awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Galois.IsFundamentalgroup (goal)
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
G : Type u_1
inst_1 : Group G
inst_2 : (X : C) → MulAction.{u_1, w} G (F.obj X).obj
inst_3 : IsNaturalSMul F G
inst_4 : GaloisCategory.{u₁, u₂} C
inst_5 : FiberFunctor F
inst_6 : TopologicalSpace G
inst_7 : IsTopologicalGroup G
inst_8 : CompactSpace G
inst_9 : ∀ (X : C), ContinuousSMul.{u_1, w} G (F.obj X).obj
h : ∀ (X : C) [IsGalois.{u₁, u₂} X], MulAction.IsPretransitive.{u_1, w} G (F.obj X).obj
t : Aut F
gi : (X : PointedGaloisObject F) → G
hgi :
  ∀ (X : PointedGaloisObject F) (x : (F.obj X.obj).obj),
    gi X • 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))
            _ _ _ (t.hom.app X.obj : F.obj X.obj ⟶ F.obj X.obj) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj))
        x
cl : (X : PointedGaloisObject F) → Set G :=
  fun (X : PointedGaloisObject F) ↦ gi X • (↑(MulAction.stabilizer G X.pt) : Set G)
c : Set G := ⋂ (i : PointedGaloisObject F), cl i
s : Finset (PointedGaloisObject F)
gs : G
hgs :
  ∀ (i : ↥s) (x : (F.obj (↑i : PointedGaloisObject F).obj).obj),
    gs • 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))
            _ _ _
            (t.hom.app (↑i : PointedGaloisObject F).obj :
              F.obj (↑i : PointedGaloisObject F).obj ⟶ F.obj (↑i : PointedGaloisObject F).obj) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (↑i : PointedGaloisObject F).obj)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (↑i : PointedGaloisObject F).obj))
        x
X : PointedGaloisObject F
hXmem : X ∈ s
h₁ : (gi X)⁻¹ * gs ∈ (↑(MulAction.stabilizer G X.pt) : Set G)
h₂ : Membership.mem (γ := Subgroup G) (MulAction.stabilizer G X.pt) ((gi X)⁻¹ * gs)
h₃ : ((gi X)⁻¹ * gs) • X.pt = X.pt
⊢ gs ∈ cl X
have h₄ : (gi X)⁻¹ • gs • X.pt = X.pt
Tactic: rw [mem_leftCoset_iff, SetLike.mem_coe, MulAction.mem_stabilizer_iff, mul_smul,
        hgs ⟨X, hXmem⟩, ← hgi X, inv_smul_smul]
-- Lean 4 proof step from Mathlib.CategoryTheory.Galois.IsFundamentalgroup (hypothesis)
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
G : Type u_1
inst_1 : Group G
inst_2 : (X : C) → MulAction.{u_1, w} G (F.obj X).obj
inst_3 : IsNaturalSMul F G
inst_4 : GaloisCategory.{u₁, u₂} C
inst_5 : FiberFunctor F
inst_6 : TopologicalSpace G
inst_7 : IsTopologicalGroup G
inst_8 : CompactSpace G
inst_9 : ∀ (X : C), ContinuousSMul.{u_1, w} G (F.obj X).obj
h : ∀ (X : C) [IsGalois.{u₁, u₂} X], MulAction.IsPretransitive.{u_1, w} G (F.obj X).obj
t : Aut F
gi : (X : PointedGaloisObject F) → G
hgi :
  ∀ (X : PointedGaloisObject F) (x : (F.obj X.obj).obj),
    gi X • 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))
            _ _ _ (t.hom.app X.obj : F.obj X.obj ⟶ F.obj X.obj) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj))
        x
cl : (X : PointedGaloisObject F) → Set G :=
  fun (X : PointedGaloisObject F) ↦ gi X • (↑(MulAction.stabilizer G X.pt) : Set G)
c : Set G := ⋂ (i : PointedGaloisObject F), cl i
⊢ ∃ (a : G), (toAut F G : (a : G) → Aut F) a = t
have hne : c.Nonempty
Tactic: have hne : c.Nonempty := by
    rw [← Set.univ_inter c]
    apply CompactSpace.isCompact_univ.inter_iInter_nonempty
    · intro X
      apply IsClosed.leftCoset
      exact Subgroup.isClosed_of_isOpen _ (stabilizer_isOpen G X.pt)
    · intro s
      rw [Set.univ_inter]
      obtain ⟨gs, hgs⟩ :=
        toAut_surjective_isGalois_finite_family F G t (fun X : s ↦ X.val.obj) h
      use gs
      simp only [Set.mem_iInter]
      intro X hXmem
      rw [mem_leftCoset_iff, SetLike.mem_coe, MulAction.mem_stabilizer_iff, mul_smul,
        hgs ⟨X, hXmem⟩, ← hgi X, inv_smul_smul]
-- Lean 4 proof step from Mathlib.CategoryTheory.Galois.IsFundamentalgroup (goal)
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
G : Type u_1
inst_1 : Group G
inst_2 : (X : C) → MulAction.{u_1, w} G (F.obj X).obj
inst_3 : IsNaturalSMul F G
inst_4 : GaloisCategory.{u₁, u₂} C
inst_5 : FiberFunctor F
inst_6 : TopologicalSpace G
inst_7 : IsTopologicalGroup G
inst_8 : CompactSpace G
inst_9 : ∀ (X : C), ContinuousSMul.{u_1, w} G (F.obj X).obj
h : ∀ (X : C) [IsGalois.{u₁, u₂} X], MulAction.IsPretransitive.{u_1, w} G (F.obj X).obj
t : Aut F
gi : (X : PointedGaloisObject F) → G
hgi :
  ∀ (X : PointedGaloisObject F) (x : (F.obj X.obj).obj),
    gi X • 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))
            _ _ _ (t.hom.app X.obj : F.obj X.obj ⟶ F.obj X.obj) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj))
        x
cl : (X : PointedGaloisObject F) → Set G :=
  fun (X : PointedGaloisObject F) ↦ gi X • (↑(MulAction.stabilizer G X.pt) : Set G)
c : Set G := ⋂ (i : PointedGaloisObject F), cl i
hne : c.Nonempty
⊢ ∃ (a : G), (toAut F G : (a : G) → Aut F) a = t
have obtain : ∀ (g : G) (hg : g ∈ c), ∃ (a : G), (toAut F G : (a : G) → Aut F) a = t
Tactic: obtain ⟨g, hg⟩ := hne
Cleared hypotheses: hne
-- Lean 4 proof step from Mathlib.CategoryTheory.Galois.IsFundamentalgroup (goal)
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
G : Type u_1
inst_1 : Group G
inst_2 : (X : C) → MulAction.{u_1, w} G (F.obj X).obj
inst_3 : IsNaturalSMul F G
inst_4 : GaloisCategory.{u₁, u₂} C
inst_5 : FiberFunctor F
inst_6 : TopologicalSpace G
inst_7 : IsTopologicalGroup G
inst_8 : CompactSpace G
inst_9 : ∀ (X : C), ContinuousSMul.{u_1, w} G (F.obj X).obj
h : ∀ (X : C) [IsGalois.{u₁, u₂} X], MulAction.IsPretransitive.{u_1, w} G (F.obj X).obj
t : Aut F
gi : (X : PointedGaloisObject F) → G
hgi :
  ∀ (X : PointedGaloisObject F) (x : (F.obj X.obj).obj),
    gi X • 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))
            _ _ _ (t.hom.app X.obj : F.obj X.obj ⟶ F.obj X.obj) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj))
        x
cl : (X : PointedGaloisObject F) → Set G :=
  fun (X : PointedGaloisObject F) ↦ gi X • (↑(MulAction.stabilizer G X.pt) : Set G)
c : Set G := ⋂ (i : PointedGaloisObject F), cl i
g : G
hg : g ∈ c
⊢ ∃ (a : G), (toAut F G : (a : G) → Aut F) a = t
have refine : ∀ (X : C), IsGalois.{u₁, u₂} X → NatTrans.app (Iso.hom ((toAut F G : (a : G) → Aut F) g)) X = t.hom.app X
Tactic: refine ⟨g, Iso.ext <| natTrans_ext_of_isGalois _ <| fun X _ ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Galois.IsFundamentalgroup (goal)
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
G : Type u_1
inst_1 : Group G
inst_2 : (X : C) → MulAction.{u_1, w} G (F.obj X).obj
inst_3 : IsNaturalSMul F G
inst_4 : GaloisCategory.{u₁, u₂} C
inst_5 : FiberFunctor F
inst_6 : TopologicalSpace G
inst_7 : IsTopologicalGroup G
inst_8 : CompactSpace G
inst_9 : ∀ (X : C), ContinuousSMul.{u_1, w} G (F.obj X).obj
h : ∀ (X : C) [IsGalois.{u₁, u₂} X], MulAction.IsPretransitive.{u_1, w} G (F.obj X).obj
t : Aut F
gi : (X : PointedGaloisObject F) → G
hgi :
  ∀ (X : PointedGaloisObject F) (x : (F.obj X.obj).obj),
    gi X • 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))
            _ _ _ (t.hom.app X.obj : F.obj X.obj ⟶ F.obj X.obj) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj))
        x
cl : (X : PointedGaloisObject F) → Set G :=
  fun (X : PointedGaloisObject F) ↦ gi X • (↑(MulAction.stabilizer G X.pt) : Set G)
c : Set G := ⋂ (i : PointedGaloisObject F), cl i
g : G
hg : g ∈ c
X : C
h_1 : IsGalois.{u₁, u₂} X
⊢ NatTrans.app (Iso.hom ((toAut F G : (a : G) → Aut F) g)) X = t.hom.app X
have h₁ : ∀ (x : (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))
            _ _ _ (NatTrans.app (Iso.hom ((toAut F G : (a : G) → Aut F) g)) X : F.obj X ⟶ F.obj X) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
        x :
      (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))
            _ _ _ (t.hom.app X) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
        x :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
Tactic: ext x
-- Lean 4 proof step from Mathlib.CategoryTheory.Galois.IsFundamentalgroup (hypothesis)
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
G : Type u_1
inst_1 : Group G
inst_2 : (X : C) → MulAction.{u_1, w} G (F.obj X).obj
inst_3 : IsNaturalSMul F G
inst_4 : GaloisCategory.{u₁, u₂} C
inst_5 : FiberFunctor F
inst_6 : TopologicalSpace G
inst_7 : IsTopologicalGroup G
inst_8 : CompactSpace G
inst_9 : ∀ (X : C), ContinuousSMul.{u_1, w} G (F.obj X).obj
h : ∀ (X : C) [IsGalois.{u₁, u₂} X], MulAction.IsPretransitive.{u_1, w} G (F.obj X).obj
t : Aut F
gi : (X : PointedGaloisObject F) → G
hgi :
  ∀ (X : PointedGaloisObject F) (x : (F.obj X.obj).obj),
    gi X • 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))
            _ _ _ (t.hom.app X.obj : F.obj X.obj ⟶ F.obj X.obj) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj))
        x
cl : (X : PointedGaloisObject F) → Set G :=
  fun (X : PointedGaloisObject F) ↦ gi X • (↑(MulAction.stabilizer G X.pt) : Set G)
c : Set G := ⋂ (i : PointedGaloisObject F), cl i
g : G
hg : g ∈ c
X : C
h_1 : IsGalois.{u₁, u₂} X
x : (F.obj X).obj
⊢ g ∈ gi { obj := X, pt := x, isGalois := ⋯ : PointedGaloisObject F } • (↑(MulAction.stabilizer G x) : Set G)
have hg₁ : ∀ (i : PointedGaloisObject F), g ∈ cl i
Tactic: simp only [Set.mem_iInter, c] at hg
Cleared hypotheses: hg
-- Lean 4 proof step from Mathlib.CategoryTheory.Galois.IsFundamentalgroup (hypothesis)
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
G : Type u_1
inst_1 : Group G
inst_2 : (X : C) → MulAction.{u_1, w} G (F.obj X).obj
inst_3 : IsNaturalSMul F G
inst_4 : GaloisCategory.{u₁, u₂} C
inst_5 : FiberFunctor F
inst_6 : TopologicalSpace G
inst_7 : IsTopologicalGroup G
inst_8 : CompactSpace G
inst_9 : ∀ (X : C), ContinuousSMul.{u_1, w} G (F.obj X).obj
h : ∀ (X : C) [IsGalois.{u₁, u₂} X], MulAction.IsPretransitive.{u_1, w} G (F.obj X).obj
t : Aut F
gi : (X : PointedGaloisObject F) → G
hgi :
  ∀ (X : PointedGaloisObject F) (x : (F.obj X.obj).obj),
    gi X • 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))
            _ _ _ (t.hom.app X.obj : F.obj X.obj ⟶ F.obj X.obj) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj))
        x
cl : (X : PointedGaloisObject F) → Set G :=
  fun (X : PointedGaloisObject F) ↦ gi X • (↑(MulAction.stabilizer G X.pt) : Set G)
c : Set G := ⋂ (i : PointedGaloisObject F), cl i
g : G
hg : g ∈ c
X : C
h_1 : IsGalois.{u₁, u₂} X
x : (F.obj X).obj
⊢ g • 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))
          _ _ _ (t.hom.app X) :
        (a : (F.obj X).obj) → (F.obj X).obj)
      x
have this : g ∈ gi { obj := X, pt := x, isGalois := sorry : PointedGaloisObject F } • (↑(MulAction.stabilizer G x) : Set G)
Tactic: have : g ∈ (gi ⟨X, x, inferInstance⟩ • MulAction.stabilizer G x : Set G) := by
    simp only [Set.mem_iInter, c] at hg
    exact hg _
-- Lean 4 proof step from Mathlib.CategoryTheory.Galois.IsFundamentalgroup (goal)
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
G : Type u_1
inst_1 : Group G
inst_2 : (X : C) → MulAction.{u_1, w} G (F.obj X).obj
inst_3 : IsNaturalSMul F G
inst_4 : GaloisCategory.{u₁, u₂} C
inst_5 : FiberFunctor F
inst_6 : TopologicalSpace G
inst_7 : IsTopologicalGroup G
inst_8 : CompactSpace G
inst_9 : ∀ (X : C), ContinuousSMul.{u_1, w} G (F.obj X).obj
h : ∀ (X : C) [IsGalois.{u₁, u₂} X], MulAction.IsPretransitive.{u_1, w} G (F.obj X).obj
t : Aut F
gi : (X : PointedGaloisObject F) → G
hgi :
  ∀ (X : PointedGaloisObject F) (x : (F.obj X.obj).obj),
    gi X • 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))
            _ _ _ (t.hom.app X.obj : F.obj X.obj ⟶ F.obj X.obj) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj))
        x
cl : (X : PointedGaloisObject F) → Set G :=
  fun (X : PointedGaloisObject F) ↦ gi X • (↑(MulAction.stabilizer G X.pt) : Set G)
c : Set G := ⋂ (i : PointedGaloisObject F), cl i
g : G
hg : g ∈ c
X : C
h_1 : IsGalois.{u₁, u₂} X
x : (F.obj X).obj
this : g ∈ gi { obj := X, pt := x, isGalois := ⋯ : PointedGaloisObject F } • (↑(MulAction.stabilizer G x) : Set G)
⊢ g • 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))
          _ _ _ (t.hom.app X) :
        (a : (F.obj X).obj) → (F.obj X).obj)
      x
have h₁ : ∀ (s : G) (hsmem : s • x = x) (hg : gi { obj := X, pt := x, isGalois := sorry : PointedGaloisObject F } • s ∈ c),
  (gi { obj := X, pt := x, isGalois := sorry : PointedGaloisObject F } • s) • 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))
          _ _ _ (t.hom.app X) :
        (a : (F.obj X).obj) → (F.obj X).obj)
      x
Tactic: obtain ⟨s, (hsmem : s • x = x), (rfl : gi ⟨X, x, inferInstance⟩ • s = _)⟩ := this
Cleared hypotheses: g, hg, this
-- Lean 4 proof step from Mathlib.CategoryTheory.Galois.IsFundamentalgroup (goal)
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
G : Type u_1
inst_1 : Group G
inst_2 : (X : C) → MulAction.{u_1, w} G (F.obj X).obj
inst_3 : IsNaturalSMul F G
inst_4 : GaloisCategory.{u₁, u₂} C
inst_5 : FiberFunctor F
inst_6 : TopologicalSpace G
inst_7 : IsTopologicalGroup G
inst_8 : CompactSpace G
inst_9 : ∀ (X : C), ContinuousSMul.{u_1, w} G (F.obj X).obj
h : ∀ (X : C) [IsGalois.{u₁, u₂} X], MulAction.IsPretransitive.{u_1, w} G (F.obj X).obj
t : Aut F
gi : (X : PointedGaloisObject F) → G
hgi :
  ∀ (X : PointedGaloisObject F) (x : (F.obj X.obj).obj),
    gi X • 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))
            _ _ _ (t.hom.app X.obj : F.obj X.obj ⟶ F.obj X.obj) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X.obj))
        x
cl : (X : PointedGaloisObject F) → Set G :=
  fun (X : PointedGaloisObject F) ↦ gi X • (↑(MulAction.stabilizer G X.pt) : Set G)
c : Set G := ⋂ (i : PointedGaloisObject F), cl i
X : C
h_1 : IsGalois.{u₁, u₂} X
x : (F.obj X).obj
s : G
hsmem : s • x = x
hg : gi { obj := X, pt := x, isGalois := ⋯ : PointedGaloisObject F } • s ∈ c
⊢ (gi { obj := X, pt := x, isGalois := ⋯ : PointedGaloisObject F } • s) • 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))
          _ _ _ (t.hom.app X) :
        (a : (F.obj X).obj) → (F.obj X).obj)
      x
have h₁ : gi { obj := X, pt := x, isGalois := sorry : PointedGaloisObject 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))
        _ _ _ (t.hom.app X) :
      (a : (F.obj X).obj) → (F.obj X).obj)
    x
Tactic: rw [smul_eq_mul, mul_smul, hsmem]
-- Lean 4 proof step from Mathlib.CategoryTheory.Galois.IsFundamentalgroup (goal)
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
G : Type u_1
inst_1 : Group G
inst_2 : (X : C) → MulAction.{u_1, w} G (F.obj X).obj
inst_3 : IsNaturalSMul F G
inst_4 : GaloisCategory.{u₁, u₂} C
inst_5 : FiberFunctor F
h : Function.Surjective (α := G) (β := Aut F) (⇑(toAut F G) : (a : G) → Aut F)
X : C
inst_6 : IsConnected.{u₁, u₂} X
x y : (F.obj X).obj
⊢ ∃ (g : G), g • x = y
have obtain : ∀ (t : Aut F) (ht : t • x = y), ∃ (g : G), g • x = y
Tactic: obtain ⟨t, ht⟩ := MulAction.exists_smul_eq (Aut F) x y
-- Lean 4 proof step from Mathlib.CategoryTheory.Galois.IsFundamentalgroup (goal)
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
G : Type u_1
inst_1 : Group G
inst_2 : (X : C) → MulAction.{u_1, w} G (F.obj X).obj
inst_3 : IsNaturalSMul F G
inst_4 : GaloisCategory.{u₁, u₂} C
inst_5 : FiberFunctor F
h : Function.Surjective (α := G) (β := Aut F) (⇑(toAut F G) : (a : G) → Aut F)
X : C
inst_6 : IsConnected.{u₁, u₂} X
x y : (F.obj X).obj
t : Aut F
ht : t • x = y
⊢ ∃ (g : G), g • x = y
have obtain : ∀ (g : G) (ht : HSMul.hSMul (α := Aut F) ((toAut F G : (a : G) → Aut F) g : Aut F) x = y), ∃ (g : G), g • x = y
Tactic: obtain ⟨g, rfl⟩ := h t
Cleared hypotheses: t, ht