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

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝ : Group G
g : G
P : Sylow p G
⊢ g • P = P ↔ g ∈ normalizer ↑P

Tactic:
by
  rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P),
      mem_set_normalizer_iff, inv_inv]
  exact
    forall_congr' fun h =>
      iff_congr Iff.rfl
        ⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
          fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩

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

Goal state before:
pp: case mp
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
⊢ vanishingIdeal s ∈ minimalPrimes R → closure s ∈ irreducibleComponents (PrimeSpectrum R)

Tactic:
rw [← zeroLocus_minimalPrimes, ← zeroLocus_vanishingIdeal_eq_closure]

Goal state after:
case mp
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
⊢ vanishingIdeal s ∈ minimalPrimes R → zeroLocus ↑(vanishingIdeal s) ∈ zeroLocus ∘ SetLike.coe '' minimalPrimes R
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic

Goal state before:
pp: G : Type u_2
G' : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : IsCyclic G'
f : G →* G'
hf : f.ker ≤ center G
⊢ IsMulCommutative G

Tactic:
refine ⟨⟨fun a b ↦ ?_⟩⟩

Goal state after:
G : Type u_2
G' : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : IsCyclic G'
f : G →* G'
hf : f.ker ≤ center G
a b : G
⊢ a * b = b * a
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : N.Normal
inst✝ : Finite (Sylow p ↥N)
P : Sylow p ↥N
g : G
x✝ : g ∈ ⊤
n : ↥N
hn : (MulAut.conj n * MulAut.conjNormal g) • P = P
⊢ ↑n * g ∈ normalizer ↑(map N.subtype ↑P)

Tactic:
← MulAut.conjNormal_val,

Goal state after:
G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : N.Normal
inst✝ : Finite (Sylow p ↥N)
P : Sylow p ↥N
g : G
x✝ : g ∈ ⊤
n : ↥N
hn : (MulAut.conjNormal ↑n * MulAut.conjNormal g) • P = P
⊢ ↑n * g ∈ normalizer ↑(map N.subtype ↑P)
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
hn : (normalizer ↑P).Normal
⊢ (↑P).Normal

Tactic:
← normalizer_eq_top_iff,

Goal state after:
G : Type u
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
hn : (normalizer ↑P).Normal
⊢ normalizer ↑↑P = ⊤
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : Fin (s.length + 1)
hx : s.toFun i ∈ s
j : Fin (s.length + 1)
hy : s.toFun j ∈ s
⊢ i ≤ j ∨ j ≤ i

Tactic:
exact le_total i j

Goal state after:
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
n : ℕ
ih :
  ∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),
    head s ≤ x →
      s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)
s : CompositionSeries X
x : X
hm : IsMaximal x (last s)
hb : head s ≤ x
hn : s.length = n + 1
h0s : 0 < s.length
hetx : ¬(eraseLast s).last = x
imxs : IsMaximal (x ⊓ (eraseLast s).last) (eraseLast s).last
t : CompositionSeries X
htb : head t = (eraseLast s).head
htl : t.length + 1 = n
htt : last t = x ⊓ (eraseLast s).last
hteqv : Equivalent (eraseLast s) (snoc t (eraseLast s).last ⋯)
⊢ (eraseLast s).last ⊓ x = last t

Tactic:
by rw [inf_comm, htt]

Goal state after:
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
n : ℕ
ih :
  ∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),
    head s ≤ x →
      s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)
s : CompositionSeries X
x : X
hm : IsMaximal x (last s)
hb : head s ≤ x
hn : s.length = n + 1
h0s : 0 < s.length
hetx : ¬(eraseLast s).last = x
imxs : IsMaximal (x ⊓ (eraseLast s).last) (eraseLast s).last
t : CompositionSeries X
htb : head t = (eraseLast s).head
htl : t.length + 1 = n
htt : last t = x ⊓ (eraseLast s).last
hteqv : Equivalent (eraseLast s) (snoc t (eraseLast s).last ⋯)
hmtx : IsMaximal (last t) x
this : s.Equivalent ((snoc t (eraseLast s).last ⋯).snoc (last s) ⋯)
⊢ (eraseLast s).last ⊓ x = last t

Tactic:
by rw [inf_comm, htt]

Goal state after:
Module: Mathlib.RingTheory.ReesAlgebra

Goal state before:
pp: R : Type u
inst✝ : CommRing R
I : Ideal R
f : R[X]
⊢ f ∈ reesAlgebra I ↔ ∀ i ∈ f.support, f.coeff i ∈ I ^ i

Tactic:
apply forall_congr'
  intro a
  rw [mem_support_iff, Iff.comm, Classical.imp_iff_right_iff, Ne, ← imp_iff_not_or]
  exact fun e => e.symm ▸ (I ^ a).zero_mem

Goal state after:
Module: Mathlib.RingTheory.Nakayama

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : Module.Finite R N
f : M →ₗ[R] N
I : Ideal R
Ile : I ≤ ⊥.jacobson
surj : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)
⊢ f.range = ⊤

Tactic:
← top_le_iff

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : Module.Finite R N
f : M →ₗ[R] N
I : Ideal R
Ile : I ≤ ⊥.jacobson
surj : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)
⊢ ⊤ ≤ f.range
Module: Mathlib.RingTheory.Ideal.Cotangent

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

Tactic:
by
      rw [← Submodule.map_le_iff_le_comap, Submodule.map_smul'', Submodule.map_top,
        Submodule.range_subtype, smul_eq_mul, pow_two]

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject

Goal state before:
pp: β : Type
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasCoproducts C
inst✝ : HasZeroMorphisms C
X Y : GradedObject β C
f g : X ⟶ Y
i : β
w :
  (Discrete.natTrans fun X ↦ (fun i ↦ f i) X.as).app { as := i } ≫
      colimit.ι (Discrete.functor fun i ↦ Y i) { as := i } =
    (Discrete.natTrans fun X ↦ (fun i ↦ g i) X.as).app { as := i } ≫
      colimit.ι (Discrete.functor fun i ↦ Y i) { as := i }
⊢ f i = g i

Tactic:
replace w : f i ≫ colimit.ι (Discrete.functor Y) ⟨i⟩ =
      g i ≫ colimit.ι (Discrete.functor Y) ⟨i⟩ := by simpa

Goal state after:
β : Type
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasCoproducts C
inst✝ : HasZeroMorphisms C
X Y : GradedObject β C
f g : X ⟶ Y
i : β
w : f i ≫ colimit.ι (Discrete.functor Y) { as := i } = g i ≫ colimit.ι (Discrete.functor Y) { as := i }
⊢ f i = g i
Module: Mathlib.CategoryTheory.Shift.Basic

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

Tactic:
shiftFunctorComm_eq C n 0 n (add_zero n)

Goal state after:
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddCommMonoid A
inst✝ : HasShift C A
X : C
n : A
⊢ (shiftFunctorZero C A).hom.app ((shiftFunctor C n).obj X) =
    ((shiftFunctorAdd' C n 0 n ⋯).symm ≪≫ shiftFunctorAdd' C 0 n n ⋯).hom.app X ≫ (shiftFunctorAdd' C 0 n n ⋯).inv.app X
Module: Mathlib.CategoryTheory.Shift.Basic

Goal state before:
pp: C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddCommMonoid A
inst✝ : HasShift C A
X : C
n : A
⊢ (shiftFunctorZero C A).inv.app ((shiftFunctor C n).obj X) =
    (shiftFunctor C n).map ((shiftFunctorZero C A).inv.app X) ≫ (shiftFunctorComm C n 0).inv.app X

Tactic:
← cancel_mono ((shiftFunctorZero C A).hom.app (X⟦n⟧)),

Goal state after:
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddCommMonoid A
inst✝ : HasShift C A
X : C
n : A
⊢ (shiftFunctorZero C A).inv.app ((shiftFunctor C n).obj X) ≫
      (shiftFunctorZero C A).hom.app ((shiftFunctor C n).obj X) =
    ((shiftFunctor C n).map ((shiftFunctorZero C A).inv.app X) ≫ (shiftFunctorComm C n 0).inv.app X) ≫
      (shiftFunctorZero C A).hom.app ((shiftFunctor C n).obj X)
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 : CochainComplex C ℕ
X : C
φ : X ⟶ K.X 0
inst✝ : HomologicalComplex.HasHomology K 0
hφ : φ ≫ K.d 0 1 = 0
⊢ IsIso (HomologicalComplex.liftCycles K φ 1 isIso_liftCycles_iff._proof_1 hφ) ↔
    { X₁ := X, X₂ := K.X 0, X₃ := K.X 1, f := φ, g := K.d 0 1, zero := hφ }.Exact ∧ Mono φ

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

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

Goal state before:
pp: case mpr
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
W : MorphismProperty C
F : C ⥤ D
⊢ W.IsInvertedBy F → W.isoClosure.IsInvertedBy F

Tactic:
intro h X Y f ⟨X', Y', f', hf', ⟨e⟩⟩

Goal state after:
case mpr
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
W : MorphismProperty C
F : C ⥤ D
h : W.IsInvertedBy F
X Y : C
f : X ⟶ Y
X' Y' : C
f' : X' ⟶ Y'
hf' : W f'
e : Arrow.mk f' ≅ Arrow.mk f
⊢ IsIso (F.map f)
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
ψ : (p : ℤ) → F.X p ⟶ G.X p
p q q' : ℤ
hpq : p + 0 = q
⊢ (ofHoms ψ).v p q hpq ≫ G.d q q' = ψ p ≫ G.d p q'

Tactic:
by
  rw [add_zero] at hpq
  subst hpq
  rw [ofHoms_v]

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
ψ : (p : ℤ) → F.X p ⟶ G.X p
p' p q : ℤ
hpq : p + 0 = q
⊢ F.d p' p ≫ (ofHoms ψ).v p q hpq = F.d p' q ≫ ψ q

Tactic:
by
  rw [add_zero] at hpq
  subst hpq
  rw [ofHoms_v]

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

Tactic:
rw [Pi.add_apply, Pi.add_apply, h.zero i j w, k.zero i j w, zero_add]

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n : ℤ
γ : Cochain F G n
p q' : ℤ
hpq : p + n = q'
⊢ γ.v p q' hpq ≫ (HomologicalComplex.XIsoOfEq G ⋯).inv = γ.v p q' ⋯

Tactic:
simp only [HomologicalComplex.XIsoOfEq, eqToIso_refl, Iso.refl_inv, comp_id]

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

Goal state before:
pp: 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)
this : Subsingleton Ω[B⁄A]
⊢ (map R A B B).ker = ⊤

Tactic:
← top_le_iff

Goal state after:
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)
this : Subsingleton Ω[B⁄A]
⊢ ⊤ ≤ (map R A B B).ker
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 : HomologicalComplex V c
k₀ : ι
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
hk₀' : ∀ (l : ι), ¬c.Rel l k₀
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ (nullHomotopicMap hom).f k₀ = 0

Tactic:
dsimp [nullHomotopicMap, dNext, prevD]
  rw [C.shape, D.shape, zero_comp, comp_zero, add_zero] <;> apply_assumption

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

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

Tactic:
rw [dif_pos (by simp only [CochainComplex.next])]

Goal state after:
case e_a
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f g : C ⟶ D
h k : D ⟶ E
i✝ : ι
P Q : CochainComplex V ℕ
e : P ⟶ Q
zero : P.X 1 ⟶ Q.X 0
comm_zero : e.f 0 = P.d 0 1 ≫ zero
one : P.X 2 ⟶ Q.X 1
comm_one : e.f 1 = zero ≫ Q.d 0 1 + P.d 1 2 ≫ one
succ :
  (n : ℕ) →
    (p :
        (f : P.X (n + 1) ⟶ Q.X n) ×'
          (f' : P.X (n + 2) ⟶ Q.X (n + 1)) ×' e.f (n + 1) = f ≫ Q.d n (n + 1) + P.d (n + 1) (n + 2) ≫ f') →
      (f'' : P.X (n + 3) ⟶ Q.X (n + 2)) ×' e.f (n + 2) = p.snd.fst ≫ Q.d (n + 1) (n + 2) + P.d (n + 2) (n + 3) ≫ f''
i : ℕ
⊢ (mkCoinductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst =
    (xNextIso P ⋯).inv ≫ (mkCoinductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: case pos
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
K L : CochainComplex C ℤ
n m : ℤ
D : Type u_2
inst✝² : Category.{v_1, u_2} D
inst✝¹ : Preadditive D
z : Cochain K L n
Φ : C ⥤ D
inst✝ : Φ.Additive
hnm : n + 1 = m
⊢ δ n m (z.map Φ) = (δ n m z).map Φ

pp: case neg
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
K L : CochainComplex C ℤ
n m : ℤ
D : Type u_2
inst✝² : Category.{v_1, u_2} D
inst✝¹ : Preadditive D
z : Cochain K L n
Φ : C ⥤ D
inst✝ : Φ.Additive
hnm : ¬n + 1 = m
⊢ δ n m (z.map Φ) = (δ n m z).map Φ

Tactic:
· ext p q hpq
    dsimp
    simp only [δ_v n m hnm _ p q hpq (q - 1) (p + 1) rfl rfl,
      Functor.map_add, Functor.map_comp, Functor.map_units_smul,
      Cochain.map_v, Functor.mapHomologicalComplex_obj_d]

Goal state after:
case neg
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
K L : CochainComplex C ℤ
n m : ℤ
D : Type u_2
inst✝² : Category.{v_1, u_2} D
inst✝¹ : Preadditive D
z : Cochain K L n
Φ : C ⥤ D
inst✝ : Φ.Additive
hnm : ¬n + 1 = m
⊢ δ n m (z.map Φ) = (δ n m z).map Φ
Module: Mathlib.CategoryTheory.Quotient

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
r : HomRel C
x✝³ x✝² : C
x✝¹ x✝ : x✝³ ⟶ x✝²
⊢ Relation.EqvGen (HomRel.CompClosure r) x✝¹ x✝ ↔ (functor r).homRel x✝¹ x✝

Tactic:
functor_homRel_eq_compClosure_eqvGen

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
r : HomRel C
x✝³ x✝² : C
x✝¹ x✝ : x✝³ ⟶ x✝²
⊢ Relation.EqvGen (HomRel.CompClosure r) x✝¹ x✝ ↔ Relation.EqvGen (HomRel.CompClosure r) x✝¹ x✝
Module: Mathlib.Algebra.Homology.HomotopyCategory.Shift

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
a b : ℤ
⊢ CategoryTheory.shiftFunctorAdd' (CochainComplex C ℤ) a b (a + b) ⋯ = shiftFunctorAdd' C a b (a + b) ⋯

Tactic:
shiftFunctorAdd'_eq

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
a b : ℤ
⊢ shiftFunctorAdd' C a b (a + b) ⋯ = shiftFunctorAdd' C a b (a + b) ⋯
Module: Mathlib.CategoryTheory.Shift.Induced

Goal state before:
pp: C : Type ?u.2
D : Type ?u.4
inst✝⁵ : Category.{v_1, ?u.2} C
inst✝⁴ : Category.{v_2, ?u.4} D
F : C ⥤ D
A : Type ?u.15
inst✝³ : AddMonoid A
inst✝² : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
inst✝¹ : ((whiskeringLeft C D D).obj F).Full
inst✝ : ((whiskeringLeft C D D).obj F).Faithful
m₁ m₂ m₃ : A
X : C
eq :
  F.map ((shiftFunctorAdd C (m₁ + m₂) m₃).hom.app X) ≫
      F.map ((shiftFunctor C m₃).map ((shiftFunctorAdd C m₁ m₂).hom.app X)) =
    F.map (eqToHom ⋯) ≫
      F.map ((shiftFunctorAdd C m₁ (m₂ + m₃)).hom.app X) ≫
        F.map ((shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X))
⊢ (i (m₁ + m₂ + m₃)).hom.app X ≫
      F.map ((shiftFunctorAdd C (m₁ + m₂) m₃).hom.app X) ≫
        (i m₃).inv.app ((shiftFunctor C (m₁ + m₂)).obj X) ≫
          (((s m₃).map (F.map ((shiftFunctorAdd C m₁ m₂).hom.app X)) ≫
                (s m₃).map ((i m₂).inv.app ((shiftFunctor C m₁).obj X))) ≫
              (s m₃).map ((s m₂).map ((i m₁).inv.app X))) ≫
            (s m₃).map ((s m₂).map ((i m₁).hom.app X)) =
    (eqToHom ⋯).app (F.obj X) ≫
      (i (m₁ + (m₂ + m₃))).hom.app X ≫
        F.map ((shiftFunctorAdd C m₁ (m₂ + m₃)).hom.app X) ≫
          (i (m₂ + m₃)).inv.app ((shiftFunctor C m₁).obj X) ≫
            (s (m₂ + m₃)).map ((i m₁).inv.app X) ≫
              (Induced.add F s i m₂ m₃).hom.app ((s m₁).obj (F.obj X)) ≫ (s m₃).map ((s m₂).map ((i m₁).hom.app X))

Tactic:
slice_lhs 6 7 =>
          erw [← Functor.map_comp, ← Functor.map_comp, Iso.inv_hom_id_app,
            (s m₂).map_id, (s m₃).map_id]

Goal state after:
C : Type ?u.2
D : Type ?u.4
inst✝⁵ : Category.{v_1, ?u.2} C
inst✝⁴ : Category.{v_2, ?u.4} D
F : C ⥤ D
A : Type ?u.15
inst✝³ : AddMonoid A
inst✝² : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
inst✝¹ : ((whiskeringLeft C D D).obj F).Full
inst✝ : ((whiskeringLeft C D D).obj F).Faithful
m₁ m₂ m₃ : A
X : C
eq :
  F.map ((shiftFunctorAdd C (m₁ + m₂) m₃).hom.app X) ≫
      F.map ((shiftFunctor C m₃).map ((shiftFunctorAdd C m₁ m₂).hom.app X)) =
    F.map (eqToHom ⋯) ≫
      F.map ((shiftFunctorAdd C m₁ (m₂ + m₃)).hom.app X) ≫
        F.map ((shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X))
⊢ (i (m₁ + m₂ + m₃)).hom.app X ≫
      F.map ((shiftFunctorAdd C (m₁ + m₂) m₃).hom.app X) ≫
        (i m₃).inv.app ((shiftFunctor C (m₁ + m₂)).obj X) ≫
          (s m₃).map (F.map ((shiftFunctorAdd C m₁ m₂).hom.app X)) ≫
            (s m₃).map ((i m₂).inv.app ((shiftFunctor C m₁).obj X)) ≫
              𝟙 ((s m₃).obj ((F ⋙ s m₂).obj ((shiftFunctor C m₁).obj X))) =
    (eqToHom ⋯).app (F.obj X) ≫
      (i (m₁ + (m₂ + m₃))).hom.app X ≫
        F.map ((shiftFunctorAdd C m₁ (m₂ + m₃)).hom.app X) ≫
          (i (m₂ + m₃)).inv.app ((shiftFunctor C m₁).obj X) ≫
            (s (m₂ + m₃)).map ((i m₁).inv.app X) ≫
              (Induced.add F s i m₂ m₃).hom.app ((s m₁).obj (F.obj X)) ≫ (s m₃).map ((s m₂).map ((i m₁).hom.app X))
Module: Mathlib.Algebra.Homology.HomotopyCategory.Shift

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
D : Type u'
inst✝³ : Category.{v', u'} D
inst✝² : Preadditive D
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
n : ℤ
X Y : HomologicalComplex C (ComplexShape.up ℤ)
r : R
f : X ⟶ Y
h₁ :
  (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) n).inv.app X ≫
      (quotient C (ComplexShape.up ℤ)).map ((shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).map f) ≫
        (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) n).hom.app Y =
    (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).map ((quotient C (ComplexShape.up ℤ)).map f)
h₂ :
  (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) n).inv.app X ≫
      (quotient C (ComplexShape.up ℤ)).map ((shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).map (r • f)) ≫
        (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) n).hom.app Y =
    (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).map ((quotient C (ComplexShape.up ℤ)).map (r • f))
⊢ (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).map (r • (quotient C (ComplexShape.up ℤ)).map f) =
    r • (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).map ((quotient C (ComplexShape.up ℤ)).map f)

Tactic:
← Functor.map_smul,

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
D : Type u'
inst✝³ : Category.{v', u'} D
inst✝² : Preadditive D
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
n : ℤ
X Y : HomologicalComplex C (ComplexShape.up ℤ)
r : R
f : X ⟶ Y
h₁ :
  (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) n).inv.app X ≫
      (quotient C (ComplexShape.up ℤ)).map ((shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).map f) ≫
        (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) n).hom.app Y =
    (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).map ((quotient C (ComplexShape.up ℤ)).map f)
h₂ :
  (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) n).inv.app X ≫
      (quotient C (ComplexShape.up ℤ)).map ((shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).map (r • f)) ≫
        (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) n).hom.app Y =
    (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).map ((quotient C (ComplexShape.up ℤ)).map (r • f))
⊢ (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).map ((quotient C (ComplexShape.up ℤ)).map (r • f)) =
    r • (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).map ((quotient C (ComplexShape.up ℤ)).map f)
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 : ℤ
x✝¹ : R
x✝ : Cochain K L n
⊢ (↑(shiftAddHom K L n a)).toFun (x✝¹ • x✝) = (RingHom.id R) x✝¹ • (↑(shiftAddHom K L n a)).toFun x✝

Tactic:
dsimp; simp only [shift_smul]

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g : T₁ ⟶ T₂
⊢ f + g = g + f

Tactic:
by ext <;> apply add_comm

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
F₁ F₂ : C ⥤ D
τ : F₁ ⟶ F₂
A : Type u_5
inst✝⁶ : AddMonoid A
inst✝⁵ : HasShift C A
inst✝⁴ : HasShift D A
inst✝³ : F₁.CommShift A
inst✝² : F₂.CommShift A
inst✝¹ : IsIso τ
inst✝ : CommShift τ A
⊢ CommShift (inv τ) A

Tactic:
have : NatTrans.CommShift (asIso τ).hom A := by assumption
  change NatTrans.CommShift (asIso τ).inv A
  infer_instance

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✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F G : C ⥤ D
e : F ≅ G
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
⊢ (shiftFunctor C (a + b)).isoWhiskerLeft e.symm ≪≫
      commShiftIso F (a + b) ≪≫ isoWhiskerRight e (shiftFunctor D (a + b)) =
    isoAdd ((shiftFunctor C a).isoWhiskerLeft e.symm ≪≫ commShiftIso F a ≪≫ isoWhiskerRight e (shiftFunctor D a))
      ((shiftFunctor C b).isoWhiskerLeft e.symm ≪≫ commShiftIso F b ≪≫ isoWhiskerRight e (shiftFunctor D b))

Tactic:
ext X
    simp only [comp_obj, F.commShiftIso_add, Iso.trans_hom, isoWhiskerLeft_hom,
      Iso.symm_hom, isoWhiskerRight_hom, NatTrans.comp_app, whiskerLeft_app,
      isoAdd_hom_app, whiskerRight_app, assoc, map_comp, NatTrans.naturality_assoc,
      NatIso.cancel_natIso_inv_left]
    simp only [← Functor.map_comp_assoc, e.hom_inv_id_app_assoc]
    simp only [← NatTrans.naturality, comp_obj, comp_map, map_comp, assoc]

Goal state after: