l3lab/ntp-mathlib huggingface dataset:
theorem degree_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
    (finSuccEquiv R n f).degree = degreeOf 0 f := by
  -- TODO: these should be lemmas
  have h₀ : ∀ {α β : Type _} (f : α → β), (fun x => x) ∘ f = f := fun f => rfl
  have h₁ : ∀ {α β : Type _} (f : α → β), f ∘ (fun x => x) = f := fun f => rfl
  have h₂ : WithBot.some = Nat.cast := rfl

  have h' : ((finSuccEquiv R n f).support.sup fun x => x) = degreeOf 0 f := by
    rw [degreeOf_eq_sup, finSuccEquiv_support f, Finset.sup_image, h₀]
  

Proof state:
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
h : f ≠ 0
h₀ : ∀ {α β : Type} (f : α → β), (fun x => x) ∘ f = f
h₁ : ∀ {α : Type ?u.1456922} {β : Type ?u.1456925} (f : α → β), (f ∘ fun x => x) = f
h₂ : WithBot.some = Nat.cast
h' : (Finset.sup (Polynomial.support ((finSuccEquiv R n) f)) fun x => x) = degreeOf 0 f
⊢ degree ((finSuccEquiv R n) f) = ↑(degreeOf 0 f)

Next tactic:
rw [Polynomial.degree, ← h', ← h₂, Finset.coe_sup_of_nonempty (support_finSuccEquiv_nonempty h),
    Finset.max_eq_sup_coe, h₁]
theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
    (finSuccEquiv R n f).natDegree = degreeOf 0 f := by
  

Proof state:
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
⊢ natDegree ((finSuccEquiv R n) f) = degreeOf 0 f

Next tactic:
by_cases c : f = 0
theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
    (finSuccEquiv R n f).natDegree = degreeOf 0 f := by
  by_cases c : f = 0
  · 

Proof state:
case pos
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
c : f = 0
⊢ natDegree ((finSuccEquiv R n) f) = degreeOf 0 f

Next tactic:
rw [c, (finSuccEquiv R n).map_zero, Polynomial.natDegree_zero, degreeOf_zero]
theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
    (finSuccEquiv R n f).natDegree = degreeOf 0 f := by
  by_cases c : f = 0
  · rw [c, (finSuccEquiv R n).map_zero, Polynomial.natDegree_zero, degreeOf_zero]
  · 

Proof state:
case neg
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
c : ¬f = 0
⊢ natDegree ((finSuccEquiv R n) f) = degreeOf 0 f

Next tactic:
rw [Polynomial.natDegree, degree_finSuccEquiv (by simpa only [Ne.def] )]
theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
    (finSuccEquiv R n f).natDegree = degreeOf 0 f := by
  by_cases c : f = 0
  · rw [c, (finSuccEquiv R n).map_zero, Polynomial.natDegree_zero, degreeOf_zero]
  · rw [Polynomial.natDegree, degree_finSuccEquiv (by 

Proof state:
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
c : ¬f = 0
⊢ f ≠ 0

Next tactic:
simpa only [Ne.def]
theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
    (finSuccEquiv R n f).natDegree = degreeOf 0 f := by
  by_cases c : f = 0
  · rw [c, (finSuccEquiv R n).map_zero, Polynomial.natDegree_zero, degreeOf_zero]
  · rw [Polynomial.natDegree, degree_finSuccEquiv (by simpa only [Ne.def] )]
    

Proof state:
case neg
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
c : ¬f = 0
⊢ WithBot.unbot' 0 ↑(degreeOf 0 f) = degreeOf 0 f

Next tactic:
erw [WithBot.unbot'_coe]
theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
    (finSuccEquiv R n f).natDegree = degreeOf 0 f := by
  by_cases c : f = 0
  · rw [c, (finSuccEquiv R n).map_zero, Polynomial.natDegree_zero, degreeOf_zero]
  · rw [Polynomial.natDegree, degree_finSuccEquiv (by simpa only [Ne.def] )]
    erw [WithBot.unbot'_coe]
    

Proof state:
case neg
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
c : ¬f = 0
⊢ ↑(degreeOf 0 f) = degreeOf 0 f

Next tactic:
simp
theorem degreeOf_coeff_finSuccEquiv (p : MvPolynomial (Fin (n + 1)) R) (j : Fin n) (i : ℕ) :
    degreeOf j (Polynomial.coeff (finSuccEquiv R n p) i) ≤ degreeOf j.succ p := by
  

Proof state:
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
j : Fin n
i : ℕ
⊢ degreeOf j (Polynomial.coeff ((finSuccEquiv R n) p) i) ≤ degreeOf (Fin.succ j) p

Next tactic:
rw [degreeOf_eq_sup, degreeOf_eq_sup, Finset.sup_le_iff]
theorem degreeOf_coeff_finSuccEquiv (p : MvPolynomial (Fin (n + 1)) R) (j : Fin n) (i : ℕ) :
    degreeOf j (Polynomial.coeff (finSuccEquiv R n p) i) ≤ degreeOf j.succ p := by
  rw [degreeOf_eq_sup, degreeOf_eq_sup, Finset.sup_le_iff]
  

Proof state:
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
j : Fin n
i : ℕ
⊢ ∀ b ∈ support (Polynomial.coeff ((finSuccEquiv R n) p) i), b j ≤ Finset.sup (support p) fun m => m (Fin.succ j)

Next tactic:
intro m hm
theorem degreeOf_coeff_finSuccEquiv (p : MvPolynomial (Fin (n + 1)) R) (j : Fin n) (i : ℕ) :
    degreeOf j (Polynomial.coeff (finSuccEquiv R n p) i) ≤ degreeOf j.succ p := by
  rw [degreeOf_eq_sup, degreeOf_eq_sup, Finset.sup_le_iff]
  intro m hm
  

Proof state:
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
j : Fin n
i : ℕ
m : Fin n →₀ ℕ
hm : m ∈ support (Polynomial.coeff ((finSuccEquiv R n) p) i)
⊢ m j ≤ Finset.sup (support p) fun m => m (Fin.succ j)

Next tactic:
rw [← Finsupp.cons_succ j i m]
theorem degreeOf_coeff_finSuccEquiv (p : MvPolynomial (Fin (n + 1)) R) (j : Fin n) (i : ℕ) :
    degreeOf j (Polynomial.coeff (finSuccEquiv R n p) i) ≤ degreeOf j.succ p := by
  rw [degreeOf_eq_sup, degreeOf_eq_sup, Finset.sup_le_iff]
  intro m hm
  rw [← Finsupp.cons_succ j i m]
  

Proof state:
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
j : Fin n
i : ℕ
m : Fin n →₀ ℕ
hm : m ∈ support (Polynomial.coeff ((finSuccEquiv R n) p) i)
⊢ (cons i m) (Fin.succ j) ≤ Finset.sup (support p) fun m => m (Fin.succ j)

Next tactic:
exact Finset.le_sup
    (f := fun (g : Fin (Nat.succ n) →₀ ℕ) => g (Fin.succ j))
    (support_coeff_finSuccEquiv.1 hm)
theorem germ_res_apply (F : X.Presheaf C) {U V : Opens X} (i : U ⟶ V) (x : U) [ConcreteCategory C]
    (s) : germ F x (F.map i.op s) = germ F (i x) s := by 

Proof state:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
U V : Opens ↑X
i : U ⟶ V
x : ↥U
inst✝ : ConcreteCategory C
s : (forget C).obj (F.obj (op V))
⊢ (germ F x) ((F.map i.op) s) = (germ F ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑V) }) x)) s

Next tactic:
rw [← comp_apply, germ_res]
/-- A morphism from the stalk of `F` at `x` to some object `Y` is completely determined by its
composition with the `germ` morphisms.
-/
@[ext]
theorem stalk_hom_ext (F : X.Presheaf C) {x} {Y : C} {f₁ f₂ : F.stalk x ⟶ Y}
    (ih : ∀ (U : Opens X) (hxU : x ∈ U), F.germ ⟨x, hxU⟩ ≫ f₁ = F.germ ⟨x, hxU⟩ ≫ f₂) : f₁ = f₂ :=
  colimit.hom_ext fun U => by
    

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y✝ Z : TopCat
F : Presheaf C X
x : ↑X
Y : C
f₁ f₂ : stalk F x ⟶ Y
ih :
  ∀ (U : Opens ↑X) (hxU : x ∈ U), germ F { val := x, property := hxU } ≫ f₁ = germ F { val := x, property := hxU } ≫ f₂
U : (OpenNhds x)ᵒᵖ
⊢ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U ≫ f₁ =
    colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) U ≫ f₂

Next tactic:
induction' U using Opposite.rec with U
/-- A morphism from the stalk of `F` at `x` to some object `Y` is completely determined by its
composition with the `germ` morphisms.
-/
@[ext]
theorem stalk_hom_ext (F : X.Presheaf C) {x} {Y : C} {f₁ f₂ : F.stalk x ⟶ Y}
    (ih : ∀ (U : Opens X) (hxU : x ∈ U), F.germ ⟨x, hxU⟩ ≫ f₁ = F.germ ⟨x, hxU⟩ ≫ f₂) : f₁ = f₂ :=
  colimit.hom_ext fun U => by
    induction' U using Opposite.rec with U; 

Proof state:
case mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y✝ Z : TopCat
F : Presheaf C X
x : ↑X
Y : C
f₁ f₂ : stalk F x ⟶ Y
ih :
  ∀ (U : Opens ↑X) (hxU : x ∈ U), germ F { val := x, property := hxU } ≫ f₁ = germ F { val := x, property := hxU } ≫ f₂
U : OpenNhds x
⊢ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) { unop := U } ≫ f₁ =
    colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) { unop := U } ≫ f₂

Next tactic:
cases' U with U hxU
/-- A morphism from the stalk of `F` at `x` to some object `Y` is completely determined by its
composition with the `germ` morphisms.
-/
@[ext]
theorem stalk_hom_ext (F : X.Presheaf C) {x} {Y : C} {f₁ f₂ : F.stalk x ⟶ Y}
    (ih : ∀ (U : Opens X) (hxU : x ∈ U), F.germ ⟨x, hxU⟩ ≫ f₁ = F.germ ⟨x, hxU⟩ ≫ f₂) : f₁ = f₂ :=
  colimit.hom_ext fun U => by
    induction' U using Opposite.rec with U; cases' U with U hxU; 

Proof state:
case mk.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y✝ Z : TopCat
F : Presheaf C X
x : ↑X
Y : C
f₁ f₂ : stalk F x ⟶ Y
ih :
  ∀ (U : Opens ↑X) (hxU : x ∈ U), germ F { val := x, property := hxU } ≫ f₁ = germ F { val := x, property := hxU } ≫ f₂
U : Opens ↑X
hxU : x ∈ U
⊢ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F)
        { unop := { obj := U, property := hxU } } ≫
      f₁ =
    colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F)
        { unop := { obj := U, property := hxU } } ≫
      f₂

Next tactic:
exact ih U hxU
/-- For a presheaf `F` on a space `X`, a continuous map `f : X ⟶ Y` induces a morphisms between the
stalk of `f _ * F` at `f x` and the stalk of `F` at `x`.
-/
def stalkPushforward (f : X ⟶ Y) (F : X.Presheaf C) (x : X) : (f _* F).stalk (f x) ⟶ F.stalk x := by
  -- This is a hack; Lean doesn't like to elaborate the term written directly.
  -- Porting note: The original proof was `trans; swap`, but `trans` does nothing.
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C X
x : ↑X
⊢ stalk (f _* F) (f x) ⟶ stalk F x

Next tactic:
refine' ?_ ≫ colimit.pre _ (OpenNhds.map f x).op
/-- For a presheaf `F` on a space `X`, a continuous map `f : X ⟶ Y` induces a morphisms between the
stalk of `f _ * F` at `f x` and the stalk of `F` at `x`.
-/
def stalkPushforward (f : X ⟶ Y) (F : X.Presheaf C) (x : X) : (f _* F).stalk (f x) ⟶ F.stalk x := by
  -- This is a hack; Lean doesn't like to elaborate the term written directly.
  -- Porting note: The original proof was `trans; swap`, but `trans` does nothing.
  refine' ?_ ≫ colimit.pre _ (OpenNhds.map f x).op
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C X
x : ↑X
⊢ stalk (f _* F) (f x) ⟶
    colimit
      ((OpenNhds.map f x).op ⋙ ((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F)

Next tactic:
exact colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F)
@[reassoc (attr := simp), elementwise (attr := simp)]
theorem stalkPushforward_germ (f : X ⟶ Y) (F : X.Presheaf C) (U : Opens Y)
    (x : (Opens.map f).obj U) :
      (f _* F).germ ⟨(f : X → Y) (x : X), x.2⟩ ≫ F.stalkPushforward C f x = F.germ x := by
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C X
U : Opens ↑Y
x : ↥((Opens.map f).obj U)
⊢ germ (f _* F) { val := f ↑x, property := (_ : ↑x ∈ (Opens.map f).obj U) } ≫ stalkPushforward C f F ↑x = germ F x

Next tactic:
rw [stalkPushforward, germ, colimit.ι_map_assoc, colimit.ι_pre, whiskerRight_app]
@[reassoc (attr := simp), elementwise (attr := simp)]
theorem stalkPushforward_germ (f : X ⟶ Y) (F : X.Presheaf C) (U : Opens Y)
    (x : (Opens.map f).obj U) :
      (f _* F).germ ⟨(f : X → Y) (x : X), x.2⟩ ≫ F.stalkPushforward C f x = F.germ x := by
  rw [stalkPushforward, germ, colimit.ι_map_assoc, colimit.ι_pre, whiskerRight_app]
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C X
U : Opens ↑Y
x : ↥((Opens.map f).obj U)
⊢ F.map
        ((NatTrans.op (OpenNhds.inclusionMapIso f ↑x).inv).app
          (op { obj := U, property := (_ : ↑{ val := f ↑x, property := (_ : ↑x ∈ (Opens.map f).obj U) } ∈ U) })) ≫
      colimit.ι (((whiskeringLeft (OpenNhds ↑x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion ↑x).op).obj F)
        ((OpenNhds.map f ↑x).op.obj
          (op { obj := U, property := (_ : ↑{ val := f ↑x, property := (_ : ↑x ∈ (Opens.map f).obj U) } ∈ U) })) =
    germ F x

Next tactic:
erw [CategoryTheory.Functor.map_id, Category.id_comp]
@[reassoc (attr := simp), elementwise (attr := simp)]
theorem stalkPushforward_germ (f : X ⟶ Y) (F : X.Presheaf C) (U : Opens Y)
    (x : (Opens.map f).obj U) :
      (f _* F).germ ⟨(f : X → Y) (x : X), x.2⟩ ≫ F.stalkPushforward C f x = F.germ x := by
  rw [stalkPushforward, germ, colimit.ι_map_assoc, colimit.ι_pre, whiskerRight_app]
  erw [CategoryTheory.Functor.map_id, Category.id_comp]
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C X
U : Opens ↑Y
x : ↥((Opens.map f).obj U)
⊢ colimit.ι (((whiskeringLeft (OpenNhds ↑x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion ↑x).op).obj F)
      ((OpenNhds.map f ↑x).op.obj
        (op { obj := U, property := (_ : ↑{ val := f ↑x, property := (_ : ↑x ∈ (Opens.map f).obj U) } ∈ U) })) =
    germ F x

Next tactic:
rfl
@[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
    ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom := by
  -- Porting note: We need to this to help ext tactic.
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
x : ↑X
⊢ stalkPushforward C (𝟙 X) ℱ x = (stalkFunctor C x).map (Pushforward.id ℱ).hom

Next tactic:
change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
@[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
    ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom := by
  -- Porting note: We need to this to help ext tactic.
  change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
x : ↑X
⊢ stalkPushforward C (𝟙 X) ℱ x = (stalkFunctor C x).map (Pushforward.id ℱ).hom

Next tactic:
ext1 j
@[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
    ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom := by
  -- Porting note: We need to this to help ext tactic.
  change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
  ext1 j
  

Proof state:
case w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
x : ↑X
j : (OpenNhds ((𝟙 X) x))ᵒᵖ
⊢ colimit.ι
        (((whiskeringLeft (OpenNhds ((𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion ((𝟙 X) x)).op).obj (𝟙 X _* ℱ))
        j ≫
      stalkPushforward C (𝟙 X) ℱ x =
    colimit.ι
        (((whiskeringLeft (OpenNhds ((𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion ((𝟙 X) x)).op).obj (𝟙 X _* ℱ))
        j ≫
      (stalkFunctor C x).map (Pushforward.id ℱ).hom

Next tactic:
induction' j with j
@[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
    ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom := by
  -- Porting note: We need to this to help ext tactic.
  change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
  ext1 j
  induction' j with j
  

Proof state:
case w.h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
x : ↑X
j : OpenNhds ((𝟙 X) x)
⊢ colimit.ι
        (((whiskeringLeft (OpenNhds ((𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion ((𝟙 X) x)).op).obj (𝟙 X _* ℱ))
        (op j) ≫
      stalkPushforward C (𝟙 X) ℱ x =
    colimit.ι
        (((whiskeringLeft (OpenNhds ((𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion ((𝟙 X) x)).op).obj (𝟙 X _* ℱ))
        (op j) ≫
      (stalkFunctor C x).map (Pushforward.id ℱ).hom

Next tactic:
rcases j with ⟨⟨_, _⟩, _⟩
@[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
    ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom := by
  -- Porting note: We need to this to help ext tactic.
  change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
  ext1 j
  induction' j with j
  rcases j with ⟨⟨_, _⟩, _⟩
  

Proof state:
case w.h.mk.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
x : ↑X
carrier✝ : Set ↑X
is_open'✝ : IsOpen carrier✝
property✝ : (𝟙 X) x ∈ { carrier := carrier✝, is_open' := is_open'✝ }
⊢ colimit.ι
        (((whiskeringLeft (OpenNhds ((𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion ((𝟙 X) x)).op).obj (𝟙 X _* ℱ))
        (op { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ }) ≫
      stalkPushforward C (𝟙 X) ℱ x =
    colimit.ι
        (((whiskeringLeft (OpenNhds ((𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion ((𝟙 X) x)).op).obj (𝟙 X _* ℱ))
        (op { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ }) ≫
      (stalkFunctor C x).map (Pushforward.id ℱ).hom

Next tactic:
erw [colimit.ι_map_assoc]
@[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
    ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom := by
  -- Porting note: We need to this to help ext tactic.
  change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
  ext1 j
  induction' j with j
  rcases j with ⟨⟨_, _⟩, _⟩
  erw [colimit.ι_map_assoc]
  

Proof state:
case w.h.mk.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
x : ↑X
carrier✝ : Set ↑X
is_open'✝ : IsOpen carrier✝
property✝ : (𝟙 X) x ∈ { carrier := carrier✝, is_open' := is_open'✝ }
⊢ (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso (𝟙 X) x).inv) ℱ).app
        (op { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ }) ≫
      colimit.ι
          ((OpenNhds.map (𝟙 X) x).op ⋙
            ((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj ℱ)
          (op { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ }) ≫
        colimit.pre (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj ℱ)
          (OpenNhds.map (𝟙 X) x).op =
    colimit.ι
        (((whiskeringLeft (OpenNhds ((𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion ((𝟙 X) x)).op).obj (𝟙 X _* ℱ))
        (op { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ }) ≫
      (stalkFunctor C x).map (Pushforward.id ℱ).hom

Next tactic:
simp [stalkFunctor, stalkPushforward]
@[simp]
theorem comp (ℱ : X.Presheaf C) (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
    ℱ.stalkPushforward C (f ≫ g) x =
      (f _* ℱ).stalkPushforward C g (f x) ≫ ℱ.stalkPushforward C f x := by
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
f : X ⟶ Y
g : Y ⟶ Z
x : ↑X
⊢ stalkPushforward C (f ≫ g) ℱ x = stalkPushforward C g (f _* ℱ) (f x) ≫ stalkPushforward C f ℱ x

Next tactic:
change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
@[simp]
theorem comp (ℱ : X.Presheaf C) (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
    ℱ.stalkPushforward C (f ≫ g) x =
      (f _* ℱ).stalkPushforward C g (f x) ≫ ℱ.stalkPushforward C f x := by
  change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
f : X ⟶ Y
g : Y ⟶ Z
x : ↑X
⊢ stalkPushforward C (f ≫ g) ℱ x = stalkPushforward C g (f _* ℱ) (f x) ≫ stalkPushforward C f ℱ x

Next tactic:
ext U
@[simp]
theorem comp (ℱ : X.Presheaf C) (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
    ℱ.stalkPushforward C (f ≫ g) x =
      (f _* ℱ).stalkPushforward C g (f x) ≫ ℱ.stalkPushforward C f x := by
  change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
  ext U
  

Proof state:
case w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
f : X ⟶ Y
g : Y ⟶ Z
x : ↑X
U : (OpenNhds ((f ≫ g) x))ᵒᵖ
⊢ colimit.ι
        (((whiskeringLeft (OpenNhds ((f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj (OpenNhds.inclusion ((f ≫ g) x)).op).obj
          ((f ≫ g) _* ℱ))
        U ≫
      stalkPushforward C (f ≫ g) ℱ x =
    colimit.ι
        (((whiskeringLeft (OpenNhds ((f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj (OpenNhds.inclusion ((f ≫ g) x)).op).obj
          ((f ≫ g) _* ℱ))
        U ≫
      stalkPushforward C g (f _* ℱ) (f x) ≫ stalkPushforward C f ℱ x

Next tactic:
rcases U with ⟨⟨_, _⟩, _⟩
@[simp]
theorem comp (ℱ : X.Presheaf C) (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
    ℱ.stalkPushforward C (f ≫ g) x =
      (f _* ℱ).stalkPushforward C g (f x) ≫ ℱ.stalkPushforward C f x := by
  change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
  ext U
  rcases U with ⟨⟨_, _⟩, _⟩
  

Proof state:
case w.mk.mk.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
f : X ⟶ Y
g : Y ⟶ Z
x : ↑X
carrier✝ : Set ↑Z
is_open'✝ : IsOpen carrier✝
property✝ : (f ≫ g) x ∈ { carrier := carrier✝, is_open' := is_open'✝ }
⊢ colimit.ι
        (((whiskeringLeft (OpenNhds ((f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj (OpenNhds.inclusion ((f ≫ g) x)).op).obj
          ((f ≫ g) _* ℱ))
        { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫
      stalkPushforward C (f ≫ g) ℱ x =
    colimit.ι
        (((whiskeringLeft (OpenNhds ((f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj (OpenNhds.inclusion ((f ≫ g) x)).op).obj
          ((f ≫ g) _* ℱ))
        { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫
      stalkPushforward C g (f _* ℱ) (f x) ≫ stalkPushforward C f ℱ x

Next tactic:
simp [stalkFunctor, stalkPushforward]
theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
    (x : X) : IsIso (F.stalkPushforward _ f x) := by
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
⊢ IsIso (stalkPushforward C f F x)

Next tactic:
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
    (x : X) : IsIso (F.stalkPushforward _ f x) := by
  haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
⊢ IsIso (stalkPushforward C f F x)

Next tactic:
convert IsIso.of_iso
      ((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
              ((OpenNhds.inclusion (f x)).op ⋙ f _* F) :
            _).symm ≪≫
        colim.mapIso _)