mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Condensed.Epi

Goal state before:
pp: A : Type u'
inst✝⁹ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type v'
inst✝⁸ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝⁷ : ConcreteCategory A FA
inst✝⁶ : HasFunctorialSurjectiveInjectiveFactorization A
X Y : Condensed A
f : X ⟶ Y
inst✝⁵ : PreservesFiniteProducts (CategoryTheory.forget A)
inst✝⁴ : ∀ (X : CompHausᵒᵖ), HasLimitsOfShape (StructuredArrow X Stonean.toCompHaus.op) A
inst✝³ : (extensiveTopology Stonean).WEqualsLocallyBijective A
inst✝² : HasSheafify (extensiveTopology Stonean) A
inst✝¹ : (extensiveTopology Stonean).HasSheafCompose (CategoryTheory.forget A)
inst✝ : Balanced (Sheaf (extensiveTopology Stonean) A)
⊢ IsLocallySurjective
      (Presheaf.coherentExtensiveEquivalence.functor.map ((StoneanCompHaus.equivalence A).inverse.map f)) ↔
    ∀ (S : Stonean), Function.Surjective ⇑(hom (f.hom.app (op S.compHaus)))

Tactic:
]

Goal state after:
A : Type u'
inst✝⁹ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type v'
inst✝⁸ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝⁷ : ConcreteCategory A FA
inst✝⁶ : HasFunctorialSurjectiveInjectiveFactorization A
X Y : Condensed A
f : X ⟶ Y
inst✝⁵ : PreservesFiniteProducts (CategoryTheory.forget A)
inst✝⁴ : ∀ (X : CompHausᵒᵖ), HasLimitsOfShape (StructuredArrow X Stonean.toCompHaus.op) A
inst✝³ : (extensiveTopology Stonean).WEqualsLocallyBijective A
inst✝² : HasSheafify (extensiveTopology Stonean) A
inst✝¹ : (extensiveTopology Stonean).HasSheafCompose (CategoryTheory.forget A)
inst✝ : Balanced (Sheaf (extensiveTopology Stonean) A)
⊢ IsLocallySurjective
      (Presheaf.coherentExtensiveEquivalence.functor.map ((StoneanCompHaus.equivalence A).inverse.map f)) ↔
    ∀ (S : Stonean), Function.Surjective ⇑(hom (f.hom.app (op S.compHaus)))
Module: Mathlib.Condensed.Discrete.LocallyConstant

Goal state before:
pp: P : TopCat → Prop
inst✝⁴ : ∀ (S : CompHausLike P) (p : ↑S.toTop → Prop), HasProp P (Subtype p)
S : CompHausLike P
Y : (CompHausLike P)ᵒᵖ ⥤ Type (max u w)
inst✝³ : HasProp P PUnit.{u + 1}
f : LocallyConstant (↑S.toTop) (Y.obj (op (of P PUnit.{u + 1})))
T : CompHausLike P
g : T ⟶ S
X✝ : TopCat
inst✝² : HasExplicitFiniteCoproducts P
inst✝¹ : HasExplicitPullbacks P
hs : ∀ ⦃X Y : CompHausLike P⦄ (f : X ⟶ Y), EffectiveEpi f → Surjective ⇑(ConcreteCategory.hom f)
inst✝ : HasExplicitFiniteCoproducts P
X : Sheaf (coherentTopology (CompHausLike P)) (Type (max u w))
x : X.obj.obj (op (of P PUnit.{u + 1}))
this✝ : Preregular (CompHausLike P)
this : PreservesFiniteProducts ((sheafToPresheaf (coherentTopology (CompHausLike P)) (Type (max u w))).obj X) :=
  @CompHausLike.LocallyConstant.adjunction._proof_2 P inst✝² inst✝¹ hs X this✝
a : Fiber ⇑((ConcreteCategory.hom ((unit P hs).app (X.obj.obj (op (of P PUnit.{u + 1}))))) x)
⊢ (ConcreteCategory.hom
        (X.obj.map
          (isTerminalPUnit.from
              (fiber ((ConcreteCategory.hom ((unit P hs).app (X.obj.obj (op (of P PUnit.{u + 1}))))) x) a)).op))
      (((ConcreteCategory.hom ((unit P hs).app (X.obj.obj (op (of P PUnit.{u + 1}))))) x) ↑⟨PUnit.unit, ⋯⟩) =
    (ConcreteCategory.hom
        (X.obj.map (sigmaIncl ((ConcreteCategory.hom ((unit P hs).app (X.obj.obj (op (of P PUnit.{u + 1}))))) x) a).op))
      x

Tactic:
]

Goal state after:
P : TopCat → Prop
inst✝⁴ : ∀ (S : CompHausLike P) (p : ↑S.toTop → Prop), HasProp P (Subtype p)
S : CompHausLike P
Y : (CompHausLike P)ᵒᵖ ⥤ Type (max u w)
inst✝³ : HasProp P PUnit.{u + 1}
f : LocallyConstant (↑S.toTop) (Y.obj (op (of P PUnit.{u + 1})))
T : CompHausLike P
g : T ⟶ S
X✝ : TopCat
inst✝² : HasExplicitFiniteCoproducts P
inst✝¹ : HasExplicitPullbacks P
hs : ∀ ⦃X Y : CompHausLike P⦄ (f : X ⟶ Y), EffectiveEpi f → Surjective ⇑(ConcreteCategory.hom f)
inst✝ : HasExplicitFiniteCoproducts P
X : Sheaf (coherentTopology (CompHausLike P)) (Type (max u w))
x : X.obj.obj (op (of P PUnit.{u + 1}))
this✝ : Preregular (CompHausLike P)
this : PreservesFiniteProducts ((sheafToPresheaf (coherentTopology (CompHausLike P)) (Type (max u w))).obj X) :=
  @CompHausLike.LocallyConstant.adjunction._proof_2 P inst✝² inst✝¹ hs X this✝
a : Fiber ⇑((ConcreteCategory.hom ((unit P hs).app (X.obj.obj (op (of P PUnit.{u + 1}))))) x)
⊢ (ConcreteCategory.hom
        (X.obj.map
          (isTerminalPUnit.from
              (fiber ((ConcreteCategory.hom ((unit P hs).app (X.obj.obj (op (of P PUnit.{u + 1}))))) x) a)).op))
      (((ConcreteCategory.hom ((unit P hs).app (X.obj.obj (op (of P PUnit.{u + 1}))))) x) ↑⟨PUnit.unit, ⋯⟩) =
    (ConcreteCategory.hom
        (X.obj.map (sigmaIncl ((ConcreteCategory.hom ((unit P hs).app (X.obj.obj (op (of P PUnit.{u + 1}))))) x) a).op))
      x
Module: Mathlib.Condensed.Discrete.Module

Goal state before:
pp: P : TopCat → Prop
R : Type u
inst✝ : Ring R
X✝ Y✝ : ModuleCat R
f : X✝ ⟶ Y✝
⊢ (functorIsoDiscreteComponents R X✝).hom ≫ (functor R).map f =
    (presheafToSheaf (coherentTopology LightProfinite) (ModuleCat R)).map ((Functor.const LightProfiniteᵒᵖ).map f) ≫
      (functorIsoDiscreteComponents R Y✝).hom

Tactic:
]

Goal state after:
P : TopCat → Prop
R : Type u
inst✝ : Ring R
X✝ Y✝ : ModuleCat R
f : X✝ ⟶ Y✝
⊢ (functorIsoDiscreteComponents R X✝).hom ≫ (functor R).map f =
    (presheafToSheaf (coherentTopology LightProfinite) (ModuleCat R)).map ((Functor.const LightProfiniteᵒᵖ).map f) ≫
      (functorIsoDiscreteComponents R Y✝).hom
Module: Mathlib.Condensed.Explicit

Goal state before:
pp: A : Type u_1
inst✝² : Category.{v_1, u_1} A
inst✝¹ : ∀ (X : CompHausᵒᵖ), HasLimitsOfShape (StructuredArrow X Stonean.toCompHaus.op) A
F : Stoneanᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts F
⊢ PreservesFiniteProducts F

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : Category.{v_1, u_1} A
inst✝¹ : ∀ (X : CompHausᵒᵖ), HasLimitsOfShape (StructuredArrow X Stonean.toCompHaus.op) A
F : Stoneanᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts F
⊢ PreservesFiniteProducts F
Module: Mathlib.Condensed.Explicit

Goal state before:
pp: A : Type u_1
inst✝⁵ : Category.{v_1, u_1} A
inst✝⁴ : ∀ (X : CompHausᵒᵖ), HasLimitsOfShape (StructuredArrow X Stonean.toCompHaus.op) A
FA : A → A → Type u_2
CA : A → Type u_3
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
inst✝¹ : ReflectsFiniteProducts (CategoryTheory.forget A)
F : Stoneanᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts (F ⋙ CategoryTheory.forget A)
⊢ PreservesFiniteProducts (F ⋙ CategoryTheory.forget A)

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁵ : Category.{v_1, u_1} A
inst✝⁴ : ∀ (X : CompHausᵒᵖ), HasLimitsOfShape (StructuredArrow X Stonean.toCompHaus.op) A
FA : A → A → Type u_2
CA : A → Type u_3
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
inst✝¹ : ReflectsFiniteProducts (CategoryTheory.forget A)
F : Stoneanᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts (F ⋙ CategoryTheory.forget A)
⊢ PreservesFiniteProducts (F ⋙ CategoryTheory.forget A)
Module: Mathlib.Condensed.Explicit

Goal state before:
pp: A : Type u_1
inst✝² : Category.{v_1, u_1} A
inst✝¹ : ∀ (X : CompHausᵒᵖ), HasLimitsOfShape (StructuredArrow X profiniteToCompHaus.op) A
F : Profiniteᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts F
hF : EqualizerCondition F
⊢ PreservesFiniteProducts F ∧ EqualizerCondition F

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : Category.{v_1, u_1} A
inst✝¹ : ∀ (X : CompHausᵒᵖ), HasLimitsOfShape (StructuredArrow X profiniteToCompHaus.op) A
F : Profiniteᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts F
hF : EqualizerCondition F
⊢ PreservesFiniteProducts F ∧ EqualizerCondition F
Module: Mathlib.Condensed.Explicit

Goal state before:
pp: A : Type u_1
inst✝⁵ : Category.{v_1, u_1} A
inst✝⁴ : ∀ (X : CompHausᵒᵖ), HasLimitsOfShape (StructuredArrow X profiniteToCompHaus.op) A
FA : A → A → Type u_2
CA : A → Type u_3
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
inst✝¹ : ReflectsFiniteLimits (CategoryTheory.forget A)
F : Profiniteᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts (F ⋙ CategoryTheory.forget A)
hF : EqualizerCondition (F ⋙ CategoryTheory.forget A)
⊢ PreservesFiniteProducts (F ⋙ CategoryTheory.forget A) ∧ EqualizerCondition (F ⋙ CategoryTheory.forget A)

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁵ : Category.{v_1, u_1} A
inst✝⁴ : ∀ (X : CompHausᵒᵖ), HasLimitsOfShape (StructuredArrow X profiniteToCompHaus.op) A
FA : A → A → Type u_2
CA : A → Type u_3
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
inst✝¹ : ReflectsFiniteLimits (CategoryTheory.forget A)
F : Profiniteᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts (F ⋙ CategoryTheory.forget A)
hF : EqualizerCondition (F ⋙ CategoryTheory.forget A)
⊢ PreservesFiniteProducts (F ⋙ CategoryTheory.forget A) ∧ EqualizerCondition (F ⋙ CategoryTheory.forget A)
Module: Mathlib.Condensed.Explicit

Goal state before:
pp: A : Type u_1
inst✝¹ : Category.{v_1, u_1} A
F : CompHausᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts F
hF : EqualizerCondition F
⊢ PreservesFiniteProducts F ∧ EqualizerCondition F

Tactic:
]

Goal state after:
A : Type u_1
inst✝¹ : Category.{v_1, u_1} A
F : CompHausᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts F
hF : EqualizerCondition F
⊢ PreservesFiniteProducts F ∧ EqualizerCondition F
Module: Mathlib.Condensed.Discrete.Module

Goal state before:
pp: P : TopCat → Prop
R : Type u
inst✝ : Ring R
X✝ Y✝ : ModuleCat R
f : X✝ ⟶ Y✝
⊢ (discrete (ModuleCat R)).map ((underlying (ModuleCat R)).map ((functor R).map f)) ≫
      (discreteUnderlyingAdj (ModuleCat R)).counit.app ((functor R).obj Y✝) =
    (functorIsoDiscreteAux₂ R X✝).inv ≫
      (presheafToSheaf (coherentTopology LightProfinite) (ModuleCat R)).map ((Functor.const LightProfiniteᵒᵖ).map f) ≫
        (functorIsoDiscreteAux₂ R Y✝).hom ≫ (discreteUnderlyingAdj (ModuleCat R)).counit.app ((functor R).obj Y✝)

Tactic:
]

Goal state after:
P : TopCat → Prop
R : Type u
inst✝ : Ring R
X✝ Y✝ : ModuleCat R
f : X✝ ⟶ Y✝
⊢ (discrete (ModuleCat R)).map ((underlying (ModuleCat R)).map ((functor R).map f)) ≫
      (discreteUnderlyingAdj (ModuleCat R)).counit.app ((functor R).obj Y✝) =
    (functorIsoDiscreteAux₂ R X✝).inv ≫
      (presheafToSheaf (coherentTopology LightProfinite) (ModuleCat R)).map ((Functor.const LightProfiniteᵒᵖ).map f) ≫
        (functorIsoDiscreteAux₂ R Y✝).hom ≫ (discreteUnderlyingAdj (ModuleCat R)).counit.app ((functor R).obj Y✝)
Module: Mathlib.Condensed.EffectiveEpi

Goal state before:
pp: X✝ Y✝ : CompHaus
f : X✝ ⟶ Y✝
hf : Epi f
⊢ ∀ (S : CompHaus) (y : (compHausToCondensed.obj Y✝).obj.obj (Opposite.op S)),
    ∃ S' φ,
      ∃ (_ : Function.Surjective ⇑(ConcreteCategory.hom φ)),
        ∃ x,
          (ConcreteCategory.hom ((compHausToCondensed.map f).hom.app (Opposite.op S'))) x =
            (ConcreteCategory.hom ((compHausToCondensed.obj Y✝).obj.map (Opposite.op φ))) y

Tactic:
]

Goal state after:
X✝ Y✝ : CompHaus
f : X✝ ⟶ Y✝
hf : Epi f
⊢ ∀ (S : CompHaus) (y : (compHausToCondensed.obj Y✝).obj.obj (Opposite.op S)),
    ∃ S' φ,
      ∃ (_ : Function.Surjective ⇑(ConcreteCategory.hom φ)),
        ∃ x,
          (ConcreteCategory.hom ((compHausToCondensed.map f).hom.app (Opposite.op S'))) x =
            (ConcreteCategory.hom ((compHausToCondensed.obj Y✝).obj.map (Opposite.op φ))) y
Module: Mathlib.Condensed.EffectiveEpi

Goal state before:
pp: X✝ Y✝ : CompHaus
f : X✝ ⟶ Y✝
hf : Function.Surjective ⇑(ConcreteCategory.hom f)
S : CompHaus
g : (compHausToCondensed.obj Y✝).obj.obj (Opposite.op S)
y : ↑S.toTop
⊢ ∃ a, (ConcreteCategory.hom (pullback.snd f g.down)) a = y

Tactic:
]

Goal state after:
X✝ Y✝ : CompHaus
f : X✝ ⟶ Y✝
hf : Function.Surjective ⇑(ConcreteCategory.hom f)
S : CompHaus
g : (compHausToCondensed.obj Y✝).obj.obj (Opposite.op S)
y : ↑S.toTop
⊢ ∃ a, (ConcreteCategory.hom (pullback.snd f g.down)) a = y
Module: Mathlib.Condensed.Explicit

Goal state before:
pp: A : Type u_1
inst✝⁴ : Category.{v_1, u_1} A
FA : A → A → Type u_2
CA : A → Type u_3
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
inst✝¹ : ReflectsFiniteLimits (CategoryTheory.forget A)
F : CompHausᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts (F ⋙ CategoryTheory.forget A)
hF : EqualizerCondition (F ⋙ CategoryTheory.forget A)
⊢ PreservesFiniteProducts (F ⋙ CategoryTheory.forget A) ∧ EqualizerCondition (F ⋙ CategoryTheory.forget A)

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁴ : Category.{v_1, u_1} A
FA : A → A → Type u_2
CA : A → Type u_3
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
inst✝¹ : ReflectsFiniteLimits (CategoryTheory.forget A)
F : CompHausᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts (F ⋙ CategoryTheory.forget A)
hF : EqualizerCondition (F ⋙ CategoryTheory.forget A)
⊢ PreservesFiniteProducts (F ⋙ CategoryTheory.forget A) ∧ EqualizerCondition (F ⋙ CategoryTheory.forget A)
Module: Mathlib.Condensed.Discrete.Module

Goal state before:
pp: case e_a
P : TopCat → Prop
R : Type u
inst✝ : Ring R
X✝ Y✝ : ModuleCat R
f : X✝ ⟶ Y✝
⊢ (discrete (ModuleCat R)).map ((underlying (ModuleCat R)).map ((functor R).map f)) ≫
      (functorIsoDiscreteAux₂ R Y✝).inv =
    (functorIsoDiscreteAux₂ R X✝).inv ≫
      (presheafToSheaf (coherentTopology LightProfinite) (ModuleCat R)).map ((Functor.const LightProfiniteᵒᵖ).map f)

Tactic:
]

Goal state after:
case e_a
P : TopCat → Prop
R : Type u
inst✝ : Ring R
X✝ Y✝ : ModuleCat R
f : X✝ ⟶ Y✝
⊢ (discrete (ModuleCat R)).map ((underlying (ModuleCat R)).map ((functor R).map f)) ≫
      (functorIsoDiscreteAux₂ R Y✝).inv =
    (functorIsoDiscreteAux₂ R X✝).inv ≫
      (presheafToSheaf (coherentTopology LightProfinite) (ModuleCat R)).map ((Functor.const LightProfiniteᵒᵖ).map f)
Module: Mathlib.Topology.Category.LightProfinite.Sequence

Goal state before:
pp: case refine_1.some.some
n m : ℕ
h : n = m
⊢ Option.some m = Option.some m

Tactic:
]

Goal state after:
Module: Mathlib.Condensed.Discrete.Colimit

Goal state before:
pp: case refine_2
I : Type u
inst✝² : Category.{u, u} I
inst✝¹ : IsCofiltered I
F : I ⥤ FintypeCat
c : Cone (F ⋙ toProfinite)
X : Type (u + 1)
hc : IsLimit c
inst✝ : ∀ (i : I), Epi (c.π.app i)
i j : I
fi : LocallyConstant (↑((F ⋙ toProfinite).obj (Opposite.unop (Opposite.op i))).toTop) X
fj : LocallyConstant (↑((F ⋙ toProfinite).obj (Opposite.unop (Opposite.op j))).toTop) X
h :
  LocallyConstant.comap (TopCat.Hom.hom (c.π.app i).hom) fi = LocallyConstant.comap (TopCat.Hom.hom (c.π.app j).hom) fj
k : I
ki : k ⟶ i
kj : k ⟶ j
h✝ : True
x✝ : ↑(toProfinite.obj (F.obj k)).toTop
x : ↑(((const I).obj c.pt).obj k).toTop
hx : (ConcreteCategory.hom (c.π.app k)) x = x✝
⊢ ({ toFun := fun g ↦ LocallyConstant.comap (TopCat.Hom.hom (toProfinite.map (F.map ki)).hom) g } fi)
      ((ConcreteCategory.hom (c.π.app k)) x) =
    ({ toFun := fun g ↦ LocallyConstant.comap (TopCat.Hom.hom (toProfinite.map (F.map kj)).hom) g } fj)
      ((ConcreteCategory.hom (c.π.app k)) x)

Tactic:
]

Goal state after:
case refine_2
I : Type u
inst✝² : Category.{u, u} I
inst✝¹ : IsCofiltered I
F : I ⥤ FintypeCat
c : Cone (F ⋙ toProfinite)
X : Type (u + 1)
hc : IsLimit c
inst✝ : ∀ (i : I), Epi (c.π.app i)
i j : I
fi : LocallyConstant (↑((F ⋙ toProfinite).obj (Opposite.unop (Opposite.op i))).toTop) X
fj : LocallyConstant (↑((F ⋙ toProfinite).obj (Opposite.unop (Opposite.op j))).toTop) X
h :
  LocallyConstant.comap (TopCat.Hom.hom (c.π.app i).hom) fi = LocallyConstant.comap (TopCat.Hom.hom (c.π.app j).hom) fj
k : I
ki : k ⟶ i
kj : k ⟶ j
h✝ : True
x✝ : ↑(toProfinite.obj (F.obj k)).toTop
x : ↑(((const I).obj c.pt).obj k).toTop
hx : (ConcreteCategory.hom (c.π.app k)) x = x✝
⊢ ({ toFun := fun g ↦ LocallyConstant.comap (TopCat.Hom.hom (toProfinite.map (F.map ki)).hom) g } fi)
      ((ConcreteCategory.hom (c.π.app k)) x) =
    ({ toFun := fun g ↦ LocallyConstant.comap (TopCat.Hom.hom (toProfinite.map (F.map kj)).hom) g } fj)
      ((ConcreteCategory.hom (c.π.app k)) x)
Module: Mathlib.Condensed.Discrete.Colimit

Goal state before:
pp: case refine_2
I : Type u
inst✝² : Category.{u, u} I
inst✝¹ : IsCofiltered I
F : I ⥤ FintypeCat
c : Cone (F ⋙ toProfinite)
X : Type (u + 1)
hc : IsLimit c
inst✝ : ∀ (i : I), Epi (c.π.app i)
i j : I
fi : LocallyConstant (↑((F ⋙ toProfinite).obj (Opposite.unop (Opposite.op i))).toTop) X
fj : LocallyConstant (↑((F ⋙ toProfinite).obj (Opposite.unop (Opposite.op j))).toTop) X
h✝¹ :
  LocallyConstant.comap (TopCat.Hom.hom (c.π.app i).hom) fi = LocallyConstant.comap (TopCat.Hom.hom (c.π.app j).hom) fj
k : I
ki : k ⟶ i
kj : k ⟶ j
h✝ : True
x✝ : ↑(toProfinite.obj (F.obj k)).toTop
x : ↑(((const I).obj c.pt).obj k).toTop
hx : (ConcreteCategory.hom (c.π.app k)) x = x✝
h :
  (LocallyConstant.comap (TopCat.Hom.hom (c.π.app i).hom) fi) x =
    (LocallyConstant.comap (TopCat.Hom.hom (c.π.app j).hom) fj) x
⊢ fi ((ConcreteCategory.hom (c.π.app i)) x) = fj ((ConcreteCategory.hom (c.π.app j)) x)

Tactic:
]

Goal state after:
case refine_2
I : Type u
inst✝² : Category.{u, u} I
inst✝¹ : IsCofiltered I
F : I ⥤ FintypeCat
c : Cone (F ⋙ toProfinite)
X : Type (u + 1)
hc : IsLimit c
inst✝ : ∀ (i : I), Epi (c.π.app i)
i j : I
fi : LocallyConstant (↑((F ⋙ toProfinite).obj (Opposite.unop (Opposite.op i))).toTop) X
fj : LocallyConstant (↑((F ⋙ toProfinite).obj (Opposite.unop (Opposite.op j))).toTop) X
h✝¹ :
  LocallyConstant.comap (TopCat.Hom.hom (c.π.app i).hom) fi = LocallyConstant.comap (TopCat.Hom.hom (c.π.app j).hom) fj
k : I
ki : k ⟶ i
kj : k ⟶ j
h✝ : True
x✝ : ↑(toProfinite.obj (F.obj k)).toTop
x : ↑(((const I).obj c.pt).obj k).toTop
hx : (ConcreteCategory.hom (c.π.app k)) x = x✝
h :
  (LocallyConstant.comap (TopCat.Hom.hom (c.π.app i).hom) fi) x =
    (LocallyConstant.comap (TopCat.Hom.hom (c.π.app j).hom) fj) x
⊢ fi ((ConcreteCategory.hom (c.π.app i)) x) = fj ((ConcreteCategory.hom (c.π.app j)) x)
Module: Mathlib.Condensed.Light.TopCatAdjunction

Goal state before:
pp: X : LightCondSet
S : LightProfinite
x : X.obj.obj (Opposite.op S)
⊢ Continuous X.coinducingCoprod

Tactic:
]

Goal state after:
X : LightCondSet
S : LightProfinite
x : X.obj.obj (Opposite.op S)
⊢ Continuous X.coinducingCoprod
Module: Mathlib.Condensed.Light.TopCatAdjunction

Goal state before:
pp: X Y : LightCondSet
f : X ⟶ Y
⊢ Continuous (⇑(ConcreteCategory.hom (f.hom.app (Opposite.op (of PUnit.{u + 1})))) ∘ LightCondSet.coinducingCoprod✝ X)

Tactic:
]

Goal state after:
X Y : LightCondSet
f : X ⟶ Y
⊢ Continuous (⇑(ConcreteCategory.hom (f.hom.app (Opposite.op (of PUnit.{u + 1})))) ∘ LightCondSet.coinducingCoprod✝ X)
Module: Mathlib.Condensed.Light.TopCatAdjunction

Goal state before:
pp: X Y : LightCondSet
f : X ⟶ Y
S : LightProfinite
x : X.obj.obj (Opposite.op S)
⊢ Continuous fun a ↦
    (ConcreteCategory.hom (Y.obj.map (CompHausLike.const (of PUnit.{u + 1}) a).op))
      ((ConcreteCategory.hom (f.hom.app (Opposite.op S))) x)

Tactic:
]

Goal state after:
X Y : LightCondSet
f : X ⟶ Y
S : LightProfinite
x : X.obj.obj (Opposite.op S)
⊢ Continuous fun a ↦
    (ConcreteCategory.hom (Y.obj.map (CompHausLike.const (of PUnit.{u + 1}) a).op))
      ((ConcreteCategory.hom (f.hom.app (Opposite.op S))) x)
Module: Mathlib.Condensed.Discrete.Colimit

Goal state before:
pp: I : Type u
inst✝² : Category.{u, u} I
inst✝¹ : IsCofiltered I
F : I ⥤ FintypeCat
c : Cone (F ⋙ toProfinite)
X : Type (u + 1)
hc : IsLimit c
inst✝ : ∀ (i : I), Epi (c.π.app i)
s : Cocone ((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X)
i : I
f : LocallyConstant (↑(toProfinite.obj (F.obj i)).toTop) X
⊢ (ConcreteCategory.hom (s.ι.app (Opposite.op i))) f = (ConcreteCategory.hom (s.ι.app (Opposite.op i))) f

Tactic:
]

Goal state after:
Module: Mathlib.Condensed.Light.TopCatAdjunction

Goal state before:
pp: X✝ Y : LightCondSet
f : X✝ ⟶ Y
X : TopCat
⊢ Continuous ((fun x ↦ x.toFun PUnit.unit) ∘ LightCondSet.coinducingCoprod✝ X.toLightCondSet)

Tactic:
]

Goal state after:
X✝ Y : LightCondSet
f : X✝ ⟶ Y
X : TopCat
⊢ Continuous ((fun x ↦ x.toFun PUnit.unit) ∘ LightCondSet.coinducingCoprod✝ X.toLightCondSet)
Module: Mathlib.Condensed.Light.TopCatAdjunction

Goal state before:
pp: X✝ Y : LightCondSet
f : X✝ ⟶ Y
X : LightCondSet
S : LightProfiniteᵒᵖ
x : X.obj.obj S
⊢ Continuous (LightCondSet.coinducingCoprod✝ X)

Tactic:
]

Goal state after:
X✝ Y : LightCondSet
f : X✝ ⟶ Y
X : LightCondSet
S : LightProfiniteᵒᵖ
x : X.obj.obj S
⊢ Continuous (LightCondSet.coinducingCoprod✝ X)
Module: Mathlib.Condensed.Light.TopCatAdjunction

Goal state before:
pp: X✝ Y : LightCondSet
f : X✝ ⟶ Y
X : TopCat
⊢ Function.Surjective ⇑(ConcreteCategory.hom (topCatAdjunction.counit.app X))

Tactic:
]

Goal state after:
X✝ Y : LightCondSet
f : X✝ ⟶ Y
X : TopCat
⊢ Function.Surjective ⇑(ConcreteCategory.hom (topCatAdjunction.counit.app X))
Module: Mathlib.Condensed.Light.TopCatAdjunction

Goal state before:
pp: X✝ Y : LightCondSet
f✝ : X✝ ⟶ Y
X : TopCat
inst✝ : SequentialSpace ↑X
f : ℕ → ↑X
p : ↑X
h : Filter.Tendsto f Filter.atTop (nhds p)
g : OnePoint ℕ → ↑X.toLightCondSet.toTopCat :=
  (topCatAdjunctionCounitEquiv X).invFun ∘ ⇑(OnePoint.continuousMapMkNat f p h)
⊢ Continuous g

Tactic:
]

Goal state after:
X✝ Y : LightCondSet
f✝ : X✝ ⟶ Y
X : TopCat
inst✝ : SequentialSpace ↑X
f : ℕ → ↑X
p : ↑X
h : Filter.Tendsto f Filter.atTop (nhds p)
g : OnePoint ℕ → ↑X.toLightCondSet.toTopCat :=
  (topCatAdjunctionCounitEquiv X).invFun ∘ ⇑(OnePoint.continuousMapMkNat f p h)
⊢ Continuous g
Module: Mathlib.Condensed.Light.TopCatAdjunction

Goal state before:
pp: X Y : LightCondSet
f : X ⟶ Y
⊢ ∀ (X : Sequential), IsIso (sequentialAdjunction.counit.app X)

Tactic:
]

Goal state after:
X Y : LightCondSet
f : X ⟶ Y
⊢ ∀ (X : Sequential), IsIso (sequentialAdjunction.counit.app X)
Module: Mathlib.Condensed.Discrete.Colimit

Goal state before:
pp: S : Profinite
F : Profiniteᵒᵖ ⥤ Type (u + 1)
X : Type (u + 1)
⊢ Presheaf.IsSheaf (coherentTopology Profinite) (locallyConstantPresheaf X)

Tactic:
]

Goal state after:
S : Profinite
F : Profiniteᵒᵖ ⥤ Type (u + 1)
X : Type (u + 1)
⊢ Presheaf.IsSheaf (coherentTopology Profinite) (locallyConstantPresheaf X)
Module: Mathlib.Condensed.Discrete.Colimit

Goal state before:
pp: F : Profiniteᵒᵖ ⥤ Type (u + 1)
inst✝² : PreservesFiniteProducts F
X Y : Profinite
inst✝¹ : Finite ↑X.toTop
inst✝ : Finite ↑Y.toTop
f : ↑Y.toTop → F.obj (Opposite.op (Profinite.of PUnit.{u + 1}))
g : X ⟶ Y
x : ↑X.toTop
⊢ (f ∘ ⇑(ConcreteCategory.hom g)) x =
    (ConcreteCategory.hom (F.map (CompHausLike.const (Profinite.of PUnit.{u + 1}) x).op))
      ((ConcreteCategory.hom (F.map g.op)) ((ConcreteCategory.hom (isoFinYonedaComponents F Y).inv) f))

Tactic:
]

Goal state after:
F : Profiniteᵒᵖ ⥤ Type (u + 1)
inst✝² : PreservesFiniteProducts F
X Y : Profinite
inst✝¹ : Finite ↑X.toTop
inst✝ : Finite ↑Y.toTop
f : ↑Y.toTop → F.obj (Opposite.op (Profinite.of PUnit.{u + 1}))
g : X ⟶ Y
x : ↑X.toTop
⊢ (f ∘ ⇑(ConcreteCategory.hom g)) x =
    (ConcreteCategory.hom (F.map (CompHausLike.const (Profinite.of PUnit.{u + 1}) x).op))
      ((ConcreteCategory.hom (F.map g.op)) ((ConcreteCategory.hom (isoFinYonedaComponents F Y).inv) f))
Module: Mathlib.Condensed.Discrete.Colimit

Goal state before:
pp: X : Profiniteᵒᵖ ⥤ Type (u + 1)
inst✝ : PreservesFiniteProducts X
hX : (S : Profinite) → IsColimit (X.mapCocone S.asLimitCone.op)
⊢ (lanPresheafExt (isoFinYoneda X ≪≫ (locallyConstantIsoFinYoneda X).symm)).inv ≫ (lanPresheafNatIso hX).hom =
    (lanPresheafNatIso fun x ↦
          isColimitLocallyConstantPresheafDiagram (X.obj (Opposite.op (Profinite.of PUnit.{u + 1}))) x).hom ≫
      counitApp X

Tactic:
]

Goal state after:
X : Profiniteᵒᵖ ⥤ Type (u + 1)
inst✝ : PreservesFiniteProducts X
hX : (S : Profinite) → IsColimit (X.mapCocone S.asLimitCone.op)
⊢ (lanPresheafExt (isoFinYoneda X ≪≫ (locallyConstantIsoFinYoneda X).symm)).inv ≫ (lanPresheafNatIso hX).hom =
    (lanPresheafNatIso fun x ↦
          isColimitLocallyConstantPresheafDiagram (X.obj (Opposite.op (Profinite.of PUnit.{u + 1}))) x).hom ≫
      counitApp X