mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.IntegralClosure.IntegrallyClosed

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
hx : (algebraMap R K) (b ^ n) = (algebraMap R K) (a ^ n * x)
⊢ (algebraMap R K) b ^ n / (algebraMap R K) a ^ n - (algebraMap R K) x = 0

Tactic:
map_pow

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
hx : (algebraMap R K) b ^ n = (algebraMap R K) (a ^ n * x)
⊢ (algebraMap R K) b ^ n / (algebraMap R K) a ^ n - (algebraMap R K) x = 0
Module: Mathlib.Algebra.GCDMonoid.IntegrallyClosed

Goal state before:
pp: case h
R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
X : FractionRing R
x✝ : IsIntegral R X
p : R[X]
hp₁ : p.Monic
hp₂ : Polynomial.eval₂ (algebraMap R (FractionRing R)) X p = 0
a✝ : GCDMonoid R
x y : R
hg : IsUnit (gcd x y)
he : X * (algebraMap R (FractionRing R)) y = (algebraMap R (FractionRing R)) x
this✝ : y ∣ x ^ p.natDegree
this : IsUnit y
coe_map_inv : ↑((Units.map ↑(algebraMap R (FractionRing R))) this.unit)⁻¹ = (algebraMap R (FractionRing R)) ↑this.unit⁻¹
⊢ (algebraMap R (FractionRing R)) x * (algebraMap R (FractionRing R)) ↑this.unit⁻¹ = X

Tactic:
rw [← coe_map_inv, eq_comm, Units.eq_mul_inv_iff_mul_eq]

Goal state after:
case h
R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
X : FractionRing R
x✝ : IsIntegral R X
p : R[X]
hp₁ : p.Monic
hp₂ : Polynomial.eval₂ (algebraMap R (FractionRing R)) X p = 0
a✝ : GCDMonoid R
x y : R
hg : IsUnit (gcd x y)
he : X * (algebraMap R (FractionRing R)) y = (algebraMap R (FractionRing R)) x
this✝ : y ∣ x ^ p.natDegree
this : IsUnit y
coe_map_inv : ↑((Units.map ↑(algebraMap R (FractionRing R))) this.unit)⁻¹ = (algebraMap R (FractionRing R)) ↑this.unit⁻¹
⊢ X * ↑((Units.map ↑(algebraMap R (FractionRing R))) this.unit) = (algebraMap R (FractionRing R)) x
Module: Mathlib.Algebra.Group.Action.Equidecomp

Goal state before:
pp: case right
X : Type u_1
G : Type u_2
inst✝¹ : Monoid G
inst✝ : MulAction G X
g f : X → X
B A : Set X
T S : Finset G
hg : IsDecompOn g B T
hf : IsDecompOn f A S
a✝ : X
aA : a✝ ∈ A
aB : a✝ ∈ f ⁻¹' B
γ : G
γ_mem : γ ∈ S
hγ : f a✝ = γ • a✝
δ : G
δ_mem : δ ∈ T
hδ : g (f a✝) = δ • f a✝
⊢ (g ∘ f) a✝ = (δ * γ) • a✝

Tactic:
rwa [mul_smul, ← hγ]

Goal state after:
Module: Mathlib.Algebra.GroupWithZero.TransferInstance

Goal state before:
pp: α : Type u
β : Type v
e : α ≃ β
inst✝ : MulZeroOneClass β
zero : Zero α := e.zero
one : One α := e.one
mul : Mul α := e.mul
⊢ MulZeroOneClass α

Tactic:
apply e.injective.mulZeroOneClass _

Goal state after:
case zero
α : Type u
β : Type v
e : α ≃ β
inst✝ : MulZeroOneClass β
zero : Zero α := ⋯
one : One α := ⋯
mul : Mul α := ⋯
⊢ e 0 = 0

case one
α : Type u
β : Type v
e : α ≃ β
inst✝ : MulZeroOneClass β
zero : Zero α := ⋯
one : One α := ⋯
mul : Mul α := ⋯
⊢ e 1 = 1

case mul
α : Type u
β : Type v
e : α ≃ β
inst✝ : MulZeroOneClass β
zero : Zero α := ⋯
one : One α := ⋯
mul : Mul α := ⋯
⊢ ∀ (a b : α), e (a * b) = e a * e b
Module: Mathlib.AlgebraicTopology.AlternatingFaceMapComplex

Goal state before:
pp: case i_surj
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
P : Type := Fin (n + 2) × Fin (n + 3)
S : Finset P := {ij | ↑ij.2 ≤ ↑ij.1}
φ : (ij : P) → ij ∈ S → P := fun ij hij ↦ (ij.2.castLT ⋯, ij.1.succ)
i' : Fin (n + 2)
j' : Fin (n + 3)
hij' : (i', j') ∈ Sᶜ
⊢ ∃ a, ∃ (ha : a ∈ S), φ a ha = (i', j')

Tactic:
simp_rw [S, Finset.compl_filter, Finset.mem_filter_univ, not_le] at hij'

Goal state after:
case i_surj
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
P : Type := Fin (n + 2) × Fin (n + 3)
S : Finset P := {ij | ↑ij.2 ≤ ↑ij.1}
φ : (ij : P) → ij ∈ S → P := fun ij hij ↦ (ij.2.castLT ⋯, ij.1.succ)
i' : Fin (n + 2)
j' : Fin (n + 3)
hij' : ↑i' < ↑j'
⊢ ∃ a, ∃ (ha : a ∈ S), φ a ha = (i', j')
Module: Mathlib.AlgebraicTopology.SimplicialObject.Basic

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X : SimplicialObject C
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
H : j.castSucc < i
⊢ X.σ j.castSucc ≫ X.δ i.succ = X.δ i ≫ X.σ j

Tactic:
dsimp [δ, σ]
  simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_σ_of_gt H]

Goal state after:
Module: Mathlib.AlgebraicTopology.AlternatingFaceMapComplex

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : CosimplicialObject C
n : ℕ
⊢ objD X n ≫ objD X (n + 1) = 0

Tactic:
simp only [d_eq_unop_d, ← unop_comp, AlternatingFaceMapComplex.d_squared, unop_zero]

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialObject.DeltaZeroIter

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : SimplicialObject C
n : ℕ
i : Fin (n + 2)
j m : ℕ
i' : Fin (m + 2)
h : n + j = m
hi'' : ↑i' = ↑i + j
⊢ X.map (SimplexCategory.δ i').op ≫ X.map (SimplexCategory.δ₀Iter j ⋯).op =
    X.map (SimplexCategory.δ₀Iter j ⋯).op ≫ X.map (SimplexCategory.δ i).op

Tactic:
simp only [← Functor.map_comp, ← op_comp, SimplexCategory.δ₀Iter_δ' _ _ _ _ hi'']

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: x : SimplexCategory
e : x ≅ x
h : Finset.univ.card = x.len + 1
eq₁ : RelIso.toRelEmbedding (orderIsoOfIso e) = Finset.univ.orderEmbOfFin h
⊢ e = Iso.refl x

Tactic:
have eq₂ :=
    Finset.orderEmbOfFin_unique' h fun i => Finset.mem_univ ((orderIsoOfIso (Iso.refl x)) i)

Goal state after:
x : SimplexCategory
e : x ≅ x
h : Finset.univ.card = x.len + 1
eq₁ : RelIso.toRelEmbedding (orderIsoOfIso e) = Finset.univ.orderEmbOfFin h
eq₂ : RelIso.toRelEmbedding (orderIsoOfIso (Iso.refl x)) = Finset.univ.orderEmbOfFin h
⊢ e = Iso.refl x
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: n : ℕ
Δ' : SimplexCategory
θ : ⦋n + 1⦌ ⟶ Δ'
i : Fin (n + 1)
hi : (Hom.toOrderHom θ) i.castSucc = (Hom.toOrderHom θ) i.succ
x : Fin (⦋n + 1⦌.len + 1)
h' : i.castSucc < x
⊢ x ≠ 0

Tactic:
rintro (rfl : x = 0)

Goal state after:
n : ℕ
Δ' : SimplexCategory
θ : ⦋n + 1⦌ ⟶ Δ'
i : Fin (n + 1)
hi : (Hom.toOrderHom θ) i.castSucc = (Hom.toOrderHom θ) i.succ
h' : i.castSucc < 0
⊢ False
Module: Mathlib.AlgebraicTopology.SimplexCategory.Rev

Goal state before:
pp: n : ℕ
i : Fin (n + 1)
⊢ rev.map (σ i) = σ i.rev

Tactic:
by
  ext j : 3
  simp [σ, Fin.predAbove_rev_right, Fin.rev_rev, rev_map_apply]

Goal state after:
Module: Mathlib.AlgebraicTopology.CechNerve

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ∀ (n : ℕ) (f : Arrow C), HasWidePullback f.right (fun x ↦ f.left) fun x ↦ f.hom
X : Augmented C
F : Arrow C
⊢ Function.RightInverse (equivalenceRightToLeft X F) (equivalenceLeftToRight X F)

Tactic:
intro A
    ext x : 2
    · refine WidePullback.hom_ext _ _ _ (fun j => ?_) ?_
      · simp
      · simpa using congr_app A.w.symm x
    · simp

Goal state after:
Module: Mathlib.AlgebraicTopology.CechNerve

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ∀ (n : ℕ) (f : Arrow C), HasWidePushout f.left (fun x ↦ f.right) fun x ↦ f.hom
F : Arrow C
X : Augmented C
G : F ⟶ Augmented.toArrow.obj X
x : SimplexCategory
j : Fin (x.len + 1)
t : X.hom.app x = X.hom.app ⦋0⦌ ≫ X.right.map (⦋0⦌.const x j)
⊢ Arrow.Hom.left G ≫ X.hom.app ⦋0⦌ ≫ X.right.map (⦋0⦌.const x j) = Arrow.Hom.left G ≫ X.hom.app x

Tactic:
← t

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ∀ (n : ℕ) (f : Arrow C), HasWidePushout f.left (fun x ↦ f.right) fun x ↦ f.hom
F : Arrow C
X : Augmented C
G : F ⟶ Augmented.toArrow.obj X
x : SimplexCategory
j : Fin (x.len + 1)
t : X.hom.app x = X.hom.app ⦋0⦌ ≫ X.right.map (⦋0⦌.const x j)
⊢ Arrow.Hom.left G ≫ X.hom.app x = Arrow.Hom.left G ≫ X.hom.app x
Module: Mathlib.Algebra.Homology.AlternatingConst

Goal state before:
pp: i j : ℕ
h : (up ℕ).Rel i j
⊢ Odd (i + j)

Tactic:
subst h
  norm_num

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

Goal state before:
pp: i j : ℕ
h : (down ℕ).Rel i j
⊢ Odd (i + j)

Tactic:
subst h
  norm_num

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n d : ℕ
s : Δ[n] _⦋d⦌
⊢ s ∈ Δ[n].nonDegenerate d ↔ StrictMono ⇑s

Tactic:
← nonDegenerate_iff_of_mono (isoNerve n).hom,

Goal state after:
n d : ℕ
s : Δ[n] _⦋d⦌
⊢ (ConcreteCategory.hom ((isoNerve.{u} n).hom.app (op ⦋d⦌))) s ∈ (nerve (ULift.{u, 0} (Fin (n + 1)))).nonDegenerate d ↔
    StrictMono ⇑s
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n : SimplexCategory
d d' : SimplexCategoryᵒᵖ
f : d ⟶ d'
g : (stdSimplex.obj n).op.obj d
i : Fin (d'.1.len + 1)
⊢ (((ConcreteCategory.hom ((stdSimplex.obj n).map (rev.map f.unop).op)) (opObjEquiv g)) i.rev).rev =
    ((ConcreteCategory.hom ((stdSimplex.obj n).map f)) (stdSimplex.opObjEquiv g)) i

Tactic:
map_rev_map_op_apply

Goal state after:
n : SimplexCategory
d d' : SimplexCategoryᵒᵖ
f : d ⟶ d'
g : (stdSimplex.obj n).op.obj d
i : Fin (d'.1.len + 1)
⊢ ((opObjEquiv g) ((ConcreteCategory.hom f.unop) i.rev.rev).rev).rev =
    ((ConcreteCategory.hom ((stdSimplex.obj n).map f)) (stdSimplex.opObjEquiv g)) i
Module: Mathlib.CategoryTheory.Shift.Twist

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
A : Type w
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
t : TwistShiftData C A
⊢ ∀ (n : A) (X : t.Category),
    (NatIso.ofComponents (fun X ↦ t.z n 0 • (shiftFunctorAdd C n 0).app X) ⋯).hom.app X =
      eqToHom ⋯ ≫ (shiftFunctorZero C A).inv.app ((shiftFunctor C n).obj X)

Tactic:
simp [shiftFunctorAdd_add_zero_hom_app, CatCenter.smul_iso_hom_eq]

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝¹⁹ : Category.{v_1, u_1} C₁
inst✝¹⁸ : Category.{v_2, u_2} C₂
inst✝¹⁷ : Category.{v_3, u_5} C₃
inst✝¹⁶ : Category.{v_4, u_6} C₄
inst✝¹⁵ : Category.{v_5, u_3} C₁₂
inst✝¹⁴ : HasZeroMorphisms C₁
inst✝¹³ : HasZeroMorphisms C₂
inst✝¹² : HasZeroMorphisms C₃
inst✝¹¹ : Preadditive C₁₂
inst✝¹⁰ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝⁹ : F₁₂.PreservesZeroMorphisms
inst✝⁸ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁷ : G.Additive
inst✝⁶ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁵ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst✝⁴ : TotalComplexShape c₁ c₂ c₁₂
inst✝³ : TotalComplexShape c₁₂ c₃ c₄
inst✝² : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝¹ : DecidableEq ι₁₂
inst✝ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
i₁ i₁' : ι₁
h₁ : c₁.Rel i₁ i₁'
i₂ : ι₂
i₃ : ι₃
j : ι₄
⊢ d₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j =
    (c₁₂.ε₁ c₃ c₄ (c₁.π c₂ c₁₂ (i₁, i₂), i₃) * c₁.ε₁ c₂ c₁₂ (i₁, i₂)) •
      (G.map ((F₁₂.map (K₁.d i₁ i₁')).app (K₂.X i₂))).app (K₃.X i₃) ≫ ιOrZero F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁' i₂ i₃ j

Tactic:
obtain rfl := c₁.next_eq' h₁

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝¹⁹ : Category.{v_1, u_1} C₁
inst✝¹⁸ : Category.{v_2, u_2} C₂
inst✝¹⁷ : Category.{v_3, u_5} C₃
inst✝¹⁶ : Category.{v_4, u_6} C₄
inst✝¹⁵ : Category.{v_5, u_3} C₁₂
inst✝¹⁴ : HasZeroMorphisms C₁
inst✝¹³ : HasZeroMorphisms C₂
inst✝¹² : HasZeroMorphisms C₃
inst✝¹¹ : Preadditive C₁₂
inst✝¹⁰ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝⁹ : F₁₂.PreservesZeroMorphisms
inst✝⁸ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁷ : G.Additive
inst✝⁶ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁵ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst✝⁴ : TotalComplexShape c₁ c₂ c₁₂
inst✝³ : TotalComplexShape c₁₂ c₃ c₄
inst✝² : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝¹ : DecidableEq ι₁₂
inst✝ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j : ι₄
h₁ : c₁.Rel i₁ (c₁.next i₁)
⊢ d₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j =
    (c₁₂.ε₁ c₃ c₄ (c₁.π c₂ c₁₂ (i₁, i₂), i₃) * c₁.ε₁ c₂ c₁₂ (i₁, i₂)) •
      (G.map ((F₁₂.map (K₁.d i₁ (c₁.next i₁))).app (K₂.X i₂))).app (K₃.X i₃) ≫
        ιOrZero F₁₂ G K₁ K₂ K₃ c₁₂ c₄ (c₁.next i₁) i₂ i₃ j
Module: Mathlib.Algebra.Homology.BifunctorAssociator

Goal state before:
pp: case neg
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝²⁰ : Category.{v_1, u_1} C₁
inst✝¹⁹ : Category.{v_2, u_2} C₂
inst✝¹⁸ : Category.{v_3, u_5} C₃
inst✝¹⁷ : Category.{v_4, u_6} C₄
inst✝¹⁶ : Category.{v_5, u_3} C₁₂
inst✝¹⁵ : HasZeroMorphisms C₁
inst✝¹⁴ : HasZeroMorphisms C₂
inst✝¹³ : HasZeroMorphisms C₃
inst✝¹² : Preadditive C₁₂
inst✝¹¹ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝¹⁰ : F₁₂.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁸ : G.Additive
inst✝⁷ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁶ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst✝⁵ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁴ : TotalComplexShape c₁₂ c₃ c₄
inst✝³ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝² : DecidableEq ι₁₂
inst✝¹ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
j j' : ι₄
inst✝ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
h₂ : ¬c₁₂.π c₃ c₄ (c₁₂.next i₁₂, i₃) = j'
h₃ : ¬c₁.Rel i₁ (c₁.next i₁)
⊢ 0 = d₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j'

Tactic:
· rw [d₁_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₃]

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²² : Category.{v_1, u_1} C₁
inst✝²¹ : Category.{v_2, u_2} C₂
inst✝²⁰ : Category.{v_3, u_5} C₃
inst✝¹⁹ : Category.{v_4, u_6} C₄
inst✝¹⁸ : Category.{v_6, u_4} C₂₃
inst✝¹⁷ : HasZeroMorphisms C₁
inst✝¹⁶ : HasZeroMorphisms C₂
inst✝¹⁵ : HasZeroMorphisms C₃
inst✝¹⁴ : Preadditive C₂₃
inst✝¹³ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹² : G₂₃.PreservesZeroMorphisms
inst✝¹¹ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁰ : F.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝⁸ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝⁷ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁶ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁵ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁴ : TotalComplexShape c₁ c₂₃ c₄
inst✝³ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝² : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝¹ : DecidableEq ι₂₃
inst✝ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
i₁ i₁' : ι₁
h₁ : c₁.Rel i₁ i₁'
i₂ : ι₂
i₃ : ι₃
j : ι₄
⊢ d₁ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j =
    c₁.ε₁ c₂₃ c₄ (i₁, c₂.π c₃ c₂₃ (i₂, i₃)) •
      (F.map (K₁.d i₁ i₁')).app ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ≫ ιOrZero F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁' i₂ i₃ j

Tactic:
obtain rfl := c₁.next_eq' h₁

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²² : Category.{v_1, u_1} C₁
inst✝²¹ : Category.{v_2, u_2} C₂
inst✝²⁰ : Category.{v_3, u_5} C₃
inst✝¹⁹ : Category.{v_4, u_6} C₄
inst✝¹⁸ : Category.{v_6, u_4} C₂₃
inst✝¹⁷ : HasZeroMorphisms C₁
inst✝¹⁶ : HasZeroMorphisms C₂
inst✝¹⁵ : HasZeroMorphisms C₃
inst✝¹⁴ : Preadditive C₂₃
inst✝¹³ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹² : G₂₃.PreservesZeroMorphisms
inst✝¹¹ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁰ : F.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝⁸ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝⁷ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁶ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁵ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁴ : TotalComplexShape c₁ c₂₃ c₄
inst✝³ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝² : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝¹ : DecidableEq ι₂₃
inst✝ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j : ι₄
h₁ : c₁.Rel i₁ (c₁.next i₁)
⊢ d₁ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j =
    c₁.ε₁ c₂₃ c₄ (i₁, c₂.π c₃ c₂₃ (i₂, i₃)) •
      (F.map (K₁.d i₁ (c₁.next i₁))).app ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ≫
        ιOrZero F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ (c₁.next i₁) i₂ i₃ j
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: ι : Type u_1
c : ComplexShape ι
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroObject C
inst✝ : Preadditive C
K L : HomologicalComplex C c
φ : (i j : ι) → K.X i ⟶ L.X j
i j : ι
⊢ homAux φ (some i) (some j) = (extend.XIso K ⋯).hom ≫ φ i j ≫ (extend.XIso L ⋯).inv

Tactic:
simp [homAux, extend.XIso, extend.X]

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
j' : ι'
i : ι
hij' : ¬c'.Rel j' (e.f i)
hj' : ¬∃ j, e.f j = j'
⊢ extend.hom e h.hom (e.f i) j' = 0

Tactic:
· exact extend.hom_eq_zero₂ _ _ _ _ (by tauto)

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
α : Cochain K L (-1)
n : ℤ
hα : (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) n
hK : HomologicalComplex.ExactAt K (n + 1)
inst✝ : Injective (L.X (n + 1))
u : K.X (n + 1) ⟶ L.X (n + 1) :=
  f.f (n + 1) - α.v (n + 1) n ⋯ ≫ L.d n (n + 1) - K.d (n + 1) (n + 2) ≫ α.v (n + 2) (n + 1) ⋯
hu : K.d n (n + 1) ≫ u = 0
⊢ (ComplexShape.up ℤ).next (n + 1) = n + 2

Tactic:
by simp; lia

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.TStructure

Goal state before:
pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X : DerivedCategory C
K : CochainComplex C ℤ
e₂ : Q.obj K ≅ X
h : (K.shortComplexTruncLE 0).ShortExact
⊢ Q.map (K.shortComplexTruncLE 0).g ≫
      Q.map (K.shortComplexTruncLEX₃ToTruncGE 0 1 DerivedCategory.TStructure.t._proof_13) =
    e₂.hom ≫ e₂.inv ≫ Q.map (K.πTruncGE 1)

Tactic:
← Q.map_comp,

Goal state after:
case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X : DerivedCategory C
K : CochainComplex C ℤ
e₂ : Q.obj K ≅ X
h : (K.shortComplexTruncLE 0).ShortExact
⊢ Q.map ((K.shortComplexTruncLE 0).g ≫ K.shortComplexTruncLEX₃ToTruncGE 0 1 DerivedCategory.TStructure.t._proof_13) =
    e₂.hom ≫ e₂.inv ≫ Q.map (K.πTruncGE 1)
Module: Mathlib.CategoryTheory.LiftingProperties.Limits

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPullback s f g t
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasLiftingProperty f' g
u : X' ⟶ X
v : Y' ⟶ Y
sq : CommSq u f' f v
⊢ sq.HasLift

Tactic:
have w : (u ≫ s) ≫ g = f' ≫ v ≫ t := by
      rw [Category.assoc, h.toCommSq.w, ← Category.assoc, ← Category.assoc, sq.w]
    exact ⟨h.lift (CommSq.mk w).lift v (by rw [CommSq.fac_right]),
      h.hom_ext (by rw [Category.assoc, h.lift_fst, CommSq.fac_left])
        (by rw [Category.assoc, h.lift_snd, sq.w]), h.lift_snd _ _ _⟩

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.TStructure

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X : DerivedCategory C
n : ℤ
⊢ X.IsGE n ↔ ∀ i < n, IsZero ((homologyFunctor C i).obj X)

Tactic:
constructor
  · rintro ⟨K, e, _⟩ i hi
    apply ((K.exactAt_of_isGE n i hi).isZero_homology).of_iso
    exact (homologyFunctor C i).mapIso e ≪≫ (homologyFunctorFactors C i).app K
  · intro hX
    have : (Q.objPreimage X).IsGE n := by
      rw [CochainComplex.isGE_iff]
      intro i hi
      rw [HomologicalComplex.exactAt_iff_isZero_homology]
      apply (hX i hi).of_iso
      exact (homologyFunctorFactors C i).symm.app _ ≪≫
        (homologyFunctor C i).mapIso (Q.objObjPreimageIso X)
    exact ⟨(Q.objPreimage X).truncGE n, (Q.objObjPreimageIso X).symm ≪≫
      asIso (Q.map ((Q.objPreimage X).πTruncGE n)), inferInstance⟩

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
n : ℤ
z : Cocycle K L n
inst✝ : L.IsKInjective
hK : HomologicalComplex.Acyclic K
m : ℤ
hm : m + 1 = n
φ : K ⟶ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L
hφ : Cochain.ofHom φ = (↑z).rightShift n 0 ⋯
h : Homotopy φ 0
f : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L) (-1)
hf : Cochain.ofHom φ = δ (-1) 0 f
⊢ (δ m n (n.negOnePow • f.rightUnshift m ⋯)).rightShift n 0 ⋯ = Cochain.ofHom φ

Tactic:
hf,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
n : ℤ
z : Cocycle K L n
inst✝ : L.IsKInjective
hK : HomologicalComplex.Acyclic K
m : ℤ
hm : m + 1 = n
φ : K ⟶ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L
hφ : Cochain.ofHom φ = (↑z).rightShift n 0 ⋯
h : Homotopy φ 0
f : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L) (-1)
hf : Cochain.ofHom φ = δ (-1) 0 f
⊢ (δ m n (n.negOnePow • f.rightUnshift m ⋯)).rightShift n 0 ⋯ = δ (-1) 0 f
Module: Mathlib.AlgebraicTopology.ModelCategory.Basic

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : CategoryWithFibrations C
inst✝⁶ : CategoryWithCofibrations C
inst✝⁵ : CategoryWithWeakEquivalences C
inst✝⁴ : HasFiniteLimits C
inst✝³ : HasFiniteColimits C
inst✝² : (weakEquivalences C).HasTwoOutOfThreeProperty
inst✝¹ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
inst✝ : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)
A B X Y : C
f : A ⟶ B
w : X ⟶ Y
h : RetractArrow f w
hw : WeakEquivalence w
hf : (trivialCofibrations C).MapFactorizationData (fibrations C) f
⊢ weakEquivalences C f

Tactic:
have : Cofibration hf.i := by
      simpa only [cofibration_iff] using hf.hi.1

Goal state after:
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : CategoryWithFibrations C
inst✝⁶ : CategoryWithCofibrations C
inst✝⁵ : CategoryWithWeakEquivalences C
inst✝⁴ : HasFiniteLimits C
inst✝³ : HasFiniteColimits C
inst✝² : (weakEquivalences C).HasTwoOutOfThreeProperty
inst✝¹ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
inst✝ : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)
A B X Y : C
f : A ⟶ B
w : X ⟶ Y
h : RetractArrow f w
hw : WeakEquivalence w
hf : (trivialCofibrations C).MapFactorizationData (fibrations C) f
this : Cofibration hf.i
⊢ weakEquivalences C f