mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Group.MinimalAxioms

Goal state before:
pp: G : Type u
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : One G
assoc : ∀ (a b c : G), a * b * c = a * (b * c)
one_mul : ∀ (a : G), 1 * a = a
inv_mul_cancel : ∀ (a : G), a⁻¹ * a = 1
a✝ a : G
⊢ 1 * (a * a⁻¹) = (a * a⁻¹)⁻¹ * (a * a⁻¹) * (a * a⁻¹)

Tactic:
by
            rw [inv_mul_cancel]

Goal state after:
Module: Mathlib.Algebra.Group.MinimalAxioms

Goal state before:
pp: G : Type u
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : One G
assoc : ∀ (a b c : G), a * b * c = a * (b * c)
mul_one : ∀ (a : G), a * 1 = a
mul_inv_cancel : ∀ (a : G), a * a⁻¹ = 1
inv_mul_cancel : ∀ (a : G), a⁻¹ * a = 1
a : G
⊢ 1 * a = a

Tactic:
by
      rw [← mul_inv_cancel a, assoc, inv_mul_cancel, mul_one]

Goal state after:
Module: Mathlib.Algebra.FiveLemma

Goal state before:
pp: M₁ : Type u_1
M₂ : Type u_2
M₃ : Type u_3
M₄ : Type u_4
N₁ : Type u_6
N₂ : Type u_7
N₃ : Type u_8
N₄ : Type u_9
inst✝⁷ : Group M₁
inst✝⁶ : Group M₂
inst✝⁵ : Group M₃
inst✝⁴ : Group M₄
inst✝³ : Group N₁
inst✝² : Group N₂
inst✝¹ : Group N₃
inst✝ : Group N₄
f₁ : M₁ →* M₂
f₂ : M₂ →* M₃
f₃ : M₃ →* M₄
g₁ : N₁ →* N₂
g₂ : N₂ →* N₃
g₃ : N₃ →* N₄
i₁ : M₁ →* N₁
i₂ : M₂ →* N₂
i₃ : M₃ →* N₃
i₄ : M₄ →* N₄
hc₁ : g₁.comp i₁ = i₂.comp f₁
hc₂ : g₂.comp i₂ = i₃.comp f₂
hc₃ : g₃.comp i₃ = i₄.comp f₃
hf₁ : Function.MulExact ⇑f₁ ⇑f₂
hf₂ : Function.MulExact ⇑f₂ ⇑f₃
hg₁ : Function.MulExact ⇑g₁ ⇑g₂
hi₁ : Function.Surjective ⇑i₁
hi₂ : Function.Injective ⇑i₂
hi₄ : Function.Injective ⇑i₄
x : M₂
hm : i₃ (f₂ x) = 1
a : M₁
hy : i₂ (f₁ a) = i₂ x
⊢ f₂ x = 1

Tactic:
apply hi₂ at hy

Goal state after:
M₁ : Type u_1
M₂ : Type u_2
M₃ : Type u_3
M₄ : Type u_4
N₁ : Type u_6
N₂ : Type u_7
N₃ : Type u_8
N₄ : Type u_9
inst✝⁷ : Group M₁
inst✝⁶ : Group M₂
inst✝⁵ : Group M₃
inst✝⁴ : Group M₄
inst✝³ : Group N₁
inst✝² : Group N₂
inst✝¹ : Group N₃
inst✝ : Group N₄
f₁ : M₁ →* M₂
f₂ : M₂ →* M₃
f₃ : M₃ →* M₄
g₁ : N₁ →* N₂
g₂ : N₂ →* N₃
g₃ : N₃ →* N₄
i₁ : M₁ →* N₁
i₂ : M₂ →* N₂
i₃ : M₃ →* N₃
i₄ : M₄ →* N₄
hc₁ : g₁.comp i₁ = i₂.comp f₁
hc₂ : g₂.comp i₂ = i₃.comp f₂
hc₃ : g₃.comp i₃ = i₄.comp f₃
hf₁ : Function.MulExact ⇑f₁ ⇑f₂
hf₂ : Function.MulExact ⇑f₂ ⇑f₃
hg₁ : Function.MulExact ⇑g₁ ⇑g₂
hi₁ : Function.Surjective ⇑i₁
hi₂ : Function.Injective ⇑i₂
hi₄ : Function.Injective ⇑i₄
x : M₂
hm : i₃ (f₂ x) = 1
a : M₁
hy : f₁ a = x
⊢ f₂ x = 1
Module: Mathlib.Algebra.FreeAlgebra.Cardinality

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
A : Type v
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
⊢ lift.{u, v} #↥(adjoin R s) ≤ max (max (lift.{v, u} #R) (lift.{u, v} #↑s)) ℵ₀

Tactic:
by
  have H := mk_range_le_lift (f := FreeAlgebra.lift R ((↑) : s → A))
  rw [lift_umax, lift_id'.{v, u}] at H
  rw [Algebra.adjoin_eq_range_freeAlgebra_lift]
  exact H.trans (FreeAlgebra.cardinalMk_le_max_lift R s)

Goal state after:
Module: Mathlib.Algebra.GCDMonoid.FinsetLemmas

Goal state before:
pp: ι : Type u_1
s : Finset ι
f : ι → ℕ
h : (↑s).Pairwise (Function.onFun Nat.Coprime f)
⊢ s.lcm f = s.prod f

Tactic:
rw [show Nat.Coprime = IsRelPrime by ext; exact Nat.coprime_iff_isRelPrime] at h

Goal state after:
ι : Type u_1
s : Finset ι
f : ι → ℕ
h : (↑s).Pairwise (Function.onFun IsRelPrime f)
⊢ s.lcm f = s.prod f
Module: Mathlib.Algebra.Quaternion

Goal state before:
pp: R : Type u_3
c₁ c₂ c₃ : R
a : ℍ[R,c₁,c₂,c₃]
inst✝ : CommRing R
⊢ a + star a = 2 * ↑a.re + ↑c₂ * ↑a.imI

Tactic:
by simp [self_add_star']

Goal state after:
Module: Mathlib.Algebra.Quaternion

Goal state before:
pp: R : Type u_3
inst✝ : CommRing R
z : ℤ
⊢ normSq ↑z = ↑z ^ 2

Tactic:
by
  rw [← coe_intCast, normSq_coe]

Goal state after:
Module: Mathlib.Algebra.Group.ForwardDiff

Goal state before:
pp: case succ
M : Type u_1
G : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : AddCommGroup G
h : M
R : Type u_3
inst✝¹ : Monoid R
inst✝ : DistribMulAction R G
r : R
n : ℕ
IH : ∀ (f : M → G), Δ_[h]^[n] (r • f) = r • Δ_[h]^[n] f
f : M → G
⊢ Δ_[h]^[n + 1] (r • f) = r • Δ_[h]^[n + 1] f

Tactic:
| succ n IH =>

Goal state after:
Module: Mathlib.Algebra.Group.ForwardDiff

Goal state before:
pp: case succ
M : Type u_3
R : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : Ring R
φ : AddChar M R
h : M
n : ℕ
IH : ∀ (x : M), Δ_[h]^[n] (⇑φ) x = (φ h - 1) ^ n * φ x
x : M
⊢ Δ_[h]^[n + 1] (⇑φ) x = (φ h - 1) ^ (n + 1) * φ x

Tactic:
| succ n IH =>

Goal state after:
Module: Mathlib.Algebra.Group.ForwardDiff

Goal state before:
pp: case succ
R : Type u_3
inst✝ : CommRing R
n : ℕ
IH : (Δ_[1]^[n] fun r ↦ r ^ n) = ↑n !
⊢ (Δ_[1]^[n + 1] fun r ↦ r ^ (n + 1)) = ↑(n + 1)!

Tactic:
| succ n IH =>

Goal state after:
Module: Mathlib.Algebra.GroupWithZero.Pointwise.Finset

Goal state before:
pp: case inl
α : Type u_1
inst✝³ : Mul α
inst✝² : Zero α
inst✝¹ : DecidableEq α
s : Finset α
inst✝ : IsLeftCancelMulZero α
hs : s.erase 0 = ∅
⊢ #s ≤ #(s * s)

Tactic:
erase_eq_empty_iff

Goal state after:
case inl
α : Type u_1
inst✝³ : Mul α
inst✝² : Zero α
inst✝¹ : DecidableEq α
s : Finset α
inst✝ : IsLeftCancelMulZero α
hs : s = ∅ ∨ s = {0}
⊢ #s ≤ #(s * s)
Module: Mathlib.Algebra.GroupWithZero.Action.Pointwise.Finset

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : DecidableEq β
inst✝¹ : GroupWithZero α
inst✝ : MulAction α β
s : Set α
t : Finset β
hs : ∀ a ∈ s, a ≠ 0
⊢ (s.PairwiseDisjoint fun x ↦ x • t) ↔ Set.InjOn (fun p ↦ p.1 • p.2) (s ×ˢ ↑t)

Tactic:
simp_rw [← pairwiseDisjoint_coe, coe_smul_finset]
  exact Set.pairwiseDisjoint_image_right_iff (fun a ha => MulAction.injective₀ (hs a ha))

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

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
a✝ : IsIdempotentComplete C
X : C
p : X ⟶ X
hp : p ≫ p = p
Y : C
i : Y ⟶ X
e : X ⟶ Y
h₁ : i ≫ e = 𝟙 Y
h₂ : e ≫ i = p
⊢ IsLimit (Fork.ofι i ⋯)

Tactic:
apply Fork.IsLimit.mk'
              intro s
              refine ⟨s.ι ≫ e, ?_⟩
              constructor
              · simp [h₂, ← Limits.Fork.condition s]
              · intro m hm
                rw [Fork.ι_ofι] at hm
                rw [← hm]
                simp only [assoc, h₁]
                exact (comp_id m).symm

Goal state after:
Module: Mathlib.CategoryTheory.Idempotents.Karoubi

Goal state before:
pp: case mk
C : Type u_1
inst✝ : Category.{v_1, u_1} C
Q : Karoubi C
X✝ : C
p✝ : X✝ ⟶ X✝
idem✝ : p✝ ≫ p✝ = p✝
h_X : { X := X✝, p := p✝, idem := idem✝ }.X = Q.X
h_p : { X := X✝, p := p✝, idem := idem✝ }.p ≫ eqToHom h_X = eqToHom h_X ≫ Q.p
⊢ { X := X✝, p := p✝, idem := idem✝ } = Q

Tactic:
cases Q

Goal state after:
case mk.mk
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X✝¹ : C
p✝¹ : X✝¹ ⟶ X✝¹
idem✝¹ : p✝¹ ≫ p✝¹ = p✝¹
X✝ : C
p✝ : X✝ ⟶ X✝
idem✝ : p✝ ≫ p✝ = p✝
h_X : { X := X✝¹, p := p✝¹, idem := idem✝¹ }.X = { X := X✝, p := p✝, idem := idem✝ }.X
h_p : { X := X✝¹, p := p✝¹, idem := idem✝¹ }.p ≫ eqToHom h_X = eqToHom h_X ≫ { X := X✝, p := p✝, idem := idem✝ }.p
⊢ { X := X✝¹, p := p✝¹, idem := idem✝¹ } = { X := X✝, p := p✝, idem := idem✝ }
Module: Mathlib.CategoryTheory.Idempotents.Basic

Goal state before:
pp: case h.right
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
D : Type u_2
inst✝ : Category.{v_2, u_2} D
ε : C ≌ D
h : IsIdempotentComplete C
X' : D
p : X' ⟶ X'
hp : p ≫ p = p
φ : X' ≅ (ε.inverse ⋙ ε.functor).obj X' := ⋯
Y : C
i : Y ⟶ ε.inverse.obj X'
e : ε.inverse.obj X' ⟶ Y
h₁ : i ≫ e = 𝟙 Y
h₂ : e ≫ i = ε.inverse.map p
⊢ ε.functor.map e ≫ ε.functor.map i = φ.inv ≫ p ≫ φ.hom

Tactic:
simp only [← ε.functor.map_comp, h₂, Equivalence.fun_inv_map]

Goal state after:
case h.right
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
D : Type u_2
inst✝ : Category.{v_2, u_2} D
ε : C ≌ D
h : IsIdempotentComplete C
X' : D
p : X' ⟶ X'
hp : p ≫ p = p
φ : X' ≅ (ε.inverse ⋙ ε.functor).obj X' := ε.counitIso.symm.app X'
Y : C
i : Y ⟶ ε.inverse.obj X'
e : ε.inverse.obj X' ⟶ Y
h₁ : i ≫ e = 𝟙 Y
h₂ : e ≫ i = ε.inverse.map p
⊢ ε.counit.app X' ≫ p ≫ ε.counitInv.app X' = φ.inv ≫ p ≫ φ.hom
Module: Mathlib.AlgebraicTopology.SimplicialObject.DeltaZeroIter

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
Y : Augmented C
n m i : ℕ
hi : n + i = m
⊢ Y.left.δ₀Iter i hi ≫ Y.hom.app (op ⦋n⦌) = Y.hom.app (op ⦋m⦌)

Tactic:
by
  simpa only [Functor.id_obj, Functor.const_obj_obj, Functor.const_obj_map, Category.comp_id] using!
    Y.hom.naturality (SimplexCategory.δ₀Iter i hi).op

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
X Y : SimplicialObject C
f g : X ⟶ Y
H : Homotopy f g
D : Type u'
inst✝ : Category.{v', u'} D
F : C ⥤ D
n✝ : ℕ
i : Fin (n✝ + 2)
j : Fin (n✝ + 1)
hij : i ≤ j.castSucc
⊢ F.map (H.h j.succ) ≫ (((whiskering C D).obj F).obj Y).δ i.castSucc =
    (((whiskering C D).obj F).obj X).δ i ≫ F.map (H.h j)

Tactic:
by
    simpa [SimplicialObject.δ] using!
      congrArg (fun k => F.map k) (H.h_succ_comp_δ_castSucc_of_lt i j hij)

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

Goal state before:
pp: n : ℕ
θ : ⦋n⦌ ⟶ ⦋n + 1⦌
inst✝ : Mono θ
⊢ ∃ i, θ = δ i

Tactic:
obtain ⟨i, θ', h⟩ := eq_comp_δ_of_not_surjective θ (by
    rw [← epi_iff_surjective]
    grind [→ le_of_epi])
  use i
  have : Mono (θ' ≫ δ i) := by
    rw [← h]
    infer_instance
  have := CategoryTheory.mono_of_mono θ' (δ i)
  rw [h, eq_id_of_mono θ', Category.id_comp]

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

Goal state before:
pp: Δ Δ'' : SimplexCategory
φ : Δ ⟶ Δ''
e : Δ ⟶ image φ
inst✝¹ : Epi e
i : image φ ⟶ Δ''
inst✝ : Mono i
fac : e ≫ i = φ
⊢ factorThruImage φ ≫ i = e ≫ i

Tactic:
fac,

Goal state after:
Δ Δ'' : SimplexCategory
φ : Δ ⟶ Δ''
e : Δ ⟶ image φ
inst✝¹ : Epi e
i : image φ ⟶ Δ''
inst✝ : Mono i
fac : e ≫ i = φ
⊢ factorThruImage φ ≫ i = φ
Module: Mathlib.AlgebraicTopology.SimplicialSet.Dimension

Goal state before:
pp: X : SSet
A B : X.Subcomplex
d : ℕ
inst✝ : X.HasDimensionLT d
h : ∀ i < d, A.obj (op ⦋i⦌) ∩ X.nonDegenerate i ⊆ B.obj (op ⦋i⦌)
⊢ A ≤ B

Tactic:
rw [le_iff_contains_nonDegenerate]

Goal state after:
X : SSet
A B : X.Subcomplex
d : ℕ
inst✝ : X.HasDimensionLT d
h : ∀ i < d, A.obj (op ⦋i⦌) ∩ X.nonDegenerate i ⊆ B.obj (op ⦋i⦌)
⊢ ∀ (n : ℕ) (x : ↑(X.nonDegenerate n)), ↑x ∈ A.obj (op ⦋n⦌) → ↑x ∈ B.obj (op ⦋n⦌)
Module: Mathlib.AlgebraicTopology.SimplicialSet.Degenerate

Goal state before:
pp: case succ
X : SSet
n✝ n : ℕ
hn : ∀ (x : X _⦋n⦌), ∃ m f, ∃ (_ : Epi f), ∃ y, x = (ConcreteCategory.hom (X.map f.op)) ↑y
x : X _⦋n + 1⦌
⊢ ∃ m f, ∃ (_ : Epi f), ∃ y, x = (ConcreteCategory.hom (X.map f.op)) ↑y

Tactic:
by_cases hx : x ∈ X.nonDegenerate (n + 1)
      · exact ⟨n + 1, 𝟙 _, inferInstance, ⟨x, hx⟩, by simp⟩
      · simp only [← mem_degenerate_iff_notMem_nonDegenerate,
          degenerate_eq_iUnion_range_σ, Set.mem_iUnion, Set.mem_range] at hx
        obtain ⟨i, y, rfl⟩ := hx
        obtain ⟨m, f, hf, z, rfl⟩ := hn y
        exact ⟨_, SimplexCategory.σ i ≫ f, inferInstance, z, by simp; rfl⟩

Goal state after:
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : Augmented C
e : X ≅ Y
ed : X.ExtraDegeneracy
n : ℕ
i : Fin (n + 2)
⊢ ((drop.mapIso e).inv.app (op ⦋n + 1⦌) ≫ ed.s (n + 1) ≫ (drop.mapIso e).hom.app (op ⦋n + 1 + 1⦌)) ≫ Y.left.δ i.succ =
    Y.left.δ i ≫ (drop.mapIso e).inv.app (op ⦋n⦌) ≫ ed.s n ≫ (drop.mapIso e).hom.app (op ⦋n + 1⦌)

Tactic:
simp [← SimplicialObject.δ_naturality, s_comp_δ_assoc,
      ← SimplicialObject.δ_naturality_assoc]

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:
by
  rw [← nonDegenerate_iff_of_mono (isoNerve n).hom,
    PartialOrder.mem_nerve_nonDegenerate_iff_strictMono]
  rfl

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

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₁₂ : ComplexShape I₁₂
inst✝⁴ : TotalComplexShape c₁ c₂ c₁₂
inst✝³ : TotalComplexShape c₂ c₁ c₁₂
inst✝² : TotalComplexShapeSymmetry c₁ c₂ c₁₂
inst✝¹ : TotalComplexShapeSymmetry c₂ c₁ c₁₂
inst✝ : TotalComplexShapeSymmetrySymmetry c₁ c₂ c₁₂
i₁ : I₁
i₂ : I₂
⊢ c₂.σ c₁ c₁₂ i₂ i₁ = c₁.σ c₂ c₁₂ i₁ i₂

Tactic:
by
  apply TotalComplexShapeSymmetrySymmetry.σ_symm

Goal state after:
Module: Mathlib.CategoryTheory.Center.NegOnePow

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
n : ℤ
X : C
⊢ (↑((-1) ^ n)).app X = n.negOnePow • 𝟙 X

Tactic:
by
  obtain ⟨n, rfl⟩ | ⟨n, rfl⟩ := Int.even_or_odd n
  · simp [zpow_add, ← mul_zpow, Int.negOnePow_even _ (Even.add_self n)]
  · rw [Int.negOnePow_odd _ (by exact odd_two_mul_add_one n)]
    simp [Units.smul_def, zpow_add, Int.two_mul, ← mul_zpow]

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

Goal state before:
pp: case neg
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
this : ∀ {X Y : D} (a b c d e f : X ⟶ Y), a = c → b = e → f = -d → a + b = c + d + (e + f)
h₃ : ¬c₁.Rel (c₁.prev i₁) i₁
h₄ : ¬c₂.Rel i₂ (c₂.next i₂)
⊢ (((F.mapBifunctorHomologicalComplex c₁ c₂).obj K₁).obj K₂).d₂ c i₁ i₂ (c.next j) ≫ hom₁ h₁ f₂ F c (c.next j) j = 0

Tactic:
rw [HomologicalComplex₂.d₂_eq_zero _ _ _ _ _ h₄, zero_comp]

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

Goal state before:
pp: case neg
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ f₂' : K₂ ⟶ L₂
h₂ : Homotopy f₂ f₂'
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
this✝² : TotalComplexShape c₂ c₁ c := TotalComplexShape.symm c₁ c₂ c
this✝¹ : TotalComplexShapeSymmetry c₁ c₂ c := TotalComplexShape.symmSymmetry c₁ c₂ c
this✝ : F.flip.Additive
this : ∀ (X₁ : C₂), (F.flip.obj X₁).Additive
H : Homotopy (mapBifunctorMap f₁ f₂ F c) (mapBifunctorMap f₁ f₂' F c) :=
  (Homotopy.ofEq ⋯).trans
    ((((mapBifunctorMapHomotopy₁ h₂ f₁ F.flip c).compRight (L₁.mapBifunctorFlipIso L₂ F c).hom).compLeft
          (K₁.mapBifunctorFlipIso K₂ F c).inv).trans
      (Homotopy.ofEq ⋯))
j j' : J
i₁ : I₁
i₂ : I₂
hj : c₁.π c₂ c (i₁, i₂) = j
hj' : ¬c₁.π c₂ c (i₁, c₂.prev i₂) = j'
⊢ c₁.ε₂ c₂ c (i₁, c₂.prev i₂) •
      (F.map (f₁.f i₁)).app (K₂.X i₂) ≫
        (F.obj (L₁.X i₁)).map (h₂.hom i₂ (c₂.prev i₂)) ≫ L₁.ιMapBifunctorOrZero L₂ F c i₁ (c₂.prev i₂) j' =
    c₁.σ c₂ c i₁ i₂ •
      c₂.ε₁ c₁ c (c₂.prev i₂, i₁) •
        (F.map (f₁.f i₁)).app (K₂.X i₂) ≫
          (F.obj (L₁.X i₁)).map (h₂.hom i₂ (c₂.prev i₂)) ≫
            L₂.ιMapBifunctorOrZero L₁ F.flip c (c₂.prev i₂) i₁ j' ≫ (L₁.mapBifunctorFlipIso L₂ F c).hom.f j'

Tactic:
rw [ιMapBifunctorOrZero_eq_zero _ _ _ _ _ _ _ hj',
        ιMapBifunctorOrZero_eq_zero _ _ _ _ _ _ _ (by rwa [ComplexShape.π_symm c₁ c₂ c]),
        comp_zero, comp_zero, smul_zero, zero_comp, comp_zero,
        comp_zero, smul_zero, smul_zero]

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

Goal state before:
pp: case pos.e_a
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'
⊢ c₁₂.ε₁ c₃ c₄ (c₁.π c₂ c₁₂ (i₁, i₂), i₃) •
      (G.map (K₁.ιMapBifunctor K₂ F₁₂ c₁₂ i₁ i₂ (c₁.π c₂ c₁₂ (i₁, i₂)) ⋯)).app (K₃.X i₃) ≫
        (G.map (mapBifunctor.D₁ K₁ K₂ F₁₂ c₁₂ (c₁.π c₂ c₁₂ (i₁, i₂)) (c₁₂.next (c₁.π c₂ c₁₂ (i₁, i₂))))).app (K₃.X i₃) ≫
          (K₁.mapBifunctor K₂ F₁₂ c₁₂).ιMapBifunctor K₃ G c₄ (c₁₂.next (c₁.π c₂ c₁₂ (i₁, i₂))) i₃ j' h₂ =
    d₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j'

Tactic:
rw [← NatTrans.comp_app_assoc, ← Functor.map_comp,
          mapBifunctor.ι_D₁]
        by_cases h₃ : c₁.Rel i₁ (c₁.next i₁)
        · have h₄ := (ComplexShape.next_π₁ c₂ c₁₂ h₃ i₂).symm
          rw [mapBifunctor.d₁_eq _ _ _ _ h₃ _ _ h₄,
            d₁_eq _ _ _ _ _ _ _ h₃,
            ιOrZero_eq _ _ _ _ _ _ _ _ _ _ _ (by rw [← h₂, ← h₄]; rfl),
            ι_eq _ _ _ _ _ _ _ _ _ _ (c₁₂.next i₁₂) _ h₄ h₂,
            Functor.map_units_smul, Functor.map_comp, NatTrans.app_units_zsmul,
            NatTrans.comp_app, Linear.units_smul_comp, assoc, smul_smul]
        · rw [d₁_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₃,
            mapBifunctor.d₁_eq_zero _ _ _ _ _ _ _ h₃,
            Functor.map_zero, zero_app, zero_comp, smul_zero]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
f g : K ⟶ L
h : Homotopy f g
p q : ℤ
hp : p + -p = 0
hq : q + -q = 0
⊢ (homotopyOp h).hom p q =
    (HomologicalComplex.XIsoOfEq L ⋯).hom.op ≫ (h.hom (-q) (-p)).op ≫ (HomologicalComplex.XIsoOfEq K ⋯).hom.op

Tactic:
simp [homotopyOp]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
f g : K ⟶ L
h : Homotopy ((opEquivalence C).functor.map f.op) ((opEquivalence C).functor.map g.op)
p q p' q' : ℤ
hp : p + p' = 0
hq : q + q' = 0
⊢ (homotopyUnop h).hom p q =
    (HomologicalComplex.XIsoOfEq K ⋯).hom ≫ (h.hom q' p').unop ≫ (HomologicalComplex.XIsoOfEq L ⋯).hom

Tactic:
by
  obtain rfl : p' = -p := by lia
  obtain rfl : q' = -q := by lia
  simp [homotopyUnop]

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
⊢ Function.Bijective ⇑toHom

Tactic:
by
  refine ⟨fun x y h ↦ ?_, fun f ↦ ?_⟩
  · obtain ⟨x, rfl⟩ := x.mk_surjective
    obtain ⟨y, rfl⟩ := y.mk_surjective
    rw [← sub_eq_zero, ← mk_sub, mk_eq_zero_iff, ← toHom_mk_eq_zero_iff,
      mk_sub, map_sub, h, sub_self]
  · obtain ⟨f, rfl⟩ := Functor.map_surjective _ f
    exact ⟨mk (Cocycle.equivHomShift f), by simp [toHom_mk]⟩

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

Goal state before:
pp: C : Type u
inst✝⁹ : Category.{v, u} C
D : Type u'
inst✝⁸ : Category.{v', u'} D
P : ObjectProperty C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : P.IsTriangulated
inst✝¹ : IsTriangulated C
X : C
hX : P.leftOrthogonal X
L : C ⥤ D
inst✝ : L.IsLocalization P.trW
Y : C
⊢ Function.Bijective L.map

Tactic:
← isColocal_trW

Goal state after:
C : Type u
inst✝⁹ : Category.{v, u} C
D : Type u'
inst✝⁸ : Category.{v', u'} D
P : ObjectProperty C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : P.IsTriangulated
inst✝¹ : IsTriangulated C
X : C
hX : P.trW.isColocal X
L : C ⥤ D
inst✝ : L.IsLocalization P.trW
Y : C
⊢ Function.Bijective L.map
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: case pos
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
i : ι
hi : ¬c.Rel i (c.next i)
hi' : c'.Rel (e.f i) (c'.next (e.f i))
⊢ (K.extendXIso e ⋯).hom ≫ 0 ≫ (L.extendXIso e ⋯).inv = (dNext (e.f i)) (extend.hom e h.hom)

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
i : ι
hi : ¬c.Rel i (c.next i)
hi' : ¬c'.Rel (e.f i) (c'.next (e.f i))
⊢ (K.extendXIso e ⋯).hom ≫ 0 ≫ (L.extendXIso e ⋯).inv = (dNext (e.f i)) (extend.hom e h.hom)

Tactic:
· simp [dNext_eq _ hi', K.extend_d_from_eq_zero _ _ _ _ rfl hi]

Goal state after:
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
i : ι
hi : ¬c.Rel i (c.next i)
hi' : ¬c'.Rel (e.f i) (c'.next (e.f i))
⊢ (K.extendXIso e ⋯).hom ≫ 0 ≫ (L.extendXIso e ⋯).inv = (dNext (e.f i)) (extend.hom e h.hom)
Module: Mathlib.CategoryTheory.Triangulated.TStructure.Basic

Goal state before:
pp: case mp
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a n n' : ℤ
hn' : a + n = n'
X : C
hX : (t.le n).shift a X
⊢ t.le n' X

Tactic:
exact ((t.le n').prop_iff_of_iso ((shiftEquiv C a).unitIso.symm.app X)).1
      (t.le_shift n (-a) n' (by lia) _ hX)

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.LiftingProperty

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
T : MorphismProperty C
J : Type u_1
⊢ T.llp.IsStableUnderCoproductsOfShape J

Tactic:
apply IsStableUnderCoproductsOfShape.mk
  intro A B _ _ f hf X Y p hp
  have := fun j ↦ hf j _ hp
  infer_instance

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

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
homRel : HomRel C
W : MorphismProperty C
h : homRel.FactorsThroughLocalization W
W' : MorphismProperty (CategoryTheory.Quotient homRel)
hW : W = W'.inverseImage (Quotient.functor homRel)
E : Type u_3
inst✝ : Category.{v_3, u_3} E
L' : CategoryTheory.Quotient homRel ⥤ D
univ : StrictUniversalPropertyFixedTarget L' W' E
x✝² x✝¹ : C
x✝ : x✝² ⟶ x✝¹
hf : W x✝
⊢ W' ((Quotient.functor homRel).map x✝)

Tactic:
hW

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
homRel : HomRel C
W : MorphismProperty C
h : homRel.FactorsThroughLocalization W
W' : MorphismProperty (CategoryTheory.Quotient homRel)
hW : W = W'.inverseImage (Quotient.functor homRel)
E : Type u_3
inst✝ : Category.{v_3, u_3} E
L' : CategoryTheory.Quotient homRel ⥤ D
univ : StrictUniversalPropertyFixedTarget L' W' E
x✝² x✝¹ : C
x✝ : x✝² ⟶ x✝¹
hf : W'.inverseImage (Quotient.functor homRel) x✝
⊢ W' ((Quotient.functor homRel).map x✝)