mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: case mp
R : Type w
inst✝ : CommRing R
a b : R
ha : a ∈ R⁰
x c : R
h : c • (a * b) = a • x
⊢ ∃ a_1, a_1 • (mk (R ∙ a * b)) b = (mk (R ∙ a * b)) x

Tactic:
smul_eq_mul,

Goal state after:
case mp
R : Type w
inst✝ : CommRing R
a b : R
ha : a ∈ R⁰
x c : R
h : c * (a * b) = a • x
⊢ ∃ a_1, a_1 • (mk (R ∙ a * b)) b = (mk (R ∙ a * b)) x
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: case mp
R : Type w
inst✝ : CommRing R
a b : R
ha : a ∈ R⁰
x c : R
h : c * (a * b) = a • x
⊢ ∃ a_1, a_1 • (mk (R ∙ a * b)) b = (mk (R ∙ a * b)) x

Tactic:
smul_eq_mul,

Goal state after:
case mp
R : Type w
inst✝ : CommRing R
a b : R
ha : a ∈ R⁰
x c : R
h : c * (a * b) = a * x
⊢ ∃ a_1, a_1 • (mk (R ∙ a * b)) b = (mk (R ∙ a * b)) x
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: case h
R : Type w
inst✝ : CommRing R
a b : R
ha : a ∈ R⁰
x c : R
h : c * b = x
⊢ Submodule.Quotient.mk (c • b) = (mk (R ∙ a * b)) (c * b)

Tactic:
smul_eq_mul,

Goal state after:
case h
R : Type w
inst✝ : CommRing R
a b : R
ha : a ∈ R⁰
x c : R
h : c * b = x
⊢ Submodule.Quotient.mk (c * b) = (mk (R ∙ a * b)) (c * b)
Module: Mathlib.Algebra.Category.ModuleCat.Differentials.Basic

Goal state before:
pp: A B : CommRingCat
f : A ⟶ B
M : ModuleCat ↑B
α β : KaehlerDifferential f ⟶ M
h : ∀ (b : ↑B), (ConcreteCategory.hom α) (d b) = (ConcreteCategory.hom β) (d b)
this : ⊤ ≤ (ModuleCat.Hom.hom (α - β)).ker
⊢ α - β = 0

Tactic:
rw [top_le_iff, LinearMap.ker_eq_top] at this

Goal state after:
A B : CommRingCat
f : A ⟶ B
M : ModuleCat ↑B
α β : KaehlerDifferential f ⟶ M
h : ∀ (b : ↑B), (ConcreteCategory.hom α) (d b) = (ConcreteCategory.hom β) (d b)
this : ModuleCat.Hom.hom (α - β) = 0
⊢ α - β = 0
Module: Mathlib.RingTheory.Ideal.Cotangent

Goal state before:
pp: R : Type u
inst✝ : CommRing R
I : Ideal R
⊢ I • Submodule.map (Submodule.subtype I) ⊤ = I * I

Tactic:
smul_eq_mul,

Goal state after:
R : Type u
inst✝ : CommRing R
I : Ideal R
⊢ I * Submodule.map (Submodule.subtype I) ⊤ = I * I
Module: Mathlib.RingTheory.Ideal.Cotangent

Goal state before:
pp: R : Type u
S : Type v
S' : Type w
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra S R
inst✝³ : CommSemiring S'
inst✝² : Algebra S' R
inst✝¹ : Algebra S S'
inst✝ : IsScalarTower S S' R
I : Ideal R
⊢ I • I ≤ I ^ 2

Tactic:
smul_eq_mul,

Goal state after:
R : Type u
S : Type v
S' : Type w
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra S R
inst✝³ : CommSemiring S'
inst✝² : Algebra S' R
inst✝¹ : Algebra S S'
inst✝ : IsScalarTower S S' R
I : Ideal R
⊢ I * I ≤ I ^ 2
Module: Mathlib.RingTheory.Ideal.Cotangent

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsLocalRing R
inst✝ : IsNoetherianRing R
M : Submodule R ↥(maximalIdeal R)
H : Submodule.map (maximalIdeal R).toCotangent M = ⊤
⊢ maximalIdeal R ≤ Submodule.map (Submodule.subtype (maximalIdeal R)) M ⊔ maximalIdeal R • maximalIdeal R

Tactic:
smul_eq_mul,

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsLocalRing R
inst✝ : IsNoetherianRing R
M : Submodule R ↥(maximalIdeal R)
H : Submodule.map (maximalIdeal R).toCotangent M = ⊤
⊢ maximalIdeal R ≤ Submodule.map (Submodule.subtype (maximalIdeal R)) M ⊔ maximalIdeal R * maximalIdeal R
Module: Mathlib.RingTheory.Filtration

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
I : Ideal R
inst✝¹ : IsNoetherianRing R
inst✝ : IsLocalRing R
h : I ≠ ⊤
x✝ : ℕ
i : R
⊢ i ∈ I ^ x✝ ↔ i ∈ I ^ x✝ • ⊤

Tactic:
smul_eq_mul,

Goal state after:
R : Type u_1
inst✝² : CommRing R
I : Ideal R
inst✝¹ : IsNoetherianRing R
inst✝ : IsLocalRing R
h : I ≠ ⊤
x✝ : ℕ
i : R
⊢ i ∈ I ^ x✝ ↔ i ∈ I ^ x✝ * ⊤
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:
rw [Set.mem_preimage, Set.mem_singleton_iff, hpqr, hi]

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

Goal state before:
pp: case pos
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
h : i✝ = j
⊢ eqToHom ⋯ ≫ 𝟙 A ≫ eqToHom ⋯ = 𝟙 (if i✝ = j then A else 0)

pp: case neg
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
h : ¬i✝ = j
⊢ 0 = 𝟙 (if i✝ = j then A else 0)

Tactic:
· subst h
      simp

Goal state after:
case neg
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
h : ¬i✝ = j
⊢ 0 = 𝟙 (if i✝ = j then A else 0)
Module: Mathlib.Algebra.Homology.Single

Goal state before:
pp: case pos
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✝
i✝ : ι
h : i✝ = j
⊢ eqToHom ⋯ ≫ (f ≫ g) ≫ eqToHom ⋯ = (eqToHom ⋯ ≫ f ≫ eqToHom ⋯) ≫ eqToHom ⋯ ≫ g ≫ eqToHom ⋯

pp: case neg
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✝
i✝ : ι
h : ¬i✝ = j
⊢ 0 = 0 ≫ 0

Tactic:
· subst h
      simp

Goal state after:
case neg
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✝
i✝ : ι
h : ¬i✝ = j
⊢ 0 = 0 ≫ 0
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
x y : S
⊢ x ⊗ₜ[R] y - (x * y) ⊗ₜ[R] 1 = (x • 1) ⊗ₜ[R] y - (x • y) ⊗ₜ[R] 1

Tactic:
smul_eq_mul,

Goal state after:
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x✝ : S ⊗[R] S
x y : S
⊢ x ⊗ₜ[R] y - (x * y) ⊗ₜ[R] 1 = (x * 1) ⊗ₜ[R] y - (x * y) ⊗ₜ[R] 1
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
j : ι
h : ¬c.Rel (c.prev j) j
⊢ (if h : ∃ j_1, c.Rel j_1 j then h.choose else j) = j

Tactic:
rw [dif_neg]

Goal state after:
case hnc
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C : HomologicalComplex V c
j : ι
h : ¬c.Rel (c.prev j) j
⊢ ¬∃ j_1, c.Rel j_1 j
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 : ι
h : ¬c.Rel i (c.next i)
⊢ (if h : ∃ j, c.Rel i j then h.choose else i) = i

Tactic:
rw [dif_neg]

Goal state after:
case hnc
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C : HomologicalComplex V c
i : ι
h : ¬c.Rel i (c.next i)
⊢ ¬∃ j, c.Rel i j
Module: Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
i : ι
inst✝ : K.HasHomology i
A : C
k : A ⟶ K.X i
j : ι
hj : c.next i = j
i' : ι
x : A ⟶ K.X i'
hx : k = x ≫ K.d i' i
⊢ K.liftCycles k j hj ⋯ ≫ K.homologyπ i = 0

Tactic:
by_cases h : c.Rel i' i

Goal state after:
case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
i : ι
inst✝ : K.HasHomology i
A : C
k : A ⟶ K.X i
j : ι
hj : c.next i = j
i' : ι
x : A ⟶ K.X i'
hx : k = x ≫ K.d i' i
h : c.Rel i' i
⊢ K.liftCycles k j hj ⋯ ≫ K.homologyπ i = 0

case neg
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
i : ι
inst✝ : K.HasHomology i
A : C
k : A ⟶ K.X i
j : ι
hj : c.next i = j
i' : ι
x : A ⟶ K.X i'
hx : k = x ≫ K.d i' i
h : ¬c.Rel i' i
⊢ K.liftCycles k j hj ⋯ ≫ K.homologyπ i = 0
Module: Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
iso : K ≅ L
ψ : L ⟶ M
i j k : ι
inst✝ : K.HasHomology i
⊢ Mono (K.homologyι i)

Tactic:
dsimp only [homologyι]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
iso : K ≅ L
ψ : L ⟶ M
i j k : ι
inst✝ : K.HasHomology i
⊢ Mono (K.sc i).homologyι
Module: Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
i j : ι
inst✝ : K.HasHomology i
hij : ¬c.Rel i j
⊢ K.pOpcycles i ≫ K.fromOpcycles i j = K.pOpcycles i ≫ 0

Tactic:
p_fromOpcycles,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
i j : ι
inst✝ : K.HasHomology i
hij : ¬c.Rel i j
⊢ K.d i j = K.pOpcycles i ≫ 0
Module: Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
iso : K ≅ L
ψ : L ⟶ M
i j k : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
⊢ homologyMap iso.hom i ≫ homologyMap iso.inv i = 𝟙 (K.homology i)

Tactic:
simp [← homologyMap_comp]

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
iso : K ≅ L
ψ : L ⟶ M
i j k : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
⊢ homologyMap iso.inv i ≫ homologyMap iso.hom i = 𝟙 (L.homology i)

Tactic:
simp [← homologyMap_comp]

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
p q : ℤ
hpq : p + n₁₂ = q
⊢ (z₁.comp (-z₂) h).v p q hpq = (-z₁.comp z₂ h).v p q hpq

Tactic:
simp only [comp_v _ _ h p _ q rfl (by lia), neg_v, comp_neg]

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n m : ℤ
hnm : n + 1 = m
z : Cochain F G n
p q : ℤ
hpq : p + m = q
q₁ q₂ : ℤ
hq₁ : q₁ = q - 1
hq₂ : p + 1 = q₂
⊢ (δ n m z).v p q hpq = z.v p q₁ ⋯ ≫ G.d q₁ q + m.negOnePow • F.d p q₂ ≫ z.v q₂ q ⋯

Tactic:
obtain rfl : q₁ = p + n := by lia

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n m : ℤ
hnm : n + 1 = m
z : Cochain F G n
p q : ℤ
hpq : p + m = q
q₂ : ℤ
hq₂ : p + 1 = q₂
hq₁ : p + n = q - 1
⊢ (δ n m z).v p q hpq = z.v p (p + n) ⋯ ≫ G.d (p + n) q + m.negOnePow • F.d p q₂ ≫ z.v q₂ q ⋯
Module: Mathlib.Algebra.Homology.Homotopy

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f g : C ⟶ D
h k : D ⟶ E
i✝ : ι
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
  (n : ℕ) →
    (p :
        (f : P.X n ⟶ Q.X (n + 1)) ×'
          (f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
      (f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
i j : ℕ
w : ¬(ComplexShape.down ℕ).Rel j i
⊢ (if h : i + 1 = j then (mkInductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst ≫ (xPrevIso Q h).hom else 0) = 0

Tactic:
rw [dif_neg]

Goal state after:
case hnc
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f g : C ⟶ D
h k : D ⟶ E
i✝ : ι
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
  (n : ℕ) →
    (p :
        (f : P.X n ⟶ Q.X (n + 1)) ×'
          (f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
      (f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
i j : ℕ
w : ¬(ComplexShape.down ℕ).Rel j i
⊢ ¬i + 1 = j
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
r : R
z : Cocycle F G n
hz : ↑z ∈ cocycle F G n
⊢ r • ↑z ∈ cocycle F G n

Tactic:
rw [mem_iff n (n + 1) rfl] at hz ⊢

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
r : R
z : Cocycle F G n
hz : δ n (n + 1) ↑z = 0
⊢ δ n (n + 1) (r • ↑z) = 0
Module: Mathlib.Algebra.Homology.HomotopyCategory.MappingCone

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
h : n + 1 = m
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + n = p₂
h₂₃ : p₂ + 1 = p₃
⊢ (liftCochain φ α β h).v p₁ p₂ h₁₂ ≫ (↑(fst φ)).v p₂ p₃ h₂₃ = α.v p₁ p₃ ⋯

Tactic:
by
  simpa only [Cochain.comp_v _ _ h p₁ p₂ p₃ h₁₂ h₂₃]
    using Cochain.congr_v (liftCochain_fst φ α β h) p₁ p₃ (by lia)

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

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f g : C ⟶ D
h k : D ⟶ E
i✝ : ι
P Q : CochainComplex V ℕ
e : P ⟶ Q
zero : P.X 1 ⟶ Q.X 0
comm_zero : e.f 0 = P.d 0 1 ≫ zero
one : P.X 2 ⟶ Q.X 1
comm_one : e.f 1 = zero ≫ Q.d 0 1 + P.d 1 2 ≫ one
succ :
  (n : ℕ) →
    (p :
        (f : P.X (n + 1) ⟶ Q.X n) ×'
          (f' : P.X (n + 2) ⟶ Q.X (n + 1)) ×' e.f (n + 1) = f ≫ Q.d n (n + 1) + P.d (n + 1) (n + 2) ≫ f') →
      (f'' : P.X (n + 3) ⟶ Q.X (n + 2)) ×' e.f (n + 2) = p.snd.fst ≫ Q.d (n + 1) (n + 2) + P.d (n + 2) (n + 3) ≫ f''
i j : ℕ
w : ¬(ComplexShape.up ℕ).Rel j i
⊢ (if h : j + 1 = i then (xNextIso P h).inv ≫ (mkCoinductiveAux₂ e zero comm_zero one comm_one succ j).snd.fst else 0) =
    0

Tactic:
rw [dif_neg]

Goal state after:
case hnc
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f g : C ⟶ D
h k : D ⟶ E
i✝ : ι
P Q : CochainComplex V ℕ
e : P ⟶ Q
zero : P.X 1 ⟶ Q.X 0
comm_zero : e.f 0 = P.d 0 1 ≫ zero
one : P.X 2 ⟶ Q.X 1
comm_one : e.f 1 = zero ≫ Q.d 0 1 + P.d 1 2 ≫ one
succ :
  (n : ℕ) →
    (p :
        (f : P.X (n + 1) ⟶ Q.X n) ×'
          (f' : P.X (n + 2) ⟶ Q.X (n + 1)) ×' e.f (n + 1) = f ≫ Q.d n (n + 1) + P.d (n + 1) (n + 2) ≫ f') →
      (f'' : P.X (n + 3) ⟶ Q.X (n + 2)) ×' e.f (n + 2) = p.snd.fst ≫ Q.d (n + 1) (n + 2) + P.d (n + 2) (n + 3) ≫ f''
i j : ℕ
w : ¬(ComplexShape.up ℕ).Rel j i
⊢ ¬j + 1 = i
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
p q : ℤ
f : K.X p ⟶ L.X q
n p' q' : ℤ
hpq' : p' + n = q'
hp' : p' ≠ p
⊢ (if h : p = p' ∧ q = q' then (HomologicalComplex.XIsoOfEq K ⋯).inv ≫ f ≫ (HomologicalComplex.XIsoOfEq L ⋯).hom
    else 0) =
    0

Tactic:
rw [dif_neg]

Goal state after:
case hnc
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
p q : ℤ
f : K.X p ⟶ L.X q
n p' q' : ℤ
hpq' : p' + n = q'
hp' : p' ≠ p
⊢ ¬(p = p' ∧ q = q')
Module: Mathlib.Algebra.Homology.Homotopy

Goal state before:
pp: ι✝ : Type u_1
V : Type u
inst✝⁶ : Category.{v, u} V
inst✝⁵ : Preadditive V
c✝ : ComplexShape ι✝
C✝ D E : HomologicalComplex V c✝
f✝ g✝ : C✝ ⟶ D
h✝ k : D ⟶ E
i✝ : ι✝
C : Type u_2
inst✝⁴ : Category.{v_1, u_2} C
inst✝³ : Preadditive C
ι : Type ?u.42
c : ComplexShape ι
inst✝² : DecidableRel c.Rel
K L : HomologicalComplex C c
f g : K ⟶ L
h : HomotopyEquiv K L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
⊢ homologyMap h.inv i ≫ homologyMap h.hom i = 𝟙 (L.homology i)

Tactic:
rw [← homologyMap_comp, h.homotopyInvHomId.homologyMap_eq, homologyMap_id]

Goal state after:
Module: Mathlib.CategoryTheory.Quotient.Linear

Goal state before:
pp: R : Type u_1
C : Type u_2
inst✝⁶ : Semiring R
inst✝⁵ : Category.{v_1, u_2} C
inst✝⁴ : Preadditive C
inst✝³ : Linear R C
r : HomRel C
inst✝² : Congruence r
hr : ∀ (a : R) ⦃X Y : C⦄ (f₁ f₂ : X ⟶ Y), r f₁ f₂ → r (a • f₁) (a • f₂)
inst✝¹ : Preadditive (Quotient r)
inst✝ : (functor r).Additive
⊢ Functor.Linear R (functor r)

Tactic:
let := linear R r hr

Goal state after:
R : Type u_1
C : Type u_2
inst✝⁶ : Semiring R
inst✝⁵ : Category.{v_1, u_2} C
inst✝⁴ : Preadditive C
inst✝³ : Linear R C
r : HomRel C
inst✝² : Congruence r
hr : ∀ (a : R) ⦃X Y : C⦄ (f₁ f₂ : X ⟶ Y), r f₁ f₂ → r (a • f₁) (a • f₂)
inst✝¹ : Preadditive (Quotient r)
inst✝ : (functor r).Additive
this : Linear R (Quotient r) := linear R r hr
⊢ Functor.Linear R (functor r)
Module: Mathlib.CategoryTheory.Shift.Induced

Goal state before:
pp: C : Type u_3
D : Type u_1
inst✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : Category.{v_2, u_1} D
F : C ⥤ D
A : Type u_2
inst✝³ : AddMonoid A
inst✝² : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
inst✝¹ : ((whiskeringLeft C D D).obj F).Full
inst✝ : ((whiskeringLeft C D D).obj F).Faithful
X : C
⊢ (zero F s i).hom.app (F.obj X) = (i 0).hom.app X ≫ F.map ((shiftFunctorZero C A).hom.app X)

Tactic:
by
  have h : whiskerLeft F (zero F s i).hom = _ :=
    ((whiskeringLeft C D D).obj F).map_preimage _
  exact (NatTrans.congr_app h X).trans (by simp)

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Shift

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
n₁ n₂ n₁₂ : ℤ
h : n₁ + n₂ = n₁₂
K : CochainComplex C ℤ
x✝² x✝¹ : ℤ
x✝ : (ComplexShape.up ℤ).Rel x✝² x✝¹
⊢ (K.shiftFunctorObjXIso n₁₂ x✝² (x✝² + n₂ + n₁) ⋯).hom ≫ ((shiftFunctor C n₁ ⋙ shiftFunctor C n₂).obj K).d x✝² x✝¹ =
    ((shiftFunctor C n₁₂).obj K).d x✝² x✝¹ ≫ (K.shiftFunctorObjXIso n₁₂ x✝¹ (x✝¹ + n₂ + n₁) ⋯).hom

Tactic:
by
      subst h
      dsimp
      simp only [add_comm n₁ n₂, Int.negOnePow_add, Linear.units_smul_comp,
        Linear.comp_units_smul, d_comp_XIsoOfEq_hom, smul_smul, XIsoOfEq_hom_comp_d]

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: case mpr
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
⊢ T.mor₁ = 0 ∧ T.mor₂ = 0 → IsZero T.obj₂

Tactic:
· intro ⟨h₁, h₂⟩
    obtain ⟨f, hf⟩ := coyoneda_exact₂ T hT (𝟙 _) (by rw [h₂, comp_zero])
    rw [IsZero.iff_id_eq_zero, hf, h₁, comp_zero]

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this : IsIso φ'.hom₁
⊢ productTriangle T ∈ distinguishedTriangles

Tactic:
have : IsIso φ'.hom₂ := by rw [h₂]; infer_instance

Goal state after:
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
⊢ productTriangle T ∈ distinguishedTriangles