mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Abelian.Projective.Extend

Goal state before:
pp: case pos
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : Preadditive C
X : C
R : ProjectiveResolution X
n : ℤ
hn : n ≤ 0
⊢ Projective (R.cochainComplex.X n)

pp: case neg
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : Preadditive C
X : C
R : ProjectiveResolution X
n : ℤ
hn : ¬n ≤ 0
⊢ Projective (R.cochainComplex.X n)

Tactic:
·

Goal state after:
case pos
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : Preadditive C
X : C
R : ProjectiveResolution X
n : ℤ
hn : n ≤ 0
⊢ Projective (R.cochainComplex.X n)
Module: Mathlib.CategoryTheory.Abelian.Projective.Extend

Goal state before:
pp: case neg
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : Preadditive C
X : C
R : ProjectiveResolution X
n : ℤ
hn : ¬n ≤ 0
⊢ Projective (R.cochainComplex.X n)

Tactic:
·

Goal state after:
case neg
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : Preadditive C
X : C
R : ProjectiveResolution X
n : ℤ
hn : ¬n ≤ 0
⊢ Projective (R.cochainComplex.X n)
Module: Mathlib.CategoryTheory.Abelian.Preradical.Colon

Goal state before:
pp: case mp
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
Φ Ψ : Preradical C
X : C
h : IsIso ((Over.Hom.left (Φ.toColon Ψ).hom).app X)
⊢ IsZero (Ψ.r.obj (Φ.quotient.obj X))

pp: case mpr
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
Φ Ψ : Preradical C
X : C
h : IsZero (Ψ.r.obj (Φ.quotient.obj X))
⊢ IsIso ((Over.Hom.left (Φ.toColon Ψ).hom).app X)

Tactic:
·

Goal state after:
case mp
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
Φ Ψ : Preradical C
X : C
h : IsIso ((Over.Hom.left (Φ.toColon Ψ).hom).app X)
⊢ IsZero (Ψ.r.obj (Φ.quotient.obj X))
Module: Mathlib.CategoryTheory.Abelian.Preradical.Colon

Goal state before:
pp: case mpr
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
Φ Ψ : Preradical C
X : C
h : IsZero (Ψ.r.obj (Φ.quotient.obj X))
⊢ IsIso ((Over.Hom.left (Φ.toColon Ψ).hom).app X)

Tactic:
·

Goal state after:
case mpr
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
Φ Ψ : Preradical C
X : C
h : IsZero (Ψ.r.obj (Φ.quotient.obj X))
⊢ IsIso ((Over.Hom.left (Φ.toColon Ψ).hom).app X)
Module: Mathlib.CategoryTheory.Abelian.Preradical.Colon

Goal state before:
pp: case mpr.refine_1
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
Φ Ψ : Preradical C
X : C
h : IsZero (Ψ.r.obj (Φ.quotient.obj X))
inv : (Φ.colon Ψ).r.obj X ⟶ (KernelFork.ofι (Φ.ι.app X) ⋯).pt
hinv : inv ≫ Φ.ι.app X = (Φ.colon Ψ).ι.app X
⊢ (Over.Hom.left (Φ.toColon Ψ).hom).app X ≫ inv = 𝟙 (Φ.obj.left.obj X)

pp: case mpr.refine_2
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
Φ Ψ : Preradical C
X : C
h : IsZero (Ψ.r.obj (Φ.quotient.obj X))
inv : (Φ.colon Ψ).r.obj X ⟶ (KernelFork.ofι (Φ.ι.app X) ⋯).pt
hinv : inv ≫ Φ.ι.app X = (Φ.colon Ψ).ι.app X
⊢ inv ≫ (Over.Hom.left (Φ.toColon Ψ).hom).app X = 𝟙 ((Φ.colon Ψ).obj.left.obj X)

Tactic:
·

Goal state after:
case mpr.refine_1
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
Φ Ψ : Preradical C
X : C
h : IsZero (Ψ.r.obj (Φ.quotient.obj X))
inv : (Φ.colon Ψ).r.obj X ⟶ (KernelFork.ofι (Φ.ι.app X) ⋯).pt
hinv : inv ≫ Φ.ι.app X = (Φ.colon Ψ).ι.app X
⊢ (Over.Hom.left (Φ.toColon Ψ).hom).app X ≫ inv = 𝟙 (Φ.obj.left.obj X)
Module: Mathlib.CategoryTheory.Abelian.Preradical.Colon

Goal state before:
pp: case mpr.refine_2
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
Φ Ψ : Preradical C
X : C
h : IsZero (Ψ.r.obj (Φ.quotient.obj X))
inv : (Φ.colon Ψ).r.obj X ⟶ (KernelFork.ofι (Φ.ι.app X) ⋯).pt
hinv : inv ≫ Φ.ι.app X = (Φ.colon Ψ).ι.app X
⊢ inv ≫ (Over.Hom.left (Φ.toColon Ψ).hom).app X = 𝟙 ((Φ.colon Ψ).obj.left.obj X)

Tactic:
·

Goal state after:
case mpr.refine_2
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
Φ Ψ : Preradical C
X : C
h : IsZero (Ψ.r.obj (Φ.quotient.obj X))
inv : (Φ.colon Ψ).r.obj X ⟶ (KernelFork.ofι (Φ.ι.app X) ⋯).pt
hinv : inv ≫ Φ.ι.app X = (Φ.colon Ψ).ι.app X
⊢ inv ≫ (Over.Hom.left (Φ.toColon Ψ).hom).app X = 𝟙 ((Φ.colon Ψ).obj.left.obj X)
Module: Mathlib.CategoryTheory.Abelian.Injective.Ext

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y : C
R : InjectiveResolution Y
n✝ n : ℕ
f : X ⟶ R.cocomplex.X n
m : ℕ
hm : n + 1 = m
hf : f ≫ R.cocomplex.d n m = 0
⊢ ↑n + 1 = ↑m

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Injective.Resolution

Goal state before:
pp: case zero
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
⊢ QuasiIsoAt (((ofCocomplex Z).fromSingle₀Equiv Z).symm ⟨Injective.ι Z, ⋯⟩) 0

pp: case succ
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
n✝ : ℕ
⊢ QuasiIsoAt (((ofCocomplex Z).fromSingle₀Equiv Z).symm ⟨Injective.ι Z, ⋯⟩) (n✝ + 1)

Tactic:
·

Goal state after:
case zero
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
⊢ QuasiIsoAt (((ofCocomplex Z).fromSingle₀Equiv Z).symm ⟨Injective.ι Z, ⋯⟩) 0
Module: Mathlib.CategoryTheory.Abelian.Injective.Ext

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y : C
R : InjectiveResolution Y
n : ℕ
f : X ⟶ R.cocomplex.X n
m : ℕ
hm : n + 1 = m
hf : f ≫ R.cocomplex.d n m = 0
⊢ -R.extMk f m hm hf = R.extMk (-f) m hm ⋯

Tactic:
by
  dsimp [extMk]
  simp only [Preadditive.neg_comp]
  rw [Cocycle.fromSingleMk_neg _ _ _ _
    (by simp [cochainComplex_d _ _ _ n m rfl rfl, reassoc_of% hf])]
  simp

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Injective.Resolution

Goal state before:
pp: case zero
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
⊢ {
        X₁ :=
          ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1).obj
              ((CochainComplex.single₀ C).obj Z)).X₂,
        X₂ := ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1).obj (ofCocomplex Z)).X₂,
        X₃ := ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1).obj (ofCocomplex Z)).X₃,
        f :=
          ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1).map
              (((ofCocomplex Z).fromSingle₀Equiv Z).symm ⟨Injective.ι Z, ⋯⟩)).τ₂,
        g := ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1).obj (ofCocomplex Z)).g,
        zero := ⋯ }.Exact ∧
    Mono
      ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1).map
          (((ofCocomplex Z).fromSingle₀Equiv Z).symm ⟨Injective.ι Z, ⋯⟩)).τ₂

pp: case zero.hf₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
⊢ ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1).obj ((CochainComplex.single₀ C).obj Z)).f = 0

pp: case zero.hg₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
⊢ ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1).obj ((CochainComplex.single₀ C).obj Z)).g = 0

pp: case zero.hf₂
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
⊢ ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1).obj (ofCocomplex Z)).f = 0

Tactic:
·

Goal state after:
case zero
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
⊢ {
        X₁ :=
          ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1).obj
              ((CochainComplex.single₀ C).obj Z)).X₂,
        X₂ := ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1).obj (ofCocomplex Z)).X₂,
        X₃ := ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1).obj (ofCocomplex Z)).X₃,
        f :=
          ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1).map
              (((ofCocomplex Z).fromSingle₀Equiv Z).symm ⟨Injective.ι Z, ⋯⟩)).τ₂,
        g := ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1).obj (ofCocomplex Z)).g,
        zero := ⋯ }.Exact ∧
    Mono
      ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1).map
          (((ofCocomplex Z).fromSingle₀Equiv Z).symm ⟨Injective.ι Z, ⋯⟩)).τ₂
Module: Mathlib.CategoryTheory.Abelian.Injective.Ext

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X Y : C
R : InjectiveResolution Y
n✝ : ℕ
inst✝ : HasDerivedCategory C
n : ℕ
f : X ⟶ R.cocomplex.X n
m : ℕ
hm : n + 1 = m
hf : f ≫ R.cocomplex.d n m = 0
⊢ ↑n + 1 = ↑m

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Injective.Ext

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y : C
R : InjectiveResolution Y
n : ℕ
f : X ⟶ R.cocomplex.X n
m : ℕ
hm : n + 1 = m
hf : f ≫ R.cocomplex.d n m = 0
p : ℕ
hp : p + 1 = n
⊢ ↑p + 1 = ↑n

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Injective.Resolution

Goal state before:
pp: case succ
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
n✝ : ℕ
⊢ QuasiIsoAt (((ofCocomplex Z).fromSingle₀Equiv Z).symm ⟨Injective.ι Z, ⋯⟩) (n✝ + 1)

Tactic:
·

Goal state after:
case succ
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
n✝ : ℕ
⊢ QuasiIsoAt (((ofCocomplex Z).fromSingle₀Equiv Z).symm ⟨Injective.ι Z, ⋯⟩) (n✝ + 1)
Module: Mathlib.CategoryTheory.Abelian.Injective.Resolution

Goal state before:
pp: case succ
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
n✝ : ℕ
⊢ HomologicalComplex.ExactAt (ofCocomplex Z) (n✝ + 1)

pp: case succ.hK
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
n✝ : ℕ
⊢ HomologicalComplex.ExactAt ((CochainComplex.single₀ C).obj Z) (n✝ + 1)

Tactic:
·

Goal state after:
case succ
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
n✝ : ℕ
⊢ HomologicalComplex.ExactAt (ofCocomplex Z) (n✝ + 1)
Module: Mathlib.CategoryTheory.Abelian.Injective.Resolution

Goal state before:
pp: case succ.hK
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
n✝ : ℕ
⊢ HomologicalComplex.ExactAt ((CochainComplex.single₀ C).obj Z) (n✝ + 1)

Tactic:
·

Goal state after:
case succ.hK
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
n✝ : ℕ
⊢ HomologicalComplex.ExactAt ((CochainComplex.single₀ C).obj Z) (n✝ + 1)
Module: Mathlib.CategoryTheory.Abelian.Injective.Ext

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y : C
R : InjectiveResolution Y
n m : ℕ
hm : n + 1 = m
x : Cocycle ((singleFunctor C 0).obj X) R.cochainComplex ↑n
⊢ ↑n + 1 = ↑m

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Injective.Ext

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y : C
R : InjectiveResolution Y
n : ℕ
f : X ⟶ R.cocomplex.X n
m : ℕ
hm : n + 1 = m
hf : f ≫ R.cocomplex.d n m = 0
X' : C
g : X' ⟶ X
this : HasDerivedCategory C
⊢ ↑n + 1 = ↑m

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Injective.Ext

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y : C
R : InjectiveResolution Y
n : ℕ
f : X ⟶ R.cocomplex.X n
m : ℕ
hm : n + 1 = m
hf : f ≫ R.cocomplex.d n m = 0
Y' : C
R' : InjectiveResolution Y'
g : Y ⟶ Y'
φ : R.Hom R' g
this : HasDerivedCategory C
⊢ ↑n = ↑n

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Injective.Ext

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y : C
R : InjectiveResolution Y
n : ℕ
f : X ⟶ R.cocomplex.X n
m : ℕ
hm : n + 1 = m
hf : f ≫ R.cocomplex.d n m = 0
Y' : C
R' : InjectiveResolution Y'
g : Y ⟶ Y'
φ : R.Hom R' g
this✝ : HasDerivedCategory C
this : (f ≫ φ.hom.f n) ≫ (R'.cochainComplexXIso (↑n) n ⋯).inv = (f ≫ (R.cochainComplexXIso (↑n) n ⋯).inv) ≫ φ.hom'.f ↑n
⊢ ↑n + 1 = ↑m

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Projective.Ext

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y : C
R : ProjectiveResolution X
n✝ n : ℕ
f : R.complex.X n ⟶ Y
m : ℕ
hm : n + 1 = m
hf : R.complex.d m n ≫ f = 0
⊢ -↑m + 1 = -↑n

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Projective.Ext

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X Y : C
R : ProjectiveResolution X
n✝ : ℕ
inst✝ : HasDerivedCategory C
n : ℕ
f : R.complex.X n ⟶ Y
m : ℕ
hm : n + 1 = m
hf : R.complex.d m n ≫ f = 0
⊢ -↑m + 1 = -↑n

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Projective.Ext

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y : C
R : ProjectiveResolution X
n : ℕ
f : R.complex.X n ⟶ Y
m : ℕ
hm : n + 1 = m
hf : R.complex.d m n ≫ f = 0
p : ℕ
hp : p + 1 = n
⊢ -↑n + 1 = -↑p

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Projective.Ext

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y : C
R : ProjectiveResolution X
n m : ℕ
hm : n + 1 = m
x : Cocycle R.cochainComplex ((singleFunctor C 0).obj Y) ↑n
⊢ -↑m + 1 = -↑n

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Bousfield

Goal state before:
pp: case left
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Abelian C
inst✝² : Abelian D
G : D ⥤ C
inst✝¹ : PreservesFiniteLimits G
inst✝ : PreservesFiniteColimits G
X Y : D
f : X ⟶ Y
hf : IsIso (G.map f)
⊢ G.kernel.monoModSerre f

pp: case right
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Abelian C
inst✝² : Abelian D
G : D ⥤ C
inst✝¹ : PreservesFiniteLimits G
inst✝ : PreservesFiniteColimits G
X Y : D
f : X ⟶ Y
hf : IsIso (G.map f)
⊢ G.kernel.epiModSerre f

Tactic:
·

Goal state after:
case left
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Abelian C
inst✝² : Abelian D
G : D ⥤ C
inst✝¹ : PreservesFiniteLimits G
inst✝ : PreservesFiniteColimits G
X Y : D
f : X ⟶ Y
hf : IsIso (G.map f)
⊢ G.kernel.monoModSerre f
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Bousfield

Goal state before:
pp: case right
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Abelian C
inst✝² : Abelian D
G : D ⥤ C
inst✝¹ : PreservesFiniteLimits G
inst✝ : PreservesFiniteColimits G
X Y : D
f : X ⟶ Y
hf : IsIso (G.map f)
⊢ G.kernel.epiModSerre f

Tactic:
·

Goal state after:
case right
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Abelian C
inst✝² : Abelian D
G : D ⥤ C
inst✝¹ : PreservesFiniteLimits G
inst✝ : PreservesFiniteColimits G
X Y : D
f : X ⟶ Y
hf : IsIso (G.map f)
⊢ G.kernel.epiModSerre f
Module: Mathlib.CategoryTheory.Abelian.Pseudoelements

Goal state before:
pp: case refine_1
R : Type u_1
inst✝ : Ring R
G : ModuleCat R
x y : Over G
P : ModuleCat R
p : P ⟶ x.left
q : P ⟶ y.left
hp : Epi p
hq : Epi q
H : p ≫ x.hom = q ≫ y.hom
a : ↑G
ha : a ∈ (ModuleCat.Hom.hom x.hom).range
⊢ a ∈ (ModuleCat.Hom.hom y.hom).range

pp: case refine_2
R : Type u_1
inst✝ : Ring R
G : ModuleCat R
x y : Over G
P : ModuleCat R
p : P ⟶ x.left
q : P ⟶ y.left
hp : Epi p
hq : Epi q
H : p ≫ x.hom = q ≫ y.hom
a : ↑G
ha : a ∈ (ModuleCat.Hom.hom y.hom).range
⊢ a ∈ (ModuleCat.Hom.hom x.hom).range

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
inst✝ : Ring R
G : ModuleCat R
x y : Over G
P : ModuleCat R
p : P ⟶ x.left
q : P ⟶ y.left
hp : Epi p
hq : Epi q
H : p ≫ x.hom = q ≫ y.hom
a : ↑G
ha : a ∈ (ModuleCat.Hom.hom x.hom).range
⊢ a ∈ (ModuleCat.Hom.hom y.hom).range
Module: Mathlib.CategoryTheory.Abelian.Pseudoelements

Goal state before:
pp: case refine_2
R : Type u_1
inst✝ : Ring R
G : ModuleCat R
x y : Over G
P : ModuleCat R
p : P ⟶ x.left
q : P ⟶ y.left
hp : Epi p
hq : Epi q
H : p ≫ x.hom = q ≫ y.hom
a : ↑G
ha : a ∈ (ModuleCat.Hom.hom y.hom).range
⊢ a ∈ (ModuleCat.Hom.hom x.hom).range

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
inst✝ : Ring R
G : ModuleCat R
x y : Over G
P : ModuleCat R
p : P ⟶ x.left
q : P ⟶ y.left
hp : Epi p
hq : Epi q
H : p ≫ x.hom = q ≫ y.hom
a : ↑G
ha : a ∈ (ModuleCat.Hom.hom y.hom).range
⊢ a ∈ (ModuleCat.Hom.hom x.hom).range
Module: Mathlib.CategoryTheory.Abelian.Pseudoelements

Goal state before:
pp: case refine_2
R : Type u_1
inst✝ : Ring R
G : ModuleCat R
x y : Over G
P : ModuleCat R
p : P ⟶ x.left
q : P ⟶ y.left
hp : Epi p
hq : Epi q
H : p ≫ x.hom = q ≫ y.hom
a : ↑G
a' : ↑y.left
ha' : (ModuleCat.Hom.hom y.hom) a' = a
a'' : ↑P
ha'' : (ConcreteCategory.hom q) a'' = a'
⊢ a ∈ (ModuleCat.Hom.hom x.hom).range

Tactic:
refine ⟨p a'', ?_⟩

Goal state after:
case refine_2
R : Type u_1
inst✝ : Ring R
G : ModuleCat R
x y : Over G
P : ModuleCat R
p : P ⟶ x.left
q : P ⟶ y.left
hp : Epi p
hq : Epi q
H : p ≫ x.hom = q ≫ y.hom
a : ↑G
a' : ↑y.left
ha' : (ModuleCat.Hom.hom y.hom) a' = a
a'' : ↑P
ha'' : (ConcreteCategory.hom q) a'' = a'
⊢ (ModuleCat.Hom.hom x.hom) ((ConcreteCategory.hom p) a'') = a
Module: Mathlib.CategoryTheory.Abelian.Projective.Ext

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y : C
R : ProjectiveResolution X
n : ℕ
f : R.complex.X n ⟶ Y
m : ℕ
hm : n + 1 = m
hf : R.complex.d m n ≫ f = 0
X' : C
R' : ProjectiveResolution X'
g : X' ⟶ X
φ : R'.Hom R g
this : HasDerivedCategory C
⊢ -↑n = -↑n

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Projective.Ext

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y : C
R : ProjectiveResolution X
n : ℕ
f : R.complex.X n ⟶ Y
m : ℕ
hm : n + 1 = m
hf : R.complex.d m n ≫ f = 0
X' : C
R' : ProjectiveResolution X'
g : X' ⟶ X
φ : R'.Hom R g
this✝ : HasDerivedCategory C
this : (R'.cochainComplexXIso (-↑n) n ⋯).hom ≫ φ.hom.f n = φ.hom'.f (-↑n) ≫ (R.cochainComplexXIso (-↑n) n ⋯).hom
⊢ -↑m + 1 = -↑n

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
⊢ (∃ X' s, ∃ (_ : P.epiModSerre s), s ≫ f = 0) → P (Abelian.image f)

pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
hf : P (Abelian.image f)
⊢ ∃ X' s, ∃ (_ : P.epiModSerre s), s ≫ f = 0

Tactic:
·

Goal state after:
case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
⊢ (∃ X' s, ∃ (_ : P.epiModSerre s), s ≫ f = 0) → P (Abelian.image f)
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
hf : P (Abelian.image f)
⊢ ∃ X' s, ∃ (_ : P.epiModSerre s), s ≫ f = 0

Tactic:
·

Goal state after:
case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
hf : P (Abelian.image f)
⊢ ∃ X' s, ∃ (_ : P.epiModSerre s), s ≫ f = 0
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
⊢ (∃ X' s, ∃ (_ : P.isoModSerre s), s ≫ f = 0) → P (Abelian.image f)

pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
hf : P (Abelian.image f)
⊢ ∃ X' s, ∃ (_ : P.isoModSerre s), s ≫ f = 0

Tactic:
·

Goal state after:
case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
⊢ (∃ X' s, ∃ (_ : P.isoModSerre s), s ≫ f = 0) → P (Abelian.image f)
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
hf : P (Abelian.image f)
⊢ ∃ X' s, ∃ (_ : P.isoModSerre s), s ≫ f = 0

Tactic:
·

Goal state after:
case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
hf : P (Abelian.image f)
⊢ ∃ X' s, ∃ (_ : P.isoModSerre s), s ≫ f = 0
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
⊢ (∃ Y' s, ∃ (_ : P.monoModSerre s), f ≫ s = 0) → P (Abelian.image f)

pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
hf : P (Abelian.image f)
⊢ ∃ Y' s, ∃ (_ : P.monoModSerre s), f ≫ s = 0

Tactic:
·

Goal state after:
case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
⊢ (∃ Y' s, ∃ (_ : P.monoModSerre s), f ≫ s = 0) → P (Abelian.image f)
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
hf : P (Abelian.image f)
⊢ ∃ Y' s, ∃ (_ : P.monoModSerre s), f ≫ s = 0

Tactic:
·

Goal state after:
case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
hf : P (Abelian.image f)
⊢ ∃ Y' s, ∃ (_ : P.monoModSerre s), f ≫ s = 0
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
⊢ (∃ Y' s, ∃ (_ : P.isoModSerre s), f ≫ s = 0) → P (Abelian.image f)

pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
hf : P (Abelian.image f)
⊢ ∃ Y' s, ∃ (_ : P.isoModSerre s), f ≫ s = 0

Tactic:
·

Goal state after:
case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
⊢ (∃ Y' s, ∃ (_ : P.isoModSerre s), f ≫ s = 0) → P (Abelian.image f)
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
hf : P (Abelian.image f)
⊢ ∃ Y' s, ∃ (_ : P.isoModSerre s), f ≫ s = 0

Tactic:
·

Goal state after:
case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
hf : P (Abelian.image f)
⊢ ∃ Y' s, ∃ (_ : P.isoModSerre s), f ≫ s = 0