mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hfg : ¬f + g ≈ 0
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
hfgne✝ : padicNorm p (↑f (max ?v1✝ (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))
hfgne :
  padicNorm p (↑f (max ?v1✝ (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))
⊢ padicNorm p (↑(f + g) (stationaryPoint hfg)) =
    max (padicNorm p (↑f (stationaryPoint hf))) (padicNorm p (↑g (stationaryPoint hg)))

pp: case v1
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hfg : ¬f + g ≈ 0
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfgne✝ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
hfgne : padicNorm p (↑f (max ?v1✝ (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))
⊢ ℕ

pp: case v2
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hfg : ¬f + g ≈ 0
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfgne✝ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
hfgne : padicNorm p (↑f (max ?v1✝ (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))
⊢ ℕ

pp: case v1
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hfg : ¬f + g ≈ 0
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
⊢ ℕ

pp: case v3
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hfg : ¬f + g ≈ 0
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
⊢ ℕ

Tactic:
·

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hfg : ¬f + g ≈ 0
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
hfgne✝ : padicNorm p (↑f (max ?v1✝ (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))
hfgne :
  padicNorm p (↑f (max ?v1✝ (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))
⊢ padicNorm p (↑(f + g) (stationaryPoint hfg)) =
    max (padicNorm p (↑f (stationaryPoint hf))) (padicNorm p (↑g (stationaryPoint hg)))
Module: Mathlib.AlgebraicGeometry.Sites.EtalePoint

Goal state before:
pp: case refine_2
X S : Scheme
f : X ⟶ S
inst✝² : Etale f
Ω : Type u
inst✝¹ : Field Ω
inst✝ : IsSepClosed Ω
x : ↥X
a : S.residueField (f x) ⟶ CommRingCat.of Ω
hx : f x = ((SpecToEquivOfField Ω S).symm ⟨f x, a⟩) default
m : ↑(S.residueField (f x)) →+* ↑(X.residueField x) := CommRingCat.Hom.hom (Hom.residueFieldMap f x)
algInst✝¹ : Algebra ↑(S.residueField (f x)) ↑(X.residueField x) := m.toAlgebra
algInst✝ : Algebra ↑(S.residueField (f x)) ↑(CommRingCat.of Ω) := (CommRingCat.Hom.hom a).toAlgebra
b : ↑(X.residueField x) →ₐ[↑(S.residueField (f x))] Ω := IsSepClosed.lift
this : Hom.residueFieldMap f x ≫ CommRingCat.ofHom b.toRingHom = a
⊢ (Spec.map (CommRingCat.ofHom b.toRingHom) ≫ X.fromSpecResidueField x) default = x

Tactic:
·

Goal state after:
case refine_2
X S : Scheme
f : X ⟶ S
inst✝² : Etale f
Ω : Type u
inst✝¹ : Field Ω
inst✝ : IsSepClosed Ω
x : ↥X
a : S.residueField (f x) ⟶ CommRingCat.of Ω
hx : f x = ((SpecToEquivOfField Ω S).symm ⟨f x, a⟩) default
m : ↑(S.residueField (f x)) →+* ↑(X.residueField x) := CommRingCat.Hom.hom (Hom.residueFieldMap f x)
algInst✝¹ : Algebra ↑(S.residueField (f x)) ↑(X.residueField x) := m.toAlgebra
algInst✝ : Algebra ↑(S.residueField (f x)) ↑(CommRingCat.of Ω) := (CommRingCat.Hom.hom a).toAlgebra
b : ↑(X.residueField x) →ₐ[↑(S.residueField (f x))] Ω := IsSepClosed.lift
this : Hom.residueFieldMap f x ≫ CommRingCat.ofHom b.toRingHom = a
⊢ (Spec.map (CommRingCat.ofHom b.toRingHom) ≫ X.fromSpecResidueField x) default = x
Module: Mathlib.CategoryTheory.Sites.Point.Conservative

Goal state before:
pp: case refine_1
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝¹ : LocallySmall.{w, v, u} C
hP :
  ∀ ⦃X : C⦄ (S : Sieve X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),
        ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x) →
      S ∈ J X
F₁ : Cᵒᵖ ⥤ Type w
U : C
f : F₁ ⟶ shrinkYoneda.{w, v, u}.obj U
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
Φ : P.FullSubcategory
u : Φ.obj.fiber.obj U
V : C
v : Φ.obj.fiber.obj V
y : ToType (F₁.obj (op V))
hx₁ :
  (ConcreteCategory.hom (Φ.obj.presheafFiber.map f)) ((ConcreteCategory.hom (Φ.obj.toPresheafFiber V v F₁)) y) =
    (ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app U)) u
t : unop (op V) ⟶ U
ht : shrinkYonedaObjObjEquiv.symm t = (ConcreteCategory.hom (f.app (op V))) y
⊢ shrinkYonedaObjObjEquiv.symm t =
    (ConcreteCategory.hom ((shrinkYoneda.{w, v, u}.obj U).map t.op)) (shrinkYonedaObjObjEquiv.symm (𝟙 U))

pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝¹ : LocallySmall.{w, v, u} C
hP :
  ∀ ⦃X : C⦄ (S : Sieve X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),
        ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x) →
      S ∈ J X
F₁ : Cᵒᵖ ⥤ Type w
U : C
f : F₁ ⟶ shrinkYoneda.{w, v, u}.obj U
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
Φ : P.FullSubcategory
u : Φ.obj.fiber.obj U
V : C
v : Φ.obj.fiber.obj V
y : ToType (F₁.obj (op V))
hx₁ :
  (ConcreteCategory.hom (Φ.obj.presheafFiber.map f)) ((ConcreteCategory.hom (Φ.obj.toPresheafFiber V v F₁)) y) =
    (ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app U)) u
t : unop (op V) ⟶ U
ht : shrinkYonedaObjObjEquiv.symm t = (ConcreteCategory.hom (f.app (op V))) y
⊢ (ConcreteCategory.hom (Φ.obj.fiber.map t)) v = u

Tactic:
·

Goal state after:
case refine_1
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝¹ : LocallySmall.{w, v, u} C
hP :
  ∀ ⦃X : C⦄ (S : Sieve X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),
        ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x) →
      S ∈ J X
F₁ : Cᵒᵖ ⥤ Type w
U : C
f : F₁ ⟶ shrinkYoneda.{w, v, u}.obj U
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
Φ : P.FullSubcategory
u : Φ.obj.fiber.obj U
V : C
v : Φ.obj.fiber.obj V
y : ToType (F₁.obj (op V))
hx₁ :
  (ConcreteCategory.hom (Φ.obj.presheafFiber.map f)) ((ConcreteCategory.hom (Φ.obj.toPresheafFiber V v F₁)) y) =
    (ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app U)) u
t : unop (op V) ⟶ U
ht : shrinkYonedaObjObjEquiv.symm t = (ConcreteCategory.hom (f.app (op V))) y
⊢ shrinkYonedaObjObjEquiv.symm t =
    (ConcreteCategory.hom ((shrinkYoneda.{w, v, u}.obj U).map t.op)) (shrinkYonedaObjObjEquiv.symm (𝟙 U))
Module: Mathlib.CategoryTheory.Sites.Point.Conservative

Goal state before:
pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝¹ : LocallySmall.{w, v, u} C
hP :
  ∀ ⦃X : C⦄ (S : Sieve X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),
        ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x) →
      S ∈ J X
F₁ : Cᵒᵖ ⥤ Type w
U : C
f : F₁ ⟶ shrinkYoneda.{w, v, u}.obj U
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
Φ : P.FullSubcategory
u : Φ.obj.fiber.obj U
V : C
v : Φ.obj.fiber.obj V
y : ToType (F₁.obj (op V))
hx₁ :
  (ConcreteCategory.hom (Φ.obj.presheafFiber.map f)) ((ConcreteCategory.hom (Φ.obj.toPresheafFiber V v F₁)) y) =
    (ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app U)) u
t : unop (op V) ⟶ U
ht : shrinkYonedaObjObjEquiv.symm t = (ConcreteCategory.hom (f.app (op V))) y
⊢ (ConcreteCategory.hom (Φ.obj.fiber.map t)) v = u

Tactic:
·

Goal state after:
case refine_2
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝¹ : LocallySmall.{w, v, u} C
hP :
  ∀ ⦃X : C⦄ (S : Sieve X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),
        ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x) →
      S ∈ J X
F₁ : Cᵒᵖ ⥤ Type w
U : C
f : F₁ ⟶ shrinkYoneda.{w, v, u}.obj U
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
Φ : P.FullSubcategory
u : Φ.obj.fiber.obj U
V : C
v : Φ.obj.fiber.obj V
y : ToType (F₁.obj (op V))
hx₁ :
  (ConcreteCategory.hom (Φ.obj.presheafFiber.map f)) ((ConcreteCategory.hom (Φ.obj.toPresheafFiber V v F₁)) y) =
    (ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app U)) u
t : unop (op V) ⟶ U
ht : shrinkYonedaObjObjEquiv.symm t = (ConcreteCategory.hom (f.app (op V))) y
⊢ (ConcreteCategory.hom (Φ.obj.fiber.map t)) v = u
Module: Mathlib.CategoryTheory.Sites.Point.Conservative

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝¹ : LocallySmall.{w, v, u} C
hP :
  ∀ ⦃X : C⦄ (S : Sieve X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),
        ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x) →
      S ∈ J X
F₁ : Cᵒᵖ ⥤ Type w
U : C
f : F₁ ⟶ shrinkYoneda.{w, v, u}.obj U
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
Φ : P.FullSubcategory
u : Φ.obj.fiber.obj U
V : C
v : Φ.obj.fiber.obj V
y : ToType (F₁.obj (op V))
hx₁ :
  (ConcreteCategory.hom (Φ.obj.presheafFiber.map f)) ((ConcreteCategory.hom (Φ.obj.toPresheafFiber V v F₁)) y) =
    (ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app U)) u
t : unop (op V) ⟶ U
ht : shrinkYonedaObjObjEquiv.symm t = (ConcreteCategory.hom (f.app (op V))) y
⊢ (ConcreteCategory.hom (Φ.obj.toPresheafFiber V v (shrinkYoneda.{w, v, u}.obj U))) (shrinkYonedaObjObjEquiv.symm t) =
    (ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app U)) u

Tactic:
·

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝¹ : LocallySmall.{w, v, u} C
hP :
  ∀ ⦃X : C⦄ (S : Sieve X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),
        ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x) →
      S ∈ J X
F₁ : Cᵒᵖ ⥤ Type w
U : C
f : F₁ ⟶ shrinkYoneda.{w, v, u}.obj U
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
Φ : P.FullSubcategory
u : Φ.obj.fiber.obj U
V : C
v : Φ.obj.fiber.obj V
y : ToType (F₁.obj (op V))
hx₁ :
  (ConcreteCategory.hom (Φ.obj.presheafFiber.map f)) ((ConcreteCategory.hom (Φ.obj.toPresheafFiber V v F₁)) y) =
    (ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app U)) u
t : unop (op V) ⟶ U
ht : shrinkYonedaObjObjEquiv.symm t = (ConcreteCategory.hom (f.app (op V))) y
⊢ (ConcreteCategory.hom (Φ.obj.toPresheafFiber V v (shrinkYoneda.{w, v, u}.obj U))) (shrinkYonedaObjObjEquiv.symm t) =
    (ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app U)) u
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ_[p]
⊢ Quotient.lift PadicSeq.norm ⋯ (q + r) ≤ max (Quotient.lift PadicSeq.norm ⋯ q) (Quotient.lift PadicSeq.norm ⋯ r)

pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ_[p]
⊢ max (Quotient.lift PadicSeq.norm ⋯ q) (Quotient.lift PadicSeq.norm ⋯ r) ≤
    Quotient.lift PadicSeq.norm ⋯ q + Quotient.lift PadicSeq.norm ⋯ r

Tactic:
·

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ_[p]
⊢ Quotient.lift PadicSeq.norm ⋯ (q + r) ≤ max (Quotient.lift PadicSeq.norm ⋯ q) (Quotient.lift PadicSeq.norm ⋯ r)
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ_[p]
⊢ max (Quotient.lift PadicSeq.norm ⋯ q) (Quotient.lift PadicSeq.norm ⋯ r) ≤
    Quotient.lift PadicSeq.norm ⋯ q + Quotient.lift PadicSeq.norm ⋯ r

Tactic:
·

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ_[p]
⊢ max (Quotient.lift PadicSeq.norm ⋯ q) (Quotient.lift PadicSeq.norm ⋯ r) ≤
    Quotient.lift PadicSeq.norm ⋯ q + Quotient.lift PadicSeq.norm ⋯ r
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case ha
p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ_[p]
⊢ 0 ≤ Quotient.lift PadicSeq.norm ⋯ q

pp: case hb
p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ_[p]
⊢ 0 ≤ Quotient.lift PadicSeq.norm ⋯ r

Tactic:
·

Goal state after:
case ha
p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ_[p]
⊢ 0 ≤ Quotient.lift PadicSeq.norm ⋯ q
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case hb
p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ_[p]
⊢ 0 ≤ Quotient.lift PadicSeq.norm ⋯ r

Tactic:
·

Goal state after:
case hb
p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ_[p]
⊢ 0 ≤ Quotient.lift PadicSeq.norm ⋯ r
Module: Mathlib.CategoryTheory.MorphismProperty.Representable

Goal state before:
pp: case refine_1
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
X Y Y' X' : D
f : X ⟶ X'
g : Y ⟶ X'
f' : Y' ⟶ Y
g' : Y' ⟶ X
P₁ : IsPullback f' g' g f
hg : F.relativelyRepresentable g
a : C
h : F.obj a ⟶ X
⊢ F.obj (hg.pullback (h ≫ f)) ⟶ Y'

pp: case refine_2
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
X Y Y' X' : D
f : X ⟶ X'
g : Y ⟶ X'
f' : Y' ⟶ Y
g' : Y' ⟶ X
P₁ : IsPullback f' g' g f
hg : F.relativelyRepresentable g
a : C
h : F.obj a ⟶ X
⊢ IsPullback ?refine_1 (F.map (hg.snd (h ≫ f))) g' h

Tactic:
·

Goal state after:
case refine_1
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
X Y Y' X' : D
f : X ⟶ X'
g : Y ⟶ X'
f' : Y' ⟶ Y
g' : Y' ⟶ X
P₁ : IsPullback f' g' g f
hg : F.relativelyRepresentable g
a : C
h : F.obj a ⟶ X
⊢ F.obj (hg.pullback (h ≫ f)) ⟶ Y'
Module: Mathlib.CategoryTheory.MorphismProperty.Representable

Goal state before:
pp: case refine_2
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
X Y Y' X' : D
f : X ⟶ X'
g : Y ⟶ X'
f' : Y' ⟶ Y
g' : Y' ⟶ X
P₁ : IsPullback f' g' g f
hg : F.relativelyRepresentable g
a : C
h : F.obj a ⟶ X
⊢ IsPullback (P₁.lift (hg.fst (h ≫ f)) (F.map (hg.snd (h ≫ f)) ≫ h) ⋯) (F.map (hg.snd (h ≫ f))) g' h

Tactic:
·

Goal state after:
case refine_2
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
X Y Y' X' : D
f : X ⟶ X'
g : Y ⟶ X'
f' : Y' ⟶ Y
g' : Y' ⟶ X
P₁ : IsPullback f' g' g f
hg : F.relativelyRepresentable g
a : C
h : F.obj a ⟶ X
⊢ IsPullback (P₁.lift (hg.fst (h ≫ f)) (F.map (hg.snd (h ≫ f)) ≫ h) ⋯) (F.map (hg.snd (h ≫ f))) g' h
Module: Mathlib.CategoryTheory.MorphismProperty.Representable

Goal state before:
pp: case refine_1
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F : C ⥤ D
X Y : D
P : MorphismProperty C
inst✝² : F.Faithful
inst✝¹ : F.Full
inst✝ : P.RespectsIso
f : X ⟶ Y
h₀ : ∀ ⦃a : C⦄ (g : F.obj a ⟶ Y), ∃ b fst snd, ∃ (_ : IsPullback fst (F.map snd) f g), P snd
a : C
g : F.obj a ⟶ Y
c : C
g_fst : F.obj c ⟶ X
g_snd : c ⟶ a
BC : IsPullback g_fst (F.map g_snd) f g
H : P g_snd
⊢ ∃ b snd fst, IsPullback fst (F.map snd) f g

pp: case refine_2
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F : C ⥤ D
X Y : D
P : MorphismProperty C
inst✝² : F.Faithful
inst✝¹ : F.Full
inst✝ : P.RespectsIso
f : X ⟶ Y
h₀ : ∀ ⦃a : C⦄ (g : F.obj a ⟶ Y), ∃ b fst snd, ∃ (_ : IsPullback fst (F.map snd) f g), P snd
a b : C
g : F.obj a ⟶ Y
fst : F.obj b ⟶ X
snd : b ⟶ a
h : IsPullback fst (F.map snd) f g
c : C
g_fst : F.obj c ⟶ X
g_snd : c ⟶ a
BC : IsPullback g_fst (F.map g_snd) f g
H : P g_snd
⊢ P snd

Tactic:
·

Goal state after:
case refine_1
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F : C ⥤ D
X Y : D
P : MorphismProperty C
inst✝² : F.Faithful
inst✝¹ : F.Full
inst✝ : P.RespectsIso
f : X ⟶ Y
h₀ : ∀ ⦃a : C⦄ (g : F.obj a ⟶ Y), ∃ b fst snd, ∃ (_ : IsPullback fst (F.map snd) f g), P snd
a : C
g : F.obj a ⟶ Y
c : C
g_fst : F.obj c ⟶ X
g_snd : c ⟶ a
BC : IsPullback g_fst (F.map g_snd) f g
H : P g_snd
⊢ ∃ b snd fst, IsPullback fst (F.map snd) f g
Module: Mathlib.AlgebraicGeometry.Sites.EtalePoint

Goal state before:
pp: S : Scheme
⊢ ⋃ i,
      Set.range
        ⇑((SpecToEquivOfField (SeparableClosure ↑(S.residueField i)) S).invFun
            ⟨i, CommRingCat.ofHom (algebraMap (↑(S.residueField i)) (SeparableClosure ↑(S.residueField i)))⟩) =
    Set.univ

Tactic:
by
    ext s
    simp only [Equiv.invFun_as_coe, Set.mem_iUnion, Set.mem_range, Set.mem_univ, iff_true]
    exact ⟨s, default, by simp [SpecToEquivOfField]⟩

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

Goal state before:
pp: case refine_2
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F : C ⥤ D
X Y : D
P : MorphismProperty C
inst✝² : F.Faithful
inst✝¹ : F.Full
inst✝ : P.RespectsIso
f : X ⟶ Y
h₀ : ∀ ⦃a : C⦄ (g : F.obj a ⟶ Y), ∃ b fst snd, ∃ (_ : IsPullback fst (F.map snd) f g), P snd
a b : C
g : F.obj a ⟶ Y
fst : F.obj b ⟶ X
snd : b ⟶ a
h : IsPullback fst (F.map snd) f g
c : C
g_fst : F.obj c ⟶ X
g_snd : c ⟶ a
BC : IsPullback g_fst (F.map g_snd) f g
H : P g_snd
⊢ P snd

Tactic:
·

Goal state after:
case refine_2
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F : C ⥤ D
X Y : D
P : MorphismProperty C
inst✝² : F.Faithful
inst✝¹ : F.Full
inst✝ : P.RespectsIso
f : X ⟶ Y
h₀ : ∀ ⦃a : C⦄ (g : F.obj a ⟶ Y), ∃ b fst snd, ∃ (_ : IsPullback fst (F.map snd) f g), P snd
a b : C
g : F.obj a ⟶ Y
fst : F.obj b ⟶ X
snd : b ⟶ a
h : IsPullback fst (F.map snd) f g
c : C
g_fst : F.obj c ⟶ X
g_snd : c ⟶ a
BC : IsPullback g_fst (F.map g_snd) f g
H : P g_snd
⊢ P snd
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case inl
p : ℕ
inst✝ : Fact (Nat.Prime p)
q : ℚ_[p]
ε : ℚ
hε : 0 < ε
q' : CauSeq ℚ (padicNorm p)
this : ∃ N, ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
N : ℕ
hN : ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
heq : q' - const (padicNorm p) (↑q' N) ≈ 0
⊢ (q' - const (padicNorm p) (↑q' N)).norm < ε

pp: case inr
p : ℕ
inst✝ : Fact (Nat.Prime p)
q : ℚ_[p]
ε : ℚ
hε : 0 < ε
q' : CauSeq ℚ (padicNorm p)
this : ∃ N, ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
N : ℕ
hN : ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
hne' : ¬q' - const (padicNorm p) (↑q' N) ≈ 0
⊢ (q' - const (padicNorm p) (↑q' N)).norm < ε

Tactic:
·

Goal state after:
case inl
p : ℕ
inst✝ : Fact (Nat.Prime p)
q : ℚ_[p]
ε : ℚ
hε : 0 < ε
q' : CauSeq ℚ (padicNorm p)
this : ∃ N, ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
N : ℕ
hN : ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
heq : q' - const (padicNorm p) (↑q' N) ≈ 0
⊢ (q' - const (padicNorm p) (↑q' N)).norm < ε
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case inr
p : ℕ
inst✝ : Fact (Nat.Prime p)
q : ℚ_[p]
ε : ℚ
hε : 0 < ε
q' : CauSeq ℚ (padicNorm p)
this : ∃ N, ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
N : ℕ
hN : ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
hne' : ¬q' - const (padicNorm p) (↑q' N) ≈ 0
⊢ (q' - const (padicNorm p) (↑q' N)).norm < ε

Tactic:
·

Goal state after:
case inr
p : ℕ
inst✝ : Fact (Nat.Prime p)
q : ℚ_[p]
ε : ℚ
hε : 0 < ε
q' : CauSeq ℚ (padicNorm p)
this : ∃ N, ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
N : ℕ
hN : ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
hne' : ¬q' - const (padicNorm p) (↑q' N) ≈ 0
⊢ (q' - const (padicNorm p) (↑q' N)).norm < ε
Module: Mathlib.AlgebraicGeometry.Sites.Representability

Goal state before:
pp: F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i : ι
⊢ (ConcreteCategory.hom ((yonedaGluedToSheaf hf).hom.app (op (X i)))) (toGlued hf i) = yonedaEquiv (f i)

Tactic:
by
  rw [← yoneda_toGlued_yonedaGluedToSheaf hf i, yonedaEquiv_comp,
    yonedaEquiv_yoneda_map]

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case inr.inl
p : ℕ
inst✝ : Fact (Nat.Prime p)
q : ℚ_[p]
ε : ℚ
hε : 0 < ε
q' : CauSeq ℚ (padicNorm p)
this : ∃ N, ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
N : ℕ
hN : ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
hne' : ¬q' - const (padicNorm p) (↑q' N) ≈ 0
hle : stationaryPoint hne' ≤ N
⊢ padicNorm p (↑q' (stationaryPoint hne') - ↑q' N) < ε

pp: case inr.inr
p : ℕ
inst✝ : Fact (Nat.Prime p)
q : ℚ_[p]
ε : ℚ
hε : 0 < ε
q' : CauSeq ℚ (padicNorm p)
this : ∃ N, ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
N : ℕ
hN : ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
hne' : ¬q' - const (padicNorm p) (↑q' N) ≈ 0
hle : ¬stationaryPoint hne' ≤ N
⊢ padicNorm p (↑q' (stationaryPoint hne') - ↑q' N) < ε

Tactic:
·

Goal state after:
case inr.inl
p : ℕ
inst✝ : Fact (Nat.Prime p)
q : ℚ_[p]
ε : ℚ
hε : 0 < ε
q' : CauSeq ℚ (padicNorm p)
this : ∃ N, ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
N : ℕ
hN : ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
hne' : ¬q' - const (padicNorm p) (↑q' N) ≈ 0
hle : stationaryPoint hne' ≤ N
⊢ padicNorm p (↑q' (stationaryPoint hne') - ↑q' N) < ε
Module: Mathlib.CategoryTheory.MorphismProperty.Representable

Goal state before:
pp: case refine_1
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F✝ : C ⥤ D
X✝ Y : D
P : MorphismProperty C
inst✝² : F✝.Faithful
inst✝¹ : F✝.Full
inst✝ : P.IsStableUnderComposition
F G H : D
f : F ⟶ G
g : G ⟶ H
hf : relative F✝ P f
hg : relative F✝ P g
Z X : C
p : F✝.obj Z ⟶ H
fst : F✝.obj X ⟶ F
snd : X ⟶ Z
h : IsPullback fst (F✝.map snd) (f ≫ g) p
⊢ IsPullback fst (F✝.map (⋯.lift (fst ≫ f) snd ⋯) ≫ F✝.map (⋯.snd p)) (f ≫ g) p

pp: case refine_2
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F✝ : C ⥤ D
X✝ Y : D
P : MorphismProperty C
inst✝² : F✝.Faithful
inst✝¹ : F✝.Full
inst✝ : P.IsStableUnderComposition
F G H : D
f : F ⟶ G
g : G ⟶ H
hf : relative F✝ P f
hg : relative F✝ P g
Z X : C
p : F✝.obj Z ⟶ H
fst : F✝.obj X ⟶ F
snd : X ⟶ Z
h : IsPullback fst (F✝.map snd) (f ≫ g) p
⊢ fst ≫ f = F✝.map (⋯.lift (fst ≫ f) snd ⋯) ≫ ⋯.fst p

Tactic:
·

Goal state after:
case refine_1
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F✝ : C ⥤ D
X✝ Y : D
P : MorphismProperty C
inst✝² : F✝.Faithful
inst✝¹ : F✝.Full
inst✝ : P.IsStableUnderComposition
F G H : D
f : F ⟶ G
g : G ⟶ H
hf : relative F✝ P f
hg : relative F✝ P g
Z X : C
p : F✝.obj Z ⟶ H
fst : F✝.obj X ⟶ F
snd : X ⟶ Z
h : IsPullback fst (F✝.map snd) (f ≫ g) p
⊢ IsPullback fst (F✝.map (⋯.lift (fst ≫ f) snd ⋯) ≫ F✝.map (⋯.snd p)) (f ≫ g) p
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case inr.inr
p : ℕ
inst✝ : Fact (Nat.Prime p)
q : ℚ_[p]
ε : ℚ
hε : 0 < ε
q' : CauSeq ℚ (padicNorm p)
this : ∃ N, ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
N : ℕ
hN : ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
hne' : ¬q' - const (padicNorm p) (↑q' N) ≈ 0
hle : ¬stationaryPoint hne' ≤ N
⊢ padicNorm p (↑q' (stationaryPoint hne') - ↑q' N) < ε

Tactic:
·

Goal state after:
case inr.inr
p : ℕ
inst✝ : Fact (Nat.Prime p)
q : ℚ_[p]
ε : ℚ
hε : 0 < ε
q' : CauSeq ℚ (padicNorm p)
this : ∃ N, ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
N : ℕ
hN : ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
hne' : ¬q' - const (padicNorm p) (↑q' N) ≈ 0
hle : ¬stationaryPoint hne' ≤ N
⊢ padicNorm p (↑q' (stationaryPoint hne') - ↑q' N) < ε