l3lab/ntp-mathlib huggingface dataset:
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  dsimp at wUx; erw [wUx] at e; clear wUx
  have wVx := wV ⟨x, mV⟩
  dsimp at wVx; 

Proof state:
case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
e : germ F { val := x, property := mU } gU = fV { val := x, property := (_ : x ∈ V.obj) }
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
wVx : fV { val := x, property := (_ : ↑{ val := x, property := mV } ∈ ↑V.obj) } = germ F { val := x, property := mV } gV
⊢ ∃ W iU iV,
    ∀ (w : ↥W.obj),
      fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.obj) }) w) =
        fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.obj) }) w)

Next tactic:
erw [wVx] at e
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  dsimp at wUx; erw [wUx] at e; clear wUx
  have wVx := wV ⟨x, mV⟩
  dsimp at wVx; erw [wVx] at e; 

Proof state:
case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
e : germ F { val := x, property := mU } gU = germ F { val := x, property := mV } gV
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
wVx : fV { val := x, property := (_ : ↑{ val := x, property := mV } ∈ ↑V.obj) } = germ F { val := x, property := mV } gV
⊢ ∃ W iU iV,
    ∀ (w : ↥W.obj),
      fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.obj) }) w) =
        fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.obj) }) w)

Next tactic:
clear wVx
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  dsimp at wUx; erw [wUx] at e; clear wUx
  have wVx := wV ⟨x, mV⟩
  dsimp at wVx; erw [wVx] at e; clear wVx
  

Proof state:
case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
e : germ F { val := x, property := mU } gU = germ F { val := x, property := mV } gV
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
⊢ ∃ W iU iV,
    ∀ (w : ↥W.obj),
      fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.obj) }) w) =
        fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.obj) }) w)

Next tactic:
rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', (e' : F.map iU'.op gU = F.map iV'.op gV)⟩
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  dsimp at wUx; erw [wUx] at e; clear wUx
  have wVx := wV ⟨x, mV⟩
  dsimp at wVx; erw [wVx] at e; clear wVx
  rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', (e' : F.map iU'.op gU = F.map iV'.op gV)⟩
  

Proof state:
case w.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
e : germ F { val := x, property := mU } gU = germ F { val := x, property := mV } gV
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ ∃ W iU iV,
    ∀ (w : ↥W.obj),
      fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.obj) }) w) =
        fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.obj) }) w)

Next tactic:
use ⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  dsimp at wUx; erw [wUx] at e; clear wUx
  have wVx := wV ⟨x, mV⟩
  dsimp at wVx; erw [wVx] at e; clear wVx
  rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', (e' : F.map iU'.op gU = F.map iV'.op gV)⟩
  use ⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩
  

Proof state:
case h
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
e : germ F { val := x, property := mU } gU = germ F { val := x, property := mV } gV
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ ∃ iU iV,
    ∀ (w : ↥{ obj := W ⊓ (U' ⊓ V'), property := (_ : x ∈ ↑W ∧ x ∈ ↑(U' ⊓ V')) }.obj),
      fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.obj) }) w) =
        fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.obj) }) w)

Next tactic:
refine' ⟨_, _, _⟩
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  dsimp at wUx; erw [wUx] at e; clear wUx
  have wVx := wV ⟨x, mV⟩
  dsimp at wVx; erw [wVx] at e; clear wVx
  rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', (e' : F.map iU'.op gU = F.map iV'.op gV)⟩
  use ⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩
  refine' ⟨_, _, _⟩
  · 

Proof state:
case h.refine'_1
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
e : germ F { val := x, property := mU } gU = germ F { val := x, property := mV } gV
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ { obj := W ⊓ (U' ⊓ V'), property := (_ : x ∈ ↑W ∧ x ∈ ↑(U' ⊓ V')) } ⟶ U

Next tactic:
change W ⊓ (U' ⊓ V') ⟶ U.obj
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  dsimp at wUx; erw [wUx] at e; clear wUx
  have wVx := wV ⟨x, mV⟩
  dsimp at wVx; erw [wVx] at e; clear wVx
  rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', (e' : F.map iU'.op gU = F.map iV'.op gV)⟩
  use ⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩
  refine' ⟨_, _, _⟩
  · change W ⊓ (U' ⊓ V') ⟶ U.obj
    

Proof state:
case h.refine'_1
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
e : germ F { val := x, property := mU } gU = germ F { val := x, property := mV } gV
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ W ⊓ (U' ⊓ V') ⟶ U.obj

Next tactic:
exact Opens.infLERight _ _ ≫ Opens.infLELeft _ _ ≫ iU
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  dsimp at wUx; erw [wUx] at e; clear wUx
  have wVx := wV ⟨x, mV⟩
  dsimp at wVx; erw [wVx] at e; clear wVx
  rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', (e' : F.map iU'.op gU = F.map iV'.op gV)⟩
  use ⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩
  refine' ⟨_, _, _⟩
  · change W ⊓ (U' ⊓ V') ⟶ U.obj
    exact Opens.infLERight _ _ ≫ Opens.infLELeft _ _ ≫ iU
  · 

Proof state:
case h.refine'_2
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
e : germ F { val := x, property := mU } gU = germ F { val := x, property := mV } gV
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ { obj := W ⊓ (U' ⊓ V'), property := (_ : x ∈ ↑W ∧ x ∈ ↑(U' ⊓ V')) } ⟶ V

Next tactic:
change W ⊓ (U' ⊓ V') ⟶ V.obj
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  dsimp at wUx; erw [wUx] at e; clear wUx
  have wVx := wV ⟨x, mV⟩
  dsimp at wVx; erw [wVx] at e; clear wVx
  rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', (e' : F.map iU'.op gU = F.map iV'.op gV)⟩
  use ⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩
  refine' ⟨_, _, _⟩
  · change W ⊓ (U' ⊓ V') ⟶ U.obj
    exact Opens.infLERight _ _ ≫ Opens.infLELeft _ _ ≫ iU
  · change W ⊓ (U' ⊓ V') ⟶ V.obj
    

Proof state:
case h.refine'_2
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
e : germ F { val := x, property := mU } gU = germ F { val := x, property := mV } gV
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ W ⊓ (U' ⊓ V') ⟶ V.obj

Next tactic:
exact Opens.infLERight _ _ ≫ Opens.infLERight _ _ ≫ iV
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  dsimp at wUx; erw [wUx] at e; clear wUx
  have wVx := wV ⟨x, mV⟩
  dsimp at wVx; erw [wVx] at e; clear wVx
  rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', (e' : F.map iU'.op gU = F.map iV'.op gV)⟩
  use ⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩
  refine' ⟨_, _, _⟩
  · change W ⊓ (U' ⊓ V') ⟶ U.obj
    exact Opens.infLERight _ _ ≫ Opens.infLELeft _ _ ≫ iU
  · change W ⊓ (U' ⊓ V') ⟶ V.obj
    exact Opens.infLERight _ _ ≫ Opens.infLERight _ _ ≫ iV
  · 

Proof state:
case h.refine'_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
e : germ F { val := x, property := mU } gU = germ F { val := x, property := mV } gV
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ ∀ (w : ↥{ obj := W ⊓ (U' ⊓ V'), property := (_ : x ∈ ↑W ∧ x ∈ ↑(U' ⊓ V')) }.obj),
    fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.obj) }) w) =
      fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.obj) }) w)

Next tactic:
intro w
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  dsimp at wUx; erw [wUx] at e; clear wUx
  have wVx := wV ⟨x, mV⟩
  dsimp at wVx; erw [wVx] at e; clear wVx
  rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', (e' : F.map iU'.op gU = F.map iV'.op gV)⟩
  use ⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩
  refine' ⟨_, _, _⟩
  · change W ⊓ (U' ⊓ V') ⟶ U.obj
    exact Opens.infLERight _ _ ≫ Opens.infLELeft _ _ ≫ iU
  · change W ⊓ (U' ⊓ V') ⟶ V.obj
    exact Opens.infLERight _ _ ≫ Opens.infLERight _ _ ≫ iV
  · intro w
    

Proof state:
case h.refine'_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
wU :
  ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑U.obj) }) x_2)) x_1 = germ F x_1 gU
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
e : germ F { val := x, property := mU } gU = germ F { val := x, property := mV } gV
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
w : ↥{ obj := W ⊓ (U' ⊓ V'), property := (_ : x ∈ ↑W ∧ x ∈ ↑(U' ⊓ V')) }.obj
⊢ fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.obj) }) w) =
    fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.obj) }) w)

Next tactic:
specialize wU ⟨w.1, w.2.2.1⟩
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  dsimp at wUx; erw [wUx] at e; clear wUx
  have wVx := wV ⟨x, mV⟩
  dsimp at wVx; erw [wVx] at e; clear wVx
  rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', (e' : F.map iU'.op gU = F.map iV'.op gV)⟩
  use ⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩
  refine' ⟨_, _, _⟩
  · change W ⊓ (U' ⊓ V') ⟶ U.obj
    exact Opens.infLERight _ _ ≫ Opens.infLELeft _ _ ≫ iU
  · change W ⊓ (U' ⊓ V') ⟶ V.obj
    exact Opens.infLERight _ _ ≫ Opens.infLERight _ _ ≫ iV
  · intro w
    specialize wU ⟨w.1, w.2.2.1⟩
    

Proof state:
case h.refine'_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
e : germ F { val := x, property := mU } gU = germ F { val := x, property := mV } gV
wV :
  ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => { val := ↑x_3, property := (_ : ↑x_3 ∈ ↑V.obj) }) x_2)) x_1 = germ F x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
w : ↥{ obj := W ⊓ (U' ⊓ V'), property := (_ : x ∈ ↑W ∧ x ∈ ↑(U' ⊓ V')) }.obj
wU :
  (fun x_1 => fU ((fun x_2 => { val := ↑x_2, property := (_ : ↑x_2 ∈ ↑U.obj) }) x_1))
      { val := ↑w, property := (_ : ↑w ∈ ↑U') } =
    germ F { val := ↑w, property := (_ : ↑w ∈ ↑U') } gU
⊢ fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.obj) }) w) =
    fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.obj) }) w)

Next tactic:
specialize wV ⟨w.1, w.2.2.2⟩
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by
  apply TopCat.stalkToFiber_injective
  intro U V fU hU fV hV e
  rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
  rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
  have wUx := wU ⟨x, mU⟩
  dsimp at wUx; erw [wUx] at e; clear wUx
  have wVx := wV ⟨x, mV⟩
  dsimp at wVx; erw [wVx] at e; clear wVx
  rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', (e' : F.map iU'.op gU = F.map iV'.op gV)⟩
  use ⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩
  refine' ⟨_, _, _⟩
  · change W ⊓ (U' ⊓ V') ⟶ U.obj
    exact Opens.infLERight _ _ ≫ Opens.infLELeft _ _ ≫ iU
  · change W ⊓ (U' ⊓ V') ⟶ V.obj
    exact Opens.infLERight _ _ ≫ Opens.infLERight _ _ ≫ iV
  · intro w
    specialize wU ⟨w.1, w.2.2.1⟩
    specialize wV ⟨w.1, w.2.2.2⟩
    

Proof state:
case h.refine'_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → stalk F ↑y
hU : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fU
fV : (y : ↥V.obj) → stalk F ↑y
hV : PrelocalPredicate.pred (Sheafify.isLocallyGerm F).toPrelocalPredicate fV
U' : Opens ↑X
mU : ↑{ val := x, property := (_ : x ∈ U.obj) } ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
V' : Opens ↑X
mV : ↑{ val := x, property := (_ : x ∈ V.obj) } ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
e : germ F { val := x, property := mU } gU = germ F { val := x, property := mV } gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
w : ↥{ obj := W ⊓ (U' ⊓ V'), property := (_ : x ∈ ↑W ∧ x ∈ ↑(U' ⊓ V')) }.obj
wU :
  (fun x_1 => fU ((fun x_2 => { val := ↑x_2, property := (_ : ↑x_2 ∈ ↑U.obj) }) x_1))
      { val := ↑w, property := (_ : ↑w ∈ ↑U') } =
    germ F { val := ↑w, property := (_ : ↑w ∈ ↑U') } gU
wV :
  (fun x_1 => fV ((fun x_2 => { val := ↑x_2, property := (_ : ↑x_2 ∈ ↑V.obj) }) x_1))
      { val := ↑w, property := (_ : ↑w ∈ ↑V') } =
    germ F { val := ↑w, property := (_ : ↑w ∈ ↑V') } gV
⊢ fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.obj) }) w) =
    fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.obj) }) w)

Next tactic:
dsimp at wU wV ⊢