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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set R
⊢ (∀ ⦃x : M⦄ ⦃a : ↑s⦄, ↑a • x = 0) ↔ ∀ x ∈ s, x ∈ annihilator R M

Tactic:
mem_annihilator

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set R
⊢ (∀ ⦃x : M⦄ ⦃a : ↑s⦄, ↑a • x = 0) ↔ ∀ x ∈ s, ∀ (m : M), x • m = 0
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set R
x : M
hx : ∀ (a : ↑s), ↑a • x = 0
⊢ s ⊆ ↑(Ideal.torsionOf R M x)

Tactic:
exact fun a ha => hx ⟨a, ha⟩

Goal state after:
Module: Mathlib.Algebra.Polynomial.Module.AEval

Goal state before:
pp: R : Type u_2
A : Type u_3
M : Type u_1
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
a : A
inst✝⁴ : Algebra R A
inst✝³ : AddCommMonoid M
inst✝² : Module A M
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
m : M
n : ℕ
⊢ X ^ n • (of R M a) m = (of R M a) (a ^ n • m)

Tactic:
rw [← of_aeval_smul, aeval_X_pow]

Goal state after:
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: case succ
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
n : ℕ
ih : ∀ (s₁ s₂ : CompositionSeries X), head s₁ = head s₂ → last s₁ = last s₂ → s₁.length = n → s₁.Equivalent s₂
s₁ s₂ : CompositionSeries X
hb : head s₁ = head s₂
ht : last s₁ = last s₂
hle : s₁.length = n + 1
h0s₂ : 0 < s₂.length
t : CompositionSeries X
htb : head t = head s₁
htl : t.length + 1 = s₁.length
htt : last t = (eraseLast s₂).last
hteq : s₁.Equivalent (snoc t (last s₁) ⋯)
this : t.Equivalent (eraseLast s₂)
⊢ Equivalent (snoc t (last s₁) ⋯) ((eraseLast s₂).snoc (last s₂) ⋯)

Tactic:
simp only [ht]

Goal state after:
case succ
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
n : ℕ
ih : ∀ (s₁ s₂ : CompositionSeries X), head s₁ = head s₂ → last s₁ = last s₂ → s₁.length = n → s₁.Equivalent s₂
s₁ s₂ : CompositionSeries X
hb : head s₁ = head s₂
ht : last s₁ = last s₂
hle : s₁.length = n + 1
h0s₂ : 0 < s₂.length
t : CompositionSeries X
htb : head t = head s₁
htl : t.length + 1 = s₁.length
htt : last t = (eraseLast s₂).last
hteq : s₁.Equivalent (snoc t (last s₁) ⋯)
this : t.Equivalent (eraseLast s₂)
⊢ Equivalent (snoc t (last s₂) ⋯) ((eraseLast s₂).snoc (last s₂) ⋯)
Module: Mathlib.Algebra.Polynomial.Module.Basic

Goal state before:
pp: case add
R : Type u_2
M : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
i : ℕ
r : R
n : ℕ
p q : PolynomialModule R M
hp : ((monomial i) r • p).coeff n = if i ≤ n then r • p.coeff (n - i) else 0
hq : ((monomial i) r • q).coeff n = if i ≤ n then r • q.coeff (n - i) else 0
⊢ ((monomial i) r • (p + q)).coeff n = if i ≤ n then r • (p + q).coeff (n - i) else 0

Tactic:
simp [smul_add, hp, hq, ite_add_ite]

Goal state after:
Module: Mathlib.RingTheory.Filtration

Goal state before:
pp: case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R M
x : M
N : Submodule R M := ⋯
hN : ∀ (k : ℕ), (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N
⊢ x ∈ ⨅ i, I ^ i • ⊤ → ∃ r, ↑r • x = x

Tactic:
obtain ⟨r, hr₁, hr₂⟩ :=
      Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
        obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N)
        have := hk k (le_refl _)
        rw [hN, hN] at this
        exact le_of_eq this.symm)

Goal state after:
case mp
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R M
x : M
N : Submodule R M := ⨅ i, I ^ i • ⊤
hN : ∀ (k : ℕ), (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N
r : R
hr₁ : r ∈ I
hr₂ : ∀ n ∈ N, r • n = n
⊢ x ∈ ⨅ i, I ^ i • ⊤ → ∃ r, ↑r • x = x
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: case refine_1
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
D : Derivation R S M
x y : S ⊗[R] S
⊢ 0 = 0 • D.tensorProductTo y + (TensorProduct.lmul' R) y • 0

Tactic:
zero_smul,

Goal state after:
case refine_1
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
D : Derivation R S M
x y : S ⊗[R] S
⊢ 0 = 0 + (TensorProduct.lmul' R) y • 0
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: case refine_1
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
D : Derivation R S M
x y : S ⊗[R] S
⊢ D.tensorProductTo (0 * y) =
    (TensorProduct.lmul' R) 0 • D.tensorProductTo y + (TensorProduct.lmul' R) y • D.tensorProductTo 0

pp: case refine_2
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
D : Derivation R S M
x y : S ⊗[R] S
⊢ ∀ (x y_1 : S),
    D.tensorProductTo (x ⊗ₜ[R] y_1 * y) =
      (TensorProduct.lmul' R) (x ⊗ₜ[R] y_1) • D.tensorProductTo y +
        (TensorProduct.lmul' R) y • D.tensorProductTo (x ⊗ₜ[R] y_1)

pp: case refine_3
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
D : Derivation R S M
x y : S ⊗[R] S
⊢ ∀ (x y_1 : S ⊗[R] S),
    D.tensorProductTo (x * y) =
        (TensorProduct.lmul' R) x • D.tensorProductTo y + (TensorProduct.lmul' R) y • D.tensorProductTo x →
      D.tensorProductTo (y_1 * y) =
          (TensorProduct.lmul' R) y_1 • D.tensorProductTo y + (TensorProduct.lmul' R) y • D.tensorProductTo y_1 →
        D.tensorProductTo ((x + y_1) * y) =
          (TensorProduct.lmul' R) (x + y_1) • D.tensorProductTo y +
            (TensorProduct.lmul' R) y • D.tensorProductTo (x + y_1)

Tactic:
· rw [zero_mul, map_zero, map_zero, zero_smul, smul_zero, add_zero]

Goal state after:
case refine_2
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
D : Derivation R S M
x y : S ⊗[R] S
⊢ ∀ (x y_1 : S),
    D.tensorProductTo (x ⊗ₜ[R] y_1 * y) =
      (TensorProduct.lmul' R) (x ⊗ₜ[R] y_1) • D.tensorProductTo y +
        (TensorProduct.lmul' R) y • D.tensorProductTo (x ⊗ₜ[R] y_1)

case refine_3
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
D : Derivation R S M
x y : S ⊗[R] S
⊢ ∀ (x y_1 : S ⊗[R] S),
    D.tensorProductTo (x * y) =
        (TensorProduct.lmul' R) x • D.tensorProductTo y + (TensorProduct.lmul' R) y • D.tensorProductTo x →
      D.tensorProductTo (y_1 * y) =
          (TensorProduct.lmul' R) y_1 • D.tensorProductTo y + (TensorProduct.lmul' R) y • D.tensorProductTo y_1 →
        D.tensorProductTo ((x + y_1) * y) =
          (TensorProduct.lmul' R) (x + y_1) • D.tensorProductTo y +
            (TensorProduct.lmul' R) y • D.tensorProductTo (x + y_1)
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: I : Type u_1
J : Type u_2
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
X Y : GradedObject I C
e : X ≅ Y
p : I → J
inst✝ : X.HasMap p
j : J
⊢ HasCoproduct (Y.mapObjFun p j)

Tactic:
have α : Discrete.functor (X.mapObjFun p j) ≅ Discrete.functor (Y.mapObjFun p j) :=
    Discrete.natIso (fun ⟨i, _⟩ => (GradedObject.eval i).mapIso e)
  exact hasColimit_of_iso α.symm

Goal state after:
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: case refine_2.refine_1
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
D : Derivation R S M
x y : S ⊗[R] S
x₁ x₂ : S
⊢ 0 = (TensorProduct.lmul' R) (x₁ ⊗ₜ[R] x₂) • 0 + 0 • D.tensorProductTo (x₁ ⊗ₜ[R] x₂)

Tactic:
zero_smul,

Goal state after:
case refine_2.refine_1
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
D : Derivation R S M
x y : S ⊗[R] S
x₁ x₂ : S
⊢ 0 = (TensorProduct.lmul' R) (x₁ ⊗ₜ[R] x₂) • 0 + 0
Module: Mathlib.CategoryTheory.Shift.Basic

Goal state before:
pp: C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
a : A
X : C
⊢ (shiftFunctorAdd' C 0 a a ⋯).inv.app X = (shiftFunctor C a).map ((shiftFunctorZero C A).hom.app X)

Tactic:
simpa using NatTrans.congr_app (congr_arg Iso.inv (shiftFunctorAdd'_zero_add C a)) X

Goal state after:
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: case refine_2.refine_1
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
D : Derivation R S M
x✝ : ↥(KaehlerDifferential.ideal R S)
hx✝ : x✝ ∈ Submodule.restrictScalars S (KaehlerDifferential.ideal R S • ⊤)
x : S ⊗[R] S
hx : (TensorProduct.lmul' R) x = 0
y : ↥(KaehlerDifferential.ideal R S)
⊢ 0 • D.tensorProductTo ↑y + 0 • D.tensorProductTo x = 0

Tactic:
zero_smul,

Goal state after:
case refine_2.refine_1
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
D : Derivation R S M
x✝ : ↥(KaehlerDifferential.ideal R S)
hx✝ : x✝ ∈ Submodule.restrictScalars S (KaehlerDifferential.ideal R S • ⊤)
x : S ⊗[R] S
hx : (TensorProduct.lmul' R) x = 0
y : ↥(KaehlerDifferential.ideal R S)
⊢ 0 + 0 • D.tensorProductTo x = 0
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: case refine_2.refine_1
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
D : Derivation R S M
x✝ : ↥(KaehlerDifferential.ideal R S)
hx✝ : x✝ ∈ Submodule.restrictScalars S (KaehlerDifferential.ideal R S • ⊤)
x : S ⊗[R] S
hx : (TensorProduct.lmul' R) x = 0
y : ↥(KaehlerDifferential.ideal R S)
⊢ 0 + 0 • D.tensorProductTo x = 0

Tactic:
zero_smul,

Goal state after:
case refine_2.refine_1
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
D : Derivation R S M
x✝ : ↥(KaehlerDifferential.ideal R S)
hx✝ : x✝ ∈ Submodule.restrictScalars S (KaehlerDifferential.ideal R S • ⊤)
x : S ⊗[R] S
hx : (TensorProduct.lmul' R) x = 0
y : ↥(KaehlerDifferential.ideal R S)
⊢ 0 + 0 = 0
Module: Mathlib.Algebra.Homology.Linear

Goal state before:
pp: R : Type u_1
inst✝⁶ : Semiring R
C : Type u_2
D : Type u_3
inst✝⁵ : Category.{v_1, u_2} C
inst✝⁴ : Preadditive C
inst✝³ : Category.{v_2, u_3} D
inst✝² : Preadditive D
inst✝¹ : Linear R C
inst✝ : Linear R D
ι : Type u_4
c : ComplexShape ι
X✝ Y✝ X Y : HomologicalComplex C c
x✝² x✝¹ : R
x✝ : X ⟶ Y
⊢ (x✝² * x✝¹) • x✝ = x✝² • x✝¹ • x✝

Tactic:
by ext; apply mul_smul

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
iso : K ≅ L
ψ : L ⟶ M
j k : ι
inst✝¹ : K.HasHomology j
inst✝ : K.HasHomology (c.prev j)
⊢ IsColimit (CokernelCofork.ofπ (K.pOpcycles j) ⋯)

Tactic:
exact (K.sc j).opcyclesIsCokernel

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

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
c : ComplexShape ι
K L : HomologicalComplex C c
φ ψ : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
⊢ ShortComplex.homologyMap ((shortComplexFunctor C c i).map (φ - ψ)) =
    ShortComplex.homologyMap ((shortComplexFunctor C c i).map φ) -
      ShortComplex.homologyMap ((shortComplexFunctor C c i).map ψ)

Tactic:
← ShortComplex.homologyMap_sub

Goal state after:
C : Type u_1
ι : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
c : ComplexShape ι
K L : HomologicalComplex C c
φ ψ : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
⊢ ShortComplex.homologyMap ((shortComplexFunctor C c i).map (φ - ψ)) =
    ShortComplex.homologyMap ((shortComplexFunctor C c i).map φ - (shortComplexFunctor C c i).map ψ)
Module: Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K : ChainComplex C ℕ
X : C
φ : K.X 0 ⟶ X
inst✝ : HomologicalComplex.HasHomology K 0
hφ : K.d 1 0 ≫ φ = 0
⊢ IsIso (HomologicalComplex.descOpcycles K φ 1 isIso_descOpcycles_iff._proof_1 hφ) ↔
    { X₁ := K.X 1, X₂ := K.X 0, X₃ := X, f := K.d 1 0, g := φ, zero := hφ }.Exact ∧ Epi φ

Tactic:
by
  suffices ∀ (i : ℕ) (hx : (ComplexShape.down ℕ).prev 0 = i)
    (hφ : K.d i 0 ≫ φ = 0), IsIso (K.descOpcycles φ i hx hφ) ↔
      (ShortComplex.mk _ _ hφ).Exact ∧ Epi φ from this 1 (by simp) hφ
  rintro _ rfl hφ
  let α : K.sc 0 ⟶ ShortComplex.mk (0 : X ⟶ X) (0 : X ⟶ X) (by simp) :=
    { τ₁ := 0
      τ₂ := φ
      τ₃ := 0 }
  exact (ShortComplex.quasiIso_iff_isIso_descOpcycles α (by simp) rfl rfl).symm.trans
    (ShortComplex.quasiIso_iff_of_zeros' α (by simp) rfl rfl)

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
f g : C ⟶ D
h : Homotopy f g
i j : ι
w : ¬c.Rel j i
⊢ (-h.hom) i j = 0

Tactic:
by rw [Pi.neg_apply, Pi.neg_apply, h.zero i j w, neg_zero]

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝¹ : HasHomotopyCofiber φ
inst✝ : DecidableRel c.Rel
i j : ι
hij : c.Rel j i
⊢ biprod.inr ≫ (XIsoBiprod φ j i hij).inv = inrX φ j

Tactic:
by
  rw [← inrX_XIsoBiprod_hom φ i j hij, Category.assoc, Iso.hom_inv_id, Category.comp_id]

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

Goal state before:
pp: case neg
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : ¬n₁ + 1 = n₂
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0

Tactic:
rw [δ_shape _ _ h₁₂]

Goal state after:
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
n : ℤ
⊢ (mappingCone φ).d (n - 1) n ≫ (snd φ).v n n ⋯ =
    (↑(fst φ)).v (n - 1) n ⋯ ≫ φ.f n + (snd φ).v (n - 1) (n - 1) ⋯ ≫ G.d (n - 1) n

Tactic:
by
  apply d_snd_v

Goal state after:
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
hm : n + 1 = m
s : Fork ((K.HomComplex L).d n m) 0
⊢ ⟨(ConcreteCategory.hom s.ι) 0, ⋯⟩ = 0

Tactic:
simp +instances only [HomComplex_X, map_zero]

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
hm : n + 1 = m
s : Fork ((K.HomComplex L).d n m) 0
⊢ ⟨0, ⋯⟩ = 0
Module: Mathlib.Algebra.Homology.Homotopy

Goal state before:
pp: case zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
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''
⊢ (xNextIso P ⋯).inv ≫ (mkCoinductiveAux₂ e zero comm_zero one comm_one succ 0).snd.fst =
    (mkCoinductiveAux₂ e zero comm_zero one comm_one succ (0 + 1)).fst ≫ (xPrevIso Q ⋯).hom

Tactic:
simp [mkCoinductiveAux₂]

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

Goal state before:
pp: case succ.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
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''
⊢ (xNextIso P ⋯).inv ≫ (mkCoinductiveAux₂ e zero comm_zero one comm_one succ (0 + 1)).snd.fst =
    (mkCoinductiveAux₂ e zero comm_zero one comm_one succ (0 + 1 + 1)).fst ≫ (xPrevIso Q ⋯).hom

Tactic:
simp [mkCoinductiveAux₂]

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

Goal state before:
pp: case succ.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
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 : ℕ
⊢ (xNextIso P ⋯).inv ≫ (mkCoinductiveAux₂ e zero comm_zero one comm_one succ (i + 1 + 1)).snd.fst =
    (mkCoinductiveAux₂ e zero comm_zero one comm_one succ (i + 1 + 1 + 1)).fst ≫ (xPrevIso Q ⋯).hom

Tactic:
simp [mkCoinductiveAux₂]

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
X Y : C
smul : SMul R ((functor r).obj X ⟶ (functor r).obj Y) := Linear.smul r hr ((functor r).obj X) ((functor r).obj Y)
a : R
f g : X ⟶ Y
⊢ a • ((functor r).map f + (functor r).map g) = (functor r).map (a • f) + (functor r).map (a • g)

Tactic:
rw [← (functor r).map_add, smul_eq, ← (functor r).map_add, smul_add]

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
X Y : C
smul : SMul R ((functor r).obj X ⟶ (functor r).obj Y) := Linear.smul r hr ((functor r).obj X) ((functor r).obj Y)
f : X ⟶ Y
⊢ (functor r).map (0 • f) = 0

Tactic:
zero_smul,

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
X Y : C
smul : SMul R ((functor r).obj X ⟶ (functor r).obj Y) := Linear.smul r hr ((functor r).obj X) ((functor r).obj Y)
f : X ⟶ Y
⊢ (functor r).map 0 = 0
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
n : ℤ
K : CochainComplex C ℤ
i j : ℤ
hij : ¬(ComplexShape.up ℤ).Rel i j
⊢ n.negOnePow • K.d (i + n) (j + n) = 0

Tactic:
rw [K.shape, smul_zero]

Goal state after:
case a
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
n : ℤ
K : CochainComplex C ℤ
i j : ℤ
hij : ¬(ComplexShape.up ℤ).Rel i j
⊢ ¬(ComplexShape.up ℤ).Rel (i + n) (j + n)
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
a p q : ℤ
hpq : p + n = q
⊢ (γ.shift a).v p q hpq = γ.v (p + a) (q + a) ⋯

Tactic:
simp only [shift_v γ a p q hpq _ _ rfl rfl, shiftFunctor_obj_X, shiftFunctorObjXIso,
    HomologicalComplex.XIsoOfEq_rfl, Iso.refl_hom, Iso.refl_inv, comp_id, id_comp]

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

Goal state before:
pp: case neg
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
a m : ℤ
hnm : ¬n + 1 = m
⊢ 0 = a.negOnePow • shift 0 a

Tactic:
shift_zero,

Goal state after:
case neg
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
a m : ℤ
hnm : ¬n + 1 = m
⊢ 0 = a.negOnePow • 0
Module: Mathlib.Algebra.Homology.HomologySequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Mono S.f
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
⊢ cyclesMap S.f i ≫ cyclesMap S.g i = 0

Tactic:
rw [← cyclesMap_comp, S.zero, cyclesMap_zero]

Goal state after:
Module: Mathlib.CategoryTheory.PathCategory.Basic

Goal state before:
pp: case h_map.cons
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
X Y b✝ c✝ : Paths V
g : Quiver.Path X b✝
e : b✝ ⟶ c✝
ih : F.map g = eqToHom ⋯ ≫ G.map g ≫ eqToHom ⋯
⊢ (eqToHom ⋯ ≫ G.map g ≫ eqToHom ⋯) ≫ eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯ =
    eqToHom ⋯ ≫ (G.map g ≫ G.map e.toPath) ≫ eqToHom ⋯

Tactic:
simp only [Category.id_comp, eqToHom_refl, eqToHom_trans_assoc, Category.assoc]

Goal state after:
Module: Mathlib.CategoryTheory.Localization.Predicate

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
L : C ⥤ D
W : MorphismProperty C
E : Type u_3
inst✝ : Category.{v_3, u_3} E
⊢ W.Q ⋙ Localization.Construction.lift W.Q ⋯ = W.Q ⋙ 𝟭 W.Localization

Tactic:
simp only [Localization.Construction.fac]

Goal state after:
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
E : Type u_3
inst✝ : Category.{v_3, u_3} E
⊢ W.Q = W.Q ⋙ 𝟭 W.Localization
Module: Mathlib.CategoryTheory.Localization.Construction

Goal state before:
pp: case mp
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
P : MorphismProperty W.Localization
inst✝ : P.IsStableUnderComposition
hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)
hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)
X Y : W.Localization
f : X ⟶ Y
a✝ : P f
⊢ ⊤ f

Tactic:
apply MorphismProperty.top_apply

Goal state after: