mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: G : Type u
inst✝ : Monoid G
p : ℕ
hp : Nat.Prime p
⊢ ∃ g, orderOf g = p ^ (exponent G).factorization p

Tactic:
by
  have := Fact.mk hp
  rcases eq_or_ne ((exponent G).factorization p) 0 with (h | h)
  · refine ⟨1, by rw [h, pow_zero, orderOf_one]⟩
  have he : 0 < exponent G :=
    Ne.bot_lt fun ht => by
      rw [ht] at h
      apply h
      rw [bot_eq_zero, Nat.factorization_zero, Finsupp.zero_apply]
  rw [← Finsupp.mem_support_iff] at h
  obtain ⟨g, hg⟩ : ∃ g : G, g ^ (exponent G / p) ≠ 1 := by
    suffices key : ¬exponent G ∣ exponent G / p by
      rwa [exponent_dvd_iff_forall_pow_eq_one, not_forall] at key
    exact fun hd =>
      hp.one_lt.not_ge
        ((mul_le_iff_le_one_left he).mp <|
          Nat.le_of_dvd he <| Nat.mul_dvd_of_dvd_div (Nat.dvd_of_mem_primeFactors h) hd)
  obtain ⟨k, hk : exponent G = p ^ _ * k⟩ := Nat.ordProj_dvd _ _
  obtain ⟨t, ht⟩ := Nat.exists_eq_succ_of_ne_zero (Finsupp.mem_support_iff.mp h)
  refine ⟨g ^ k, ?_⟩
  rw [ht]
  apply orderOf_eq_prime_pow
  · rwa [hk, mul_comm, ht, pow_succ, ← mul_assoc, Nat.mul_div_cancel _ hp.pos, pow_mul] at hg
  · rw [← Nat.succ_eq_add_one, ← ht, ← pow_mul, mul_comm, ← hk]
    exact pow_exponent_eq_one g

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic

Goal state before:
pp: α : Type u_1
G : Type u_2
G' : Type u_3
a : α
inst✝³ : Group α
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : IsCyclic α
H : Subgroup α
this : (a : Prop) → Decidable a
g : α
hg : ∀ (x : α), x ∈ zpowers g
hx : ¬∃ x ∈ H, x ≠ 1
x : α
h : x ∈ H
⊢ x ∈ ⊥

Tactic:
simp at *; tauto

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: case refine_1
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
x : R
hx : x ∈ I ^ n
y : R
hy : y ∈ J ^ n
add : x + y = 1
mul : x * y = 0
⊢ s ⊆ ↑(basicOpen x)

Tactic:
rw [← hJ', basicOpen_eq_zeroLocus_of_mul_add _ _ mul add]

Goal state after:
case refine_1
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
x : R
hx : x ∈ I ^ n
y : R
hy : y ∈ J ^ n
add : x + y = 1
mul : x * y = 0
⊢ zeroLocus ↑J ⊆ zeroLocus {y}
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: case H
G : Type u
inst✝ : CommMonoid G
h : ∀ (g : G), 0 < orderOf g
he : ExponentExists G
⊢ ∀ (a : ℕ), (∃ y, orderOf y = a) → a ≤ exponent G

Tactic:
forall_exists_index,

Goal state after:
case H
G : Type u
inst✝ : CommMonoid G
h : ∀ (g : G), 0 < orderOf g
he : ExponentExists G
⊢ ∀ (a : ℕ) (x : G), orderOf x = a → a ≤ exponent G
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: case pos
G : Type u
inst✝ : CommMonoid G
h : ∃ g, orderOf g = 0
⊢ exponent G = 0

pp: case neg
G : Type u
inst✝ : CommMonoid G
h : ¬∃ g, orderOf g = 0
⊢ exponent G = ⨆ g, orderOf g

Tactic:
· obtain ⟨g, hg⟩ := h
    exact exponent_eq_zero_of_order_zero hg

Goal state after:
case neg
G : Type u
inst✝ : CommMonoid G
h : ¬∃ g, orderOf g = 0
⊢ exponent G = ⨆ g, orderOf g
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: case refine_2
M₁ : Type u_1
M₂ : Type u_2
inst✝¹ : Monoid M₁
inst✝ : Monoid M₂
⊢ exponent M₁ ∣ exponent (M₁ × M₂)

Tactic:
exact MonoidHom.exponent_dvd (f := MonoidHom.fst M₁ M₂) Prod.fst_surjective

Goal state after:
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: G : Type u
inst✝¹ : Monoid G
inst✝ : IsCancelMul G
a b : G
h : ∀ (g : G), g ^ 2 = 1
⊢ a * (a * b) * b = a ^ 2 * b ^ 2

Tactic:
simp [pow_two, mul_assoc]

Goal state after:
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: G : Type u
inst✝¹ : Monoid G
inst✝ : IsCancelMul G
a b : G
h : ∀ (g : G), g ^ 2 = 1
⊢ (a * b) ^ 2 = a * (b * a) * b

Tactic:
simp [pow_two, mul_assoc]

Goal state after:
Module: Mathlib.Algebra.Module.ZMod

Goal state before:
pp: n : ℕ
M : Type u_1
S : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module (ZMod n) M
inst✝¹ : SetLike S M
inst✝ : AddSubgroupClass S M
x : M
K : S
hx : x ∈ K
c : ZMod n
⊢ ↑c.cast • x ∈ K

Tactic:
Int.cast_smul_eq_zsmul

Goal state after:
n : ℕ
M : Type u_1
S : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module (ZMod n) M
inst✝¹ : SetLike S M
inst✝ : AddSubgroupClass S M
x : M
K : S
hx : x ∈ K
c : ZMod n
⊢ c.cast • x ∈ K
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic

Goal state before:
pp: α : Type u_1
inst✝¹ : CommGroup α
inst✝ : IsSimpleGroup α
⊢ Nat.Prime (Nat.card α)

Tactic:
have hα : Nontrivial α := IsSimpleGroup.toNontrivial

Goal state after:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : IsSimpleGroup α
hα : Nontrivial α
⊢ Nat.Prime (Nat.card α)
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic

Goal state before:
pp: M : Type u_4
N : Type u_5
inst✝² : CommGroup M
inst✝¹ : Group N
f : M →* N
h : (Nat.card ↥f.ker).Coprime (Nat.card ↥f.range)
inst✝ : IsCyclic ↥f.ker
hN : IsCyclic ↥f.range
hf : Function.Surjective ⇑f
⊢ IsCyclic M

Tactic:
exact isCyclic_of_coprime_card_range_card_ker f h

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

Goal state before:
pp: R : Type u_2
M : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
p : R[X]
i : ℕ
m : M
⊢ (comp p) (single R i m) = p ^ i • single R 0 m

Tactic:
rw [comp_apply, map_single, eval_single]

Goal state after:
R : Type u_2
M : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
p : R[X]
i : ℕ
m : M
⊢ p ^ i • (lsingle R 0) m = p ^ i • single R 0 m
Module: Mathlib.RingTheory.Filtration

Goal state before:
pp: case mpr
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
  Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), single R i '' ↑(F.N i))
hF : ∀ n ≥ n₀, I • F.N n = F.N (n + 1)
i : ℕ
this : ∀ i ≤ n₀, single R i '' ↑(F.N i) ⊆ ↑F'
⊢ single R i '' ↑(F.N i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), single R i '' ↑(F.N i)))

Tactic:
induction i with
    | zero => exact this _ zero_le
    | succ j hj => ?_

Goal state after:
case mpr.succ
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
  Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), single R i '' ↑(F.N i))
hF : ∀ n ≥ n₀, I • F.N n = F.N (n + 1)
this : ∀ i ≤ n₀, single R i '' ↑(F.N i) ⊆ ↑F'
j : ℕ
hj : single R j '' ↑(F.N j) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), single R i '' ↑(F.N i)))
⊢ single R (j + 1) '' ↑(F.N (j + 1)) ⊆
    ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), single R i '' ↑(F.N i)))
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).toCotangent.ker)

Tactic:
← Submodule.comap_map_eq,

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))
      (Submodule.comap (maximalIdeal R).toCotangent (Submodule.map (maximalIdeal R).toCotangent M))
Module: Mathlib.RingTheory.Ideal.Cotangent

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsLocalRing R
inst✝ : IsNoetherianRing R
⊢ (∃ x, span (ResidueField R) (⇑(maximalIdeal R).toCotangent '' {x}) = ⊤) ↔ ∃ a, maximalIdeal R = R ∙ a

Tactic:
CotangentSpace.span_image_eq_top_iff,

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsLocalRing R
inst✝ : IsNoetherianRing R
⊢ (∃ x, R ∙ x = ⊤) ↔ ∃ a, maximalIdeal R = R ∙ a
Module: Mathlib.Algebra.Category.ModuleCat.Projective

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
P : ModuleCat R
inst✝ : Module.Projective R ↑P
⊢ CategoryTheory.Projective P

Tactic:
by
  refine ⟨fun E X epi => ?_⟩
  obtain ⟨f, h⟩ := Module.projective_lifting_property X.hom E.hom
    ((ModuleCat.epi_iff_surjective _).mp epi)
  exact ⟨ofHom f, hom_ext h⟩

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

Goal state before:
pp: ι : Type u_1
c : ComplexShape ι
i j : ι
h : c.Rel i j
⊢ c.next i = j

Tactic:
by
  apply c.next_eq _ h
  rw [next]
  rw [dif_pos]
  exact Exists.choose_spec ⟨j, h⟩

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Presheaf

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
M✝ M₁ M₂ : PresheafOfModules R
M : PresheafOfModules R
s : M.sections
⊢ (fun f ↦ sectionsMk (fun X ↦ (ConcreteCategory.hom (f.app X)) 1) ⋯)
      ((fun s ↦
          { app := fun X ↦ ModuleCat.ofHom ((ringLmapEquivSelf ↑(R.obj X) ℤ ↑(M.obj X)).symm (↑s X)), naturality := ⋯ })
        s) =
    s

Tactic:
by
    ext X
    exact (LinearMap.ringLmapEquivSelf (R.obj X) ℤ (M.obj X)).apply_symm_apply (s.val X)

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

Goal state before:
pp: 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
K : HomologicalComplex V c
j : ι
A : V
f g : (single V c j).obj A ⟶ K
hfg : f.f j = g.f j
⊢ f = g

Tactic:
ext i
  by_cases h : i = j
  · subst h
    exact hfg
  · apply (isZero_single_obj_X c j A i h).eq_of_src

Goal state after:
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 j j' : ι
rij : c.Rel i j
rij' : c.Rel i j'
⊢ C.d i j' ≫ eqToHom ⋯ = C.d i j

Tactic:
obtain rfl := c.next_eq rij rij'
  simp only [eqToHom_refl, comp_id]

Goal state after:
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 a 0).hom.app X = eqToHom ⋯ ≫ (shiftFunctorZero C A).inv.app ((shiftFunctor C a).obj X)

Tactic:
simp [← shiftFunctorAdd'_add_zero_hom_app, shiftFunctorAdd']

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

Goal state before:
pp: case neg
ι : Type u_1
V : Type u
inst✝¹¹ : Category.{v, u} V
inst✝¹⁰ : Preadditive V
W : Type u_2
inst✝⁹ : Category.{v_1, u_2} W
inst✝⁸ : Preadditive W
W₁ : Type u_3
W₂ : Type u_4
inst✝⁷ : Category.{v_2, u_3} W₁
inst✝⁶ : Category.{v_3, u_4} W₂
inst✝⁵ : HasZeroMorphisms W₁
inst✝⁴ : HasZeroMorphisms W₂
c✝ : ComplexShape ι
C D : HomologicalComplex V c✝
f : C ⟶ D
i✝ : ι
inst✝³ : HasZeroObject W₁
inst✝² : HasZeroObject W₂
F : W₁ ⥤ W₂
inst✝¹ : F.PreservesZeroMorphisms
c : ComplexShape ι
inst✝ : DecidableEq ι
j : ι
X : W₁
i : ι
h : ¬i = j
⊢ 0 ≫ 0 = 𝟙 (F.obj (((single W₁ c j).obj X).X i))

Tactic:
rw [zero_comp, ← F.map_id,
              (isZero_single_obj_X c j X _ h).eq_of_src (𝟙 _) 0, F.map_zero]

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

Goal state before:
pp: case mp
C : Type u_1
ι : Type u_2
c : ComplexShape ι
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex (HomologicalComplex C c)
⊢ S.Exact → ∀ (i : ι), (S.map (eval C c i)).Exact

pp: case mpr
C : Type u_1
ι : Type u_2
c : ComplexShape ι
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex (HomologicalComplex C c)
⊢ (∀ (i : ι), (S.map (eval C c i)).Exact) → S.Exact

Tactic:
· intro hS i
    exact hS.map (eval C c i)

Goal state after:
case mpr
C : Type u_1
ι : Type u_2
c : ComplexShape ι
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex (HomologicalComplex C c)
⊢ (∀ (i : ι), (S.map (eval C c i)).Exact) → S.Exact
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
⊢ Epi (K.homologyπ i)

Tactic:
dsimp only [homologyπ]

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
⊢ Epi (K.sc i).homologyπ
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
⊢ (K.sc i).Exact ↔ IsZero (K.sc i).homology

Tactic:
ShortComplex.exact_iff_isZero_homology

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 : ι
inst✝ : K.HasHomology i
⊢ IsZero (K.sc i).homology ↔ IsZero (K.sc i).homology
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: case mpr
R : Type u
inst✝⁶ : CommRing R
A : Type u_2
B : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra R A
inst✝² : Algebra A B
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R A B
h : Function.Surjective ⇑(algebraMap A B)
x : A →₀ A
hx : x ∈ ↑(mapRange.linearMap (Algebra.linearMap A B) ∘ₗ lmapDomain A A ⇑(algebraMap A B)).ker
⊢ 1 ⊗ₜ[A] (linearCombination A ⇑(D R A)) (x.sum single) ∈ (kerCotangentToTensor R A B).range

Tactic:
Finsupp.sum,

Goal state after:
case mpr
R : Type u
inst✝⁶ : CommRing R
A : Type u_2
B : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra R A
inst✝² : Algebra A B
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R A B
h : Function.Surjective ⇑(algebraMap A B)
x : A →₀ A
hx : x ∈ ↑(mapRange.linearMap (Algebra.linearMap A B) ∘ₗ lmapDomain A A ⇑(algebraMap A B)).ker
⊢ 1 ⊗ₜ[A] (linearCombination A ⇑(D R A)) (∑ a ∈ x.support, single a (x a)) ∈ (kerCotangentToTensor R A B).range
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: case mpr
R : Type u
inst✝⁶ : CommRing R
A : Type u_2
B : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra R A
inst✝² : Algebra A B
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R A B
h : Function.Surjective ⇑(algebraMap A B)
x : A →₀ A
hx : x ∈ ↑(mapRange.linearMap (Algebra.linearMap A B) ∘ₗ lmapDomain A A ⇑(algebraMap A B)).ker
c : B
a✝ : c ∈ Finset.image (⇑(algebraMap A B)) x.support
⊢ ∃ y, (kerCotangentToTensor R A B) y = 1 ⊗ₜ[A] ∑ x_1 ∈ x.support with (algebraMap A B) x_1 = c, x x_1 • (D R A) x_1

Tactic:
have : ∑ i ∈ x.support with algebraMap A B i = c, x i ∈ RingHom.ker (algebraMap A B) := by
      simpa [Finsupp.mapDomain, Finsupp.sum, Finsupp.finsetSum_apply, RingHom.mem_ker,
        Finsupp.single_apply, ← Finset.sum_filter] using DFunLike.congr_fun hx c

Goal state after:
case mpr
R : Type u
inst✝⁶ : CommRing R
A : Type u_2
B : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra R A
inst✝² : Algebra A B
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R A B
h : Function.Surjective ⇑(algebraMap A B)
x : A →₀ A
hx : x ∈ ↑(mapRange.linearMap (Algebra.linearMap A B) ∘ₗ lmapDomain A A ⇑(algebraMap A B)).ker
c : B
a✝ : c ∈ Finset.image (⇑(algebraMap A B)) x.support
this : ∑ i ∈ x.support with (algebraMap A B) i = c, x i ∈ RingHom.ker (algebraMap A B)
⊢ ∃ y, (kerCotangentToTensor R A B) y = 1 ⊗ₜ[A] ∑ x_1 ∈ x.support with (algebraMap A B) x_1 = c, x x_1 • (D R A) x_1
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift

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
K L M : CochainComplex C ℤ
n✝ : ℤ
γ✝ γ₁ γ₂ : Cochain K L n✝
n a n' : ℤ
hn' : n + a = n'
γ γ' : Cochain K L n
⊢ (γ + γ').leftShift a n' hn' = γ.leftShift a n' hn' + γ'.leftShift a n' hn'

Tactic:
by simp only [leftShift_add]

Goal state after:
Module: Mathlib.CategoryTheory.Shift.CommShift

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
A : Type u_4
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : F.CommShift A
a b : A
⊢ commShiftIso F (a + b) = CommShift.isoAdd' ⋯ (commShiftIso F a) (commShiftIso F b)

Tactic:
simp only [commShiftIso_add, CommShift.isoAdd]

Goal state after:
Module: Mathlib.CategoryTheory.Shift.CommShift

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
A : Type u_4
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : F.CommShift A
X : C
a b : A
h : a + b = 0
⊢ F.map ((shiftFunctorCompIsoId C a b h).inv.app X) =
    (shiftFunctorCompIsoId D a b h).inv.app (F.obj X) ≫
      (shiftFunctor D b).map ((commShiftIso F a).inv.app X) ≫ (commShiftIso F b).inv.app ((shiftFunctor C a).obj X)

Tactic:
← cancel_epi (F.map ((shiftFunctorCompIsoId C a b h).hom.app X)),

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
A : Type u_4
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : F.CommShift A
X : C
a b : A
h : a + b = 0
⊢ F.map ((shiftFunctorCompIsoId C a b h).hom.app X) ≫ F.map ((shiftFunctorCompIsoId C a b h).inv.app X) =
    F.map ((shiftFunctorCompIsoId C a b h).hom.app X) ≫
      (shiftFunctorCompIsoId D a b h).inv.app (F.obj X) ≫
        (shiftFunctor D b).map ((commShiftIso F a).inv.app X) ≫ (commShiftIso F b).inv.app ((shiftFunctor C a).obj X)
Module: Mathlib.CategoryTheory.Shift.CommShift

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
A : Type u_4
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : F.CommShift A
X : C
a b : A
h : a + b = 0
⊢ F.map ((shiftFunctorCompIsoId C a b h).inv.app X) =
    (shiftFunctorCompIsoId D a b h).inv.app (F.obj X) ≫
      (shiftFunctor D b).map ((commShiftIso F a).inv.app X) ≫ (commShiftIso F b).inv.app ((shiftFunctor C a).obj X)

Tactic:
rw [← cancel_epi (F.map ((shiftFunctorCompIsoId C a b h).hom.app X)), ← F.map_comp,
    Iso.hom_inv_id_app, F.map_id, map_shiftFunctorCompIsoId_hom_app]
  simp only [comp_obj, id_obj, Category.assoc, Iso.hom_inv_id_app_assoc,
    ← Functor.map_comp_assoc, Iso.hom_inv_id_app, Functor.map_id, Category.id_comp]

Goal state after:
Module: Mathlib.CategoryTheory.Shift.CommShift

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
F₁ F₂ : C ⥤ D
τ : F₁ ⟶ F₂
A : Type u_5
inst✝⁴ : AddMonoid A
inst✝³ : HasShift C A
inst✝² : HasShift D A
inst✝¹ : F₁.CommShift A
inst✝ : F₂.CommShift A
a : A
hτ : CommShiftCore τ a
X : C
⊢ (shiftFunctor D a).map (τ.app X) =
    (Functor.commShiftIso F₁ a).inv.app X ≫ τ.app ((shiftFunctor C a).obj X) ≫ (Functor.commShiftIso F₂ a).hom.app X

Tactic:
by
  rw [← hτ.shift_app_comm, Iso.inv_hom_id_app_assoc]

Goal state after:
Module: Mathlib.CategoryTheory.Shift.CommShift

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Category.{v_3, u_3} E
A : Type u_4
inst✝¹ : AddMonoid A
inst✝ : HasShift D A
F : C ⥤ D
hF : F.FullyFaithful
s : A → C ⥤ C
i : (i : A) → s i ⋙ F ≅ F ⋙ shiftFunctor D i
⊢ F.CommShift A

Tactic:
by
  letI := hF.hasShift s i
  exact
  { commShiftIso := i
    commShiftIso_zero := by
      ext X
      simp [ShiftMkCore.shiftFunctorZero_eq]
    commShiftIso_add := fun a b => by
      ext X
      simp [ShiftMkCore.shiftFunctorAdd_eq, ShiftMkCore.shiftFunctor_eq,
        ← Functor.map_comp_assoc] }

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

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝²² : Category.{v_1, u_1} C
inst✝²¹ : Category.{v_2, u_2} D
inst✝²⁰ : Category.{v_3, u_3} E
inst✝¹⁹ : HasShift C ℤ
inst✝¹⁸ : HasShift D ℤ
inst✝¹⁷ : HasShift E ℤ
F : C ⥤ D
inst✝¹⁶ : F.CommShift ℤ
G : D ⥤ E
inst✝¹⁵ : G.CommShift ℤ
inst✝¹⁴ : HasZeroObject C
inst✝¹³ : HasZeroObject D
inst✝¹² : HasZeroObject E
inst✝¹¹ : Preadditive C
inst✝¹⁰ : Preadditive D
inst✝⁹ : Preadditive E
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor E n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
inst✝³ : Pretriangulated E
inst✝² : (F ⋙ G).IsTriangulated
inst✝¹ : F.IsTriangulated
inst✝ : F.mapArrow.EssSurj
T : Triangle D
hT : T ∈ distinguishedTriangles
⊢ G.mapTriangle.obj T ∈ distinguishedTriangles

Tactic:
by
    obtain ⟨T', hT', ⟨e⟩⟩ := F.mem_mapTriangle_essImage_of_distinguished T hT
    exact isomorphic_distinguished _ ((F ⋙ G).map_distinguished T' hT') _
      (G.mapTriangle.mapIso e.symm ≪≫ (mapTriangleCompIso F G).symm.app _)

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasShift C ℤ
inst✝¹² : HasShift D ℤ
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasZeroObject D
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
inst✝³ : F.CommShift ℤ
inst✝² : F.IsTriangulated
inst✝¹ : (F.mapComposableArrows 2).EssSurj
inst✝ : IsTriangulated C
Y₁ Y₂ Y₃ Z₁₂ Z₂₃ Z₁₃ : D
u₁₂ : Y₁ ⟶ Y₂
⊢ ∀ {u₂₃ : Y₂ ⟶ Y₃} {u₁₃ : Y₁ ⟶ Y₃} (comm : u₁₂ ≫ u₂₃ = u₁₃) {v₁₂ : Y₂ ⟶ Z₁₂} {w₁₂ : Z₁₂ ⟶ (shiftFunctor D 1).obj Y₁}
    (h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles) {v₂₃ : Y₃ ⟶ Z₂₃} {w₂₃ : Z₂₃ ⟶ (shiftFunctor D 1).obj Y₂}
    (h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles) {v₁₃ : Y₃ ⟶ Z₁₃} {w₁₃ : Z₁₃ ⟶ (shiftFunctor D 1).obj Y₁}
    (h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ distinguishedTriangles), Nonempty (Octahedron comm h₁₂ h₂₃ h₁₃)

Tactic:
u₂₃

Goal state after:
C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasShift C ℤ
inst✝¹² : HasShift D ℤ
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasZeroObject D
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
inst✝³ : F.CommShift ℤ
inst✝² : F.IsTriangulated
inst✝¹ : (F.mapComposableArrows 2).EssSurj
inst✝ : IsTriangulated C
Y₁ Y₂ Y₃ Z₁₂ Z₂₃ Z₁₃ : D
u₁₂ : Y₁ ⟶ Y₂
u₂₃ : Y₂ ⟶ Y₃
⊢ ∀ {u₁₃ : Y₁ ⟶ Y₃} (comm : u₁₂ ≫ u₂₃ = u₁₃) {v₁₂ : Y₂ ⟶ Z₁₂} {w₁₂ : Z₁₂ ⟶ (shiftFunctor D 1).obj Y₁}
    (h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles) {v₂₃ : Y₃ ⟶ Z₂₃} {w₂₃ : Z₂₃ ⟶ (shiftFunctor D 1).obj Y₂}
    (h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles) {v₁₃ : Y₃ ⟶ Z₁₃} {w₁₃ : Z₁₃ ⟶ (shiftFunctor D 1).obj Y₁}
    (h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ distinguishedTriangles), Nonempty (Octahedron comm h₁₂ h₂₃ h₁₃)