awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Galois.Decomposition
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
inst_2 : FiberFunctor F
X : C
t : (F.obj X).obj
rw :
  (@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
            (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
          _ _ _
          (CategoryStruct.comp (Y := ∏ᶜ fun (b : (F.obj X).obj) ↦ F.obj X) (piComparison F fun (x : (F.obj X).obj) ↦ X)
              (Pi.π (fun (b : (F.obj X).obj) ↦ F.obj X) t) :
            F.obj (piObj.{w, u₂, u₁} fun (x : (F.obj X).obj) ↦ X) ⟶ F.obj X) :
        (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (∏ᶜ fun (x : (F.obj X).obj) ↦ X))) →
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
      (mkSelfProdFib F X : (F.obj (selfProd F X)).obj) =
    t
⊢ (@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
            (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
          _ _ _
          (F.map (Pi.π (fun (x : (F.obj X).obj) ↦ X) t) :
            F.obj (piObj.{w, u₂, u₁} fun (x : (F.obj X).obj) ↦ X) ⟶ F.obj X) :
        (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (∏ᶜ fun (x : (F.obj X).obj) ↦ X))) →
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
      (mkSelfProdFib F X : (F.obj (selfProd F X)).obj) =
    t

Tactic:
rw [← piComparison_comp_π]

Premises used: CategoryTheory.Limits.piComparison_comp_π
-- Module: Mathlib.CategoryTheory.Galois.Decomposition
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
inst_2 : FiberFunctor F
X : C
t : (F.obj X).obj
⊢ (@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
            (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
          _ _ _
          (CategoryStruct.comp (Y := ∏ᶜ fun (b : (F.obj X).obj) ↦ F.obj X) (piComparison F fun (x : (F.obj X).obj) ↦ X)
              (Pi.π (fun (b : (F.obj X).obj) ↦ F.obj X) t) :
            F.obj (piObj.{w, u₂, u₁} fun (x : (F.obj X).obj) ↦ X) ⟶ F.obj X) :
        (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (∏ᶜ fun (x : (F.obj X).obj) ↦ X))) →
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
      (mkSelfProdFib F X : (F.obj (selfProd F X)).obj) =
    t

Tactic:
simp [← PreservesProduct.iso_hom, mkSelfProdFib]

Premises used: CategoryTheory.Limits.PreservesProduct.iso_hom, _private.0.CategoryTheory.PreGaloisCategory.mkSelfProdFib
-- Module: Mathlib.CategoryTheory.Galois.Decomposition
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
inst_2 : FiberFunctor F
X A : C
u : A ⟶ selfProd F X
a : (F.obj A).obj
h :
  Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _ (F.map u) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
        a :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    (mkSelfProdFib F X : (F.obj (selfProd F X)).obj)
x : (F.obj X).obj
⊢ (@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
            (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
          _ _ _ (F.map (selfProdProj u x)) :
        (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
      a =
    x

Tactic:
simp_all
-- Module: Mathlib.CategoryTheory.Galois.Decomposition
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
inst_2 : FiberFunctor F
X A : C
u : A ⟶ selfProd F X
a : (F.obj A).obj
h :
  Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _ (F.map u) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
        a :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    (mkSelfProdFib F X : (F.obj (selfProd F X)).obj)
inst_3 : IsConnected.{u₁, u₂} A
b : (F.obj A).obj
σ : (t : (F.obj X).obj) → (F.obj X).obj :=
  fun (t : (F.obj X).obj) ↦
    (@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
            (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
          _ _ _ (F.map (selfProdProj u t)) :
        (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
      b
apply : Function.Bijective σ
⊢ (F.obj X).obj ≃ (F.obj X).obj

Tactic:
apply Equiv.ofBijective σ

Premises used: Equiv.ofBijective
-- Module: Mathlib.CategoryTheory.Galois.Decomposition
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
inst_2 : FiberFunctor F
X A : C
u : A ⟶ selfProd F X
a : (F.obj A).obj
h :
  Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _ (F.map u) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
        a :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    (mkSelfProdFib F X : (F.obj (selfProd F X)).obj)
inst_3 : IsConnected.{u₁, u₂} A
b : (F.obj A).obj
σ : (t : (F.obj X).obj) → (F.obj X).obj :=
  fun (t : (F.obj X).obj) ↦
    (@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
            (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
          _ _ _ (F.map (selfProdProj u t)) :
        (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
      b
apply : Function.Injective σ
⊢ Function.Bijective σ

Tactic:
apply Finite.injective_iff_bijective.mp

Premises used: Finite.injective_iff_bijective, Iff.mp
-- Module: Mathlib.CategoryTheory.Galois.Decomposition
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
inst_2 : FiberFunctor F
X A : C
u : A ⟶ selfProd F X
a : (F.obj A).obj
h :
  Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _ (F.map u) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
        a :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    (mkSelfProdFib F X : (F.obj (selfProd F X)).obj)
inst_3 : IsConnected.{u₁, u₂} A
inst_4 : Mono u
b : (F.obj A).obj
⊢ Mono (selfProdPermIncl h b)

Tactic:
mono_comp _ _

Premises used: CategoryTheory.mono_comp, _private.0.CategoryTheory.PreGaloisCategory.selfProdPermIncl
-- Module: Mathlib.CategoryTheory.Galois.Decomposition
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
inst_2 : FiberFunctor F
X A : C
u : A ⟶ selfProd F X
a : (F.obj A).obj
h :
  Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _ (F.map u) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
        a :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    (mkSelfProdFib F X : (F.obj (selfProd F X)).obj)
inst_3 : IsConnected.{u₁, u₂} A
b : (F.obj A).obj
t : (F.obj X).obj
⊢ Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _ (F.map (selfProdProj u t)) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
        b :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
    ((fiberPerm h b : (a : (F.obj X).obj) → (F.obj X).obj) t : (F.obj X).obj)

Tactic:
rfl
-- Module: Mathlib.CategoryTheory.Galois.Decomposition
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
inst_2 : FiberFunctor F
X A : C
u : A ⟶ selfProd F X
a : (F.obj A).obj
h :
  Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _ (F.map u) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
        a :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    (mkSelfProdFib F X : (F.obj (selfProd F X)).obj)
inst_3 : IsConnected.{u₁, u₂} A
inst_4 : Mono u
b : (F.obj A).obj
⊢ Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _ (F.map u) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
        b :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _ (F.map (selfProdPermIncl h b)) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
        a :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))

Tactic:
selfProdTermIncl_fib_eq h b

Premises used: _private.0.CategoryTheory.PreGaloisCategory.selfProdTermIncl_fib_eq
-- Module: Mathlib.CategoryTheory.Galois.Decomposition
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
inst_2 : FiberFunctor F
X A : C
u : A ⟶ selfProd F X
a : (F.obj A).obj
h :
  Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _ (F.map u) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
        a :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    (mkSelfProdFib F X : (F.obj (selfProd F X)).obj)
inst_3 : IsConnected.{u₁, u₂} A
inst_4 : Mono u
b : (F.obj A).obj
⊢ Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _ (F.map u) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
        b :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _ (F.map (selfProdPermIncl h b)) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
        a :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))

Tactic:
exact selfProdTermIncl_fib_eq h b

Premises used: _private.0.CategoryTheory.PreGaloisCategory.selfProdTermIncl_fib_eq
-- Module: Mathlib.CategoryTheory.Galois.Decomposition
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
inst_2 : FiberFunctor F
X : C
obtain :
  ∀ (A : C) (u : A ⟶ selfProd F X) (a : (F.obj A).obj)
    (h1 :
      Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
        ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
                (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
                (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
                (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
                  (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
                _ _ _ (F.map u) :
              (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
                (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
            a :
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
        (mkSelfProdFib F X : (F.obj (selfProd F X)).obj))
    (h2 : IsConnected.{u₁, u₂} A) (h3 : Mono u),
    ∃ (A : C) (a : (F.obj A).obj),
      IsGalois.{u₁, u₂} A ∧
        Function.Bijective.{u₂ + 1, w + 1} (β := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
          fun (f : A ⟶ X) ↦
          (@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
                (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
                (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
                (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
                  (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
                _ _ _ (F.map f) :
              (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
                (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
            a
⊢ ∃ (A : C) (a : (F.obj A).obj),
    IsGalois.{u₁, u₂} A ∧
      Function.Bijective.{u₂ + 1, w + 1} (β := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
        fun (f : A ⟶ X) ↦
        (@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
              (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
              (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
                (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
              _ _ _ (F.map f) :
            (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
          a

Tactic:
obtain ⟨A, u, a, h1, h2, h3⟩ := fiber_in_connected_component F (selfProd F X)
    (mkSelfProdFib F X)

Premises used: _private.0.CategoryTheory.PreGaloisCategory.selfProd, _private.0.CategoryTheory.PreGaloisCategory.mkSelfProdFib, CategoryTheory.PreGaloisCategory.fiber_in_connected_component
-- Module: Mathlib.CategoryTheory.Galois.Decomposition
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
inst_2 : FiberFunctor F
X A : C
u : A ⟶ selfProd F X
a : (F.obj A).obj
h1 :
  Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _ (F.map u) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
        a :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    (mkSelfProdFib F X : (F.obj (selfProd F X)).obj)
h2 : IsConnected.{u₁, u₂} A
h3 : Mono u
h :
  ∃ (a : (F.obj A).obj),
    IsGalois.{u₁, u₂} A ∧
      Function.Bijective.{u₂ + 1, w + 1} (β := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
        fun (f : A ⟶ X) ↦
        (@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
              (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
              (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
                (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
              _ _ _ (F.map f) :
            (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
          a
⊢ ∃ (A : C) (a : (F.obj A).obj),
    IsGalois.{u₁, u₂} A ∧
      Function.Bijective.{u₂ + 1, w + 1} (β := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
        fun (f : A ⟶ X) ↦
        (@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
              (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
              (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
                (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
              _ _ _ (F.map f) :
            (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
              (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
          a

Tactic:
use A