mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.MorphismProperty.Representable

Goal state before:
pp: case h₀
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝² : F.Full
A₁ A₂ A₃ : C
X : D
f₁ : F.obj A₁ ⟶ X
hf₁ : F.relativelyRepresentable f₁
f₂ : F.obj A₂ ⟶ X
f₃ : F.obj A₃ ⟶ X
inst✝¹ : HasPullback (hf₁.fst' f₂) (hf₁.fst' f₃)
inst✝ : F.Faithful
Z : C
φ φ' : Z ⟶ hf₁.pullback₃ f₂ f₃
h₁ : φ ≫ p₁ hf₁ f₂ f₃ = φ' ≫ p₁ hf₁ f₂ f₃
h₂ : φ ≫ p₂ hf₁ f₂ f₃ = φ' ≫ p₂ hf₁ f₂ f₃
h₃ : φ ≫ p₃ hf₁ f₂ f₃ = φ' ≫ p₃ hf₁ f₂ f₃
⊢ φ ≫ pullback.fst (hf₁.fst' f₂) (hf₁.fst' f₃) = φ' ≫ pullback.fst (hf₁.fst' f₂) (hf₁.fst' f₃)

Tactic:
ext

Goal state after:
case h₀.h₁
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝² : F.Full
A₁ A₂ A₃ : C
X : D
f₁ : F.obj A₁ ⟶ X
hf₁ : F.relativelyRepresentable f₁
f₂ : F.obj A₂ ⟶ X
f₃ : F.obj A₃ ⟶ X
inst✝¹ : HasPullback (hf₁.fst' f₂) (hf₁.fst' f₃)
inst✝ : F.Faithful
Z : C
φ φ' : Z ⟶ hf₁.pullback₃ f₂ f₃
h₁ : φ ≫ p₁ hf₁ f₂ f₃ = φ' ≫ p₁ hf₁ f₂ f₃
h₂ : φ ≫ p₂ hf₁ f₂ f₃ = φ' ≫ p₂ hf₁ f₂ f₃
h₃ : φ ≫ p₃ hf₁ f₂ f₃ = φ' ≫ p₃ hf₁ f₂ f₃
⊢ (φ ≫ pullback.fst (hf₁.fst' f₂) (hf₁.fst' f₃)) ≫ hf₁.fst' f₂ =
    (φ' ≫ pullback.fst (hf₁.fst' f₂) (hf₁.fst' f₃)) ≫ hf₁.fst' f₂

case h₂
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝² : F.Full
A₁ A₂ A₃ : C
X : D
f₁ : F.obj A₁ ⟶ X
hf₁ : F.relativelyRepresentable f₁
f₂ : F.obj A₂ ⟶ X
f₃ : F.obj A₃ ⟶ X
inst✝¹ : HasPullback (hf₁.fst' f₂) (hf₁.fst' f₃)
inst✝ : F.Faithful
Z : C
φ φ' : Z ⟶ hf₁.pullback₃ f₂ f₃
h₁ : φ ≫ p₁ hf₁ f₂ f₃ = φ' ≫ p₁ hf₁ f₂ f₃
h₂ : φ ≫ p₂ hf₁ f₂ f₃ = φ' ≫ p₂ hf₁ f₂ f₃
h₃ : φ ≫ p₃ hf₁ f₂ f₃ = φ' ≫ p₃ hf₁ f₂ f₃
⊢ (φ ≫ pullback.fst (hf₁.fst' f₂) (hf₁.fst' f₃)) ≫ hf₁.snd f₂ =
    (φ' ≫ pullback.fst (hf₁.fst' f₂) (hf₁.fst' f₃)) ≫ hf₁.snd f₂
Module: Mathlib.AlgebraicGeometry.Sites.Representability

Goal state before:
pp: F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i j k : ι
inst✝ : Presheaf.IsLocallySurjective zariskiTopology (Sigma.desc f)
⊢ Sheaf.IsLocallyInjective (yonedaGluedToSheaf hf) ∧ Sheaf.IsLocallySurjective (yonedaGluedToSheaf hf)

Tactic:
constructor

Goal state after:
case left
F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i j k : ι
inst✝ : Presheaf.IsLocallySurjective zariskiTopology (Sigma.desc f)
⊢ Sheaf.IsLocallyInjective (yonedaGluedToSheaf hf)

case right
F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i j k : ι
inst✝ : Presheaf.IsLocallySurjective zariskiTopology (Sigma.desc f)
⊢ Sheaf.IsLocallySurjective (yonedaGluedToSheaf hf)
Module: Mathlib.CategoryTheory.MorphismProperty.Representable

Goal state before:
pp: case h₁
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝² : F.Full
A₁ A₂ A₃ : C
X : D
f₁ : F.obj A₁ ⟶ X
hf₁ : F.relativelyRepresentable f₁
f₂ : F.obj A₂ ⟶ X
f₃ : F.obj A₃ ⟶ X
inst✝¹ : HasPullback (hf₁.fst' f₂) (hf₁.fst' f₃)
inst✝ : F.Faithful
Z : C
φ φ' : Z ⟶ hf₁.pullback₃ f₂ f₃
h₁ : φ ≫ p₁ hf₁ f₂ f₃ = φ' ≫ p₁ hf₁ f₂ f₃
h₂ : φ ≫ p₂ hf₁ f₂ f₃ = φ' ≫ p₂ hf₁ f₂ f₃
h₃ : φ ≫ p₃ hf₁ f₂ f₃ = φ' ≫ p₃ hf₁ f₂ f₃
⊢ φ ≫ pullback.snd (hf₁.fst' f₂) (hf₁.fst' f₃) = φ' ≫ pullback.snd (hf₁.fst' f₂) (hf₁.fst' f₃)

Tactic:
ext

Goal state after:
case h₁.h₁
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝² : F.Full
A₁ A₂ A₃ : C
X : D
f₁ : F.obj A₁ ⟶ X
hf₁ : F.relativelyRepresentable f₁
f₂ : F.obj A₂ ⟶ X
f₃ : F.obj A₃ ⟶ X
inst✝¹ : HasPullback (hf₁.fst' f₂) (hf₁.fst' f₃)
inst✝ : F.Faithful
Z : C
φ φ' : Z ⟶ hf₁.pullback₃ f₂ f₃
h₁ : φ ≫ p₁ hf₁ f₂ f₃ = φ' ≫ p₁ hf₁ f₂ f₃
h₂ : φ ≫ p₂ hf₁ f₂ f₃ = φ' ≫ p₂ hf₁ f₂ f₃
h₃ : φ ≫ p₃ hf₁ f₂ f₃ = φ' ≫ p₃ hf₁ f₂ f₃
⊢ (φ ≫ pullback.snd (hf₁.fst' f₂) (hf₁.fst' f₃)) ≫ hf₁.fst' f₃ =
    (φ' ≫ pullback.snd (hf₁.fst' f₂) (hf₁.fst' f₃)) ≫ hf₁.fst' f₃

case h₂
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝² : F.Full
A₁ A₂ A₃ : C
X : D
f₁ : F.obj A₁ ⟶ X
hf₁ : F.relativelyRepresentable f₁
f₂ : F.obj A₂ ⟶ X
f₃ : F.obj A₃ ⟶ X
inst✝¹ : HasPullback (hf₁.fst' f₂) (hf₁.fst' f₃)
inst✝ : F.Faithful
Z : C
φ φ' : Z ⟶ hf₁.pullback₃ f₂ f₃
h₁ : φ ≫ p₁ hf₁ f₂ f₃ = φ' ≫ p₁ hf₁ f₂ f₃
h₂ : φ ≫ p₂ hf₁ f₂ f₃ = φ' ≫ p₂ hf₁ f₂ f₃
h₃ : φ ≫ p₃ hf₁ f₂ f₃ = φ' ≫ p₃ hf₁ f₂ f₃
⊢ (φ ≫ pullback.snd (hf₁.fst' f₂) (hf₁.fst' f₃)) ≫ hf₁.snd f₃ =
    (φ' ≫ pullback.snd (hf₁.fst' f₂) (hf₁.fst' f₃)) ≫ hf₁.snd f₃
Module: Mathlib.AlgebraicTopology.DoldKan.Homotopies

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
q n m : ℕ
hnm : c.Rel m n
X Y : SimplicialObject C
f : X ⟶ Y
⊢ f.app (op ⦋n⦌) ≫ hσ' q n m hnm = hσ' q n m hnm ≫ f.app (op ⦋m⦌)

Tactic:
by
  obtain rfl : n + 1 = m := hnm
  -- `simp? [hσ', hσ]` says:
  simp only [AlternatingFaceMapComplex.obj_X, hσ', hσ, Int.reduceNeg, eqToHom_refl, comp_id]
  split_ifs
  · rw [zero_comp, comp_zero]
  · simp

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
⊢ φ ≫ (P q).f (n + 1) = φ ↔ HigherFacesVanish q φ

Tactic:
constructor

Goal state after:
case mp
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
⊢ φ ≫ (P q).f (n + 1) = φ → HigherFacesVanish q φ

case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
⊢ HigherFacesVanish q φ → φ ≫ (P q).f (n + 1) = φ
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : ℤ
⊢ ‖↑k‖ < 1 ↔ ↑p ∣ k

Tactic:
constructor

Goal state after:
case mp
p : ℕ
hp : Fact (Nat.Prime p)
k : ℤ
⊢ ‖↑k‖ < 1 → ↑p ∣ k

case mpr
p : ℕ
hp : Fact (Nat.Prime p)
k : ℤ
⊢ ↑p ∣ k → ‖↑k‖ < 1
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : ℤ
h : ¬↑p ∣ k
⊢ ‖↑k‖ = ‖↑↑k‖

Tactic:
by norm_cast

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
a : ℝ
ha : 0 < a
hf : ∀ (i : ℕ), ‖↑f i‖ ≤ a
⊢ ‖f.lim‖ ≤ a

Tactic:
by
  obtain ⟨N, hN⟩ := Setoid.symm (CauSeq.equiv_lim f) _ ha
  calc
    ‖f.lim‖ = ‖f.lim - f N + f N‖ := by simp
    _ ≤ max ‖f.lim - f N‖ ‖f N‖ := nonarchimedean _ _
    _ ≤ a := max_le (le_of_lt (hN _ le_rfl)) (hf _)

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
⊢ Valuation.comap (Rat.castHom ℚ_[p]) mulValuation = Rat.padicValuation p

Tactic:
ext

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
r✝ : ℚ
⊢ (Valuation.comap (Rat.castHom ℚ_[p]) mulValuation) r✝ = (Rat.padicValuation p) r✝
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
⊢ Valuation.comap (Int.castRingHom ℚ_[p]) mulValuation = Int.padicValuation p

Tactic:
ext

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
r✝ : ℤ
⊢ (Valuation.comap (Int.castRingHom ℚ_[p]) mulValuation) r✝ = (Int.padicValuation p) r✝
Module: Mathlib.AlgebraicTopology.SimplicialObject.Split

Goal state before:
pp: Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
⊢ A.EqId ↔ A.fst = Δ

Tactic:
constructor

Goal state after:
case mp
Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
⊢ A.EqId → A.fst = Δ

case mpr
Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
⊢ A.fst = Δ → A.EqId
Module: Mathlib.AlgebraicTopology.DoldKan.Faces

Goal state before:
pp: case neg.inr.H
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
q a m : ℕ
φ : Y ⟶ X _⦋m + 1 + 1⦌
v : HigherFacesVanish q φ
j : Fin (m + 1 + 1)
hj₁ : m + 1 + 1 ≤ ↑j + (q + 1)
hqn : q ≤ m + 1
ha : q + a = m + 1
hj₂ : ¬a = ↑j
haj : a < ↑j
ham : a ≤ m
ham'' : a = m
⊢ ⟨a + 1, ⋯⟩ = j.castSucc

Tactic:
ext

Goal state after:
case neg.inr.H
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
q a m : ℕ
φ : Y ⟶ X _⦋m + 1 + 1⦌
v : HigherFacesVanish q φ
j : Fin (m + 1 + 1)
hj₁ : m + 1 + 1 ≤ ↑j + (q + 1)
hqn : q ≤ m + 1
ha : q + a = m + 1
hj₂ : ¬a = ↑j
haj : a < ↑j
ham : a ≤ m
ham'' : a = m
⊢ ↑⟨a + 1, ⋯⟩ = ↑j.castSucc
Module: Mathlib.AlgebraicTopology.SimplicialObject.Split

Goal state before:
pp: Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
⊢ A.fst = Δ ↔ (unop A.fst).len = (unop Δ).len

Tactic:
constructor

Goal state after:
case mp
Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
⊢ A.fst = Δ → (unop A.fst).len = (unop Δ).len

case mpr
Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
⊢ (unop A.fst).len = (unop Δ).len → A.fst = Δ
Module: Mathlib.AlgebraicTopology.SimplicialObject.Split

Goal state before:
pp: case mpr
Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
h : (unop A.fst).len = (unop Δ).len
⊢ unop A.fst = unop Δ

Tactic:
ext

Goal state after:
case mpr
Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
h : (unop A.fst).len = (unop Δ).len
⊢ (unop A.fst).len = (unop Δ).len
Module: Mathlib.AlgebraicTopology.SimplicialObject.Split

Goal state before:
pp: Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
⊢ (unop A.fst).len = (unop Δ).len ↔ (unop Δ).len ≤ (unop A.fst).len

Tactic:
constructor

Goal state after:
case mp
Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
⊢ (unop A.fst).len = (unop Δ).len → (unop Δ).len ≤ (unop A.fst).len

case mpr
Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
⊢ (unop Δ).len ≤ (unop A.fst).len → (unop A.fst).len = (unop Δ).len
Module: Mathlib.AlgebraicTopology.SimplicialObject.Split

Goal state before:
pp: Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
⊢ A.EqId ↔ Mono A.e

Tactic:
constructor

Goal state after:
case mp
Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
⊢ A.EqId → Mono A.e

case mpr
Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
⊢ Mono A.e → A.EqId
Module: Mathlib.AlgebraicTopology.DoldKan.FunctorGamma

Goal state before:
pp: j : ℕ
i : Fin (j + 2)
⊢ Isδ₀ (SimplexCategory.δ i) ↔ i = 0

Tactic:
constructor

Goal state after:
case mp
j : ℕ
i : Fin (j + 2)
⊢ Isδ₀ (SimplexCategory.δ i) → i = 0

case mpr
j : ℕ
i : Fin (j + 2)
⊢ i = 0 → Isδ₀ (SimplexCategory.δ i)
Module: Mathlib.AlgebraicTopology.DoldKan.FunctorGamma

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : ChainComplex C ℕ
Δ Δ' : SimplexCategory
i : Δ' ⟶ Δ
inst✝ : Mono i
hi : Isδ₀ i
this : Δ ≠ Δ'
⊢ (if h : Δ = Δ' then eqToHom ⋯ else if h : Isδ₀ i then K.d Δ.len Δ'.len else 0) = K.d Δ.len Δ'.len

Tactic:
simp only [dif_neg this, dif_pos hi]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q n : ℕ
hq : ∀ (i : Fin (n + 1 + 1)), n + 1 + 1 ≤ ↑i + q → X.σ i ≫ (P q).f (n + 1 + 1) = 0
i : Fin (n + 1 + 1)
h : ¬n + 1 + 1 ≤ ↑i + q
hi : n + 1 = ↑i + q
v : HigherFacesVanish q ((P q).f n.succ ≫ X.σ i)
j : Fin (n + 1)
hj : True ∧ ↑j < q
k : ℕ
hk : k + ↑j = n
⊢ i = ⟨↑i, ⋯⟩.castSucc

Tactic:
ext

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q n : ℕ
hq : ∀ (i : Fin (n + 1 + 1)), n + 1 + 1 ≤ ↑i + q → X.σ i ≫ (P q).f (n + 1 + 1) = 0
i : Fin (n + 1 + 1)
h : ¬n + 1 + 1 ≤ ↑i + q
hi : n + 1 = ↑i + q
v : HigherFacesVanish q ((P q).f n.succ ≫ X.σ i)
j : Fin (n + 1)
hj : True ∧ ↑j < q
k : ℕ
hk : k + ↑j = n
⊢ ↑i = ↑⟨↑i, ⋯⟩.castSucc
Module: Mathlib.AlgebraicTopology.SimplicialObject.Split

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op ⦋n⦌) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
f₂ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op ⦋n⦌) = f₂ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₂, comm := c₂ }.f n
⊢ f₁ = f₂

Tactic:
ext

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op ⦋n⦌) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
f₂ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op ⦋n⦌) = f₂ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₂, comm := c₂ }.f n
x✝ : ℕ
⊢ f₁ x✝ = f₂ x✝
Module: Mathlib.AlgebraicTopology.DoldKan.Degeneracies

Goal state before:
pp: case zero
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Δ' : SimplexCategory
θ : ⦋0⦌ ⟶ Δ'
hθ : ¬Function.Injective ⇑(SimplexCategory.Hom.toOrderHom θ)
x y : Fin (⦋0⦌.len + 1)
h : (SimplexCategory.Hom.toOrderHom θ) x = (SimplexCategory.Hom.toOrderHom θ) y
⊢ x = y

Tactic:
fin_cases x

Goal state after:
case zero.«0»
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Δ' : SimplexCategory
θ : ⦋0⦌ ⟶ Δ'
hθ : ¬Function.Injective ⇑(SimplexCategory.Hom.toOrderHom θ)
y : Fin (⦋0⦌.len + 1)
h : (SimplexCategory.Hom.toOrderHom θ) ((fun i ↦ i) ⟨0, ⋯⟩) = (SimplexCategory.Hom.toOrderHom θ) y
⊢ (fun i ↦ i) ⟨0, ⋯⟩ = y
Module: Mathlib.AlgebraicTopology.DoldKan.Normalized

Goal state before:
pp: A : Type u_1
inst✝¹ : Category.{v_1, u_1} A
inst✝ : Abelian A
X✝ X : SimplicialObject A
n : ℕ
⊢ (NormalizedMooreComplex.objX X (n + 1)).factorThru (PInfty.f (n + 1)) ⋯ ≫
      (inclusionOfMooreComplexMap X).f (n + 1) ≫ ((alternatingFaceMapComplex A).obj X).d (n + 1) n =
    K[X].d (n + 1) n ≫ PInfty.f n

Tactic:
inclusionOfMooreComplexMap_f,

Goal state after:
A : Type u_1
inst✝¹ : Category.{v_1, u_1} A
inst✝ : Abelian A
X✝ X : SimplicialObject A
n : ℕ
⊢ (NormalizedMooreComplex.objX X (n + 1)).factorThru (PInfty.f (n + 1)) ⋯ ≫
      (NormalizedMooreComplex.objX X (n + 1)).arrow ≫ ((alternatingFaceMapComplex A).obj X).d (n + 1) n =
    K[X].d (n + 1) n ≫ PInfty.f n
Module: Mathlib.AlgebraicTopology.DoldKan.Normalized

Goal state before:
pp: A : Type u_1
inst✝¹ : Category.{v_1, u_1} A
inst✝ : Abelian A
X : SimplicialObject A
Z✝ : ChainComplex A ℕ
x✝¹ x✝ : Z✝ ⟶ (normalizedMooreComplex A).obj X
hf : x✝¹ ≫ inclusionOfMooreComplexMap X = x✝ ≫ inclusionOfMooreComplexMap X
n : ℕ
⊢ x✝¹.f n = x✝.f n

Tactic:
ext

Goal state after:
A : Type u_1
inst✝¹ : Category.{v_1, u_1} A
inst✝ : Abelian A
X : SimplicialObject A
Z✝ : ChainComplex A ℕ
x✝¹ x✝ : Z✝ ⟶ (normalizedMooreComplex A).obj X
hf : x✝¹ ≫ inclusionOfMooreComplexMap X = x✝ ≫ inclusionOfMooreComplexMap X
n : ℕ
⊢ x✝¹.f n ≫ (NormalizedMooreComplex.objX X n).arrow = x✝.f n ≫ (NormalizedMooreComplex.objX X n).arrow
Module: Mathlib.CategoryTheory.Idempotents.HomologicalComplex

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
ι : Type u_2
c : ComplexShape ι
P Q : Karoubi (HomologicalComplex C c)
φ : P ⟶ Q
⊢ (𝟭 (Karoubi (HomologicalComplex C c))).map φ ≫ { f := { f := fun n ↦ Q.p.f n, comm' := ⋯ }, comm := ⋯ } =
    { f := { f := fun n ↦ P.p.f n, comm' := ⋯ }, comm := ⋯ } ≫ (functor ⋙ inverse).map φ

Tactic:
ext

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
ι : Type u_2
c : ComplexShape ι
P Q : Karoubi (HomologicalComplex C c)
φ : P ⟶ Q
i✝ : ι
⊢ ((𝟭 (Karoubi (HomologicalComplex C c))).map φ ≫ { f := { f := fun n ↦ Q.p.f n, comm' := ⋯ }, comm := ⋯ }).f.f i✝ =
    ({ f := { f := fun n ↦ P.p.f n, comm' := ⋯ }, comm := ⋯ } ≫ (functor ⋙ inverse).map φ).f.f i✝
Module: Mathlib.AlgebraicTopology.DoldKan.GammaCompN

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
n : ℕ
⊢ K.d (n + 1) n =
    ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫
      ((-1) ^ ↑0 • (Γ₀.obj K).δ 0) ≫ (Γ₀.splitting K).πSummand (Splitting.IndexSet.id (op ⦋n⦌))

Tactic:
simp only [Fin.val_zero, pow_zero, one_zsmul]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
n : ℕ
⊢ K.d (n + 1) n =
    ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫
      (Γ₀.obj K).δ 0 ≫ (Γ₀.splitting K).πSummand (Splitting.IndexSet.id (op ⦋n⦌))
Module: Mathlib.CategoryTheory.Idempotents.HomologicalComplex

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
ι : Type u_2
c : ComplexShape ι
P Q : Karoubi (HomologicalComplex C c)
φ : P ⟶ Q
⊢ (functor ⋙ inverse).map φ ≫ { f := { f := fun n ↦ Q.p.f n, comm' := ⋯ }, comm := ⋯ } =
    { f := { f := fun n ↦ P.p.f n, comm' := ⋯ }, comm := ⋯ } ≫ (𝟭 (Karoubi (HomologicalComplex C c))).map φ

Tactic:
ext

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
ι : Type u_2
c : ComplexShape ι
P Q : Karoubi (HomologicalComplex C c)
φ : P ⟶ Q
i✝ : ι
⊢ ((functor ⋙ inverse).map φ ≫ { f := { f := fun n ↦ Q.p.f n, comm' := ⋯ }, comm := ⋯ }).f.f i✝ =
    ({ f := { f := fun n ↦ P.p.f n, comm' := ⋯ }, comm := ⋯ } ≫ (𝟭 (Karoubi (HomologicalComplex C c))).map φ).f.f i✝
Module: Mathlib.CategoryTheory.Idempotents.HomologicalComplex

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
ι : Type u_2
c : ComplexShape ι
⊢ { app := fun P ↦ { f := { f := fun n ↦ P.p.f n, comm' := ⋯ }, comm := ⋯ }, naturality := ⋯ } ≫
      { app := fun P ↦ { f := { f := fun n ↦ P.p.f n, comm' := ⋯ }, comm := ⋯ }, naturality := ⋯ } =
    𝟙 (𝟭 (Karoubi (HomologicalComplex C c)))

Tactic:
ext

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
ι : Type u_2
c : ComplexShape ι
x✝ : Karoubi (HomologicalComplex C c)
i✝ : ι
⊢ (({ app := fun P ↦ { f := { f := fun n ↦ P.p.f n, comm' := ⋯ }, comm := ⋯ }, naturality := ⋯ } ≫
                { app := fun P ↦ { f := { f := fun n ↦ P.p.f n, comm' := ⋯ }, comm := ⋯ }, naturality := ⋯ }).app
            x✝).f.f
      i✝ =
    ((𝟙 (𝟭 (Karoubi (HomologicalComplex C c)))).app x✝).f.f i✝
Module: Mathlib.CategoryTheory.Idempotents.HomologicalComplex

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
ι : Type u_2
c : ComplexShape ι
⊢ { app := fun P ↦ { f := { f := fun n ↦ P.p.f n, comm' := ⋯ }, comm := ⋯ }, naturality := ⋯ } ≫
      { app := fun P ↦ { f := { f := fun n ↦ P.p.f n, comm' := ⋯ }, comm := ⋯ }, naturality := ⋯ } =
    𝟙 (functor ⋙ inverse)

Tactic:
ext

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
ι : Type u_2
c : ComplexShape ι
x✝ : Karoubi (HomologicalComplex C c)
i✝ : ι
⊢ (({ app := fun P ↦ { f := { f := fun n ↦ P.p.f n, comm' := ⋯ }, comm := ⋯ }, naturality := ⋯ } ≫
                { app := fun P ↦ { f := { f := fun n ↦ P.p.f n, comm' := ⋯ }, comm := ⋯ }, naturality := ⋯ }).app
            x✝).f.f
      i✝ =
    ((𝟙 (functor ⋙ inverse)).app x✝).f.f i✝
Module: Mathlib.AlgebraicTopology.DoldKan.GammaCompN

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
⊢ N₁Γ₀.app K =
    (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.symm ≪≫ (toKaroubi (ChainComplex C ℕ)).mapIso (Γ₀NondegComplexIso K)

Tactic:
ext

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
i✝ : ℕ
⊢ (N₁Γ₀.app K).hom.f.f i✝ =
    ((Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.symm ≪≫
              (toKaroubi (ChainComplex C ℕ)).mapIso (Γ₀NondegComplexIso K)).hom.f.f
      i✝
Module: Mathlib.AlgebraicTopology.DoldKan.FunctorGamma

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
K : ChainComplex C ℕ
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
⊢ ((splitting K).cofan Δ).inj A ≫ (obj K).map θ =
    Termwise.mapMono K i ≫ ((splitting K).cofan Δ').inj (Splitting.IndexSet.mk e)

Tactic:
dsimp [Splitting.cofan]
  change (_ ≫ (Γ₀.obj K).map A.e.op) ≫ (Γ₀.obj K).map θ = _
  rw [assoc, ← Functor.map_comp]
  dsimp [splitting]
  rw [Γ₀.Obj.map_on_summand₀ K (Splitting.IndexSet.id A.1)
    (show e ≫ i = ((Splitting.IndexSet.e A).op ≫ θ).unop ≫ 𝟙 _ by rw [comp_id, fac]; rfl)]
  dsimp only [Splitting.IndexSet.id_fst, Splitting.IndexSet.mk, op_unop, Splitting.IndexSet.e]
  rw [Γ₀.Obj.map_on_summand₀ K (Splitting.IndexSet.id (op Δ''))
      (show e ≫ 𝟙 Δ'' = e.op.unop ≫ 𝟙 _ by simp), Termwise.mapMono_id]
  dsimp only [Splitting.IndexSet.id_fst]
  rw [id_comp]
  rfl

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : ChainComplex C ℕ
inst✝ : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
⊢ factorThruImage (θ.unop ≫ A.e) ≫ image.ι (θ.unop ≫ A.e) = θ.unop ≫ A.e

Tactic:
apply image.fac

Goal state after:
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
Z : C
n : ℕ
f : Z ⟶ X _⦋n⦌
⊢ f ≫ PInfty.f n = 0 ↔ f ≫ s.πSummand (IndexSet.id (op ⦋n⦌)) = 0

Tactic:
constructor

Goal state after:
case mp
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
Z : C
n : ℕ
f : Z ⟶ X _⦋n⦌
⊢ f ≫ PInfty.f n = 0 → f ≫ s.πSummand (IndexSet.id (op ⦋n⦌)) = 0

case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
Z : C
n : ℕ
f : Z ⟶ X _⦋n⦌
⊢ f ≫ s.πSummand (IndexSet.id (op ⦋n⦌)) = 0 → f ≫ PInfty.f n = 0
Module: Mathlib.AlgebraicTopology.DoldKan.NReflectsIso

Goal state before:
pp: case refine_1.refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
P : Karoubi (SimplicialObject C)
n : ℕ
⊢ ((N₂ ⋙ (karoubiChainComplexEquivalence C ℕ).functor).obj P).d (n + 1) n ≫ eqToHom ⋯ =
    eqToHom ⋯ ≫
      ((karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C ⋙
                N₁ ⋙
                  (karoubiChainComplexEquivalence (Karoubi C) ℕ).functor ⋙
                    (KaroubiKaroubi.equivalence C).inverse.mapHomologicalComplex (ComplexShape.down ℕ)).obj
            P).d
        (n + 1) n

Tactic:
ext

Goal state after:
case refine_1.refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
P : Karoubi (SimplicialObject C)
n : ℕ
⊢ (((N₂ ⋙ (karoubiChainComplexEquivalence C ℕ).functor).obj P).d (n + 1) n ≫ eqToHom ⋯).f =
    (eqToHom ⋯ ≫
        ((karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C ⋙
                  N₁ ⋙
                    (karoubiChainComplexEquivalence (Karoubi C) ℕ).functor ⋙
                      (KaroubiKaroubi.equivalence C).inverse.mapHomologicalComplex (ComplexShape.down ℕ)).obj
              P).d
          (n + 1) n).f
Module: Mathlib.AlgebraicTopology.DoldKan.EquivalencePseudoabelian

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
⊢ (N₂.map (isoΓ₀.hom.app X)).f = PInfty

Tactic:
ext

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
i✝ : ℕ
⊢ (N₂.map (isoΓ₀.hom.app X)).f.f i✝ = PInfty.f i✝
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
m k : ℕ
i : ⦋m⦌ ⟶ ⦋m + k + 1 + 1⦌
h₁ : ⦋m⦌.len ≠ m + k + 1 + 1
h : Function.Surjective ⇑(Hom.toOrderHom i)
⊢ False

Tactic:
by
        rw [← SimplexCategory.epi_iff_surjective] at h
        grind [→ le_of_epi]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
m k : ℕ
h₁ : ⦋m⦌.len ≠ m + k + 1 + 1
j₁ : Fin (m + k + 2 + 1)
i : ⦋m⦌ ⟶ ⦋m + k + 1⦌
h : Function.Surjective ⇑(Hom.toOrderHom i)
⊢ False

Tactic:
by
        rw [← SimplexCategory.epi_iff_surjective] at h
        grind [→ le_of_epi]

Goal state after: