mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: case neg
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X : SimplicialObject C
n n' : ℕ
i : ⦋n⦌ ⟶ ⦋n'⦌
inst✝ : Mono i
h : ¬n = n'
hi : ¬Isδ₀ i
⊢ 0 = 0

pp: case neg.h₁
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X : SimplicialObject C
n n' : ℕ
i : ⦋n⦌ ⟶ ⦋n'⦌
inst✝ : Mono i
h : ¬n = n'
hi : ¬Isδ₀ i
⊢ ⦋n⦌.len ≠ n'

pp: case neg.h₂
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X : SimplicialObject C
n n' : ℕ
i : ⦋n⦌ ⟶ ⦋n'⦌
inst✝ : Mono i
h : ¬n = n'
hi : ¬Isδ₀ i
⊢ ¬Isδ₀ i

Tactic:
]

Goal state after:
case neg.h₁
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X : SimplicialObject C
n n' : ℕ
i : ⦋n⦌ ⟶ ⦋n'⦌
inst✝ : Mono i
h : ¬n = n'
hi : ¬Isδ₀ i
⊢ ⦋n⦌.len ≠ n'

case neg.h₂
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X : SimplicialObject C
n n' : ℕ
i : ⦋n⦌ ⟶ ⦋n'⦌
inst✝ : Mono i
h : ¬n = n'
hi : ¬Isδ₀ i
⊢ ¬Isδ₀ i
Module: Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
i j : ℕ
x✝ : (ComplexShape.down ℕ).Rel i j
⊢ s.πSummand (IndexSet.id (op ⦋i⦌)) ≫
      (s.cofan (op ⦋i⦌)).inj (IndexSet.id (op ⦋i⦌)) ≫ K[X].d i j ≫ s.πSummand (IndexSet.id (op ⦋j⦌)) =
    s.πSummand (IndexSet.id (op ⦋i⦌)) ≫
      (s.cofan (op ⦋i⦌)).inj (IndexSet.id (op ⦋i⦌)) ≫ K[X].d i j ≫ s.πSummand (IndexSet.id (op ⦋j⦌))

pp: case h₀
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
i j : ℕ
x✝ : (ComplexShape.down ℕ).Rel i j
⊢ ∀ b ∈ Finset.univ,
    b ≠ IndexSet.id (op ⦋i⦌) →
      ((s.πSummand b ≫ (s.cofan (op ⦋i⦌)).inj b) ≫ K[X].d i j) ≫ s.πSummand (IndexSet.id (op ⦋j⦌)) = 0

pp: case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
i j : ℕ
x✝ : (ComplexShape.down ℕ).Rel i j
⊢ IndexSet.id (op ⦋i⦌) ∉ Finset.univ →
    ((s.πSummand (IndexSet.id (op ⦋i⦌)) ≫ (s.cofan (op ⦋i⦌)).inj (IndexSet.id (op ⦋i⦌))) ≫ K[X].d i j) ≫
        s.πSummand (IndexSet.id (op ⦋j⦌)) =
      0

Tactic:
]

Goal state after:
case h₀
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
i j : ℕ
x✝ : (ComplexShape.down ℕ).Rel i j
⊢ ∀ b ∈ Finset.univ,
    b ≠ IndexSet.id (op ⦋i⦌) →
      ((s.πSummand b ≫ (s.cofan (op ⦋i⦌)).inj b) ≫ K[X].d i j) ≫ s.πSummand (IndexSet.id (op ⦋j⦌)) = 0

case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
i j : ℕ
x✝ : (ComplexShape.down ℕ).Rel i j
⊢ IndexSet.id (op ⦋i⦌) ∉ Finset.univ →
    ((s.πSummand (IndexSet.id (op ⦋i⦌)) ≫ (s.cofan (op ⦋i⦌)).inj (IndexSet.id (op ⦋i⦌))) ≫ K[X].d i j) ≫
        s.πSummand (IndexSet.id (op ⦋j⦌)) =
      0
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C
Δ Δ' : SimplexCategoryᵒᵖ
θ : Δ ⟶ Δ'
A : Splitting.IndexSet Δ
⊢ PInfty.f (unop A.fst).len ≫ X.map (image.ι (θ.unop ≫ A.e)).op ≫ X.map (A.pull θ).e.op =
    PInfty.f (unop A.fst).len ≫ X.map A.e.op ≫ ((toKaroubi (SimplicialObject C)).obj X).X.map θ

Tactic:
]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C
Δ Δ' : SimplexCategoryᵒᵖ
θ : Δ ⟶ Δ'
A : Splitting.IndexSet Δ
⊢ PInfty.f (unop A.fst).len ≫ X.map (image.ι (θ.unop ≫ A.e)).op ≫ X.map (A.pull θ).e.op =
    PInfty.f (unop A.fst).len ≫ X.map A.e.op ≫ ((toKaroubi (SimplicialObject C)).obj X).X.map θ
Module: Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
⊢ (𝟙 ((toKaroubi (ChainComplex C ℕ)).obj s.nondegComplex)).f = 𝟙 s.nondegComplex

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
⊢ (𝟙 ((toKaroubi (ChainComplex C ℕ)).obj s.nondegComplex)).f = 𝟙 s.nondegComplex
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C
⊢ Γ₂N₁.natTrans.app X =
    (Γ₂N₂ToKaroubiIso.app X).inv ≫
      (N₂ ⋙ Γ₂).map ((toKaroubi (SimplicialObject C)).obj X).decompId_i ≫
        (Γ₂N₂ToKaroubiIso.hom ≫ Γ₂N₁.natTrans).app ((toKaroubi (SimplicialObject C)).obj X).X ≫
          ((toKaroubi (SimplicialObject C)).obj X).decompId_p

Tactic:
]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C
⊢ Γ₂N₁.natTrans.app X =
    (Γ₂N₂ToKaroubiIso.app X).inv ≫
      (N₂ ⋙ Γ₂).map ((toKaroubi (SimplicialObject C)).obj X).decompId_i ≫
        (Γ₂N₂ToKaroubiIso.hom ≫ Γ₂N₁.natTrans).app ((toKaroubi (SimplicialObject C)).obj X).X ≫
          ((toKaroubi (SimplicialObject C)).obj X).decompId_p
Module: Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
S₁ S₂ : Split C
Φ : S₁ ⟶ S₂
i j : ℕ
x✝ : (ComplexShape.down ℕ).Rel i j
⊢ (S₁.s.cofan (op ⦋i⦌)).inj (Splitting.IndexSet.id (op ⦋i⦌)) ≫
      ((alternatingFaceMapComplex C).obj S₁.X).d i j ≫
        ((alternatingFaceMapComplex C).map Φ.F).f j ≫ S₂.s.πSummand (Splitting.IndexSet.id (op ⦋j⦌)) =
    ((S₁.s.cofan (op ⦋i⦌)).inj (Splitting.IndexSet.id (op ⦋i⦌)) ≫
        K[S₁.X].d i j ≫ S₁.s.πSummand (Splitting.IndexSet.id (op ⦋j⦌))) ≫
      Φ.f j

Tactic:
]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
S₁ S₂ : Split C
Φ : S₁ ⟶ S₂
i j : ℕ
x✝ : (ComplexShape.down ℕ).Rel i j
⊢ (S₁.s.cofan (op ⦋i⦌)).inj (Splitting.IndexSet.id (op ⦋i⦌)) ≫
      ((alternatingFaceMapComplex C).obj S₁.X).d i j ≫
        ((alternatingFaceMapComplex C).map Φ.F).f j ≫ S₂.s.πSummand (Splitting.IndexSet.id (op ⦋j⦌)) =
    ((S₁.s.cofan (op ⦋i⦌)).inj (Splitting.IndexSet.id (op ⦋i⦌)) ≫
        K[S₁.X].d i j ≫ S₁.s.πSummand (Splitting.IndexSet.id (op ⦋j⦌))) ≫
      Φ.f j
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C
⊢ Γ₂N₁.natTrans.app X =
    Γ₂N₂ToKaroubiIso.inv.app X ≫
      𝟙 (Γ₂.obj (N₂.obj ((toKaroubi (SimplicialObject C)).obj X))) ≫
        (Γ₂N₂ToKaroubiIso.hom.app ((toKaroubi (SimplicialObject C)).obj X).X ≫
            Γ₂N₁.natTrans.app ((toKaroubi (SimplicialObject C)).obj X).X) ≫
          𝟙
            { X := ((toKaroubi (SimplicialObject C)).obj X).X, p := 𝟙 ((toKaroubi (SimplicialObject C)).obj X).X,
              idem := ⋯ }

Tactic:
]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C
⊢ Γ₂N₁.natTrans.app X =
    Γ₂N₂ToKaroubiIso.inv.app X ≫
      𝟙 (Γ₂.obj (N₂.obj ((toKaroubi (SimplicialObject C)).obj X))) ≫
        (Γ₂N₂ToKaroubiIso.hom.app ((toKaroubi (SimplicialObject C)).obj X).X ≫
            Γ₂N₁.natTrans.app ((toKaroubi (SimplicialObject C)).obj X).X) ≫
          𝟙
            { X := ((toKaroubi (SimplicialObject C)).obj X).X, p := 𝟙 ((toKaroubi (SimplicialObject C)).obj X).X,
              idem := ⋯ }
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C
⊢ Γ₂N₁.natTrans.app X =
    Γ₂N₂ToKaroubiIso.inv.app X ≫ (Γ₂N₂ToKaroubiIso.hom.app X ≫ Γ₂N₁.natTrans.app X) ≫ 𝟙 { X := X, p := 𝟙 X, idem := ⋯ }

Tactic:
]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C
⊢ Γ₂N₁.natTrans.app X =
    Γ₂N₂ToKaroubiIso.inv.app X ≫ (Γ₂N₂ToKaroubiIso.hom.app X ≫ Γ₂N₁.natTrans.app X) ≫ 𝟙 { X := X, p := 𝟙 X, idem := ⋯ }
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C
⊢ Γ₂N₁.natTrans.app X = Γ₂N₁.natTrans.app X

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Goal state before:
pp: case e_a
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
S₁ S₂ : Split C
Φ : S₁ ⟶ S₂
i j : ℕ
x✝ : (ComplexShape.down ℕ).Rel i j
A : Splitting.IndexSet (op ⦋j⦌)
⊢ Φ.f (unop A.fst).len ≫ (S₂.s.cofan (op ⦋j⦌)).inj A ≫ S₂.s.πSummand (Splitting.IndexSet.id (op ⦋j⦌)) =
    (S₁.s.cofan (op ⦋j⦌)).inj A ≫ S₁.s.πSummand (Splitting.IndexSet.id (op ⦋j⦌)) ≫ Φ.f j

Tactic:
]

Goal state after:
case e_a
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
S₁ S₂ : Split C
Φ : S₁ ⟶ S₂
i j : ℕ
x✝ : (ComplexShape.down ℕ).Rel i j
A : Splitting.IndexSet (op ⦋j⦌)
⊢ Φ.f (unop A.fst).len ≫ (S₂.s.cofan (op ⦋j⦌)).inj A ≫ S₂.s.πSummand (Splitting.IndexSet.id (op ⦋j⦌)) =
    (S₁.s.cofan (op ⦋j⦌)).inj A ≫ S₁.s.πSummand (Splitting.IndexSet.id (op ⦋j⦌)) ≫ Φ.f j
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
eq₁ :
  (N₂Γ₂.inv.app (N₂.obj P)).f.f n =
    PInfty.f n ≫ P.p.app (op ⦋n⦌) ≫ ((Γ₀.splitting (N₂.obj P).X).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌))
⊢ ((Γ₀.splitting (N₂.obj P).X).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌)) ≫
      ((N₂ ⋙ Γ₂).map P.decompId_i ≫ (Γ₂N₂ToKaroubiIso.hom ≫ Γ₂N₁.natTrans).app P.X ≫ P.decompId_p).f.app (op ⦋n⦌) =
    PInfty.f n ≫ P.p.app (op ⦋n⦌)

Tactic:
]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
eq₁ :
  (N₂Γ₂.inv.app (N₂.obj P)).f.f n =
    PInfty.f n ≫ P.p.app (op ⦋n⦌) ≫ ((Γ₀.splitting (N₂.obj P).X).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌))
⊢ ((Γ₀.splitting (N₂.obj P).X).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌)) ≫
      ((N₂ ⋙ Γ₂).map P.decompId_i ≫ (Γ₂N₂ToKaroubiIso.hom ≫ Γ₂N₁.natTrans).app P.X ≫ P.decompId_p).f.app (op ⦋n⦌) =
    PInfty.f n ≫ P.p.app (op ⦋n⦌)
Module: Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Goal state before:
pp: case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
S₁ S₂ : Split C
Φ : S₁ ⟶ S₂
i j : ℕ
x✝ : (ComplexShape.down ℕ).Rel i j
⊢ Φ.f (unop (Splitting.IndexSet.id (op ⦋j⦌)).fst).len ≫
      𝟙 (Splitting.summand S₂.s.N (op ⦋j⦌) (Splitting.IndexSet.id (op ⦋j⦌))) =
    (S₁.s.cofan (op ⦋j⦌)).inj (Splitting.IndexSet.id (op ⦋j⦌)) ≫ S₁.s.πSummand (Splitting.IndexSet.id (op ⦋j⦌)) ≫ Φ.f j

Tactic:
]

Goal state after:
case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
S₁ S₂ : Split C
Φ : S₁ ⟶ S₂
i j : ℕ
x✝ : (ComplexShape.down ℕ).Rel i j
⊢ Φ.f (unop (Splitting.IndexSet.id (op ⦋j⦌)).fst).len ≫
      𝟙 (Splitting.summand S₂.s.N (op ⦋j⦌) (Splitting.IndexSet.id (op ⦋j⦌))) =
    (S₁.s.cofan (op ⦋j⦌)).inj (Splitting.IndexSet.id (op ⦋j⦌)) ≫ S₁.s.πSummand (Splitting.IndexSet.id (op ⦋j⦌)) ≫ Φ.f j
Module: Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Goal state before:
pp: case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
S₁ S₂ : Split C
Φ : S₁ ⟶ S₂
i j : ℕ
x✝ : (ComplexShape.down ℕ).Rel i j
⊢ Φ.f j = Φ.f j

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Goal state before:
pp: case neg
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
S₁ S₂ : Split C
Φ : S₁ ⟶ S₂
i j : ℕ
x✝ : (ComplexShape.down ℕ).Rel i j
A : Splitting.IndexSet (op ⦋j⦌)
h : ¬A.EqId
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X✝ Y✝ : Split C
Φ : X✝ ⟶ Y✝
n : ℕ
⊢ (X✝.s.cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌)) ≫ Φ.F.app (op ⦋n⦌) ≫ PInfty.f n =
    (X✝.s.cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌)) ≫ PInfty.f n ≫ Φ.F.app (op ⦋n⦌)

Tactic:
]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X✝ Y✝ : Split C
Φ : X✝ ⟶ Y✝
n : ℕ
⊢ (X✝.s.cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌)) ≫ Φ.F.app (op ⦋n⦌) ≫ PInfty.f n =
    (X✝.s.cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌)) ≫ PInfty.f n ≫ Φ.F.app (op ⦋n⦌)
Module: Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X✝ Y✝ : Split C
Φ : X✝ ⟶ Y✝
n : ℕ
⊢ (X✝.s.cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌)) ≫ PInfty.f n ≫ Φ.F.app (op ⦋n⦌) =
    (X✝.s.cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌)) ≫ PInfty.f n ≫ Φ.F.app (op ⦋n⦌)

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
eq₁ :
  (N₂Γ₂.inv.app (N₂.obj P)).f.f n =
    PInfty.f n ≫ P.p.app (op ⦋n⦌) ≫ ((Γ₀.splitting (N₂.obj P).X).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌))
⊢ PInfty.f (unop (Splitting.IndexSet.id (op ⦋n⦌)).fst).len ≫
      P.p.app (Splitting.IndexSet.id (op ⦋n⦌)).fst ≫
        ((Γ₀.splitting (N₂.obj { X := P.X, p := 𝟙 P.X, idem := ⋯ }).X).cofan (op ⦋n⦌)).inj
            (Splitting.IndexSet.id (op ⦋n⦌)) ≫
          (Γ₂.map (toKaroubiCompN₂IsoN₁.hom.app P.X)).f.app (op ⦋n⦌) ≫
            ((Γ₀.splitting K[P.X]).desc (op ⦋n⦌) fun A ↦ PInfty.f (unop A.fst).len ≫ P.X.map A.e.op) ≫
              P.p.app (op ⦋n⦌) =
    PInfty.f n ≫ P.p.app (op ⦋n⦌)

Tactic:
]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
eq₁ :
  (N₂Γ₂.inv.app (N₂.obj P)).f.f n =
    PInfty.f n ≫ P.p.app (op ⦋n⦌) ≫ ((Γ₀.splitting (N₂.obj P).X).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌))
⊢ PInfty.f (unop (Splitting.IndexSet.id (op ⦋n⦌)).fst).len ≫
      P.p.app (Splitting.IndexSet.id (op ⦋n⦌)).fst ≫
        ((Γ₀.splitting (N₂.obj { X := P.X, p := 𝟙 P.X, idem := ⋯ }).X).cofan (op ⦋n⦌)).inj
            (Splitting.IndexSet.id (op ⦋n⦌)) ≫
          (Γ₂.map (toKaroubiCompN₂IsoN₁.hom.app P.X)).f.app (op ⦋n⦌) ≫
            ((Γ₀.splitting K[P.X]).desc (op ⦋n⦌) fun A ↦ PInfty.f (unop A.fst).len ≫ P.X.map A.e.op) ≫
              P.p.app (op ⦋n⦌) =
    PInfty.f n ≫ P.p.app (op ⦋n⦌)
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
eq₁ :
  (N₂Γ₂.inv.app (N₂.obj P)).f.f n =
    PInfty.f n ≫ P.p.app (op ⦋n⦌) ≫ ((Γ₀.splitting (N₂.obj P).X).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌))
⊢ PInfty.f n ≫
      P.p.app (op ⦋n⦌) ≫
        (PInfty.f (unop (Splitting.IndexSet.id (op ⦋n⦌)).fst).len ≫
            ((Γ₀.splitting K[P.X]).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌))) ≫
          ((Γ₀.splitting K[P.X]).desc (op ⦋n⦌) fun A ↦ PInfty.f (unop A.fst).len ≫ P.X.map A.e.op) ≫ P.p.app (op ⦋n⦌) =
    PInfty.f n ≫ P.p.app (op ⦋n⦌)

Tactic:
]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
eq₁ :
  (N₂Γ₂.inv.app (N₂.obj P)).f.f n =
    PInfty.f n ≫ P.p.app (op ⦋n⦌) ≫ ((Γ₀.splitting (N₂.obj P).X).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌))
⊢ PInfty.f n ≫
      P.p.app (op ⦋n⦌) ≫
        (PInfty.f (unop (Splitting.IndexSet.id (op ⦋n⦌)).fst).len ≫
            ((Γ₀.splitting K[P.X]).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌))) ≫
          ((Γ₀.splitting K[P.X]).desc (op ⦋n⦌) fun A ↦ PInfty.f (unop A.fst).len ≫ P.X.map A.e.op) ≫ P.p.app (op ⦋n⦌) =
    PInfty.f n ≫ P.p.app (op ⦋n⦌)
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
⊢ 𝟙 (N₂.obj P) = 𝟙 (N₂.obj P)

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
h : N₂.map (Γ₂N₂.natTrans.app P) = inv (N₂Γ₂.inv.app (N₂.obj P))
⊢ IsIso (N₂.map (Γ₂N₂.natTrans.app P))

Tactic:
]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
h : N₂.map (Γ₂N₂.natTrans.app P) = inv (N₂Γ₂.inv.app (N₂.obj P))
⊢ IsIso (N₂.map (Γ₂N₂.natTrans.app P))
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
h : N₂.map (Γ₂N₂.natTrans.app P) = inv (N₂Γ₂.inv.app (N₂.obj P))
⊢ IsIso (inv (N₂Γ₂.inv.app (N₂.obj P)))

Tactic:
]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
h : N₂.map (Γ₂N₂.natTrans.app P) = inv (N₂Γ₂.inv.app (N₂.obj P))
⊢ IsIso (inv (N₂Γ₂.inv.app (N₂.obj P)))
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C
⊢ IsIso ((Γ₂N₂ToKaroubiIso.app X).inv ≫ Γ₂N₂.natTrans.app ((toKaroubi (SimplicialObject C)).obj X))

Tactic:
]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C
⊢ IsIso ((Γ₂N₂ToKaroubiIso.app X).inv ≫ Γ₂N₂.natTrans.app ((toKaroubi (SimplicialObject C)).obj X))
Module: Mathlib.Combinatorics.Quiver.Cast

Goal state before:
pp: U : Type u_1
inst✝ : Quiver U
u v u' v' : U
hu : u = u'
hv : v = v'
e : u ⟶ v
⊢ (u' ⟶ v') = (u' ⟶ v')

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Quiver.Cast

Goal state before:
pp: U : Type u_1
inst✝ : Quiver U
u v u' v' : U
hu : u = u'
hv : v = v'
e : u ⟶ v
e' : u' ⟶ v'
⊢ _root_.cast ⋯ e = e' ↔ e ≍ e'

Tactic:
]

Goal state after:
U : Type u_1
inst✝ : Quiver U
u v u' v' : U
hu : u = u'
hv : v = v'
e : u ⟶ v
e' : u' ⟶ v'
⊢ _root_.cast ⋯ e = e' ↔ e ≍ e'
Module: Mathlib.Combinatorics.Quiver.Cast

Goal state before:
pp: U : Type u_1
inst✝ : Quiver U
u v u' v' : U
hu : u = u'
hv : v = v'
e : u ⟶ v
e' : u' ⟶ v'
⊢ _root_.cast ⋯ e = e' ↔ e ≍ e'

Tactic:
exact _root_.cast_eq_iff_heq

Goal state after:
Module: Mathlib.Combinatorics.Quiver.Cast

Goal state before:
pp: U : Type u_1
inst✝ : Quiver U
u v u' v' : U
hu : u = u'
hv : v = v'
e : u ⟶ v
e' : u' ⟶ v'
⊢ e ≍ e' ↔ e' ≍ e

Tactic:
]

Goal state after:
U : Type u_1
inst✝ : Quiver U
u v u' v' : U
hu : u = u'
hv : v = v'
e : u ⟶ v
e' : u' ⟶ v'
⊢ e ≍ e' ↔ e' ≍ e
Module: Mathlib.Combinatorics.Quiver.Cast

Goal state before:
pp: U : Type u_1
inst✝ : Quiver U
u v u' v' : U
hu : u = u'
hv : v = v'
p : Path u v
⊢ Path u' v' = Path u' v'

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Quiver.Cast

Goal state before:
pp: U : Type u_1
inst✝ : Quiver U
u v u' v' : U
hu : u = u'
hv : v = v'
p : Path u v
⊢ _root_.cast ⋯ p ≍ p

Tactic:
]

Goal state after:
U : Type u_1
inst✝ : Quiver U
u v u' v' : U
hu : u = u'
hv : v = v'
p : Path u v
⊢ _root_.cast ⋯ p ≍ p
Module: Mathlib.Combinatorics.Quiver.Cast

Goal state before:
pp: U : Type u_1
inst✝ : Quiver U
u v u' v' : U
hu : u = u'
hv : v = v'
p : Path u v
p' : Path u' v'
⊢ _root_.cast ⋯ p = p' ↔ p ≍ p'

Tactic:
]

Goal state after:
U : Type u_1
inst✝ : Quiver U
u v u' v' : U
hu : u = u'
hv : v = v'
p : Path u v
p' : Path u' v'
⊢ _root_.cast ⋯ p = p' ↔ p ≍ p'
Module: Mathlib.Combinatorics.Quiver.Cast

Goal state before:
pp: U : Type u_1
inst✝ : Quiver U
u v u' v' : U
hu : u = u'
hv : v = v'
p : Path u v
p' : Path u' v'
⊢ _root_.cast ⋯ p = p' ↔ p ≍ p'

Tactic:
exact _root_.cast_eq_iff_heq

Goal state after:
Module: Mathlib.Combinatorics.Quiver.Cast

Goal state before:
pp: U : Type u_1
inst✝ : Quiver U
u v v' w : U
p : Path u v
p' : Path u v'
e : v ⟶ w
e' : v' ⟶ w
h : p.cons e = p'.cons e'
⊢ p ≍ p'

Tactic:
]

Goal state after:
U : Type u_1
inst✝ : Quiver U
u v v' w : U
p : Path u v
p' : Path u v'
e : v ⟶ w
e' : v' ⟶ w
h : p.cons e = p'.cons e'
⊢ p ≍ p'
Module: Mathlib.Combinatorics.Quiver.Cast

Goal state before:
pp: U : Type u_1
inst✝ : Quiver U
u v v' w : U
p : Path u v
p' : Path u v'
e : v ⟶ w
e' : v' ⟶ w
h : p.cons e = p'.cons e'
⊢ e ≍ e'

Tactic:
]

Goal state after:
U : Type u_1
inst✝ : Quiver U
u v v' w : U
p : Path u v
p' : Path u v'
e : v ⟶ w
e' : v' ⟶ w
h : p.cons e = p'.cons e'
⊢ e ≍ e'
Module: Mathlib.Combinatorics.Quiver.SingleObj

Goal state before:
pp: case cons
α : Type u_1
x b✝ c✝ : SingleObj α
a✝¹ : Path (star α) b✝
a✝ : b✝ ⟶ c✝
ih : listToPath (pathToList a✝¹) = a✝¹
⊢ a✝¹.cons a✝ = a✝¹.cons a✝

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Quiver.SingleObj

Goal state before:
pp: case cons
α : Type u_1
a : α
l : List α
ih : pathToList (listToPath l) = l
⊢ a :: l = a :: l

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.SingleObj

Goal state before:
pp: G : Type u
inst✝ : Group G
x y : SingleObj G
f : x ⟶ y
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.SingleObj

Goal state before:
pp: M G : Type u
inst✝³ : Monoid M
inst✝² : Group G
N : Type v
inst✝¹ : Monoid N
C : Type v
inst✝ : Category.{w, v} C
f : C → G
X✝ Y✝ Z✝ : C
f✝ : X✝ ⟶ Y✝
g✝ : Y✝ ⟶ Z✝
⊢ f Z✝ = f Z✝

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.SingleObj

Goal state before:
pp: M : Type u
N : Type v
inst✝¹ : Monoid M
inst✝ : Monoid N
e : M ≃* N
⊢ (MonoidHom.id M).toFunctor = (e.symm.toMonoidHom.comp e.toMonoidHom).toFunctor

Tactic:
]

Goal state after:
M : Type u
N : Type v
inst✝¹ : Monoid M
inst✝ : Monoid N
e : M ≃* N
⊢ (MonoidHom.id M).toFunctor = (e.symm.toMonoidHom.comp e.toMonoidHom).toFunctor
Module: Mathlib.CategoryTheory.SingleObj

Goal state before:
pp: M : Type u
N : Type v
inst✝¹ : Monoid M
inst✝ : Monoid N
e : M ≃* N
⊢ (e.toMonoidHom.comp e.symm.toMonoidHom).toFunctor = (MonoidHom.id N).toFunctor

Tactic:
]

Goal state after:
M : Type u
N : Type v
inst✝¹ : Monoid M
inst✝ : Monoid N
e : M ≃* N
⊢ (e.toMonoidHom.comp e.symm.toMonoidHom).toFunctor = (MonoidHom.id N).toFunctor
Module: Mathlib.Topology.Homotopy.Basic

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f₀ f₁ : C(X, Y)
F : f₀.Homotopy f₁
t : ℝ
ht : t ≤ 0
x : X
⊢ (F.extend t) x = F (0, x)

Tactic:
]

Goal state after:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f₀ f₁ : C(X, Y)
F : f₀.Homotopy f₁
t : ℝ
ht : t ≤ 0
x : X
⊢ (F.extend t) x = F (0, x)
Module: Mathlib.Topology.Homotopy.Basic

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f₀ f₁ : C(X, Y)
F : f₀.Homotopy f₁
t : ℝ
ht : 1 ≤ t
x : X
⊢ (F.extend t) x = F (1, x)

Tactic:
]

Goal state after:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f₀ f₁ : C(X, Y)
F : f₀.Homotopy f₁
t : ℝ
ht : 1 ≤ t
x : X
⊢ (F.extend t) x = F (1, x)
Module: Mathlib.CategoryTheory.Groupoid.Grpd.Basic

Goal state before:
pp: J : Type u
F : J → Grpd
s✝ : Limits.Fan F
j✝ : J
⊢ (Functor.pi' fun j ↦ s✝.proj j) ≫ (piLimitFan F).proj j✝ = s✝.proj j✝

Tactic:
dsimp only [piLimitFan]

Goal state after:
J : Type u
F : J → Grpd
s✝ : Limits.Fan F
j✝ : J
⊢ (Functor.pi' fun j ↦ s✝.proj j) ≫ (Limits.Fan.mk (of ((j : J) → ↑(F j))) fun j ↦ Pi.eval (fun j ↦ ↑(F j)) j).proj j✝ =
    s✝.proj j✝