l3lab/ntp-mathlib huggingface dataset:
/-- Sending objects to cochain complexes supported at `0` then taking `0`-th homology
is the same as doing nothing.
-/
noncomputable def homologyFunctor0Single₀ : single₀ C ⋙ homology'Functor C _ 0 ≅ 𝟭 C :=
  NatIso.ofComponents (fun X => homology'.congr _ _ (by simp) (by simp) ≪≫ homology'ZeroZero)
    fun f => by
      -- Porting note: why can't `aesop_cat` do this?
      

Proof state:
C : Type u
inst✝⁷ : Category.{v, u} C
inst✝⁶ : HasZeroMorphisms C
inst✝⁵ : HasZeroObject C
ι : Type u_1
inst✝⁴ : DecidableEq ι
c : ComplexShape ι
j : ι
inst✝³ : HasEqualizers C
inst✝² : HasCokernels C
inst✝¹ : HasImages C
inst✝ : HasImageMaps C
X✝ Y✝ : C
f : X✝ ⟶ Y✝
⊢ (single₀ C ⋙ homology'Functor C (ComplexShape.up ℕ) 0).map f ≫
      ((fun X =>
            homology'.congr
                (_ : HomologicalComplex.dTo ((single₀ C).obj X) 0 ≫ HomologicalComplex.dFrom ((single₀ C).obj X) 0 = 0)
                (_ : 0 ≫ 0 = 0)
                (_ : HomologicalComplex.d ((single₀ C).obj X) (ComplexShape.prev (ComplexShape.up ℕ) 0) 0 = 0)
                (_ : 0 = 0) ≪≫
              homology'ZeroZero)
          Y✝).hom =
    ((fun X =>
            homology'.congr
                (_ : HomologicalComplex.dTo ((single₀ C).obj X) 0 ≫ HomologicalComplex.dFrom ((single₀ C).obj X) 0 = 0)
                (_ : 0 ≫ 0 = 0)
                (_ : HomologicalComplex.d ((single₀ C).obj X) (ComplexShape.prev (ComplexShape.up ℕ) 0) 0 = 0)
                (_ : 0 = 0) ≪≫
              homology'ZeroZero)
          X✝).hom ≫
      (𝟭 C).map f

Next tactic:
dsimp
/-- Sending objects to cochain complexes supported at `0` then taking `0`-th homology
is the same as doing nothing.
-/
noncomputable def homologyFunctor0Single₀ : single₀ C ⋙ homology'Functor C _ 0 ≅ 𝟭 C :=
  NatIso.ofComponents (fun X => homology'.congr _ _ (by simp) (by simp) ≪≫ homology'ZeroZero)
    fun f => by
      -- Porting note: why can't `aesop_cat` do this?
      dsimp
      

Proof state:
C : Type u
inst✝⁷ : Category.{v, u} C
inst✝⁶ : HasZeroMorphisms C
inst✝⁵ : HasZeroObject C
ι : Type u_1
inst✝⁴ : DecidableEq ι
c : ComplexShape ι
j : ι
inst✝³ : HasEqualizers C
inst✝² : HasCokernels C
inst✝¹ : HasImages C
inst✝ : HasImageMaps C
X✝ Y✝ : C
f : X✝ ⟶ Y✝
⊢ homology'.map
        (_ : HomologicalComplex.dTo ((single₀ C).obj X✝) 0 ≫ HomologicalComplex.dFrom ((single₀ C).obj X✝) 0 = 0)
        (_ : HomologicalComplex.dTo ((single₀ C).obj Y✝) 0 ≫ HomologicalComplex.dFrom ((single₀ C).obj Y✝) 0 = 0)
        (HomologicalComplex.Hom.sqTo ((single₀ C).map f) 0) (HomologicalComplex.Hom.sqFrom ((single₀ C).map f) 0)
        (_ :
          (HomologicalComplex.Hom.sqTo ((single₀ C).map f) 0).right =
            (HomologicalComplex.Hom.sqTo ((single₀ C).map f) 0).right) ≫
      homology'.map
          (_ : HomologicalComplex.dTo ((single₀ C).obj Y✝) 0 ≫ HomologicalComplex.dFrom ((single₀ C).obj Y✝) 0 = 0)
          (_ : 0 ≫ 0 = 0) (CommaMorphism.mk (𝟙 (HomologicalComplex.xPrev ((single₀ C).obj Y✝) 0)) (𝟙 Y✝))
          (CommaMorphism.mk (𝟙 Y✝) (𝟙 (HomologicalComplex.xNext ((single₀ C).obj Y✝) 0)))
          (_ :
            (CommaMorphism.mk (𝟙 (Arrow.mk 0).left) (𝟙 (Arrow.mk 0).right)).right =
              (CommaMorphism.mk (𝟙 (Arrow.mk 0).left) (𝟙 (Arrow.mk 0).right)).right) ≫
        homology'.desc 0 0 (_ : 0 ≫ 0 = 0) (Subobject.arrow (kernelSubobject 0))
          (_ : imageToKernel 0 0 (_ : 0 ≫ 0 = 0) ≫ Subobject.arrow (kernelSubobject 0) = 0) =
    (homology'.map
          (_ : HomologicalComplex.dTo ((single₀ C).obj X✝) 0 ≫ HomologicalComplex.dFrom ((single₀ C).obj X✝) 0 = 0)
          (_ : 0 ≫ 0 = 0) (CommaMorphism.mk (𝟙 (HomologicalComplex.xPrev ((single₀ C).obj X✝) 0)) (𝟙 X✝))
          (CommaMorphism.mk (𝟙 X✝) (𝟙 (HomologicalComplex.xNext ((single₀ C).obj X✝) 0)))
          (_ :
            (CommaMorphism.mk (𝟙 (Arrow.mk 0).left) (𝟙 (Arrow.mk 0).right)).right =
              (CommaMorphism.mk (𝟙 (Arrow.mk 0).left) (𝟙 (Arrow.mk 0).right)).right) ≫
        homology'.desc 0 0 (_ : 0 ≫ 0 = 0) (Subobject.arrow (kernelSubobject 0))
          (_ : imageToKernel 0 0 (_ : 0 ≫ 0 = 0) ≫ Subobject.arrow (kernelSubobject 0) = 0)) ≫
      f

Next tactic:
ext
/-- Sending objects to cochain complexes supported at `0` then taking `0`-th homology
is the same as doing nothing.
-/
noncomputable def homologyFunctor0Single₀ : single₀ C ⋙ homology'Functor C _ 0 ≅ 𝟭 C :=
  NatIso.ofComponents (fun X => homology'.congr _ _ (by simp) (by simp) ≪≫ homology'ZeroZero)
    fun f => by
      -- Porting note: why can't `aesop_cat` do this?
      dsimp
      ext
      

Proof state:
case p
C : Type u
inst✝⁷ : Category.{v, u} C
inst✝⁶ : HasZeroMorphisms C
inst✝⁵ : HasZeroObject C
ι : Type u_1
inst✝⁴ : DecidableEq ι
c : ComplexShape ι
j : ι
inst✝³ : HasEqualizers C
inst✝² : HasCokernels C
inst✝¹ : HasImages C
inst✝ : HasImageMaps C
X✝ Y✝ : C
f : X✝ ⟶ Y✝
⊢ homology'.π (HomologicalComplex.dTo ((single₀ C).obj X✝) 0) (HomologicalComplex.dFrom ((single₀ C).obj X✝) 0)
        (_ : HomologicalComplex.dTo ((single₀ C).obj X✝) 0 ≫ HomologicalComplex.dFrom ((single₀ C).obj X✝) 0 = 0) ≫
      homology'.map
          (_ : HomologicalComplex.dTo ((single₀ C).obj X✝) 0 ≫ HomologicalComplex.dFrom ((single₀ C).obj X✝) 0 = 0)
          (_ : HomologicalComplex.dTo ((single₀ C).obj Y✝) 0 ≫ HomologicalComplex.dFrom ((single₀ C).obj Y✝) 0 = 0)
          (HomologicalComplex.Hom.sqTo ((single₀ C).map f) 0) (HomologicalComplex.Hom.sqFrom ((single₀ C).map f) 0)
          (_ :
            (HomologicalComplex.Hom.sqTo ((single₀ C).map f) 0).right =
              (HomologicalComplex.Hom.sqTo ((single₀ C).map f) 0).right) ≫
        homology'.map
            (_ : HomologicalComplex.dTo ((single₀ C).obj Y✝) 0 ≫ HomologicalComplex.dFrom ((single₀ C).obj Y✝) 0 = 0)
            (_ : 0 ≫ 0 = 0) (CommaMorphism.mk (𝟙 (HomologicalComplex.xPrev ((single₀ C).obj Y✝) 0)) (𝟙 Y✝))
            (CommaMorphism.mk (𝟙 Y✝) (𝟙 (HomologicalComplex.xNext ((single₀ C).obj Y✝) 0)))
            (_ :
              (CommaMorphism.mk (𝟙 (Arrow.mk 0).left) (𝟙 (Arrow.mk 0).right)).right =
                (CommaMorphism.mk (𝟙 (Arrow.mk 0).left) (𝟙 (Arrow.mk 0).right)).right) ≫
          homology'.desc 0 0 (_ : 0 ≫ 0 = 0) (Subobject.arrow (kernelSubobject 0))
            (_ : imageToKernel 0 0 (_ : 0 ≫ 0 = 0) ≫ Subobject.arrow (kernelSubobject 0) = 0) =
    homology'.π (HomologicalComplex.dTo ((single₀ C).obj X✝) 0) (HomologicalComplex.dFrom ((single₀ C).obj X✝) 0)
        (_ : HomologicalComplex.dTo ((single₀ C).obj X✝) 0 ≫ HomologicalComplex.dFrom ((single₀ C).obj X✝) 0 = 0) ≫
      (homology'.map
            (_ : HomologicalComplex.dTo ((single₀ C).obj X✝) 0 ≫ HomologicalComplex.dFrom ((single₀ C).obj X✝) 0 = 0)
            (_ : 0 ≫ 0 = 0) (CommaMorphism.mk (𝟙 (HomologicalComplex.xPrev ((single₀ C).obj X✝) 0)) (𝟙 X✝))
            (CommaMorphism.mk (𝟙 X✝) (𝟙 (HomologicalComplex.xNext ((single₀ C).obj X✝) 0)))
            (_ :
              (CommaMorphism.mk (𝟙 (Arrow.mk 0).left) (𝟙 (Arrow.mk 0).right)).right =
                (CommaMorphism.mk (𝟙 (Arrow.mk 0).left) (𝟙 (Arrow.mk 0).right)).right) ≫
          homology'.desc 0 0 (_ : 0 ≫ 0 = 0) (Subobject.arrow (kernelSubobject 0))
            (_ : imageToKernel 0 0 (_ : 0 ≫ 0 = 0) ≫ Subobject.arrow (kernelSubobject 0) = 0)) ≫
        f

Next tactic:
simp
/-- Sending objects to cochain complexes supported at `0` then taking `(n+1)`-st homology
is the same as the zero functor.
-/
noncomputable def homology'FunctorSuccSingle₀ (n : ℕ) :
    single₀ C ⋙ homology'Functor C _ (n + 1) ≅ 0 :=
  NatIso.ofComponents
    (fun X =>
      homology'.congr _ _ (by 

Proof state:
C : Type u
inst✝⁷ : Category.{v, u} C
inst✝⁶ : HasZeroMorphisms C
inst✝⁵ : HasZeroObject C
ι : Type u_1
inst✝⁴ : DecidableEq ι
c : ComplexShape ι
j : ι
inst✝³ : HasEqualizers C
inst✝² : HasCokernels C
inst✝¹ : HasImages C
inst✝ : HasImageMaps C
n : ℕ
X : C
⊢ HomologicalComplex.dTo ((single₀ C).obj X) (n + 1) = 0

Next tactic:
simp
/-- Sending objects to cochain complexes supported at `0` then taking `(n+1)`-st homology
is the same as the zero functor.
-/
noncomputable def homology'FunctorSuccSingle₀ (n : ℕ) :
    single₀ C ⋙ homology'Functor C _ (n + 1) ≅ 0 :=
  NatIso.ofComponents
    (fun X =>
      homology'.congr _ _ (by simp) (by 

Proof state:
C : Type u
inst✝⁷ : Category.{v, u} C
inst✝⁶ : HasZeroMorphisms C
inst✝⁵ : HasZeroObject C
ι : Type u_1
inst✝⁴ : DecidableEq ι
c : ComplexShape ι
j : ι
inst✝³ : HasEqualizers C
inst✝² : HasCokernels C
inst✝¹ : HasImages C
inst✝ : HasImageMaps C
n : ℕ
X : C
⊢ HomologicalComplex.dFrom ((single₀ C).obj X) (n + 1) = 0

Next tactic:
simp
lemma exactAt_succ_single_obj (A : C) (n : ℕ) :
    ExactAt ((single₀ C).obj A) (n + 1) :=
  exactAt_single_obj _ _ _ _ (by 

Proof state:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : C
n : ℕ
⊢ n + 1 ≠ 0

Next tactic:
simp
instance (H : Type u) [TopologicalSpace H] : SetLike (StructureGroupoid H) (PartialHomeomorph H H)
    where
  coe s := s.members
  coe_injective' N O h := by 

Proof state:
H✝ : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝¹ : TopologicalSpace H✝
H : Type u
inst✝ : TopologicalSpace H
N O : StructureGroupoid H
h : (fun s => s.members) N = (fun s => s.members) O
⊢ N = O

Next tactic:
cases N
instance (H : Type u) [TopologicalSpace H] : SetLike (StructureGroupoid H) (PartialHomeomorph H H)
    where
  coe s := s.members
  coe_injective' N O h := by cases N; 

Proof state:
case mk
H✝ : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝¹ : TopologicalSpace H✝
H : Type u
inst✝ : TopologicalSpace H
O : StructureGroupoid H
members✝ : Set (PartialHomeomorph H H)
trans'✝ : ∀ (e e' : PartialHomeomorph H H), e ∈ members✝ → e' ∈ members✝ → e ≫ₕ e' ∈ members✝
symm'✝ : ∀ e ∈ members✝, PartialHomeomorph.symm e ∈ members✝
id_mem'✝ : PartialHomeomorph.refl H ∈ members✝
locality'✝ :
  ∀ (e : PartialHomeomorph H H),
    (∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ members✝) → e ∈ members✝
eq_on_source'✝ : ∀ (e e' : PartialHomeomorph H H), e ∈ members✝ → e' ≈ e → e' ∈ members✝
h :
  (fun s => s.members)
      { members := members✝, trans' := trans'✝, symm' := symm'✝, id_mem' := id_mem'✝, locality' := locality'✝,
        eq_on_source' := eq_on_source'✝ } =
    (fun s => s.members) O
⊢ { members := members✝, trans' := trans'✝, symm' := symm'✝, id_mem' := id_mem'✝, locality' := locality'✝,
      eq_on_source' := eq_on_source'✝ } =
    O

Next tactic:
cases O
instance (H : Type u) [TopologicalSpace H] : SetLike (StructureGroupoid H) (PartialHomeomorph H H)
    where
  coe s := s.members
  coe_injective' N O h := by cases N; cases O; 

Proof state:
case mk.mk
H✝ : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝¹ : TopologicalSpace H✝
H : Type u
inst✝ : TopologicalSpace H
members✝¹ : Set (PartialHomeomorph H H)
trans'✝¹ : ∀ (e e' : PartialHomeomorph H H), e ∈ members✝¹ → e' ∈ members✝¹ → e ≫ₕ e' ∈ members✝¹
symm'✝¹ : ∀ e ∈ members✝¹, PartialHomeomorph.symm e ∈ members✝¹
id_mem'✝¹ : PartialHomeomorph.refl H ∈ members✝¹
locality'✝¹ :
  ∀ (e : PartialHomeomorph H H),
    (∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ members✝¹) → e ∈ members✝¹
eq_on_source'✝¹ : ∀ (e e' : PartialHomeomorph H H), e ∈ members✝¹ → e' ≈ e → e' ∈ members✝¹
members✝ : Set (PartialHomeomorph H H)
trans'✝ : ∀ (e e' : PartialHomeomorph H H), e ∈ members✝ → e' ∈ members✝ → e ≫ₕ e' ∈ members✝
symm'✝ : ∀ e ∈ members✝, PartialHomeomorph.symm e ∈ members✝
id_mem'✝ : PartialHomeomorph.refl H ∈ members✝
locality'✝ :
  ∀ (e : PartialHomeomorph H H),
    (∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ members✝) → e ∈ members✝
eq_on_source'✝ : ∀ (e e' : PartialHomeomorph H H), e ∈ members✝ → e' ≈ e → e' ∈ members✝
h :
  (fun s => s.members)
      { members := members✝¹, trans' := trans'✝¹, symm' := symm'✝¹, id_mem' := id_mem'✝¹, locality' := locality'✝¹,
        eq_on_source' := eq_on_source'✝¹ } =
    (fun s => s.members)
      { members := members✝, trans' := trans'✝, symm' := symm'✝, id_mem' := id_mem'✝, locality' := locality'✝,
        eq_on_source' := eq_on_source'✝ }
⊢ { members := members✝¹, trans' := trans'✝¹, symm' := symm'✝¹, id_mem' := id_mem'✝¹, locality' := locality'✝¹,
      eq_on_source' := eq_on_source'✝¹ } =
    { members := members✝, trans' := trans'✝, symm' := symm'✝, id_mem' := id_mem'✝, locality' := locality'✝,
      eq_on_source' := eq_on_source'✝ }

Next tactic:
congr
instance : Inf (StructureGroupoid H) :=
  ⟨fun G G' => StructureGroupoid.mk
    (members := G.members ∩ G'.members)
    (trans' := fun e e' he he' =>
      ⟨G.trans' e e' he.left he'.left, G'.trans' e e' he.right he'.right⟩)
    (symm' := fun e he => ⟨G.symm' e he.left, G'.symm' e he.right⟩)
    (id_mem' := ⟨G.id_mem', G'.id_mem'⟩)
    (locality' := by
      

Proof state:
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
G G' : StructureGroupoid H
⊢ ∀ (e : PartialHomeomorph H H),
    (∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members ∩ G'.members) →
      e ∈ G.members ∩ G'.members

Next tactic:
intro e hx
instance : Inf (StructureGroupoid H) :=
  ⟨fun G G' => StructureGroupoid.mk
    (members := G.members ∩ G'.members)
    (trans' := fun e e' he he' =>
      ⟨G.trans' e e' he.left he'.left, G'.trans' e e' he.right he'.right⟩)
    (symm' := fun e he => ⟨G.symm' e he.left, G'.symm' e he.right⟩)
    (id_mem' := ⟨G.id_mem', G'.id_mem'⟩)
    (locality' := by
      intro e hx
      

Proof state:
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
G G' : StructureGroupoid H
e : PartialHomeomorph H H
hx : ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members ∩ G'.members
⊢ e ∈ G.members ∩ G'.members

Next tactic:
apply (mem_inter_iff e G.members G'.members).mpr
instance : Inf (StructureGroupoid H) :=
  ⟨fun G G' => StructureGroupoid.mk
    (members := G.members ∩ G'.members)
    (trans' := fun e e' he he' =>
      ⟨G.trans' e e' he.left he'.left, G'.trans' e e' he.right he'.right⟩)
    (symm' := fun e he => ⟨G.symm' e he.left, G'.symm' e he.right⟩)
    (id_mem' := ⟨G.id_mem', G'.id_mem'⟩)
    (locality' := by
      intro e hx
      apply (mem_inter_iff e G.members G'.members).mpr
      

Proof state:
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
G G' : StructureGroupoid H
e : PartialHomeomorph H H
hx : ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members ∩ G'.members
⊢ e ∈ G.members ∧ e ∈ G'.members

Next tactic:
refine And.intro (G.locality' e ?_) (G'.locality' e ?_)
instance : Inf (StructureGroupoid H) :=
  ⟨fun G G' => StructureGroupoid.mk
    (members := G.members ∩ G'.members)
    (trans' := fun e e' he he' =>
      ⟨G.trans' e e' he.left he'.left, G'.trans' e e' he.right he'.right⟩)
    (symm' := fun e he => ⟨G.symm' e he.left, G'.symm' e he.right⟩)
    (id_mem' := ⟨G.id_mem', G'.id_mem'⟩)
    (locality' := by
      intro e hx
      apply (mem_inter_iff e G.members G'.members).mpr
      refine And.intro (G.locality' e ?_) (G'.locality' e ?_)
      

Proof state:
case refine_1
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
G G' : StructureGroupoid H
e : PartialHomeomorph H H
hx : ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members ∩ G'.members
⊢ ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members
case refine_2
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
G G' : StructureGroupoid H
e : PartialHomeomorph H H
hx : ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members ∩ G'.members
⊢ ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G'.members

Next tactic:
all_goals
        intro x hex
        rcases hx x hex with ⟨s, hs⟩
        use s
        refine And.intro hs.left (And.intro hs.right.left ?_)
instance : Inf (StructureGroupoid H) :=
  ⟨fun G G' => StructureGroupoid.mk
    (members := G.members ∩ G'.members)
    (trans' := fun e e' he he' =>
      ⟨G.trans' e e' he.left he'.left, G'.trans' e e' he.right he'.right⟩)
    (symm' := fun e he => ⟨G.symm' e he.left, G'.symm' e he.right⟩)
    (id_mem' := ⟨G.id_mem', G'.id_mem'⟩)
    (locality' := by
      intro e hx
      apply (mem_inter_iff e G.members G'.members).mpr
      refine And.intro (G.locality' e ?_) (G'.locality' e ?_)
      all_goals
        

Proof state:
case refine_1
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
G G' : StructureGroupoid H
e : PartialHomeomorph H H
hx : ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members ∩ G'.members
⊢ ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members

Next tactic:
intro x hex
instance : Inf (StructureGroupoid H) :=
  ⟨fun G G' => StructureGroupoid.mk
    (members := G.members ∩ G'.members)
    (trans' := fun e e' he he' =>
      ⟨G.trans' e e' he.left he'.left, G'.trans' e e' he.right he'.right⟩)
    (symm' := fun e he => ⟨G.symm' e he.left, G'.symm' e he.right⟩)
    (id_mem' := ⟨G.id_mem', G'.id_mem'⟩)
    (locality' := by
      intro e hx
      apply (mem_inter_iff e G.members G'.members).mpr
      refine And.intro (G.locality' e ?_) (G'.locality' e ?_)
      all_goals
        intro x hex
        

Proof state:
case refine_1
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
G G' : StructureGroupoid H
e : PartialHomeomorph H H
hx : ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members ∩ G'.members
x : H
hex : x ∈ e.source
⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members

Next tactic:
rcases hx x hex with ⟨s, hs⟩
instance : Inf (StructureGroupoid H) :=
  ⟨fun G G' => StructureGroupoid.mk
    (members := G.members ∩ G'.members)
    (trans' := fun e e' he he' =>
      ⟨G.trans' e e' he.left he'.left, G'.trans' e e' he.right he'.right⟩)
    (symm' := fun e he => ⟨G.symm' e he.left, G'.symm' e he.right⟩)
    (id_mem' := ⟨G.id_mem', G'.id_mem'⟩)
    (locality' := by
      intro e hx
      apply (mem_inter_iff e G.members G'.members).mpr
      refine And.intro (G.locality' e ?_) (G'.locality' e ?_)
      all_goals
        intro x hex
        rcases hx x hex with ⟨s, hs⟩
        

Proof state:
case refine_1.intro
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
G G' : StructureGroupoid H
e : PartialHomeomorph H H
hx : ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members ∩ G'.members
x : H
hex : x ∈ e.source
s : Set H
hs : IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members ∩ G'.members
⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members

Next tactic:
use s
instance : Inf (StructureGroupoid H) :=
  ⟨fun G G' => StructureGroupoid.mk
    (members := G.members ∩ G'.members)
    (trans' := fun e e' he he' =>
      ⟨G.trans' e e' he.left he'.left, G'.trans' e e' he.right he'.right⟩)
    (symm' := fun e he => ⟨G.symm' e he.left, G'.symm' e he.right⟩)
    (id_mem' := ⟨G.id_mem', G'.id_mem'⟩)
    (locality' := by
      intro e hx
      apply (mem_inter_iff e G.members G'.members).mpr
      refine And.intro (G.locality' e ?_) (G'.locality' e ?_)
      all_goals
        intro x hex
        rcases hx x hex with ⟨s, hs⟩
        use s
        

Proof state:
case h
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
G G' : StructureGroupoid H
e : PartialHomeomorph H H
hx : ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members ∩ G'.members
x : H
hex : x ∈ e.source
s : Set H
hs : IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members ∩ G'.members
⊢ IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members

Next tactic:
refine And.intro hs.left (And.intro hs.right.left ?_)
instance : Inf (StructureGroupoid H) :=
  ⟨fun G G' => StructureGroupoid.mk
    (members := G.members ∩ G'.members)
    (trans' := fun e e' he he' =>
      ⟨G.trans' e e' he.left he'.left, G'.trans' e e' he.right he'.right⟩)
    (symm' := fun e he => ⟨G.symm' e he.left, G'.symm' e he.right⟩)
    (id_mem' := ⟨G.id_mem', G'.id_mem'⟩)
    (locality' := by
      intro e hx
      apply (mem_inter_iff e G.members G'.members).mpr
      refine And.intro (G.locality' e ?_) (G'.locality' e ?_)
      all_goals
        

Proof state:
case refine_2
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
G G' : StructureGroupoid H
e : PartialHomeomorph H H
hx : ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members ∩ G'.members
⊢ ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G'.members

Next tactic:
intro x hex
instance : Inf (StructureGroupoid H) :=
  ⟨fun G G' => StructureGroupoid.mk
    (members := G.members ∩ G'.members)
    (trans' := fun e e' he he' =>
      ⟨G.trans' e e' he.left he'.left, G'.trans' e e' he.right he'.right⟩)
    (symm' := fun e he => ⟨G.symm' e he.left, G'.symm' e he.right⟩)
    (id_mem' := ⟨G.id_mem', G'.id_mem'⟩)
    (locality' := by
      intro e hx
      apply (mem_inter_iff e G.members G'.members).mpr
      refine And.intro (G.locality' e ?_) (G'.locality' e ?_)
      all_goals
        intro x hex
        

Proof state:
case refine_2
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
G G' : StructureGroupoid H
e : PartialHomeomorph H H
hx : ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members ∩ G'.members
x : H
hex : x ∈ e.source
⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G'.members

Next tactic:
rcases hx x hex with ⟨s, hs⟩
instance : Inf (StructureGroupoid H) :=
  ⟨fun G G' => StructureGroupoid.mk
    (members := G.members ∩ G'.members)
    (trans' := fun e e' he he' =>
      ⟨G.trans' e e' he.left he'.left, G'.trans' e e' he.right he'.right⟩)
    (symm' := fun e he => ⟨G.symm' e he.left, G'.symm' e he.right⟩)
    (id_mem' := ⟨G.id_mem', G'.id_mem'⟩)
    (locality' := by
      intro e hx
      apply (mem_inter_iff e G.members G'.members).mpr
      refine And.intro (G.locality' e ?_) (G'.locality' e ?_)
      all_goals
        intro x hex
        rcases hx x hex with ⟨s, hs⟩
        

Proof state:
case refine_2.intro
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
G G' : StructureGroupoid H
e : PartialHomeomorph H H
hx : ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members ∩ G'.members
x : H
hex : x ∈ e.source
s : Set H
hs : IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members ∩ G'.members
⊢ ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G'.members

Next tactic:
use s
instance : Inf (StructureGroupoid H) :=
  ⟨fun G G' => StructureGroupoid.mk
    (members := G.members ∩ G'.members)
    (trans' := fun e e' he he' =>
      ⟨G.trans' e e' he.left he'.left, G'.trans' e e' he.right he'.right⟩)
    (symm' := fun e he => ⟨G.symm' e he.left, G'.symm' e he.right⟩)
    (id_mem' := ⟨G.id_mem', G'.id_mem'⟩)
    (locality' := by
      intro e hx
      apply (mem_inter_iff e G.members G'.members).mpr
      refine And.intro (G.locality' e ?_) (G'.locality' e ?_)
      all_goals
        intro x hex
        rcases hx x hex with ⟨s, hs⟩
        use s
        

Proof state:
case h
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝ : TopologicalSpace H
G G' : StructureGroupoid H
e : PartialHomeomorph H H
hx : ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members ∩ G'.members
x : H
hex : x ∈ e.source
s : Set H
hs : IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G.members ∩ G'.members
⊢ IsOpen s ∧ x ∈ s ∧ PartialHomeomorph.restr e s ∈ G'.members

Next tactic:
refine And.intro hs.left (And.intro hs.right.left ?_)