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

Goal state before:
pp: C₁ : Type u₁
inst✝⁷ : Category.{v₁, u₁} C₁
W₁ : MorphismProperty C₁
C₂ : Type u₂
inst✝⁶ : Category.{v₂, u₂} C₂
W₂ : MorphismProperty C₂
D₁ : Type u₃
inst✝⁵ : Category.{v₃, u₃} D₁
D₂ : Type u₄
inst✝⁴ : Category.{v₄, u₄} D₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝² : L₂.IsLocalization W₂
X Y : C₁
inst✝¹ : HasSmallLocalizedHom W₁ X Y
inst✝ : HasSmallLocalizedHom W₂ (Φ.functor.obj X) (Φ.functor.obj Y)
G : D₁ ⥤ D₂
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G
g : W₁.Q.obj X ⟶ W₁.Q.obj Y
G' : W₁.Localization ⥤ W₂.Localization := Φ.localizedFunctor W₁.Q W₂.Q
β : Φ.functor ⋙ W₂.Q ≅ W₁.Q ⋙ G' := CatCommSq.iso Φ.functor W₁.Q W₂.Q G'
⊢ (SmallHom.equiv W₂ L₂)
      ((SmallHom.equiv W₂ W₂.Q).symm
        ((((CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).symm.app X).homCongr
            ((CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).symm.app Y))
          ((Φ.localizedFunctor W₁.Q W₂.Q).map g))) =
    e.hom.app X ≫ G.map ((SmallHom.equiv W₁ L₁) ((SmallHom.equiv W₁ W₁.Q).symm g)) ≫ e.inv.app Y

Tactic:
let E₁ := (uniq W₁.Q L₁ W₁).functor

Goal state after:
C₁ : Type u₁
inst✝⁷ : Category.{v₁, u₁} C₁
W₁ : MorphismProperty C₁
C₂ : Type u₂
inst✝⁶ : Category.{v₂, u₂} C₂
W₂ : MorphismProperty C₂
D₁ : Type u₃
inst✝⁵ : Category.{v₃, u₃} D₁
D₂ : Type u₄
inst✝⁴ : Category.{v₄, u₄} D₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝² : L₂.IsLocalization W₂
X Y : C₁
inst✝¹ : HasSmallLocalizedHom W₁ X Y
inst✝ : HasSmallLocalizedHom W₂ (Φ.functor.obj X) (Φ.functor.obj Y)
G : D₁ ⥤ D₂
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G
g : W₁.Q.obj X ⟶ W₁.Q.obj Y
G' : W₁.Localization ⥤ W₂.Localization := Φ.localizedFunctor W₁.Q W₂.Q
β : Φ.functor ⋙ W₂.Q ≅ W₁.Q ⋙ G' := CatCommSq.iso Φ.functor W₁.Q W₂.Q G'
E₁ : W₁.Localization ⥤ D₁ := (uniq W₁.Q L₁ W₁).functor
⊢ (SmallHom.equiv W₂ L₂)
      ((SmallHom.equiv W₂ W₂.Q).symm
        ((((CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).symm.app X).homCongr
            ((CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).symm.app Y))
          ((Φ.localizedFunctor W₁.Q W₂.Q).map g))) =
    e.hom.app X ≫ G.map ((SmallHom.equiv W₁ L₁) ((SmallHom.equiv W₁ W₁.Q).symm g)) ≫ e.inv.app Y
Module: Mathlib.CategoryTheory.Localization.SmallHom

Goal state before:
pp: C₁ : Type u₁
inst✝³ : Category.{v₁, u₁} C₁
W₁ : MorphismProperty C₁
C₂ : Type u₂
inst✝² : Category.{v₂, u₂} C₂
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
X Y : C₁
inst✝¹ : HasSmallLocalizedHom W₁ X Y
inst✝ : HasSmallLocalizedHom W₂ (Φ.functor.obj X) (Φ.functor.obj Y)
f : X ⟶ Y
⊢ Φ.smallHomMap (SmallHom.mk W₁ f) = SmallHom.mk W₂ (Φ.functor.map f)

Tactic:
apply (SmallHom.equiv W₂ W₂.Q).injective
  simp [Φ.equiv_smallHomMap W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q) (CatCommSq.iso _ _ _ _)]

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

Goal state before:
pp: C : Type u₁
inst✝¹¹ : Category.{v₁, u₁} C
W : MorphismProperty C
M : Type w'
inst✝¹⁰ : AddMonoid M
inst✝⁹ : HasShift C M
inst✝⁸ : W.IsCompatibleWithShift M
X Y Z T : C
a₁ a₂ a₃ a₁₂ a₂₃ a : M
inst✝⁷ : HasSmallLocalizedShiftedHom W M X Y
inst✝⁶ : HasSmallLocalizedShiftedHom W M X Z
inst✝⁵ : HasSmallLocalizedShiftedHom W M X T
inst✝⁴ : HasSmallLocalizedShiftedHom W M Y Z
inst✝³ : HasSmallLocalizedShiftedHom W M Y T
inst✝² : HasSmallLocalizedShiftedHom W M Z T
inst✝¹ : HasSmallLocalizedShiftedHom W M Z Z
inst✝ : HasSmallLocalizedShiftedHom W M T T
α : SmallShiftedHom W X Y a₁
β : SmallShiftedHom W Y Z a₂
γ : SmallShiftedHom W Z T a₃
h₁₂ : a₂ + a₁ = a₁₂
h₂₃ : a₃ + a₂ = a₂₃
h : a₃ + a₂ + a₁ = a
⊢ (α.comp β h₁₂).comp γ ⋯ = α.comp (β.comp γ h₂₃) ⋯

Tactic:
apply (equiv W W.Q).injective

Goal state after:
C : Type u₁
inst✝¹¹ : Category.{v₁, u₁} C
W : MorphismProperty C
M : Type w'
inst✝¹⁰ : AddMonoid M
inst✝⁹ : HasShift C M
inst✝⁸ : W.IsCompatibleWithShift M
X Y Z T : C
a₁ a₂ a₃ a₁₂ a₂₃ a : M
inst✝⁷ : HasSmallLocalizedShiftedHom W M X Y
inst✝⁶ : HasSmallLocalizedShiftedHom W M X Z
inst✝⁵ : HasSmallLocalizedShiftedHom W M X T
inst✝⁴ : HasSmallLocalizedShiftedHom W M Y Z
inst✝³ : HasSmallLocalizedShiftedHom W M Y T
inst✝² : HasSmallLocalizedShiftedHom W M Z T
inst✝¹ : HasSmallLocalizedShiftedHom W M Z Z
inst✝ : HasSmallLocalizedShiftedHom W M T T
α : SmallShiftedHom W X Y a₁
β : SmallShiftedHom W Y Z a₂
γ : SmallShiftedHom W Z T a₃
h₁₂ : a₂ + a₁ = a₁₂
h₂₃ : a₃ + a₂ = a₂₃
h : a₃ + a₂ + a₁ = a
⊢ (equiv W W.Q) ((α.comp β h₁₂).comp γ ⋯) = (equiv W W.Q) (α.comp (β.comp γ h₂₃) ⋯)
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
S S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
i : ι
⊢ (composableArrows₂ S₁ i).map' 0 1 homMk₁._proof_4 _proof_246✝ ≫ homologyMap φ.τ₂ i =
    homologyMap φ.τ₁ i ≫ (composableArrows₂ S₂ i).map' 0 1 homMk₁._proof_4 _proof_246✝

Tactic:
by
    dsimp
    simp only [← homologyMap_comp, φ.comm₁₂]

Goal state after:
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.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0
hR₁' : Epi (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_epi_mono'._proof_1)
hR₂ : R₂.Exact
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)
ψ : mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 ⟶ mk₃ (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) 0 :=
  homMk₃ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) (𝟙 ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).obj' 3 _proof_300✝)) ⋯ ⋯
    ⋯
⊢ Mono (app' ψ 3 mono_of_epi_of_mono_of_mono'._proof_7)

Tactic:
dsimp [ψ]

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.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0
hR₁' : Epi (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_epi_mono'._proof_1)
hR₂ : R₂.Exact
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)
ψ : mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 ⟶ mk₃ (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) 0 :=
  homMk₃ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) (𝟙 ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).obj' 3 _proof_300✝)) ⋯ ⋯
    ⋯
⊢ Mono (𝟙 (Precomp.obj (mk₂ (R₁.map (homOfLE ⋯)) 0) (R₁.obj 0) 3))
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.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0
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)
ψ : mk₃ 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) ⟶ mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) :=
  homMk₃ (𝟙 ((mk₃ 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).obj' 0 _proof_297✝)) (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯
    ⋯
⊢ Epi (app' ψ 0 mono_of_epi_of_mono_of_mono'._proof_13)

Tactic:
dsimp [ψ]

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.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0
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)
ψ : mk₃ 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) ⟶ mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) :=
  homMk₃ (𝟙 ((mk₃ 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).obj' 0 _proof_297✝)) (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯
    ⋯
⊢ Epi (𝟙 (R₁.obj 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' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₀ : Mono (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_2)
h₁ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₃ 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) ⟶ mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) :=
  homMk₃ (𝟙 ((mk₃ 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).obj' 0 _proof_297✝)) (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯
    ⋯
⊢ Epi (app' ψ 0 mono_of_epi_of_mono_of_mono'._proof_13)

Tactic:
dsimp [ψ]

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' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₀ : Mono (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_2)
h₁ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₃ 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) ⟶ mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) :=
  homMk₃ (𝟙 ((mk₃ 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).obj' 0 _proof_297✝)) (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯
    ⋯
⊢ Epi (𝟙 ((mk₃ 0 (R₁.map (homOfLE ⋯)) (R₁.map (homOfLE ⋯))).obj 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
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 : Epi (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 ⟶ mk₃ (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) 0 :=
  homMk₃ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) (𝟙 ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).obj' 3 _proof_300✝)) ⋯ ⋯
    ⋯
⊢ Mono (app' ψ 3 mono_of_epi_of_mono_of_mono'._proof_7)

Tactic:
dsimp [ψ]

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
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 : Epi (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 ⟶ mk₃ (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) 0 :=
  homMk₃ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) (𝟙 ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).obj' 3 _proof_300✝)) ⋯ ⋯
    ⋯
⊢ Mono (𝟙 ((mk₃ (R₁.map (homOfLE ⋯)) (R₁.map (homOfLE ⋯)) 0).obj 3))
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughProjectives

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
P : C
inst✝² : Projective P
K : CochainComplex C ℤ
i : ℤ
π : K ⟶ (singleFunctor C i).obj P
inst✝¹ : K.IsStrictlyLE i
inst✝ : QuasiIsoAt π i
e : cycles K i ≅ K.X i := iCyclesIso K i (i + 1) ⋯ ⋯
α : K.X i ⟶ P := e.inv ≫ homologyπ K i ≫ homologyMap π i ≫ (singleObjHomologySelfIso (ComplexShape.up ℤ) i P).hom
⊢ iCycles K i ≫ π.f i =
    cyclesMap π i ≫ (singleObjCyclesSelfIso (ComplexShape.up ℤ) i P).hom ≫ (singleObjXSelf (ComplexShape.up ℤ) i P).inv

Tactic:
← singleObjCyclesSelfIso_inv_iCycles,

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
P : C
inst✝² : Projective P
K : CochainComplex C ℤ
i : ℤ
π : K ⟶ (singleFunctor C i).obj P
inst✝¹ : K.IsStrictlyLE i
inst✝ : QuasiIsoAt π i
e : cycles K i ≅ K.X i := iCyclesIso K i (i + 1) ⋯ ⋯
α : K.X i ⟶ P := e.inv ≫ homologyπ K i ≫ homologyMap π i ≫ (singleObjHomologySelfIso (ComplexShape.up ℤ) i P).hom
⊢ iCycles K i ≫ π.f i =
    cyclesMap π i ≫
      (singleObjCyclesSelfIso (ComplexShape.up ℤ) i P).hom ≫
        (singleObjCyclesSelfIso (ComplexShape.up ℤ) i P).inv ≫ ((single C (ComplexShape.up ℤ) i).obj P).iCycles i
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughProjectives

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
inst✝² : HasDerivedCategory C
P : C
inst✝¹ : Projective P
L : CochainComplex C ℤ
i n : ℤ
hn : n < i
inst✝ : L.IsStrictlyLE n
K : CochainComplex C ℤ
w✝ : K.IsStrictlyLE i
π : K ⟶ (CochainComplex.singleFunctor C i).obj P
h : IsIso (Q.map π)
g : K ⟶ L
hπ : IsSplitEpi π
h₁ : inv (Q.map π) = Q.map (section_ π)
h₂ : section_ π ≫ g = 0
⊢ Q.map 0 = 0

Tactic:
Q.map_zero

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
inst✝² : HasDerivedCategory C
P : C
inst✝¹ : Projective P
L : CochainComplex C ℤ
i n : ℤ
hn : n < i
inst✝ : L.IsStrictlyLE n
K : CochainComplex C ℤ
w✝ : K.IsStrictlyLE i
π : K ⟶ (CochainComplex.singleFunctor C i).obj P
h : IsIso (Q.map π)
g : K ⟶ L
hπ : IsSplitEpi π
h₁ : inv (Q.map π) = Q.map (section_ π)
h₂ : section_ π ≫ g = 0
⊢ 0 = 0
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
inst✝² : HasDerivedCategory C
I : C
inst✝¹ : Injective I
K : CochainComplex C ℤ
i n : ℤ
hn : i < n
inst✝ : K.IsStrictlyGE n
L : CochainComplex C ℤ
w✝ : L.IsStrictlyGE i
g : K ⟶ L
ι : (CochainComplex.singleFunctor C i).obj I ⟶ L
h : IsIso (Q.map ι)
hπ : IsSplitMono ι
h₁ : inv (Q.map ι) = Q.map (retraction ι)
h₂ : g ≫ retraction ι = 0
⊢ Q.map 0 = 0

Tactic:
Q.map_zero

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
inst✝² : HasDerivedCategory C
I : C
inst✝¹ : Injective I
K : CochainComplex C ℤ
i n : ℤ
hn : i < n
inst✝ : K.IsStrictlyGE n
L : CochainComplex C ℤ
w✝ : L.IsStrictlyGE i
g : K ⟶ L
ι : (CochainComplex.singleFunctor C i).obj I ⟶ L
h : IsIso (Q.map ι)
hπ : IsSplitMono ι
h₁ : inv (Q.map ι) = Q.map (retraction ι)
h₂ : g ≫ retraction ι = 0
⊢ 0 = 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
⊢ solution.π.ker = Submodule.span A (Set.range relations.relation)

Tactic:
← solution.injective_fromQuotient_iff_ker_π_eq_span

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.RingTheory.IntegralDomain

Goal state before:
pp: case neg
R : Type u_1
G : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : Group G
inst✝¹ : Fintype G
f : G →* R
inst✝ : Decidable (f = 1)
h : ¬f = 1
⊢ ∑ g, f g = ↑0

Tactic:
rw [Nat.cast_zero]
    exact sum_hom_units_eq_zero f h

Goal state after:
Module: Mathlib.RingTheory.IntegralDomain

Goal state before:
pp: R : Type u_1
G : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : Group G
inst✝¹ : Fintype G
f : G →* R
inst✝ : Decidable (f = 1)
⊢ ∑ g, f g = ↑(if f = 1 then Fintype.card G else 0)

Tactic:
by
  split_ifs with h
  · simp [h]
  · rw [Nat.cast_zero]
    exact sum_hom_units_eq_zero f h

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs

Goal state before:
pp: n : Type u
inst✝⁸ : DecidableEq n
inst✝⁷ : Fintype n
R : Type v
inst✝⁶ : CommRing R
S : Type u_1
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u_2
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
g : SpecialLinearGroup n R
⊢ (GeneralLinearGroup.map (algebraMap S T)) ((mapGL S) g) = (mapGL T) g

Tactic:
by
  ext
  simp [IsScalarTower.algebraMap_apply R S T]

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

Goal state before:
pp: case inr.inr.refine_2
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
S : Type u_2
inst✝² : CommRing S
inst✝¹ : LinearOrder S
inst✝ : IsStrictOrderedRing S
A : Matrix n n S
h1 : Pairwise fun i j ↦ A i j < 0
h✝ : Nonempty n
h2 : A.det = 0
v : n → S
h_vnz : v ≠ 0
h_vA : v ᵥ* A = 0
this :
  ∀ {n : Type u_1} [inst : Fintype n] [inst_1 : DecidableEq n] {S : Type u_2} [inst_2 : CommRing S]
    [inst_3 : LinearOrder S] [IsStrictOrderedRing S] {A : Matrix n n S},
    (Pairwise fun i j ↦ A i j < 0) →
      ∀ (h : Nonempty n), A.det = 0 → ∀ (v : n → S), v ≠ 0 → v ᵥ* A = 0 → 0 < Finset.univ.sup' ⋯ v → ∃ j, ∑ i, A i j ≤ 0
h_sup : ¬0 < Finset.univ.sup' ⋯ v
⊢ 0 < Finset.univ.sup' ⋯ (-1 • v)

Tactic:
obtain ⟨i, hi⟩ := Function.ne_iff.mp h_vnz
        simp_rw [Finset.lt_sup'_iff, Finset.mem_univ, true_and] at h_sup ⊢
        simp_rw [not_exists, not_lt] at h_sup
        refine ⟨i, ?_⟩
        rw [Pi.smul_apply, neg_smul, one_smul, Left.neg_pos_iff]
        exact Ne.lt_of_le hi (h_sup i)

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

Goal state before:
pp: n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
S : Type u_2
inst✝² : CommRing S
inst✝¹ : LinearOrder S
inst✝ : IsStrictOrderedRing S
A : Matrix n n S
h1 : Pairwise fun i j ↦ A i j < 0
h2 : ∀ (i : n), 0 < ∑ j, A i j
⊢ A.det ≠ 0

Tactic:
by
  rw [← Matrix.det_transpose]
  refine det_ne_zero_of_sum_col_pos ?_ ?_
  · simp_rw [Matrix.transpose_apply]
    exact fun i j h => h1 h.symm
  · simp_rw [Matrix.transpose_apply]
    exact h2

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

Goal state before:
pp: n : Type u
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type v
inst✝ : CommRing R
A : SpecialLinearGroup n R
⊢ A ∈ center (SpecialLinearGroup n R) ↔ ∃ r, r ^ Fintype.card n = 1 ∧ (scalar n) r = ↑A

Tactic:
rcases isEmpty_or_nonempty n with hn | ⟨⟨i⟩⟩; · exact ⟨by aesop, by simp [Subsingleton.elim A 1]⟩
  refine ⟨fun h ↦ ⟨A i i, ?_, ?_⟩, fun ⟨r, _, hr⟩ ↦ Subgroup.mem_center_iff.mpr fun B ↦ ?_⟩
  · have : det ((scalar n) (A i i)) = 1 := (scalar_eq_self_of_mem_center h i).symm ▸ A.property
    simpa using! this
  · exact scalar_eq_self_of_mem_center h i
  · suffices ↑ₘ(B * A) = ↑ₘ(A * B) from Subtype.val_injective this
    simpa only [coe_mul, ← hr] using! (scalar_commute (n := n) r (Commute.all r) B).symm

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

Goal state before:
pp: case neg
n : Type u
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type v
inst✝ : CommRing R
A : ↥(center (SpecialLinearGroup n R))
h : ¬IsEmpty n
pf✝ : Nonempty n
⊢ ↑↑A = (scalar n) ↑↑((⋯ ▸ center_equiv_rootsOfUnity' (Classical.arbitrary n)) A)

Tactic:
center_equiv_rootsOfUnity'_apply,

Goal state after:
case neg
n : Type u
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type v
inst✝ : CommRing R
A : ↥(center (SpecialLinearGroup n R))
h : ¬IsEmpty n
pf✝ : Nonempty n
⊢ ↑↑A = (scalar n) ↑↑(rootsOfUnity.mkOfPowEq (↑↑A (Classical.arbitrary n) (Classical.arbitrary n)) ⋯)
Module: Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup

Goal state before:
pp: R : Type u_2
inst✝ : Field R
g : SL(2, R)
hg : ↑g 1 0 = 0
⊢ ∃ a b, ∃ (h : a ≠ 0), g = ⟨!![a, b; 0, a⁻¹], ⋯⟩

Tactic:
induction g using Matrix.SpecialLinearGroup.fin_two_induction with

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
v : Fin 2 → R
hab : IsCoprime (v 0) (v 1)
A : SL(2, R)
⊢ IsCoprime ((v ᵥ* ↑A) 0) ((v ᵥ* ↑A) 1)

Tactic:
obtain ⟨g, hg⟩ := hab.exists_SL2_row 0
  have : v = g 0 := funext fun t ↦ by { fin_cases t <;> tauto }
  simpa only [this] using! isCoprime_row (g * A) 0

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

Goal state before:
pp: F : Type u_1
inst✝ : Field F
a : F
ha : a ≠ 0
k : Fin 2
⊢ (diag2 a ha • Pi.single 0 1) k = (a • Pi.single 0 1) k

Tactic:
fin_cases k

Goal state after:
case «0»
F : Type u_1
inst✝ : Field F
a : F
ha : a ≠ 0
⊢ (diag2 a ha • Pi.single 0 1) ((fun i ↦ i) ⟨0, ⋯⟩) = (a • Pi.single 0 1) ((fun i ↦ i) ⟨0, ⋯⟩)

case «1»
F : Type u_1
inst✝ : Field F
a : F
ha : a ≠ 0
⊢ (diag2 a ha • Pi.single 0 1) ((fun i ↦ i) ⟨1, ⋯⟩) = (a • Pi.single 0 1) ((fun i ↦ i) ⟨1, ⋯⟩)
Module: Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup

Goal state before:
pp: F : Type u_1
inst✝ : Field F
a : F
ha : a ≠ 0
k : Fin 2
⊢ (diag2 a ha • Pi.single 1 1) k = (a⁻¹ • Pi.single 1 1) k

Tactic:
fin_cases k

Goal state after:
case «0»
F : Type u_1
inst✝ : Field F
a : F
ha : a ≠ 0
⊢ (diag2 a ha • Pi.single 1 1) ((fun i ↦ i) ⟨0, ⋯⟩) = (a⁻¹ • Pi.single 1 1) ((fun i ↦ i) ⟨0, ⋯⟩)

case «1»
F : Type u_1
inst✝ : Field F
a : F
ha : a ≠ 0
⊢ (diag2 a ha • Pi.single 1 1) ((fun i ↦ i) ⟨1, ⋯⟩) = (a⁻¹ • Pi.single 1 1) ((fun i ↦ i) ⟨1, ⋯⟩)
Module: Mathlib.LinearAlgebra.Matrix.SesquilinearForm

Goal state before:
pp: R : Type u_1
R₁ : Type u_2
S₁ : Type u_3
R₂ : Type u_4
S₂ : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₁' : Type u_8
M₂' : Type u_9
N₂ : Type u_10
n : Type u_11
m : Type u_12
n' : Type u_13
m' : Type u_14
ι : Type u_15
inst✝⁹ : Semiring R₁
inst✝⁸ : Semiring S₁
inst✝⁷ : Semiring R₂
inst✝⁶ : Semiring S₂
inst✝⁵ : AddCommMonoid N₂
inst✝⁴ : Module S₁ N₂
inst✝³ : Module S₂ N₂
inst✝² : SMulCommClass S₂ S₁ N₂
inst✝¹ : Fintype n
inst✝ : Fintype m
σ₁ : R₁ →+* S₁
σ₂ : R₂ →+* S₂
f : Matrix n m N₂
x✝² : n → R₁
x✝¹ x✝ : m → R₂
⊢ ∑ i, ∑ j, σ₂ ((x✝¹ + x✝) j) •> σ₁ (x✝² i) •> f i j =
    ∑ i, ∑ j, σ₂ (x✝¹ j) •> σ₁ (x✝² i) •> f i j + ∑ i, ∑ j, σ₂ (x✝ j) •> σ₁ (x✝² i) •> f i j

Tactic:
by simp only [Pi.add_apply, map_add, add_smul, sum_add_distrib]

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

Goal state before:
pp: R : Type u_1
R₁ : Type u_2
S₁ : Type u_3
R₂ : Type u_4
S₂ : Type u_5
N₂ : Type u_10
n : Type u_11
m : Type u_12
inst✝¹⁵ : CommSemiring R
inst✝¹⁴ : Semiring R₁
inst✝¹³ : Semiring S₁
inst✝¹² : Semiring R₂
inst✝¹¹ : Semiring S₂
inst✝¹⁰ : AddCommMonoid N₂
inst✝⁹ : Module R N₂
inst✝⁸ : Module S₁ N₂
inst✝⁷ : Module S₂ N₂
inst✝⁶ : SMulCommClass S₁ R N₂
inst✝⁵ : SMulCommClass S₂ R N₂
inst✝⁴ : SMulCommClass S₂ S₁ N₂
σ₁ : R₁ →+* S₁
σ₂ : R₂ →+* S₂
inst✝³ : Fintype n
inst✝² : Fintype m
inst✝¹ : DecidableEq n
inst✝ : DecidableEq m
f : (n → R₁) →ₛₗ[σ₁] (m → R₂) →ₛₗ[σ₂] N₂
⊢ toLinearMap₂'Aux σ₁ σ₂ ((toMatrix₂Aux R (fun i ↦ Pi.single i 1) fun j ↦ Pi.single j 1) f) = f

Tactic:
refine ext_basis (Pi.basisFun R₁ n) (Pi.basisFun R₂ m) fun i j => ?_

Goal state after:
R : Type u_1
R₁ : Type u_2
S₁ : Type u_3
R₂ : Type u_4
S₂ : Type u_5
N₂ : Type u_10
n : Type u_11
m : Type u_12
inst✝¹⁵ : CommSemiring R
inst✝¹⁴ : Semiring R₁
inst✝¹³ : Semiring S₁
inst✝¹² : Semiring R₂
inst✝¹¹ : Semiring S₂
inst✝¹⁰ : AddCommMonoid N₂
inst✝⁹ : Module R N₂
inst✝⁸ : Module S₁ N₂
inst✝⁷ : Module S₂ N₂
inst✝⁶ : SMulCommClass S₁ R N₂
inst✝⁵ : SMulCommClass S₂ R N₂
inst✝⁴ : SMulCommClass S₂ S₁ N₂
σ₁ : R₁ →+* S₁
σ₂ : R₂ →+* S₂
inst✝³ : Fintype n
inst✝² : Fintype m
inst✝¹ : DecidableEq n
inst✝ : DecidableEq m
f : (n → R₁) →ₛₗ[σ₁] (m → R₂) →ₛₗ[σ₂] N₂
i : n
j : m
⊢ ((toLinearMap₂'Aux σ₁ σ₂ ((toMatrix₂Aux R (fun i ↦ Pi.single i 1) fun j ↦ Pi.single j 1) f)) ((Pi.basisFun R₁ n) i))
      ((Pi.basisFun R₂ m) j) =
    (f ((Pi.basisFun R₁ n) i)) ((Pi.basisFun R₂ m) j)
Module: Mathlib.Algebra.DirectSum.Decomposition

Goal state before:
pp: ι : Type u_1
M : Type u_3
σ : Type u_4
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid M
inst✝² : SetLike σ M
inst✝¹ : AddSubmonoidClass σ M
ℳ : ι → σ
inst✝ : Decomposition ℳ
motive : M → Prop
zero : motive 0
homogeneous : ∀ {i : ι} (m : ↥(ℳ i)), motive ↑m
add : ∀ (m m' : M), motive m → motive m' → motive (m + m')
ℳ' : ι → AddSubmonoid M := fun i ↦ { carrier := ↑(ℳ i), add_mem' := ⋯, zero_mem' := ⋯ }
t : Decomposition ℳ'
⊢ ∀ (m : M), motive m

Tactic:
have mem : ∀ m, m ∈ iSup ℳ' := fun _m ↦
    (DirectSum.IsInternal.addSubmonoid_iSup_eq_top ℳ' (Decomposition.isInternal ℳ')).symm ▸ trivial

Goal state after:
ι : Type u_1
M : Type u_3
σ : Type u_4
inst✝⁴ : DecidableEq ι
inst✝³ : AddCommMonoid M
inst✝² : SetLike σ M
inst✝¹ : AddSubmonoidClass σ M
ℳ : ι → σ
inst✝ : Decomposition ℳ
motive : M → Prop
zero : motive 0
homogeneous : ∀ {i : ι} (m : ↥(ℳ i)), motive ↑m
add : ∀ (m m' : M), motive m → motive m' → motive (m + m')
ℳ' : ι → AddSubmonoid M := fun i ↦ { carrier := ↑(ℳ i), add_mem' := ⋯, zero_mem' := ⋯ }
t : Decomposition ℳ'
mem : ∀ (m : M), m ∈ iSup ℳ'
⊢ ∀ (m : M), motive m