awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.AlgebraicGeometry.StructureSheaf
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
hU : IsCompact (↑U : Set (↑(PrimeSpectrum.Top R) : Type u))
s :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
g : (x : ↥U) → R
hxg :
  ∀ (x : ↥U), Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen (g x)) (↑x : (↑(PrimeSpectrum.Top R) : Type u))
igU : ∀ (x : ↥U), basicOpen (g x) ≤ U
f : (x : ↥U) → M
H :
  ∀ (x : ↥U),
    Eq (α :=
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen (g x))))
      (const (f x) (g x) (basicOpen (g x)) ⋯)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              ⋯.hom.op) :
          (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g x)))))
        s)
i j : ↥U
⊢ Eq (α := LocalizedModule (Submonoid.powers (g i * g j)) M)
    (LocalizedModule.mk (g i • f j)
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (Submonoid.powers (g i * g j)) x) (g i * g j)
        ⋯))
    (LocalizedModule.mk (g j • f i)
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (Submonoid.powers (g i * g j)) x) (g i * g j)
        ⋯))

Tactic:
toBasicOpenₗ_injective (g i * g j) ?_

Premises used: HMul.hMul, _private.0.AlgebraicGeometry.StructureSheaf.toBasicOpenₗ_injective
-- Module: Mathlib.AlgebraicGeometry.StructureSheaf
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
hU : IsCompact (↑U : Set (↑(PrimeSpectrum.Top R) : Type u))
s :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
g : (x : ↥U) → R
hxg :
  ∀ (x : ↥U), Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen (g x)) (↑x : (↑(PrimeSpectrum.Top R) : Type u))
igU : ∀ (x : ↥U), basicOpen (g x) ≤ U
f : (x : ↥U) → M
H :
  ∀ (x : ↥U),
    Eq (α :=
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen (g x))))
      (const (f x) (g x) (basicOpen (g x)) ⋯)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              ⋯.hom.op) :
          (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g x)))))
        s)
i j : ↥U
⊢ Eq (α :=
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen (g i))))
    (const (f i) (g i) (basicOpen (g i)) ⋯)
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (CategoryTheory.Functor.map
            (ObjectProperty.FullSubcategory.obj (P :=
              CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
              (structureSheafInType R M))
            ⋯.hom.op) :
        (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
          (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g i)))))
      s)

Tactic:
have := H i
-- Module: Mathlib.AlgebraicGeometry.StructureSheaf
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
hU : IsCompact (↑U : Set (↑(PrimeSpectrum.Top R) : Type u))
s :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
g : (x : ↥U) → R
hxg :
  ∀ (x : ↥U), Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen (g x)) (↑x : (↑(PrimeSpectrum.Top R) : Type u))
igU : ∀ (x : ↥U), basicOpen (g x) ≤ U
f : (x : ↥U) → M
H :
  ∀ (x : ↥U),
    Eq (α :=
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen (g x))))
      (const (f x) (g x) (basicOpen (g x)) ⋯)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              ⋯.hom.op) :
          (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g x)))))
        s)
i j : ↥U
this :
  Eq (α :=
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen (g i))))
    (const (f i) (g i) (basicOpen (g i)) ⋯)
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (CategoryTheory.Functor.map
            (ObjectProperty.FullSubcategory.obj (P :=
              CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
              (structureSheafInType R M))
            ⋯.hom.op) :
        (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
          (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g i)))))
      s)
a : ↥(unop (op (basicOpen (g i * g j))))
⊢ (g j * g i) • f j = (g i * g j) • f j

Tactic:
ring_nf
-- Module: Mathlib.AlgebraicGeometry.StructureSheaf
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
hU : IsCompact (↑U : Set (↑(PrimeSpectrum.Top R) : Type u))
s :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
g : (x : ↥U) → R
hxg :
  ∀ (x : ↥U), Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen (g x)) (↑x : (↑(PrimeSpectrum.Top R) : Type u))
igU : ∀ (x : ↥U), basicOpen (g x) ≤ U
f : (x : ↥U) → M
H :
  ∀ (x : ↥U),
    Eq (α :=
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen (g x))))
      (const (f x) (g x) (basicOpen (g x)) ⋯)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              ⋯.hom.op) :
          (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g x)))))
        s)
i j : ↥U
this :
  Eq (α :=
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen (g i))))
    (const (f i) (g i) (basicOpen (g i)) ⋯)
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (CategoryTheory.Functor.map
            (ObjectProperty.FullSubcategory.obj (P :=
              CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
              (structureSheafInType R M))
            ⋯.hom.op) :
        (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
          (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g i)))))
      s)
⊢ basicOpen (g i * g j) ≤ U

Tactic:
((PrimeSpectrum.basicOpen_mul (g i) (g j)).trans_le inf_le_right).trans (igU _)

Premises used: Set.Mem, PrimeSpectrum.basicOpen_mul, inf_le_right, Eq.trans_le, LE.le.trans
-- Module: Mathlib.AlgebraicGeometry.StructureSheaf
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
hU : IsCompact (↑U : Set (↑(PrimeSpectrum.Top R) : Type u))
s :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
g : (x : ↥U) → R
hxg :
  ∀ (x : ↥U), Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen (g x)) (↑x : (↑(PrimeSpectrum.Top R) : Type u))
igU : ∀ (x : ↥U), basicOpen (g x) ≤ U
f : (x : ↥U) → M
H :
  ∀ (x : ↥U),
    Eq (α :=
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen (g x))))
      (const (f x) (g x) (basicOpen (g x)) ⋯)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              ⋯.hom.op) :
          (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g x)))))
        s)
i j : ↥U
this :
  Eq (α :=
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen (g i))))
    (const (f i) (g i) (basicOpen (g i)) ⋯)
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (CategoryTheory.Functor.map
            (ObjectProperty.FullSubcategory.obj (P :=
              CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
              (structureSheafInType R M))
            ⋯.hom.op) :
        (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
          (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g i)))))
      s)
⊢ basicOpen (g i * g j) ≤ U

Tactic:
exact ((PrimeSpectrum.basicOpen_mul (g i) (g j)).trans_le inf_le_right).trans (igU _)

Premises used: PrimeSpectrum.basicOpen_mul, inf_le_right, Eq.trans_le, LE.le.trans
-- Module: Mathlib.AlgebraicGeometry.StructureSheaf
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
hU : IsCompact (↑U : Set (↑(PrimeSpectrum.Top R) : Type u))
s :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
g : (x : ↥U) → R
hxg :
  ∀ (x : ↥U), Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen (g x)) (↑x : (↑(PrimeSpectrum.Top R) : Type u))
igU : ∀ (x : ↥U), basicOpen (g x) ≤ U
f : (x : ↥U) → M
H :
  ∀ (x : ↥U),
    Eq (α :=
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen (g x))))
      (const (f x) (g x) (basicOpen (g x)) ⋯)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              ⋯.hom.op) :
          (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g x)))))
        s)
⊢ ∀ (i j : ↥U),
    Eq (α := LocalizedModule (Submonoid.powers (g i * g j)) M)
      (LocalizedModule.mk (g i • f j)
        (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (Submonoid.powers (g i * g j)) x) (g i * g j)
          ⋯))
      (LocalizedModule.mk (g j • f i)
        (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (Submonoid.powers (g i * g j)) x) (g i * g j)
          ⋯))

Tactic:
have (i j : _) : LocalizedModule.mk (g i • f j) ⟨g i * g j, Submonoid.mem_powers _⟩ =
      LocalizedModule.mk (g j • f i) ⟨g i * g j, Submonoid.mem_powers _⟩ := by
    refine toBasicOpenₗ_injective (g i * g j) ?_
    simp only [toBasicOpenₗ_mk]
    have := H i
    trans (structureSheafInType R M).obj.map (homOfLE ?_).op s
    · refine .trans (Subtype.ext <| funext fun a ↦ ?_) congr((structureSheafInType R M).obj.map
        (homOfLE ((PrimeSpectrum.basicOpen_mul (g i) (g j)).trans_le inf_le_right)).op $(H j))
      exact LocalizedModule.mk_eq.mpr ⟨1, by simp [Submonoid.smul_def, ← smul_assoc]; ring_nf⟩
    · refine congr((structureSheafInType R M).obj.map (homOfLE ((PrimeSpectrum.basicOpen_mul (g i)
        (g j)).trans_le inf_le_left)).op $(H i)).symm.trans (Subtype.ext <| funext fun a ↦ ?_)
      exact LocalizedModule.mk_eq.mpr ⟨1, by simp [Submonoid.smul_def, ← smul_assoc]⟩
    · exact ((PrimeSpectrum.basicOpen_mul (g i) (g j)).trans_le inf_le_right).trans (igU _)
-- Module: Mathlib.AlgebraicGeometry.StructureSheaf
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
hU : IsCompact (↑U : Set (↑(PrimeSpectrum.Top R) : Type u))
s :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
g : (x : ↥U) → R
hxg :
  ∀ (x : ↥U), Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen (g x)) (↑x : (↑(PrimeSpectrum.Top R) : Type u))
igU : ∀ (x : ↥U), basicOpen (g x) ≤ U
f : (x : ↥U) → M
H :
  ∀ (x : ↥U),
    Eq (α :=
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen (g x))))
      (const (f x) (g x) (basicOpen (g x)) ⋯)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              ⋯.hom.op) :
          (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g x)))))
        s)
this :
  ∀ (i j : ↥U),
    Eq (α := LocalizedModule (Submonoid.powers (g i * g j)) M)
      (LocalizedModule.mk (g i • f j)
        (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (Submonoid.powers (g i * g j)) x) (g i * g j)
          ⋯))
      (LocalizedModule.mk (g j • f i)
        (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (Submonoid.powers (g i * g j)) x) (g i * g j)
          ⋯))
⊢ ∀ (i j : ↥U), ∃ (a : ℕ), ((g i * g j) ^ (a + 1) * g i) • f j = ((g i * g j) ^ (a + 1) * g j) • f i

Tactic:
simp only [LocalizedModule.mk_eq, Submonoid.smul_def, Subtype.exists, Submonoid.mem_powers_iff,
    exists_prop, exists_exists_eq_and, ← mul_smul, ← pow_succ, ← mul_assoc _ (_ * _)] at this

Premises used: HMul.hMul, LocalizedModule.mk_eq, pow_succ, mul_assoc, Subtype.exists, exists_prop, Submonoid.smul_def, exists_exists_eq_and, Submonoid.mem_powers_iff, SemigroupAction.mul_smul
-- Module: Mathlib.AlgebraicGeometry.StructureSheaf
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
hU : IsCompact (↑U : Set (↑(PrimeSpectrum.Top R) : Type u))
s :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
g : (x : ↥U) → R
hxg :
  ∀ (x : ↥U), Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen (g x)) (↑x : (↑(PrimeSpectrum.Top R) : Type u))
igU : ∀ (x : ↥U), basicOpen (g x) ≤ U
f : (x : ↥U) → M
H :
  ∀ (x : ↥U),
    Eq (α :=
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen (g x))))
      (const (f x) (g x) (basicOpen (g x)) ⋯)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              ⋯.hom.op) :
          (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g x)))))
        s)
n : (i j : ↥U) → ℕ
hn : ∀ (i j : ↥U), ((g i * g j) ^ (n i j + 1) * g i) • f j = ((g i * g j) ^ (n i j + 1) * g j) • f i
t : Finset ↥U
ht : (↑U : Set (↑(PrimeSpectrum.Top R) : Type u)) ⊆ ⋃ i ∈ t, (↑(basicOpen (g i)) : Set (PrimeSpectrum R))
N : ℕ := (t ×ˢ t).sup fun (x : ↥U × ↥U) ↦ n x.1 x.2 + 1
x : ↥t
⊢ basicOpen ((fun (i : ↥t) ↦ g (↑i : ↥U) ^ (N + 1)) x) ≤ U

Tactic:
simpa using igU x.1

Premises used: Subtype.val
-- Module: Mathlib.AlgebraicGeometry.StructureSheaf
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
hU : IsCompact (↑U : Set (↑(PrimeSpectrum.Top R) : Type u))
s :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
g : (x : ↥U) → R
hxg :
  ∀ (x : ↥U), Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen (g x)) (↑x : (↑(PrimeSpectrum.Top R) : Type u))
igU : ∀ (x : ↥U), basicOpen (g x) ≤ U
f : (x : ↥U) → M
H :
  ∀ (x : ↥U),
    Eq (α :=
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen (g x))))
      (const (f x) (g x) (basicOpen (g x)) ⋯)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              ⋯.hom.op) :
          (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g x)))))
        s)
n : (i j : ↥U) → ℕ
hn : ∀ (i j : ↥U), ((g i * g j) ^ (n i j + 1) * g i) • f j = ((g i * g j) ^ (n i j + 1) * g j) • f i
t : Finset ↥U
ht : (↑U : Set (↑(PrimeSpectrum.Top R) : Type u)) ⊆ ⋃ i ∈ t, (↑(basicOpen (g i)) : Set (PrimeSpectrum R))
N : ℕ := (t ×ˢ t).sup fun (x : ↥U × ↥U) ↦ n x.1 x.2 + 1
x : (↑(PrimeSpectrum.Top R) : Type u)
hx : x ∈ U
⊢ Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u))
    (⨆ (i : ↥t), basicOpen ((fun (i : ↥t) ↦ g (↑i : ↥U) ^ (N + 1)) i)) x

Tactic:
simpa using ht hx
-- Module: Mathlib.AlgebraicGeometry.StructureSheaf
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
hU : IsCompact (↑U : Set (↑(PrimeSpectrum.Top R) : Type u))
s :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
g : (x : ↥U) → R
hxg :
  ∀ (x : ↥U), Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen (g x)) (↑x : (↑(PrimeSpectrum.Top R) : Type u))
igU : ∀ (x : ↥U), basicOpen (g x) ≤ U
f : (x : ↥U) → M
H :
  ∀ (x : ↥U),
    Eq (α :=
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen (g x))))
      (const (f x) (g x) (basicOpen (g x)) ⋯)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              ⋯.hom.op) :
          (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g x)))))
        s)
n : (i j : ↥U) → ℕ
hn : ∀ (i j : ↥U), ((g i * g j) ^ (n i j + 1) * g i) • f j = ((g i * g j) ^ (n i j + 1) * g j) • f i
t : Finset ↥U
ht : (↑U : Set (↑(PrimeSpectrum.Top R) : Type u)) ⊆ ⋃ i ∈ t, (↑(basicOpen (g i)) : Set (PrimeSpectrum R))
N : ℕ := (t ×ˢ t).sup fun (x : ↥U × ↥U) ↦ n x.1 x.2 + 1
i j : ↥t
refine_1 : g (↑j : ↥U) ^ (N + 1) • g (↑i : ↥U) ^ N • f (↑i : ↥U) = g (↑i : ↥U) ^ (N + 1) • g (↑j : ↥U) ^ N • f (↑j : ↥U)
⊢ HSMul.hSMul (α := R) (β := M) ((fun (i : ↥t) ↦ g (↑i : ↥U) ^ (N + 1)) j)
      ((fun (i : ↥t) ↦ g (↑i : ↥U) ^ N • f (↑i : ↥U)) i) =
    HSMul.hSMul (α := R) (β := M) ((fun (i : ↥t) ↦ g (↑i : ↥U) ^ (N + 1)) i)
      ((fun (i : ↥t) ↦ g (↑i : ↥U) ^ N • f (↑i : ↥U)) j)

Tactic:
dsimp