mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝² : e.IsTruncGE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : HasZeroObject C
j : ι
j' : ι'
hj' : e.f j = j'
hj : ¬e.BoundaryGE j
φ : (shortComplexFunctor C c' j').obj K ⟶ (shortComplexFunctor C c' j').obj (K.truncGE e) :=
  (shortComplexFunctor C c' j').map (K.πTruncGE e)
this✝ : Epi φ.τ₁
this : IsIso φ.τ₂
⊢ c'.next j' = e.f (c.next j)

Tactic:
simpa only [← hj'] using e.next_f rfl

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

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✝⁷ : Category.{v_1, u_3} C
inst✝⁶ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁵ : e.IsTruncGE
inst✝⁴ : ∀ (i' : ι'), K.HasHomology i'
inst✝³ : ∀ (i' : ι'), L.HasHomology i'
inst✝² : ∀ (i' : ι'), M.HasHomology i'
inst✝¹ : HasZeroObject C
ι'' : Type u_4
c'' : ComplexShape ι''
e' : c''.Embedding c'
inst✝ : K.IsStrictlySupported e'
i' : ι'
hi' : ∀ (i : ι''), e'.f i ≠ i'
i : ι
hi : e.f i = i'
⊢ IsZero ((K.truncGE e).X i')

Tactic:
by_cases hi''' : e.BoundaryGE i

Goal state after:
case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁷ : Category.{v_1, u_3} C
inst✝⁶ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁵ : e.IsTruncGE
inst✝⁴ : ∀ (i' : ι'), K.HasHomology i'
inst✝³ : ∀ (i' : ι'), L.HasHomology i'
inst✝² : ∀ (i' : ι'), M.HasHomology i'
inst✝¹ : HasZeroObject C
ι'' : Type u_4
c'' : ComplexShape ι''
e' : c''.Embedding c'
inst✝ : K.IsStrictlySupported e'
i' : ι'
hi' : ∀ (i : ι''), e'.f i ≠ i'
i : ι
hi : e.f i = i'
hi''' : e.BoundaryGE i
⊢ IsZero ((K.truncGE e).X i')

case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁷ : Category.{v_1, u_3} C
inst✝⁶ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁵ : e.IsTruncGE
inst✝⁴ : ∀ (i' : ι'), K.HasHomology i'
inst✝³ : ∀ (i' : ι'), L.HasHomology i'
inst✝² : ∀ (i' : ι'), M.HasHomology i'
inst✝¹ : HasZeroObject C
ι'' : Type u_4
c'' : ComplexShape ι''
e' : c''.Embedding c'
inst✝ : K.IsStrictlySupported e'
i' : ι'
hi' : ∀ (i : ι''), e'.f i ≠ i'
i : ι
hi : e.f i = i'
hi''' : ¬e.BoundaryGE i
⊢ IsZero ((K.truncGE e).X i')
Module: Mathlib.Algebra.Homology.Embedding.AreComplementary

Goal state before:
pp: n₀ n₁ : ℤ
h : n₀ + 1 = n₁
i₁ i₂ : ℕ
⊢ (embeddingUpIntLE n₀).f i₁ ≠ (embeddingUpIntGE n₁).f i₂

Tactic:
by dsimp; lia

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

Goal state before:
pp: n₁ i : ℤ
k : ℕ
h : i + ↑k + 1 = n₁
hi : i ≤ i + ↑k
⊢ (embeddingUpIntLE (i + ↑k)).f k = i

Tactic:
by dsimp; lia

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
K L : CochainComplex C ℤ
φ : K ⟶ L
inst✝² : HasZeroObject C
inst✝¹ : ∀ (i : ℤ), HasHomology K i
inst✝ : ∀ (i : ℤ), HasHomology L i
n : ℤ
h : ∀ (i : ℤ), n ≤ i → QuasiIsoAt φ i
i : ℕ
⊢ n ≤ (embeddingUpIntGE n).f i

Tactic:
by dsimp; lia

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
K L : CochainComplex C ℤ
φ : K ⟶ L
inst✝² : HasZeroObject C
inst✝¹ : ∀ (i : ℤ), HasHomology K i
inst✝ : ∀ (i : ℤ), HasHomology L i
i : ℤ
k : ℕ
h : ∀ (i_1 : ℕ) (i' : ℤ), (embeddingUpIntLE (i + ↑k)).f i_1 = i' → QuasiIsoAt φ i'
hi : i ≤ i + ↑k
⊢ (embeddingUpIntLE (i + ↑k)).f k = i

Tactic:
by dsimp; lia

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
K L : CochainComplex C ℤ
φ : K ⟶ L
inst✝² : HasZeroObject C
inst✝¹ : ∀ (i : ℤ), HasHomology K i
inst✝ : ∀ (i : ℤ), HasHomology L i
n : ℤ
h : ∀ i ≤ n, QuasiIsoAt φ i
i : ℕ
⊢ (embeddingUpIntLE n).f i ≤ n

Tactic:
by dsimp; lia

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: case mp
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
⊢ (∀ (X Y : C) (a b : ℤ),
      Small.{w, w'}
        ((shiftFunctor (DerivedCategory C) a).obj (Q.obj ((CochainComplex.singleFunctor C 0).obj X)) ⟶
          (shiftFunctor (DerivedCategory C) b).obj (Q.obj ((CochainComplex.singleFunctor C 0).obj Y)))) →
    ∀ (X Y : C) (n : ℤ),
      0 ≤ n →
        Small.{w, w'} ((singleFunctor C 0).obj X ⟶ (shiftFunctor (DerivedCategory C) n).obj ((singleFunctor C 0).obj Y))

Tactic:
intro h X Y n hn
    exact (small_congr ((shiftFunctorZero _ ℤ).app
      ((singleFunctor C 0).obj X)).homFromEquiv).1 (h X Y 0 n)

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
⊢ (homologyFunctor C (up ℤ) 0).homologySequenceδ
        (Triangle.mk ((quotient C (up ℤ)).map S.f) ((quotient C (up ℤ)).map (inr S.f))
          ((quotient C (up ℤ)).map (triangle S.f).mor₃ ≫ (Functor.commShiftIso (quotient C (up ℤ)) 1).hom.app S.X₁))
        n₀ n₁ h ≫
      (homologyFunctorFactors C (up ℤ) n₁).hom.app S.X₁ =
    (homologyFunctorFactors C (up ℤ) n₀).hom.app (mappingCone S.f) ≫
      HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫
        hS.δ n₀ n₁ h ≫
          (homologyFunctorFactors C (up ℤ) n₁).inv.app S.X₁ ≫ (homologyFunctorFactors C (up ℤ) n₁).hom.app S.X₁

Tactic:
Iso.inv_hom_id_app,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
⊢ (homologyFunctor C (up ℤ) 0).homologySequenceδ
        (Triangle.mk ((quotient C (up ℤ)).map S.f) ((quotient C (up ℤ)).map (inr S.f))
          ((quotient C (up ℤ)).map (triangle S.f).mor₃ ≫ (Functor.commShiftIso (quotient C (up ℤ)) 1).hom.app S.X₁))
        n₀ n₁ h ≫
      (homologyFunctorFactors C (up ℤ) n₁).hom.app S.X₁ =
    (homologyFunctorFactors C (up ℤ) n₀).hom.app (mappingCone S.f) ≫
      HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫
        hS.δ n₀ n₁ h ≫ 𝟙 ((HomologicalComplex.homologyFunctor C (up ℤ) n₁).obj S.X₁)
Module: Mathlib.CategoryTheory.Shift.Opposite

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : Type u_2
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
X : OppositeShift C A
⊢ (shiftFunctorZero (OppositeShift C A) A).hom.app X = ((shiftFunctorZero C A).inv.app (Opposite.unop X)).op

Tactic:
rw [← cancel_mono ((shiftFunctorZero (OppositeShift C A) A).inv.app X),
    Iso.hom_inv_id_app, oppositeShiftFunctorZero_inv_app, ← op_comp,
    Iso.hom_inv_id_app, op_id]
  rfl

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
inst✝ : HasShift C B
φ : A →+ B
X : PullbackShift C φ
⊢ (shiftFunctorZero (PullbackShift C φ) A).inv.app X ≫ (shiftFunctorZero (PullbackShift C φ) A).hom.app X =
    (shiftFunctorZero (PullbackShift C φ) A).inv.app X ≫
      (pullbackShiftIso C φ 0 0 ⋯).hom.app X ≫ (shiftFunctorZero C B).hom.app X

Tactic:
Iso.inv_hom_id_app,

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
inst✝ : HasShift C B
φ : A →+ B
X : PullbackShift C φ
⊢ 𝟙 ((𝟭 (PullbackShift C φ)).obj X) =
    (shiftFunctorZero (PullbackShift C φ) A).inv.app X ≫
      (pullbackShiftIso C φ 0 0 ⋯).hom.app X ≫ (shiftFunctorZero C B).hom.app X
Module: Mathlib.CategoryTheory.Shift.Pullback

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
inst✝ : HasShift C B
φ : A →+ B
X : PullbackShift C φ
⊢ (shiftFunctorZero (PullbackShift C φ) A).inv.app X ≫ (shiftFunctorZero (PullbackShift C φ) A).hom.app X =
    (shiftFunctorZero (PullbackShift C φ) A).inv.app X ≫
      (pullbackShiftIso C φ 0 (φ 0) ⋯).hom.app X ≫ (shiftFunctorZero' C (φ 0) ⋯).hom.app X

Tactic:
Iso.inv_hom_id_app,

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
inst✝ : HasShift C B
φ : A →+ B
X : PullbackShift C φ
⊢ 𝟙 ((𝟭 (PullbackShift C φ)).obj X) =
    (shiftFunctorZero (PullbackShift C φ) A).inv.app X ≫
      (pullbackShiftIso C φ 0 (φ 0) ⋯).hom.app X ≫ (shiftFunctorZero' C (φ 0) ⋯).hom.app X
Module: Mathlib.CategoryTheory.Shift.Adjunction

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
h : CompatibilityCounit adj e₁ e₂
X : C
this :
  F.map (e₂.inv.app (F.obj X)) ≫ adj.counit.app ((shiftFunctor D a).obj (F.obj X)) =
    F.map (e₂.inv.app (F.obj X) ≫ e₂.hom.app (F.obj X)) ≫
      e₁.hom.app (G.obj (F.obj X)) ≫ (shiftFunctor D a).map (adj.counit.app (F.obj X))
⊢ e₁.hom.app X =
    F.map ((shiftFunctor C a).map (adj.unit.app X)) ≫
      F.map (e₂.inv.app (F.obj X)) ≫ adj.counit.app ((shiftFunctor D a).obj (F.obj X))

Tactic:
Iso.inv_hom_id_app,

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
h : CompatibilityCounit adj e₁ e₂
X : C
this :
  F.map (e₂.inv.app (F.obj X)) ≫ adj.counit.app ((shiftFunctor D a).obj (F.obj X)) =
    F.map (𝟙 ((G ⋙ shiftFunctor C a).obj (F.obj X))) ≫
      e₁.hom.app (G.obj (F.obj X)) ≫ (shiftFunctor D a).map (adj.counit.app (F.obj X))
⊢ e₁.hom.app X =
    F.map ((shiftFunctor C a).map (adj.unit.app X)) ≫
      F.map (e₂.inv.app (F.obj X)) ≫ adj.counit.app ((shiftFunctor D a).obj (F.obj X))
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Basic

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
X : Cᵒᵖ
a₁ a₂ a₃ : ℤ
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : ℤ
h₁ : a₁ + b₁ = 0
h₂ : a₂ + b₂ = 0
h₃ : a₃ + b₃ = 0
⊢ b₁ = (AddMonoidHom.mk' (fun n ↦ -n) ⋯) a₁

Tactic:
by dsimp; lia

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
X : Cᵒᵖ
a₁ a₂ a₃ : ℤ
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : ℤ
h₁ : a₁ + b₁ = 0
h₂ : a₂ + b₂ = 0
h₃ : a₃ + b₃ = 0
⊢ b₂ = (AddMonoidHom.mk' (fun n ↦ -n) ⋯) a₂

Tactic:
by dsimp; lia

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
X : Cᵒᵖ
a₁ a₂ a₃ : ℤ
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : ℤ
h₁ : a₁ + b₁ = 0
h₂ : a₂ + b₂ = 0
h₃ : a₃ + b₃ = 0
⊢ b₃ = (AddMonoidHom.mk' (fun n ↦ -n) ⋯) a₃

Tactic:
by dsimp; lia

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

Goal state before:
pp: case e_a
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
inst✝ : HasShift C B
φ : A →+ B
X : PullbackShift C φ
a₁ a₂ : A
h₃ : φ a₁ + φ a₂ = φ (a₁ + a₂)
⊢ ((shiftMonoidalFunctor C B).map (Functor.LaxMonoidal.μ (Discrete.addMonoidalFunctor φ) { as := a₁ } { as := a₂ })).app
      X =
    (pullbackShiftIso C φ (a₁ + a₂) (φ a₁ + φ a₂) h₃).inv.app X

Tactic:
Discrete.addMonoidalFunctor_μ

Goal state after:
case e_a
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
inst✝ : HasShift C B
φ : A →+ B
X : PullbackShift C φ
a₁ a₂ : A
h₃ : φ a₁ + φ a₂ = φ (a₁ + a₂)
⊢ ((shiftMonoidalFunctor C B).map (Discrete.eqToHom ⋯)).app X =
    (pullbackShiftIso C φ (a₁ + a₂) (φ a₁ + φ a₂) h₃).inv.app X
Module: Mathlib.CategoryTheory.Shift.Pullback

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
inst✝ : HasShift C B
φ : A →+ B
X : PullbackShift C φ
a₁ a₂ a₃ : A
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : B
h₁ : b₁ = φ a₁
h₂ : b₂ = φ a₂
h₃ : b₃ = φ a₃
⊢ (shiftFunctorAdd' (PullbackShift C φ) a₁ a₂ a₃ h).inv.app X ≫
      (shiftFunctorAdd' (PullbackShift C φ) a₁ a₂ a₃ h).hom.app X =
    (shiftFunctorAdd' (PullbackShift C φ) a₁ a₂ a₃ h).inv.app X ≫
      (pullbackShiftIso C φ a₃ b₃ h₃).hom.app X ≫
        (shiftFunctorAdd' C b₁ b₂ b₃ ⋯).hom.app X ≫
          (pullbackShiftIso C φ a₂ b₂ h₂).inv.app ((shiftFunctor C b₁).obj X) ≫
            (shiftFunctor (PullbackShift C φ) a₂).map ((pullbackShiftIso C φ a₁ b₁ h₁).inv.app X)

Tactic:
Iso.inv_hom_id_app,

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
inst✝ : HasShift C B
φ : A →+ B
X : PullbackShift C φ
a₁ a₂ a₃ : A
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : B
h₁ : b₁ = φ a₁
h₂ : b₂ = φ a₂
h₃ : b₃ = φ a₃
⊢ 𝟙 ((shiftFunctor (PullbackShift C φ) a₁ ⋙ shiftFunctor (PullbackShift C φ) a₂).obj X) =
    (shiftFunctorAdd' (PullbackShift C φ) a₁ a₂ a₃ h).inv.app X ≫
      (pullbackShiftIso C φ a₃ b₃ h₃).hom.app X ≫
        (shiftFunctorAdd' C b₁ b₂ b₃ ⋯).hom.app X ≫
          (pullbackShiftIso C φ a₂ b₂ h₂).inv.app ((shiftFunctor C b₁).obj X) ≫
            (shiftFunctor (PullbackShift C φ) a₂).map ((pullbackShiftIso C φ a₁ b₁ h₁).inv.app X)
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Basic

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
X : Cᵒᵖ
m n p : ℤ
h : m + n = p
⊢ (opShiftFunctorEquivalence C p).unitIso.inv.app X ≫ (opShiftFunctorEquivalence C p).unitIso.hom.app X =
    (((shiftFunctor C p).map ((shiftFunctorAdd' Cᵒᵖ n m p ⋯).hom.app X).unop).op ≫
        ((shiftFunctorAdd' C m n p h).inv.app (Opposite.unop ((shiftFunctor Cᵒᵖ n ⋙ shiftFunctor Cᵒᵖ m).obj X))).op ≫
          ((shiftFunctor C n).map
                ((opShiftFunctorEquivalence C m).unitIso.inv.app ((shiftFunctor Cᵒᵖ n).obj X)).unop).op ≫
            (opShiftFunctorEquivalence C n).unitIso.inv.app X) ≫
      (opShiftFunctorEquivalence C p).unitIso.hom.app X

Tactic:
Iso.inv_hom_id_app,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
X : Cᵒᵖ
m n p : ℤ
h : m + n = p
⊢ 𝟙 (((opShiftFunctorEquivalence C p).functor ⋙ (opShiftFunctorEquivalence C p).inverse).obj X) =
    (((shiftFunctor C p).map ((shiftFunctorAdd' Cᵒᵖ n m p ⋯).hom.app X).unop).op ≫
        ((shiftFunctorAdd' C m n p h).inv.app (Opposite.unop ((shiftFunctor Cᵒᵖ n ⋙ shiftFunctor Cᵒᵖ m).obj X))).op ≫
          ((shiftFunctor C n).map
                ((opShiftFunctorEquivalence C m).unitIso.inv.app ((shiftFunctor Cᵒᵖ n).obj X)).unop).op ≫
            (opShiftFunctorEquivalence C n).unitIso.inv.app X) ≫
      (opShiftFunctorEquivalence C p).unitIso.hom.app X
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X : C
S : ShortComplex C
hS : S.ShortExact
n₀ : ℕ
x₃ : Ext X S.X₃ n₀
n₁ : ℕ
hn₁ : n₀ + 1 = n₁
hx₃ : x₃.comp hS.extClass hn₁ = 0
this :
  ∀
    (x₂ :
      ↑{ X₁ := AddCommGrpCat.of (Ext X S.X₂ n₀), X₂ := AddCommGrpCat.of (Ext X S.X₃ n₀),
            X₃ := AddCommGrpCat.of (Ext X S.X₁ n₁), f := AddCommGrpCat.ofHom ((mk₀ S.g).postcomp X ⋯),
            g := AddCommGrpCat.ofHom (hS.extClass.postcomp X hn₁), zero := ⋯ }.X₂),
    (ConcreteCategory.hom
            { X₁ := AddCommGrpCat.of (Ext X S.X₂ n₀), X₂ := AddCommGrpCat.of (Ext X S.X₃ n₀),
                X₃ := AddCommGrpCat.of (Ext X S.X₁ n₁), f := AddCommGrpCat.ofHom ((mk₀ S.g).postcomp X ⋯),
                g := AddCommGrpCat.ofHom (hS.extClass.postcomp X hn₁), zero := ⋯ }.g)
          x₂ =
        0 →
      ∃ x₁,
        (ConcreteCategory.hom
              { X₁ := AddCommGrpCat.of (Ext X S.X₂ n₀), X₂ := AddCommGrpCat.of (Ext X S.X₃ n₀),
                  X₃ := AddCommGrpCat.of (Ext X S.X₁ n₁), f := AddCommGrpCat.ofHom ((mk₀ S.g).postcomp X ⋯),
                  g := AddCommGrpCat.ofHom (hS.extClass.postcomp X hn₁), zero := ⋯ }.f)
            x₁ =
          x₂
⊢ ∃ x₂, x₂.comp (mk₀ S.g) ⋯ = x₃

Tactic:
exact this x₃ hx₃

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
S : ShortComplex C
hS : S.ShortExact
Y : C
n₁ : ℕ
x₃ : Ext S.X₃ Y n₁
hx₃ : (mk₀ S.g).comp x₃ ⋯ = 0
n₀ : ℕ
hn₀ : 1 + n₀ = n₁
this :
  ∀
    (x₂ :
      ↑{ X₁ := AddCommGrpCat.of (Ext S.X₁ Y n₀), X₂ := AddCommGrpCat.of (Ext S.X₃ Y n₁),
            X₃ := AddCommGrpCat.of (Ext S.X₂ Y n₁), f := AddCommGrpCat.ofHom (hS.extClass.precomp Y hn₀),
            g := AddCommGrpCat.ofHom ((mk₀ S.g).precomp Y ⋯), zero := ⋯ }.X₂),
    (ConcreteCategory.hom
            { X₁ := AddCommGrpCat.of (Ext S.X₁ Y n₀), X₂ := AddCommGrpCat.of (Ext S.X₃ Y n₁),
                X₃ := AddCommGrpCat.of (Ext S.X₂ Y n₁), f := AddCommGrpCat.ofHom (hS.extClass.precomp Y hn₀),
                g := AddCommGrpCat.ofHom ((mk₀ S.g).precomp Y ⋯), zero := ⋯ }.g)
          x₂ =
        0 →
      ∃ x₁,
        (ConcreteCategory.hom
              { X₁ := AddCommGrpCat.of (Ext S.X₁ Y n₀), X₂ := AddCommGrpCat.of (Ext S.X₃ Y n₁),
                  X₃ := AddCommGrpCat.of (Ext S.X₂ Y n₁), f := AddCommGrpCat.ofHom (hS.extClass.precomp Y hn₀),
                  g := AddCommGrpCat.ofHom ((mk₀ S.g).precomp Y ⋯), zero := ⋯ }.f)
            x₁ =
          x₂
⊢ ∃ x₁, hS.extClass.comp x₁ hn₀ = x₃

Tactic:
exact this x₃ hx₃

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup

Goal state before:
pp: case succ
n : ℕ
h : ↑(T ^ ↑n) = !![1, ↑n; 0, 1]
⊢ ↑(T ^ ↑n * T ^ 1) = !![1, ↑n + 1; 0, 1]

Tactic:
zpow_one,

Goal state after:
case succ
n : ℕ
h : ↑(T ^ ↑n) = !![1, ↑n; 0, 1]
⊢ ↑(T ^ ↑n * T) = !![1, ↑n + 1; 0, 1]
Module: Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup

Goal state before:
pp: case pred
n : ℕ
h : ↑(T ^ (-↑n)) = !![1, -↑n; 0, 1]
⊢ ↑(T ^ (-↑n) * (T ^ 1)⁻¹) = !![1, -↑n - 1; 0, 1]

Tactic:
zpow_one,

Goal state after:
case pred
n : ℕ
h : ↑(T ^ (-↑n)) = !![1, -↑n; 0, 1]
⊢ ↑(T ^ (-↑n) * T⁻¹) = !![1, -↑n - 1; 0, 1]
Module: Mathlib.RingTheory.GradedAlgebra.Basic

Goal state before:
pp: ι : Type u_1
A : Type u_3
σ : Type u_4
inst✝¹¹ : Semiring A
inst✝¹⁰ : DecidableEq ι
inst✝⁹ : AddCommMonoid ι
inst✝⁸ : PartialOrder ι
inst✝⁷ : CanonicallyOrderedAdd ι
inst✝⁶ : SetLike σ A
inst✝⁵ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝⁴ : GradedRing 𝒜
a : A
i : ι
inst✝³ : Sub ι
inst✝² : OrderedSub ι
inst✝¹ : AddLeftReflectLE ι
n : ι
inst✝ : Decidable (i ≤ n)
b : ↥(𝒜 i)
⊢ ↑(((decompose 𝒜) (a * ↑b)) n) = if i ≤ n then ↑(((decompose 𝒜) a) (n - i)) * ↑b else 0

Tactic:
rw [decompose_mul, decompose_coe, coe_mul_of_apply]

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
M : Type u_4
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Invertible 2
Q : QuadraticMap R M R
hB : Q.Anisotropic
x : M
hx : ∀ (y : M), ((associated' Q) x) y = 0
⊢ Q x = 0

Tactic:
rw [← hx x]

Goal state after:
R : Type u_3
M : Type u_4
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Invertible 2
Q : QuadraticMap R M R
hB : Q.Anisotropic
x : M
hx : ∀ (y : M), ((associated' Q) x) y = 0
⊢ Q x = ((associated' Q) x) x
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: M : Type u_4
N : Type u_5
R₂ : Type u
inst✝⁵ : CommSemiring R₂
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R₂ M
inst✝² : PartialOrder N
inst✝¹ : AddCommMonoid N
inst✝ : Module R₂ N
Q : QuadraticMap R₂ M N
hQ : Q.PosDef
x : M
⊢ Q x ≤ 0 ↔ x = 0

Tactic:
by
  refine ⟨fun h ↦ ?_, fun h ↦ by simp [h]⟩
  have : Q x = 0 := le_antisymm h (hQ.nonneg x)
  rwa [← hQ.anisotropic]

Goal state after:
Module: Mathlib.LinearAlgebra.ExteriorAlgebra.Basic

Goal state before:
pp: case neg
R✝ : Type u1
inst✝⁵ : CommRing R✝
M✝ : Type u2
inst✝⁴ : AddCommGroup M✝
inst✝³ : Module R✝ M✝
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
n : ℕ
hn : ∀ (f : Fin n → M) (x y : Fin n), f x = f y → x < y → (List.ofFn fun i ↦ (ι R) (f i)).prod = 0
f : Fin (n + 1) → M
x y : Fin (n + 1)
hfxy : f x = f y
h : x < y
hx : ¬x = 0
⊢ (ι R) (f 0) * (List.ofFn fun i ↦ (ι R) (f i.succ)).prod = 0

Tactic:
· convert! mul_zero (ι R (f 0))
          refine
            hn
              (fun i => f <| Fin.succ i) (x.pred hx)
              (y.pred (ne_of_lt <| lt_of_le_of_lt x.zero_le h).symm) ?_
              (Fin.pred_lt_pred_iff.mpr h)
          simp only [Fin.succ_pred]
          exact hfxy

Goal state after:
Module: Mathlib.LinearAlgebra.ExteriorAlgebra.Basic

Goal state before:
pp: R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
m n : ℕ
a : Fin m → M
b : Fin n → M
⊢ (List.ofFn fun i ↦ (ι R) (a i)).prod * (List.ofFn fun i ↦ (ι R) (b i)).prod =
    (List.ofFn (⇑(ι R) ∘ Fin.append a b)).prod

Tactic:
rw [← List.map_ofFn, List.ofFn_fin_append, List.map_append, List.prod_append]

Goal state after:
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
m n : ℕ
a : Fin m → M
b : Fin n → M
⊢ (List.ofFn fun i ↦ (ι R) (a i)).prod * (List.ofFn fun i ↦ (ι R) (b i)).prod =
    (List.map (⇑(ι R)) (List.ofFn a)).prod * (List.map (⇑(ι R)) (List.ofFn b)).prod
Module: Mathlib.LinearAlgebra.ExteriorAlgebra.Basic

Goal state before:
pp: case neg
R : Type u1
inst✝³ : CommRing R
M : Type u2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
m n : ℕ
I : Type u_1
inst✝ : LinearOrder I
v : I → M
s : ↑(powersetCard I m)
t : ↑(powersetCard I n)
h : Disjoint ↑s ↑t
i : Fin (m + n)
e : Fin (m + n) ≃o ↥↑(disjUnion h) := orderIsoOfFin (disjUnion h)
hi : m ≤ ↑i
⊢ Fin.append (v ∘ ⇑(ofFinEmbEquiv.symm s)) (v ∘ ⇑(ofFinEmbEquiv.symm t)) i =
    v
      ↑(e
          (e.symm
            ((Equiv.Finset.disjUnionEquiv (↑s) (↑t) h)
              (↑((orderIsoOfFin s).sumCongr (orderIsoOfFin t)) (finSumFinEquiv.symm i)))))

Tactic:
rw [← Fin.natAdd_subNat_cast hi, Fin.append_right, OrderIso.apply_symm_apply,
      finSumFinEquiv_symm_apply_natAdd]
    aesop

Goal state after: