-- Mathlib/AlgebraicTopology/DoldKan/Faces.lean
theorem comp_Hσ_eq {Y : C} {n a q : ℕ} {φ : Y ⟶ X _⦋n + 1⦌} (v : HigherFacesVanish q φ)
    (hnaq : n = a + q) :
    φ ≫ (Hσ q).f (n + 1) =
      -φ ≫ X.δ ⟨a + 1, Nat.succ_lt_succ (Nat.lt_succ_iff.mpr (Nat.le.intro hnaq.symm))⟩ ≫
        X.σ ⟨a, Nat.lt_succ_iff.mpr (Nat.le.intro hnaq.symm)⟩

Proof state:
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
⊢ ∑ i : Fin (a + 2),
        ((-1) ^ a * (-1) ^ ↑(Fin.cast ⋯ (Fin.castLE ⋯ i))) • (φ ≫ X.δ (Fin.cast ⋯ (Fin.castLE ⋯ i))) ≫ X.σ ⟨a, ⋯⟩ +
      ∑ x : Fin (n + 1 + 2), ((-1) ^ ↑x * (-1) ^ (a + 1)) • (φ ≫ X.σ ⟨a + 1, ⋯⟩) ≫ X.δ x =
    -(φ ≫ X.δ ⟨a + 1, ⋯⟩) ≫ X.σ ⟨a, ⋯⟩

case hf
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
⊢ ∀ (j : Fin q),
    ((-1) ^ a * (-1) ^ ↑(Fin.cast ⋯ (Fin.natAdd (a + 2) j))) •
        (φ ≫ X.δ (Fin.cast ⋯ (Fin.natAdd (a + 2) j))) ≫ X.σ ⟨a, ⋯⟩ =
      0

Tactic:
swap

Result:
case hf
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
⊢ ∀ (j : Fin q),
    ((-1) ^ a * (-1) ^ ↑(Fin.cast ⋯ (Fin.natAdd (a + 2) j))) •
        (φ ≫ X.δ (Fin.cast ⋯ (Fin.natAdd (a + 2) j))) ≫ X.σ ⟨a, ⋯⟩ =
      0

C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
⊢ ∑ i : Fin (a + 2),
        ((-1) ^ a * (-1) ^ ↑(Fin.cast ⋯ (Fin.castLE ⋯ i))) • (φ ≫ X.δ (Fin.cast ⋯ (Fin.castLE ⋯ i))) ≫ X.σ ⟨a, ⋯⟩ +
      ∑ x : Fin (n + 1 + 2), ((-1) ^ ↑x * (-1) ^ (a + 1)) • (φ ≫ X.σ ⟨a + 1, ⋯⟩) ≫ X.δ x =
    -(φ ≫ X.δ ⟨a + 1, ⋯⟩) ≫ X.σ ⟨a, ⋯⟩

Proof state:
case a.h.mk
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
simplif : ∀ (a b c d e f : Y ⟶ X _⦋n + 1⦌), b = f → d + e = 0 → c + a = 0 → a + b + (c + d + e) = f
i : ℕ
hi : i < a + 1
a✝ : ⟨i, hi⟩ ∈ Finset.univ
⊢ ((-1) ^ ↑⟨i, hi⟩ * (-1) ^ (a + 1)) • φ ≫ X.σ ⟨a + 1, ⋯⟩ ≫ X.δ ⟨↑⟨i, hi⟩, ⋯⟩ +
      ((-1) ^ a * (-1) ^ ↑⟨i, hi⟩) • φ ≫ X.δ ⟨↑⟨i, hi⟩, ⋯⟩ ≫ X.σ ⟨a, ⋯⟩ =
    0

Tactic:
simp only

Result:
case a.h.mk
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
simplif : ∀ (a b c d e f : Y ⟶ X _⦋n + 1⦌), b = f → d + e = 0 → c + a = 0 → a + b + (c + d + e) = f
i : ℕ
hi : i < a + 1
a✝ : ⟨i, hi⟩ ∈ Finset.univ
⊢ ((-1) ^ i * (-1) ^ (a + 1)) • φ ≫ X.σ ⟨a + 1, ⋯⟩ ≫ X.δ ⟨i, ⋯⟩ + ((-1) ^ a * (-1) ^ i) • φ ≫ X.δ ⟨i, ⋯⟩ ≫ X.σ ⟨a, ⋯⟩ =
    0

Proof state:
case hf.mk.hj₁
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
k : ℕ
hk : k < q
h : ↑(Fin.cast ⋯ (Fin.natAdd (a + 3) ⟨k, hk⟩)) = ↑(Fin.succ 0)
⊢ False

Tactic:
dsimp [Fin.cast] at h

Result:
case hf.mk.hj₁
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
k : ℕ
hk : k < q
h : a + 3 + k = 1
⊢ False

Proof state:
case a.h.mk
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
simplif : ∀ (a b c d e f : Y ⟶ X _⦋n + 1⦌), b = f → d + e = 0 → c + a = 0 → a + b + (c + d + e) = f
i : ℕ
hi : i < a + 1
a✝ : ⟨i, hi⟩ ∈ Finset.univ
hia : ⟨i, ⋯⟩ ≤ ⟨a, ⋯⟩.castSucc
⊢ ((-1) ^ i * (-1) ^ (a + 1)) • φ ≫ X.δ ⟨i, ⋯⟩ ≫ X.σ ⟨a, ⋯⟩ = -((-1) ^ a * (-1) ^ i) • φ ≫ X.δ ⟨i, ⋯⟩ ≫ X.σ ⟨a, ⋯⟩

Tactic:
congr 2

Result:
case a.h.mk.e_a
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
simplif : ∀ (a b c d e f : Y ⟶ X _⦋n + 1⦌), b = f → d + e = 0 → c + a = 0 → a + b + (c + d + e) = f
i : ℕ
hi : i < a + 1
a✝ : ⟨i, hi⟩ ∈ Finset.univ
hia : ⟨i, ⋯⟩ ≤ ⟨a, ⋯⟩.castSucc
⊢ (-1) ^ i * (-1) ^ (a + 1) = -((-1) ^ a * (-1) ^ i)

Proof state:
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
⊢ ∑ i : Fin (a + 2),
        ((-1) ^ a * (-1) ^ ↑(Fin.cast ⋯ (Fin.castLE ⋯ i))) • (φ ≫ X.δ (Fin.cast ⋯ (Fin.castLE ⋯ i))) ≫ X.σ ⟨a, ⋯⟩ +
      ∑ i : Fin (a + 3),
        ((-1) ^ ↑(Fin.cast ⋯ (Fin.castLE ⋯ i)) * (-1) ^ (a + 1)) •
          (φ ≫ X.σ ⟨a + 1, ⋯⟩) ≫ X.δ (Fin.cast ⋯ (Fin.castLE ⋯ i)) =
    -(φ ≫ X.δ ⟨a + 1, ⋯⟩) ≫ X.σ ⟨a, ⋯⟩

case hf
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
⊢ ∀ (j : Fin q),
    ((-1) ^ ↑(Fin.cast ⋯ (Fin.natAdd (a + 3) j)) * (-1) ^ (a + 1)) •
        (φ ≫ X.σ ⟨a + 1, ⋯⟩) ≫ X.δ (Fin.cast ⋯ (Fin.natAdd (a + 3) j)) =
      0

Tactic:
swap

Result:
case hf
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
⊢ ∀ (j : Fin q),
    ((-1) ^ ↑(Fin.cast ⋯ (Fin.natAdd (a + 3) j)) * (-1) ^ (a + 1)) •
        (φ ≫ X.σ ⟨a + 1, ⋯⟩) ≫ X.δ (Fin.cast ⋯ (Fin.natAdd (a + 3) j)) =
      0

C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
⊢ ∑ i : Fin (a + 2),
        ((-1) ^ a * (-1) ^ ↑(Fin.cast ⋯ (Fin.castLE ⋯ i))) • (φ ≫ X.δ (Fin.cast ⋯ (Fin.castLE ⋯ i))) ≫ X.σ ⟨a, ⋯⟩ +
      ∑ i : Fin (a + 3),
        ((-1) ^ ↑(Fin.cast ⋯ (Fin.castLE ⋯ i)) * (-1) ^ (a + 1)) •
          (φ ≫ X.σ ⟨a + 1, ⋯⟩) ≫ X.δ (Fin.cast ⋯ (Fin.castLE ⋯ i)) =
    -(φ ≫ X.δ ⟨a + 1, ⋯⟩) ≫ X.σ ⟨a, ⋯⟩

Proof state:
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
simplif : ∀ (a b c d e f : Y ⟶ X _⦋n + 1⦌), b = f → d + e = 0 → c + a = 0 → a + b + (c + d + e) = f
i : ℕ
hi : i < a + 1
a✝ : ⟨i, hi⟩ ∈ Finset.univ
⊢ i < n + 2

Tactic:
omega

Result:
no goals