awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Sheaf.Smooth (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
EM : Type u_2
inst_1 : NormedAddCommGroup EM
inst_2 : NormedSpace 𝕜 EM
HM : Type u_3
inst_3 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace 𝕜 E
H : Type u_5
inst_6 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_6
inst_7 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E H'
M : Type u
inst_8 : TopologicalSpace M
inst_9 : ChartedSpace HM M
N G A A' R : Type u
inst_10 : TopologicalSpace N
inst_11 : ChartedSpace H N
inst_12 : TopologicalSpace G
inst_13 : ChartedSpace H G
inst_14 : TopologicalSpace A
inst_15 : ChartedSpace H A
inst_16 : TopologicalSpace A'
inst_17 : ChartedSpace H' A'
inst_18 : TopologicalSpace R
inst_19 : ChartedSpace H R
EP : Type u_7
inst_20 : NormedAddCommGroup EP
inst_21 : NormedSpace 𝕜 EP
HP : Type u_8
inst_22 : TopologicalSpace HP
IP : ModelWithCorners 𝕜 EP HP
P : Type u
inst_23 : TopologicalSpace P
inst_24 : ChartedSpace HP P
inst_25 : Group G
inst_26 : LieGroup I ∞ G
⊢ CategoryTheory.Presheaf.IsSheaf (C := Opens (↑(TopCat.of M) : Type u))
    (Opens.grothendieckTopology (↑(TopCat.of M) : Type u)) (smoothPresheafGroup IM I M G)
have rw : CategoryTheory.Presheaf.IsSheaf (C := Opens (↑(TopCat.of M) : Type u))
  (Opens.grothendieckTopology (↑(TopCat.of M) : Type u))
  (CategoryTheory.Functor.comp (smoothPresheafGroup IM I M G)
    (CategoryTheory.forget GrpCat (FC := fun (x1 x2 : GrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
      GrpCat.carrier)))
Tactic: rw [CategoryTheory.Presheaf.isSheaf_iff_isSheaf_forget _ _ (CategoryTheory.forget GrpCat)]
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Sheaf.Smooth (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
EM : Type u_2
inst_1 : NormedAddCommGroup EM
inst_2 : NormedSpace 𝕜 EM
HM : Type u_3
inst_3 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace 𝕜 E
H : Type u_5
inst_6 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_6
inst_7 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E H'
M : Type u
inst_8 : TopologicalSpace M
inst_9 : ChartedSpace HM M
N G A A' R : Type u
inst_10 : TopologicalSpace N
inst_11 : ChartedSpace H N
inst_12 : TopologicalSpace G
inst_13 : ChartedSpace H G
inst_14 : TopologicalSpace A
inst_15 : ChartedSpace H A
inst_16 : TopologicalSpace A'
inst_17 : ChartedSpace H' A'
inst_18 : TopologicalSpace R
inst_19 : ChartedSpace H R
EP : Type u_7
inst_20 : NormedAddCommGroup EP
inst_21 : NormedSpace 𝕜 EP
HP : Type u_8
inst_22 : TopologicalSpace HP
IP : ModelWithCorners 𝕜 EP HP
P : Type u
inst_23 : TopologicalSpace P
inst_24 : ChartedSpace HP P
inst_25 : CommGroup A
inst_26 : CommGroup A'
inst_27 : LieGroup I ∞ A
inst_28 : LieGroup I' ∞ A'
⊢ CategoryTheory.Presheaf.IsSheaf (C := Opens (↑(TopCat.of M) : Type u))
    (Opens.grothendieckTopology (↑(TopCat.of M) : Type u)) (smoothPresheafCommGroup IM I M A)
have rw : CategoryTheory.Presheaf.IsSheaf (C := Opens (↑(TopCat.of M) : Type u))
  (Opens.grothendieckTopology (↑(TopCat.of M) : Type u))
  (CategoryTheory.Functor.comp (smoothPresheafCommGroup IM I M A)
    (CategoryTheory.forget CommGrpCat (FC := fun (x1 x2 : CommGrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
      CommGrpCat.carrier)))
Tactic: rw [CategoryTheory.Presheaf.isSheaf_iff_isSheaf_forget _ _
        (CategoryTheory.forget CommGrpCat)]
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Sheaf.Smooth (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
EM : Type u_2
inst_1 : NormedAddCommGroup EM
inst_2 : NormedSpace 𝕜 EM
HM : Type u_3
inst_3 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace 𝕜 E
H : Type u_5
inst_6 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_6
inst_7 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E H'
M : Type u
inst_8 : TopologicalSpace M
inst_9 : ChartedSpace HM M
N G A A' R : Type u
inst_10 : TopologicalSpace N
inst_11 : ChartedSpace H N
inst_12 : TopologicalSpace G
inst_13 : ChartedSpace H G
inst_14 : TopologicalSpace A
inst_15 : ChartedSpace H A
inst_16 : TopologicalSpace A'
inst_17 : ChartedSpace H' A'
inst_18 : TopologicalSpace R
inst_19 : ChartedSpace H R
EP : Type u_7
inst_20 : NormedAddCommGroup EP
inst_21 : NormedSpace 𝕜 EP
HP : Type u_8
inst_22 : TopologicalSpace HP
IP : ModelWithCorners 𝕜 EP HP
P : Type u
inst_23 : TopologicalSpace P
inst_24 : ChartedSpace HP P
inst_25 : Ring R
inst_26 : ContMDiffRing I ∞ R
⊢ CategoryTheory.Presheaf.IsSheaf (C := Opens (↑(TopCat.of M) : Type u))
    (Opens.grothendieckTopology (↑(TopCat.of M) : Type u)) (smoothPresheafRing IM I M R)
have rw : CategoryTheory.Presheaf.IsSheaf (C := Opens (↑(TopCat.of M) : Type u))
  (Opens.grothendieckTopology (↑(TopCat.of M) : Type u))
  (CategoryTheory.Functor.comp (smoothPresheafRing IM I M R)
    (CategoryTheory.forget RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
      RingCat.carrier)))
Tactic: rw [CategoryTheory.Presheaf.isSheaf_iff_isSheaf_forget _ _ (CategoryTheory.forget RingCat)]
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Sheaf.Smooth (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
EM : Type u_2
inst_1 : NormedAddCommGroup EM
inst_2 : NormedSpace 𝕜 EM
HM : Type u_3
inst_3 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace 𝕜 E
H : Type u_5
inst_6 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_6
inst_7 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E H'
M : Type u
inst_8 : TopologicalSpace M
inst_9 : ChartedSpace HM M
N G A A' R : Type u
inst_10 : TopologicalSpace N
inst_11 : ChartedSpace H N
inst_12 : TopologicalSpace G
inst_13 : ChartedSpace H G
inst_14 : TopologicalSpace A
inst_15 : ChartedSpace H A
inst_16 : TopologicalSpace A'
inst_17 : ChartedSpace H' A'
inst_18 : TopologicalSpace R
inst_19 : ChartedSpace H R
EP : Type u_7
inst_20 : NormedAddCommGroup EP
inst_21 : NormedSpace 𝕜 EP
HP : Type u_8
inst_22 : TopologicalSpace HP
IP : ModelWithCorners 𝕜 EP HP
P : Type u
inst_23 : TopologicalSpace P
inst_24 : ChartedSpace HP P
inst_25 : CommRing R
inst_26 : ContMDiffRing I ∞ R
⊢ CategoryTheory.Presheaf.IsSheaf (C := Opens (↑(TopCat.of M) : Type u))
    (Opens.grothendieckTopology (↑(TopCat.of M) : Type u)) (smoothPresheafCommRing IM I M R)
have rw : CategoryTheory.Presheaf.IsSheaf (C := Opens (↑(TopCat.of M) : Type u))
  (Opens.grothendieckTopology (↑(TopCat.of M) : Type u))
  (CategoryTheory.Functor.comp (smoothPresheafCommRing IM I M R)
    (CategoryTheory.forget CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
      CommRingCat.carrier)))
Tactic: rw [CategoryTheory.Presheaf.isSheaf_iff_isSheaf_forget _ _
      (CategoryTheory.forget CommRingCat)]
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Sheaf.Smooth (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
EM : Type u_2
inst_1 : NormedAddCommGroup EM
inst_2 : NormedSpace 𝕜 EM
HM : Type u_3
inst_3 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace 𝕜 E
H : Type u_5
inst_6 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u
inst_7 : TopologicalSpace M
inst_8 : ChartedSpace HM M
R : Type u
inst_9 : TopologicalSpace R
inst_10 : ChartedSpace H R
inst_11 : CommRing R
inst_12 : ContMDiffRing I ∞ R
x : (↑(TopCat.of M) : Type u)
U : (OpenNhds x)ᵒᵖ
⊢ Eq (α :=
    ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheaf IM I M R)).obj U ⟶
      (↑(TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) x) : Type u))
    (colimit.ι ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheaf IM I M R)) U ≫
      Iso.inv (forgetStalk IM I M R x))
    (↾(⇑(CommRingCat.Hom.hom
            (colimit.ι ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) U)) :
        (a : (↑(((OpenNhds.inclusion x).op ⋙ (smoothSheafCommRing IM I M R).presheaf).obj U) : Type u)) →
          (↑(colimit ((OpenNhds.inclusion x).op ⋙ (smoothSheafCommRing IM I M R).presheaf)) : Type u)))
have dsimp : Eq (α :=
  Functor.obj (TopCat.Sheaf.presheaf (smoothSheaf IM I M R)) (op ((OpenNhds.inclusion x).obj (unop U))) ⟶
    (↑(TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) x) : Type u))
  (colimit.ι ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheaf IM I M R)) U ≫
    Iso.inv (forgetStalk IM I M R x))
  (↾(⇑(CommRingCat.Hom.hom
          (colimit.ι ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) U)) :
      (a : (↑((smoothSheafCommRing IM I M R).presheaf.obj (op ((OpenNhds.inclusion x).obj (unop U)))) : Type u)) →
        (↑(colimit ((OpenNhds.inclusion x).op ⋙ (smoothSheafCommRing IM I M R).presheaf)) : Type u)))
Tactic: dsimp
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Sheaf.Smooth (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
EM : Type u_2
inst_1 : NormedAddCommGroup EM
inst_2 : NormedSpace 𝕜 EM
HM : Type u_3
inst_3 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace 𝕜 E
H : Type u_5
inst_6 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u
inst_7 : TopologicalSpace M
inst_8 : ChartedSpace HM M
R : Type u
inst_9 : TopologicalSpace R
inst_10 : ChartedSpace H R
inst_11 : CommRing R
inst_12 : ContMDiffRing I ∞ R
x : (↑(TopCat.of M) : Type u)
U : (OpenNhds x)ᵒᵖ
⊢ Eq (α :=
    Functor.obj (TopCat.Sheaf.presheaf (smoothSheaf IM I M R)) (op ((OpenNhds.inclusion x).obj (unop U))) ⟶
      (↑(TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) x) : Type u))
    (colimit.ι ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheaf IM I M R)) U ≫
      Iso.inv (forgetStalk IM I M R x))
    (↾(⇑(CommRingCat.Hom.hom
            (colimit.ι ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) U)) :
        (a : (↑((smoothSheafCommRing IM I M R).presheaf.obj (op ((OpenNhds.inclusion x).obj (unop U)))) : Type u)) →
          (↑(colimit ((OpenNhds.inclusion x).op ⋙ (smoothSheafCommRing IM I M R).presheaf)) : Type u)))
have rw : Eq (α :=
  Functor.obj (TopCat.Sheaf.presheaf (smoothSheaf IM I M R)) (op ((OpenNhds.inclusion x).obj (unop U))) ⟶
    colimit ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheaf IM I M R)))
  (CategoryStruct.comp (Y :=
    (↑(colimit ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R))) : Type u))
    (↾(⇑(CommRingCat.Hom.hom
            (colimit.ι ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) U)) :
        (a : (↑((smoothSheafCommRing IM I M R).presheaf.obj (op ((OpenNhds.inclusion x).obj (unop U)))) : Type u)) →
          (↑(colimit ((OpenNhds.inclusion x).op ⋙ (smoothSheafCommRing IM I M R).presheaf)) : Type u)))
    (Iso.hom (forgetStalk IM I M R x)))
  (CategoryStruct.comp (Y :=
    (↑(TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) x) : Type u))
    (↾(⇑(CommRingCat.Hom.hom
            (colimit.ι ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) U)) :
        (a : (↑((smoothSheafCommRing IM I M R).presheaf.obj (op ((OpenNhds.inclusion x).obj (unop U)))) : Type u)) →
          (↑(colimit ((OpenNhds.inclusion x).op ⋙ (smoothSheafCommRing IM I M R).presheaf)) : Type u)))
    (Iso.hom (forgetStalk IM I M R x)))
Tactic: rw [Iso.comp_inv_eq, ← smoothSheafCommRing.ι_forgetStalk_hom]
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Sheaf.Smooth (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
EM : Type u_2
inst_1 : NormedAddCommGroup EM
inst_2 : NormedSpace 𝕜 EM
HM : Type u_3
inst_3 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace 𝕜 E
H : Type u_5
inst_6 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_6
inst_7 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E H'
M : Type u
inst_8 : TopologicalSpace M
inst_9 : ChartedSpace HM M
N G A A' R : Type u
inst_10 : TopologicalSpace N
inst_11 : ChartedSpace H N
inst_12 : TopologicalSpace G
inst_13 : ChartedSpace H G
inst_14 : TopologicalSpace A
inst_15 : ChartedSpace H A
inst_16 : TopologicalSpace A'
inst_17 : ChartedSpace H' A'
inst_18 : TopologicalSpace R
inst_19 : ChartedSpace H R
EP : Type u_7
inst_20 : NormedAddCommGroup EP
inst_21 : NormedSpace 𝕜 EP
HP : Type u_8
inst_22 : TopologicalSpace HP
IP : ModelWithCorners 𝕜 EP HP
P : Type u
inst_23 : TopologicalSpace P
inst_24 : ChartedSpace HP P
inst_25 : CommRing R
inst_26 : ContMDiffRing I ∞ R
x : (↑(TopCat.of M) : Type u)
⊢ TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) x ⟶ CommRingCat.of R
have refine_1 : (U : (OpenNhds x)ᵒᵖ) →
  (Functor.obj
          (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑(TopCat.of M) : Type u))ᵒᵖ CommRingCat)
            (OpenNhds.inclusion x).op)
          (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R))).obj
      U ⟶
    Functor.obj (Functor.obj (Functor.const (OpenNhds x)ᵒᵖ) (CommRingCat.of R)) U
Tactic: refine CategoryTheory.Limits.colimit.desc _ ⟨_, ⟨fun U ↦ ?_, ?_⟩⟩
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Sheaf.Smooth (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
EM : Type u_2
inst_1 : NormedAddCommGroup EM
inst_2 : NormedSpace 𝕜 EM
HM : Type u_3
inst_3 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace 𝕜 E
H : Type u_5
inst_6 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_6
inst_7 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E H'
M : Type u
inst_8 : TopologicalSpace M
inst_9 : ChartedSpace HM M
N G A A' R : Type u
inst_10 : TopologicalSpace N
inst_11 : ChartedSpace H N
inst_12 : TopologicalSpace G
inst_13 : ChartedSpace H G
inst_14 : TopologicalSpace A
inst_15 : ChartedSpace H A
inst_16 : TopologicalSpace A'
inst_17 : ChartedSpace H' A'
inst_18 : TopologicalSpace R
inst_19 : ChartedSpace H R
EP : Type u_7
inst_20 : NormedAddCommGroup EP
inst_21 : NormedSpace 𝕜 EP
HP : Type u_8
inst_22 : TopologicalSpace HP
IP : ModelWithCorners 𝕜 EP HP
P : Type u
inst_23 : TopologicalSpace P
inst_24 : ChartedSpace HP P
inst_25 : CommRing R
inst_26 : ContMDiffRing I ∞ R
x : (↑(TopCat.of M) : Type u)
refine_1 :
  (U : (OpenNhds x)ᵒᵖ) →
    (Functor.obj
            (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑(TopCat.of M) : Type u))ᵒᵖ CommRingCat)
              (OpenNhds.inclusion x).op)
            (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R))).obj
        U ⟶
      Functor.obj (Functor.obj (Functor.const (OpenNhds x)ᵒᵖ) (CommRingCat.of R)) U
⊢ TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) x ⟶ CommRingCat.of R
have refine_2 : ∀ ⦃X Y : (OpenNhds x)ᵒᵖ⦄ (f : X ⟶ Y),
  Eq (α :=
    (Functor.obj
            (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑(TopCat.of M) : Type u))ᵒᵖ CommRingCat)
              (OpenNhds.inclusion x).op)
            (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R))).obj
        X ⟶
      Functor.obj (Functor.obj (Functor.const (OpenNhds x)ᵒᵖ) (CommRingCat.of R)) Y)
    ((Functor.obj
            (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑(TopCat.of M) : Type u))ᵒᵖ CommRingCat)
              (OpenNhds.inclusion x).op)
            (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R))).map
        f ≫
      (fun (U : (OpenNhds x)ᵒᵖ) ↦
          (evalAt IM I M R x (unop U) :
            Functor.obj (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R))
                (op (↑(unop U) : Opens (↑(TopCat.of M) : Type u))) ⟶
              CommRingCat.of R))
        Y)
    ((fun (U : (OpenNhds x)ᵒᵖ) ↦
          (evalAt IM I M R x (unop U) :
            Functor.obj (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R))
                (op (↑(unop U) : Opens (↑(TopCat.of M) : Type u))) ⟶
              CommRingCat.of R))
        X ≫
      CategoryTheory.Functor.map (Functor.obj (Functor.const (OpenNhds x)ᵒᵖ) (CommRingCat.of R)) f)
Tactic: refine CategoryTheory.Limits.colimit.desc _ ⟨_, ⟨fun U ↦ ?_, ?_⟩⟩
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Sheaf.Smooth (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
EM : Type u_2
inst_1 : NormedAddCommGroup EM
inst_2 : NormedSpace 𝕜 EM
HM : Type u_3
inst_3 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace 𝕜 E
H : Type u_5
inst_6 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u
inst_7 : TopologicalSpace M
inst_8 : ChartedSpace HM M
R : Type u
inst_9 : TopologicalSpace R
inst_10 : ChartedSpace H R
inst_11 : CommRing R
inst_12 : ContMDiffRing I ∞ R
x : (↑(TopCat.of M) : Type u)
⊢ Eq (α := TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (smoothSheaf IM I M R)) x ⟶ (↑(CommRingCat.of R) : Type u))
    (CategoryStruct.comp (Y :=
      (↑(TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) x) : Type u))
      (Iso.inv (forgetStalk IM I M R x))
      (↾(⇑(CommRingCat.Hom.hom (evalHom IM I M R x)) :
          (a : (↑((smoothSheafCommRing IM I M R).presheaf.stalk x) : Type u)) → (↑(CommRingCat.of R) : Type u))))
    (smoothSheaf.evalHom IM I R x)
have w : ∀ (j : (OpenNhds x)ᵒᵖ),
  colimit.ι
        (Functor.obj
          (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑(TopCat.of M) : Type u))ᵒᵖ (Type u))
            (OpenNhds.inclusion x).op)
          (TopCat.Sheaf.presheaf (smoothSheaf IM I M R)))
        j ≫
      CategoryStruct.comp (Y :=
        (↑(TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) x) : Type u))
        (Iso.inv (forgetStalk IM I M R x))
        (↾(⇑(CommRingCat.Hom.hom (evalHom IM I M R x)) :
            (a : (↑((smoothSheafCommRing IM I M R).presheaf.stalk x) : Type u)) → (↑(CommRingCat.of R) : Type u))) =
    colimit.ι
        (Functor.obj
          (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑(TopCat.of M) : Type u))ᵒᵖ (Type u))
            (OpenNhds.inclusion x).op)
          (TopCat.Sheaf.presheaf (smoothSheaf IM I M R)))
        j ≫
      smoothSheaf.evalHom IM I R x
Tactic: apply Limits.colimit.hom_ext
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Sheaf.Smooth (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
EM : Type u_2
inst_1 : NormedAddCommGroup EM
inst_2 : NormedSpace 𝕜 EM
HM : Type u_3
inst_3 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace 𝕜 E
H : Type u_5
inst_6 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u
inst_7 : TopologicalSpace M
inst_8 : ChartedSpace HM M
R : Type u
inst_9 : TopologicalSpace R
inst_10 : ChartedSpace H R
inst_11 : CommRing R
inst_12 : ContMDiffRing I ∞ R
x : (↑(TopCat.of M) : Type u)
U : (OpenNhds x)ᵒᵖ
⊢ colimit.ι ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheaf IM I M R)) U ≫
      CategoryStruct.comp (Y :=
        (↑(TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) x) : Type u))
        (Iso.inv (forgetStalk IM I M R x))
        (↾(⇑(CommRingCat.Hom.hom (evalHom IM I M R x)) :
            (a : (↑((smoothSheafCommRing IM I M R).presheaf.stalk x) : Type u)) → (↑(CommRingCat.of R) : Type u))) =
    colimit.ι ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheaf IM I M R)) U ≫
      smoothSheaf.evalHom IM I R x
have w : Eq (α :=
  ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheaf IM I M R)).obj U ⟶ (↑(CommRingCat.of R) : Type u))
  (CategoryStruct.comp (Y :=
    (↑(TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) x) : Type u))
    (↾(⇑(CommRingCat.Hom.hom
            (colimit.ι ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) U)) :
        (a : (↑((smoothSheafCommRing IM I M R).presheaf.obj ((OpenNhds.inclusion x).op.obj U)) : Type u)) →
          (↑(colimit ((OpenNhds.inclusion x).op ⋙ (smoothSheafCommRing IM I M R).presheaf)) : Type u)))
    (↾(⇑(CommRingCat.Hom.hom (evalHom IM I M R x)) :
        (a : (↑((smoothSheafCommRing IM I M R).presheaf.stalk x) : Type u)) → (↑(CommRingCat.of R) : Type u))))
  (↾smoothSheaf.evalAt IM I (↑(CommRingCat.of R) : Type u) x (unop U))
Tactic: rw [smoothSheafCommRing.ι_forgetStalk_inv_assoc, smoothSheaf.ι_evalHom]
-- Lean 4 proof step from Mathlib.Geometry.Manifold.Sheaf.Smooth (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
EM : Type u_2
inst_1 : NormedAddCommGroup EM
inst_2 : NormedSpace 𝕜 EM
HM : Type u_3
inst_3 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace 𝕜 E
H : Type u_5
inst_6 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u
inst_7 : TopologicalSpace M
inst_8 : ChartedSpace HM M
R : Type u
inst_9 : TopologicalSpace R
inst_10 : ChartedSpace H R
inst_11 : CommRing R
inst_12 : ContMDiffRing I ∞ R
x : (↑(TopCat.of M) : Type u)
U : (OpenNhds x)ᵒᵖ
⊢ Eq (α :=
    ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheaf IM I M R)).obj U ⟶ (↑(CommRingCat.of R) : Type u))
    (CategoryStruct.comp (Y :=
      (↑(TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) x) : Type u))
      (↾(⇑(CommRingCat.Hom.hom
              (colimit.ι ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) U)) :
          (a : (↑((smoothSheafCommRing IM I M R).presheaf.obj ((OpenNhds.inclusion x).op.obj U)) : Type u)) →
            (↑(colimit ((OpenNhds.inclusion x).op ⋙ (smoothSheafCommRing IM I M R).presheaf)) : Type u)))
      (↾(⇑(CommRingCat.Hom.hom (evalHom IM I M R x)) :
          (a : (↑((smoothSheafCommRing IM I M R).presheaf.stalk x) : Type u)) → (↑(CommRingCat.of R) : Type u))))
    (↾smoothSheaf.evalAt IM I (↑(CommRingCat.of R) : Type u) x (unop U))
have w : ∀ (x_1 : Functor.obj (TopCat.Sheaf.presheaf (smoothSheaf IM I M R)) ((OpenNhds.inclusion x).op.obj U)),
  TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (CategoryStruct.comp (Y :=
          (↑(TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) x) : Type u))
          (↾(⇑(CommRingCat.Hom.hom
                  (colimit.ι ((OpenNhds.inclusion x).op ⋙ TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R)) U)) :
              (a : (↑((smoothSheafCommRing IM I M R).presheaf.obj ((OpenNhds.inclusion x).op.obj U)) : Type u)) →
                (↑(colimit ((OpenNhds.inclusion x).op ⋙ (smoothSheafCommRing IM I M R).presheaf)) : Type u)))
          (↾(⇑(CommRingCat.Hom.hom (evalHom IM I M R x)) :
              (a : (↑((smoothSheafCommRing IM I M R).presheaf.stalk x) : Type u)) → (↑(CommRingCat.of R) : Type u)))))
      x_1 =
    TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun) (↾smoothSheaf.evalAt IM I (↑(CommRingCat.of R) : Type u) x (unop U)))
      x_1
Tactic: ext x