mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Localization.HomEquiv

Goal state before:
pp: C₁ : Type u_2
D₁ : Type u_5
inst✝² : Category.{v_2, u_2} C₁
inst✝¹ : Category.{v_5, u_5} D₁
W₁ : MorphismProperty C₁
L₁ : C₁ ⥤ D₁
inst✝ : L₁.IsLocalization W₁
X Y : C₁
f : L₁.obj X ⟶ L₁.obj Y
⊢ (id W₁).homMap L₁ L₁ f = f

Tactic:
simpa using

Goal state after:
C₁ : Type u_2
D₁ : Type u_5
inst✝² : Category.{v_2, u_2} C₁
inst✝¹ : Category.{v_5, u_5} D₁
W₁ : MorphismProperty C₁
L₁ : C₁ ⥤ D₁
inst✝ : L₁.IsLocalization W₁
X Y : C₁
f : L₁.obj X ⟶ L₁.obj Y
⊢ (id W₁).homMap L₁ L₁ f = f
Module: Mathlib.Algebra.Homology.HomologySequenceLemmas

Goal state before:
pp: case neg
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
i : ι
h₁ : Epi (homologyMap φ.τ₁ i)
h₂ : Mono (homologyMap φ.τ₂ i)
h₃ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₁ j)
hi : ¬∃ j, c.Rel i j
this : Epi S₁.g
⊢ ∀ (j : ι), ¬c.Rel i j

Tactic:
simpa using

Goal state after:
case neg
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
i : ι
h₁ : Epi (homologyMap φ.τ₁ i)
h₂ : Mono (homologyMap φ.τ₂ i)
h₃ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₁ j)
hi : ¬∃ j, c.Rel i j
this : Epi S₁.g
⊢ ∀ (j : ι), ¬c.Rel i j
Module: Mathlib.Algebra.Homology.HomologySequenceLemmas

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
i : ι
h₁ : Epi (homologyMap φ.τ₂ i)
h₂ : ∀ (j : ι), c.Rel i j → Epi (homologyMap φ.τ₁ j)
h₃ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₂ j)
hi : ¬∃ j, c.Rel i j
this : Epi S₂.g
eq : homologyMap φ.τ₂ i ≫ homologyMap S₂.g i = homologyMap S₁.g i ≫ homologyMap φ.τ₃ i
⊢ ∀ (j : ι), ¬c.Rel i j

Tactic:
simpa using

Goal state after:
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
i : ι
h₁ : Epi (homologyMap φ.τ₂ i)
h₂ : ∀ (j : ι), c.Rel i j → Epi (homologyMap φ.τ₁ j)
h₃ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₂ j)
hi : ¬∃ j, c.Rel i j
this : Epi S₂.g
eq : homologyMap φ.τ₂ i ≫ homologyMap S₂.g i = homologyMap S₁.g i ≫ homologyMap φ.τ₃ i
⊢ ∀ (j : ι), ¬c.Rel i j
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X Y : C
a b : ℤ
inst✝ : HasDerivedCategory C
this :
  ∀ (a b : ℤ),
    Small.{w, w'}
      ((shiftFunctor (DerivedCategory C) a).obj ((singleFunctor C 0).obj X) ⟶
        (shiftFunctor (DerivedCategory C) b).obj ((singleFunctor C 0).obj Y))
φ₁ φ₂ : (singleFunctor C a).obj X ⟶ (singleFunctor C b).obj Y
h :
  (fun φ ↦ ((singleFunctors C).shiftIso (-a) a 0 ⋯).hom.app X ≫ φ ≫ ((singleFunctors C).shiftIso (-b) b 0 ⋯).inv.app Y)
      φ₁ =
    (fun φ ↦
        ((singleFunctors C).shiftIso (-a) a 0 ⋯).hom.app X ≫ φ ≫ ((singleFunctors C).shiftIso (-b) b 0 ⋯).inv.app Y)
      φ₂
⊢ φ₁ = φ₂

Tactic:
simpa using

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X Y : C
a b : ℤ
inst✝ : HasDerivedCategory C
this :
  ∀ (a b : ℤ),
    Small.{w, w'}
      ((shiftFunctor (DerivedCategory C) a).obj ((singleFunctor C 0).obj X) ⟶
        (shiftFunctor (DerivedCategory C) b).obj ((singleFunctor C 0).obj Y))
φ₁ φ₂ : (singleFunctor C a).obj X ⟶ (singleFunctor C b).obj Y
h :
  (fun φ ↦ ((singleFunctors C).shiftIso (-a) a 0 ⋯).hom.app X ≫ φ ≫ ((singleFunctors C).shiftIso (-b) b 0 ⋯).inv.app Y)
      φ₁ =
    (fun φ ↦
        ((singleFunctors C).shiftIso (-a) a 0 ⋯).hom.app X ≫ φ ≫ ((singleFunctors C).shiftIso (-b) b 0 ⋯).inv.app Y)
      φ₂
⊢ φ₁ = φ₂
Module: Mathlib.Algebra.Homology.Refinements

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
K : HomologicalComplex C c
A : C
i : ι
z : A ⟶ K.X i
j : ι
hj : c.prev i = j
⊢ z ≫ K.pOpcycles i = 0 ↔ ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ z = x ≫ K.d j i

Tactic:
by
  subst hj
  apply (K.sc i).comp_pOpcycles_eq_zero_iff_up_to_refinements

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
S₁ S₂ : ShortComplex (CochainComplex C ℤ)
f : S₁ ⟶ S₂
i : ℤ
⊢ (map S₁.f S₂.f f.τ₁ f.τ₂ ⋯ ≫ descShortComplex S₂).f i = (descShortComplex S₁ ≫ f.τ₃).f i

Tactic:
simpa using

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex (CochainComplex C ℤ)
f : S₁ ⟶ S₂
i : ℤ
⊢ f.τ₂.f i ≫ S₂.g.f i = S₁.g.f i ≫ f.τ₃.f i
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
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
⊢ (up ℤ).next n₀ = n₁

Tactic:
simpa using

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
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
⊢ n₀ + 1 = n₁
Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 3
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₁ : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11)
h₃ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
⊢ R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_mono_of_mono'._proof_4 = 0

Tactic:
simpa using

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 3
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₁ : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11)
h₃ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
⊢ R₁.map' 0 1 ⋯ ⋯ ≫ R₁.map' 1 2 ⋯ ⋯ = 0
Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 3
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₂ : Epi (app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4)
h₃ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
⊢ R₂.map' 1 3 mono_of_epi_of_mono_of_mono'._proof_11 mono_of_epi_of_mono_of_mono'._proof_7 = 0

Tactic:
simpa using

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 3
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₂ : Epi (app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4)
h₃ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
⊢ R₂.map' 1 2 ⋯ ⋯ ≫ R₂.map' 2 3 ⋯ ⋯ = 0
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
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
A' : C
π : A' ⟶ A
w✝¹ : Epi π
x' : A' ⟶ (mappingCone S.f).X n₀
w✝ : x' ≫ (mappingCone S.f).d n₀ n₁ = 0
hx' :
  π ≫ x = HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫ HomologicalComplex.homologyπ (mappingCone S.f) n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w : a ≫ S.X₁.d n₁ (n₁ + 1) = 0 ∧ a ≫ S.f.f n₁ + b ≫ S.X₂.d n₀ n₁ = 0
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀
⊢ (up ℤ).Rel n₀ n₁

Tactic:
simpa using

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
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
A' : C
π : A' ⟶ A
w✝¹ : Epi π
x' : A' ⟶ (mappingCone S.f).X n₀
w✝ : x' ≫ (mappingCone S.f).d n₀ n₁ = 0
hx' :
  π ≫ x = HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫ HomologicalComplex.homologyπ (mappingCone S.f) n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w : a ≫ S.X₁.d n₁ (n₁ + 1) = 0 ∧ a ≫ S.f.f n₁ + b ≫ S.X₂.d n₀ n₁ = 0
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀
⊢ n₀ + 1 = n₁
Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₁' : Epi (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_epi_mono'._proof_1)
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_2)
h₁ : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
⊢ R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0

Tactic:
simpa using

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₁' : Epi (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_epi_mono'._proof_1)
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_2)
h₁ : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
⊢ R₁.map' 0 1 ⋯ ⋯ ≫ R₁.map' 1 2 ⋯ ⋯ = 0
Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₀ : Epi (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
⊢ R₂.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0

Tactic:
simpa using

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₀ : Epi (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
⊢ R₂.map' 0 1 ⋯ ⋯ ≫ R₂.map' 1 2 ⋯ ⋯ = 0
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
⊢ (shiftFunctorZero Cᵒᵖ ℤ).inv.app X =
    ((shiftFunctorZero C ℤ).hom.app (Opposite.unop X)).op ≫ (shiftFunctorOpIso C 0 0 ⋯).inv.app X

Tactic:
rw [← cancel_epi ((shiftFunctorZero Cᵒᵖ ℤ).hom.app X), Iso.hom_inv_id_app,
    shiftFunctorZero_op_hom_app, assoc, ← op_comp_assoc, Iso.hom_inv_id_app, op_id,
    id_comp, Iso.hom_inv_id_app]

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝⁵ : AddMonoid A
inst✝⁴ : HasShift C A
inst✝³ : HasShift D A
inst✝² : F.CommShift A
inst✝¹ : G.CommShift A
inst✝ : adj.CommShift A
a : A
X : C
⊢ adj.unit.app ((shiftFunctor C a).obj X) ≫ (Functor.commShiftIso (F ⋙ G) a).hom.app X =
    (shiftFunctor C a).map (adj.unit.app X)

Tactic:
simpa using

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝⁵ : AddMonoid A
inst✝⁴ : HasShift C A
inst✝³ : HasShift D A
inst✝² : F.CommShift A
inst✝¹ : G.CommShift A
inst✝ : adj.CommShift A
a : A
X : C
⊢ adj.unit.app ((shiftFunctor C a).obj X) ≫ (Functor.commShiftIso (F ⋙ G) a).hom.app X =
    (shiftFunctor C a).map (adj.unit.app X)
Module: Mathlib.CategoryTheory.Shift.Adjunction

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝⁵ : AddMonoid A
inst✝⁴ : HasShift C A
inst✝³ : HasShift D A
inst✝² : F.CommShift A
inst✝¹ : G.CommShift A
inst✝ : adj.CommShift A
a : A
Y : D
⊢ (Functor.commShiftIso (G ⋙ F) a).hom.app Y ≫ (shiftFunctor D a).map (adj.counit.app Y) =
    adj.counit.app ((shiftFunctor D a).obj Y)

Tactic:
simpa using

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝⁵ : AddMonoid A
inst✝⁴ : HasShift C A
inst✝³ : HasShift D A
inst✝² : F.CommShift A
inst✝¹ : G.CommShift A
inst✝ : adj.CommShift A
a : A
Y : D
⊢ (Functor.commShiftIso (G ⋙ F) a).hom.app Y ≫ (shiftFunctor D a).map (adj.counit.app Y) =
    adj.counit.app ((shiftFunctor D a).obj Y)
Module: Mathlib.CategoryTheory.Shift.Adjunction

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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝⁴ : AddMonoid A
inst✝³ : HasShift C A
inst✝² : HasShift D A
inst✝¹ : F.CommShift A
inst✝ : G.CommShift A
x✝¹ : NatTrans.CommShift adj.unit A
a : A
x✝ : D
X : C
⊢ (shiftFunctor C a).map (adj.unit.app X) =
    adj.unit.app ((shiftFunctor C a).obj X) ≫
      G.map ((Functor.commShiftIso F a).hom.app X) ≫ (Functor.commShiftIso G a).hom.app (F.obj X)

Tactic:
simpa using

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝⁴ : AddMonoid A
inst✝³ : HasShift C A
inst✝² : HasShift D A
inst✝¹ : F.CommShift A
inst✝ : G.CommShift A
x✝¹ : NatTrans.CommShift adj.unit A
a : A
x✝ : D
X : C
⊢ (shiftFunctor C a).map (adj.unit.app X) =
    adj.unit.app ((shiftFunctor C a).obj X) ≫
      G.map ((Functor.commShiftIso F a).hom.app X) ≫ (Functor.commShiftIso G a).hom.app (F.obj X)
Module: Mathlib.CategoryTheory.Shift.ShiftedHomOpposite

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
X Y : C
a : ℤ
f : ShiftedHom (Opposite.op X) (Opposite.op Y) a
b : ℤ
Z : C
z : ShiftedHom X Z b
c : ℤ
h : b + a = c
⊢ ((opEquiv a).symm f).comp z h = (opEquiv a).symm (Quiver.Hom.op z ≫ f) ≫ (shiftFunctorAdd' C b a c h).inv.app Z

Tactic:
by
  rw [ShiftedHom.opEquiv_symm_apply, ShiftedHom.opEquiv_symm_apply,
    ShiftedHom.comp]
  dsimp
  simp only [assoc, Functor.map_comp]

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝⁵ : AddMonoid A
inst✝⁴ : HasShift C A
inst✝³ : HasShift D A
inst✝² : F.CommShift A
inst✝¹ : G.CommShift A
inst✝ : adj.CommShift A
a : A
X : C
⊢ (shiftFunctor C a).map (adj.unit.app X) =
    adj.unit.app ((shiftFunctor C a).obj X) ≫
      G.map ((Functor.commShiftIso F a).hom.app X) ≫ (Functor.commShiftIso G a).hom.app (F.obj X)

Tactic:
simpa using

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝⁵ : AddMonoid A
inst✝⁴ : HasShift C A
inst✝³ : HasShift D A
inst✝² : F.CommShift A
inst✝¹ : G.CommShift A
inst✝ : adj.CommShift A
a : A
X : C
⊢ (shiftFunctor C a).map (adj.unit.app X) =
    adj.unit.app ((shiftFunctor C a).obj X) ≫
      G.map ((Functor.commShiftIso F a).hom.app X) ≫ (Functor.commShiftIso G a).hom.app (F.obj X)
Module: Mathlib.CategoryTheory.Shift.Adjunction

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
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
a b : A
h : a + b = 0
inst✝ : G.CommShift A
X : C
⊢ b + a = 0

Tactic:
simp [eq_neg_of_add_eq_zero_left h]

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

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
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
a b : A
h : a + b = 0
inst✝ : G.CommShift A
Y : C
⊢ b + a = 0

Tactic:
simp [eq_neg_of_add_eq_zero_left h]

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
B : C
n m a a' a'' : ℤ
ha' : n + a = a'
ha'' : m + a' = a''
⊢ NatIso.ofComponents
      (fun A ↦
        (let __Equiv := Quiver.Hom.opEquiv.trans (ShiftedHom.opEquiv' (m + n) a a'' ⋯).symm;
          { toEquiv := __Equiv, map_add' := ⋯ }).toAddCommGrpIso)
      ⋯ =
    Functor.isoWhiskerRight (shiftFunctorAdd Cᵒᵖ m n) (preadditiveYoneda.obj ((shiftFunctor C a).obj B)) ≪≫
      (shiftFunctor Cᵒᵖ m).associator (shiftFunctor Cᵒᵖ n) (preadditiveYoneda.obj ((shiftFunctor C a).obj B)) ≪≫
        (shiftFunctor Cᵒᵖ m).isoWhiskerLeft
            (NatIso.ofComponents
              (fun A ↦
                (let __Equiv := Quiver.Hom.opEquiv.trans (ShiftedHom.opEquiv' n a a' ha').symm;
                  { toEquiv := __Equiv, map_add' := ⋯ }).toAddCommGrpIso)
              ⋯) ≪≫
          NatIso.ofComponents
            (fun A ↦
              (let __Equiv := Quiver.Hom.opEquiv.trans (ShiftedHom.opEquiv' m a' a'' ha'').symm;
                { toEquiv := __Equiv, map_add' := ⋯ }).toAddCommGrpIso)
            ⋯

Tactic:
by
    ext _ x
    exact ShiftedHom.opEquiv'_add_symm n m a a' a'' ha' ha'' x.op

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
L M N : C
f : M ⟶ N
hf : Mono f
g : L ⟶ M
hx : addEquiv₀.symm g ∈ ((mk₀ f).postcomp L ⋯).ker
⊢ addEquiv₀.symm g = 0

Tactic:
simpa using

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
L M N : C
f : M ⟶ N
hf : Mono f
g : L ⟶ M
hx : addEquiv₀.symm g ∈ ((mk₀ f).postcomp L ⋯).ker
⊢ g ≫ f = 0
Module: Mathlib.LinearAlgebra.SesquilinearForm.Orthogonal

Goal state before:
pp: K : Type u_7
K₁ : Type u_8
V₁ : Type u_10
V₂ : Type u_11
inst✝⁵ : Field K
inst✝⁴ : Field K₁
inst✝³ : AddCommGroup V₁
inst✝² : Module K₁ V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
J₁ J₁' : K₁ →+* K
B : V₁ →ₛₗ[J₁] V₁ →ₛₗ[J₁'] V₂
x : V₁
hx : (B x) x ≠ 0
μ : V₁ → K₁
h : J₁' (μ x) • (B x) x = 0
y : J₁' (μ x) = 0
⊢ μ x = 0

Tactic:
simpa using

Goal state after:
K : Type u_7
K₁ : Type u_8
V₁ : Type u_10
V₂ : Type u_11
inst✝⁵ : Field K
inst✝⁴ : Field K₁
inst✝³ : AddCommGroup V₁
inst✝² : Module K₁ V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
J₁ J₁' : K₁ →+* K
B : V₁ →ₛₗ[J₁] V₁ →ₛₗ[J₁'] V₂
x : V₁
hx : (B x) x ≠ 0
μ : V₁ → K₁
h : J₁' (μ x) • (B x) x = 0
y : J₁' (μ x) = 0
⊢ μ x = 0
Module: Mathlib.LinearAlgebra.SesquilinearForm.Orthogonal

Goal state before:
pp: R : Type u_1
M : Type u_4
M₁ : Type u_5
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup M₁
inst✝ : Module R M₁
B : M →ₗ[R] M →ₗ[R] M₁
hB : B.IsRefl
W : Submodule R M
hW : Disjoint W (Submodule.orthogonalBilin B W)
⊢ (B.domRestrict₁₂ W W).Nondegenerate

Tactic:
rw [(hB.domRestrict W).nondegenerate_iff_separatingLeft]

Goal state after:
R : Type u_1
M : Type u_4
M₁ : Type u_5
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup M₁
inst✝ : Module R M₁
B : M →ₗ[R] M →ₗ[R] M₁
hB : B.IsRefl
W : Submodule R M
hW : Disjoint W (Submodule.orthogonalBilin B W)
⊢ (B.domRestrict₁₂ W W).SeparatingLeft
Module: Mathlib.Algebra.Module.Presentation.Basic

Goal state before:
pp: A : Type u
inst✝² : Ring A
relations : Relations A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
π : (relations.G →₀ A) →ₗ[A] M
hπ : π ∘ₗ relations.map = 0
r : relations.R
⊢ π (relations.relation r) = 0

Tactic:
simpa using

Goal state after:
A : Type u
inst✝² : Ring A
relations : Relations A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
π : (relations.G →₀ A) →ₗ[A] M
hπ : π ∘ₗ relations.map = 0
r : relations.R
⊢ π (relations.relation r) = 0
Module: Mathlib.Algebra.Module.Presentation.Basic

Goal state before:
pp: A : Type u
inst✝² : Ring A
relations : Relations A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
solution : relations.Solution M
h : solution.π.ker = Submodule.span A (Set.range relations.relation)
x : relations.G →₀ A
hx : relations.toQuotient x ∈ solution.fromQuotient.ker
⊢ x ∈ solution.π.ker

Tactic:
simpa using

Goal state after:
A : Type u
inst✝² : Ring A
relations : Relations A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
solution : relations.Solution M
h : solution.π.ker = Submodule.span A (Set.range relations.relation)
x : relations.G →₀ A
hx : relations.toQuotient x ∈ solution.fromQuotient.ker
⊢ solution.π x = 0
Module: Mathlib.Algebra.Module.Presentation.Basic

Goal state before:
pp: A : Type u
inst✝² : Ring A
relations : Relations A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
solution : relations.Solution M
h : solution.IsPresentation
⊢ Function.Surjective ⇑solution.π

Tactic:
simpa using

Goal state after:
A : Type u
inst✝² : Ring A
relations : Relations A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
solution : relations.Solution M
h : solution.IsPresentation
⊢ Function.Surjective ⇑solution.fromQuotient
Module: Mathlib.Algebra.Module.Presentation.Basic

Goal state before:
pp: A : Type u
inst✝² : Ring A
relations : Relations A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
solution : relations.Solution M
h : solution.IsPresentation
⊢ solution.π.ker = Submodule.span A (Set.range relations.relation)

Tactic:
simpa using

Goal state after:
A : Type u
inst✝² : Ring A
relations : Relations A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
solution : relations.Solution M
h : solution.IsPresentation
⊢ Function.Injective ⇑solution.fromQuotient
Module: Mathlib.Algebra.Module.Presentation.Basic

Goal state before:
pp: A : Type u
inst✝⁴ : Ring A
relations : Relations A
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module A M
solution : relations.Solution M
h : solution.IsPresentation
N : Type v'
inst✝¹ : AddCommGroup N
inst✝ : Module A N
f f' : M →ₗ[A] N
h' : solution.postcomp f = solution.postcomp f'
g : relations.G
⊢ (f ∘ₗ solution.fromQuotient) (relations.toQuotient (Finsupp.single g 1)) =
    (f' ∘ₗ solution.fromQuotient) (relations.toQuotient (Finsupp.single g 1))

Tactic:
simpa using

Goal state after:
A : Type u
inst✝⁴ : Ring A
relations : Relations A
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module A M
solution : relations.Solution M
h : solution.IsPresentation
N : Type v'
inst✝¹ : AddCommGroup N
inst✝ : Module A N
f f' : M →ₗ[A] N
h' : solution.postcomp f = solution.postcomp f'
g : relations.G
⊢ f (solution.var g) = f' (solution.var g)
Module: Mathlib.Algebra.Module.Presentation.Basic

Goal state before:
pp: A : Type u
inst✝ : Ring A
relations : Relations A
⊢ Function.Bijective ⇑(ofQuotient relations).fromQuotient

Tactic:
simpa using

Goal state after:
A : Type u
inst✝ : Ring A
relations : Relations A
⊢ Function.Bijective id
Module: Mathlib.LinearAlgebra.Matrix.Nondegenerate

Goal state before:
pp: m : Type u_1
n : Type u_2
R : Type u_3
inst✝² : CommSemiring R
M : Matrix m n R
inst✝¹ : Fintype m
inst✝ : Fintype n
⊢ M.SeparatingRight ↔ ∀ (w : n → R), (∀ (v : m → R), v ⬝ᵥ M *ᵥ w = 0) → w = 0

Tactic:
refine forall_congr' fun w ↦ ⟨fun hM hw ↦ hM ?_, fun hM hw ↦ hM ?_⟩ <;>
  convert! hw

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

Goal state before:
pp: case refine_2
m : Type u_1
n : Type u_2
R : Type u_3
inst✝² : CommSemiring R
M : Matrix m n R
inst✝¹ : Finite m
inst✝ : Fintype n
this : Fintype m
h : ∀ (v : n → R), M *ᵥ v = 0 → v = 0
w : n → R
hw : ∀ (v : m → R), v ⬝ᵥ M *ᵥ w = 0
i : m
⊢ (M *ᵥ w) i = 0 i

Tactic:
simpa using

Goal state after:
case refine_2
m : Type u_1
n : Type u_2
R : Type u_3
inst✝² : CommSemiring R
M : Matrix m n R
inst✝¹ : Finite m
inst✝ : Fintype n
this : Fintype m
h : ∀ (v : n → R), M *ᵥ v = 0 → v = 0
w : n → R
hw : ∀ (v : m → R), v ⬝ᵥ M *ᵥ w = 0
i : m
⊢ (M *ᵥ w) i = 0