mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.RingHomProperties

Goal state before:
pp: P Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
h₂ :
  ∀ ⦃R S T : Type u⦄ [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]
    [inst_4 : Algebra R T], Q (algebraMap R S) → P (algebraMap S (S ⊗[R] T)) → P (algebraMap R T)
R S R' S' : Type u
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing R'
inst✝⁷ : CommRing S'
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R R'
inst✝⁴ : Algebra R S'
inst✝³ : Algebra S S'
inst✝² : Algebra R' S'
inst✝¹ : IsScalarTower R S S'
inst✝ : IsScalarTower R R' S'
h : Algebra.IsPushout R S R' S'
hQ : Q (algebraMap R R')
H : P (algebraMap R' S')
e : R' ⊗[R] S ≃ₐ[R'] S' := Algebra.IsPushout.equiv R R' S S'
⊢ (↑e.symm).comp (algebraMap R' S') = algebraMap R' (R' ⊗[R] S)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
X Y Z : C
inst✝⁴ : HasPullbacks C
S T : C
f✝ : X ⟶ T
g✝ : Y ⟶ T
i : T ⟶ S
inst✝³ : HasPullback i i
inst✝² : HasPullback f✝ g✝
inst✝¹ : HasPullback (f✝ ≫ i) (g✝ ≫ i)
inst✝ : HasPullback (diagonal i) (map (f✝ ≫ i) (g✝ ≫ i) i i f✝ g✝ (𝟙 S) ⋯ ⋯)
g : Y ⟶ X
f : X ⟶ S
⊢ 𝟙 Y ≫ g ≫ f = g ≫ f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
X Y Z : C
inst✝⁴ : HasPullbacks C
S T : C
f✝ : X ⟶ T
g✝ : Y ⟶ T
i : T ⟶ S
inst✝³ : HasPullback i i
inst✝² : HasPullback f✝ g✝
inst✝¹ : HasPullback (f✝ ≫ i) (g✝ ≫ i)
inst✝ : HasPullback (diagonal i) (map (f✝ ≫ i) (g✝ ≫ i) i i f✝ g✝ (𝟙 S) ⋯ ⋯)
g : Y ⟶ X
f : X ⟶ S
⊢ (g ≫ f) ≫ 𝟙 S = g ≫ f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
X Y Z : C
inst✝⁴ : HasPullbacks C
S T : C
f✝ : X ⟶ T
g✝ : Y ⟶ T
i : T ⟶ S
inst✝³ : HasPullback i i
inst✝² : HasPullback f✝ g✝
inst✝¹ : HasPullback (f✝ ≫ i) (g✝ ≫ i)
inst✝ : HasPullback (diagonal i) (map (f✝ ≫ i) (g✝ ≫ i) i i f✝ g✝ (𝟙 S) ⋯ ⋯)
g : Y ⟶ X
f : X ⟶ S
⊢ f ≫ 𝟙 S = 𝟙 X ≫ f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y : C
inst✝ : HasPullbacks C
S : C
g : Y ⟶ X
f : X ⟶ S
⊢ f = 𝟙 X ≫ f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.RingHomProperties

Goal state before:
pp: P Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
h₂ :
  ∀ ⦃R S T : Type u⦄ [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]
    [inst_4 : Algebra R T], Q (algebraMap R S) → P (algebraMap S (S ⊗[R] T)) → P (algebraMap R T)
⊢ CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ P) fun {R S} [CommRing R] [CommRing S] ↦ Q

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.RingHomProperties

Goal state before:
pp: P Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
R S T : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hPQ : CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ P) fun {R S} [CommRing R] [CommRing S] ↦ Q
h : Q (algebraMap R T)
H : P Algebra.TensorProduct.includeRight.toRingHom
⊢ P (algebraMap R S)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y : C
inst✝ : HasPullbacks C
S : C
g : Y ⟶ X
f : X ⟶ S
i : pullback (g ≫ f) f ≅ pullback (g ≫ f) (𝟙 X ≫ f) := congrHom ⋯ ⋯
⊢ (g ≫ f) ≫ 𝟙 S = g ≫ f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝ : Category.{v_1, u_1} J
⊢ W.IsStableUnderLimitsOfShape J ↔ W.limitsOfShape J ≤ W

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y : C
inst✝ : HasPullbacks C
S : C
g : Y ⟶ X
f : X ⟶ S
i : pullback (g ≫ f) f ≅ pullback (g ≫ f) (𝟙 X ≫ f) := congrHom ⋯ ⋯
⊢ f ≫ 𝟙 S = 𝟙 X ≫ f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝¹ : Category.{v_1, u_1} J
inst✝ : W.IsStableUnderLimitsOfShape J
⊢ W.limitsOfShape J ≤ W

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝ : Category.{v_1, u_1} J
X₁ X₂ : J ⥤ C
c₁ : Cocone X₁
c₂ : Cocone X₂
h₁ : IsColimit c₁
h₂ : IsColimit c₂
f : X₁ ⟶ X₂
hf : W.functorCategory J f
φ : c₁.pt ⟶ c₂.pt
hφ : ∀ (j : J), c₁.ι.app j ≫ φ = f.app j ≫ c₂.ι.app j
j : J
⊢ c₁.ι.app j ≫ φ = c₁.ι.app j ≫ h₁.desc { pt := c₂.pt, ι := f ≫ c₂.ι }

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y : C
inst✝ : HasPullbacks C
S : C
g : Y ⟶ X
f : X ⟶ S
i : pullback (g ≫ f) f ≅ pullback (g ≫ f) (𝟙 X ≫ f) := congrHom ⋯ ⋯
⊢ (𝟙 X ≫ f) ≫ 𝟙 S = 𝟙 X ≫ f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Presentation

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : Type w
inst✝ : Category.{t, w} J
X : C
pres : ColimitPresentation J X
i j : J
f : i ⟶ j
⊢ pres.diag.map f ≫ pres.ι.app j = pres.ι.app i

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Presentation

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : Type w
inst✝² : Category.{t, w} J
X : C
P : ColimitPresentation J X
D : Type u_1
inst✝¹ : Category.{v_1, u_1} D
F : C ⥤ D
inst✝ : PreservesColimitsOfShape J F
⊢ ∀ (j : J),
    (F.mapCocone { pt := X, ι := P.ι }).ι.app j ≫ (Iso.refl (F.mapCocone { pt := X, ι := P.ι }).pt).hom =
      { pt := F.obj X, ι := Functor.whiskerRight P.ι F ≫ (Functor.constComp J X F).hom }.ι.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Presentation

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : Type w
inst✝ : Category.{t, w} J
X : C
pres : LimitPresentation J X
i j : J
f : i ⟶ j
⊢ pres.π.app i ≫ pres.diag.map f = pres.π.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Presentation

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
J✝ : Type w
inst✝³ : Category.{t, w} J✝
X✝ : C
J : Type w
inst✝² : Category.{t, w} J
X : C
P : LimitPresentation J X
D : Type u_1
inst✝¹ : Category.{v_1, u_1} D
F : C ⥤ D
inst✝ : PreservesLimitsOfShape J F
⊢ ∀ (j : J),
    (F.mapCone { pt := X, π := P.π }).π.app j =
      (Iso.refl (F.mapCone { pt := X, π := P.π }).pt).hom ≫
        { pt := F.obj X, π := (Functor.constComp J X F).inv ≫ Functor.whiskerRight P.π F }.π.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : ObjectProperty C
J : Type u'
inst✝ : Category.{v', u'} J
Q : ObjectProperty C
h : P ≤ Q
⊢ P.strictLimitsOfShape J ≤ Q.strictLimitsOfShape J

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
J : Type u'
inst✝¹ : Category.{v', u'} J
inst✝ : Nonempty J
⊢ ⊥.strictLimitsOfShape J = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
P : ObjectProperty C
J : Type u'
inst✝¹ : Category.{v', u'} J
J' : Type u''
inst✝ : Category.{v'', u''} J'
X : C
p : P.LimitOfShape J X
j : J
⊢ X ⟶ P.ι.obj { obj := p.diag.obj j, property := ⋯ }

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
P : ObjectProperty C
J : Type u'
inst✝¹ : Category.{v', u'} J
J' : Type u''
inst✝ : Category.{v'', u''} J'
X : C
p : P.LimitOfShape J X
X✝ Y✝ : J
f : X✝ ⟶ Y✝
⊢ (StructuredArrow.mk (p.π.app X✝)).right.obj ⟶ (StructuredArrow.mk (p.π.app Y✝)).right.obj

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
P : ObjectProperty C
J : Type u'
inst✝¹ : Category.{v', u'} J
J' : Type u''
inst✝ : Category.{v'', u''} J'
X : C
p : P.LimitOfShape J X
X✝ Y✝ : J
f : X✝ ⟶ Y✝
⊢ (StructuredArrow.mk (p.π.app X✝)).hom ≫ P.ι.map (homMk (p.diag.map f)) = (StructuredArrow.mk (p.π.app Y✝)).hom

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : ObjectProperty C
J : Type u'
inst✝ : Category.{v', u'} J
⊢ P.strictLimitsOfShape J ≤ P.limitsOfShape J

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
J : Type u'
inst✝¹ : Category.{v', u'} J
inst✝ : Nonempty J
⊢ ⊥.limitsOfShape J = ⊥

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝ : Category.{v_1, u_1} J
X₁ X₂ : J ⥤ C
c₁ : Cocone X₁
c₂ : Cocone X₂
h₁ : IsColimit c₁
h₂ : IsColimit c₂
f : X₁ ⟶ X₂
hf : W.functorCategory J f
φ : c₁.pt ⟶ c₂.pt
hφ : ∀ (j : J), c₁.ι.app j ≫ φ = f.app j ≫ c₂.ι.app j
⊢ W.colimitsOfShape J φ

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y : C
inst✝ : HasPullbacks C
S : C
g : Y ⟶ X
f : X ⟶ S
i : pullback (g ≫ f) f ≅ pullback (g ≫ f) (𝟙 X ≫ f) := congrHom ⋯ ⋯
⊢ diagonal f ≫ 𝟙 (diagonalObj f) = 𝟙 X ≫ diagonal f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
P : ObjectProperty C
J : Type u'
inst✝¹ : Category.{v', u'} J
J' : Type u''
inst✝ : Category.{v'', u''} J'
⊢ ∀ {X Y : C} (x : X ≅ Y), P.limitsOfShape J X → P.limitsOfShape J Y

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W₁ W₂ : MorphismProperty C
h : W₁ ≤ W₂
J : Type u_2
inst✝ : Category.{v_2, u_2} J
⊢ W₁.colimitsOfShape J ≤ W₂.colimitsOfShape J

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y : C
inst✝ : HasPullbacks C
S : C
g : Y ⟶ X
f : X ⟶ S
i : pullback (g ≫ f) f ≅ pullback (g ≫ f) (𝟙 X ≫ f) := congrHom ⋯ ⋯
⊢ map (g ≫ f) (𝟙 X ≫ f) f f g (𝟙 X) (𝟙 S) ⋯ ⋯ ≫ 𝟙 (diagonalObj f) = i.inv ≫ map (g ≫ f) f f f g (𝟙 X) (𝟙 S) ⋯ ⋯

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : ObjectProperty C
J : Type u'
inst✝ : Category.{v', u'} J
⊢ (P.strictLimitsOfShape J).isoClosure = P.limitsOfShape J

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : ObjectProperty C
J : Type u'
inst✝ : Category.{v', u'} J
Q : ObjectProperty C
hPQ : P ≤ Q
⊢ P.limitsOfShape J ≤ Q.limitsOfShape J

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : ObjectProperty C
J : Type u'
inst✝ : Category.{v', u'} J
⊢ P.isoClosure.limitsOfShape J = P.limitsOfShape J

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
P : ObjectProperty C
J : Type u'
inst✝⁶ : Category.{v', u'} J
J' : Type u''
inst✝⁵ : Category.{v'', u''} J'
inst✝⁴ : ObjectProperty.Small.{w, v_1, u_1} P
inst✝³ : LocallySmall.{w, v_1, u_1} C
inst✝² : Small.{w, u'} J
inst✝¹ : LocallySmall.{w, v', u'} J
F : J ⥤ C
inst✝ : HasLimit F
hF : ∀ (j : J), P (F.obj j)
⊢ HasLimit (P.lift F hF ⋙ P.ι)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

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
P : ObjectProperty C
J : Type u'
inst✝⁵ : Category.{v', u'} J
J' : Type u''
inst✝⁴ : Category.{v'', u''} J'
inst✝³ : ObjectProperty.Small.{w, v_1, u_1} P
inst✝² : LocallySmall.{w, v_1, u_1} C
inst✝¹ : Small.{w, u'} J
inst✝ : LocallySmall.{w, v', u'} J
⊢ ObjectProperty.Small.{w, v_1, u_1} (P.strictLimitsOfShape J)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝² : Category.{v_1, u_1} J
J' : Type u_2
inst✝¹ : Category.{v_2, u_2} J'
F : J ⥤ J'
inst✝ : F.Final
X✝ Y✝ : C
f✝ : X✝ ⟶ Y✝
X₁ X₂ : J' ⥤ C
c₁ : Cocone X₁
c₂ : Cocone X₂
h₁ : IsColimit c₁
h₂ : IsColimit c₂
f : X₁ ⟶ X₂
hf : W.functorCategory J' f
h₁' : IsColimit (Cocone.whisker F c₁)
h₂' : IsColimit (Cocone.whisker F c₂)
j : J
⊢ (Cocone.whisker F c₁).ι.app j ≫ h₁.desc { pt := c₂.pt, ι := f ≫ c₂.ι } =
    (Cocone.whisker F c₁).ι.app j ≫ h₁'.desc { pt := c₂.pt, ι := F.whiskerLeft f ≫ (Cocone.whisker F c₂).ι }

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
D : Type u'
inst✝³ : Category.{v', u'} D
F : C ⥤ D
inst✝² : PreservesLimitsOfShape (Discrete WalkingPair) F
inst✝¹ : PreservesLimitsOfShape (Discrete PEmpty.{1}) F
inst✝ : HasFiniteProducts C
n : ℕ
⊢ ∀ (f : Fin (n + 1) → C), PreservesLimit (Discrete.functor f) F

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

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
P : ObjectProperty C
J : Type u'
inst✝⁵ : Category.{v', u'} J
J' : Type u''
inst✝⁴ : Category.{v'', u''} J'
inst✝³ : ObjectProperty.Small.{w, v_1, u_1} P
inst✝² : LocallySmall.{w, v_1, u_1} C
inst✝¹ : Small.{w, u'} J
inst✝ : LocallySmall.{w, v', u'} J
⊢ ObjectProperty.EssentiallySmall.{w, v_1, u_1} (P.limitsOfShape J)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
D : Type u'
inst✝³ : Category.{v', u'} D
F : C ⥤ D
inst✝² : PreservesLimitsOfShape (Discrete WalkingPair) F
inst✝¹ : PreservesLimitsOfShape (Discrete PEmpty.{1}) F
inst✝ : HasFiniteProducts C
n : ℕ
⊢ PreservesLimitsOfShape (Discrete (Fin n)) F

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
P : ObjectProperty C
J : Type u'
inst✝¹ : Category.{v', u'} J
inst✝ : P.IsClosedUnderIsomorphisms
h : P.strictLimitsOfShape J ≤ P
⊢ P.limitsOfShape J ≤ P

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

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
P : ObjectProperty C
J : Type u'
inst✝² : Category.{v', u'} J
J' : Type u''
inst✝¹ : Category.{v'', u''} J'
inst✝ : Nonempty J
⊢ ⊥.limitsOfShape J ≤ ⊥

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
P : ObjectProperty C
J : Type u'
inst✝¹ : Category.{v', u'} J
J' : Type u''
inst✝ : Category.{v'', u''} J'
x✝¹ : C
x✝ : ⊤.limitsOfShape J x✝¹
⊢ ⊤ x✝¹

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝² : Category.{v_1, u_1} J
J' : Type u_2
inst✝¹ : Category.{v_2, u_2} J'
F : J ⥤ J'
inst✝ : F.Final
⊢ W.colimitsOfShape J' ≤ W.colimitsOfShape J

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
P : ObjectProperty C
J : Type u'
inst✝¹ : Category.{v', u'} J
J' : Type u''
inst✝ : Category.{v'', u''} J'
e : J ≌ J'
⊢ P.IsClosedUnderLimitsOfShape J ↔ P.IsClosedUnderLimitsOfShape J'

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
P : ObjectProperty C
J : Type u'
inst✝² : Category.{v', u'} J
J' : Type u''
inst✝¹ : Category.{v'', u''} J'
e : J ≌ J'
inst✝ : P.IsClosedUnderLimitsOfShape J
⊢ P.IsClosedUnderLimitsOfShape J'

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts

Goal state before:
pp: J : Type v
inst✝² : SmallCategory J
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i ↦ f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
⊢ (b : Fin (n + 1)) → f b ⟶ c₂.pt

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
J : Type u'
inst✝¹ : Category.{v', u'} J
P : ObjectProperty D
inst✝ : P.IsClosedUnderIsomorphisms
e : C ≌ D
⊢ (P.inverseImage e.functor).IsClosedUnderLimitsOfShape J ↔ P.IsClosedUnderLimitsOfShape J

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y : C
inst✝ : HasPullbacks C
S : C
g : Y ⟶ X
f : X ⟶ S
⊢ IsPullback g (lift (𝟙 Y) g ⋯) (diagonal f) (map (g ≫ f) f f f g (𝟙 X) (𝟙 S) ⋯ ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts

Goal state before:
pp: J : Type v
inst✝² : SmallCategory J
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i ↦ f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsColimit c₁
t₂ : IsColimit c₂
s : Cocone (Discrete.functor f)
⊢ (extendCofan c₁ c₂).pt ⟶ s.pt

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ContainsZero

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
P : ObjectProperty C
inst✝¹ : P.ContainsZero
inst✝ : P.IsClosedUnderIsomorphisms
Z : C
hZ : IsZero Z
⊢ P Z

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ContainsZero

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
P Q : ObjectProperty C
inst✝ : HasZeroObject C
⊢ ⊤ 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ContainsZero

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u'
inst✝⁴ : Category.{v', u'} D
P Q : ObjectProperty C
inst✝³ : P.ContainsZero
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroMorphisms D
F : C ⥤ D
inst✝ : F.PreservesZeroMorphisms
⊢ ∃ Z, IsZero Z ∧ P.map F Z

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ContainsZero

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
P Q : ObjectProperty C
inst✝ : P.ContainsZero
⊢ ∃ Z, IsZero Z ∧ P.isoClosure Z

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ContainsZero

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
D : Type u'
inst✝³ : Category.{v', u'} D
P Q : ObjectProperty C
inst✝² : P.ContainsZero
inst✝¹ : P.IsClosedUnderIsomorphisms
inst✝ : Q.ContainsZero
⊢ ∃ Z, IsZero Z ∧ (P ⊓ Q) Z

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ContainsZero

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
P Q : ObjectProperty C
inst✝ : P.ContainsZero
⊢ ∃ Z, IsZero Z ∧ P.op Z

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ContainsZero

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
P✝ Q : ObjectProperty C
P : ObjectProperty Cᵒᵖ
inst✝ : P.ContainsZero
⊢ ∃ Z, IsZero Z ∧ P.unop Z

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ContainsZero

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
P Q : ObjectProperty C
inst✝ : P.ContainsZero
⊢ ∃ X, IsZero X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ContainsZero

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
D : Type u'
inst✝³ : Category.{v', u'} D
P Q : ObjectProperty C
inst✝² : P.ContainsZero
inst✝¹ : Q.ContainsZero
inst✝ : Q.IsClosedUnderIsomorphisms
⊢ ∃ Z, IsZero Z ∧ (P ⊓ Q) Z

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
X✝ Y✝ Z : C
inst✝¹ : HasPullbacks C
X Y S X' Y' S' : C
f : X ⟶ S
g : Y ⟶ S
f' : X' ⟶ S'
g' : Y' ⟶ S'
i₁ : X ⟶ X'
i₂ : Y ⟶ Y'
i₃ : S ⟶ S'
e₁ : f ≫ i₃ = i₁ ≫ f'
e₂ : g ≫ i₃ = i₂ ≫ g'
inst✝ : Mono i₃
⊢ (fst (fst (fst f' g') i₁) (fst (snd f' g') i₂) ≫ snd (fst f' g') i₁) ≫ f =
    (snd (fst (fst f' g') i₁) (fst (snd f' g') i₂) ≫ snd (snd f' g') i₂) ≫ g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Diagonal

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
X✝ Y✝ Z : C
inst✝¹ : HasPullbacks C
X Y S X' Y' S' : C
f : X ⟶ S
g : Y ⟶ S
f' : X' ⟶ S'
g' : Y' ⟶ S'
i₁ : X ⟶ X'
i₂ : Y ⟶ Y'
i₃ : S ⟶ S'
e₁ : f ≫ i₃ = i₁ ≫ f'
e₂ : g ≫ i₃ = i₂ ≫ g'
inst✝ : Mono i₃
⊢ lift (map f g f' g' i₁ i₂ i₃ e₁ e₂) (fst f g) ⋯ ≫ fst (fst f' g') i₁ =
    lift (map f g f' g' i₁ i₂ i₃ e₁ e₂) (snd f g) ⋯ ≫ fst (snd f' g') i₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts

Goal state before:
pp: J : Type v
inst✝² : SmallCategory J
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i ↦ f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsColimit c₁
t₂ : IsColimit c₂
s : Cocone (Discrete.functor f)
⊢ ∀ (j : Discrete (Fin (n + 1))),
    (extendCofan c₁ c₂).ι.app j ≫
        ↑(BinaryCofan.IsColimit.desc' t₂ (s.ι.app { as := 0 })
            (t₁.desc { pt := s.pt, ι := Discrete.natTrans fun i ↦ s.ι.app { as := i.as.succ } })) =
      s.ι.app j

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝ : Category.{v_1, u_1} J
Y₁ Y₂ : C
g : (𝟭 C).obj Y₁ ⟶ (𝟭 C).obj Y₂
X Y : C
X₁ X₂ : J ⥤ C
c₁ : Cocone X₁
c₂ : Cocone X₂
h₁ : IsColimit c₁
h₂ : IsColimit c₂
f : X₁ ⟶ X₂
hf : W.functorCategory J f
e : { left := c₁.1, right := c₂.1, hom := h₁.1 { pt := c₂.pt, ι := f ≫ c₂.ι } } ≅ { left := Y₁, right := Y₂, hom := g }
e₁ : Arrow.leftFunc.obj { left := c₁.1, right := c₂.1, hom := h₁.1 { pt := c₂.pt, ι := f ≫ c₂.ι } } ≅
  Arrow.leftFunc.obj { left := Y₁, right := Y₂, hom := g } :=
  Arrow.leftFunc.mapIso e
e₂ : Arrow.rightFunc.obj { left := c₁.1, right := c₂.1, hom := h₁.1 { pt := c₂.pt, ι := f ≫ c₂.ι } } ≅
  Arrow.rightFunc.obj { left := Y₁, right := Y₂, hom := g } :=
  Arrow.rightFunc.mapIso e
fac : e₁.hom ≫ g = h₁.desc { pt := c₂.pt, ι := f ≫ c₂.ι } ≫ e₂.hom
c₁' : Cocone X₁ := { pt := Y₁, ι := c₁.ι ≫ (Functor.const J).map e₁.hom }
c₂' : Cocone X₂ := { pt := Y₂, ι := c₂.ι ≫ (Functor.const J).map e₂.hom }
h₁' : IsColimit c₁'
h₂' : IsColimit c₂'
j : J
⊢ c₁'.ι.app j ≫ h₁'.desc { pt := c₂'.pt, ι := f ≫ c₂'.ι } = c₁'.ι.app j ≫ g

Tactic:
by

Goal state after: