algebraic_geometry.projective_spectrum.scheme ⟷ Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -169,9 +169,9 @@ theorem MemCarrier.clear_denominator' [DecidableEq (Away f)] {z : Localization.A
           (∑ i in c.support.attach,
             acd (c i) (Finset.mem_image.mpr ⟨i, ⟨i.2, rfl⟩⟩) * i.1.2.some) :=
   by
-  rw [← submodule_span_eq, Finsupp.span_eq_range_total, LinearMap.mem_range] at hz 
+  rw [← submodule_span_eq, Finsupp.span_eq_range_total, LinearMap.mem_range] at hz
   rcases hz with ⟨c, eq1⟩
-  rw [Finsupp.total_apply, Finsupp.sum] at eq1 
+  rw [Finsupp.total_apply, Finsupp.sum] at eq1
   obtain ⟨⟨_, N, rfl⟩, hN⟩ :=
     IsLocalization.exist_integer_multiples_of_finset (Submonoid.powers f) (c.support.image c)
   choose acd hacd using hN
@@ -200,13 +200,13 @@ theorem MemCarrier.clear_denominator [DecidableEq (Away f)] {z : A⁰_ f} (hz :
 theorem disjoint : Disjoint (x.1.asHomogeneousIdeal.toIdeal : Set A) (Submonoid.powers f : Set A) :=
   by
   by_contra rid
-  rw [Set.not_disjoint_iff] at rid 
+  rw [Set.not_disjoint_iff] at rid
   choose g hg using rid
   obtain ⟨hg1, ⟨k, rfl⟩⟩ := hg
   by_cases k_ineq : 0 < k
-  · erw [x.1.IsPrime.pow_mem_iff_mem _ k_ineq] at hg1 
+  · erw [x.1.IsPrime.pow_mem_iff_mem _ k_ineq] at hg1
     exact x.2 hg1
-  · erw [show k = 0 by linarith, pow_zero, ← Ideal.eq_top_iff_one] at hg1 
+  · erw [show k = 0 by linarith, pow_zero, ← Ideal.eq_top_iff_one] at hg1
     apply x.1.IsPrime.1
     exact hg1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.disjoint AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.disjoint
@@ -219,12 +219,12 @@ theorem carrier_ne_top : carrier 𝒜 x ≠ ⊤ :=
   classical
   contrapose! eq_top
   obtain ⟨c, N, acd, eq1⟩ := mem_carrier.clear_denominator _ x ((Ideal.eq_top_iff_one _).mp eq_top)
-  rw [Algebra.smul_def, HomogeneousLocalization.one_val, mul_one] at eq1 
-  change Localization.mk (f ^ N) 1 = mk (∑ _, _) 1 at eq1 
-  simp only [mk_eq_mk', IsLocalization.eq] at eq1 
+  rw [Algebra.smul_def, HomogeneousLocalization.one_val, mul_one] at eq1
+  change Localization.mk (f ^ N) 1 = mk (∑ _, _) 1 at eq1
+  simp only [mk_eq_mk', IsLocalization.eq] at eq1
   rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
-  erw [one_mul, one_mul] at eq1 
-  change f ^ _ * f ^ _ = f ^ _ * _ at eq1 
+  erw [one_mul, one_mul] at eq1
+  change f ^ _ * f ^ _ = f ^ _ * _ at eq1
   rw [Set.not_disjoint_iff_nonempty_inter]
   refine'
     ⟨f ^ M * f ^ N, eq1.symm ▸ mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _),
@@ -246,20 +246,20 @@ def toFun (x : Proj.T| pbo f) : Spec.T A⁰_ f :=
     simp only [mem_carrier_iff] at hx12 ⊢
     let J := span (⇑(algebraMap A (away f)) '' x.val.as_homogeneous_ideal)
     suffices h : ∀ x y : Localization.Away f, x * y ∈ J → x ∈ J ∨ y ∈ J
-    · rw [HomogeneousLocalization.mul_val] at hx12 ; exact h x1.val x2.val hx12
+    · rw [HomogeneousLocalization.mul_val] at hx12; exact h x1.val x2.val hx12
     clear x1 x2 hx12
     intro x1 x2 hx12
     induction' x1 using Localization.induction_on with data_x1
     induction' x2 using Localization.induction_on with data_x2
     rcases data_x1, data_x2 with ⟨⟨a1, _, ⟨n1, rfl⟩⟩, ⟨a2, _, ⟨n2, rfl⟩⟩⟩
     rcases mem_carrier.clear_denominator' x hx12 with ⟨c, N, acd, eq1⟩
-    simp only [Algebra.smul_def] at eq1 
-    change Localization.mk (f ^ N) 1 * (mk _ _ * mk _ _) = mk (∑ _, _) _ at eq1 
-    simp only [Localization.mk_mul, one_mul] at eq1 
-    simp only [mk_eq_mk', IsLocalization.eq] at eq1 
+    simp only [Algebra.smul_def] at eq1
+    change Localization.mk (f ^ N) 1 * (mk _ _ * mk _ _) = mk (∑ _, _) _ at eq1
+    simp only [Localization.mk_mul, one_mul] at eq1
+    simp only [mk_eq_mk', IsLocalization.eq] at eq1
     rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
-    rw [Submonoid.coe_one, one_mul] at eq1 
-    change f ^ _ * (_ * _) = f ^ _ * (f ^ _ * f ^ _ * _) at eq1 
+    rw [Submonoid.coe_one, one_mul] at eq1
+    change f ^ _ * (_ * _) = f ^ _ * (f ^ _ * f ^ _ * _) at eq1
     rcases x.1.IsPrime.mem_or_mem (show a1 * a2 * f ^ N * f ^ M ∈ _ from _) with (h1 | rid2)
     rcases x.1.IsPrime.mem_or_mem h1 with (h1 | rid1)
     rcases x.1.IsPrime.mem_or_mem h1 with (h1 | h2)
@@ -296,7 +296,7 @@ theorem preimage_eq (a b : A) (k : ℕ) (a_mem : a ∈ 𝒜 k) (b_mem1 : b ∈ 
   ext1 y
   constructor <;> intro hy
   · refine' ⟨y.2, _⟩
-    rw [Set.mem_preimage, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen] at hy 
+    rw [Set.mem_preimage, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen] at hy
     rw [ProjectiveSpectrum.mem_coe_basicOpen]
     intro a_mem_y
     apply hy
@@ -305,18 +305,18 @@ theorem preimage_eq (a b : A) (k : ℕ) (a_mem : a ∈ 𝒜 k) (b_mem1 : b ∈ 
     simp only [show (mk a ⟨b, ⟨k, hk⟩⟩ : away f) = mk 1 ⟨f ^ k, ⟨_, rfl⟩⟩ * mk a 1 by
         rw [mk_mul, one_mul, mul_one]; congr; rw [hk]]
     exact Ideal.mul_mem_left _ _ (Ideal.subset_span ⟨_, a_mem_y, rfl⟩)
-  · change y.1 ∈ _ at hy 
+  · change y.1 ∈ _ at hy
     rcases hy with ⟨hy1, hy2⟩
-    rw [ProjectiveSpectrum.mem_coe_basicOpen] at hy1 hy2 
+    rw [ProjectiveSpectrum.mem_coe_basicOpen] at hy1 hy2
     rw [Set.mem_preimage, to_fun, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen]
-    intro rid; dsimp at rid 
+    intro rid; dsimp at rid
     rcases mem_carrier.clear_denominator 𝒜 _ rid with ⟨c, N, acd, eq1⟩
-    rw [Algebra.smul_def] at eq1 
-    change Localization.mk (f ^ N) 1 * mk _ _ = mk (∑ _, _) _ at eq1 
-    rw [mk_mul, one_mul, mk_eq_mk', IsLocalization.eq] at eq1 
+    rw [Algebra.smul_def] at eq1
+    change Localization.mk (f ^ N) 1 * mk _ _ = mk (∑ _, _) _ at eq1
+    rw [mk_mul, one_mul, mk_eq_mk', IsLocalization.eq] at eq1
     rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
-    rw [Submonoid.coe_one, one_mul] at eq1 
-    simp only [Subtype.coe_mk] at eq1 
+    rw [Submonoid.coe_one, one_mul] at eq1
+    simp only [Subtype.coe_mk] at eq1
     rcases y.1.IsPrime.mem_or_mem (show a * f ^ N * f ^ M ∈ _ from _) with (H1 | H3)
     rcases y.1.IsPrime.mem_or_mem H1 with (H1 | H2)
     · exact hy2 H1
@@ -434,7 +434,7 @@ theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
     (by
       constructor <;> intro h i <;> specialize h i
       · rw [Set.mem_image]; refine' ⟨_, h, rfl⟩
-      · rw [Set.mem_image] at h ; rcases h with ⟨x, h, hx⟩
+      · rw [Set.mem_image] at h; rcases h with ⟨x, h, hx⟩
         convert h; rw [ext_iff_val, val_mk']; dsimp only [Subtype.coe_mk]; rw [← hx]; rfl)
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_carrier_iff' AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -217,6 +217,20 @@ theorem carrier_ne_top : carrier 𝒜 x ≠ ⊤ :=
   by
   have eq_top := Disjoint x
   classical
+  contrapose! eq_top
+  obtain ⟨c, N, acd, eq1⟩ := mem_carrier.clear_denominator _ x ((Ideal.eq_top_iff_one _).mp eq_top)
+  rw [Algebra.smul_def, HomogeneousLocalization.one_val, mul_one] at eq1 
+  change Localization.mk (f ^ N) 1 = mk (∑ _, _) 1 at eq1 
+  simp only [mk_eq_mk', IsLocalization.eq] at eq1 
+  rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
+  erw [one_mul, one_mul] at eq1 
+  change f ^ _ * f ^ _ = f ^ _ * _ at eq1 
+  rw [Set.not_disjoint_iff_nonempty_inter]
+  refine'
+    ⟨f ^ M * f ^ N, eq1.symm ▸ mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _),
+      ⟨M + N, by rw [pow_add]⟩⟩
+  generalize_proofs h₁ h₂
+  exact (Classical.choose_spec h₂).1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.carrier_ne_top AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.carrier_ne_top
 -/
 
@@ -227,7 +241,41 @@ variable (f)
 `Spec A⁰_f`. This is bundled into a continuous map in `Top_component.forward`.
 -/
 def toFun (x : Proj.T| pbo f) : Spec.T A⁰_ f :=
-  ⟨carrier 𝒜 x, carrier_ne_top x, fun x1 x2 hx12 => by classical⟩
+  ⟨carrier 𝒜 x, carrier_ne_top x, fun x1 x2 hx12 => by
+    classical
+    simp only [mem_carrier_iff] at hx12 ⊢
+    let J := span (⇑(algebraMap A (away f)) '' x.val.as_homogeneous_ideal)
+    suffices h : ∀ x y : Localization.Away f, x * y ∈ J → x ∈ J ∨ y ∈ J
+    · rw [HomogeneousLocalization.mul_val] at hx12 ; exact h x1.val x2.val hx12
+    clear x1 x2 hx12
+    intro x1 x2 hx12
+    induction' x1 using Localization.induction_on with data_x1
+    induction' x2 using Localization.induction_on with data_x2
+    rcases data_x1, data_x2 with ⟨⟨a1, _, ⟨n1, rfl⟩⟩, ⟨a2, _, ⟨n2, rfl⟩⟩⟩
+    rcases mem_carrier.clear_denominator' x hx12 with ⟨c, N, acd, eq1⟩
+    simp only [Algebra.smul_def] at eq1 
+    change Localization.mk (f ^ N) 1 * (mk _ _ * mk _ _) = mk (∑ _, _) _ at eq1 
+    simp only [Localization.mk_mul, one_mul] at eq1 
+    simp only [mk_eq_mk', IsLocalization.eq] at eq1 
+    rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
+    rw [Submonoid.coe_one, one_mul] at eq1 
+    change f ^ _ * (_ * _) = f ^ _ * (f ^ _ * f ^ _ * _) at eq1 
+    rcases x.1.IsPrime.mem_or_mem (show a1 * a2 * f ^ N * f ^ M ∈ _ from _) with (h1 | rid2)
+    rcases x.1.IsPrime.mem_or_mem h1 with (h1 | rid1)
+    rcases x.1.IsPrime.mem_or_mem h1 with (h1 | h2)
+    · left;
+      simp only [show (mk a1 ⟨f ^ n1, _⟩ : away f) = mk a1 1 * mk 1 ⟨f ^ n1, ⟨n1, rfl⟩⟩ by
+          rw [Localization.mk_mul, mul_one, one_mul]]
+      exact Ideal.mul_mem_right _ _ (Ideal.subset_span ⟨_, h1, rfl⟩)
+    · right;
+      simp only [show (mk a2 ⟨f ^ n2, _⟩ : away f) = mk a2 1 * mk 1 ⟨f ^ n2, ⟨n2, rfl⟩⟩ by
+          rw [Localization.mk_mul, mul_one, one_mul]]
+      exact Ideal.mul_mem_right _ _ (Ideal.subset_span ⟨_, h2, rfl⟩)
+    · exact False.elim (x.2 (x.1.IsPrime.mem_of_pow_mem N rid1))
+    · exact False.elim (x.2 (x.1.IsPrime.mem_of_pow_mem M rid2))
+    · rw [← mul_comm (f ^ M), ← mul_comm (f ^ N), eq1]
+      refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _))
+      generalize_proofs h₁ h₂; exact (Classical.choose_spec h₂).1⟩
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.to_fun AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.toFun
 -/
 
@@ -243,7 +291,40 @@ theorem preimage_eq (a b : A) (k : ℕ) (a_mem : a ∈ 𝒜 k) (b_mem1 : b ∈ 
         (@PrimeSpectrum.basicOpen (A⁰_ f) _ (Quotient.mk'' ⟨k, ⟨a, a_mem⟩, ⟨b, b_mem1⟩, b_mem2⟩) :
           Set (PrimeSpectrum (HomogeneousLocalization.Away 𝒜 f))) =
       {x | x.1 ∈ (pbo f) ⊓ pbo a} :=
-  by classical
+  by
+  classical
+  ext1 y
+  constructor <;> intro hy
+  · refine' ⟨y.2, _⟩
+    rw [Set.mem_preimage, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen] at hy 
+    rw [ProjectiveSpectrum.mem_coe_basicOpen]
+    intro a_mem_y
+    apply hy
+    rw [to_fun, mem_carrier_iff, HomogeneousLocalization.val_mk'', Subtype.coe_mk]
+    dsimp; rcases b_mem2 with ⟨k, hk⟩
+    simp only [show (mk a ⟨b, ⟨k, hk⟩⟩ : away f) = mk 1 ⟨f ^ k, ⟨_, rfl⟩⟩ * mk a 1 by
+        rw [mk_mul, one_mul, mul_one]; congr; rw [hk]]
+    exact Ideal.mul_mem_left _ _ (Ideal.subset_span ⟨_, a_mem_y, rfl⟩)
+  · change y.1 ∈ _ at hy 
+    rcases hy with ⟨hy1, hy2⟩
+    rw [ProjectiveSpectrum.mem_coe_basicOpen] at hy1 hy2 
+    rw [Set.mem_preimage, to_fun, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen]
+    intro rid; dsimp at rid 
+    rcases mem_carrier.clear_denominator 𝒜 _ rid with ⟨c, N, acd, eq1⟩
+    rw [Algebra.smul_def] at eq1 
+    change Localization.mk (f ^ N) 1 * mk _ _ = mk (∑ _, _) _ at eq1 
+    rw [mk_mul, one_mul, mk_eq_mk', IsLocalization.eq] at eq1 
+    rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
+    rw [Submonoid.coe_one, one_mul] at eq1 
+    simp only [Subtype.coe_mk] at eq1 
+    rcases y.1.IsPrime.mem_or_mem (show a * f ^ N * f ^ M ∈ _ from _) with (H1 | H3)
+    rcases y.1.IsPrime.mem_or_mem H1 with (H1 | H2)
+    · exact hy2 H1
+    · exact y.2 (y.1.IsPrime.mem_of_pow_mem N H2)
+    · exact y.2 (y.1.IsPrime.mem_of_pow_mem M H3)
+    · rw [mul_comm _ (f ^ N), mul_comm _ (f ^ M), eq1]
+      refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _))
+      generalize_proofs h₁ h₂; exact (Classical.choose_spec h₂).1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.preimage_eq AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.preimage_eq
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -217,20 +217,6 @@ theorem carrier_ne_top : carrier 𝒜 x ≠ ⊤ :=
   by
   have eq_top := Disjoint x
   classical
-  contrapose! eq_top
-  obtain ⟨c, N, acd, eq1⟩ := mem_carrier.clear_denominator _ x ((Ideal.eq_top_iff_one _).mp eq_top)
-  rw [Algebra.smul_def, HomogeneousLocalization.one_val, mul_one] at eq1 
-  change Localization.mk (f ^ N) 1 = mk (∑ _, _) 1 at eq1 
-  simp only [mk_eq_mk', IsLocalization.eq] at eq1 
-  rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
-  erw [one_mul, one_mul] at eq1 
-  change f ^ _ * f ^ _ = f ^ _ * _ at eq1 
-  rw [Set.not_disjoint_iff_nonempty_inter]
-  refine'
-    ⟨f ^ M * f ^ N, eq1.symm ▸ mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _),
-      ⟨M + N, by rw [pow_add]⟩⟩
-  generalize_proofs h₁ h₂
-  exact (Classical.choose_spec h₂).1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.carrier_ne_top AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.carrier_ne_top
 -/
 
@@ -241,41 +227,7 @@ variable (f)
 `Spec A⁰_f`. This is bundled into a continuous map in `Top_component.forward`.
 -/
 def toFun (x : Proj.T| pbo f) : Spec.T A⁰_ f :=
-  ⟨carrier 𝒜 x, carrier_ne_top x, fun x1 x2 hx12 => by
-    classical
-    simp only [mem_carrier_iff] at hx12 ⊢
-    let J := span (⇑(algebraMap A (away f)) '' x.val.as_homogeneous_ideal)
-    suffices h : ∀ x y : Localization.Away f, x * y ∈ J → x ∈ J ∨ y ∈ J
-    · rw [HomogeneousLocalization.mul_val] at hx12 ; exact h x1.val x2.val hx12
-    clear x1 x2 hx12
-    intro x1 x2 hx12
-    induction' x1 using Localization.induction_on with data_x1
-    induction' x2 using Localization.induction_on with data_x2
-    rcases data_x1, data_x2 with ⟨⟨a1, _, ⟨n1, rfl⟩⟩, ⟨a2, _, ⟨n2, rfl⟩⟩⟩
-    rcases mem_carrier.clear_denominator' x hx12 with ⟨c, N, acd, eq1⟩
-    simp only [Algebra.smul_def] at eq1 
-    change Localization.mk (f ^ N) 1 * (mk _ _ * mk _ _) = mk (∑ _, _) _ at eq1 
-    simp only [Localization.mk_mul, one_mul] at eq1 
-    simp only [mk_eq_mk', IsLocalization.eq] at eq1 
-    rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
-    rw [Submonoid.coe_one, one_mul] at eq1 
-    change f ^ _ * (_ * _) = f ^ _ * (f ^ _ * f ^ _ * _) at eq1 
-    rcases x.1.IsPrime.mem_or_mem (show a1 * a2 * f ^ N * f ^ M ∈ _ from _) with (h1 | rid2)
-    rcases x.1.IsPrime.mem_or_mem h1 with (h1 | rid1)
-    rcases x.1.IsPrime.mem_or_mem h1 with (h1 | h2)
-    · left;
-      simp only [show (mk a1 ⟨f ^ n1, _⟩ : away f) = mk a1 1 * mk 1 ⟨f ^ n1, ⟨n1, rfl⟩⟩ by
-          rw [Localization.mk_mul, mul_one, one_mul]]
-      exact Ideal.mul_mem_right _ _ (Ideal.subset_span ⟨_, h1, rfl⟩)
-    · right;
-      simp only [show (mk a2 ⟨f ^ n2, _⟩ : away f) = mk a2 1 * mk 1 ⟨f ^ n2, ⟨n2, rfl⟩⟩ by
-          rw [Localization.mk_mul, mul_one, one_mul]]
-      exact Ideal.mul_mem_right _ _ (Ideal.subset_span ⟨_, h2, rfl⟩)
-    · exact False.elim (x.2 (x.1.IsPrime.mem_of_pow_mem N rid1))
-    · exact False.elim (x.2 (x.1.IsPrime.mem_of_pow_mem M rid2))
-    · rw [← mul_comm (f ^ M), ← mul_comm (f ^ N), eq1]
-      refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _))
-      generalize_proofs h₁ h₂; exact (Classical.choose_spec h₂).1⟩
+  ⟨carrier 𝒜 x, carrier_ne_top x, fun x1 x2 hx12 => by classical⟩
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.to_fun AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.toFun
 -/
 
@@ -291,40 +243,7 @@ theorem preimage_eq (a b : A) (k : ℕ) (a_mem : a ∈ 𝒜 k) (b_mem1 : b ∈ 
         (@PrimeSpectrum.basicOpen (A⁰_ f) _ (Quotient.mk'' ⟨k, ⟨a, a_mem⟩, ⟨b, b_mem1⟩, b_mem2⟩) :
           Set (PrimeSpectrum (HomogeneousLocalization.Away 𝒜 f))) =
       {x | x.1 ∈ (pbo f) ⊓ pbo a} :=
-  by
-  classical
-  ext1 y
-  constructor <;> intro hy
-  · refine' ⟨y.2, _⟩
-    rw [Set.mem_preimage, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen] at hy 
-    rw [ProjectiveSpectrum.mem_coe_basicOpen]
-    intro a_mem_y
-    apply hy
-    rw [to_fun, mem_carrier_iff, HomogeneousLocalization.val_mk'', Subtype.coe_mk]
-    dsimp; rcases b_mem2 with ⟨k, hk⟩
-    simp only [show (mk a ⟨b, ⟨k, hk⟩⟩ : away f) = mk 1 ⟨f ^ k, ⟨_, rfl⟩⟩ * mk a 1 by
-        rw [mk_mul, one_mul, mul_one]; congr; rw [hk]]
-    exact Ideal.mul_mem_left _ _ (Ideal.subset_span ⟨_, a_mem_y, rfl⟩)
-  · change y.1 ∈ _ at hy 
-    rcases hy with ⟨hy1, hy2⟩
-    rw [ProjectiveSpectrum.mem_coe_basicOpen] at hy1 hy2 
-    rw [Set.mem_preimage, to_fun, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen]
-    intro rid; dsimp at rid 
-    rcases mem_carrier.clear_denominator 𝒜 _ rid with ⟨c, N, acd, eq1⟩
-    rw [Algebra.smul_def] at eq1 
-    change Localization.mk (f ^ N) 1 * mk _ _ = mk (∑ _, _) _ at eq1 
-    rw [mk_mul, one_mul, mk_eq_mk', IsLocalization.eq] at eq1 
-    rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
-    rw [Submonoid.coe_one, one_mul] at eq1 
-    simp only [Subtype.coe_mk] at eq1 
-    rcases y.1.IsPrime.mem_or_mem (show a * f ^ N * f ^ M ∈ _ from _) with (H1 | H3)
-    rcases y.1.IsPrime.mem_or_mem H1 with (H1 | H2)
-    · exact hy2 H1
-    · exact y.2 (y.1.IsPrime.mem_of_pow_mem N H2)
-    · exact y.2 (y.1.IsPrime.mem_of_pow_mem M H3)
-    · rw [mul_comm _ (f ^ N), mul_comm _ (f ^ M), eq1]
-      refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _))
-      generalize_proofs h₁ h₂; exact (Classical.choose_spec h₂).1
+  by classical
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.preimage_eq AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.preimage_eq
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,9 +3,9 @@ Copyright (c) 2022 Jujian Zhang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jujian Zhang
 -/
-import Mathbin.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
-import Mathbin.AlgebraicGeometry.Spec
-import Mathbin.RingTheory.GradedAlgebra.Radical
+import AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
+import AlgebraicGeometry.Spec
+import RingTheory.GradedAlgebra.Radical
 
 #align_import algebraic_geometry.projective_spectrum.scheme from "leanprover-community/mathlib"@"2fe465deb81bcd7ccafa065bb686888a82f15372"
 
@@ -363,8 +363,8 @@ open _Root_.HomogeneousLocalization
 
 variable {𝒜} {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m)
 
-/- ./././Mathport/Syntax/Translate/Expr.lean:336:4: warning: unsupported (TODO): `[tacs] -/
-/- ./././Mathport/Syntax/Translate/Expr.lean:336:4: warning: unsupported (TODO): `[tacs] -/
+/- ./././Mathport/Syntax/Translate/Expr.lean:337:4: warning: unsupported (TODO): `[tacs] -/
+/- ./././Mathport/Syntax/Translate/Expr.lean:337:4: warning: unsupported (TODO): `[tacs] -/
 private unsafe def mem_tac : tactic Unit :=
   let b : tactic Unit := sorry
   b <|> sorry

mathlib commit https://github.com/leanprover-community/mathlib/commit/63721b2c3eba6c325ecf8ae8cca27155a4f6306f

@@ -369,8 +369,8 @@ private unsafe def mem_tac : tactic Unit :=
   let b : tactic Unit := sorry
   b <|> sorry
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2444063103.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2444063103.mem_tac -/
 #print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier /-
 /-- The function from `Spec A⁰_f` to `Proj|D(f)` is defined by `q ↦ {a | aᵢᵐ/fⁱ ∈ q}`, i.e. sending
 `q` a prime ideal in `A⁰_f` to the homogeneous prime relevant ideal containing only and all the
@@ -402,8 +402,8 @@ def carrier (q : Spec.T A⁰_ f) : Set A :=
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier
 -/
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2444063103.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2444063103.mem_tac -/
 #print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff /-
 theorem mem_carrier_iff (q : Spec.T A⁰_ f) (a : A) :
     a ∈ carrier f_deg q ↔
@@ -439,14 +439,14 @@ theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_carrier_iff' AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'
 -/
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2444063103.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2444063103.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2444063103.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2444063103.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2444063103.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2444063103.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2444063103.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2444063103.mem_tac -/
 #print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.add_mem /-
 theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
     (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q :=
@@ -531,8 +531,8 @@ theorem carrier.zero_mem : (0 : A) ∈ carrier f_deg q := fun i =>
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.zero_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.zero_mem
 -/
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2444063103.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2444063103.mem_tac -/
 #print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.smul_mem /-
 theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q :=
   by

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,16 +2,13 @@
 Copyright (c) 2022 Jujian Zhang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jujian Zhang
-
-! This file was ported from Lean 3 source module algebraic_geometry.projective_spectrum.scheme
-! leanprover-community/mathlib commit 2fe465deb81bcd7ccafa065bb686888a82f15372
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
 import Mathbin.AlgebraicGeometry.Spec
 import Mathbin.RingTheory.GradedAlgebra.Radical
 
+#align_import algebraic_geometry.projective_spectrum.scheme from "leanprover-community/mathlib"@"2fe465deb81bcd7ccafa065bb686888a82f15372"
+
 /-!
 # Proj as a scheme
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/2fe465deb81bcd7ccafa065bb686888a82f15372

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jujian Zhang
 
 ! This file was ported from Lean 3 source module algebraic_geometry.projective_spectrum.scheme
-! leanprover-community/mathlib commit d39590fc8728fbf6743249802486f8c91ffe07bc
+! leanprover-community/mathlib commit 2fe465deb81bcd7ccafa065bb686888a82f15372
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.RingTheory.GradedAlgebra.Radical
 /-!
 # Proj as a scheme
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file is to prove that `Proj` is a scheme.
 
 ## Notation
@@ -369,8 +372,8 @@ private unsafe def mem_tac : tactic Unit :=
   let b : tactic Unit := sorry
   b <|> sorry
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
 #print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier /-
 /-- The function from `Spec A⁰_f` to `Proj|D(f)` is defined by `q ↦ {a | aᵢᵐ/fⁱ ∈ q}`, i.e. sending
 `q` a prime ideal in `A⁰_f` to the homogeneous prime relevant ideal containing only and all the
@@ -402,8 +405,8 @@ def carrier (q : Spec.T A⁰_ f) : Set A :=
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier
 -/
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
 #print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff /-
 theorem mem_carrier_iff (q : Spec.T A⁰_ f) (a : A) :
     a ∈ carrier f_deg q ↔
@@ -439,14 +442,14 @@ theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_carrier_iff' AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'
 -/
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
 #print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.add_mem /-
 theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
     (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q :=
@@ -531,8 +534,8 @@ theorem carrier.zero_mem : (0 : A) ∈ carrier f_deg q := fun i =>
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.zero_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.zero_mem
 -/
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2357112569.mem_tac -/
 #print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.smul_mem /-
 theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q :=
   by

mathlib commit https://github.com/leanprover-community/mathlib/commit/8905e5ed90859939681a725b00f6063e65096d95

@@ -142,6 +142,7 @@ open Ideal
 -- and we need this correspondence to be continuous in their Zariski topology.
 variable {𝒜} {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (x : Proj| pbo f)
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.carrier /-
 /--
 For any `x` in `Proj| (pbo f)`, the corresponding ideal in `Spec A⁰_f`. This fact that this ideal
 is prime is proven in `Top_component.forward.to_fun`-/
@@ -149,12 +150,16 @@ def carrier : Ideal (A⁰_ f) :=
   Ideal.comap (algebraMap (A⁰_ f) (Away f))
     (Ideal.span <| algebraMap A (Away f) '' x.val.asHomogeneousIdeal)
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.carrier AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.carrier
+-/
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.mem_carrier_iff /-
 theorem mem_carrier_iff (z : A⁰_ f) :
     z ∈ carrier 𝒜 x ↔ z.val ∈ Ideal.span (algebraMap A (Away f) '' x.1.asHomogeneousIdeal) :=
   Iff.rfl
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.mem_carrier_iff AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.mem_carrier_iff
+-/
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.MemCarrier.clear_denominator' /-
 theorem MemCarrier.clear_denominator' [DecidableEq (Away f)] {z : Localization.Away f}
     (hz : z ∈ span (algebraMap A (Away f) '' x.val.asHomogeneousIdeal)) :
     ∃ (c : algebraMap A (Away f) '' x.1.asHomogeneousIdeal →₀ Away f) (N : ℕ) (acd :
@@ -177,7 +182,9 @@ theorem MemCarrier.clear_denominator' [DecidableEq (Away f)] {z : Localization.A
   rw [_root_.map_mul, hacd, (Classical.choose_spec i.1.2).2, smul_eq_mul, smul_mul_assoc]
   rfl
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.mem_carrier.clear_denominator' AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.MemCarrier.clear_denominator'
+-/
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.MemCarrier.clear_denominator /-
 theorem MemCarrier.clear_denominator [DecidableEq (Away f)] {z : A⁰_ f} (hz : z ∈ carrier 𝒜 x) :
     ∃ (c : algebraMap A (Away f) '' x.1.asHomogeneousIdeal →₀ Away f) (N : ℕ) (acd :
       ∀ y ∈ c.support.image c, A),
@@ -187,7 +194,9 @@ theorem MemCarrier.clear_denominator [DecidableEq (Away f)] {z : A⁰_ f} (hz :
             acd (c i) (Finset.mem_image.mpr ⟨i, ⟨i.2, rfl⟩⟩) * i.1.2.some) :=
   MemCarrier.clear_denominator' x <| (mem_carrier_iff 𝒜 x z).mpr hz
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.mem_carrier.clear_denominator AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.MemCarrier.clear_denominator
+-/
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.disjoint /-
 theorem disjoint : Disjoint (x.1.asHomogeneousIdeal.toIdeal : Set A) (Submonoid.powers f : Set A) :=
   by
   by_contra rid
@@ -201,7 +210,9 @@ theorem disjoint : Disjoint (x.1.asHomogeneousIdeal.toIdeal : Set A) (Submonoid.
     apply x.1.IsPrime.1
     exact hg1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.disjoint AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.disjoint
+-/
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.carrier_ne_top /-
 theorem carrier_ne_top : carrier 𝒜 x ≠ ⊤ :=
   by
   have eq_top := Disjoint x
@@ -221,9 +232,11 @@ theorem carrier_ne_top : carrier 𝒜 x ≠ ⊤ :=
   generalize_proofs h₁ h₂
   exact (Classical.choose_spec h₂).1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.carrier_ne_top AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.carrier_ne_top
+-/
 
 variable (f)
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.toFun /-
 /-- The function between the basic open set `D(f)` in `Proj` to the corresponding basic open set in
 `Spec A⁰_f`. This is bundled into a continuous map in `Top_component.forward`.
 -/
@@ -264,7 +277,9 @@ def toFun (x : Proj.T| pbo f) : Spec.T A⁰_ f :=
       refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _))
       generalize_proofs h₁ h₂; exact (Classical.choose_spec h₂).1⟩
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.to_fun AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.toFun
+-/
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.preimage_eq /-
 /-
 The preimage of basic open set `D(a/f^n)` in `Spec A⁰_f` under the forward map from `Proj A` to
 `Spec A⁰_f` is the basic open set `D(a) ∩ D(f)` in  `Proj A`. This lemma is used to prove that the
@@ -311,6 +326,7 @@ theorem preimage_eq (a b : A) (k : ℕ) (a_mem : a ∈ 𝒜 k) (b_mem1 : b ∈ 
       refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _))
       generalize_proofs h₁ h₂; exact (Classical.choose_spec h₂).1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.preimage_eq AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.preimage_eq
+-/
 
 end ToSpec
 
@@ -318,6 +334,7 @@ section
 
 variable {𝒜}
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec /-
 /-- The continuous function between the basic open set `D(f)` in `Proj` to the corresponding basic
 open set in `Spec A⁰_f`.
 -/
@@ -332,6 +349,7 @@ def toSpec {f : A} : (Proj.T| pbo f) ⟶ Spec.T A⁰_ f
     refine' is_open_induced_iff.mpr ⟨(pbo f).1 ⊓ (pbo a).1, IsOpen.inter (pbo f).2 (pbo a).2, _⟩
     ext z; constructor <;> intro hz <;> simpa [Set.mem_preimage]
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec
+-/
 
 end
 
@@ -353,6 +371,7 @@ private unsafe def mem_tac : tactic Unit :=
 
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier /-
 /-- The function from `Spec A⁰_f` to `Proj|D(f)` is defined by `q ↦ {a | aᵢᵐ/fⁱ ∈ q}`, i.e. sending
 `q` a prime ideal in `A⁰_f` to the homogeneous prime relevant ideal containing only and all the
 elements `a : A` such that for every `i`, the degree 0 element formed by dividing the `m`-th power
@@ -381,9 +400,11 @@ def carrier (q : Spec.T A⁰_ f) : Set A :=
           A⁰_ f) ∈
         q.1}
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier
+-/
 
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff /-
 theorem mem_carrier_iff (q : Spec.T A⁰_ f) (a : A) :
     a ∈ carrier f_deg q ↔
       ∀ i,
@@ -401,7 +422,9 @@ theorem mem_carrier_iff (q : Spec.T A⁰_ f) (a : A) :
           q.1 :=
   Iff.rfl
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_carrier_iff AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff
+-/
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff' /-
 theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
     a ∈ carrier f_deg q ↔
       ∀ i,
@@ -414,6 +437,7 @@ theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
       · rw [Set.mem_image] at h ; rcases h with ⟨x, h, hx⟩
         convert h; rw [ext_iff_val, val_mk']; dsimp only [Subtype.coe_mk]; rw [← hx]; rfl)
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_carrier_iff' AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'
+-/
 
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
@@ -423,6 +447,7 @@ theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.add_mem /-
 theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
     (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q :=
   by
@@ -493,18 +518,22 @@ theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg
   · rw [mul_assoc, ← pow_add, add_comm (m - j), Nat.add_sub_assoc h1]; · simp_rw [pow_add]; rfl
   · rw [← mul_assoc, ← pow_add, Nat.add_sub_of_le (le_of_not_le h1)]; · simp_rw [pow_add]; rfl
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.add_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.add_mem
+-/
 
 variable (hm : 0 < m) (q : Spec.T A⁰_ f)
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.zero_mem /-
 theorem carrier.zero_mem : (0 : A) ∈ carrier f_deg q := fun i =>
   by
   convert Submodule.zero_mem q.1 using 1
   rw [ext_iff_val, val_mk', zero_val]; simp_rw [map_zero, zero_pow hm]
   convert Localization.mk_zero _ using 1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.zero_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.zero_mem
+-/
 
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.smul_mem /-
 theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q :=
   by
   revert c
@@ -532,7 +561,9 @@ theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ ca
     · simp_rw [zero_pow hm]; convert carrier.zero_mem f_deg hm q i; rw [map_zero, zero_pow hm]
   · simp_rw [add_smul]; exact fun _ _ => carrier.add_mem f_deg q
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.smul_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.smul_mem
+-/
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal /-
 /-- For a prime ideal `q` in `A⁰_f`, the set `{a | aᵢᵐ/fⁱ ∈ q}` as an ideal.
 -/
 def carrier.asIdeal : Ideal A where
@@ -541,7 +572,9 @@ def carrier.asIdeal : Ideal A where
   add_mem' a b := carrier.add_mem f_deg q
   smul_mem' := carrier.smul_mem f_deg hm q
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.as_ideal AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal
+-/
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.homogeneous /-
 theorem carrier.asIdeal.homogeneous : (carrier.asIdeal f_deg hm q).Homogeneous 𝒜 := fun i a ha j =>
   (em (i = j)).elim (fun h => h ▸ by simpa only [proj_apply, decompose_coe, of_eq_same] using ha _)
     fun h =>
@@ -550,13 +583,17 @@ theorem carrier.asIdeal.homogeneous : (carrier.asIdeal f_deg hm q).Homogeneous 
       zero_pow hm]
     convert carrier.zero_mem f_deg hm q j; rw [map_zero, zero_pow hm]
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.as_ideal.homogeneous AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.homogeneous
+-/
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asHomogeneousIdeal /-
 /-- For a prime ideal `q` in `A⁰_f`, the set `{a | aᵢᵐ/fⁱ ∈ q}` as a homogeneous ideal.
 -/
 def carrier.asHomogeneousIdeal : HomogeneousIdeal 𝒜 :=
   ⟨carrier.asIdeal f_deg hm q, carrier.asIdeal.homogeneous f_deg hm q⟩
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.as_homogeneous_ideal AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asHomogeneousIdeal
+-/
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.denom_not_mem /-
 theorem carrier.denom_not_mem : f ∉ carrier.asIdeal f_deg hm q := fun rid =>
   q.IsPrime.ne_top <|
     (Ideal.eq_top_iff_one _).mpr
@@ -565,15 +602,21 @@ theorem carrier.denom_not_mem : f ∉ carrier.asIdeal f_deg hm q := fun rid =>
         simpa only [ext_iff_val, one_val, proj_apply, decompose_of_mem_same _ f_deg, val_mk'] using
           (mk_self (⟨_, m, rfl⟩ : Submonoid.powers f)).symm)
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.denom_not_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.denom_not_mem
+-/
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.relevant /-
 theorem carrier.relevant : ¬HomogeneousIdeal.irrelevant 𝒜 ≤ carrier.asHomogeneousIdeal f_deg hm q :=
   fun rid => carrier.denom_not_mem f_deg hm q <| rid <| DirectSum.decompose_of_mem_ne 𝒜 f_deg hm.ne'
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.relevant AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.relevant
+-/
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.ne_top /-
 theorem carrier.asIdeal.ne_top : carrier.asIdeal f_deg hm q ≠ ⊤ := fun rid =>
   carrier.denom_not_mem f_deg hm q (rid.symm ▸ Submodule.mem_top)
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.as_ideal.ne_top AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.ne_top
+-/
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.prime /-
 theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
   (carrier.asIdeal.homogeneous f_deg hm q).isPrime_of_homogeneous_mem_or_mem
     (carrier.asIdeal.ne_top f_deg hm q) fun x y ⟨nx, hnx⟩ ⟨ny, hny⟩ hxy =>
@@ -593,9 +636,11 @@ theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
           | exact hnx
           | exact hny
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.as_ideal.prime AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.prime
+-/
 
 variable (f_deg)
 
+#print AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.toFun /-
 /-- The function `Spec A⁰_f → Proj|D(f)` by sending `q` to `{a | aᵢᵐ/fⁱ ∈ q}`.
 -/
 def toFun : (Spec.T A⁰_ f) → Proj.T| pbo f := fun q =>
@@ -603,6 +648,7 @@ def toFun : (Spec.T A⁰_ f) → Proj.T| pbo f := fun q =>
       carrier.relevant f_deg hm q⟩,
     (ProjectiveSpectrum.mem_basicOpen _ f _).mp <| carrier.denom_not_mem f_deg hm q⟩
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.to_fun AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.toFun
+-/
 
 end FromSpec
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -98,40 +98,31 @@ open CategoryTheory Opposite
 
 open ProjectiveSpectrum.StructureSheaf
 
--- mathport name: exprProj
 local notation "Proj" => Proj.toLocallyRingedSpace 𝒜
 
--- mathport name: «exprProj.T»
 -- `Proj` as a locally ringed space
 local notation "Proj.T" => Proj.1.1.1
 
--- mathport name: «exprProj| »
 -- the underlying topological space of `Proj`
 local notation "Proj| " U => Proj.restrict (Opens.openEmbedding (U : Opens Proj.T))
 
--- mathport name: «exprProj.T| »
 -- `Proj` restrict to some open set
 local notation "Proj.T| " U =>
   (Proj.restrict (Opens.openEmbedding (U : Opens Proj.T))).toSheafedSpace.toPresheafedSpace.1
 
--- mathport name: «exprpbo »
 -- the underlying topological space of `Proj` restricted to some open set
 local notation "pbo " x => ProjectiveSpectrum.basicOpen 𝒜 x
 
--- mathport name: «exprsbo »
 -- basic open sets in `Proj`
 local notation "sbo " f => PrimeSpectrum.basicOpen f
 
--- mathport name: «exprSpec »
 -- basic open sets in `Spec`
 local notation "Spec " ring => Spec.locallyRingedSpaceObj (CommRingCat.of Ring)
 
--- mathport name: «exprSpec.T »
 -- `Spec` as a locally ringed space
 local notation "Spec.T " ring =>
   (Spec.locallyRingedSpaceObj (CommRingCat.of Ring)).toSheafedSpace.toPresheafedSpace.1
 
--- mathport name: «exprA⁰_ »
 -- the underlying topological space of `Spec`
 local notation "A⁰_ " f => HomogeneousLocalization.Away 𝒜 f
 
@@ -360,8 +351,6 @@ private unsafe def mem_tac : tactic Unit :=
   let b : tactic Unit := sorry
   b <|> sorry
 
-include f_deg
-
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
 /-- The function from `Spec A⁰_f` to `Proj|D(f)` is defined by `q ↦ {a | aᵢᵐ/fⁱ ∈ q}`, i.e. sending
@@ -507,8 +496,6 @@ theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg
 
 variable (hm : 0 < m) (q : Spec.T A⁰_ f)
 
-include hm
-
 theorem carrier.zero_mem : (0 : A) ∈ carrier f_deg q := fun i =>
   by
   convert Submodule.zero_mem q.1 using 1

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -489,7 +489,7 @@ theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg
     run_tac
       mem_tac;
     rw [← add_smul]; congr; zify [le_of_not_lt h2, le_of_not_le h1]; abel
-  convert_to (∑ i in range (m + m + 1), g i) ∈ q.1; swap
+  convert_to ∑ i in range (m + m + 1), g i ∈ q.1; swap
   · refine' q.1.sum_mem fun j hj => nsmul_mem _ _; split_ifs
     exacts [q.1.zero_mem, q.1.mul_mem_left _ (hb i), q.1.mul_mem_right _ (ha i)]
   rw [ext_iff_val, val_mk']

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -354,8 +354,8 @@ open _Root_.HomogeneousLocalization
 
 variable {𝒜} {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m)
 
-/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/
-/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/
+/- ./././Mathport/Syntax/Translate/Expr.lean:336:4: warning: unsupported (TODO): `[tacs] -/
+/- ./././Mathport/Syntax/Translate/Expr.lean:336:4: warning: unsupported (TODO): `[tacs] -/
 private unsafe def mem_tac : tactic Unit :=
   let b : tactic Unit := sorry
   b <|> sorry

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -215,21 +215,20 @@ theorem carrier_ne_top : carrier 𝒜 x ≠ ⊤ :=
   by
   have eq_top := Disjoint x
   classical
-    contrapose! eq_top
-    obtain ⟨c, N, acd, eq1⟩ :=
-      mem_carrier.clear_denominator _ x ((Ideal.eq_top_iff_one _).mp eq_top)
-    rw [Algebra.smul_def, HomogeneousLocalization.one_val, mul_one] at eq1 
-    change Localization.mk (f ^ N) 1 = mk (∑ _, _) 1 at eq1 
-    simp only [mk_eq_mk', IsLocalization.eq] at eq1 
-    rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
-    erw [one_mul, one_mul] at eq1 
-    change f ^ _ * f ^ _ = f ^ _ * _ at eq1 
-    rw [Set.not_disjoint_iff_nonempty_inter]
-    refine'
-      ⟨f ^ M * f ^ N, eq1.symm ▸ mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _),
-        ⟨M + N, by rw [pow_add]⟩⟩
-    generalize_proofs h₁ h₂
-    exact (Classical.choose_spec h₂).1
+  contrapose! eq_top
+  obtain ⟨c, N, acd, eq1⟩ := mem_carrier.clear_denominator _ x ((Ideal.eq_top_iff_one _).mp eq_top)
+  rw [Algebra.smul_def, HomogeneousLocalization.one_val, mul_one] at eq1 
+  change Localization.mk (f ^ N) 1 = mk (∑ _, _) 1 at eq1 
+  simp only [mk_eq_mk', IsLocalization.eq] at eq1 
+  rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
+  erw [one_mul, one_mul] at eq1 
+  change f ^ _ * f ^ _ = f ^ _ * _ at eq1 
+  rw [Set.not_disjoint_iff_nonempty_inter]
+  refine'
+    ⟨f ^ M * f ^ N, eq1.symm ▸ mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _),
+      ⟨M + N, by rw [pow_add]⟩⟩
+  generalize_proofs h₁ h₂
+  exact (Classical.choose_spec h₂).1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.carrier_ne_top AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.carrier_ne_top
 
 variable (f)
@@ -240,39 +239,39 @@ variable (f)
 def toFun (x : Proj.T| pbo f) : Spec.T A⁰_ f :=
   ⟨carrier 𝒜 x, carrier_ne_top x, fun x1 x2 hx12 => by
     classical
-      simp only [mem_carrier_iff] at hx12 ⊢
-      let J := span (⇑(algebraMap A (away f)) '' x.val.as_homogeneous_ideal)
-      suffices h : ∀ x y : Localization.Away f, x * y ∈ J → x ∈ J ∨ y ∈ J
-      · rw [HomogeneousLocalization.mul_val] at hx12 ; exact h x1.val x2.val hx12
-      clear x1 x2 hx12
-      intro x1 x2 hx12
-      induction' x1 using Localization.induction_on with data_x1
-      induction' x2 using Localization.induction_on with data_x2
-      rcases data_x1, data_x2 with ⟨⟨a1, _, ⟨n1, rfl⟩⟩, ⟨a2, _, ⟨n2, rfl⟩⟩⟩
-      rcases mem_carrier.clear_denominator' x hx12 with ⟨c, N, acd, eq1⟩
-      simp only [Algebra.smul_def] at eq1 
-      change Localization.mk (f ^ N) 1 * (mk _ _ * mk _ _) = mk (∑ _, _) _ at eq1 
-      simp only [Localization.mk_mul, one_mul] at eq1 
-      simp only [mk_eq_mk', IsLocalization.eq] at eq1 
-      rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
-      rw [Submonoid.coe_one, one_mul] at eq1 
-      change f ^ _ * (_ * _) = f ^ _ * (f ^ _ * f ^ _ * _) at eq1 
-      rcases x.1.IsPrime.mem_or_mem (show a1 * a2 * f ^ N * f ^ M ∈ _ from _) with (h1 | rid2)
-      rcases x.1.IsPrime.mem_or_mem h1 with (h1 | rid1)
-      rcases x.1.IsPrime.mem_or_mem h1 with (h1 | h2)
-      · left;
-        simp only [show (mk a1 ⟨f ^ n1, _⟩ : away f) = mk a1 1 * mk 1 ⟨f ^ n1, ⟨n1, rfl⟩⟩ by
-            rw [Localization.mk_mul, mul_one, one_mul]]
-        exact Ideal.mul_mem_right _ _ (Ideal.subset_span ⟨_, h1, rfl⟩)
-      · right;
-        simp only [show (mk a2 ⟨f ^ n2, _⟩ : away f) = mk a2 1 * mk 1 ⟨f ^ n2, ⟨n2, rfl⟩⟩ by
-            rw [Localization.mk_mul, mul_one, one_mul]]
-        exact Ideal.mul_mem_right _ _ (Ideal.subset_span ⟨_, h2, rfl⟩)
-      · exact False.elim (x.2 (x.1.IsPrime.mem_of_pow_mem N rid1))
-      · exact False.elim (x.2 (x.1.IsPrime.mem_of_pow_mem M rid2))
-      · rw [← mul_comm (f ^ M), ← mul_comm (f ^ N), eq1]
-        refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _))
-        generalize_proofs h₁ h₂; exact (Classical.choose_spec h₂).1⟩
+    simp only [mem_carrier_iff] at hx12 ⊢
+    let J := span (⇑(algebraMap A (away f)) '' x.val.as_homogeneous_ideal)
+    suffices h : ∀ x y : Localization.Away f, x * y ∈ J → x ∈ J ∨ y ∈ J
+    · rw [HomogeneousLocalization.mul_val] at hx12 ; exact h x1.val x2.val hx12
+    clear x1 x2 hx12
+    intro x1 x2 hx12
+    induction' x1 using Localization.induction_on with data_x1
+    induction' x2 using Localization.induction_on with data_x2
+    rcases data_x1, data_x2 with ⟨⟨a1, _, ⟨n1, rfl⟩⟩, ⟨a2, _, ⟨n2, rfl⟩⟩⟩
+    rcases mem_carrier.clear_denominator' x hx12 with ⟨c, N, acd, eq1⟩
+    simp only [Algebra.smul_def] at eq1 
+    change Localization.mk (f ^ N) 1 * (mk _ _ * mk _ _) = mk (∑ _, _) _ at eq1 
+    simp only [Localization.mk_mul, one_mul] at eq1 
+    simp only [mk_eq_mk', IsLocalization.eq] at eq1 
+    rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
+    rw [Submonoid.coe_one, one_mul] at eq1 
+    change f ^ _ * (_ * _) = f ^ _ * (f ^ _ * f ^ _ * _) at eq1 
+    rcases x.1.IsPrime.mem_or_mem (show a1 * a2 * f ^ N * f ^ M ∈ _ from _) with (h1 | rid2)
+    rcases x.1.IsPrime.mem_or_mem h1 with (h1 | rid1)
+    rcases x.1.IsPrime.mem_or_mem h1 with (h1 | h2)
+    · left;
+      simp only [show (mk a1 ⟨f ^ n1, _⟩ : away f) = mk a1 1 * mk 1 ⟨f ^ n1, ⟨n1, rfl⟩⟩ by
+          rw [Localization.mk_mul, mul_one, one_mul]]
+      exact Ideal.mul_mem_right _ _ (Ideal.subset_span ⟨_, h1, rfl⟩)
+    · right;
+      simp only [show (mk a2 ⟨f ^ n2, _⟩ : away f) = mk a2 1 * mk 1 ⟨f ^ n2, ⟨n2, rfl⟩⟩ by
+          rw [Localization.mk_mul, mul_one, one_mul]]
+      exact Ideal.mul_mem_right _ _ (Ideal.subset_span ⟨_, h2, rfl⟩)
+    · exact False.elim (x.2 (x.1.IsPrime.mem_of_pow_mem N rid1))
+    · exact False.elim (x.2 (x.1.IsPrime.mem_of_pow_mem M rid2))
+    · rw [← mul_comm (f ^ M), ← mul_comm (f ^ N), eq1]
+      refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _))
+      generalize_proofs h₁ h₂; exact (Classical.choose_spec h₂).1⟩
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.to_fun AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.toFun
 
 /-
@@ -285,41 +284,41 @@ theorem preimage_eq (a b : A) (k : ℕ) (a_mem : a ∈ 𝒜 k) (b_mem1 : b ∈ 
     toFun 𝒜 f ⁻¹'
         (@PrimeSpectrum.basicOpen (A⁰_ f) _ (Quotient.mk'' ⟨k, ⟨a, a_mem⟩, ⟨b, b_mem1⟩, b_mem2⟩) :
           Set (PrimeSpectrum (HomogeneousLocalization.Away 𝒜 f))) =
-      { x | x.1 ∈ (pbo f) ⊓ pbo a } :=
+      {x | x.1 ∈ (pbo f) ⊓ pbo a} :=
   by
   classical
-    ext1 y
-    constructor <;> intro hy
-    · refine' ⟨y.2, _⟩
-      rw [Set.mem_preimage, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen] at hy 
-      rw [ProjectiveSpectrum.mem_coe_basicOpen]
-      intro a_mem_y
-      apply hy
-      rw [to_fun, mem_carrier_iff, HomogeneousLocalization.val_mk'', Subtype.coe_mk]
-      dsimp; rcases b_mem2 with ⟨k, hk⟩
-      simp only [show (mk a ⟨b, ⟨k, hk⟩⟩ : away f) = mk 1 ⟨f ^ k, ⟨_, rfl⟩⟩ * mk a 1 by
-          rw [mk_mul, one_mul, mul_one]; congr; rw [hk]]
-      exact Ideal.mul_mem_left _ _ (Ideal.subset_span ⟨_, a_mem_y, rfl⟩)
-    · change y.1 ∈ _ at hy 
-      rcases hy with ⟨hy1, hy2⟩
-      rw [ProjectiveSpectrum.mem_coe_basicOpen] at hy1 hy2 
-      rw [Set.mem_preimage, to_fun, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen]
-      intro rid; dsimp at rid 
-      rcases mem_carrier.clear_denominator 𝒜 _ rid with ⟨c, N, acd, eq1⟩
-      rw [Algebra.smul_def] at eq1 
-      change Localization.mk (f ^ N) 1 * mk _ _ = mk (∑ _, _) _ at eq1 
-      rw [mk_mul, one_mul, mk_eq_mk', IsLocalization.eq] at eq1 
-      rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
-      rw [Submonoid.coe_one, one_mul] at eq1 
-      simp only [Subtype.coe_mk] at eq1 
-      rcases y.1.IsPrime.mem_or_mem (show a * f ^ N * f ^ M ∈ _ from _) with (H1 | H3)
-      rcases y.1.IsPrime.mem_or_mem H1 with (H1 | H2)
-      · exact hy2 H1
-      · exact y.2 (y.1.IsPrime.mem_of_pow_mem N H2)
-      · exact y.2 (y.1.IsPrime.mem_of_pow_mem M H3)
-      · rw [mul_comm _ (f ^ N), mul_comm _ (f ^ M), eq1]
-        refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _))
-        generalize_proofs h₁ h₂; exact (Classical.choose_spec h₂).1
+  ext1 y
+  constructor <;> intro hy
+  · refine' ⟨y.2, _⟩
+    rw [Set.mem_preimage, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen] at hy 
+    rw [ProjectiveSpectrum.mem_coe_basicOpen]
+    intro a_mem_y
+    apply hy
+    rw [to_fun, mem_carrier_iff, HomogeneousLocalization.val_mk'', Subtype.coe_mk]
+    dsimp; rcases b_mem2 with ⟨k, hk⟩
+    simp only [show (mk a ⟨b, ⟨k, hk⟩⟩ : away f) = mk 1 ⟨f ^ k, ⟨_, rfl⟩⟩ * mk a 1 by
+        rw [mk_mul, one_mul, mul_one]; congr; rw [hk]]
+    exact Ideal.mul_mem_left _ _ (Ideal.subset_span ⟨_, a_mem_y, rfl⟩)
+  · change y.1 ∈ _ at hy 
+    rcases hy with ⟨hy1, hy2⟩
+    rw [ProjectiveSpectrum.mem_coe_basicOpen] at hy1 hy2 
+    rw [Set.mem_preimage, to_fun, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen]
+    intro rid; dsimp at rid 
+    rcases mem_carrier.clear_denominator 𝒜 _ rid with ⟨c, N, acd, eq1⟩
+    rw [Algebra.smul_def] at eq1 
+    change Localization.mk (f ^ N) 1 * mk _ _ = mk (∑ _, _) _ at eq1 
+    rw [mk_mul, one_mul, mk_eq_mk', IsLocalization.eq] at eq1 
+    rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
+    rw [Submonoid.coe_one, one_mul] at eq1 
+    simp only [Subtype.coe_mk] at eq1 
+    rcases y.1.IsPrime.mem_or_mem (show a * f ^ N * f ^ M ∈ _ from _) with (H1 | H3)
+    rcases y.1.IsPrime.mem_or_mem H1 with (H1 | H2)
+    · exact hy2 H1
+    · exact y.2 (y.1.IsPrime.mem_of_pow_mem N H2)
+    · exact y.2 (y.1.IsPrime.mem_of_pow_mem M H3)
+    · rw [mul_comm _ (f ^ N), mul_comm _ (f ^ M), eq1]
+      refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _))
+      generalize_proofs h₁ h₂; exact (Classical.choose_spec h₂).1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.preimage_eq AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.preimage_eq
 
 end ToSpec
@@ -378,7 +377,7 @@ The set `{a | aᵢᵐ/fⁱ ∈ q}`
 * is relevant, as proved in `carrier.relevant`.
 -/
 def carrier (q : Spec.T A⁰_ f) : Set A :=
-  { a |
+  {a |
     ∀ i,
       (Quotient.mk''
             ⟨m * i,
@@ -391,7 +390,7 @@ def carrier (q : Spec.T A⁰_ f) : Set A :=
                     mem_tac⟩,
               ⟨_, rfl⟩⟩ :
           A⁰_ f) ∈
-        q.1 }
+        q.1}
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier
 
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
@@ -490,7 +489,7 @@ theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg
     run_tac
       mem_tac;
     rw [← add_smul]; congr; zify [le_of_not_lt h2, le_of_not_le h1]; abel
-  convert_to(∑ i in range (m + m + 1), g i) ∈ q.1; swap
+  convert_to (∑ i in range (m + m + 1), g i) ∈ q.1; swap
   · refine' q.1.sum_mem fun j hj => nsmul_mem _ _; split_ifs
     exacts [q.1.zero_mem, q.1.mul_mem_left _ (hb i), q.1.mul_mem_right _ (ha i)]
   rw [ext_iff_val, val_mk']
@@ -527,14 +526,16 @@ theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ ca
   · rintro n ⟨a, ha⟩ i
     simp_rw [Subtype.coe_mk, proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
     split_ifs
-    · convert_to(Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, _⟩, ⟨n, rfl⟩⟩ *
+    · convert_to
+        (Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, _⟩, ⟨n, rfl⟩⟩ *
               Quotient.mk''
                 ⟨_,
                   ⟨proj 𝒜 (i - n) x ^ m, by
                     run_tac
                       mem_tac⟩,
                   ⟨_, _⟩, ⟨i - n, rfl⟩⟩ :
-            A⁰_ f) ∈ q.1
+            A⁰_ f) ∈
+          q.1
       · erw [ext_iff_val, val_mk', mul_val, val_mk', val_mk', Subtype.coe_mk]
         simp_rw [mul_pow, Subtype.coe_mk]; rw [Localization.mk_mul]
         congr; erw [← pow_add, Nat.add_sub_of_le h]

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -166,16 +166,16 @@ theorem mem_carrier_iff (z : A⁰_ f) :
 
 theorem MemCarrier.clear_denominator' [DecidableEq (Away f)] {z : Localization.Away f}
     (hz : z ∈ span (algebraMap A (Away f) '' x.val.asHomogeneousIdeal)) :
-    ∃ (c : algebraMap A (Away f) '' x.1.asHomogeneousIdeal →₀ Away f)(N : ℕ)(acd :
+    ∃ (c : algebraMap A (Away f) '' x.1.asHomogeneousIdeal →₀ Away f) (N : ℕ) (acd :
       ∀ y ∈ c.support.image c, A),
       f ^ N • z =
         algebraMap A (Away f)
           (∑ i in c.support.attach,
             acd (c i) (Finset.mem_image.mpr ⟨i, ⟨i.2, rfl⟩⟩) * i.1.2.some) :=
   by
-  rw [← submodule_span_eq, Finsupp.span_eq_range_total, LinearMap.mem_range] at hz
+  rw [← submodule_span_eq, Finsupp.span_eq_range_total, LinearMap.mem_range] at hz 
   rcases hz with ⟨c, eq1⟩
-  rw [Finsupp.total_apply, Finsupp.sum] at eq1
+  rw [Finsupp.total_apply, Finsupp.sum] at eq1 
   obtain ⟨⟨_, N, rfl⟩, hN⟩ :=
     IsLocalization.exist_integer_multiples_of_finset (Submonoid.powers f) (c.support.image c)
   choose acd hacd using hN
@@ -188,7 +188,7 @@ theorem MemCarrier.clear_denominator' [DecidableEq (Away f)] {z : Localization.A
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.mem_carrier.clear_denominator' AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.MemCarrier.clear_denominator'
 
 theorem MemCarrier.clear_denominator [DecidableEq (Away f)] {z : A⁰_ f} (hz : z ∈ carrier 𝒜 x) :
-    ∃ (c : algebraMap A (Away f) '' x.1.asHomogeneousIdeal →₀ Away f)(N : ℕ)(acd :
+    ∃ (c : algebraMap A (Away f) '' x.1.asHomogeneousIdeal →₀ Away f) (N : ℕ) (acd :
       ∀ y ∈ c.support.image c, A),
       f ^ N • z.val =
         algebraMap A (Away f)
@@ -200,13 +200,13 @@ theorem MemCarrier.clear_denominator [DecidableEq (Away f)] {z : A⁰_ f} (hz :
 theorem disjoint : Disjoint (x.1.asHomogeneousIdeal.toIdeal : Set A) (Submonoid.powers f : Set A) :=
   by
   by_contra rid
-  rw [Set.not_disjoint_iff] at rid
+  rw [Set.not_disjoint_iff] at rid 
   choose g hg using rid
   obtain ⟨hg1, ⟨k, rfl⟩⟩ := hg
   by_cases k_ineq : 0 < k
-  · erw [x.1.IsPrime.pow_mem_iff_mem _ k_ineq] at hg1
+  · erw [x.1.IsPrime.pow_mem_iff_mem _ k_ineq] at hg1 
     exact x.2 hg1
-  · erw [show k = 0 by linarith, pow_zero, ← Ideal.eq_top_iff_one] at hg1
+  · erw [show k = 0 by linarith, pow_zero, ← Ideal.eq_top_iff_one] at hg1 
     apply x.1.IsPrime.1
     exact hg1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.disjoint AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.disjoint
@@ -218,12 +218,12 @@ theorem carrier_ne_top : carrier 𝒜 x ≠ ⊤ :=
     contrapose! eq_top
     obtain ⟨c, N, acd, eq1⟩ :=
       mem_carrier.clear_denominator _ x ((Ideal.eq_top_iff_one _).mp eq_top)
-    rw [Algebra.smul_def, HomogeneousLocalization.one_val, mul_one] at eq1
-    change Localization.mk (f ^ N) 1 = mk (∑ _, _) 1 at eq1
-    simp only [mk_eq_mk', IsLocalization.eq] at eq1
+    rw [Algebra.smul_def, HomogeneousLocalization.one_val, mul_one] at eq1 
+    change Localization.mk (f ^ N) 1 = mk (∑ _, _) 1 at eq1 
+    simp only [mk_eq_mk', IsLocalization.eq] at eq1 
     rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
-    erw [one_mul, one_mul] at eq1
-    change f ^ _ * f ^ _ = f ^ _ * _ at eq1
+    erw [one_mul, one_mul] at eq1 
+    change f ^ _ * f ^ _ = f ^ _ * _ at eq1 
     rw [Set.not_disjoint_iff_nonempty_inter]
     refine'
       ⟨f ^ M * f ^ N, eq1.symm ▸ mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _),
@@ -240,23 +240,23 @@ variable (f)
 def toFun (x : Proj.T| pbo f) : Spec.T A⁰_ f :=
   ⟨carrier 𝒜 x, carrier_ne_top x, fun x1 x2 hx12 => by
     classical
-      simp only [mem_carrier_iff] at hx12⊢
+      simp only [mem_carrier_iff] at hx12 ⊢
       let J := span (⇑(algebraMap A (away f)) '' x.val.as_homogeneous_ideal)
       suffices h : ∀ x y : Localization.Away f, x * y ∈ J → x ∈ J ∨ y ∈ J
-      · rw [HomogeneousLocalization.mul_val] at hx12; exact h x1.val x2.val hx12
+      · rw [HomogeneousLocalization.mul_val] at hx12 ; exact h x1.val x2.val hx12
       clear x1 x2 hx12
       intro x1 x2 hx12
       induction' x1 using Localization.induction_on with data_x1
       induction' x2 using Localization.induction_on with data_x2
       rcases data_x1, data_x2 with ⟨⟨a1, _, ⟨n1, rfl⟩⟩, ⟨a2, _, ⟨n2, rfl⟩⟩⟩
       rcases mem_carrier.clear_denominator' x hx12 with ⟨c, N, acd, eq1⟩
-      simp only [Algebra.smul_def] at eq1
-      change Localization.mk (f ^ N) 1 * (mk _ _ * mk _ _) = mk (∑ _, _) _ at eq1
-      simp only [Localization.mk_mul, one_mul] at eq1
-      simp only [mk_eq_mk', IsLocalization.eq] at eq1
+      simp only [Algebra.smul_def] at eq1 
+      change Localization.mk (f ^ N) 1 * (mk _ _ * mk _ _) = mk (∑ _, _) _ at eq1 
+      simp only [Localization.mk_mul, one_mul] at eq1 
+      simp only [mk_eq_mk', IsLocalization.eq] at eq1 
       rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
-      rw [Submonoid.coe_one, one_mul] at eq1
-      change f ^ _ * (_ * _) = f ^ _ * (f ^ _ * f ^ _ * _) at eq1
+      rw [Submonoid.coe_one, one_mul] at eq1 
+      change f ^ _ * (_ * _) = f ^ _ * (f ^ _ * f ^ _ * _) at eq1 
       rcases x.1.IsPrime.mem_or_mem (show a1 * a2 * f ^ N * f ^ M ∈ _ from _) with (h1 | rid2)
       rcases x.1.IsPrime.mem_or_mem h1 with (h1 | rid1)
       rcases x.1.IsPrime.mem_or_mem h1 with (h1 | h2)
@@ -291,27 +291,27 @@ theorem preimage_eq (a b : A) (k : ℕ) (a_mem : a ∈ 𝒜 k) (b_mem1 : b ∈ 
     ext1 y
     constructor <;> intro hy
     · refine' ⟨y.2, _⟩
-      rw [Set.mem_preimage, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen] at hy
+      rw [Set.mem_preimage, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen] at hy 
       rw [ProjectiveSpectrum.mem_coe_basicOpen]
       intro a_mem_y
       apply hy
       rw [to_fun, mem_carrier_iff, HomogeneousLocalization.val_mk'', Subtype.coe_mk]
       dsimp; rcases b_mem2 with ⟨k, hk⟩
       simp only [show (mk a ⟨b, ⟨k, hk⟩⟩ : away f) = mk 1 ⟨f ^ k, ⟨_, rfl⟩⟩ * mk a 1 by
-          rw [mk_mul, one_mul, mul_one]; congr ; rw [hk]]
+          rw [mk_mul, one_mul, mul_one]; congr; rw [hk]]
       exact Ideal.mul_mem_left _ _ (Ideal.subset_span ⟨_, a_mem_y, rfl⟩)
-    · change y.1 ∈ _ at hy
+    · change y.1 ∈ _ at hy 
       rcases hy with ⟨hy1, hy2⟩
-      rw [ProjectiveSpectrum.mem_coe_basicOpen] at hy1 hy2
+      rw [ProjectiveSpectrum.mem_coe_basicOpen] at hy1 hy2 
       rw [Set.mem_preimage, to_fun, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen]
-      intro rid; dsimp at rid
+      intro rid; dsimp at rid 
       rcases mem_carrier.clear_denominator 𝒜 _ rid with ⟨c, N, acd, eq1⟩
-      rw [Algebra.smul_def] at eq1
-      change Localization.mk (f ^ N) 1 * mk _ _ = mk (∑ _, _) _ at eq1
-      rw [mk_mul, one_mul, mk_eq_mk', IsLocalization.eq] at eq1
+      rw [Algebra.smul_def] at eq1 
+      change Localization.mk (f ^ N) 1 * mk _ _ = mk (∑ _, _) _ at eq1 
+      rw [mk_mul, one_mul, mk_eq_mk', IsLocalization.eq] at eq1 
       rcases eq1 with ⟨⟨_, ⟨M, rfl⟩⟩, eq1⟩
-      rw [Submonoid.coe_one, one_mul] at eq1
-      simp only [Subtype.coe_mk] at eq1
+      rw [Submonoid.coe_one, one_mul] at eq1 
+      simp only [Subtype.coe_mk] at eq1 
       rcases y.1.IsPrime.mem_or_mem (show a * f ^ N * f ^ M ∈ _ from _) with (H1 | H3)
       rcases y.1.IsPrime.mem_or_mem H1 with (H1 | H2)
       · exact hy2 H1
@@ -423,7 +423,7 @@ theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
     (by
       constructor <;> intro h i <;> specialize h i
       · rw [Set.mem_image]; refine' ⟨_, h, rfl⟩
-      · rw [Set.mem_image] at h; rcases h with ⟨x, h, hx⟩
+      · rw [Set.mem_image] at h ; rcases h with ⟨x, h, hx⟩
         convert h; rw [ext_iff_val, val_mk']; dsimp only [Subtype.coe_mk]; rw [← hx]; rfl)
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_carrier_iff' AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'
 
@@ -489,10 +489,10 @@ theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg
   · rw [(_ : m * i = _)];
     run_tac
       mem_tac;
-    rw [← add_smul]; congr ; zify [le_of_not_lt h2, le_of_not_le h1] ; abel
+    rw [← add_smul]; congr; zify [le_of_not_lt h2, le_of_not_le h1]; abel
   convert_to(∑ i in range (m + m + 1), g i) ∈ q.1; swap
   · refine' q.1.sum_mem fun j hj => nsmul_mem _ _; split_ifs
-    exacts[q.1.zero_mem, q.1.mul_mem_left _ (hb i), q.1.mul_mem_right _ (ha i)]
+    exacts [q.1.zero_mem, q.1.mul_mem_left _ (hb i), q.1.mul_mem_right _ (ha i)]
   rw [ext_iff_val, val_mk']
   change _ = (algebraMap (HomogeneousLocalization.Away 𝒜 f) (Localization.Away f)) _
   dsimp only [Subtype.coe_mk]; rw [map_sum, mk_sum]
@@ -537,7 +537,7 @@ theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ ca
             A⁰_ f) ∈ q.1
       · erw [ext_iff_val, val_mk', mul_val, val_mk', val_mk', Subtype.coe_mk]
         simp_rw [mul_pow, Subtype.coe_mk]; rw [Localization.mk_mul]
-        congr ; erw [← pow_add, Nat.add_sub_of_le h]
+        congr; erw [← pow_add, Nat.add_sub_of_le h]
       · exact Ideal.mul_mem_left _ _ (hx _); rw [smul_eq_mul, mul_comm];
         run_tac
           mem_tac
@@ -600,7 +600,10 @@ theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
         intro n hn; convert q.1.zero_mem using 1
         rw [ext_iff_val, val_mk', zero_val]; simp_rw [proj_apply, Subtype.coe_mk]
         convert mk_zero _; rw [decompose_of_mem_ne 𝒜 _ hn.symm, zero_pow hm]
-        · first |exact hnx|exact hny
+        ·
+          first
+          | exact hnx
+          | exact hny
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.as_ideal.prime AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.prime
 
 variable (f_deg)

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -78,7 +78,7 @@ noncomputable section
 
 namespace AlgebraicGeometry
 
-open DirectSum BigOperators Pointwise BigOperators
+open scoped DirectSum BigOperators Pointwise BigOperators
 
 open DirectSum SetLike.GradedMonoid Localization
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -243,8 +243,7 @@ def toFun (x : Proj.T| pbo f) : Spec.T A⁰_ f :=
       simp only [mem_carrier_iff] at hx12⊢
       let J := span (⇑(algebraMap A (away f)) '' x.val.as_homogeneous_ideal)
       suffices h : ∀ x y : Localization.Away f, x * y ∈ J → x ∈ J ∨ y ∈ J
-      · rw [HomogeneousLocalization.mul_val] at hx12
-        exact h x1.val x2.val hx12
+      · rw [HomogeneousLocalization.mul_val] at hx12; exact h x1.val x2.val hx12
       clear x1 x2 hx12
       intro x1 x2 hx12
       induction' x1 using Localization.induction_on with data_x1
@@ -261,11 +260,11 @@ def toFun (x : Proj.T| pbo f) : Spec.T A⁰_ f :=
       rcases x.1.IsPrime.mem_or_mem (show a1 * a2 * f ^ N * f ^ M ∈ _ from _) with (h1 | rid2)
       rcases x.1.IsPrime.mem_or_mem h1 with (h1 | rid1)
       rcases x.1.IsPrime.mem_or_mem h1 with (h1 | h2)
-      · left
+      · left;
         simp only [show (mk a1 ⟨f ^ n1, _⟩ : away f) = mk a1 1 * mk 1 ⟨f ^ n1, ⟨n1, rfl⟩⟩ by
             rw [Localization.mk_mul, mul_one, one_mul]]
         exact Ideal.mul_mem_right _ _ (Ideal.subset_span ⟨_, h1, rfl⟩)
-      · right
+      · right;
         simp only [show (mk a2 ⟨f ^ n2, _⟩ : away f) = mk a2 1 * mk 1 ⟨f ^ n2, ⟨n2, rfl⟩⟩ by
             rw [Localization.mk_mul, mul_one, one_mul]]
         exact Ideal.mul_mem_right _ _ (Ideal.subset_span ⟨_, h2, rfl⟩)
@@ -273,8 +272,7 @@ def toFun (x : Proj.T| pbo f) : Spec.T A⁰_ f :=
       · exact False.elim (x.2 (x.1.IsPrime.mem_of_pow_mem M rid2))
       · rw [← mul_comm (f ^ M), ← mul_comm (f ^ N), eq1]
         refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _))
-        generalize_proofs h₁ h₂
-        exact (Classical.choose_spec h₂).1⟩
+        generalize_proofs h₁ h₂; exact (Classical.choose_spec h₂).1⟩
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.to_fun AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.toFun
 
 /-
@@ -298,20 +296,15 @@ theorem preimage_eq (a b : A) (k : ℕ) (a_mem : a ∈ 𝒜 k) (b_mem1 : b ∈ 
       intro a_mem_y
       apply hy
       rw [to_fun, mem_carrier_iff, HomogeneousLocalization.val_mk'', Subtype.coe_mk]
-      dsimp
-      rcases b_mem2 with ⟨k, hk⟩
-      simp only [show (mk a ⟨b, ⟨k, hk⟩⟩ : away f) = mk 1 ⟨f ^ k, ⟨_, rfl⟩⟩ * mk a 1
-          by
-          rw [mk_mul, one_mul, mul_one]
-          congr
-          rw [hk]]
+      dsimp; rcases b_mem2 with ⟨k, hk⟩
+      simp only [show (mk a ⟨b, ⟨k, hk⟩⟩ : away f) = mk 1 ⟨f ^ k, ⟨_, rfl⟩⟩ * mk a 1 by
+          rw [mk_mul, one_mul, mul_one]; congr ; rw [hk]]
       exact Ideal.mul_mem_left _ _ (Ideal.subset_span ⟨_, a_mem_y, rfl⟩)
     · change y.1 ∈ _ at hy
       rcases hy with ⟨hy1, hy2⟩
       rw [ProjectiveSpectrum.mem_coe_basicOpen] at hy1 hy2
       rw [Set.mem_preimage, to_fun, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen]
-      intro rid
-      dsimp at rid
+      intro rid; dsimp at rid
       rcases mem_carrier.clear_denominator 𝒜 _ rid with ⟨c, N, acd, eq1⟩
       rw [Algebra.smul_def] at eq1
       change Localization.mk (f ^ N) 1 * mk _ _ = mk (∑ _, _) _ at eq1
@@ -326,8 +319,7 @@ theorem preimage_eq (a b : A) (k : ℕ) (a_mem : a ∈ 𝒜 k) (b_mem1 : b ∈ 
       · exact y.2 (y.1.IsPrime.mem_of_pow_mem M H3)
       · rw [mul_comm _ (f ^ N), mul_comm _ (f ^ M), eq1]
         refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i hi => mul_mem_left _ _ _))
-        generalize_proofs h₁ h₂
-        exact (Classical.choose_spec h₂).1
+        generalize_proofs h₁ h₂; exact (Classical.choose_spec h₂).1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.preimage_eq AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.preimage_eq
 
 end ToSpec
@@ -430,15 +422,9 @@ theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
   (mem_carrier_iff f_deg q a).trans
     (by
       constructor <;> intro h i <;> specialize h i
-      · rw [Set.mem_image]
-        refine' ⟨_, h, rfl⟩
-      · rw [Set.mem_image] at h
-        rcases h with ⟨x, h, hx⟩
-        convert h
-        rw [ext_iff_val, val_mk']
-        dsimp only [Subtype.coe_mk]
-        rw [← hx]
-        rfl)
+      · rw [Set.mem_image]; refine' ⟨_, h, rfl⟩
+      · rw [Set.mem_image] at h; rcases h with ⟨x, h, hx⟩
+        convert h; rw [ext_iff_val, val_mk']; dsimp only [Subtype.coe_mk]; rw [← hx]; rfl)
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_carrier_iff' AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'
 
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
@@ -496,21 +482,16 @@ theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg
                       mem_tac⟩,
                 ⟨i, rfl⟩⟩
   rotate_left
-  · rw [(_ : m * i = _)]
+  · rw [(_ : m * i = _)];
     run_tac
-      mem_tac
-    rw [← add_smul, Nat.add_sub_of_le h1]
-    rfl
-  · rw [(_ : m * i = _)]
+      mem_tac;
+    rw [← add_smul, Nat.add_sub_of_le h1]; rfl
+  · rw [(_ : m * i = _)];
     run_tac
-      mem_tac
-    rw [← add_smul]
-    congr
-    zify [le_of_not_lt h2, le_of_not_le h1]
-    abel
+      mem_tac;
+    rw [← add_smul]; congr ; zify [le_of_not_lt h2, le_of_not_le h1] ; abel
   convert_to(∑ i in range (m + m + 1), g i) ∈ q.1; swap
-  · refine' q.1.sum_mem fun j hj => nsmul_mem _ _
-    split_ifs
+  · refine' q.1.sum_mem fun j hj => nsmul_mem _ _; split_ifs
     exacts[q.1.zero_mem, q.1.mul_mem_left _ (hb i), q.1.mul_mem_right _ (ha i)]
   rw [ext_iff_val, val_mk']
   change _ = (algebraMap (HomogeneousLocalization.Away 𝒜 f) (Localization.Away f)) _
@@ -521,12 +502,8 @@ theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg
   split_ifs with h2 h1
   · exact ((Finset.mem_range.1 hj).not_le h2).elim
   all_goals simp only [mul_val, zero_val, val_mk', Subtype.coe_mk, mk_mul, ← smul_mk]; congr 2
-  · rw [mul_assoc, ← pow_add, add_comm (m - j), Nat.add_sub_assoc h1];
-  · simp_rw [pow_add]
-    rfl
-  · rw [← mul_assoc, ← pow_add, Nat.add_sub_of_le (le_of_not_le h1)];
-  · simp_rw [pow_add]
-    rfl
+  · rw [mul_assoc, ← pow_add, add_comm (m - j), Nat.add_sub_assoc h1]; · simp_rw [pow_add]; rfl
+  · rw [← mul_assoc, ← pow_add, Nat.add_sub_of_le (le_of_not_le h1)]; · simp_rw [pow_add]; rfl
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.add_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.add_mem
 
 variable (hm : 0 < m) (q : Spec.T A⁰_ f)
@@ -546,8 +523,7 @@ theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ ca
   by
   revert c
   refine' DirectSum.Decomposition.inductionOn 𝒜 _ _ _
-  · rw [zero_smul]
-    exact carrier.zero_mem f_deg hm _
+  · rw [zero_smul]; exact carrier.zero_mem f_deg hm _
   · rintro n ⟨a, ha⟩ i
     simp_rw [Subtype.coe_mk, proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
     split_ifs
@@ -560,19 +536,13 @@ theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ ca
                   ⟨_, _⟩, ⟨i - n, rfl⟩⟩ :
             A⁰_ f) ∈ q.1
       · erw [ext_iff_val, val_mk', mul_val, val_mk', val_mk', Subtype.coe_mk]
-        simp_rw [mul_pow, Subtype.coe_mk]
-        rw [Localization.mk_mul]
-        congr
-        erw [← pow_add, Nat.add_sub_of_le h]
-      · exact Ideal.mul_mem_left _ _ (hx _)
-        rw [smul_eq_mul, mul_comm]
+        simp_rw [mul_pow, Subtype.coe_mk]; rw [Localization.mk_mul]
+        congr ; erw [← pow_add, Nat.add_sub_of_le h]
+      · exact Ideal.mul_mem_left _ _ (hx _); rw [smul_eq_mul, mul_comm];
         run_tac
           mem_tac
-    · simp_rw [zero_pow hm]
-      convert carrier.zero_mem f_deg hm q i
-      rw [map_zero, zero_pow hm]
-  · simp_rw [add_smul]
-    exact fun _ _ => carrier.add_mem f_deg q
+    · simp_rw [zero_pow hm]; convert carrier.zero_mem f_deg hm q i; rw [map_zero, zero_pow hm]
+  · simp_rw [add_smul]; exact fun _ _ => carrier.add_mem f_deg q
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.smul_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.smul_mem
 
 /-- For a prime ideal `q` in `A⁰_f`, the set `{a | aᵢᵐ/fⁱ ∈ q}` as an ideal.
@@ -622,8 +592,7 @@ theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
     show (∀ i, _ ∈ _) ∨ ∀ i, _ ∈ _
       by
       rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
-      · apply q.2.mem_or_mem
-        convert hxy (nx + ny) using 1
+      · apply q.2.mem_or_mem; convert hxy (nx + ny) using 1
         simp_rw [proj_apply, decompose_of_mem_same 𝒜 hnx, decompose_of_mem_same 𝒜 hny,
           decompose_of_mem_same 𝒜 (mul_mem hnx hny), mul_pow, pow_add]
         simpa only [ext_iff_val, val_mk', mul_val, mk_mul]

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -368,7 +368,6 @@ variable {𝒜} {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m)
 private unsafe def mem_tac : tactic Unit :=
   let b : tactic Unit := sorry
   b <|> sorry
-#align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_tac algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_tac
 
 include f_deg
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8b8ba04e2f326f3f7cf24ad129beda58531ada61

@@ -372,8 +372,8 @@ private unsafe def mem_tac : tactic Unit :=
 
 include f_deg
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
 /-- The function from `Spec A⁰_f` to `Proj|D(f)` is defined by `q ↦ {a | aᵢᵐ/fⁱ ∈ q}`, i.e. sending
 `q` a prime ideal in `A⁰_f` to the homogeneous prime relevant ideal containing only and all the
 elements `a : A` such that for every `i`, the degree 0 element formed by dividing the `m`-th power
@@ -403,8 +403,8 @@ def carrier (q : Spec.T A⁰_ f) : Set A :=
         q.1 }
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
 theorem mem_carrier_iff (q : Spec.T A⁰_ f) (a : A) :
     a ∈ carrier f_deg q ↔
       ∀ i,
@@ -442,14 +442,14 @@ theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
         rfl)
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_carrier_iff' AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
 theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
     (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q :=
   by
@@ -541,8 +541,8 @@ theorem carrier.zero_mem : (0 : A) ∈ carrier f_deg q := fun i =>
   convert Localization.mk_zero _ using 1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.zero_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.zero_mem
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1624094475.mem_tac -/
 theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q :=
   by
   revert c

mathlib commit https://github.com/leanprover-community/mathlib/commit/9b2b58d6b14b895b2f375108e765cb47de71aebd

@@ -372,8 +372,8 @@ private unsafe def mem_tac : tactic Unit :=
 
 include f_deg
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
 /-- The function from `Spec A⁰_f` to `Proj|D(f)` is defined by `q ↦ {a | aᵢᵐ/fⁱ ∈ q}`, i.e. sending
 `q` a prime ideal in `A⁰_f` to the homogeneous prime relevant ideal containing only and all the
 elements `a : A` such that for every `i`, the degree 0 element formed by dividing the `m`-th power
@@ -403,8 +403,8 @@ def carrier (q : Spec.T A⁰_ f) : Set A :=
         q.1 }
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
 theorem mem_carrier_iff (q : Spec.T A⁰_ f) (a : A) :
     a ∈ carrier f_deg q ↔
       ∀ i,
@@ -442,14 +442,14 @@ theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
         rfl)
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_carrier_iff' AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
 theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
     (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q :=
   by
@@ -541,8 +541,8 @@ theorem carrier.zero_mem : (0 : A) ∈ carrier f_deg q := fun i =>
   convert Localization.mk_zero _ using 1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.zero_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.zero_mem
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1492521391.mem_tac -/
 theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q :=
   by
   revert c

mathlib commit https://github.com/leanprover-community/mathlib/commit/3cacc945118c8c637d89950af01da78307f59325

@@ -372,8 +372,8 @@ private unsafe def mem_tac : tactic Unit :=
 
 include f_deg
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
 /-- The function from `Spec A⁰_f` to `Proj|D(f)` is defined by `q ↦ {a | aᵢᵐ/fⁱ ∈ q}`, i.e. sending
 `q` a prime ideal in `A⁰_f` to the homogeneous prime relevant ideal containing only and all the
 elements `a : A` such that for every `i`, the degree 0 element formed by dividing the `m`-th power
@@ -403,8 +403,8 @@ def carrier (q : Spec.T A⁰_ f) : Set A :=
         q.1 }
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
 theorem mem_carrier_iff (q : Spec.T A⁰_ f) (a : A) :
     a ∈ carrier f_deg q ↔
       ∀ i,
@@ -442,14 +442,14 @@ theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
         rfl)
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_carrier_iff' AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
 theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
     (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q :=
   by
@@ -541,8 +541,8 @@ theorem carrier.zero_mem : (0 : A) ∈ carrier f_deg q := fun i =>
   convert Localization.mk_zero _ using 1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.zero_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.zero_mem
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.2197349699.mem_tac -/
 theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q :=
   by
   revert c

mathlib commit https://github.com/leanprover-community/mathlib/commit/1f4705ccdfe1e557fc54a0ce081a05e33d2e6240

@@ -372,8 +372,8 @@ private unsafe def mem_tac : tactic Unit :=
 
 include f_deg
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
 /-- The function from `Spec A⁰_f` to `Proj|D(f)` is defined by `q ↦ {a | aᵢᵐ/fⁱ ∈ q}`, i.e. sending
 `q` a prime ideal in `A⁰_f` to the homogeneous prime relevant ideal containing only and all the
 elements `a : A` such that for every `i`, the degree 0 element formed by dividing the `m`-th power
@@ -403,8 +403,8 @@ def carrier (q : Spec.T A⁰_ f) : Set A :=
         q.1 }
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
 theorem mem_carrier_iff (q : Spec.T A⁰_ f) (a : A) :
     a ∈ carrier f_deg q ↔
       ∀ i,
@@ -442,14 +442,14 @@ theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
         rfl)
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_carrier_iff' AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
 theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
     (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q :=
   by
@@ -541,8 +541,8 @@ theorem carrier.zero_mem : (0 : A) ∈ carrier f_deg q := fun i =>
   convert Localization.mk_zero _ using 1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.zero_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.zero_mem
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.4090330619.mem_tac -/
 theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q :=
   by
   revert c

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

@@ -509,7 +509,7 @@ theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg
     congr
     zify [le_of_not_lt h2, le_of_not_le h1]
     abel
-  convert_to (∑ i in range (m + m + 1), g i) ∈ q.1; swap
+  convert_to(∑ i in range (m + m + 1), g i) ∈ q.1; swap
   · refine' q.1.sum_mem fun j hj => nsmul_mem _ _
     split_ifs
     exacts[q.1.zero_mem, q.1.mul_mem_left _ (hb i), q.1.mul_mem_right _ (ha i)]
@@ -552,16 +552,14 @@ theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ ca
   · rintro n ⟨a, ha⟩ i
     simp_rw [Subtype.coe_mk, proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
     split_ifs
-    · convert_to
-        (Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, _⟩, ⟨n, rfl⟩⟩ *
+    · convert_to(Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, _⟩, ⟨n, rfl⟩⟩ *
               Quotient.mk''
                 ⟨_,
                   ⟨proj 𝒜 (i - n) x ^ m, by
                     run_tac
                       mem_tac⟩,
                   ⟨_, _⟩, ⟨i - n, rfl⟩⟩ :
-            A⁰_ f) ∈
-          q.1
+            A⁰_ f) ∈ q.1
       · erw [ext_iff_val, val_mk', mul_val, val_mk', val_mk', Subtype.coe_mk]
         simp_rw [mul_pow, Subtype.coe_mk]
         rw [Localization.mk_mul]

mathlib commit https://github.com/leanprover-community/mathlib/commit/1a313d8bba1bad05faba71a4a4e9742ab5bd9efd

@@ -372,8 +372,8 @@ private unsafe def mem_tac : tactic Unit :=
 
 include f_deg
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
 /-- The function from `Spec A⁰_f` to `Proj|D(f)` is defined by `q ↦ {a | aᵢᵐ/fⁱ ∈ q}`, i.e. sending
 `q` a prime ideal in `A⁰_f` to the homogeneous prime relevant ideal containing only and all the
 elements `a : A` such that for every `i`, the degree 0 element formed by dividing the `m`-th power
@@ -403,8 +403,8 @@ def carrier (q : Spec.T A⁰_ f) : Set A :=
         q.1 }
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
 theorem mem_carrier_iff (q : Spec.T A⁰_ f) (a : A) :
     a ∈ carrier f_deg q ↔
       ∀ i,
@@ -442,14 +442,14 @@ theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
         rfl)
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_carrier_iff' AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
 theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
     (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q :=
   by
@@ -541,8 +541,8 @@ theorem carrier.zero_mem : (0 : A) ∈ carrier f_deg q := fun i =>
   convert Localization.mk_zero _ using 1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.zero_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.zero_mem
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1705891745.mem_tac -/
 theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q :=
   by
   revert c

mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442

@@ -363,8 +363,8 @@ open _Root_.HomogeneousLocalization
 
 variable {𝒜} {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m)
 
-/- ./././Mathport/Syntax/Translate/Expr.lean:334:4: warning: unsupported (TODO): `[tacs] -/
-/- ./././Mathport/Syntax/Translate/Expr.lean:334:4: warning: unsupported (TODO): `[tacs] -/
+/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/
+/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/
 private unsafe def mem_tac : tactic Unit :=
   let b : tactic Unit := sorry
   b <|> sorry
@@ -372,8 +372,8 @@ private unsafe def mem_tac : tactic Unit :=
 
 include f_deg
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
 /-- The function from `Spec A⁰_f` to `Proj|D(f)` is defined by `q ↦ {a | aᵢᵐ/fⁱ ∈ q}`, i.e. sending
 `q` a prime ideal in `A⁰_f` to the homogeneous prime relevant ideal containing only and all the
 elements `a : A` such that for every `i`, the degree 0 element formed by dividing the `m`-th power
@@ -403,8 +403,8 @@ def carrier (q : Spec.T A⁰_ f) : Set A :=
         q.1 }
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
 theorem mem_carrier_iff (q : Spec.T A⁰_ f) (a : A) :
     a ∈ carrier f_deg q ↔
       ∀ i,
@@ -442,14 +442,14 @@ theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
         rfl)
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_carrier_iff' AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
 theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
     (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q :=
   by
@@ -541,8 +541,8 @@ theorem carrier.zero_mem : (0 : A) ∈ carrier f_deg q := fun i =>
   convert Localization.mk_zero _ using 1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.zero_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.zero_mem
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
 theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q :=
   by
   revert c

mathlib commit https://github.com/leanprover-community/mathlib/commit/22131150f88a2d125713ffa0f4693e3355b1eb49

@@ -372,8 +372,8 @@ private unsafe def mem_tac : tactic Unit :=
 
 include f_deg
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
 /-- The function from `Spec A⁰_f` to `Proj|D(f)` is defined by `q ↦ {a | aᵢᵐ/fⁱ ∈ q}`, i.e. sending
 `q` a prime ideal in `A⁰_f` to the homogeneous prime relevant ideal containing only and all the
 elements `a : A` such that for every `i`, the degree 0 element formed by dividing the `m`-th power
@@ -403,8 +403,8 @@ def carrier (q : Spec.T A⁰_ f) : Set A :=
         q.1 }
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
 theorem mem_carrier_iff (q : Spec.T A⁰_ f) (a : A) :
     a ∈ carrier f_deg q ↔
       ∀ i,
@@ -442,14 +442,14 @@ theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
         rfl)
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_carrier_iff' AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
 theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
     (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q :=
   by
@@ -541,8 +541,8 @@ theorem carrier.zero_mem : (0 : A) ∈ carrier f_deg q := fun i =>
   convert Localization.mk_zero _ using 1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.zero_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.zero_mem
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.1627101143.mem_tac -/
 theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q :=
   by
   revert c

mathlib commit https://github.com/leanprover-community/mathlib/commit/3ade05ac9447ae31a22d2ea5423435e054131240

@@ -372,8 +372,8 @@ private unsafe def mem_tac : tactic Unit :=
 
 include f_deg
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.2341749097.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.2341749097.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
 /-- The function from `Spec A⁰_f` to `Proj|D(f)` is defined by `q ↦ {a | aᵢᵐ/fⁱ ∈ q}`, i.e. sending
 `q` a prime ideal in `A⁰_f` to the homogeneous prime relevant ideal containing only and all the
 elements `a : A` such that for every `i`, the degree 0 element formed by dividing the `m`-th power
@@ -403,8 +403,8 @@ def carrier (q : Spec.T A⁰_ f) : Set A :=
         q.1 }
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.2341749097.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.2341749097.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
 theorem mem_carrier_iff (q : Spec.T A⁰_ f) (a : A) :
     a ∈ carrier f_deg q ↔
       ∀ i,
@@ -442,14 +442,14 @@ theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
         rfl)
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.mem_carrier_iff' AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.2341749097.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.2341749097.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.2341749097.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.2341749097.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.2341749097.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.2341749097.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.2341749097.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.2341749097.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
 theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
     (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q :=
   by
@@ -541,8 +541,8 @@ theorem carrier.zero_mem : (0 : A) ∈ carrier f_deg q := fun i =>
   convert Localization.mk_zero _ using 1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.zero_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.zero_mem
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.2341749097.mem_tac -/
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.2341749097.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:72:18: unsupported non-interactive tactic _private.3982521519.mem_tac -/
 theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q :=
   by
   revert c

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

A PR analogous to #12338: reformatting proofs following the multiple goals linter of #12339.

@@ -295,21 +295,21 @@ def toFun (x : Proj.T| pbo f) : Spec.T A⁰_ f :=
     rw [Submonoid.coe_one, one_mul] at eq1
     change f ^ _ * (_ * _) = f ^ _ * (f ^ _ * f ^ _ * _) at eq1
     have that : a1 * a2 * f ^ N * f ^ M ∈ x.val.asHomogeneousIdeal.toIdeal := ?_
-    rcases x.1.isPrime.mem_or_mem (show a1 * a2 * f ^ N * f ^ M ∈ _ from that) with (h1 | rid2)
-    rcases x.1.isPrime.mem_or_mem h1 with (h1 | rid1)
-    rcases x.1.isPrime.mem_or_mem h1 with (h1 | h2)
-    · left;
-      simp only [show (Localization.mk a1 ⟨f ^ n1, _⟩ : Away f) =
-        Localization.mk a1 1 * Localization.mk 1 (⟨f ^ n1, ⟨n1, rfl⟩⟩ : Submonoid.powers f) by
-          rw [Localization.mk_mul, mul_one, one_mul]]
-      exact Ideal.mul_mem_right _ _ (Ideal.subset_span ⟨_, h1, rfl⟩)
-    · right;
-      simp only [show (mk a2 ⟨f ^ n2, _⟩ : Away f) =
-        mk a2 1 * Localization.mk 1 (⟨f ^ n2, ⟨n2, rfl⟩⟩ : Submonoid.powers f) by
-          rw [Localization.mk_mul, mul_one, one_mul]]
-      exact Ideal.mul_mem_right _ _ (Ideal.subset_span ⟨_, h2, rfl⟩)
-    · exact False.elim (x.2 (x.1.isPrime.mem_of_pow_mem N rid1))
-    · exact False.elim (x.2 (x.1.isPrime.mem_of_pow_mem M rid2))
+    · rcases x.1.isPrime.mem_or_mem (show a1 * a2 * f ^ N * f ^ M ∈ _ from that) with (h1 | rid2)
+      · rcases x.1.isPrime.mem_or_mem h1 with (h1 | rid1)
+        · rcases x.1.isPrime.mem_or_mem h1 with (h1 | h2)
+          · left;
+            simp only [show (Localization.mk a1 ⟨f ^ n1, _⟩ : Away f) =
+              Localization.mk a1 1 * Localization.mk 1 (⟨f ^ n1, ⟨n1, rfl⟩⟩ : Submonoid.powers f) by
+                rw [Localization.mk_mul, mul_one, one_mul]]
+            exact Ideal.mul_mem_right _ _ (Ideal.subset_span ⟨_, h1, rfl⟩)
+          · right;
+            simp only [show (mk a2 ⟨f ^ n2, _⟩ : Away f) =
+              mk a2 1 * Localization.mk 1 (⟨f ^ n2, ⟨n2, rfl⟩⟩ : Submonoid.powers f) by
+                rw [Localization.mk_mul, mul_one, one_mul]]
+            exact Ideal.mul_mem_right _ _ (Ideal.subset_span ⟨_, h2, rfl⟩)
+        · exact False.elim (x.2 (x.1.isPrime.mem_of_pow_mem N rid1))
+      · exact False.elim (x.2 (x.1.isPrime.mem_of_pow_mem M rid2))
     · rw [← mul_comm (f ^ M), ← mul_comm (f ^ N), eq1]
       refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i _ => mul_mem_left _ _ _))
       generalize_proofs h₁ h₂; exact (Classical.choose_spec h₂).1⟩
@@ -358,9 +358,9 @@ theorem preimage_eq (a b : A) (k : ℕ) (a_mem : a ∈ 𝒜 k) (b_mem1 : b ∈ 
       refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i _ => mul_mem_left _ _ _))
       generalize_proofs h₁ h₂; exact (Classical.choose_spec h₂).1
     rcases y.1.isPrime.mem_or_mem this with (H1 | H3)
-    rcases y.1.isPrime.mem_or_mem H1 with (H1 | H2)
-    · exact hy2 H1
-    · exact y.2 (y.1.isPrime.mem_of_pow_mem N H2)
+    · rcases y.1.isPrime.mem_or_mem H1 with (H1 | H2)
+      · exact hy2 H1
+      · exact y.2 (y.1.isPrime.mem_of_pow_mem N H2)
     · exact y.2 (y.1.isPrime.mem_of_pow_mem M H3)
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.preimage_eq AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.preimage_eq
 
@@ -524,18 +524,22 @@ theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ ca
     let product : A⁰_ f :=
       Mul.mul (Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, ?_⟩, ⟨n, rfl⟩⟩ : A⁰_ f)
           (Quotient.mk'' ⟨_, ⟨proj 𝒜 (i - n) x ^ m, by mem_tac⟩, ⟨_, ?_⟩, ⟨i - n, rfl⟩⟩ : A⁰_ f)
-    split_ifs with h
-    · convert_to product ∈ q.1
-      · dsimp [product]
-        erw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
-          HomogeneousLocalization.mul_val, HomogeneousLocalization.val_mk'',
-          HomogeneousLocalization.val_mk'']
-        simp_rw [mul_pow]; rw [Localization.mk_mul]
-        congr; erw [← pow_add, Nat.add_sub_of_le h]
-        · rw [(_ : m • n = _)]; mem_tac; simp only [smul_eq_mul, mul_comm]
-        · rw [(_ : m • (i - n) = _)]; mem_tac; simp only [smul_eq_mul, mul_comm]
-      · apply Ideal.mul_mem_left (α := A⁰_ f) _ _ (hx _)
-    · simpa only [map_zero, zero_pow hm.ne'] using zero_mem f_deg hm q i
+    · split_ifs with h
+      · convert_to product ∈ q.1
+        · dsimp [product]
+          erw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
+            HomogeneousLocalization.mul_val, HomogeneousLocalization.val_mk'',
+            HomogeneousLocalization.val_mk'']
+          · simp_rw [mul_pow]; rw [Localization.mk_mul]
+            · congr; erw [← pow_add, Nat.add_sub_of_le h]
+            · rw [(_ : m • n = _)]
+              · mem_tac
+              · simp only [smul_eq_mul, mul_comm]
+            · rw [(_ : m • (i - n) = _)]
+              · mem_tac
+              · simp only [smul_eq_mul, mul_comm]
+        · apply Ideal.mul_mem_left (α := A⁰_ f) _ _ (hx _)
+      · simpa only [map_zero, zero_pow hm.ne'] using zero_mem f_deg hm q i
   · simp_rw [add_smul]; exact fun _ _ => carrier.add_mem f_deg q
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.smul_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.smul_mem
 

Now that I am defining NNRat.cast, I want a definitive answer to this naming issue. Plenty of lemmas in mathlib already use natCast/intCast/ratCast over nat_cast/int_cast/rat_cast, and this matches with the general expectation that underscore-separated name parts correspond to a single declaration.

@@ -407,7 +407,7 @@ variable {𝒜} {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m)
 open Lean Meta Elab Tactic
 
 macro "mem_tac_aux" : tactic =>
-  `(tactic| first | exact pow_mem_graded _ (Submodule.coe_mem _) | exact nat_cast_mem_graded _ _ |
+  `(tactic| first | exact pow_mem_graded _ (Submodule.coe_mem _) | exact natCast_mem_graded _ _ |
     exact pow_mem_graded _ f_deg)
 
 macro "mem_tac" : tactic =>

Purely automatic replacement. If this is in any way controversial; I'm happy to just close this PR.

@@ -737,8 +737,8 @@ lemma image_basicOpen_eq_basicOpen (a : A) (i : ℕ) :
 end toSpec
 
 variable {𝒜} in
-/-- The continuous function `Spec A⁰_f → Proj|D(f)` by sending `q` to `{a | aᵢᵐ/fⁱ ∈ q}` where
-`m` is the degree of `f`.-/
+/-- The continuous function `Spec A⁰_f → Proj|D(f)` sending `q` to `{a | aᵢᵐ/fⁱ ∈ q}` where
+`m` is the degree of `f` -/
 def fromSpec {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) :
     (Spec.T (A⁰_ f)) ⟶ (Proj.T| (pbo f)) where
   toFun := FromSpec.toFun f_deg hm

Also do the same for "/-A". This is a purely aesthetic change (and exhaustive).

@@ -737,7 +737,7 @@ lemma image_basicOpen_eq_basicOpen (a : A) (i : ℕ) :
 end toSpec
 
 variable {𝒜} in
-/--The continuous function `Spec A⁰_f → Proj|D(f)` by sending `q` to `{a | aᵢᵐ/fⁱ ∈ q}` where
+/-- The continuous function `Spec A⁰_f → Proj|D(f)` by sending `q` to `{a | aᵢᵐ/fⁱ ∈ q}` where
 `m` is the degree of `f`.-/
 def fromSpec {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) :
     (Spec.T (A⁰_ f)) ⟶ (Proj.T| (pbo f)) where

and tweak formatting of some doc comments

Extracted from #11905.

@@ -103,32 +103,32 @@ open ProjectiveSpectrum.StructureSheaf
 -- Porting note: currently require lack of hygiene to use in variable declarations
 -- maybe all make into notation3?
 set_option hygiene false
--- `Proj` as a locally ringed space
+/-- `Proj` as a locally ringed space -/
 local notation3 "Proj" => Proj.toLocallyRingedSpace 𝒜
 
--- the underlying topological space of `Proj`
+/-- The underlying topological space of `Proj` -/
 local notation3 "Proj.T" => PresheafedSpace.carrier <| SheafedSpace.toPresheafedSpace
   <| LocallyRingedSpace.toSheafedSpace <| Proj.toLocallyRingedSpace 𝒜
 
--- `Proj` restrict to some open set
+/-- `Proj` restrict to some open set -/
 macro "Proj| " U:term : term =>
   `((Proj.toLocallyRingedSpace 𝒜).restrict (Opens.openEmbedding (X := Proj.T) ($U : Opens Proj.T)))
 
--- the underlying topological space of `Proj` restricted to some open set
+/-- the underlying topological space of `Proj` restricted to some open set -/
 local notation "Proj.T| " U => PresheafedSpace.carrier <| SheafedSpace.toPresheafedSpace
   <| LocallyRingedSpace.toSheafedSpace
     <| (LocallyRingedSpace.restrict Proj (Opens.openEmbedding (X := Proj.T) (U : Opens Proj.T)))
 
--- basic open sets in `Proj`
+/-- basic open sets in `Proj` -/
 local notation "pbo " x => ProjectiveSpectrum.basicOpen 𝒜 x
 
--- basic open sets in `Spec`
+/-- basic open sets in `Spec` -/
 local notation "sbo " f => PrimeSpectrum.basicOpen f
 
--- `Spec` as a locally ringed space
+/-- `Spec` as a locally ringed space -/
 local notation3 "Spec " ring => Spec.locallyRingedSpaceObj (CommRingCat.of ring)
 
--- the underlying topological space of `Spec`
+/-- the underlying topological space of `Spec` -/
 local notation "Spec.T " ring =>
   (Spec.locallyRingedSpaceObj (CommRingCat.of ring)).toSheafedSpace.toPresheafedSpace.1
 
@@ -194,10 +194,8 @@ theorem MemCarrier.clear_denominator [DecidableEq (Away f)] {z : A⁰_ f} (hz :
   MemCarrier.clear_denominator' x <| (mem_carrier_iff x z).mpr hz
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.mem_carrier.clear_denominator AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.MemCarrier.clear_denominator
 
-/--
-For `x : pbo f`
-The ideal `A⁰_f ∩ span {g / 1 | g ∈ x}` is equal to `span {a/f^n | a ∈ x and deg(a) = deg(f^n)}`
--/
+/-- For `x : pbo f`, the ideal `A⁰_f ∩ span {g / 1 | g ∈ x}`
+is equal to `span {a/f^n | a ∈ x and deg(a) = deg(f^n)}`. -/
 lemma carrier_eq_span :
     carrier x =
     Ideal.span { z : HomogeneousLocalization.Away 𝒜 f |
@@ -370,9 +368,8 @@ end ToSpec
 
 section
 
-/-- The continuous function between the basic open set `D(f)` in `Proj` to the corresponding basic
-open set in `Spec A⁰_f`.
--/
+/-- The continuous function from the basic open set `D(f)` in `Proj`
+to the corresponding basic open set in `Spec A⁰_f`. -/
 def toSpec (f : A) : (Proj.T| pbo f) ⟶ Spec.T A⁰_ f where
   toFun := ToSpec.toFun f
   continuous_toFun := by
@@ -608,8 +605,7 @@ theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
         · first | exact hnx | exact hny
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.as_ideal.prime AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.prime
 
-/-- The function `Spec A⁰_f → Proj|D(f)` by sending `q` to `{a | aᵢᵐ/fⁱ ∈ q}`.
--/
+/-- The function `Spec A⁰_f → Proj|D(f)` sending `q` to `{a | aᵢᵐ/fⁱ ∈ q}`. -/
 def toFun : (Spec.T A⁰_ f) → Proj.T| pbo f := fun q =>
   ⟨⟨carrier.asHomogeneousIdeal f_deg hm q, carrier.asIdeal.prime f_deg hm q,
       carrier.relevant f_deg hm q⟩,

Co-authored-by: Joël Riou <joel.riou@universite-paris-saclay.fr> Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com>

@@ -53,20 +53,26 @@ open sets in `Proj`, more specifically:
     * This ideal is prime, the proof can be found in
       `ProjIsoSpecTopComponent.FromSpec.carrier.prime`.
     Hence we have a well defined function `Spec.T A⁰_f → Proj.T | (pbo f)`, this function is called
-    `ProjIsoSpecTopComponent.FromSpec.toFun`. But to prove the continuity of this function,
-    we need to prove `fromSpec ∘ toSpec` and `toSpec ∘ fromSpec` are both identities (TBC).
+    `ProjIsoSpecTopComponent.FromSpec.toFun`. But to prove the continuity of this function, we need
+    to prove `fromSpec ∘ toSpec` and `toSpec ∘ fromSpec` are both identities; these are achieved in
+    `ProjIsoSpecTopComponent.fromSpec_toSpec` and `ProjIsoSpecTopComponent.toSpec_fromSpec`.
 
 ## Main Definitions and Statements
 
-* `degreeZeroPart`: the degree zero part of the localized ring `Aₓ` where `x` is a homogeneous
-  element of degree `n` is the subring of elements of the form `a/f^m` where `a` has degree `mn`.
-
-For a homogeneous element `f` of degree `n`
-* `ProjIsoSpecTopComponent.toSpec`: `forward f` is the
-  continuous map between `Proj.T| pbo f` and `Spec.T A⁰_f`
+For a homogeneous element `f` of degree `m`
+* `ProjIsoSpecTopComponent.toSpec`: the continuous map between `Proj.T| pbo f` and `Spec.T A⁰_f`
+  defined by sending `x : Proj| (pbo f)` to `A⁰_f ∩ span {g / 1 | g ∈ x}`. We also denote this map
+  as `ψ`.
 * `ProjIsoSpecTopComponent.ToSpec.preimage_eq`: for any `a: A`, if `a/f^m` has degree zero,
   then the preimage of `sbo a/f^m` under `to_Spec f` is `pbo f ∩ pbo a`.
 
+If we further assume `m` is positive
+* `ProjIsoSpecTopComponent.fromSpec`: the continuous map between `Spec.T A⁰_f` and `Proj.T| pbo f`
+  defined by sending `q` to `{a | aᵢᵐ/fⁱ ∈ q}` where `aᵢ` is the `i`-th coordinate of `a`.
+  We also denote this map as `φ`
+* `projIsoSpecTopComponent`: the homeomorphism `Proj.T| pbo f ≅ Spec.T A⁰_f` obtained by `φ` and
+  `ψ`.
+## Reference
 * [Robin Hartshorne, *Algebraic Geometry*][Har77]: Chapter II.2 Proposition 2.5
 -/
 
@@ -364,12 +370,10 @@ end ToSpec
 
 section
 
-variable {𝒜}
-
 /-- The continuous function between the basic open set `D(f)` in `Proj` to the corresponding basic
 open set in `Spec A⁰_f`.
 -/
-def toSpec {f : A} : (Proj.T| pbo f) ⟶ Spec.T A⁰_ f where
+def toSpec (f : A) : (Proj.T| pbo f) ⟶ Spec.T A⁰_ f where
   toFun := ToSpec.toFun f
   continuous_toFun := by
     rw [PrimeSpectrum.isTopologicalBasis_basic_opens.continuous_iff]
@@ -381,6 +385,15 @@ def toSpec {f : A} : (Proj.T| pbo f) ⟶ Spec.T A⁰_ f where
     · intro hz; simpa only [Set.inf_eq_inter,Set.mem_inter_iff,Set.mem_preimage]
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec
 
+variable {𝒜} in
+lemma toSpec_preimage_eq {f : A} (a b : A) (k : ℕ) (a_mem : a ∈ 𝒜 k) (b_mem1 : b ∈ 𝒜 k)
+    (b_mem2 : b ∈ Submonoid.powers f) :
+    toSpec 𝒜 f ⁻¹'
+        (@PrimeSpectrum.basicOpen (A⁰_ f) _ (Quotient.mk'' ⟨k, ⟨a, a_mem⟩, ⟨b, b_mem1⟩, b_mem2⟩) :
+          Set (PrimeSpectrum (HomogeneousLocalization.Away 𝒜 f))) =
+      {x | x.1 ∈ (pbo f) ⊓ pbo a} :=
+  ToSpec.preimage_eq f a b k a_mem b_mem1 b_mem2
+
 end
 
 namespace FromSpec
@@ -607,8 +620,8 @@ end FromSpec
 
 section toSpecFromSpec
 
-lemma toSpec_fromSpec {f : A} {m : ℕ} (hm : 0 < m) (f_deg : f ∈ 𝒜 m) (x : Spec.T (A⁰_ f)) :
-    toSpec (FromSpec.toFun f_deg hm x) = x := show _ = (_ : PrimeSpectrum _) by
+lemma toSpec_fromSpec {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (x : Spec.T (A⁰_ f)) :
+    toSpec 𝒜 f (FromSpec.toFun f_deg hm x) = x := show _ = (_ : PrimeSpectrum _) by
   ext (z : A⁰_ f); fconstructor <;> intro hz
   · change z ∈ ToSpec.carrier _ at hz
     erw [ToSpec.carrier_eq_span, mem_span_set] at hz
@@ -664,8 +677,8 @@ end toSpecFromSpec
 
 section fromSpecToSpec
 
-lemma fromSpec_toSpec {f : A} {m : ℕ} (hm : 0 < m) (f_deg : f ∈ 𝒜 m) (x : Proj.T| pbo f) :
-    FromSpec.toFun f_deg hm (toSpec x) = x := by
+lemma fromSpec_toSpec {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (x : Proj.T| pbo f) :
+    FromSpec.toFun f_deg hm (toSpec 𝒜 f x) = x := by
   classical
   refine Subtype.ext <| ProjectiveSpectrum.ext _ _ <| HomogeneousIdeal.ext <| Ideal.ext fun z ↦ ?_
   constructor <;> intro hz
@@ -694,23 +707,81 @@ lemma fromSpec_toSpec {f : A} {m : ℕ} (hm : 0 < m) (f_deg : f ∈ 𝒜 m) (x :
     exact Ideal.mul_mem_right _ _ <|
       Ideal.subset_span ⟨_, ⟨Ideal.pow_mem_of_mem _ (x.1.asHomogeneousIdeal.2 i hz) _ hm, rfl⟩⟩
 
-lemma toSpec_injective {f : A} {m : ℕ} (hm : 0 < m) (f_deg : f ∈ 𝒜 m):
-    Function.Injective (toSpec (𝒜 := 𝒜) (f := f)) := by
+lemma toSpec_injective {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) :
+    Function.Injective (toSpec 𝒜 f) := by
   intro x₁ x₂ h
   have := congr_arg (FromSpec.toFun f_deg hm) h
   rwa [fromSpec_toSpec, fromSpec_toSpec] at this
 
-lemma toSpec_surjective {f : A} {m : ℕ} (hm : 0 < m) (f_deg : f ∈ 𝒜 m):
-    Function.Surjective (toSpec (𝒜 := 𝒜) (f := f)) :=
+lemma toSpec_surjective {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) :
+    Function.Surjective (toSpec 𝒜 f) :=
   Function.surjective_iff_hasRightInverse |>.mpr
-    ⟨FromSpec.toFun f_deg hm, toSpec_fromSpec 𝒜 hm f_deg⟩
+    ⟨FromSpec.toFun f_deg hm, toSpec_fromSpec 𝒜 f_deg hm⟩
 
-lemma toSpec_bijective {f : A} {m : ℕ} (hm : 0 < m) (f_deg : f ∈ 𝒜 m):
+lemma toSpec_bijective {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m):
     Function.Bijective (toSpec (𝒜 := 𝒜) (f := f)) :=
-  ⟨toSpec_injective 𝒜 hm f_deg, toSpec_surjective 𝒜 hm f_deg⟩
+  ⟨toSpec_injective 𝒜 f_deg hm, toSpec_surjective 𝒜 f_deg hm⟩
 
 end fromSpecToSpec
 
+namespace toSpec
+
+variable {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m)
+
+variable {𝒜} in
+lemma image_basicOpen_eq_basicOpen (a : A) (i : ℕ) :
+    toSpec 𝒜 f '' {x | x.1 ∈ (pbo f) ⊓ pbo (decompose 𝒜 a i)} =
+    (PrimeSpectrum.basicOpen (R := A⁰_ f) <|
+      Quotient.mk'' ⟨m * i, ⟨decompose 𝒜 a i ^ m, SetLike.pow_mem_graded _ (Submodule.coe_mem _)⟩,
+        ⟨f^i, by rw [mul_comm]; exact SetLike.pow_mem_graded _ f_deg⟩, ⟨i, rfl⟩⟩).1 :=
+  Set.preimage_injective.mpr (toSpec_surjective 𝒜 f_deg hm) <|
+    Set.preimage_image_eq _ (toSpec_injective 𝒜 f_deg hm) ▸ by
+  rw [Opens.carrier_eq_coe, toSpec_preimage_eq, ProjectiveSpectrum.basicOpen_pow 𝒜 _ m hm]
+
+end toSpec
+
+variable {𝒜} in
+/--The continuous function `Spec A⁰_f → Proj|D(f)` by sending `q` to `{a | aᵢᵐ/fⁱ ∈ q}` where
+`m` is the degree of `f`.-/
+def fromSpec {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) :
+    (Spec.T (A⁰_ f)) ⟶ (Proj.T| (pbo f)) where
+  toFun := FromSpec.toFun f_deg hm
+  continuous_toFun := by
+    rw [isTopologicalBasis_subtype (ProjectiveSpectrum.isTopologicalBasis_basic_opens 𝒜) (pbo f).1
+      |>.continuous_iff]
+    rintro s ⟨_, ⟨a, rfl⟩, rfl⟩
+    have h₁ : Subtype.val (p := (pbo f).1) ⁻¹' (pbo a) =
+        ⋃ i : ℕ, {x | x.1 ∈ (pbo f) ⊓ pbo (decompose 𝒜 a i)} := by
+      ext ⟨x, (hx : f ∉ x.asHomogeneousIdeal)⟩
+      rw [ProjectiveSpectrum.basicOpen_eq_union_of_projection 𝒜 a]
+      change (∃ _, _) ↔ (∃ _, _)
+      simp only [GradedAlgebra.proj_apply, Set.mem_range, Set.iUnion_exists, exists_exists_eq_and,
+        Set.mem_iUnion, SetLike.mem_coe, exists_prop, exists_and_right,
+        ProjectiveSpectrum.mem_basicOpen, Opens.carrier_eq_coe, Set.mem_setOf_eq]
+      exact exists_congr fun n ↦ show _ ↔ (_ ∉ _ ∧ _ ∉ _) by tauto
+
+    let e : _ ≃ _ :=
+      ⟨FromSpec.toFun f_deg hm, ToSpec.toFun f, toSpec_fromSpec _ _ _, fromSpec_toSpec _ _ _⟩
+    change IsOpen <| e ⁻¹' _
+    rw [Set.preimage_equiv_eq_image_symm, h₁, Set.image_iUnion]
+    exact isOpen_iUnion fun i ↦ toSpec.image_basicOpen_eq_basicOpen f_deg hm a i ▸
+      PrimeSpectrum.isOpen_basicOpen
+
 end ProjIsoSpecTopComponent
 
+variable {𝒜} in
+/--
+The homeomorphism `Proj|D(f) ≅ Spec A⁰_f` defined by
+- `φ : Proj|D(f) ⟶ Spec A⁰_f` by sending `x` to `A⁰_f ∩ span {g / 1 | g ∈ x}`
+- `ψ : Spec A⁰_f ⟶ Proj|D(f)` by sending `q` to `{a | aᵢᵐ/fⁱ ∈ q}`.
+-/
+def projIsoSpecTopComponent {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) :
+    (Proj.T| (pbo f)) ≅ (Spec.T (A⁰_ f))  where
+  hom := ProjIsoSpecTopComponent.toSpec 𝒜 f
+  inv := ProjIsoSpecTopComponent.fromSpec f_deg hm
+  hom_inv_id := ConcreteCategory.hom_ext _ _
+    (ProjIsoSpecTopComponent.fromSpec_toSpec 𝒜 f_deg hm)
+  inv_hom_id := ConcreteCategory.hom_ext _ _
+    (ProjIsoSpecTopComponent.toSpec_fromSpec 𝒜 f_deg hm)
+
 end AlgebraicGeometry

From #9618, #10525 and #10614.

@@ -657,6 +657,9 @@ lemma toSpec_fromSpec {f : A} {m : ℕ} (hm : 0 < m) (f_deg : f ∈ 𝒜 m) (x :
       rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
         HomogeneousLocalization.zero_val, Localization.mk_zero]
 
+@[deprecated] -- 2024-03-02
+alias toSpecFromSpec := toSpec_fromSpec
+
 end toSpecFromSpec
 
 section fromSpecToSpec

@@ -685,7 +685,7 @@ lemma fromSpec_toSpec {f : A} {m : ℕ} (hm : 0 < m) (f_deg : f ∈ 𝒜 m) (x :
   · intro i
     erw [ToSpec.mem_carrier_iff, HomogeneousLocalization.val_mk'']
     dsimp only [GradedAlgebra.proj_apply]
-    rw [show (mk (decompose 𝒜 z i ^ m) ⟨f^i, ⟨i, rfl⟩⟩: Away f) =
+    rw [show (mk (decompose 𝒜 z i ^ m) ⟨f^i, ⟨i, rfl⟩⟩ : Away f) =
       (decompose 𝒜 z i ^ m : A) • (mk 1 ⟨f^i, ⟨i, rfl⟩⟩ : Away f) by
       · rw [smul_mk, smul_eq_mul, mul_one], Algebra.smul_def]
     exact Ideal.mul_mem_right _ _ <|

Empty lines were removed by executing the following Python script twice

import os
import re


# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
  for filename in files:
    if filename.endswith('.lean'):
      file_path = os.path.join(dir_path, filename)

      # Open the file and read its contents
      with open(file_path, 'r') as file:
        content = file.read()

      # Use a regular expression to replace sequences of "variable" lines separated by empty lines
      # with sequences without empty lines
      modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)

      # Write the modified content back to the file
      with open(file_path, 'w') as file:
        file.write(modified_content)

@@ -84,11 +84,8 @@ open DirectSum SetLike.GradedMonoid Localization
 open Finset hiding mk_zero
 
 variable {R A : Type*}
-
 variable [CommRing R] [CommRing A] [Algebra R A]
-
 variable (𝒜 : ℕ → Submodule R A)
-
 variable [GradedAlgebra 𝒜]
 
 open TopCat TopologicalSpace

I ran tryAtEachStep on all files under Mathlib to find all locations where omega succeeds. For each that was a linarith without an only, I tried replacing it with omega, and I verified that elaboration time got smaller. (In almost all cases, there was a noticeable speedup.) I also replaced some slow aesops along the way.

@@ -242,7 +242,7 @@ theorem disjoint :
   · erw [x.1.isPrime.pow_mem_iff_mem _ k_ineq] at hg1
     exact x.2 hg1
   · dsimp at hg1
-    erw [show k = 0 by linarith, pow_zero, ← Ideal.eq_top_iff_one] at hg1
+    erw [show k = 0 by omega, pow_zero, ← Ideal.eq_top_iff_one] at hg1
     apply x.1.isPrime.1
     exact hg1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.disjoint AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.disjoint

Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com>

@@ -610,7 +610,7 @@ end FromSpec
 
 section toSpecFromSpec
 
-lemma toSpecFromSpec {f : A} {m : ℕ} (hm : 0 < m) (f_deg : f ∈ 𝒜 m) (x : Spec.T (A⁰_ f)) :
+lemma toSpec_fromSpec {f : A} {m : ℕ} (hm : 0 < m) (f_deg : f ∈ 𝒜 m) (x : Spec.T (A⁰_ f)) :
     toSpec (FromSpec.toFun f_deg hm x) = x := show _ = (_ : PrimeSpectrum _) by
   ext (z : A⁰_ f); fconstructor <;> intro hz
   · change z ∈ ToSpec.carrier _ at hz
@@ -662,6 +662,55 @@ lemma toSpecFromSpec {f : A} {m : ℕ} (hm : 0 < m) (f_deg : f ∈ 𝒜 m) (x :
 
 end toSpecFromSpec
 
+section fromSpecToSpec
+
+lemma fromSpec_toSpec {f : A} {m : ℕ} (hm : 0 < m) (f_deg : f ∈ 𝒜 m) (x : Proj.T| pbo f) :
+    FromSpec.toFun f_deg hm (toSpec x) = x := by
+  classical
+  refine Subtype.ext <| ProjectiveSpectrum.ext _ _ <| HomogeneousIdeal.ext <| Ideal.ext fun z ↦ ?_
+  constructor <;> intro hz
+  · rw [← DirectSum.sum_support_decompose 𝒜 z]
+    refine Ideal.sum_mem _ fun i _ ↦ ?_
+    obtain ⟨c, N, acd, eq1⟩ := ToSpec.MemCarrier.clear_denominator x (hz i)
+    rw [HomogeneousLocalization.val_mk'', smul_mk, ← mk_one_eq_algebraMap, mk_eq_mk_iff,
+      r_iff_exists, OneMemClass.coe_one, one_mul] at eq1
+    obtain ⟨⟨_, ⟨k, rfl⟩⟩, eq1⟩ := eq1
+    dsimp at eq1
+    rw [← mul_assoc, ← mul_assoc, ← pow_add, ← pow_add] at eq1
+    suffices mem : f^(k + i) * ∑ i in c.support.attach, acd (c i) _ * _ ∈
+      x.1.asHomogeneousIdeal from
+      x.1.isPrime.mem_of_pow_mem _ <| x.1.isPrime.mem_or_mem (eq1.symm ▸ mem)
+        |>.resolve_left fun r ↦ ProjectiveSpectrum.mem_basicOpen 𝒜 _ _
+        |>.mp x.2 <| x.1.isPrime.mem_of_pow_mem _ r
+    exact Ideal.mul_mem_left _ _ <| Ideal.sum_mem _ fun i _ ↦
+      Ideal.mul_mem_left _ _ i.1.2.choose_spec.1
+
+  · intro i
+    erw [ToSpec.mem_carrier_iff, HomogeneousLocalization.val_mk'']
+    dsimp only [GradedAlgebra.proj_apply]
+    rw [show (mk (decompose 𝒜 z i ^ m) ⟨f^i, ⟨i, rfl⟩⟩: Away f) =
+      (decompose 𝒜 z i ^ m : A) • (mk 1 ⟨f^i, ⟨i, rfl⟩⟩ : Away f) by
+      · rw [smul_mk, smul_eq_mul, mul_one], Algebra.smul_def]
+    exact Ideal.mul_mem_right _ _ <|
+      Ideal.subset_span ⟨_, ⟨Ideal.pow_mem_of_mem _ (x.1.asHomogeneousIdeal.2 i hz) _ hm, rfl⟩⟩
+
+lemma toSpec_injective {f : A} {m : ℕ} (hm : 0 < m) (f_deg : f ∈ 𝒜 m):
+    Function.Injective (toSpec (𝒜 := 𝒜) (f := f)) := by
+  intro x₁ x₂ h
+  have := congr_arg (FromSpec.toFun f_deg hm) h
+  rwa [fromSpec_toSpec, fromSpec_toSpec] at this
+
+lemma toSpec_surjective {f : A} {m : ℕ} (hm : 0 < m) (f_deg : f ∈ 𝒜 m):
+    Function.Surjective (toSpec (𝒜 := 𝒜) (f := f)) :=
+  Function.surjective_iff_hasRightInverse |>.mpr
+    ⟨FromSpec.toFun f_deg hm, toSpec_fromSpec 𝒜 hm f_deg⟩
+
+lemma toSpec_bijective {f : A} {m : ℕ} (hm : 0 < m) (f_deg : f ∈ 𝒜 m):
+    Function.Bijective (toSpec (𝒜 := 𝒜) (f := f)) :=
+  ⟨toSpec_injective 𝒜 hm f_deg, toSpec_surjective 𝒜 hm f_deg⟩
+
+end fromSpecToSpec
+
 end ProjIsoSpecTopComponent
 
 end AlgebraicGeometry

Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com>

@@ -39,7 +39,7 @@ open sets in `Proj`, more specifically:
   - forward direction `toSpec`:
     for any `x : pbo f`, i.e. a relevant homogeneous prime ideal `x`, send it to
     `A⁰_f ∩ span {g / 1 | g ∈ x}` (see `ProjIsoSpecTopComponent.IoSpec.carrier`). This ideal is
-    prime, the proof is in `ProjIsoSpecTopcomponent.ToSpec.toFun`. The fact that this function
+    prime, the proof is in `ProjIsoSpecTopComponent.ToSpec.toFun`. The fact that this function
     is continuous is found in `ProjIsoSpecTopComponent.toSpec`
   - backward direction `fromSpec`:
     for any `q : Spec A⁰_f`, we send it to `{a | ∀ i, aᵢᵐ/fⁱ ∈ q}`; we need this to be a
@@ -191,6 +191,47 @@ theorem MemCarrier.clear_denominator [DecidableEq (Away f)] {z : A⁰_ f} (hz :
   MemCarrier.clear_denominator' x <| (mem_carrier_iff x z).mpr hz
 #align algebraic_geometry.Proj_iso_Spec_Top_component.to_Spec.mem_carrier.clear_denominator AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.MemCarrier.clear_denominator
 
+/--
+For `x : pbo f`
+The ideal `A⁰_f ∩ span {g / 1 | g ∈ x}` is equal to `span {a/f^n | a ∈ x and deg(a) = deg(f^n)}`
+-/
+lemma carrier_eq_span :
+    carrier x =
+    Ideal.span { z : HomogeneousLocalization.Away 𝒜 f |
+      ∃ (s F : A) (_ : s ∈ x.1.asHomogeneousIdeal) (n : ℕ)
+        (s_mem : s ∈ 𝒜 n) (F_mem1 : F ∈ 𝒜 n) (F_mem2 : F ∈ Submonoid.powers f),
+        z = Quotient.mk'' ⟨n, ⟨s, s_mem⟩, ⟨F, F_mem1⟩,F_mem2⟩ } := by
+  classical
+  refine le_antisymm ?_ <| Ideal.span_le.mpr ?_
+  · intro z hz
+    let k : ℕ := z.den_mem.choose
+    have hk : f^k = z.den := z.den_mem.choose_spec
+    suffices mem1 : z.num ∈ x.1.asHomogeneousIdeal by
+      refine Ideal.subset_span ⟨_, _, mem1, _, z.num_mem_deg, z.den_mem_deg, z.den_mem, ?_⟩
+      rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
+        HomogeneousLocalization.eq_num_div_den]
+
+    obtain ⟨c, N, acd, eq1⟩ := MemCarrier.clear_denominator x hz
+    rw [z.eq_num_div_den, smul_mk, smul_eq_mul, ← mk_one_eq_algebraMap,
+      Localization.mk_eq_mk_iff, Localization.r_iff_exists] at eq1
+    obtain ⟨⟨_, ⟨l, rfl⟩⟩, eq1⟩ := eq1
+    dsimp only [OneMemClass.coe_one] at eq1
+    rw [one_mul, ← hk, ← mul_assoc, ← mul_assoc, ← pow_add, ← pow_add] at eq1
+
+    suffices f^(l + k) * ∑ i in c.support.attach, acd (c i) _ * i.1.2.choose ∈
+      x.1.asHomogeneousIdeal from
+      (x.1.isPrime.mem_or_mem (eq1.symm ▸ this)).resolve_left fun r ↦
+        (ProjectiveSpectrum.mem_basicOpen 𝒜 _ _).mp x.2 <| x.1.isPrime.mem_of_pow_mem _ r
+    exact Ideal.mul_mem_left _ _ <| Ideal.sum_mem _ (fun ⟨j, hj⟩ _ ↦
+      Ideal.mul_mem_left _ _ j.2.choose_spec.1)
+
+  · rintro _ ⟨s, _, hs, n, s_mem, F_mem, ⟨l, rfl⟩, rfl⟩
+    erw [mem_carrier_iff, HomogeneousLocalization.val_mk'']
+    erw [show (Localization.mk s ⟨f ^ l, ⟨_, rfl⟩⟩ : Localization.Away f) =
+      (Localization.mk 1 ⟨f^l, ⟨_, rfl⟩⟩ : Localization.Away f) * Localization.mk s 1 by
+      · rw [Localization.mk_mul, mul_one, one_mul]]
+    exact Ideal.mul_mem_left _ _ <| Ideal.subset_span ⟨s, hs, rfl⟩
+
 theorem disjoint :
     Disjoint (x.1.asHomogeneousIdeal.toIdeal : Set A) (Submonoid.powers f : Set A) := by
   by_contra rid
@@ -567,6 +608,60 @@ def toFun : (Spec.T A⁰_ f) → Proj.T| pbo f := fun q =>
 
 end FromSpec
 
+section toSpecFromSpec
+
+lemma toSpecFromSpec {f : A} {m : ℕ} (hm : 0 < m) (f_deg : f ∈ 𝒜 m) (x : Spec.T (A⁰_ f)) :
+    toSpec (FromSpec.toFun f_deg hm x) = x := show _ = (_ : PrimeSpectrum _) by
+  ext (z : A⁰_ f); fconstructor <;> intro hz
+  · change z ∈ ToSpec.carrier _ at hz
+    erw [ToSpec.carrier_eq_span, mem_span_set] at hz
+    obtain ⟨c, support_le, rfl⟩ := hz
+    refine Ideal.sum_mem _ fun j hj ↦ Ideal.mul_mem_left _ _ ?_
+    obtain ⟨s, _, hs, n, s_mem, F_mem, ⟨k, rfl⟩, rfl⟩ := support_le hj
+    refine x.IsPrime.mem_of_pow_mem m <| show Quotient.mk'' ⟨_, ⟨s^m, _⟩, _, _⟩ ∈ x.asIdeal from ?_
+    convert hs n using 1
+    rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
+      HomogeneousLocalization.val_mk'']
+    simp only [smul_eq_mul, SetLike.coe_gnpow, GradedAlgebra.proj_apply,
+      DirectSum.decompose_of_mem_same 𝒜 s_mem]
+    by_cases ineq : f^k = 0
+    · have := IsLocalization.uniqueOfZeroMem (M := Submonoid.powers f) (S := Localization.Away f)
+        ⟨k, ineq⟩
+      exact Subsingleton.elim _ _
+    · simp_rw [← pow_mul]; congr
+      exact DirectSum.degree_eq_of_mem_mem 𝒜 F_mem (SetLike.pow_mem_graded k f_deg) ineq |>.symm
+
+  · erw [ToSpec.mem_carrier_iff]
+    obtain ⟨k, (k_spec : f^k = z.den)⟩ := z.den_mem
+    rw [show z.val = (Localization.mk z.num ⟨f^k, ⟨k, rfl⟩⟩ : Away f) by
+        · rw [z.eq_num_div_den]; congr; exact k_spec.symm,
+      show (mk z.num ⟨f^k, ⟨k, rfl⟩⟩ : Away f) = mk z.num 1 * (mk 1 ⟨f^k, ⟨k, rfl⟩⟩ : Away f) by
+        · rw [mk_mul, mul_one, one_mul]]
+    refine Ideal.mul_mem_right _ _ <| Ideal.subset_span ⟨z.num, ?_, rfl⟩
+
+    intro j
+    by_cases ineq : z.deg = j
+    · subst ineq
+      simp only [CommRingCat.coe_of, GradedAlgebra.proj_apply,
+        DirectSum.decompose_of_mem_same 𝒜 z.num_mem_deg]
+      convert x.IsPrime.pow_mem_iff_mem m hm |>.mpr hz using 1
+      rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
+        HomogeneousLocalization.pow_val, z.eq_num_div_den, Localization.mk_pow]
+      by_cases ineq : f^k = 0
+      · have := IsLocalization.uniqueOfZeroMem (M := Submonoid.powers f) (S := Localization.Away f)
+          ⟨k, ineq⟩
+        exact Subsingleton.elim _ _
+      · dsimp; congr 2
+        rw [← k_spec, ← pow_mul, show z.deg = k * m from
+          degree_eq_of_mem_mem 𝒜 (k_spec ▸ z.den_mem_deg) (SetLike.pow_mem_graded k f_deg) ineq]
+    · simp only [CommRingCat.coe_of, GradedAlgebra.proj_apply, zero_pow (by linarith only [hm]),
+        DirectSum.decompose_of_mem_ne 𝒜 z.num_mem_deg ineq]
+      convert x.asIdeal.zero_mem
+      rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
+        HomogeneousLocalization.zero_val, Localization.mk_zero]
+
+end toSpecFromSpec
+
 end ProjIsoSpecTopComponent
 
 end AlgebraicGeometry

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -478,7 +478,7 @@ theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ ca
           (Quotient.mk'' ⟨_, ⟨proj 𝒜 (i - n) x ^ m, by mem_tac⟩, ⟨_, ?_⟩, ⟨i - n, rfl⟩⟩ : A⁰_ f)
     split_ifs with h
     · convert_to product ∈ q.1
-      · dsimp
+      · dsimp [product]
         erw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
           HomogeneousLocalization.mul_val, HomogeneousLocalization.val_mk'',
           HomogeneousLocalization.val_mk'']

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -236,8 +236,8 @@ def toFun (x : Proj.T| pbo f) : Spec.T A⁰_ f :=
     classical
     rw [mem_carrier_iff] at hx12
     let J := span ((algebraMap A (Away f) : A → (Away f)) '' x.val.asHomogeneousIdeal)
-    suffices h : ∀ x y : Localization.Away f, x * y ∈ J → x ∈ J ∨ y ∈ J
-    · rw [HomogeneousLocalization.mul_val] at hx12; exact h x1.val x2.val hx12
+    suffices h : ∀ x y : Localization.Away f, x * y ∈ J → x ∈ J ∨ y ∈ J by
+      rw [HomogeneousLocalization.mul_val] at hx12; exact h x1.val x2.val hx12
     clear x1 x2 hx12
     intro x1 x2 hx12
     induction' x1 using Localization.induction_on with data_x1

This involves moving lemmas from Algebra.GroupPower.Ring to Algebra.GroupWithZero.Basic and changing some 0 < n assumptions to n ≠ 0.

From LeanAPAP

@@ -462,7 +462,7 @@ variable (hm : 0 < m) (q : Spec.T A⁰_ f)
 theorem carrier.zero_mem : (0 : A) ∈ carrier f_deg q := fun i => by
   convert Submodule.zero_mem q.1 using 1
   rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
-    HomogeneousLocalization.zero_val]; simp_rw [map_zero, zero_pow hm]
+    HomogeneousLocalization.zero_val]; simp_rw [map_zero, zero_pow hm.ne']
   convert Localization.mk_zero (S := Submonoid.powers f) _ using 1
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.zero_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.zero_mem
 
@@ -487,7 +487,7 @@ theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ ca
         · rw [(_ : m • n = _)]; mem_tac; simp only [smul_eq_mul, mul_comm]
         · rw [(_ : m • (i - n) = _)]; mem_tac; simp only [smul_eq_mul, mul_comm]
       · apply Ideal.mul_mem_left (α := A⁰_ f) _ _ (hx _)
-    · simp_rw [zero_pow hm]; convert carrier.zero_mem f_deg hm q i; rw [map_zero, zero_pow hm]
+    · simpa only [map_zero, zero_pow hm.ne'] using zero_mem f_deg hm q i
   · simp_rw [add_smul]; exact fun _ _ => carrier.add_mem f_deg q
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.smul_mem AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.smul_mem
 
@@ -505,9 +505,8 @@ theorem carrier.asIdeal.homogeneous : (carrier.asIdeal f_deg hm q).IsHomogeneous
   fun i a ha j =>
   (em (i = j)).elim (fun h => h ▸ by simpa only [proj_apply, decompose_coe, of_eq_same] using ha _)
     fun h => by
-    simp only [proj_apply, decompose_of_mem_ne 𝒜 (Submodule.coe_mem (decompose 𝒜 a i)) h,
-      zero_pow hm]
-    convert carrier.zero_mem f_deg hm q j; rw [map_zero, zero_pow hm]
+    simpa only [proj_apply, decompose_of_mem_ne 𝒜 (Submodule.coe_mem (decompose 𝒜 a i)) h,
+      zero_pow hm.ne', map_zero] using carrier.zero_mem f_deg hm q j
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.as_ideal.homogeneous AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.homogeneous
 
 /-- For a prime ideal `q` in `A⁰_f`, the set `{a | aᵢᵐ/fⁱ ∈ q}` as a homogeneous ideal.
@@ -554,7 +553,7 @@ theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
         rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
           HomogeneousLocalization.zero_val]; simp_rw [proj_apply]
         convert mk_zero (S := Submonoid.powers f) _
-        rw [decompose_of_mem_ne 𝒜 _ hn.symm, zero_pow hm]
+        rw [decompose_of_mem_ne 𝒜 _ hn.symm, zero_pow hm.ne']
         · first | exact hnx | exact hny
 #align algebraic_geometry.Proj_iso_Spec_Top_component.from_Spec.carrier.as_ideal.prime AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.prime
 

Similarly, upgrade tendsto_nhds_generateFrom, IsTopologicalBasis.continuous, Topology.IsLower.continuous_of_Ici, and Topology.IsUpper.continuous_iff_Iic.

The old lemmas are now deprecated, and the new ones have _iff in their names. Once we remove the old lemmas, we can drop the _iff suffixes.

@@ -334,7 +334,7 @@ open set in `Spec A⁰_f`.
 def toSpec {f : A} : (Proj.T| pbo f) ⟶ Spec.T A⁰_ f where
   toFun := ToSpec.toFun f
   continuous_toFun := by
-    apply IsTopologicalBasis.continuous PrimeSpectrum.isTopologicalBasis_basic_opens
+    rw [PrimeSpectrum.isTopologicalBasis_basic_opens.continuous_iff]
     rintro _ ⟨⟨k, ⟨a, ha⟩, ⟨b, hb1⟩, ⟨k', hb2⟩⟩, rfl⟩; dsimp
     erw [ToSpec.preimage_eq f a b k ha hb1 ⟨k', hb2⟩]
     refine' isOpen_induced_iff.mpr ⟨(pbo f).1 ⊓ (pbo a).1, IsOpen.inter (pbo f).2 (pbo a).2, _⟩

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -471,7 +471,7 @@ theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ ca
   refine' DirectSum.Decomposition.inductionOn 𝒜 _ _ _
   · rw [zero_smul]; exact carrier.zero_mem f_deg hm _
   · rintro n ⟨a, ha⟩ i
-    simp_rw [Subtype.coe_mk, proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
+    simp_rw [proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
     -- Porting note: having trouble with Mul instance
     let product : A⁰_ f :=
       Mul.mul (Quotient.mk'' ⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, ?_⟩, ⟨n, rfl⟩⟩ : A⁰_ f)
@@ -481,8 +481,8 @@ theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ ca
       · dsimp
         erw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
           HomogeneousLocalization.mul_val, HomogeneousLocalization.val_mk'',
-          HomogeneousLocalization.val_mk'', Subtype.coe_mk]
-        simp_rw [mul_pow, Subtype.coe_mk]; rw [Localization.mk_mul]
+          HomogeneousLocalization.val_mk'']
+        simp_rw [mul_pow]; rw [Localization.mk_mul]
         congr; erw [← pow_add, Nat.add_sub_of_le h]
         · rw [(_ : m • n = _)]; mem_tac; simp only [smul_eq_mul, mul_comm]
         · rw [(_ : m • (i - n) = _)]; mem_tac; simp only [smul_eq_mul, mul_comm]
@@ -543,7 +543,7 @@ theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
       rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
       · apply q.2.mem_or_mem; convert hxy (nx + ny) using 1
         dsimp
-        simp_rw [proj_apply, decompose_of_mem_same 𝒜 hnx, decompose_of_mem_same 𝒜 hny,
+        simp_rw [decompose_of_mem_same 𝒜 hnx, decompose_of_mem_same 𝒜 hny,
           decompose_of_mem_same 𝒜 (SetLike.GradedMonoid.toGradedMul.mul_mem hnx hny),
           mul_pow, pow_add]
         simp only [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
@@ -552,7 +552,7 @@ theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
       all_goals
         intro n hn; convert q.1.zero_mem using 1
         rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
-          HomogeneousLocalization.zero_val]; simp_rw [proj_apply, Subtype.coe_mk]
+          HomogeneousLocalization.zero_val]; simp_rw [proj_apply]
         convert mk_zero (S := Submonoid.powers f) _
         rw [decompose_of_mem_ne 𝒜 _ hn.symm, zero_pow hm]
         · first | exact hnx | exact hny

@@ -412,21 +412,23 @@ theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg
   let g : ℕ → A⁰_ f := fun j => (m + m).choose j •
       if h2 : m + m < j then (0 : A⁰_ f)
       else
+        -- Porting note: inlining `l`, `r` causes a "can't synth HMul A⁰_ f A⁰_ f ?" error
         if h1 : j ≤ m then
-          -- Porting note : cannot use * notation since can't synth HMul A⁰_ f A⁰_ f ?
-          Mul.mul (Quotient.mk''
-              ⟨m * i, ⟨proj 𝒜 i a ^ j * proj 𝒜 i b ^ (m - j), ?_⟩,
-                ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩ : A⁰_ f)
-            (Quotient.mk''
-              ⟨m * i, ⟨proj 𝒜 i b ^ m, by mem_tac⟩,
-                ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩ : A⁰_ f)
+          letI l : A⁰_ f := Quotient.mk''
+            ⟨m * i, ⟨proj 𝒜 i a ^ j * proj 𝒜 i b ^ (m - j), ?_⟩,
+              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
+          letI r : A⁰_ f := Quotient.mk''
+            ⟨m * i, ⟨proj 𝒜 i b ^ m, by mem_tac⟩,
+              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
+          l * r
         else
-          Mul.mul (Quotient.mk''
-              ⟨m * i, ⟨proj 𝒜 i a ^ m, by mem_tac⟩,
-                ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩ : A⁰_ f)
-            (Quotient.mk''
-              ⟨m * i, ⟨proj 𝒜 i a ^ (j - m) * proj 𝒜 i b ^ (m + m - j), ?_⟩,
-                ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩ : A⁰_ f)
+          letI l : A⁰_ f := Quotient.mk''
+            ⟨m * i, ⟨proj 𝒜 i a ^ m, by mem_tac⟩,
+              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
+          letI r : A⁰_ f := Quotient.mk''
+            ⟨m * i, ⟨proj 𝒜 i a ^ (j - m) * proj 𝒜 i b ^ (m + m - j), ?_⟩,
+              ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
+          l * r
   rotate_left
   · rw [(_ : m * i = _)]
     -- Porting note: it seems unification with mul_mem is more fiddly reducing value of mem_tac

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -83,7 +83,7 @@ open DirectSum SetLike.GradedMonoid Localization
 
 open Finset hiding mk_zero
 
-variable {R A : Type _}
+variable {R A : Type*}
 
 variable [CommRing R] [CommRing A] [Algebra R A]
 

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,16 +2,13 @@
 Copyright (c) 2022 Jujian Zhang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jujian Zhang
-
-! This file was ported from Lean 3 source module algebraic_geometry.projective_spectrum.scheme
-! leanprover-community/mathlib commit d39590fc8728fbf6743249802486f8c91ffe07bc
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
 import Mathlib.AlgebraicGeometry.Spec
 import Mathlib.RingTheory.GradedAlgebra.Radical
 
+#align_import algebraic_geometry.projective_spectrum.scheme from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
+
 /-!
 # Proj as a scheme
 

Grepping for [^ .:{-] [^ :] and reviewing the results. Once I started I couldn't stop. :-)

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -278,7 +278,7 @@ def toFun (x : Proj.T| pbo f) : Spec.T A⁰_ f :=
 
 /-
 The preimage of basic open set `D(a/f^n)` in `Spec A⁰_f` under the forward map from `Proj A` to
-`Spec A⁰_f` is the basic open set `D(a) ∩ D(f)` in  `Proj A`. This lemma is used to prove that the
+`Spec A⁰_f` is the basic open set `D(a) ∩ D(f)` in `Proj A`. This lemma is used to prove that the
 forward map is continuous.
 -/
 theorem preimage_eq (a b : A) (k : ℕ) (a_mem : a ∈ 𝒜 k) (b_mem1 : b ∈ 𝒜 k)
@@ -572,4 +572,3 @@ end FromSpec
 end ProjIsoSpecTopComponent
 
 end AlgebraicGeometry
-