l3lab/ntp-mathlib huggingface dataset:
theorem skewAdjointMatricesLieSubalgebraEquiv_apply (P : Matrix n n R) (h : Invertible P)
    (A : skewAdjointMatricesLieSubalgebra J) :
    ↑(skewAdjointMatricesLieSubalgebraEquiv J P h A) = P⁻¹ * (A : Matrix n n R) * P := by
  

Proof state:
R : Type u
n : Type w
inst✝² : CommRing R
inst✝¹ : DecidableEq n
inst✝ : Fintype n
J P : Matrix n n R
h : Invertible P
A : ↥(skewAdjointMatricesLieSubalgebra J)
⊢ ↑((skewAdjointMatricesLieSubalgebraEquiv J P h) A) = P⁻¹ * ↑A * P

Next tactic:
simp [skewAdjointMatricesLieSubalgebraEquiv]
/-- An equivalence of matrix algebras commuting with the transpose endomorphisms restricts to an
equivalence of Lie algebras of skew-adjoint matrices. -/
def skewAdjointMatricesLieSubalgebraEquivTranspose {m : Type w} [DecidableEq m] [Fintype m]
    (e : Matrix n n R ≃ₐ[R] Matrix m m R) (h : ∀ A, (e A)ᵀ = e Aᵀ) :
    skewAdjointMatricesLieSubalgebra J ≃ₗ⁅R⁆ skewAdjointMatricesLieSubalgebra (e J) :=
  LieEquiv.ofSubalgebras _ _ e.toLieEquiv <| by
    

Proof state:
R : Type u
n : Type w
inst✝⁴ : CommRing R
inst✝³ : DecidableEq n
inst✝² : Fintype n
J : Matrix n n R
m : Type w
inst✝¹ : DecidableEq m
inst✝ : Fintype m
e : Matrix n n R ≃ₐ[R] Matrix m m R
h : ∀ (A : Matrix n n R), (e A)ᵀ = e Aᵀ
⊢ LieSubalgebra.map (AlgEquiv.toLieEquiv e).toLieHom (skewAdjointMatricesLieSubalgebra J) =
    skewAdjointMatricesLieSubalgebra (e J)

Next tactic:
ext A
/-- An equivalence of matrix algebras commuting with the transpose endomorphisms restricts to an
equivalence of Lie algebras of skew-adjoint matrices. -/
def skewAdjointMatricesLieSubalgebraEquivTranspose {m : Type w} [DecidableEq m] [Fintype m]
    (e : Matrix n n R ≃ₐ[R] Matrix m m R) (h : ∀ A, (e A)ᵀ = e Aᵀ) :
    skewAdjointMatricesLieSubalgebra J ≃ₗ⁅R⁆ skewAdjointMatricesLieSubalgebra (e J) :=
  LieEquiv.ofSubalgebras _ _ e.toLieEquiv <| by
    ext A
    

Proof state:
case h
R : Type u
n : Type w
inst✝⁴ : CommRing R
inst✝³ : DecidableEq n
inst✝² : Fintype n
J : Matrix n n R
m : Type w
inst✝¹ : DecidableEq m
inst✝ : Fintype m
e : Matrix n n R ≃ₐ[R] Matrix m m R
h : ∀ (A : Matrix n n R), (e A)ᵀ = e Aᵀ
A : Matrix m m R
⊢ A ∈ LieSubalgebra.map (AlgEquiv.toLieEquiv e).toLieHom (skewAdjointMatricesLieSubalgebra J) ↔
    A ∈ skewAdjointMatricesLieSubalgebra (e J)

Next tactic:
suffices J.IsSkewAdjoint (e.symm A) ↔ (e J).IsSkewAdjoint A by
      -- Porting note: Originally `simpa [this]`
      simpa [- LieSubalgebra.mem_map, LieSubalgebra.mem_map_submodule]
/-- An equivalence of matrix algebras commuting with the transpose endomorphisms restricts to an
equivalence of Lie algebras of skew-adjoint matrices. -/
def skewAdjointMatricesLieSubalgebraEquivTranspose {m : Type w} [DecidableEq m] [Fintype m]
    (e : Matrix n n R ≃ₐ[R] Matrix m m R) (h : ∀ A, (e A)ᵀ = e Aᵀ) :
    skewAdjointMatricesLieSubalgebra J ≃ₗ⁅R⁆ skewAdjointMatricesLieSubalgebra (e J) :=
  LieEquiv.ofSubalgebras _ _ e.toLieEquiv <| by
    ext A
    suffices J.IsSkewAdjoint (e.symm A) ↔ (e J).IsSkewAdjoint A by
      -- Porting note: Originally `simpa [this]`
      

Proof state:
R : Type u
n : Type w
inst✝⁴ : CommRing R
inst✝³ : DecidableEq n
inst✝² : Fintype n
J : Matrix n n R
m : Type w
inst✝¹ : DecidableEq m
inst✝ : Fintype m
e : Matrix n n R ≃ₐ[R] Matrix m m R
h : ∀ (A : Matrix n n R), (e A)ᵀ = e Aᵀ
A : Matrix m m R
this : Matrix.IsSkewAdjoint J ((AlgEquiv.symm e) A) ↔ Matrix.IsSkewAdjoint (e J) A
⊢ A ∈ LieSubalgebra.map (AlgEquiv.toLieEquiv e).toLieHom (skewAdjointMatricesLieSubalgebra J) ↔
    A ∈ skewAdjointMatricesLieSubalgebra (e J)

Next tactic:
simpa [- LieSubalgebra.mem_map, LieSubalgebra.mem_map_submodule]
/-- An equivalence of matrix algebras commuting with the transpose endomorphisms restricts to an
equivalence of Lie algebras of skew-adjoint matrices. -/
def skewAdjointMatricesLieSubalgebraEquivTranspose {m : Type w} [DecidableEq m] [Fintype m]
    (e : Matrix n n R ≃ₐ[R] Matrix m m R) (h : ∀ A, (e A)ᵀ = e Aᵀ) :
    skewAdjointMatricesLieSubalgebra J ≃ₗ⁅R⁆ skewAdjointMatricesLieSubalgebra (e J) :=
  LieEquiv.ofSubalgebras _ _ e.toLieEquiv <| by
    ext A
    suffices J.IsSkewAdjoint (e.symm A) ↔ (e J).IsSkewAdjoint A by
      -- Porting note: Originally `simpa [this]`
      simpa [- LieSubalgebra.mem_map, LieSubalgebra.mem_map_submodule]
    

Proof state:
case h
R : Type u
n : Type w
inst✝⁴ : CommRing R
inst✝³ : DecidableEq n
inst✝² : Fintype n
J : Matrix n n R
m : Type w
inst✝¹ : DecidableEq m
inst✝ : Fintype m
e : Matrix n n R ≃ₐ[R] Matrix m m R
h : ∀ (A : Matrix n n R), (e A)ᵀ = e Aᵀ
A : Matrix m m R
⊢ Matrix.IsSkewAdjoint J ((AlgEquiv.symm e) A) ↔ Matrix.IsSkewAdjoint (e J) A

Next tactic:
simp only [Matrix.IsSkewAdjoint, Matrix.IsAdjointPair, ← h,
      ← Function.Injective.eq_iff e.injective, map_mul, AlgEquiv.apply_symm_apply, map_neg]
theorem mem_skewAdjointMatricesLieSubalgebra_unit_smul (u : Rˣ) (J A : Matrix n n R) :
    A ∈ skewAdjointMatricesLieSubalgebra (u • J) ↔ A ∈ skewAdjointMatricesLieSubalgebra J := by
  

Proof state:
R : Type u
n : Type w
inst✝² : CommRing R
inst✝¹ : DecidableEq n
inst✝ : Fintype n
J✝ : Matrix n n R
u : Rˣ
J A : Matrix n n R
⊢ A ∈ skewAdjointMatricesLieSubalgebra (u • J) ↔ A ∈ skewAdjointMatricesLieSubalgebra J

Next tactic:
change A ∈ skewAdjointMatricesSubmodule (u • J) ↔ A ∈ skewAdjointMatricesSubmodule J
theorem mem_skewAdjointMatricesLieSubalgebra_unit_smul (u : Rˣ) (J A : Matrix n n R) :
    A ∈ skewAdjointMatricesLieSubalgebra (u • J) ↔ A ∈ skewAdjointMatricesLieSubalgebra J := by
  change A ∈ skewAdjointMatricesSubmodule (u • J) ↔ A ∈ skewAdjointMatricesSubmodule J
  

Proof state:
R : Type u
n : Type w
inst✝² : CommRing R
inst✝¹ : DecidableEq n
inst✝ : Fintype n
J✝ : Matrix n n R
u : Rˣ
J A : Matrix n n R
⊢ A ∈ skewAdjointMatricesSubmodule (u • J) ↔ A ∈ skewAdjointMatricesSubmodule J

Next tactic:
simp only [mem_skewAdjointMatricesSubmodule, Matrix.IsSkewAdjoint, Matrix.IsAdjointPair]
theorem mem_skewAdjointMatricesLieSubalgebra_unit_smul (u : Rˣ) (J A : Matrix n n R) :
    A ∈ skewAdjointMatricesLieSubalgebra (u • J) ↔ A ∈ skewAdjointMatricesLieSubalgebra J := by
  change A ∈ skewAdjointMatricesSubmodule (u • J) ↔ A ∈ skewAdjointMatricesSubmodule J
  simp only [mem_skewAdjointMatricesSubmodule, Matrix.IsSkewAdjoint, Matrix.IsAdjointPair]
  

Proof state:
R : Type u
n : Type w
inst✝² : CommRing R
inst✝¹ : DecidableEq n
inst✝ : Fintype n
J✝ : Matrix n n R
u : Rˣ
J A : Matrix n n R
⊢ Aᵀ * u • J = u • J * -A ↔ Aᵀ * J = J * -A

Next tactic:
constructor
theorem mem_skewAdjointMatricesLieSubalgebra_unit_smul (u : Rˣ) (J A : Matrix n n R) :
    A ∈ skewAdjointMatricesLieSubalgebra (u • J) ↔ A ∈ skewAdjointMatricesLieSubalgebra J := by
  change A ∈ skewAdjointMatricesSubmodule (u • J) ↔ A ∈ skewAdjointMatricesSubmodule J
  simp only [mem_skewAdjointMatricesSubmodule, Matrix.IsSkewAdjoint, Matrix.IsAdjointPair]
  constructor <;> 

Proof state:
case mp
R : Type u
n : Type w
inst✝² : CommRing R
inst✝¹ : DecidableEq n
inst✝ : Fintype n
J✝ : Matrix n n R
u : Rˣ
J A : Matrix n n R
⊢ Aᵀ * u • J = u • J * -A → Aᵀ * J = J * -A

Next tactic:
intro h
theorem mem_skewAdjointMatricesLieSubalgebra_unit_smul (u : Rˣ) (J A : Matrix n n R) :
    A ∈ skewAdjointMatricesLieSubalgebra (u • J) ↔ A ∈ skewAdjointMatricesLieSubalgebra J := by
  change A ∈ skewAdjointMatricesSubmodule (u • J) ↔ A ∈ skewAdjointMatricesSubmodule J
  simp only [mem_skewAdjointMatricesSubmodule, Matrix.IsSkewAdjoint, Matrix.IsAdjointPair]
  constructor <;> 

Proof state:
case mpr
R : Type u
n : Type w
inst✝² : CommRing R
inst✝¹ : DecidableEq n
inst✝ : Fintype n
J✝ : Matrix n n R
u : Rˣ
J A : Matrix n n R
⊢ Aᵀ * J = J * -A → Aᵀ * u • J = u • J * -A

Next tactic:
intro h
theorem mem_skewAdjointMatricesLieSubalgebra_unit_smul (u : Rˣ) (J A : Matrix n n R) :
    A ∈ skewAdjointMatricesLieSubalgebra (u • J) ↔ A ∈ skewAdjointMatricesLieSubalgebra J := by
  change A ∈ skewAdjointMatricesSubmodule (u • J) ↔ A ∈ skewAdjointMatricesSubmodule J
  simp only [mem_skewAdjointMatricesSubmodule, Matrix.IsSkewAdjoint, Matrix.IsAdjointPair]
  constructor <;> intro h
  · 

Proof state:
case mp
R : Type u
n : Type w
inst✝² : CommRing R
inst✝¹ : DecidableEq n
inst✝ : Fintype n
J✝ : Matrix n n R
u : Rˣ
J A : Matrix n n R
h : Aᵀ * u • J = u • J * -A
⊢ Aᵀ * J = J * -A

Next tactic:
simpa using congr_arg (fun B => u⁻¹ • B) h
theorem mem_skewAdjointMatricesLieSubalgebra_unit_smul (u : Rˣ) (J A : Matrix n n R) :
    A ∈ skewAdjointMatricesLieSubalgebra (u • J) ↔ A ∈ skewAdjointMatricesLieSubalgebra J := by
  change A ∈ skewAdjointMatricesSubmodule (u • J) ↔ A ∈ skewAdjointMatricesSubmodule J
  simp only [mem_skewAdjointMatricesSubmodule, Matrix.IsSkewAdjoint, Matrix.IsAdjointPair]
  constructor <;> intro h
  · simpa using congr_arg (fun B => u⁻¹ • B) h
  · 

Proof state:
case mpr
R : Type u
n : Type w
inst✝² : CommRing R
inst✝¹ : DecidableEq n
inst✝ : Fintype n
J✝ : Matrix n n R
u : Rˣ
J A : Matrix n n R
h : Aᵀ * J = J * -A
⊢ Aᵀ * u • J = u • J * -A

Next tactic:
simp [h]
theorem reverse_eq {l : List α} (p : Palindrome l) : reverse l = l := by
  

Proof state:
α : Type u_1
β : Type u_2
l✝ l : List α
p : Palindrome l
⊢ reverse l = l

Next tactic:
induction p
theorem reverse_eq {l : List α} (p : Palindrome l) : reverse l = l := by
  induction p <;> 

Proof state:
case nil
α : Type u_1
β : Type u_2
l✝ l : List α
⊢ reverse [] = []

Next tactic:
try (exact rfl)
theorem reverse_eq {l : List α} (p : Palindrome l) : reverse l = l := by
  induction p <;> try (

Proof state:
case nil
α : Type u_1
β : Type u_2
l✝ l : List α
⊢ reverse [] = []

Next tactic:
exact rfl
theorem reverse_eq {l : List α} (p : Palindrome l) : reverse l = l := by
  induction p <;> 

Proof state:
case singleton
α : Type u_1
β : Type u_2
l✝ l : List α
x✝ : α
⊢ reverse [x✝] = [x✝]

Next tactic:
try (exact rfl)
theorem reverse_eq {l : List α} (p : Palindrome l) : reverse l = l := by
  induction p <;> try (

Proof state:
case singleton
α : Type u_1
β : Type u_2
l✝ l : List α
x✝ : α
⊢ reverse [x✝] = [x✝]

Next tactic:
exact rfl
theorem reverse_eq {l : List α} (p : Palindrome l) : reverse l = l := by
  induction p <;> 

Proof state:
case cons_concat
α : Type u_1
β : Type u_2
l✝¹ l : List α
x✝ : α
l✝ : List α
a✝ : Palindrome l✝
a_ih✝ : reverse l✝ = l✝
⊢ reverse (x✝ :: (l✝ ++ [x✝])) = x✝ :: (l✝ ++ [x✝])

Next tactic:
try (exact rfl)
theorem reverse_eq {l : List α} (p : Palindrome l) : reverse l = l := by
  induction p <;> try (

Proof state:
case cons_concat
α : Type u_1
β : Type u_2
l✝¹ l : List α
x✝ : α
l✝ : List α
a✝ : Palindrome l✝
a_ih✝ : reverse l✝ = l✝
⊢ reverse (x✝ :: (l✝ ++ [x✝])) = x✝ :: (l✝ ++ [x✝])

Next tactic:
exact rfl
theorem reverse_eq {l : List α} (p : Palindrome l) : reverse l = l := by
  induction p <;> try (exact rfl)
  

Proof state:
case cons_concat
α : Type u_1
β : Type u_2
l✝¹ l : List α
x✝ : α
l✝ : List α
a✝ : Palindrome l✝
a_ih✝ : reverse l✝ = l✝
⊢ reverse (x✝ :: (l✝ ++ [x✝])) = x✝ :: (l✝ ++ [x✝])

Next tactic:
simp
theorem reverse_eq {l : List α} (p : Palindrome l) : reverse l = l := by
  induction p <;> try (exact rfl)
  simp; 

Proof state:
case cons_concat
α : Type u_1
β : Type u_2
l✝¹ l : List α
x✝ : α
l✝ : List α
a✝ : Palindrome l✝
a_ih✝ : reverse l✝ = l✝
⊢ reverse l✝ = l✝

Next tactic:
assumption
theorem of_reverse_eq {l : List α} : reverse l = l → Palindrome l := by
  

Proof state:
α : Type u_1
β : Type u_2
l✝ l : List α
⊢ reverse l = l → Palindrome l

Next tactic:
refine' bidirectionalRecOn l (fun _ => Palindrome.nil) (fun a _ => Palindrome.singleton a) _
theorem of_reverse_eq {l : List α} : reverse l = l → Palindrome l := by
  refine' bidirectionalRecOn l (fun _ => Palindrome.nil) (fun a _ => Palindrome.singleton a) _
  

Proof state:
α : Type u_1
β : Type u_2
l✝ l : List α
⊢ ∀ (a : α) (l : List α) (b : α),
    (reverse l = l → Palindrome l) → reverse (a :: (l ++ [b])) = a :: (l ++ [b]) → Palindrome (a :: (l ++ [b]))

Next tactic:
intro x l y hp hr
theorem of_reverse_eq {l : List α} : reverse l = l → Palindrome l := by
  refine' bidirectionalRecOn l (fun _ => Palindrome.nil) (fun a _ => Palindrome.singleton a) _
  intro x l y hp hr
  

Proof state:
α : Type u_1
β : Type u_2
l✝¹ l✝ : List α
x : α
l : List α
y : α
hp : reverse l = l → Palindrome l
hr : reverse (x :: (l ++ [y])) = x :: (l ++ [y])
⊢ Palindrome (x :: (l ++ [y]))

Next tactic:
rw [reverse_cons, reverse_append] at hr
theorem of_reverse_eq {l : List α} : reverse l = l → Palindrome l := by
  refine' bidirectionalRecOn l (fun _ => Palindrome.nil) (fun a _ => Palindrome.singleton a) _
  intro x l y hp hr
  rw [reverse_cons, reverse_append] at hr
  

Proof state:
α : Type u_1
β : Type u_2
l✝¹ l✝ : List α
x : α
l : List α
y : α
hp : reverse l = l → Palindrome l
hr : reverse [y] ++ reverse l ++ [x] = x :: (l ++ [y])
⊢ Palindrome (x :: (l ++ [y]))

Next tactic:
rw [head_eq_of_cons_eq hr]
theorem of_reverse_eq {l : List α} : reverse l = l → Palindrome l := by
  refine' bidirectionalRecOn l (fun _ => Palindrome.nil) (fun a _ => Palindrome.singleton a) _
  intro x l y hp hr
  rw [reverse_cons, reverse_append] at hr
  rw [head_eq_of_cons_eq hr]
  

Proof state:
α : Type u_1
β : Type u_2
l✝¹ l✝ : List α
x : α
l : List α
y : α
hp : reverse l = l → Palindrome l
hr : reverse [y] ++ reverse l ++ [x] = x :: (l ++ [y])
⊢ Palindrome (x :: (l ++ [x]))

Next tactic:
have : Palindrome l := hp (append_inj_left' (tail_eq_of_cons_eq hr) rfl)
theorem of_reverse_eq {l : List α} : reverse l = l → Palindrome l := by
  refine' bidirectionalRecOn l (fun _ => Palindrome.nil) (fun a _ => Palindrome.singleton a) _
  intro x l y hp hr
  rw [reverse_cons, reverse_append] at hr
  rw [head_eq_of_cons_eq hr]
  have : Palindrome l := hp (append_inj_left' (tail_eq_of_cons_eq hr) rfl)
  

Proof state:
α : Type u_1
β : Type u_2
l✝¹ l✝ : List α
x : α
l : List α
y : α
hp : reverse l = l → Palindrome l
hr : reverse [y] ++ reverse l ++ [x] = x :: (l ++ [y])
this : Palindrome l
⊢ Palindrome (x :: (l ++ [x]))

Next tactic:
exact Palindrome.cons_concat x this
theorem append_reverse (l : List α) : Palindrome (l ++ reverse l) := by
  

Proof state:
α : Type u_1
β : Type u_2
l✝ l : List α
⊢ Palindrome (l ++ reverse l)

Next tactic:
apply of_reverse_eq
theorem append_reverse (l : List α) : Palindrome (l ++ reverse l) := by
  apply of_reverse_eq
  

Proof state:
case a
α : Type u_1
β : Type u_2
l✝ l : List α
⊢ reverse (l ++ reverse l) = l ++ reverse l

Next tactic:
rw [reverse_append, reverse_reverse]
protected theorem map (f : α → β) (p : Palindrome l) : Palindrome (map f l) :=
  of_reverse_eq <| by 

Proof state:
α : Type u_1
β : Type u_2
l : List α
f : α → β
p : Palindrome l
⊢ reverse (map f l) = map f l

Next tactic:
rw [← map_reverse, p.reverse_eq]
/--
If the germ of a section `f` is a unit in the stalk at `x`, then `f` must be a unit on some small
neighborhood around `x`.
-/
theorem isUnit_res_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U)) (x : U)
    (h : IsUnit (X.presheaf.germ x f)) :
    ∃ (V : Opens X) (i : V ⟶ U) (_ : x.1 ∈ V), IsUnit (X.presheaf.map i.op f) := by
  

Proof state:
X : RingedSpace
U : Opens ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op U))
x : ↥U
h : IsUnit ((germ X.presheaf x) f)
⊢ ∃ V i, ∃ (_ : ↑x ∈ V), IsUnit ((X.presheaf.map i.op) f)

Next tactic:
obtain ⟨g', heq⟩ := h.exists_right_inv
/--
If the germ of a section `f` is a unit in the stalk at `x`, then `f` must be a unit on some small
neighborhood around `x`.
-/
theorem isUnit_res_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U)) (x : U)
    (h : IsUnit (X.presheaf.germ x f)) :
    ∃ (V : Opens X) (i : V ⟶ U) (_ : x.1 ∈ V), IsUnit (X.presheaf.map i.op f) := by
  obtain ⟨g', heq⟩ := h.exists_right_inv
  

Proof state:
case intro
X : RingedSpace
U : Opens ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op U))
x : ↥U
h : IsUnit ((germ X.presheaf x) f)
g' : (CategoryTheory.forget CommRingCat).obj (stalk X.presheaf ↑x)
heq : (germ X.presheaf x) f * g' = 1
⊢ ∃ V i, ∃ (_ : ↑x ∈ V), IsUnit ((X.presheaf.map i.op) f)

Next tactic:
obtain ⟨V, hxV, g, rfl⟩ := X.presheaf.germ_exist x.1 g'
/--
If the germ of a section `f` is a unit in the stalk at `x`, then `f` must be a unit on some small
neighborhood around `x`.
-/
theorem isUnit_res_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U)) (x : U)
    (h : IsUnit (X.presheaf.germ x f)) :
    ∃ (V : Opens X) (i : V ⟶ U) (_ : x.1 ∈ V), IsUnit (X.presheaf.map i.op f) := by
  obtain ⟨g', heq⟩ := h.exists_right_inv
  obtain ⟨V, hxV, g, rfl⟩ := X.presheaf.germ_exist x.1 g'
  

Proof state:
case intro.intro.intro.intro
X : RingedSpace
U : Opens ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op U))
x : ↥U
h : IsUnit ((germ X.presheaf x) f)
V : Opens ↑↑X.toPresheafedSpace
hxV : ↑x ∈ V
g : (CategoryTheory.forget CommRingCat).obj (X.presheaf.obj (op V))
heq : (germ X.presheaf x) f * (germ X.presheaf { val := ↑x, property := hxV }) g = 1
⊢ ∃ V i, ∃ (_ : ↑x ∈ V), IsUnit ((X.presheaf.map i.op) f)

Next tactic:
let W := U ⊓ V
/--
If the germ of a section `f` is a unit in the stalk at `x`, then `f` must be a unit on some small
neighborhood around `x`.
-/
theorem isUnit_res_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U)) (x : U)
    (h : IsUnit (X.presheaf.germ x f)) :
    ∃ (V : Opens X) (i : V ⟶ U) (_ : x.1 ∈ V), IsUnit (X.presheaf.map i.op f) := by
  obtain ⟨g', heq⟩ := h.exists_right_inv
  obtain ⟨V, hxV, g, rfl⟩ := X.presheaf.germ_exist x.1 g'
  let W := U ⊓ V
  

Proof state:
case intro.intro.intro.intro
X : RingedSpace
U : Opens ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op U))
x : ↥U
h : IsUnit ((germ X.presheaf x) f)
V : Opens ↑↑X.toPresheafedSpace
hxV : ↑x ∈ V
g : (CategoryTheory.forget CommRingCat).obj (X.presheaf.obj (op V))
heq : (germ X.presheaf x) f * (germ X.presheaf { val := ↑x, property := hxV }) g = 1
W : Opens ↑↑X.toPresheafedSpace := U ⊓ V
⊢ ∃ V i, ∃ (_ : ↑x ∈ V), IsUnit ((X.presheaf.map i.op) f)

Next tactic:
have hxW : x.1 ∈ W := ⟨x.2, hxV⟩
/--
If the germ of a section `f` is a unit in the stalk at `x`, then `f` must be a unit on some small
neighborhood around `x`.
-/
theorem isUnit_res_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U)) (x : U)
    (h : IsUnit (X.presheaf.germ x f)) :
    ∃ (V : Opens X) (i : V ⟶ U) (_ : x.1 ∈ V), IsUnit (X.presheaf.map i.op f) := by
  obtain ⟨g', heq⟩ := h.exists_right_inv
  obtain ⟨V, hxV, g, rfl⟩ := X.presheaf.germ_exist x.1 g'
  let W := U ⊓ V
  have hxW : x.1 ∈ W := ⟨x.2, hxV⟩
  -- Porting note : `erw` can't write into `HEq`, so this is replaced with another `HEq` in the
  -- desired form
  

Proof state:
case intro.intro.intro.intro
X : RingedSpace
U : Opens ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op U))
x : ↥U
h : IsUnit ((germ X.presheaf x) f)
V : Opens ↑↑X.toPresheafedSpace
hxV : ↑x ∈ V
g : (CategoryTheory.forget CommRingCat).obj (X.presheaf.obj (op V))
heq : (germ X.presheaf x) f * (germ X.presheaf { val := ↑x, property := hxV }) g = 1
W : Opens ↑↑X.toPresheafedSpace := U ⊓ V
hxW : ↑x ∈ W
⊢ ∃ V i, ∃ (_ : ↑x ∈ V), IsUnit ((X.presheaf.map i.op) f)

Next tactic:
replace heq : (X.presheaf.germ ⟨x.val, hxW⟩) ((X.presheaf.map (U.infLELeft V).op) f *
    (X.presheaf.map (U.infLERight V).op) g) = (X.presheaf.germ ⟨x.val, hxW⟩) 1
/--
If the germ of a section `f` is a unit in the stalk at `x`, then `f` must be a unit on some small
neighborhood around `x`.
-/
theorem isUnit_res_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U)) (x : U)
    (h : IsUnit (X.presheaf.germ x f)) :
    ∃ (V : Opens X) (i : V ⟶ U) (_ : x.1 ∈ V), IsUnit (X.presheaf.map i.op f) := by
  obtain ⟨g', heq⟩ := h.exists_right_inv
  obtain ⟨V, hxV, g, rfl⟩ := X.presheaf.germ_exist x.1 g'
  let W := U ⊓ V
  have hxW : x.1 ∈ W := ⟨x.2, hxV⟩
  -- Porting note : `erw` can't write into `HEq`, so this is replaced with another `HEq` in the
  -- desired form
  replace heq : (X.presheaf.germ ⟨x.val, hxW⟩) ((X.presheaf.map (U.infLELeft V).op) f *
    (X.presheaf.map (U.infLERight V).op) g) = (X.presheaf.germ ⟨x.val, hxW⟩) 1
  · 

Proof state:
case heq
X : RingedSpace
U : Opens ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op U))
x : ↥U
h : IsUnit ((germ X.presheaf x) f)
V : Opens ↑↑X.toPresheafedSpace
hxV : ↑x ∈ V
g : (CategoryTheory.forget CommRingCat).obj (X.presheaf.obj (op V))
heq : (germ X.presheaf x) f * (germ X.presheaf { val := ↑x, property := hxV }) g = 1
W : Opens ↑↑X.toPresheafedSpace := U ⊓ V
hxW : ↑x ∈ W
⊢ (germ X.presheaf { val := ↑x, property := hxW })
      ((X.presheaf.map (Opens.infLELeft U V).op) f * (X.presheaf.map (Opens.infLERight U V).op) g) =
    (germ X.presheaf { val := ↑x, property := hxW }) 1

Next tactic:
dsimp [germ]
/--
If the germ of a section `f` is a unit in the stalk at `x`, then `f` must be a unit on some small
neighborhood around `x`.
-/
theorem isUnit_res_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U)) (x : U)
    (h : IsUnit (X.presheaf.germ x f)) :
    ∃ (V : Opens X) (i : V ⟶ U) (_ : x.1 ∈ V), IsUnit (X.presheaf.map i.op f) := by
  obtain ⟨g', heq⟩ := h.exists_right_inv
  obtain ⟨V, hxV, g, rfl⟩ := X.presheaf.germ_exist x.1 g'
  let W := U ⊓ V
  have hxW : x.1 ∈ W := ⟨x.2, hxV⟩
  -- Porting note : `erw` can't write into `HEq`, so this is replaced with another `HEq` in the
  -- desired form
  replace heq : (X.presheaf.germ ⟨x.val, hxW⟩) ((X.presheaf.map (U.infLELeft V).op) f *
    (X.presheaf.map (U.infLERight V).op) g) = (X.presheaf.germ ⟨x.val, hxW⟩) 1
  · dsimp [germ]
    

Proof state:
case heq
X : RingedSpace
U : Opens ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op U))
x : ↥U
h : IsUnit ((germ X.presheaf x) f)
V : Opens ↑↑X.toPresheafedSpace
hxV : ↑x ∈ V
g : (CategoryTheory.forget CommRingCat).obj (X.presheaf.obj (op V))
heq : (germ X.presheaf x) f * (germ X.presheaf { val := ↑x, property := hxV }) g = 1
W : Opens ↑↑X.toPresheafedSpace := U ⊓ V
hxW : ↑x ∈ W
⊢ (Limits.colimit.ι ((OpenNhds.inclusion ↑x).op ⋙ X.presheaf)
        (op { obj := U ⊓ V, property := (_ : ↑{ val := ↑x, property := hxW } ∈ U ⊓ V) }))
      ((X.presheaf.map (Opens.infLELeft U V).op) f * (X.presheaf.map (Opens.infLERight U V).op) g) =
    (Limits.colimit.ι ((OpenNhds.inclusion ↑x).op ⋙ X.presheaf)
        (op { obj := U ⊓ V, property := (_ : ↑{ val := ↑x, property := hxW } ∈ U ⊓ V) }))
      1

Next tactic:
erw [map_mul, map_one, show X.presheaf.germ ⟨x, hxW⟩ ((X.presheaf.map (U.infLELeft V).op) f) =
      X.presheaf.germ x f from X.presheaf.germ_res_apply (Opens.infLELeft U V) ⟨x.1, hxW⟩ f,
      show X.presheaf.germ ⟨x, hxW⟩ (X.presheaf.map (U.infLERight V).op g) =
      X.presheaf.germ ⟨x, hxV⟩ g from X.presheaf.germ_res_apply (Opens.infLERight U V) ⟨x.1, hxW⟩ g]
/--
If the germ of a section `f` is a unit in the stalk at `x`, then `f` must be a unit on some small
neighborhood around `x`.
-/
theorem isUnit_res_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U)) (x : U)
    (h : IsUnit (X.presheaf.germ x f)) :
    ∃ (V : Opens X) (i : V ⟶ U) (_ : x.1 ∈ V), IsUnit (X.presheaf.map i.op f) := by
  obtain ⟨g', heq⟩ := h.exists_right_inv
  obtain ⟨V, hxV, g, rfl⟩ := X.presheaf.germ_exist x.1 g'
  let W := U ⊓ V
  have hxW : x.1 ∈ W := ⟨x.2, hxV⟩
  -- Porting note : `erw` can't write into `HEq`, so this is replaced with another `HEq` in the
  -- desired form
  replace heq : (X.presheaf.germ ⟨x.val, hxW⟩) ((X.presheaf.map (U.infLELeft V).op) f *
    (X.presheaf.map (U.infLERight V).op) g) = (X.presheaf.germ ⟨x.val, hxW⟩) 1
  · dsimp [germ]
    erw [map_mul, map_one, show X.presheaf.germ ⟨x, hxW⟩ ((X.presheaf.map (U.infLELeft V).op) f) =
      X.presheaf.germ x f from X.presheaf.germ_res_apply (Opens.infLELeft U V) ⟨x.1, hxW⟩ f,
      show X.presheaf.germ ⟨x, hxW⟩ (X.presheaf.map (U.infLERight V).op g) =
      X.presheaf.germ ⟨x, hxV⟩ g from X.presheaf.germ_res_apply (Opens.infLERight U V) ⟨x.1, hxW⟩ g]
    

Proof state:
case heq
X : RingedSpace
U : Opens ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op U))
x : ↥U
h : IsUnit ((germ X.presheaf x) f)
V : Opens ↑↑X.toPresheafedSpace
hxV : ↑x ∈ V
g : (CategoryTheory.forget CommRingCat).obj (X.presheaf.obj (op V))
heq : (germ X.presheaf x) f * (germ X.presheaf { val := ↑x, property := hxV }) g = 1
W : Opens ↑↑X.toPresheafedSpace := U ⊓ V
hxW : ↑x ∈ W
⊢ (germ X.presheaf x) f * (germ X.presheaf { val := ↑x, property := hxV }) g = 1

Next tactic:
exact heq