = s.π.app j

Tactic:
classical
        cases j
        simp only [sum_comp, Category.assoc, Bicone.toCone_π_app, b.ι_π, comp_dite]
        simp

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.AdditiveFunctor

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} E
inst✝⁵ : Preadditive C
inst✝⁴ : Preadditive D
inst✝³ : Preadditive E
F : C ⥤ D
G : D ⥤ E
inst✝² : G.Additive
inst✝¹ : (F ⋙ G).Additive
inst✝ : G.Faithful
x✝¹ x✝ : C
f₁ f₂ : x✝¹ ⟶ x✝
⊢ G.map (F.map (f₁ + f₂)) = G.map (F.map f₁ + F.map f₂)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
J : Type u_1
inst✝ : Fintype J
f : J → C
b : Bicone f
total : ∑ j, b.π j ≫ b.ι j = 𝟙 b.pt
s : Cone (Discrete.functor f)
j : Discrete J
⊢ (∑ j, s.π.app { as := j } ≫ b.ι j) ≫ b.toCone.π.app j = s.π.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.AdditiveFunctor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Preadditive C
inst✝² : Preadditive D
F : C ⥤ D
inst✝¹ : F.Additive
inst✝ : HasZeroObject C
⊢ IsZero (F.obj 0)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
g : K → C
e : J ≃ K
w : (j : J) → g (e j) ≅ f j
inst✝¹ : HasBiproduct f
inst✝ : HasBiproduct g
⊢ (whiskerEquiv e w).inv = lift fun j ↦ π g (e j) ≫ (w j).hom

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.AdditiveFunctor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
inst✝¹ : Preadditive C
inst✝ : Preadditive D
F : C ⥤ D
hF : IsZero F
x✝³ x✝² : C
x✝¹ x✝ : x✝³ ⟶ x✝²
⊢ IsZero (F.obj x✝²)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.AdditiveFunctor

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝¹⁰ : Category.{v_1, u_1} C
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Category.{v_3, u_3} E
inst✝⁷ : Preadditive C
inst✝⁶ : Preadditive D
inst✝⁵ : Preadditive E
F✝ : C ⥤ D
inst✝⁴ : F✝.Additive
E' : Type u_4
inst✝³ : Category.{v_4, u_4} E'
inst✝² : Preadditive E'
G : C ⥤ D ⥤ E
F : E ⥤ E'
inst✝¹ : F.Additive
inst✝ : G.Additive
⊢ ((postcompose₂.obj F).obj G).Additive

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J✝ : Type w
C✝ : Type uC
inst✝⁷ : Category.{uC', uC} C✝
inst✝⁶ : HasZeroMorphisms C✝
D : Type uD
inst✝⁵ : Category.{uD', uD} D
inst✝⁴ : HasZeroMorphisms D
F : J✝ → C✝
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
ι : Type u_3
f : ι → Type u_2
g : (i : ι) → f i → C
inst✝¹ : ∀ (i : ι), HasBiproduct (g i)
inst✝ : HasBiproduct fun i ↦ ⨁ g i
x✝¹ x✝ : (i : ι) × f i
j : ι
x : f j
j' : ι
y : f j'
⊢ (biproduct.ι (g ⟨j, x⟩.fst) ⟨j, x⟩.snd ≫ biproduct.ι (fun i ↦ ⨁ g i) ⟨j, x⟩.fst) ≫
      biproduct.π (fun i ↦ ⨁ g i) ⟨j', y⟩.fst ≫ biproduct.π (g ⟨j', y⟩.fst) ⟨j', y⟩.snd =
    if h : ⟨j, x⟩ = ⟨j', y⟩ then eqToHom ⋯ else 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.AdditiveFunctor

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Category.{v₂, u₂} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F : C ⥤ D
inst✝ : F.Additive
J : Type
x✝ : Finite J
val✝ : Fintype J
f✝ : J → C
b✝ : Bicone f✝
hb : b✝.IsBilimit
⊢ ∑ j, (F.mapBicone b✝).π j ≫ (F.mapBicone b✝).ι j = 𝟙 (F.mapBicone b✝).pt

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.AdditiveFunctor

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : Preadditive C
inst✝³ : Preadditive D
F : C ⥤ D
inst✝² : HasBinaryBiproducts C
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : PreservesBinaryBiproducts F
X Y : C
f g : X ⟶ Y
⊢ F.map (f + g) = F.map f + F.map g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.AdditiveFunctor

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : Preadditive C
inst✝³ : Preadditive D
F : C ⥤ D
inst✝² : HasBinaryProducts C
inst✝¹ : PreservesLimitsOfShape (Discrete WalkingPair) F
inst✝ : F.PreservesZeroMorphisms
⊢ F.Additive

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.AdditiveFunctor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
e : C ≌ D
inst✝ : e.functor.Additive
f g : D
f✝ g✝ : f ⟶ g
⊢ e.functor.map (e.inverse.map (f✝ + g✝)) = e.functor.map (e.inverse.map f✝ + e.inverse.map g✝)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.AdditiveFunctor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F : C ⥤ D
inst✝ : F.Additive
⊢ additiveFunctor C D F

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.AdditiveFunctor

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : Preadditive C
inst✝³ : Preadditive D
inst✝² : HasZeroObject C
inst✝¹ : HasZeroObject D
inst✝ : HasBinaryBiproducts C
F : C ⥤ D
h : leftExactFunctor C D F
⊢ additiveFunctor C D F

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J : Type w
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
f : J → C
inst✝² : HasBiproduct f
p : J → Prop
inst✝¹ : HasBiproduct (Subtype.restrict p f)
inst✝ : DecidablePred p
j : J
⊢ fromSubtype f p ≫ π f j = if h : p j then π (Subtype.restrict p f) ⟨j, h⟩ else 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.AdditiveFunctor

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : Preadditive C
inst✝³ : Preadditive D
inst✝² : HasZeroObject C
inst✝¹ : HasZeroObject D
inst✝ : HasBinaryBiproducts C
F : C ⥤ D
h : rightExactFunctor C D F
⊢ additiveFunctor C D F

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J : Type w
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
f : J → C
inst✝² : HasBiproduct f
p : J → Prop
inst✝¹ : HasBiproduct (Subtype.restrict p f)
inst✝ : DecidablePred p
⊢ ∀ (j : J), fromSubtype f p ≫ π f j = (lift fun j ↦ if h : p j then π (Subtype.restrict p f) ⟨j, h⟩ else 0) ≫ π f j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
J : Type u_1
inst✝ : Fintype J
f : J → C
b : Bicone f
total : ∑ j, b.π j ≫ b.ι j = 𝟙 b.pt
s : Cone (Discrete.functor f)
m : s.pt ⟶ b.toCone.pt
h : ∀ (j : Discrete J), m ≫ b.toCone.π.app j = s.π.app j
j : J
a✝ : j ∈ Finset.univ
⊢ m ≫ b.π j ≫ b.ι j = s.π.app { as := j } ≫ b.ι j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: C : Type uC
inst✝¹ : Category.{uC', uC} C
inst✝ : HasZeroMorphisms C
P Q : C
c c' : BinaryBicone P Q
f g : c ⟶ c'
w : f.hom = g.hom
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J : Type w
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
j : Subtype p
⊢ fromSubtype f p ≫ π f ↑j = π (Subtype.restrict p f) j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝⁴ : Category.{uC', uC} C
inst✝³ : HasZeroMorphisms C
D : Type uD
inst✝² : Category.{uD', uD} D
inst✝¹ : HasZeroMorphisms D
P Q : C
F : C ⥤ D
inst✝ : F.PreservesZeroMorphisms
A : BinaryBicone P Q
⊢ F.map A.inl ≫ F.map A.fst = 𝟙 (F.obj P)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
J : Type u_1
inst✝ : Fintype J
f : J → C
b : Bicone f
total : ∑ j, b.π j ≫ b.ι j = 𝟙 b.pt
s : Cone (Discrete.functor f)
m : s.pt ⟶ b.toCone.pt
h : ∀ (j : Discrete J), m ≫ b.toCone.π.app j = s.π.app j
⊢ m = ∑ j, s.π.app { as := j } ≫ b.ι j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝⁴ : Category.{uC', uC} C
inst✝³ : HasZeroMorphisms C
D : Type uD
inst✝² : Category.{uD', uD} D
inst✝¹ : HasZeroMorphisms D
P Q : C
F : C ⥤ D
inst✝ : F.PreservesZeroMorphisms
A : BinaryBicone P Q
⊢ F.map A.inl ≫ F.map A.snd = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝⁴ : Category.{uC', uC} C
inst✝³ : HasZeroMorphisms C
D : Type uD
inst✝² : Category.{uD', uD} D
inst✝¹ : HasZeroMorphisms D
P Q : C
F : C ⥤ D
inst✝ : F.PreservesZeroMorphisms
A : BinaryBicone P Q
⊢ F.map A.inr ≫ F.map A.fst = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
J : Type u_1
inst✝ : Fintype J
f : J → C
b : Bicone f
total : ∑ j, b.π j ≫ b.ι j = 𝟙 b.pt
s : Cocone (Discrete.functor f)
j : Discrete J
⊢ b.toCocone.ι.app j ≫ ∑ j, b.π j ≫ s.ι.app { as := j } = s.ι.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝⁴ : Category.{uC', uC} C
inst✝³ : HasZeroMorphisms C
D : Type uD
inst✝² : Category.{uD', uD} D
inst✝¹ : HasZeroMorphisms D
P Q : C
F : C ⥤ D
inst✝ : F.PreservesZeroMorphisms
A : BinaryBicone P Q
⊢ F.map A.inr ≫ F.map A.snd = 𝟙 (F.obj Q)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝⁴ : Category.{uC', uC} C
inst✝³ : HasZeroMorphisms C
D : Type uD
inst✝² : Category.{uD', uD} D
inst✝¹ : HasZeroMorphisms D
P Q : C
F : C ⥤ D
inst✝ : F.PreservesZeroMorphisms
X✝ Y✝ : BinaryBicone P Q
f : X✝ ⟶ Y✝
⊢ F.map f.hom ≫
      { pt := F.obj Y✝.pt, fst := F.map Y✝.fst, snd := F.map Y✝.snd, inl := F.map Y✝.inl, inr := F.map Y✝.inr,
          inl_fst := ⋯, inl_snd := ⋯, inr_fst := ⋯, inr_snd := ⋯ }.fst =
    { pt := F.obj X✝.pt, fst := F.map X✝.fst, snd := F.map X✝.snd, inl := F.map X✝.inl, inr := F.map X✝.inr,
        inl_fst := ⋯, inl_snd := ⋯, inr_fst := ⋯, inr_snd := ⋯ }.fst

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝⁴ : Category.{uC', uC} C
inst✝³ : HasZeroMorphisms C
D : Type uD
inst✝² : Category.{uD', uD} D
inst✝¹ : HasZeroMorphisms D
P Q : C
F : C ⥤ D
inst✝ : F.PreservesZeroMorphisms
X✝ Y✝ : BinaryBicone P Q
f : X✝ ⟶ Y✝
⊢ F.map f.hom ≫
      { pt := F.obj Y✝.pt, fst := F.map Y✝.fst, snd := F.map Y✝.snd, inl := F.map Y✝.inl, inr := F.map Y✝.inr,
          inl_fst := ⋯, inl_snd := ⋯, inr_fst := ⋯, inr_snd := ⋯ }.snd =
    { pt := F.obj X✝.pt, fst := F.map X✝.fst, snd := F.map X✝.snd, inl := F.map X✝.inl, inr := F.map X✝.inr,
        inl_fst := ⋯, inl_snd := ⋯, inr_fst := ⋯, inr_snd := ⋯ }.snd

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝⁴ : Category.{uC', uC} C
inst✝³ : HasZeroMorphisms C
D : Type uD
inst✝² : Category.{uD', uD} D
inst✝¹ : HasZeroMorphisms D
P Q : C
F : C ⥤ D
inst✝ : F.PreservesZeroMorphisms
X✝ Y✝ : BinaryBicone P Q
f : X✝ ⟶ Y✝
⊢ { pt := F.obj X✝.pt, fst := F.map X✝.fst, snd := F.map X✝.snd, inl := F.map X✝.inl, inr := F.map X✝.inr, inl_fst := ⋯,
          inl_snd := ⋯, inr_fst := ⋯, inr_snd := ⋯ }.inl ≫
      F.map f.hom =
    { pt := F.obj Y✝.pt, fst := F.map Y✝.fst, snd := F.map Y✝.snd, inl := F.map Y✝.inl, inr := F.map Y✝.inr,
        inl_fst := ⋯, inl_snd := ⋯, inr_fst := ⋯, inr_snd := ⋯ }.inl

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝⁴ : Category.{uC', uC} C
inst✝³ : HasZeroMorphisms C
D : Type uD
inst✝² : Category.{uD', uD} D
inst✝¹ : HasZeroMorphisms D
P Q : C
F : C ⥤ D
inst✝ : F.PreservesZeroMorphisms
X✝ Y✝ : BinaryBicone P Q
f : X✝ ⟶ Y✝
⊢ { pt := F.obj X✝.pt, fst := F.map X✝.fst, snd := F.map X✝.snd, inl := F.map X✝.inl, inr := F.map X✝.inr, inl_fst := ⋯,
          inl_snd := ⋯, inr_fst := ⋯, inr_snd := ⋯ }.inr ≫
      F.map f.hom =
    { pt := F.obj Y✝.pt, fst := F.map Y✝.fst, snd := F.map Y✝.snd, inl := F.map Y✝.inl, inr := F.map Y✝.inr,
        inl_fst := ⋯, inl_snd := ⋯, inr_fst := ⋯, inr_snd := ⋯ }.inr

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J : Type w
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
f : J → C
inst✝² : HasBiproduct f
p : J → Prop
inst✝¹ : HasBiproduct (Subtype.restrict p f)
inst✝ : DecidablePred p
j : J
⊢ ι f j ≫ toSubtype f p = if h : p j then ι (Subtype.restrict p f) ⟨j, h⟩ else 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝⁶ : Category.{uC', uC} C
inst✝⁵ : HasZeroMorphisms C
D : Type uD
inst✝⁴ : Category.{uD', uD} D
inst✝³ : HasZeroMorphisms D
P Q : C
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : F.Full
inst✝ : F.Faithful
X✝ Y✝ : BinaryBicone P Q
t : (functoriality P Q F).obj X✝ ⟶ (functoriality P Q F).obj Y✝
⊢ F.map (F.preimage t.hom ≫ Y✝.fst) = F.map X✝.fst

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J : Type w
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
f : J → C
inst✝² : HasBiproduct f
p : J → Prop
inst✝¹ : HasBiproduct (Subtype.restrict p f)
inst✝ : DecidablePred p
⊢ ∀ (j : J), ι f j ≫ toSubtype f p = ι f j ≫ desc fun j ↦ if h : p j then ι (Subtype.restrict p f) ⟨j, h⟩ else 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝⁶ : Category.{uC', uC} C
inst✝⁵ : HasZeroMorphisms C
D : Type uD
inst✝⁴ : Category.{uD', uD} D
inst✝³ : HasZeroMorphisms D
P Q : C
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : F.Full
inst✝ : F.Faithful
X✝ Y✝ : BinaryBicone P Q
t : (functoriality P Q F).obj X✝ ⟶ (functoriality P Q F).obj Y✝
⊢ F.map (F.preimage t.hom ≫ Y✝.snd) = F.map X✝.snd

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝⁶ : Category.{uC', uC} C
inst✝⁵ : HasZeroMorphisms C
D : Type uD
inst✝⁴ : Category.{uD', uD} D
inst✝³ : HasZeroMorphisms D
P Q : C
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : F.Full
inst✝ : F.Faithful
X✝ Y✝ : BinaryBicone P Q
t : (functoriality P Q F).obj X✝ ⟶ (functoriality P Q F).obj Y✝
⊢ F.map (X✝.inl ≫ F.preimage t.hom) = F.map Y✝.inl

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
J : Type u_1
inst✝ : Fintype J
f : J → C
b : Bicone f
total : ∑ j, b.π j ≫ b.ι j = 𝟙 b.pt
s : Cocone (Discrete.functor f)
m : b.toCocone.pt ⟶ s.pt
h : ∀ (j : Discrete J), b.toCocone.ι.app j ≫ m = s.ι.app j
⊢ m = ∑ j, b.π j ≫ s.ι.app { as := j }

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝⁶ : Category.{uC', uC} C
inst✝⁵ : HasZeroMorphisms C
D : Type uD
inst✝⁴ : Category.{uD', uD} D
inst✝³ : HasZeroMorphisms D
P Q : C
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : F.Full
inst✝ : F.Faithful
X✝ Y✝ : BinaryBicone P Q
t : (functoriality P Q F).obj X✝ ⟶ (functoriality P Q F).obj Y✝
⊢ F.map (X✝.inr ≫ F.preimage t.hom) = F.map Y✝.inr

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝⁶ : Category.{uC', uC} C
inst✝⁵ : HasZeroMorphisms C
D : Type uD
inst✝⁴ : Category.{uD', uD} D
inst✝³ : HasZeroMorphisms D
P Q : C
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : F.Full
inst✝ : F.Faithful
X✝ Y✝ : BinaryBicone P Q
t : (functoriality P Q F).obj X✝ ⟶ (functoriality P Q F).obj Y✝
⊢ (functoriality P Q F).map { hom := F.preimage t.hom, wfst := ⋯, wsnd := ⋯, winl := ⋯, winr := ⋯ } = t

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J : Type w
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
j : Subtype p
⊢ ι f ↑j ≫ toSubtype f p = ι (Subtype.restrict p f) j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
J : Type u_1
inst✝ : Fintype J
f : J → C
b : Bicone f
i : b.IsBilimit
j : Discrete J
⊢ (∑ j, b.π j ≫ b.ι j) ≫ b.toCone.π.app j = 𝟙 b.pt ≫ b.toCone.π.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J : Type w
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
⊢ fromSubtype f p ≫ toSubtype f p = 𝟙 (⨁ Subtype.restrict p f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
J : Type u_1
inst✝ : Fintype J
f : J → C
t : Bicone f
ht : IsLimit t.toCone
j : Discrete J
⊢ (∑ j, t.π j ≫ t.ι j) ≫ t.toCone.π.app j = 𝟙 t.pt ≫ t.toCone.π.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
J : Type u_1
inst✝ : Fintype J
f : J → C
t : Cone (Discrete.functor f)
ht : IsLimit t
⊢ ∀ (j : Discrete J), t.π.app j = (Iso.refl t.pt).hom ≫ (Bicone.ofLimitCone ht).toCone.π.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
P P' Q Q' X Y : C
b : BinaryBicone X Y
j j' : WalkingPair
⊢ WalkingPair.casesOn j b.inl b.inr ≫ WalkingPair.casesOn j' b.fst b.snd = if h : j = j' then eqToHom ⋯ else 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts

Goal state before:
pp: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
P P' Q Q' X Y : C
b : BinaryBicone X Y
x✝ : Discrete WalkingPair
as : WalkingPair
⊢ b.toBicone.toCone.π.app { as := as } = (Iso.refl b.toBicone.toCone.pt).hom ≫ b.toCone.π.app { as := as }

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J : Type w
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
f : J → C
inst✝² : HasBiproduct f
p : J → Prop
inst✝¹ : HasBiproduct (Subtype.restrict p f)
inst✝ : DecidablePred p
⊢ toSubtype f p ≫ fromSubtype f p = map fun j ↦ if p j then 𝟙 (f j) else 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Biproducts

Goal state before:
pp: J✝ : Type w
C✝ : Type uC
inst✝⁷ : Category.{uC', uC} C✝
inst✝⁶ : HasZeroMorphisms C✝
D : Type uD
inst✝⁵ : Category.{uD', uD} D
inst✝⁴ : HasZeroMorphisms D
F : J✝ → C✝
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
i : J
inst✝¹ : HasBiproduct f
inst✝ : HasBiproduct (Subtype.restrict (fun j ↦ j ≠ i) f)
⊢ (fromSubtype f fun j ↦ j ≠ i) ≫ π f i = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
J : Type u_1
inst✝ : Fintype J
f : J → C
t : Bicone f
ht : IsColimit t.toCocone
j : Discrete J
⊢ t.toCocone.ι.app j ≫ ∑ j, t.π j ≫ t.ι j = t.toCocone.ι.app j ≫ 𝟙 t.pt

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Biproducts

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
J : Type u_1
inst✝ : Fintype J
f : J → C
t : Cocone (Discrete.functor f)
ht : IsColimit t
⊢ ∀ (j : Discrete J), t.ι.app j ≫ (Iso.refl t.pt).hom = (Bicone.ofColimitCocone ht).toCocone.ι.app j

Tactic:
by

Goal state after: