awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Modules.Tilde (goal)
Goal:
R : CommRingCat
M : (Spec (CommRingCat.of (↑R : Type u))).Modules
U : (Spec (CommRingCat.of (↑R : Type u))).Opens
this :
  ∀ (U : (Spec (CommRingCat.of (↑R : Type u))).Opens) (hU : U = PrimeSpectrum.basicOpen 1),
    tilde.toOpen ((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤)) U ≫
        NatTrans.app (InducedCategory.Hom.hom (CategoryTheory.Functor.map modulesSpecToSheaf M.fromTildeΓ)) (op U) =
      (Functor.obj modulesSpecToSheaf M).obj (P :=
            CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
        (Quiver.Hom.op (homOfLE ⋯))
hU : ¬U = PrimeSpectrum.basicOpen 1
⊢ tilde.toOpen ((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤)) U ≫
      NatTrans.app (InducedCategory.Hom.hom (CategoryTheory.Functor.map modulesSpecToSheaf M.fromTildeΓ)) (op U) =
    (Functor.obj modulesSpecToSheaf M).obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
      (Quiver.Hom.op (homOfLE ⋯))
have inr : Eq.{u + 1} (α := (Spec (CommRingCat.of (↑R : Type u))).Opens) (PrimeSpectrum.basicOpen 1) (PrimeSpectrum.basicOpen 1)
Tactic: rw [← tilde.toOpen_res _ (PrimeSpectrum.basicOpen 1) _ (homOfLE (by simp)), Category.assoc,
      NatTrans.naturality, ← Category.assoc, this, ← Functor.map_comp, ← op_comp, homOfLE_comp]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Modules.Tilde (goal)
Goal:
R : CommRingCat
M : (Spec (CommRingCat.of (↑R : Type u))).Modules
U : (Spec (CommRingCat.of (↑R : Type u))).Opens
hU : U = PrimeSpectrum.basicOpen 1
⊢ tilde.toOpen ((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤)) U ≫
      NatTrans.app (InducedCategory.Hom.hom (CategoryTheory.Functor.map modulesSpecToSheaf M.fromTildeΓ)) (op U) =
    (Functor.obj modulesSpecToSheaf M).obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
      (Quiver.Hom.op (homOfLE ⋯))
have subst : Eq (α :=
  (Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤) ⟶
    (Functor.obj modulesSpecToSheaf M).obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
      (op (PrimeSpectrum.basicOpen 1)))
  (tilde.toOpen ((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤)) (PrimeSpectrum.basicOpen 1) ≫
    NatTrans.app (InducedCategory.Hom.hom (CategoryTheory.Functor.map modulesSpecToSheaf M.fromTildeΓ))
      (op (PrimeSpectrum.basicOpen 1)))
  ((Functor.obj modulesSpecToSheaf M).obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
    (Quiver.Hom.op (homOfLE sorry)))
Tactic: subst hU
Cleared hypotheses: U, hU
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Modules.Tilde (goal)
Goal:
R : CommRingCat
M : (Spec (CommRingCat.of (↑R : Type u))).Modules
⊢ Eq (α :=
    (Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤) ⟶
      (Functor.obj modulesSpecToSheaf M).obj (P :=
            CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
        (op (PrimeSpectrum.basicOpen 1)))
    (tilde.toOpen ((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤)) (PrimeSpectrum.basicOpen 1) ≫
      NatTrans.app (InducedCategory.Hom.hom (CategoryTheory.Functor.map modulesSpecToSheaf M.fromTildeΓ))
        (op (PrimeSpectrum.basicOpen 1)))
    ((Functor.obj modulesSpecToSheaf M).obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
      (Quiver.Hom.op (homOfLE ⋯)))
have simp : Eq (α :=
  (Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤) ⟶
    (Functor.obj modulesSpecToSheaf M).obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
      (op (PrimeSpectrum.basicOpen 1)))
  (tilde.toOpen ((Functor.obj modulesSpecToSheaf M).presheaf.obj (op ⊤)) (PrimeSpectrum.basicOpen 1) ≫
    ModuleCat.ofHom
      (IsLocalizedModule.lift (Submonoid.powers 1)
        (ModuleCat.Hom.hom
          (tilde.toOpen
            ((Functor.obj modulesSpecToSheaf M).obj (P :=
                  CategoryTheory.Presheaf.IsSheaf
                    (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
              (op ⊤))
            (PrimeSpectrum.basicOpen 1)))
        (ModuleCat.Hom.hom
          ((Functor.obj modulesSpecToSheaf M).obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
            (Quiver.Hom.op (homOfLE sorry))))
        sorry))
  ((Functor.obj modulesSpecToSheaf M).obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
    (Quiver.Hom.op (homOfLE sorry)))
Tactic: simp only [fromTildeΓ, inducedFunctor_obj,
    homOfLE_leOfHom, Functor.FullyFaithful.map_preimage, TopCat.Sheaf.extend_hom_app]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Modules.Tilde (goal)
Goal:
R : CommRingCat
M_1 M : ModuleCat (↑R : Type u)
⊢ (tilde.functor R).map (NatTrans.app (Iso.hom toTildeΓNatIso) M) ≫
      NatTrans.app fromTildeΓNatTrans ((tilde.functor R).obj M) =
    𝟙 ((tilde.functor R).obj M)
have apply : Eq (α :=
  Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M) ⟶ Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M))
  (CategoryTheory.Functor.map modulesSpecToSheaf
    ((tilde.functor R).map (NatTrans.app (Iso.hom toTildeΓNatIso) M) ≫
      NatTrans.app fromTildeΓNatTrans ((tilde.functor R).obj M)))
  (CategoryTheory.Functor.map modulesSpecToSheaf (𝟙 ((tilde.functor R).obj M)))
Tactic: apply SpecModulesToSheafFullyFaithful.map_injective
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Modules.Tilde (goal)
Goal:
R : CommRingCat
M_1 M : ModuleCat (↑R : Type u)
⊢ Eq (α :=
    Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M) ⟶ Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M))
    (CategoryTheory.Functor.map modulesSpecToSheaf
      ((tilde.functor R).map (NatTrans.app (Iso.hom toTildeΓNatIso) M) ≫
        NatTrans.app fromTildeΓNatTrans ((tilde.functor R).obj M)))
    (CategoryTheory.Functor.map modulesSpecToSheaf (𝟙 ((tilde.functor R).obj M)))
have h : Eq (α :=
  (Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M)).obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))) ⟶
    (Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M)).obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))))
  (InducedCategory.Hom.hom
    (CategoryTheory.Functor.map modulesSpecToSheaf
      ((tilde.functor R).map (NatTrans.app (Iso.hom toTildeΓNatIso) M) ≫
        NatTrans.app fromTildeΓNatTrans ((tilde.functor R).obj M))))
  (InducedCategory.Hom.hom (CategoryTheory.Functor.map modulesSpecToSheaf (𝟙 ((tilde.functor R).obj M))))
Tactic: apply CategoryTheory.Sheaf.hom_ext
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Modules.Tilde (goal)
Goal:
R : CommRingCat
M_1 M : ModuleCat (↑R : Type u)
r :
  (InducedCategory (C := (↑(CommRingCat.of (↑R : Type u)) : Type u)) (Opens ↥(Spec (CommRingCat.of (↑R : Type u))))
      basicOpen)ᵒᵖ
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
have h : ∀ (x : ↥(Submonoid.powers (unop r))),
  IsUnit (M :=
    Module.End (↑R : Type u)
      (↑(Functor.obj
            (Functor.op (inducedFunctor basicOpen) ⋙
              (Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M)).obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))))
            r) :
        Type u))
    ((algebraMap (↑R : Type u)
            (Module.End (↑R : Type u)
              (↑(Functor.obj
                    (Functor.op (inducedFunctor basicOpen) ⋙
                      (Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M)).obj (P :=
                        CategoryTheory.Presheaf.IsSheaf
                          (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))))
                    r) :
                Type u)) :
          (a : (↑R : Type u)) →
            Module.End (↑R : Type u)
              (↑(((inducedFunctor basicOpen).op ⋙ (modulesSpecToSheaf.obj ((tilde.functor R).obj M)).obj).obj r) :
                Type u))
        (↑x : (↑R : Type u)) :
      Module.End (↑R : Type u)
        (↑(Functor.obj
              (Functor.op (inducedFunctor basicOpen) ⋙
                (Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M)).obj (P :=
                  CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))))
              r) :
          Type u))
Tactic: apply IsLocalizedModule.ext (.powers (M := R) r.unop)
      (toOpen _ (PrimeSpectrum.basicOpen (R := R) r.unop)).hom
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Modules.Tilde (goal)
Goal:
R : CommRingCat
M_1 M : ModuleCat (↑R : Type u)
r :
  (InducedCategory (C := (↑(CommRingCat.of (↑R : Type u)) : Type u)) (Opens ↥(Spec (CommRingCat.of (↑R : Type u))))
      basicOpen)ᵒᵖ
h :
  ∀ (x : ↥(Submonoid.powers (unop r))),
    IsUnit (M :=
      Module.End (↑R : Type u)
        (↑(Functor.obj
              (Functor.op (inducedFunctor basicOpen) ⋙
                (Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M)).obj (P :=
                  CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))))
              r) :
          Type u))
      ((algebraMap (↑R : Type u)
              (Module.End (↑R : Type u)
                (↑(Functor.obj
                      (Functor.op (inducedFunctor basicOpen) ⋙
                        (Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M)).obj (P :=
                          CategoryTheory.Presheaf.IsSheaf
                            (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))))
                      r) :
                  Type u)) :
            (a : (↑R : Type u)) →
              Module.End (↑R : Type u)
                (↑(((inducedFunctor basicOpen).op ⋙ (modulesSpecToSheaf.obj ((tilde.functor R).obj M)).obj).obj r) :
                  Type u))
          (↑x : (↑R : Type u)) :
        Module.End (↑R : Type u)
          (↑(Functor.obj
                (Functor.op (inducedFunctor basicOpen) ⋙
                  (Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M)).obj (P :=
                    CategoryTheory.Presheaf.IsSheaf
                      (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))))
                r) :
            Type u))
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
have h₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Tactic: apply IsLocalizedModule.ext (.powers (M := R) r.unop)
      (toOpen _ (PrimeSpectrum.basicOpen (R := R) r.unop)).hom
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Modules.Tilde (goal)
Goal:
R : CommRingCat
M_1 M : ModuleCat (↑R : Type u)
r :
  (InducedCategory (C := (↑(CommRingCat.of (↑R : Type u)) : Type u)) (Opens ↥(Spec (CommRingCat.of (↑R : Type u))))
      basicOpen)ᵒᵖ
⊢ LE.le (α := Submonoid (↑R : Type u)) (Submonoid.powers (unop r))
    (Submonoid.comap (F :=
      (↑R : Type u) →+*
        Module.End (↑R : Type u)
          (↑(Functor.obj
                (Functor.op (inducedFunctor basicOpen) ⋙
                  (Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M)).obj (P :=
                    CategoryTheory.Presheaf.IsSheaf
                      (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))))
                r) :
            Type u))
      (algebraMap (↑R : Type u)
        (Module.End (↑R : Type u)
          (↑(Functor.obj
                (Functor.op (inducedFunctor basicOpen) ⋙
                  (Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M)).obj (P :=
                    CategoryTheory.Presheaf.IsSheaf
                      (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))))
                r) :
            Type u)))
      (IsUnit.submonoid
        (Module.End (↑R : Type u)
          (↑(Functor.obj
                (Functor.op (inducedFunctor basicOpen) ⋙
                  (Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M)).obj (P :=
                    CategoryTheory.Presheaf.IsSheaf
                      (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))))
                r) :
            Type u))))
have h : IsUnit (M :=
  Module.End (↑R : Type u)
    (↑(Functor.obj
          (Functor.op (inducedFunctor basicOpen) ⋙
            (Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M)).obj (P :=
              CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))))
          r) :
      Type u))
  ((algebraMap (↑R : Type u)
          (Module.End (↑R : Type u)
            (↑(Functor.obj
                  (Functor.op (inducedFunctor basicOpen) ⋙
                    (Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M)).obj (P :=
                      CategoryTheory.Presheaf.IsSheaf
                        (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))))
                  r) :
              Type u)) :
        (a : (↑R : Type u)) →
          Module.End (↑R : Type u)
            (↑(((inducedFunctor basicOpen).op ⋙ (modulesSpecToSheaf.obj ((tilde.functor R).obj M)).obj).obj r) :
              Type u))
      (unop r) :
    Module.End (↑R : Type u)
      (↑(Functor.obj
            (Functor.op (inducedFunctor basicOpen) ⋙
              (Functor.obj modulesSpecToSheaf ((tilde.functor R).obj M)).obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))))
            r) :
        Type u))
Tactic: simp only [Submonoid.powers_le, Submonoid.mem_comap, IsUnit.mem_submonoid_iff]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Modules.Tilde (goal)
Goal:
R : CommRingCat
M_1 M : ModuleCat (↑R : Type u)
r :
  (InducedCategory (C := (↑(CommRingCat.of (↑R : Type u)) : Type u)) (Opens ↥(Spec (CommRingCat.of (↑R : Type u))))
      basicOpen)ᵒᵖ
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
have h : Eq (α :=
  LinearMap (R := (↑R : Type u)) (RingHom.id (↑R : Type u)) (↑M : Type u)
    (↑((Functor.obj modulesSpecToSheaf (tilde M)).obj (P :=
              CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
          (op (basicOpen (unop r)))) :
      Type u))
  (ModuleCat.Hom.hom
      (NatTrans.app
        (InducedCategory.Hom.hom
          (CategoryTheory.Functor.map modulesSpecToSheaf (tilde.map (toOpen M ⊤) ≫ (tilde M).fromTildeΓ)))
        (op (basicOpen (unop r)))) ∘ₗ
    ModuleCat.Hom.hom (toOpen M (basicOpen (unop r))))
  (ModuleCat.Hom.hom
      (NatTrans.app (InducedCategory.Hom.hom (CategoryTheory.Functor.map modulesSpecToSheaf (𝟙 (tilde M))))
        (op (basicOpen (unop r)))) ∘ₗ
    ModuleCat.Hom.hom (toOpen M (basicOpen (unop r))))
Tactic: dsimp [toTildeΓNatIso, isoTop,
      TopCat.Sheaf.restrictHomEquivHom, Functor.IsCoverDense.restrictHomEquivHom,
      fromTildeΓNatTrans, moduleSpecΓFunctor, Sheaf.forget, sheafToPresheaf]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Modules.Tilde (goal)
Goal:
R : CommRingCat
M_1 M : ModuleCat (↑R : Type u)
r :
  (InducedCategory (C := (↑(CommRingCat.of (↑R : Type u)) : Type u)) (Opens ↥(Spec (CommRingCat.of (↑R : Type u))))
      basicOpen)ᵒᵖ
⊢ Eq (α :=
    LinearMap (R := (↑R : Type u)) (RingHom.id (↑R : Type u)) (↑M : Type u)
      (↑((Functor.obj modulesSpecToSheaf (tilde M)).obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
            (op (basicOpen (unop r)))) :
        Type u))
    (ModuleCat.Hom.hom
        (NatTrans.app
          (InducedCategory.Hom.hom
            (CategoryTheory.Functor.map modulesSpecToSheaf (tilde.map (toOpen M ⊤) ≫ (tilde M).fromTildeΓ)))
          (op (basicOpen (unop r)))) ∘ₗ
      ModuleCat.Hom.hom (toOpen M (basicOpen (unop r))))
    (ModuleCat.Hom.hom
        (NatTrans.app (InducedCategory.Hom.hom (CategoryTheory.Functor.map modulesSpecToSheaf (𝟙 (tilde M))))
          (op (basicOpen (unop r)))) ∘ₗ
      ModuleCat.Hom.hom (toOpen M (basicOpen (unop r))))
have h : Eq (α :=
  LinearMap (R := (↑R : Type u)) (RingHom.id (↑R : Type u)) (↑M : Type u)
    (↑((Functor.obj modulesSpecToSheaf (tilde M)).obj (P :=
              CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
          (op (basicOpen (unop r)))) :
      Type u))
  (ModuleCat.Hom.hom
    (toOpen M (basicOpen (unop r)) ≫
      NatTrans.app
        (InducedCategory.Hom.hom
          (CategoryTheory.Functor.map modulesSpecToSheaf (tilde.map (toOpen M ⊤)) ≫
            CategoryTheory.Functor.map modulesSpecToSheaf (tilde M).fromTildeΓ))
        (op (basicOpen (unop r)))))
  (ModuleCat.Hom.hom
    (toOpen M (basicOpen (unop r)) ≫
      NatTrans.app (InducedCategory.Hom.hom (CategoryTheory.Functor.map modulesSpecToSheaf (𝟙 (tilde M))))
        (op (basicOpen (unop r)))))
Tactic: simp only [← ModuleCat.hom_comp, Functor.map_comp]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Modules.Tilde (goal)
Goal:
R : CommRingCat
M_1 M : ModuleCat (↑R : Type u)
r :
  (InducedCategory (C := (↑(CommRingCat.of (↑R : Type u)) : Type u)) (Opens ↥(Spec (CommRingCat.of (↑R : Type u))))
      basicOpen)ᵒᵖ
⊢ Eq (α :=
    LinearMap (R := (↑R : Type u)) (RingHom.id (↑R : Type u)) (↑M : Type u)
      (↑((Functor.obj modulesSpecToSheaf (tilde M)).obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
            (op (basicOpen (unop r)))) :
        Type u))
    (ModuleCat.Hom.hom
      (toOpen M (basicOpen (unop r)) ≫
        NatTrans.app
          (InducedCategory.Hom.hom
            (CategoryTheory.Functor.map modulesSpecToSheaf (tilde.map (toOpen M ⊤)) ≫
              CategoryTheory.Functor.map modulesSpecToSheaf (tilde M).fromTildeΓ))
          (op (basicOpen (unop r)))))
    (ModuleCat.Hom.hom
      (toOpen M (basicOpen (unop r)) ≫
        NatTrans.app (InducedCategory.Hom.hom (CategoryTheory.Functor.map modulesSpecToSheaf (𝟙 (tilde M))))
          (op (basicOpen (unop r)))))
have h : Eq (α :=
  M.Hom
    ((Functor.obj modulesSpecToSheaf (tilde M)).obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
      (op (basicOpen (unop r)))))
  (toOpen M (basicOpen (unop r)) ≫
    NatTrans.app
      (InducedCategory.Hom.hom
        (CategoryTheory.Functor.map modulesSpecToSheaf (tilde.map (toOpen M ⊤)) ≫
          CategoryTheory.Functor.map modulesSpecToSheaf (tilde M).fromTildeΓ))
      (op (basicOpen (unop r))))
  (toOpen M (basicOpen (unop r)) ≫
    NatTrans.app (InducedCategory.Hom.hom (CategoryTheory.Functor.map modulesSpecToSheaf (𝟙 (tilde M))))
      (op (basicOpen (unop r))))
Tactic: congr 1
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Modules.Tilde (goal)
Goal:
R : CommRingCat
M_1 : ModuleCat (↑R : Type u)
M : (Spec (CommRingCat.of (↑R : Type u))).Modules
⊢ NatTrans.app (Iso.hom toTildeΓNatIso) (Functor.obj moduleSpecΓFunctor M) ≫
      CategoryTheory.Functor.map moduleSpecΓFunctor (NatTrans.app fromTildeΓNatTrans M) =
    𝟙 (Functor.obj moduleSpecΓFunctor M)
have dsimp : toOpen
      ((Functor.obj modulesSpecToSheaf M).obj (P :=
            CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
        (op ⊤))
      ⊤ ≫
    NatTrans.app (InducedCategory.Hom.hom (CategoryTheory.Functor.map modulesSpecToSheaf M.fromTildeΓ)) (op ⊤) =
  𝟙
    ((Functor.obj modulesSpecToSheaf M).obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
      (op ⊤))
Tactic: dsimp [toTildeΓNatIso, fromTildeΓNatTrans, tilde.isoTop, moduleSpecΓFunctor, Sheaf.forget]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Modules.Tilde (goal)
Goal:
R : CommRingCat
M_1 : ModuleCat (↑R : Type u)
M : (Spec (CommRingCat.of (↑R : Type u))).Modules
⊢ toOpen
        ((Functor.obj modulesSpecToSheaf M).obj (P :=
              CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
          (op ⊤))
        ⊤ ≫
      NatTrans.app (InducedCategory.Hom.hom (CategoryTheory.Functor.map modulesSpecToSheaf M.fromTildeΓ)) (op ⊤) =
    𝟙
      ((Functor.obj modulesSpecToSheaf M).obj (P :=
            CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
        (op ⊤))
have rw : (Functor.obj modulesSpecToSheaf M).obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).map
    (Quiver.Hom.op (homOfLE sorry)) =
  𝟙
    ((Functor.obj modulesSpecToSheaf M).obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥(Spec (CommRingCat.of (↑R : Type u))))).obj
      (op ⊤))
Tactic: rw [toOpen_fromTildeΓ_app]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Modules.Tilde (hypothesis)
Goal:
R : CommRingCat
M : ModuleCat (↑R : Type u)
s : Set (↑M : Type u)
hs : Eq (α := Submodule (↑R : Type u) (↑M : Type u)) (Submodule.span (↑R : Type u) s) ⊤
t : Set ((↑s : Type u) →₀ (↑R : Type u))
ht :
  Submodule.span (↑R : Type u) t =
    LinearMap.ker (Finsupp.linearCombination (↑R : Type u) (Subtype.val (p := fun (x : (↑M : Type u)) ↦ x ∈ s)))
⊢ SheafOfModules.Presentation (tilde M)
have H₁ : Function.Exact (M := (↑(ModuleCat.of (↑R : Type u) ((↑t : Type u) →₀ (↑R : Type u))) : Type u)) (N :=
  (↑(ModuleCat.of (↑R : Type u) ((↑s : Type u) →₀ (↑R : Type u))) : Type u)) (P :=
  (↑(ModuleCat.of (↑R : Type u) (↑M : Type u)) : Type u))
  (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat (↑R : Type u)) ↦
        LinearMap (R := (↑R : Type u)) (RingHom.id (↑R : Type u)) (↑x1 : Type u) (↑x2 : Type u)) (CC :=
        ModuleCat.carrier) (X := ModuleCat.of (↑R : Type u) ((↑t : Type u) →₀ (↑R : Type u))) (Y :=
        ModuleCat.of (↑R : Type u) ((↑s : Type u) →₀ (↑R : Type u)))
        (ModuleCat.ofHom
          (Finsupp.linearCombination (↑R : Type u)
            (Subtype.val (p := fun (x : (↑s : Type u) →₀ (↑R : Type u)) ↦ x ∈ t))))) :
    (a : (↑(ModuleCat.of (↑R : Type u) ((↑t : Type u) →₀ (↑R : Type u))) : Type u)) →
      (↑(ModuleCat.of (↑R : Type u) ((↑s : Type u) →₀ (↑R : Type u))) : Type u))
  (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat (↑R : Type u)) ↦
        LinearMap (R := (↑R : Type u)) (RingHom.id (↑R : Type u)) (↑x1 : Type u) (↑x2 : Type u)) (CC :=
        ModuleCat.carrier) (X := ModuleCat.of (↑R : Type u) ((↑s : Type u) →₀ (↑R : Type u))) (Y :=
        ModuleCat.of (↑R : Type u) (↑M : Type u))
        (ModuleCat.ofHom
          (Finsupp.linearCombination (↑R : Type u) (Subtype.val (p := fun (x : (↑M : Type u)) ↦ x ∈ s))))) :
    (a : (↑(ModuleCat.of (↑R : Type u) ((↑s : Type u) →₀ (↑R : Type u))) : Type u)) →
      (↑(ModuleCat.of (↑R : Type u) (↑M : Type u)) : Type u))
Tactic: haveI H₁ : Function.Exact
      (ModuleCat.ofHom (Finsupp.linearCombination (α := t) R (↑)))
      (ModuleCat.ofHom (Finsupp.linearCombination (α := s) (M := M) R (↑))) :=
    (LinearMap.exact_iff.mpr (by simp [Finsupp.range_linearCombination, ht]))