algebraic_geometry.projective_spectrum.topology ⟷ Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jujian Zhang, Johan Commelin
 -/
 import RingTheory.GradedAlgebra.HomogeneousIdeal
-import Topology.Category.Top.Basic
+import Topology.Category.TopCat.Basic
 import Topology.Sets.Opens
 
 #align_import algebraic_geometry.projective_spectrum.topology from "leanprover-community/mathlib"@"4280f5f32e16755ec7985ce11e189b6cd6ff6735"

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

@@ -402,7 +402,7 @@ theorem sup_vanishingIdeal_le (t t' : Set (ProjectiveSpectrum 𝒜)) :
   rw [← HomogeneousIdeal.mem_iff, HomogeneousIdeal.toIdeal_sup, mem_vanishing_ideal,
     Submodule.mem_sup]
   rintro ⟨f, hf, g, hg, rfl⟩ x ⟨hxt, hxt'⟩
-  erw [mem_vanishing_ideal] at hf hg 
+  erw [mem_vanishing_ideal] at hf hg
   apply Submodule.add_mem <;> solve_by_elim
 #align projective_spectrum.sup_vanishing_ideal_le ProjectiveSpectrum.sup_vanishingIdeal_le
 -/
@@ -461,8 +461,8 @@ theorem zeroLocus_vanishingIdeal_eq_closure (t : Set (ProjectiveSpectrum 𝒜))
   by
   apply Set.Subset.antisymm
   · rintro x hx t' ⟨ht', ht⟩
-    obtain ⟨fs, rfl⟩ : ∃ s, t' = zero_locus 𝒜 s := by rwa [is_closed_iff_zero_locus] at ht' 
-    rw [subset_zero_locus_iff_subset_vanishing_ideal] at ht 
+    obtain ⟨fs, rfl⟩ : ∃ s, t' = zero_locus 𝒜 s := by rwa [is_closed_iff_zero_locus] at ht'
+    rw [subset_zero_locus_iff_subset_vanishing_ideal] at ht
     exact Set.Subset.trans ht hx
   · rw [(is_closed_zero_locus _ _).closure_subset_iff]
     exact subset_zero_locus_vanishing_ideal 𝒜 t
@@ -474,9 +474,9 @@ theorem vanishingIdeal_closure (t : Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (closure t) = vanishingIdeal t :=
   by
   have := (gc_ideal 𝒜).u_l_u_eq_u t
-  dsimp only at this 
+  dsimp only at this
   ext1
-  erw [zero_locus_vanishing_ideal_eq_closure 𝒜 t] at this 
+  erw [zero_locus_vanishing_ideal_eq_closure 𝒜 t] at this
   exact this
 #align projective_spectrum.vanishing_ideal_closure ProjectiveSpectrum.vanishingIdeal_closure
 -/
@@ -589,7 +589,7 @@ theorem isTopologicalBasis_basic_opens :
   · rintro _ ⟨r, rfl⟩
     exact is_open_basic_open 𝒜
   · rintro p U hp ⟨s, hs⟩
-    rw [← compl_compl U, Set.mem_compl_iff, ← hs, mem_zero_locus, Set.not_subset] at hp 
+    rw [← compl_compl U, Set.mem_compl_iff, ← hs, mem_zero_locus, Set.not_subset] at hp
     obtain ⟨f, hfs, hfp⟩ := hp
     refine' ⟨basic_open 𝒜 f, ⟨f, rfl⟩, hfp, _⟩
     rw [← Set.compl_subset_compl, ← hs, basic_open_eq_zero_locus_compl, compl_compl]

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

@@ -570,7 +570,10 @@ theorem basicOpen_eq_union_of_projection (f : A) :
       · rcases show ∃ i, GradedAlgebra.proj 𝒜 i f ∉ z.as_homogeneous_ideal
             by
             contrapose! hz with H
-            classical with ⟨i, hi⟩
+            classical
+            rw [← DirectSum.sum_support_decompose 𝒜 f]
+            apply Ideal.sum_mem _ fun i hi => H i with
+          ⟨i, hi⟩
         exact ⟨basic_open 𝒜 (GradedAlgebra.proj 𝒜 i f), ⟨i, rfl⟩, by rwa [mem_basic_open]⟩
       · obtain ⟨_, ⟨i, rfl⟩, hz⟩ := hz
         exact fun rid => hz (z.1.2 i rid)

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

@@ -570,10 +570,7 @@ theorem basicOpen_eq_union_of_projection (f : A) :
       · rcases show ∃ i, GradedAlgebra.proj 𝒜 i f ∉ z.as_homogeneous_ideal
             by
             contrapose! hz with H
-            classical
-            rw [← DirectSum.sum_support_decompose 𝒜 f]
-            apply Ideal.sum_mem _ fun i hi => H i with
-          ⟨i, hi⟩
+            classical with ⟨i, hi⟩
         exact ⟨basic_open 𝒜 (GradedAlgebra.proj 𝒜 i f), ⟨i, rfl⟩, by rwa [mem_basic_open]⟩
       · obtain ⟨_, ⟨i, rfl⟩, hz⟩ := hz
         exact fun rid => hz (z.1.2 i rid)

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

@@ -585,7 +585,7 @@ theorem isTopologicalBasis_basic_opens :
     TopologicalSpace.IsTopologicalBasis
       (Set.range fun r : A => (basicOpen 𝒜 r : Set (ProjectiveSpectrum 𝒜))) :=
   by
-  apply TopologicalSpace.isTopologicalBasis_of_open_of_nhds
+  apply TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds
   · rintro _ ⟨r, rfl⟩
     exact is_open_basic_open 𝒜
   · rintro p U hp ⟨s, hs⟩

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

@@ -3,9 +3,9 @@ Copyright (c) 2020 Jujian Zhang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jujian Zhang, Johan Commelin
 -/
-import Mathbin.RingTheory.GradedAlgebra.HomogeneousIdeal
-import Mathbin.Topology.Category.Top.Basic
-import Mathbin.Topology.Sets.Opens
+import RingTheory.GradedAlgebra.HomogeneousIdeal
+import Topology.Category.Top.Basic
+import Topology.Sets.Opens
 
 #align_import algebraic_geometry.projective_spectrum.topology from "leanprover-community/mathlib"@"4280f5f32e16755ec7985ce11e189b6cd6ff6735"
 

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

@@ -2,16 +2,13 @@
 Copyright (c) 2020 Jujian Zhang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jujian Zhang, Johan Commelin
-
-! This file was ported from Lean 3 source module algebraic_geometry.projective_spectrum.topology
-! leanprover-community/mathlib commit 4280f5f32e16755ec7985ce11e189b6cd6ff6735
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.RingTheory.GradedAlgebra.HomogeneousIdeal
 import Mathbin.Topology.Category.Top.Basic
 import Mathbin.Topology.Sets.Opens
 
+#align_import algebraic_geometry.projective_spectrum.topology from "leanprover-community/mathlib"@"4280f5f32e16755ec7985ce11e189b6cd6ff6735"
+
 /-!
 # Projective spectrum of a graded ring
 

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

@@ -55,6 +55,7 @@ variable [CommSemiring R] [CommRing A] [Algebra R A]
 
 variable (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜]
 
+#print ProjectiveSpectrum /-
 /-- The projective spectrum of a graded commutative ring is the subtype of all homogenous ideals
 that are prime and do not contain the irrelevant ideal. -/
 @[ext, nolint has_nonempty_instance]
@@ -63,11 +64,13 @@ structure ProjectiveSpectrum where
   IsPrime : as_homogeneous_ideal.toIdeal.IsPrime
   not_irrelevant_le : ¬HomogeneousIdeal.irrelevant 𝒜 ≤ as_homogeneous_ideal
 #align projective_spectrum ProjectiveSpectrum
+-/
 
 attribute [instance] ProjectiveSpectrum.isPrime
 
 namespace ProjectiveSpectrum
 
+#print ProjectiveSpectrum.zeroLocus /-
 /-- The zero locus of a set `s` of elements of a commutative ring `A` is the set of all relevant
 homogeneous prime ideals of the ring that contain the set `s`.
 
@@ -78,20 +81,26 @@ of `projective_spectrum 𝒜` where all "functions" in `s` vanish simultaneously
 def zeroLocus (s : Set A) : Set (ProjectiveSpectrum 𝒜) :=
   {x | s ⊆ x.asHomogeneousIdeal}
 #align projective_spectrum.zero_locus ProjectiveSpectrum.zeroLocus
+-/
 
+#print ProjectiveSpectrum.mem_zeroLocus /-
 @[simp]
 theorem mem_zeroLocus (x : ProjectiveSpectrum 𝒜) (s : Set A) :
     x ∈ zeroLocus 𝒜 s ↔ s ⊆ x.asHomogeneousIdeal :=
   Iff.rfl
 #align projective_spectrum.mem_zero_locus ProjectiveSpectrum.mem_zeroLocus
+-/
 
+#print ProjectiveSpectrum.zeroLocus_span /-
 @[simp]
 theorem zeroLocus_span (s : Set A) : zeroLocus 𝒜 (Ideal.span s) = zeroLocus 𝒜 s := by ext x;
   exact (Submodule.gi _ _).gc s x.as_homogeneous_ideal.to_ideal
 #align projective_spectrum.zero_locus_span ProjectiveSpectrum.zeroLocus_span
+-/
 
 variable {𝒜}
 
+#print ProjectiveSpectrum.vanishingIdeal /-
 /-- The vanishing ideal of a set `t` of points of the projective spectrum of a commutative ring `R`
 is the intersection of all the relevant homogeneous prime ideals in the set `t`.
 
@@ -102,7 +111,9 @@ ideal of `A` consisting of all "functions" that vanish on all of `t`. -/
 def vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) : HomogeneousIdeal 𝒜 :=
   ⨅ (x : ProjectiveSpectrum 𝒜) (h : x ∈ t), x.asHomogeneousIdeal
 #align projective_spectrum.vanishing_ideal ProjectiveSpectrum.vanishingIdeal
+-/
 
+#print ProjectiveSpectrum.coe_vanishingIdeal /-
 theorem coe_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
     (vanishingIdeal t : Set A) =
       {f | ∀ x : ProjectiveSpectrum 𝒜, x ∈ t → f ∈ x.asHomogeneousIdeal} :=
@@ -113,34 +124,44 @@ theorem coe_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
   apply forall_congr' fun x => _
   rw [HomogeneousIdeal.toIdeal_iInf, Submodule.mem_iInf, HomogeneousIdeal.mem_iff]
 #align projective_spectrum.coe_vanishing_ideal ProjectiveSpectrum.coe_vanishingIdeal
+-/
 
+#print ProjectiveSpectrum.mem_vanishingIdeal /-
 theorem mem_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) (f : A) :
     f ∈ vanishingIdeal t ↔ ∀ x : ProjectiveSpectrum 𝒜, x ∈ t → f ∈ x.asHomogeneousIdeal := by
   rw [← SetLike.mem_coe, coe_vanishing_ideal, Set.mem_setOf_eq]
 #align projective_spectrum.mem_vanishing_ideal ProjectiveSpectrum.mem_vanishingIdeal
+-/
 
+#print ProjectiveSpectrum.vanishingIdeal_singleton /-
 @[simp]
 theorem vanishingIdeal_singleton (x : ProjectiveSpectrum 𝒜) :
     vanishingIdeal ({x} : Set (ProjectiveSpectrum 𝒜)) = x.asHomogeneousIdeal := by
   simp [vanishing_ideal]
 #align projective_spectrum.vanishing_ideal_singleton ProjectiveSpectrum.vanishingIdeal_singleton
+-/
 
+#print ProjectiveSpectrum.subset_zeroLocus_iff_le_vanishingIdeal /-
 theorem subset_zeroLocus_iff_le_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) (I : Ideal A) :
     t ⊆ zeroLocus 𝒜 I ↔ I ≤ (vanishingIdeal t).toIdeal :=
   ⟨fun h f k => (mem_vanishingIdeal _ _).mpr fun x j => (mem_zeroLocus _ _ _).mpr (h j) k, fun h =>
     fun x j =>
     (mem_zeroLocus _ _ _).mpr (le_trans h fun f h => ((mem_vanishingIdeal _ _).mp h) x j)⟩
 #align projective_spectrum.subset_zero_locus_iff_le_vanishing_ideal ProjectiveSpectrum.subset_zeroLocus_iff_le_vanishingIdeal
+-/
 
 variable (𝒜)
 
+#print ProjectiveSpectrum.gc_ideal /-
 /-- `zero_locus` and `vanishing_ideal` form a galois connection. -/
 theorem gc_ideal :
     @GaloisConnection (Ideal A) (Set (ProjectiveSpectrum 𝒜))ᵒᵈ _ _ (fun I => zeroLocus 𝒜 I) fun t =>
       (vanishingIdeal t).toIdeal :=
   fun I t => subset_zeroLocus_iff_le_vanishingIdeal t I
 #align projective_spectrum.gc_ideal ProjectiveSpectrum.gc_ideal
+-/
 
+#print ProjectiveSpectrum.gc_set /-
 /-- `zero_locus` and `vanishing_ideal` form a galois connection. -/
 theorem gc_set :
     @GaloisConnection (Set A) (Set (ProjectiveSpectrum 𝒜))ᵒᵈ _ _ (fun s => zeroLocus 𝒜 s) fun t =>
@@ -149,7 +170,9 @@ theorem gc_set :
   have ideal_gc : GaloisConnection Ideal.span coe := (Submodule.gi A _).gc
   simpa [zero_locus_span, Function.comp] using GaloisConnection.compose ideal_gc (gc_ideal 𝒜)
 #align projective_spectrum.gc_set ProjectiveSpectrum.gc_set
+-/
 
+#print ProjectiveSpectrum.gc_homogeneousIdeal /-
 theorem gc_homogeneousIdeal :
     @GaloisConnection (HomogeneousIdeal 𝒜) (Set (ProjectiveSpectrum 𝒜))ᵒᵈ _ _
       (fun I => zeroLocus 𝒜 I) fun t => vanishingIdeal t :=
@@ -157,160 +180,224 @@ theorem gc_homogeneousIdeal :
   simpa [show I.to_ideal ≤ (vanishing_ideal t).toIdeal ↔ I ≤ vanishing_ideal t from Iff.rfl] using
     subset_zero_locus_iff_le_vanishing_ideal t I.to_ideal
 #align projective_spectrum.gc_homogeneous_ideal ProjectiveSpectrum.gc_homogeneousIdeal
+-/
 
+#print ProjectiveSpectrum.subset_zeroLocus_iff_subset_vanishingIdeal /-
 theorem subset_zeroLocus_iff_subset_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) (s : Set A) :
     t ⊆ zeroLocus 𝒜 s ↔ s ⊆ vanishingIdeal t :=
   (gc_set _) s t
 #align projective_spectrum.subset_zero_locus_iff_subset_vanishing_ideal ProjectiveSpectrum.subset_zeroLocus_iff_subset_vanishingIdeal
+-/
 
+#print ProjectiveSpectrum.subset_vanishingIdeal_zeroLocus /-
 theorem subset_vanishingIdeal_zeroLocus (s : Set A) : s ⊆ vanishingIdeal (zeroLocus 𝒜 s) :=
   (gc_set _).le_u_l s
 #align projective_spectrum.subset_vanishing_ideal_zero_locus ProjectiveSpectrum.subset_vanishingIdeal_zeroLocus
+-/
 
+#print ProjectiveSpectrum.ideal_le_vanishingIdeal_zeroLocus /-
 theorem ideal_le_vanishingIdeal_zeroLocus (I : Ideal A) :
     I ≤ (vanishingIdeal (zeroLocus 𝒜 I)).toIdeal :=
   (gc_ideal _).le_u_l I
 #align projective_spectrum.ideal_le_vanishing_ideal_zero_locus ProjectiveSpectrum.ideal_le_vanishingIdeal_zeroLocus
+-/
 
+#print ProjectiveSpectrum.homogeneousIdeal_le_vanishingIdeal_zeroLocus /-
 theorem homogeneousIdeal_le_vanishingIdeal_zeroLocus (I : HomogeneousIdeal 𝒜) :
     I ≤ vanishingIdeal (zeroLocus 𝒜 I) :=
   (gc_homogeneousIdeal _).le_u_l I
 #align projective_spectrum.homogeneous_ideal_le_vanishing_ideal_zero_locus ProjectiveSpectrum.homogeneousIdeal_le_vanishingIdeal_zeroLocus
+-/
 
+#print ProjectiveSpectrum.subset_zeroLocus_vanishingIdeal /-
 theorem subset_zeroLocus_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
     t ⊆ zeroLocus 𝒜 (vanishingIdeal t) :=
   (gc_ideal _).l_u_le t
 #align projective_spectrum.subset_zero_locus_vanishing_ideal ProjectiveSpectrum.subset_zeroLocus_vanishingIdeal
+-/
 
+#print ProjectiveSpectrum.zeroLocus_anti_mono /-
 theorem zeroLocus_anti_mono {s t : Set A} (h : s ⊆ t) : zeroLocus 𝒜 t ⊆ zeroLocus 𝒜 s :=
   (gc_set _).monotone_l h
 #align projective_spectrum.zero_locus_anti_mono ProjectiveSpectrum.zeroLocus_anti_mono
+-/
 
+#print ProjectiveSpectrum.zeroLocus_anti_mono_ideal /-
 theorem zeroLocus_anti_mono_ideal {s t : Ideal A} (h : s ≤ t) :
     zeroLocus 𝒜 (t : Set A) ⊆ zeroLocus 𝒜 (s : Set A) :=
   (gc_ideal _).monotone_l h
 #align projective_spectrum.zero_locus_anti_mono_ideal ProjectiveSpectrum.zeroLocus_anti_mono_ideal
+-/
 
+#print ProjectiveSpectrum.zeroLocus_anti_mono_homogeneousIdeal /-
 theorem zeroLocus_anti_mono_homogeneousIdeal {s t : HomogeneousIdeal 𝒜} (h : s ≤ t) :
     zeroLocus 𝒜 (t : Set A) ⊆ zeroLocus 𝒜 (s : Set A) :=
   (gc_homogeneousIdeal _).monotone_l h
 #align projective_spectrum.zero_locus_anti_mono_homogeneous_ideal ProjectiveSpectrum.zeroLocus_anti_mono_homogeneousIdeal
+-/
 
+#print ProjectiveSpectrum.vanishingIdeal_anti_mono /-
 theorem vanishingIdeal_anti_mono {s t : Set (ProjectiveSpectrum 𝒜)} (h : s ⊆ t) :
     vanishingIdeal t ≤ vanishingIdeal s :=
   (gc_ideal _).monotone_u h
 #align projective_spectrum.vanishing_ideal_anti_mono ProjectiveSpectrum.vanishingIdeal_anti_mono
+-/
 
+#print ProjectiveSpectrum.zeroLocus_bot /-
 theorem zeroLocus_bot : zeroLocus 𝒜 ((⊥ : Ideal A) : Set A) = Set.univ :=
   (gc_ideal 𝒜).l_bot
 #align projective_spectrum.zero_locus_bot ProjectiveSpectrum.zeroLocus_bot
+-/
 
+#print ProjectiveSpectrum.zeroLocus_singleton_zero /-
 @[simp]
 theorem zeroLocus_singleton_zero : zeroLocus 𝒜 ({0} : Set A) = Set.univ :=
   zeroLocus_bot _
 #align projective_spectrum.zero_locus_singleton_zero ProjectiveSpectrum.zeroLocus_singleton_zero
+-/
 
+#print ProjectiveSpectrum.zeroLocus_empty /-
 @[simp]
 theorem zeroLocus_empty : zeroLocus 𝒜 (∅ : Set A) = Set.univ :=
   (gc_set 𝒜).l_bot
 #align projective_spectrum.zero_locus_empty ProjectiveSpectrum.zeroLocus_empty
+-/
 
+#print ProjectiveSpectrum.vanishingIdeal_univ /-
 @[simp]
 theorem vanishingIdeal_univ : vanishingIdeal (∅ : Set (ProjectiveSpectrum 𝒜)) = ⊤ := by
   simpa using (gc_ideal _).u_top
 #align projective_spectrum.vanishing_ideal_univ ProjectiveSpectrum.vanishingIdeal_univ
+-/
 
+#print ProjectiveSpectrum.zeroLocus_empty_of_one_mem /-
 theorem zeroLocus_empty_of_one_mem {s : Set A} (h : (1 : A) ∈ s) : zeroLocus 𝒜 s = ∅ :=
   Set.eq_empty_iff_forall_not_mem.mpr fun x hx =>
     (inferInstance : x.asHomogeneousIdeal.toIdeal.IsPrime).ne_top <|
       x.asHomogeneousIdeal.toIdeal.eq_top_iff_one.mpr <| hx h
 #align projective_spectrum.zero_locus_empty_of_one_mem ProjectiveSpectrum.zeroLocus_empty_of_one_mem
+-/
 
+#print ProjectiveSpectrum.zeroLocus_singleton_one /-
 @[simp]
 theorem zeroLocus_singleton_one : zeroLocus 𝒜 ({1} : Set A) = ∅ :=
   zeroLocus_empty_of_one_mem 𝒜 (Set.mem_singleton (1 : A))
 #align projective_spectrum.zero_locus_singleton_one ProjectiveSpectrum.zeroLocus_singleton_one
+-/
 
+#print ProjectiveSpectrum.zeroLocus_univ /-
 @[simp]
 theorem zeroLocus_univ : zeroLocus 𝒜 (Set.univ : Set A) = ∅ :=
   zeroLocus_empty_of_one_mem _ (Set.mem_univ 1)
 #align projective_spectrum.zero_locus_univ ProjectiveSpectrum.zeroLocus_univ
+-/
 
+#print ProjectiveSpectrum.zeroLocus_sup_ideal /-
 theorem zeroLocus_sup_ideal (I J : Ideal A) :
     zeroLocus 𝒜 ((I ⊔ J : Ideal A) : Set A) = zeroLocus _ I ∩ zeroLocus _ J :=
   (gc_ideal 𝒜).l_sup
 #align projective_spectrum.zero_locus_sup_ideal ProjectiveSpectrum.zeroLocus_sup_ideal
+-/
 
+#print ProjectiveSpectrum.zeroLocus_sup_homogeneousIdeal /-
 theorem zeroLocus_sup_homogeneousIdeal (I J : HomogeneousIdeal 𝒜) :
     zeroLocus 𝒜 ((I ⊔ J : HomogeneousIdeal 𝒜) : Set A) = zeroLocus _ I ∩ zeroLocus _ J :=
   (gc_homogeneousIdeal 𝒜).l_sup
 #align projective_spectrum.zero_locus_sup_homogeneous_ideal ProjectiveSpectrum.zeroLocus_sup_homogeneousIdeal
+-/
 
+#print ProjectiveSpectrum.zeroLocus_union /-
 theorem zeroLocus_union (s s' : Set A) : zeroLocus 𝒜 (s ∪ s') = zeroLocus _ s ∩ zeroLocus _ s' :=
   (gc_set 𝒜).l_sup
 #align projective_spectrum.zero_locus_union ProjectiveSpectrum.zeroLocus_union
+-/
 
+#print ProjectiveSpectrum.vanishingIdeal_union /-
 theorem vanishingIdeal_union (t t' : Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (t ∪ t') = vanishingIdeal t ⊓ vanishingIdeal t' := by
   ext1 <;> convert (gc_ideal 𝒜).u_inf
 #align projective_spectrum.vanishing_ideal_union ProjectiveSpectrum.vanishingIdeal_union
+-/
 
+#print ProjectiveSpectrum.zeroLocus_iSup_ideal /-
 theorem zeroLocus_iSup_ideal {γ : Sort _} (I : γ → Ideal A) :
     zeroLocus _ ((⨆ i, I i : Ideal A) : Set A) = ⋂ i, zeroLocus 𝒜 (I i) :=
   (gc_ideal 𝒜).l_iSup
 #align projective_spectrum.zero_locus_supr_ideal ProjectiveSpectrum.zeroLocus_iSup_ideal
+-/
 
+#print ProjectiveSpectrum.zeroLocus_iSup_homogeneousIdeal /-
 theorem zeroLocus_iSup_homogeneousIdeal {γ : Sort _} (I : γ → HomogeneousIdeal 𝒜) :
     zeroLocus _ ((⨆ i, I i : HomogeneousIdeal 𝒜) : Set A) = ⋂ i, zeroLocus 𝒜 (I i) :=
   (gc_homogeneousIdeal 𝒜).l_iSup
 #align projective_spectrum.zero_locus_supr_homogeneous_ideal ProjectiveSpectrum.zeroLocus_iSup_homogeneousIdeal
+-/
 
+#print ProjectiveSpectrum.zeroLocus_iUnion /-
 theorem zeroLocus_iUnion {γ : Sort _} (s : γ → Set A) :
     zeroLocus 𝒜 (⋃ i, s i) = ⋂ i, zeroLocus 𝒜 (s i) :=
   (gc_set 𝒜).l_iSup
 #align projective_spectrum.zero_locus_Union ProjectiveSpectrum.zeroLocus_iUnion
+-/
 
+#print ProjectiveSpectrum.zeroLocus_bUnion /-
 theorem zeroLocus_bUnion (s : Set (Set A)) :
     zeroLocus 𝒜 (⋃ s' ∈ s, s' : Set A) = ⋂ s' ∈ s, zeroLocus 𝒜 s' := by simp only [zero_locus_Union]
 #align projective_spectrum.zero_locus_bUnion ProjectiveSpectrum.zeroLocus_bUnion
+-/
 
+#print ProjectiveSpectrum.vanishingIdeal_iUnion /-
 theorem vanishingIdeal_iUnion {γ : Sort _} (t : γ → Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (⋃ i, t i) = ⨅ i, vanishingIdeal (t i) :=
   HomogeneousIdeal.toIdeal_injective <| by
     convert (gc_ideal 𝒜).u_iInf <;> exact HomogeneousIdeal.toIdeal_iInf _
 #align projective_spectrum.vanishing_ideal_Union ProjectiveSpectrum.vanishingIdeal_iUnion
+-/
 
+#print ProjectiveSpectrum.zeroLocus_inf /-
 theorem zeroLocus_inf (I J : Ideal A) :
     zeroLocus 𝒜 ((I ⊓ J : Ideal A) : Set A) = zeroLocus 𝒜 I ∪ zeroLocus 𝒜 J :=
   Set.ext fun x => x.IsPrime.inf_le
 #align projective_spectrum.zero_locus_inf ProjectiveSpectrum.zeroLocus_inf
+-/
 
+#print ProjectiveSpectrum.union_zeroLocus /-
 theorem union_zeroLocus (s s' : Set A) :
     zeroLocus 𝒜 s ∪ zeroLocus 𝒜 s' = zeroLocus 𝒜 (Ideal.span s ⊓ Ideal.span s' : Ideal A) := by
   rw [zero_locus_inf]; simp
 #align projective_spectrum.union_zero_locus ProjectiveSpectrum.union_zeroLocus
+-/
 
+#print ProjectiveSpectrum.zeroLocus_mul_ideal /-
 theorem zeroLocus_mul_ideal (I J : Ideal A) :
     zeroLocus 𝒜 ((I * J : Ideal A) : Set A) = zeroLocus 𝒜 I ∪ zeroLocus 𝒜 J :=
   Set.ext fun x => x.IsPrime.mul_le
 #align projective_spectrum.zero_locus_mul_ideal ProjectiveSpectrum.zeroLocus_mul_ideal
+-/
 
+#print ProjectiveSpectrum.zeroLocus_mul_homogeneousIdeal /-
 theorem zeroLocus_mul_homogeneousIdeal (I J : HomogeneousIdeal 𝒜) :
     zeroLocus 𝒜 ((I * J : HomogeneousIdeal 𝒜) : Set A) = zeroLocus 𝒜 I ∪ zeroLocus 𝒜 J :=
   Set.ext fun x => x.IsPrime.mul_le
 #align projective_spectrum.zero_locus_mul_homogeneous_ideal ProjectiveSpectrum.zeroLocus_mul_homogeneousIdeal
+-/
 
+#print ProjectiveSpectrum.zeroLocus_singleton_mul /-
 theorem zeroLocus_singleton_mul (f g : A) :
     zeroLocus 𝒜 ({f * g} : Set A) = zeroLocus 𝒜 {f} ∪ zeroLocus 𝒜 {g} :=
   Set.ext fun x => by simpa using x.is_prime.mul_mem_iff_mem_or_mem
 #align projective_spectrum.zero_locus_singleton_mul ProjectiveSpectrum.zeroLocus_singleton_mul
+-/
 
+#print ProjectiveSpectrum.zeroLocus_singleton_pow /-
 @[simp]
 theorem zeroLocus_singleton_pow (f : A) (n : ℕ) (hn : 0 < n) :
     zeroLocus 𝒜 ({f ^ n} : Set A) = zeroLocus 𝒜 {f} :=
   Set.ext fun x => by simpa using x.is_prime.pow_mem_iff_mem n hn
 #align projective_spectrum.zero_locus_singleton_pow ProjectiveSpectrum.zeroLocus_singleton_pow
+-/
 
+#print ProjectiveSpectrum.sup_vanishingIdeal_le /-
 theorem sup_vanishingIdeal_le (t t' : Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal t ⊔ vanishingIdeal t' ≤ vanishingIdeal (t ∩ t') :=
   by
@@ -321,12 +408,16 @@ theorem sup_vanishingIdeal_le (t t' : Set (ProjectiveSpectrum 𝒜)) :
   erw [mem_vanishing_ideal] at hf hg 
   apply Submodule.add_mem <;> solve_by_elim
 #align projective_spectrum.sup_vanishing_ideal_le ProjectiveSpectrum.sup_vanishingIdeal_le
+-/
 
+#print ProjectiveSpectrum.mem_compl_zeroLocus_iff_not_mem /-
 theorem mem_compl_zeroLocus_iff_not_mem {f : A} {I : ProjectiveSpectrum 𝒜} :
     I ∈ (zeroLocus 𝒜 {f} : Set (ProjectiveSpectrum 𝒜))ᶜ ↔ f ∉ I.asHomogeneousIdeal := by
   rw [Set.mem_compl_iff, mem_zero_locus, Set.singleton_subset_iff] <;> rfl
 #align projective_spectrum.mem_compl_zero_locus_iff_not_mem ProjectiveSpectrum.mem_compl_zeroLocus_iff_not_mem
+-/
 
+#print ProjectiveSpectrum.zariskiTopology /-
 /-- The Zariski topology on the prime spectrum of a commutative ring is defined via the closed sets
 of the topology: they are exactly those sets that are the zero locus of a subset of the ring. -/
 instance zariskiTopology : TopologicalSpace (ProjectiveSpectrum 𝒜) :=
@@ -340,24 +431,34 @@ instance zariskiTopology : TopologicalSpace (ProjectiveSpectrum 𝒜) :=
       exact ⟨_, zero_locus_Union 𝒜 _⟩)
     (by rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩; exact ⟨_, (union_zero_locus 𝒜 s t).symm⟩)
 #align projective_spectrum.zariski_topology ProjectiveSpectrum.zariskiTopology
+-/
 
+#print ProjectiveSpectrum.top /-
 /-- The underlying topology of `Proj` is the projective spectrum of graded ring `A`. -/
 def top : TopCat :=
   TopCat.of (ProjectiveSpectrum 𝒜)
 #align projective_spectrum.Top ProjectiveSpectrum.top
+-/
 
+#print ProjectiveSpectrum.isOpen_iff /-
 theorem isOpen_iff (U : Set (ProjectiveSpectrum 𝒜)) : IsOpen U ↔ ∃ s, Uᶜ = zeroLocus 𝒜 s := by
   simp only [@eq_comm _ (Uᶜ)] <;> rfl
 #align projective_spectrum.is_open_iff ProjectiveSpectrum.isOpen_iff
+-/
 
+#print ProjectiveSpectrum.isClosed_iff_zeroLocus /-
 theorem isClosed_iff_zeroLocus (Z : Set (ProjectiveSpectrum 𝒜)) :
     IsClosed Z ↔ ∃ s, Z = zeroLocus 𝒜 s := by rw [← isOpen_compl_iff, is_open_iff, compl_compl]
 #align projective_spectrum.is_closed_iff_zero_locus ProjectiveSpectrum.isClosed_iff_zeroLocus
+-/
 
+#print ProjectiveSpectrum.isClosed_zeroLocus /-
 theorem isClosed_zeroLocus (s : Set A) : IsClosed (zeroLocus 𝒜 s) := by
   rw [is_closed_iff_zero_locus]; exact ⟨s, rfl⟩
 #align projective_spectrum.is_closed_zero_locus ProjectiveSpectrum.isClosed_zeroLocus
+-/
 
+#print ProjectiveSpectrum.zeroLocus_vanishingIdeal_eq_closure /-
 theorem zeroLocus_vanishingIdeal_eq_closure (t : Set (ProjectiveSpectrum 𝒜)) :
     zeroLocus 𝒜 (vanishingIdeal t : Set A) = closure t :=
   by
@@ -369,7 +470,9 @@ theorem zeroLocus_vanishingIdeal_eq_closure (t : Set (ProjectiveSpectrum 𝒜))
   · rw [(is_closed_zero_locus _ _).closure_subset_iff]
     exact subset_zero_locus_vanishing_ideal 𝒜 t
 #align projective_spectrum.zero_locus_vanishing_ideal_eq_closure ProjectiveSpectrum.zeroLocus_vanishingIdeal_eq_closure
+-/
 
+#print ProjectiveSpectrum.vanishingIdeal_closure /-
 theorem vanishingIdeal_closure (t : Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (closure t) = vanishingIdeal t :=
   by
@@ -379,64 +482,88 @@ theorem vanishingIdeal_closure (t : Set (ProjectiveSpectrum 𝒜)) :
   erw [zero_locus_vanishing_ideal_eq_closure 𝒜 t] at this 
   exact this
 #align projective_spectrum.vanishing_ideal_closure ProjectiveSpectrum.vanishingIdeal_closure
+-/
 
 section BasicOpen
 
+#print ProjectiveSpectrum.basicOpen /-
 /-- `basic_open r` is the open subset containing all prime ideals not containing `r`. -/
 def basicOpen (r : A) : TopologicalSpace.Opens (ProjectiveSpectrum 𝒜)
     where
   carrier := {x | r ∉ x.asHomogeneousIdeal}
   is_open' := ⟨{r}, Set.ext fun x => Set.singleton_subset_iff.trans <| Classical.not_not.symm⟩
 #align projective_spectrum.basic_open ProjectiveSpectrum.basicOpen
+-/
 
+#print ProjectiveSpectrum.mem_basicOpen /-
 @[simp]
 theorem mem_basicOpen (f : A) (x : ProjectiveSpectrum 𝒜) :
     x ∈ basicOpen 𝒜 f ↔ f ∉ x.asHomogeneousIdeal :=
   Iff.rfl
 #align projective_spectrum.mem_basic_open ProjectiveSpectrum.mem_basicOpen
+-/
 
+#print ProjectiveSpectrum.mem_coe_basicOpen /-
 theorem mem_coe_basicOpen (f : A) (x : ProjectiveSpectrum 𝒜) :
     x ∈ (↑(basicOpen 𝒜 f) : Set (ProjectiveSpectrum 𝒜)) ↔ f ∉ x.asHomogeneousIdeal :=
   Iff.rfl
 #align projective_spectrum.mem_coe_basic_open ProjectiveSpectrum.mem_coe_basicOpen
+-/
 
+#print ProjectiveSpectrum.isOpen_basicOpen /-
 theorem isOpen_basicOpen {a : A} : IsOpen (basicOpen 𝒜 a : Set (ProjectiveSpectrum 𝒜)) :=
   (basicOpen 𝒜 a).IsOpen
 #align projective_spectrum.is_open_basic_open ProjectiveSpectrum.isOpen_basicOpen
+-/
 
+#print ProjectiveSpectrum.basicOpen_eq_zeroLocus_compl /-
 @[simp]
 theorem basicOpen_eq_zeroLocus_compl (r : A) :
     (basicOpen 𝒜 r : Set (ProjectiveSpectrum 𝒜)) = zeroLocus 𝒜 {r}ᶜ :=
   Set.ext fun x => by simpa only [Set.mem_compl_iff, mem_zero_locus, Set.singleton_subset_iff]
 #align projective_spectrum.basic_open_eq_zero_locus_compl ProjectiveSpectrum.basicOpen_eq_zeroLocus_compl
+-/
 
+#print ProjectiveSpectrum.basicOpen_one /-
 @[simp]
 theorem basicOpen_one : basicOpen 𝒜 (1 : A) = ⊤ :=
   TopologicalSpace.Opens.ext <| by simp
 #align projective_spectrum.basic_open_one ProjectiveSpectrum.basicOpen_one
+-/
 
+#print ProjectiveSpectrum.basicOpen_zero /-
 @[simp]
 theorem basicOpen_zero : basicOpen 𝒜 (0 : A) = ⊥ :=
   TopologicalSpace.Opens.ext <| by simp
 #align projective_spectrum.basic_open_zero ProjectiveSpectrum.basicOpen_zero
+-/
 
+#print ProjectiveSpectrum.basicOpen_mul /-
 theorem basicOpen_mul (f g : A) : basicOpen 𝒜 (f * g) = basicOpen 𝒜 f ⊓ basicOpen 𝒜 g :=
   TopologicalSpace.Opens.ext <| by simp [zero_locus_singleton_mul]
 #align projective_spectrum.basic_open_mul ProjectiveSpectrum.basicOpen_mul
+-/
 
+#print ProjectiveSpectrum.basicOpen_mul_le_left /-
 theorem basicOpen_mul_le_left (f g : A) : basicOpen 𝒜 (f * g) ≤ basicOpen 𝒜 f := by
   rw [basic_open_mul 𝒜 f g]; exact inf_le_left
 #align projective_spectrum.basic_open_mul_le_left ProjectiveSpectrum.basicOpen_mul_le_left
+-/
 
+#print ProjectiveSpectrum.basicOpen_mul_le_right /-
 theorem basicOpen_mul_le_right (f g : A) : basicOpen 𝒜 (f * g) ≤ basicOpen 𝒜 g := by
   rw [basic_open_mul 𝒜 f g]; exact inf_le_right
 #align projective_spectrum.basic_open_mul_le_right ProjectiveSpectrum.basicOpen_mul_le_right
+-/
 
+#print ProjectiveSpectrum.basicOpen_pow /-
 @[simp]
 theorem basicOpen_pow (f : A) (n : ℕ) (hn : 0 < n) : basicOpen 𝒜 (f ^ n) = basicOpen 𝒜 f :=
   TopologicalSpace.Opens.ext <| by simpa using zero_locus_singleton_pow 𝒜 f n hn
 #align projective_spectrum.basic_open_pow ProjectiveSpectrum.basicOpen_pow
+-/
 
+#print ProjectiveSpectrum.basicOpen_eq_union_of_projection /-
 theorem basicOpen_eq_union_of_projection (f : A) :
     basicOpen 𝒜 f = ⨆ i : ℕ, basicOpen 𝒜 (GradedAlgebra.proj 𝒜 i f) :=
   TopologicalSpace.Opens.ext <|
@@ -454,7 +581,9 @@ theorem basicOpen_eq_union_of_projection (f : A) :
       · obtain ⟨_, ⟨i, rfl⟩, hz⟩ := hz
         exact fun rid => hz (z.1.2 i rid)
 #align projective_spectrum.basic_open_eq_union_of_projection ProjectiveSpectrum.basicOpen_eq_union_of_projection
+-/
 
+#print ProjectiveSpectrum.isTopologicalBasis_basic_opens /-
 theorem isTopologicalBasis_basic_opens :
     TopologicalSpace.IsTopologicalBasis
       (Set.range fun r : A => (basicOpen 𝒜 r : Set (ProjectiveSpectrum 𝒜))) :=
@@ -469,6 +598,7 @@ theorem isTopologicalBasis_basic_opens :
     rw [← Set.compl_subset_compl, ← hs, basic_open_eq_zero_locus_compl, compl_compl]
     exact zero_locus_anti_mono 𝒜 (set.singleton_subset_iff.mpr hfs)
 #align projective_spectrum.is_topological_basis_basic_opens ProjectiveSpectrum.isTopologicalBasis_basic_opens
+-/
 
 end BasicOpen
 
@@ -485,18 +615,23 @@ where `x ≤ y` if and only if `y ∈ closure {x}`.
 instance : PartialOrder (ProjectiveSpectrum 𝒜) :=
   PartialOrder.lift asHomogeneousIdeal fun ⟨_, _, _⟩ ⟨_, _, _⟩ => mk.inj_eq.mpr
 
+#print ProjectiveSpectrum.as_ideal_le_as_ideal /-
 @[simp]
 theorem as_ideal_le_as_ideal (x y : ProjectiveSpectrum 𝒜) :
     x.asHomogeneousIdeal ≤ y.asHomogeneousIdeal ↔ x ≤ y :=
   Iff.rfl
 #align projective_spectrum.as_ideal_le_as_ideal ProjectiveSpectrum.as_ideal_le_as_ideal
+-/
 
+#print ProjectiveSpectrum.as_ideal_lt_as_ideal /-
 @[simp]
 theorem as_ideal_lt_as_ideal (x y : ProjectiveSpectrum 𝒜) :
     x.asHomogeneousIdeal < y.asHomogeneousIdeal ↔ x < y :=
   Iff.rfl
 #align projective_spectrum.as_ideal_lt_as_ideal ProjectiveSpectrum.as_ideal_lt_as_ideal
+-/
 
+#print ProjectiveSpectrum.le_iff_mem_closure /-
 theorem le_iff_mem_closure (x y : ProjectiveSpectrum 𝒜) :
     x ≤ y ↔ y ∈ closure ({x} : Set (ProjectiveSpectrum 𝒜)) :=
   by
@@ -504,6 +639,7 @@ theorem le_iff_mem_closure (x y : ProjectiveSpectrum 𝒜) :
     vanishing_ideal_singleton]
   simp only [coe_subset_coe, Subtype.coe_le_coe, coe_coe]
 #align projective_spectrum.le_iff_mem_closure ProjectiveSpectrum.le_iff_mem_closure
+-/
 
 end Order
 

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

@@ -76,7 +76,7 @@ At a point `x` (a homogeneous prime ideal) the function (i.e., element) `f` take
 quotient ring `A` modulo the prime ideal `x`. In this manner, `zero_locus s` is exactly the subset
 of `projective_spectrum 𝒜` where all "functions" in `s` vanish simultaneously. -/
 def zeroLocus (s : Set A) : Set (ProjectiveSpectrum 𝒜) :=
-  { x | s ⊆ x.asHomogeneousIdeal }
+  {x | s ⊆ x.asHomogeneousIdeal}
 #align projective_spectrum.zero_locus ProjectiveSpectrum.zeroLocus
 
 @[simp]
@@ -105,7 +105,7 @@ def vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) : HomogeneousIdeal 𝒜 :
 
 theorem coe_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
     (vanishingIdeal t : Set A) =
-      { f | ∀ x : ProjectiveSpectrum 𝒜, x ∈ t → f ∈ x.asHomogeneousIdeal } :=
+      {f | ∀ x : ProjectiveSpectrum 𝒜, x ∈ t → f ∈ x.asHomogeneousIdeal} :=
   by
   ext f
   rw [vanishing_ideal, SetLike.mem_coe, ← HomogeneousIdeal.mem_iff, HomogeneousIdeal.toIdeal_iInf,
@@ -252,7 +252,7 @@ theorem zeroLocus_union (s s' : Set A) : zeroLocus 𝒜 (s ∪ s') = zeroLocus _
 
 theorem vanishingIdeal_union (t t' : Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (t ∪ t') = vanishingIdeal t ⊓ vanishingIdeal t' := by
-  ext1 <;> convert(gc_ideal 𝒜).u_inf
+  ext1 <;> convert (gc_ideal 𝒜).u_inf
 #align projective_spectrum.vanishing_ideal_union ProjectiveSpectrum.vanishingIdeal_union
 
 theorem zeroLocus_iSup_ideal {γ : Sort _} (I : γ → Ideal A) :
@@ -277,7 +277,7 @@ theorem zeroLocus_bUnion (s : Set (Set A)) :
 theorem vanishingIdeal_iUnion {γ : Sort _} (t : γ → Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (⋃ i, t i) = ⨅ i, vanishingIdeal (t i) :=
   HomogeneousIdeal.toIdeal_injective <| by
-    convert(gc_ideal 𝒜).u_iInf <;> exact HomogeneousIdeal.toIdeal_iInf _
+    convert (gc_ideal 𝒜).u_iInf <;> exact HomogeneousIdeal.toIdeal_iInf _
 #align projective_spectrum.vanishing_ideal_Union ProjectiveSpectrum.vanishingIdeal_iUnion
 
 theorem zeroLocus_inf (I J : Ideal A) :
@@ -385,7 +385,7 @@ section BasicOpen
 /-- `basic_open r` is the open subset containing all prime ideals not containing `r`. -/
 def basicOpen (r : A) : TopologicalSpace.Opens (ProjectiveSpectrum 𝒜)
     where
-  carrier := { x | r ∉ x.asHomogeneousIdeal }
+  carrier := {x | r ∉ x.asHomogeneousIdeal}
   is_open' := ⟨{r}, Set.ext fun x => Set.singleton_subset_iff.trans <| Classical.not_not.symm⟩
 #align projective_spectrum.basic_open ProjectiveSpectrum.basicOpen
 
@@ -447,8 +447,8 @@ theorem basicOpen_eq_union_of_projection (f : A) :
             by
             contrapose! hz with H
             classical
-              rw [← DirectSum.sum_support_decompose 𝒜 f]
-              apply Ideal.sum_mem _ fun i hi => H i with
+            rw [← DirectSum.sum_support_decompose 𝒜 f]
+            apply Ideal.sum_mem _ fun i hi => H i with
           ⟨i, hi⟩
         exact ⟨basic_open 𝒜 (GradedAlgebra.proj 𝒜 i f), ⟨i, rfl⟩, by rwa [mem_basic_open]⟩
       · obtain ⟨_, ⟨i, rfl⟩, hz⟩ := hz

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

@@ -318,7 +318,7 @@ theorem sup_vanishingIdeal_le (t t' : Set (ProjectiveSpectrum 𝒜)) :
   rw [← HomogeneousIdeal.mem_iff, HomogeneousIdeal.toIdeal_sup, mem_vanishing_ideal,
     Submodule.mem_sup]
   rintro ⟨f, hf, g, hg, rfl⟩ x ⟨hxt, hxt'⟩
-  erw [mem_vanishing_ideal] at hf hg
+  erw [mem_vanishing_ideal] at hf hg 
   apply Submodule.add_mem <;> solve_by_elim
 #align projective_spectrum.sup_vanishing_ideal_le ProjectiveSpectrum.sup_vanishingIdeal_le
 
@@ -363,8 +363,8 @@ theorem zeroLocus_vanishingIdeal_eq_closure (t : Set (ProjectiveSpectrum 𝒜))
   by
   apply Set.Subset.antisymm
   · rintro x hx t' ⟨ht', ht⟩
-    obtain ⟨fs, rfl⟩ : ∃ s, t' = zero_locus 𝒜 s := by rwa [is_closed_iff_zero_locus] at ht'
-    rw [subset_zero_locus_iff_subset_vanishing_ideal] at ht
+    obtain ⟨fs, rfl⟩ : ∃ s, t' = zero_locus 𝒜 s := by rwa [is_closed_iff_zero_locus] at ht' 
+    rw [subset_zero_locus_iff_subset_vanishing_ideal] at ht 
     exact Set.Subset.trans ht hx
   · rw [(is_closed_zero_locus _ _).closure_subset_iff]
     exact subset_zero_locus_vanishing_ideal 𝒜 t
@@ -374,9 +374,9 @@ theorem vanishingIdeal_closure (t : Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (closure t) = vanishingIdeal t :=
   by
   have := (gc_ideal 𝒜).u_l_u_eq_u t
-  dsimp only at this
+  dsimp only at this 
   ext1
-  erw [zero_locus_vanishing_ideal_eq_closure 𝒜 t] at this
+  erw [zero_locus_vanishing_ideal_eq_closure 𝒜 t] at this 
   exact this
 #align projective_spectrum.vanishing_ideal_closure ProjectiveSpectrum.vanishingIdeal_closure
 
@@ -463,7 +463,7 @@ theorem isTopologicalBasis_basic_opens :
   · rintro _ ⟨r, rfl⟩
     exact is_open_basic_open 𝒜
   · rintro p U hp ⟨s, hs⟩
-    rw [← compl_compl U, Set.mem_compl_iff, ← hs, mem_zero_locus, Set.not_subset] at hp
+    rw [← compl_compl U, Set.mem_compl_iff, ← hs, mem_zero_locus, Set.not_subset] at hp 
     obtain ⟨f, hfs, hfp⟩ := hp
     refine' ⟨basic_open 𝒜 f, ⟨f, rfl⟩, hfp, _⟩
     rw [← Set.compl_subset_compl, ← hs, basic_open_eq_zero_locus_compl, compl_compl]

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

@@ -45,7 +45,7 @@ It is naturally endowed with a topology: the Zariski topology.
 
 noncomputable section
 
-open DirectSum BigOperators Pointwise
+open scoped DirectSum BigOperators Pointwise
 
 open DirectSum SetLike TopCat TopologicalSpace CategoryTheory Opposite
 

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

@@ -55,12 +55,6 @@ variable [CommSemiring R] [CommRing A] [Algebra R A]
 
 variable (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜]
 
-/- warning: projective_spectrum -> ProjectiveSpectrum is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜], Type.{u2}
-but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜], Type.{u2}
-Case conversion may be inaccurate. Consider using '#align projective_spectrum ProjectiveSpectrumₓ'. -/
 /-- The projective spectrum of a graded commutative ring is the subtype of all homogenous ideals
 that are prime and do not contain the irrelevant ideal. -/
 @[ext, nolint has_nonempty_instance]
@@ -74,12 +68,6 @@ attribute [instance] ProjectiveSpectrum.isPrime
 
 namespace ProjectiveSpectrum
 
-/- warning: projective_spectrum.zero_locus -> ProjectiveSpectrum.zeroLocus is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus ProjectiveSpectrum.zeroLocusₓ'. -/
 /-- The zero locus of a set `s` of elements of a commutative ring `A` is the set of all relevant
 homogeneous prime ideals of the ring that contain the set `s`.
 
@@ -91,21 +79,12 @@ def zeroLocus (s : Set A) : Set (ProjectiveSpectrum 𝒜) :=
   { x | s ⊆ x.asHomogeneousIdeal }
 #align projective_spectrum.zero_locus ProjectiveSpectrum.zeroLocus
 
-/- warning: projective_spectrum.mem_zero_locus -> ProjectiveSpectrum.mem_zeroLocus is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.mem_zero_locus ProjectiveSpectrum.mem_zeroLocusₓ'. -/
 @[simp]
 theorem mem_zeroLocus (x : ProjectiveSpectrum 𝒜) (s : Set A) :
     x ∈ zeroLocus 𝒜 s ↔ s ⊆ x.asHomogeneousIdeal :=
   Iff.rfl
 #align projective_spectrum.mem_zero_locus ProjectiveSpectrum.mem_zeroLocus
 
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 @[simp]
 theorem zeroLocus_span (s : Set A) : zeroLocus 𝒜 (Ideal.span s) = zeroLocus 𝒜 s := by ext x;
   exact (Submodule.gi _ _).gc s x.as_homogeneous_ideal.to_ideal
@@ -113,12 +92,6 @@ theorem zeroLocus_span (s : Set A) : zeroLocus 𝒜 (Ideal.span s) = zeroLocus 
 
 variable {𝒜}
 
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 /-- The vanishing ideal of a set `t` of points of the projective spectrum of a commutative ring `R`
 is the intersection of all the relevant homogeneous prime ideals in the set `t`.
 
@@ -130,9 +103,6 @@ def vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) : HomogeneousIdeal 𝒜 :
   ⨅ (x : ProjectiveSpectrum 𝒜) (h : x ∈ t), x.asHomogeneousIdeal
 #align projective_spectrum.vanishing_ideal ProjectiveSpectrum.vanishingIdeal
 
-/- warning: projective_spectrum.coe_vanishing_ideal -> ProjectiveSpectrum.coe_vanishingIdeal is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.coe_vanishing_ideal ProjectiveSpectrum.coe_vanishingIdealₓ'. -/
 theorem coe_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
     (vanishingIdeal t : Set A) =
       { f | ∀ x : ProjectiveSpectrum 𝒜, x ∈ t → f ∈ x.asHomogeneousIdeal } :=
@@ -144,32 +114,17 @@ theorem coe_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
   rw [HomogeneousIdeal.toIdeal_iInf, Submodule.mem_iInf, HomogeneousIdeal.mem_iff]
 #align projective_spectrum.coe_vanishing_ideal ProjectiveSpectrum.coe_vanishingIdeal
 
-/- warning: projective_spectrum.mem_vanishing_ideal -> ProjectiveSpectrum.mem_vanishingIdeal is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.mem_vanishing_ideal ProjectiveSpectrum.mem_vanishingIdealₓ'. -/
 theorem mem_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) (f : A) :
     f ∈ vanishingIdeal t ↔ ∀ x : ProjectiveSpectrum 𝒜, x ∈ t → f ∈ x.asHomogeneousIdeal := by
   rw [← SetLike.mem_coe, coe_vanishing_ideal, Set.mem_setOf_eq]
 #align projective_spectrum.mem_vanishing_ideal ProjectiveSpectrum.mem_vanishingIdeal
 
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-  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_3 : Algebra.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {𝒜 : Nat -> (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3))} [_inst_4 : GradedAlgebra.{0, u2, u1} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 𝒜] (x : ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), Eq.{succ u1} (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (ProjectiveSpectrum.vanishingIdeal.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u1, u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instSingletonSet.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x)) (ProjectiveSpectrum.asHomogeneousIdeal.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x)
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.vanishing_ideal_singleton ProjectiveSpectrum.vanishingIdeal_singletonₓ'. -/
 @[simp]
 theorem vanishingIdeal_singleton (x : ProjectiveSpectrum 𝒜) :
     vanishingIdeal ({x} : Set (ProjectiveSpectrum 𝒜)) = x.asHomogeneousIdeal := by
   simp [vanishing_ideal]
 #align projective_spectrum.vanishing_ideal_singleton ProjectiveSpectrum.vanishingIdeal_singleton
 
-/- warning: projective_spectrum.subset_zero_locus_iff_le_vanishing_ideal -> ProjectiveSpectrum.subset_zeroLocus_iff_le_vanishingIdeal is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] {𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))} [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (I : Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))), Iff (HasSubset.Subset.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instHasSubsetSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) t (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) I))) (LE.le.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Preorder.toLE.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Submodule.completeLattice.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))))) I (HomogeneousIdeal.toIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4 (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t)))
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.subset_zero_locus_iff_le_vanishing_ideal ProjectiveSpectrum.subset_zeroLocus_iff_le_vanishingIdealₓ'. -/
 theorem subset_zeroLocus_iff_le_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) (I : Ideal A) :
     t ⊆ zeroLocus 𝒜 I ↔ I ≤ (vanishingIdeal t).toIdeal :=
   ⟨fun h f k => (mem_vanishingIdeal _ _).mpr fun x j => (mem_zeroLocus _ _ _).mpr (h j) k, fun h =>
@@ -179,12 +134,6 @@ theorem subset_zeroLocus_iff_le_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜
 
 variable (𝒜)
 
-/- warning: projective_spectrum.gc_ideal -> ProjectiveSpectrum.gc_ideal is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align projective_spectrum.gc_ideal ProjectiveSpectrum.gc_idealₓ'. -/
 /-- `zero_locus` and `vanishing_ideal` form a galois connection. -/
 theorem gc_ideal :
     @GaloisConnection (Ideal A) (Set (ProjectiveSpectrum 𝒜))ᵒᵈ _ _ (fun I => zeroLocus 𝒜 I) fun t =>
@@ -192,9 +141,6 @@ theorem gc_ideal :
   fun I t => subset_zeroLocus_iff_le_vanishingIdeal t I
 #align projective_spectrum.gc_ideal ProjectiveSpectrum.gc_ideal
 
-/- warning: projective_spectrum.gc_set -> ProjectiveSpectrum.gc_set is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.gc_set ProjectiveSpectrum.gc_setₓ'. -/
 /-- `zero_locus` and `vanishing_ideal` form a galois connection. -/
 theorem gc_set :
     @GaloisConnection (Set A) (Set (ProjectiveSpectrum 𝒜))ᵒᵈ _ _ (fun s => zeroLocus 𝒜 s) fun t =>
@@ -204,9 +150,6 @@ theorem gc_set :
   simpa [zero_locus_span, Function.comp] using GaloisConnection.compose ideal_gc (gc_ideal 𝒜)
 #align projective_spectrum.gc_set ProjectiveSpectrum.gc_set
 
-/- warning: projective_spectrum.gc_homogeneous_ideal -> ProjectiveSpectrum.gc_homogeneousIdeal is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.gc_homogeneous_ideal ProjectiveSpectrum.gc_homogeneousIdealₓ'. -/
 theorem gc_homogeneousIdeal :
     @GaloisConnection (HomogeneousIdeal 𝒜) (Set (ProjectiveSpectrum 𝒜))ᵒᵈ _ _
       (fun I => zeroLocus 𝒜 I) fun t => vanishingIdeal t :=
@@ -215,315 +158,159 @@ theorem gc_homogeneousIdeal :
     subset_zero_locus_iff_le_vanishing_ideal t I.to_ideal
 #align projective_spectrum.gc_homogeneous_ideal ProjectiveSpectrum.gc_homogeneousIdeal
 
-/- warning: projective_spectrum.subset_zero_locus_iff_subset_vanishing_ideal -> ProjectiveSpectrum.subset_zeroLocus_iff_subset_vanishingIdeal is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.subset_zero_locus_iff_subset_vanishing_ideal ProjectiveSpectrum.subset_zeroLocus_iff_subset_vanishingIdealₓ'. -/
 theorem subset_zeroLocus_iff_subset_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) (s : Set A) :
     t ⊆ zeroLocus 𝒜 s ↔ s ⊆ vanishingIdeal t :=
   (gc_set _) s t
 #align projective_spectrum.subset_zero_locus_iff_subset_vanishing_ideal ProjectiveSpectrum.subset_zeroLocus_iff_subset_vanishingIdeal
 
-/- warning: projective_spectrum.subset_vanishing_ideal_zero_locus -> ProjectiveSpectrum.subset_vanishingIdeal_zeroLocus is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.subset_vanishing_ideal_zero_locus ProjectiveSpectrum.subset_vanishingIdeal_zeroLocusₓ'. -/
 theorem subset_vanishingIdeal_zeroLocus (s : Set A) : s ⊆ vanishingIdeal (zeroLocus 𝒜 s) :=
   (gc_set _).le_u_l s
 #align projective_spectrum.subset_vanishing_ideal_zero_locus ProjectiveSpectrum.subset_vanishingIdeal_zeroLocus
 
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 theorem ideal_le_vanishingIdeal_zeroLocus (I : Ideal A) :
     I ≤ (vanishingIdeal (zeroLocus 𝒜 I)).toIdeal :=
   (gc_ideal _).le_u_l I
 #align projective_spectrum.ideal_le_vanishing_ideal_zero_locus ProjectiveSpectrum.ideal_le_vanishingIdeal_zeroLocus
 
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-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.homogeneous_ideal_le_vanishing_ideal_zero_locus ProjectiveSpectrum.homogeneousIdeal_le_vanishingIdeal_zeroLocusₓ'. -/
 theorem homogeneousIdeal_le_vanishingIdeal_zeroLocus (I : HomogeneousIdeal 𝒜) :
     I ≤ vanishingIdeal (zeroLocus 𝒜 I) :=
   (gc_homogeneousIdeal _).le_u_l I
 #align projective_spectrum.homogeneous_ideal_le_vanishing_ideal_zero_locus ProjectiveSpectrum.homogeneousIdeal_le_vanishingIdeal_zeroLocus
 
-/- warning: projective_spectrum.subset_zero_locus_vanishing_ideal -> ProjectiveSpectrum.subset_zeroLocus_vanishingIdeal is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.subset_zero_locus_vanishing_ideal ProjectiveSpectrum.subset_zeroLocus_vanishingIdealₓ'. -/
 theorem subset_zeroLocus_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
     t ⊆ zeroLocus 𝒜 (vanishingIdeal t) :=
   (gc_ideal _).l_u_le t
 #align projective_spectrum.subset_zero_locus_vanishing_ideal ProjectiveSpectrum.subset_zeroLocus_vanishingIdeal
 
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 theorem zeroLocus_anti_mono {s t : Set A} (h : s ⊆ t) : zeroLocus 𝒜 t ⊆ zeroLocus 𝒜 s :=
   (gc_set _).monotone_l h
 #align projective_spectrum.zero_locus_anti_mono ProjectiveSpectrum.zeroLocus_anti_mono
 
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 theorem zeroLocus_anti_mono_ideal {s t : Ideal A} (h : s ≤ t) :
     zeroLocus 𝒜 (t : Set A) ⊆ zeroLocus 𝒜 (s : Set A) :=
   (gc_ideal _).monotone_l h
 #align projective_spectrum.zero_locus_anti_mono_ideal ProjectiveSpectrum.zeroLocus_anti_mono_ideal
 
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 theorem zeroLocus_anti_mono_homogeneousIdeal {s t : HomogeneousIdeal 𝒜} (h : s ≤ t) :
     zeroLocus 𝒜 (t : Set A) ⊆ zeroLocus 𝒜 (s : Set A) :=
   (gc_homogeneousIdeal _).monotone_l h
 #align projective_spectrum.zero_locus_anti_mono_homogeneous_ideal ProjectiveSpectrum.zeroLocus_anti_mono_homogeneousIdeal
 
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 theorem vanishingIdeal_anti_mono {s t : Set (ProjectiveSpectrum 𝒜)} (h : s ⊆ t) :
     vanishingIdeal t ≤ vanishingIdeal s :=
   (gc_ideal _).monotone_u h
 #align projective_spectrum.vanishing_ideal_anti_mono ProjectiveSpectrum.vanishingIdeal_anti_mono
 
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 theorem zeroLocus_bot : zeroLocus 𝒜 ((⊥ : Ideal A) : Set A) = Set.univ :=
   (gc_ideal 𝒜).l_bot
 #align projective_spectrum.zero_locus_bot ProjectiveSpectrum.zeroLocus_bot
 
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 @[simp]
 theorem zeroLocus_singleton_zero : zeroLocus 𝒜 ({0} : Set A) = Set.univ :=
   zeroLocus_bot _
 #align projective_spectrum.zero_locus_singleton_zero ProjectiveSpectrum.zeroLocus_singleton_zero
 
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 @[simp]
 theorem zeroLocus_empty : zeroLocus 𝒜 (∅ : Set A) = Set.univ :=
   (gc_set 𝒜).l_bot
 #align projective_spectrum.zero_locus_empty ProjectiveSpectrum.zeroLocus_empty
 
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 @[simp]
 theorem vanishingIdeal_univ : vanishingIdeal (∅ : Set (ProjectiveSpectrum 𝒜)) = ⊤ := by
   simpa using (gc_ideal _).u_top
 #align projective_spectrum.vanishing_ideal_univ ProjectiveSpectrum.vanishingIdeal_univ
 
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 theorem zeroLocus_empty_of_one_mem {s : Set A} (h : (1 : A) ∈ s) : zeroLocus 𝒜 s = ∅ :=
   Set.eq_empty_iff_forall_not_mem.mpr fun x hx =>
     (inferInstance : x.asHomogeneousIdeal.toIdeal.IsPrime).ne_top <|
       x.asHomogeneousIdeal.toIdeal.eq_top_iff_one.mpr <| hx h
 #align projective_spectrum.zero_locus_empty_of_one_mem ProjectiveSpectrum.zeroLocus_empty_of_one_mem
 
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 @[simp]
 theorem zeroLocus_singleton_one : zeroLocus 𝒜 ({1} : Set A) = ∅ :=
   zeroLocus_empty_of_one_mem 𝒜 (Set.mem_singleton (1 : A))
 #align projective_spectrum.zero_locus_singleton_one ProjectiveSpectrum.zeroLocus_singleton_one
 
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 @[simp]
 theorem zeroLocus_univ : zeroLocus 𝒜 (Set.univ : Set A) = ∅ :=
   zeroLocus_empty_of_one_mem _ (Set.mem_univ 1)
 #align projective_spectrum.zero_locus_univ ProjectiveSpectrum.zeroLocus_univ
 
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 theorem zeroLocus_sup_ideal (I J : Ideal A) :
     zeroLocus 𝒜 ((I ⊔ J : Ideal A) : Set A) = zeroLocus _ I ∩ zeroLocus _ J :=
   (gc_ideal 𝒜).l_sup
 #align projective_spectrum.zero_locus_sup_ideal ProjectiveSpectrum.zeroLocus_sup_ideal
 
-/- warning: projective_spectrum.zero_locus_sup_homogeneous_ideal -> ProjectiveSpectrum.zeroLocus_sup_homogeneousIdeal is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_sup_homogeneous_ideal ProjectiveSpectrum.zeroLocus_sup_homogeneousIdealₓ'. -/
 theorem zeroLocus_sup_homogeneousIdeal (I J : HomogeneousIdeal 𝒜) :
     zeroLocus 𝒜 ((I ⊔ J : HomogeneousIdeal 𝒜) : Set A) = zeroLocus _ I ∩ zeroLocus _ J :=
   (gc_homogeneousIdeal 𝒜).l_sup
 #align projective_spectrum.zero_locus_sup_homogeneous_ideal ProjectiveSpectrum.zeroLocus_sup_homogeneousIdeal
 
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-Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_union ProjectiveSpectrum.zeroLocus_unionₓ'. -/
 theorem zeroLocus_union (s s' : Set A) : zeroLocus 𝒜 (s ∪ s') = zeroLocus _ s ∩ zeroLocus _ s' :=
   (gc_set 𝒜).l_sup
 #align projective_spectrum.zero_locus_union ProjectiveSpectrum.zeroLocus_union
 
-/- warning: projective_spectrum.vanishing_ideal_union -> ProjectiveSpectrum.vanishingIdeal_union is a dubious translation:
-<too large>
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 theorem vanishingIdeal_union (t t' : Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (t ∪ t') = vanishingIdeal t ⊓ vanishingIdeal t' := by
   ext1 <;> convert(gc_ideal 𝒜).u_inf
 #align projective_spectrum.vanishing_ideal_union ProjectiveSpectrum.vanishingIdeal_union
 
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-Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_supr_ideal ProjectiveSpectrum.zeroLocus_iSup_idealₓ'. -/
 theorem zeroLocus_iSup_ideal {γ : Sort _} (I : γ → Ideal A) :
     zeroLocus _ ((⨆ i, I i : Ideal A) : Set A) = ⋂ i, zeroLocus 𝒜 (I i) :=
   (gc_ideal 𝒜).l_iSup
 #align projective_spectrum.zero_locus_supr_ideal ProjectiveSpectrum.zeroLocus_iSup_ideal
 
-/- warning: projective_spectrum.zero_locus_supr_homogeneous_ideal -> ProjectiveSpectrum.zeroLocus_iSup_homogeneousIdeal is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_supr_homogeneous_ideal ProjectiveSpectrum.zeroLocus_iSup_homogeneousIdealₓ'. -/
 theorem zeroLocus_iSup_homogeneousIdeal {γ : Sort _} (I : γ → HomogeneousIdeal 𝒜) :
     zeroLocus _ ((⨆ i, I i : HomogeneousIdeal 𝒜) : Set A) = ⋂ i, zeroLocus 𝒜 (I i) :=
   (gc_homogeneousIdeal 𝒜).l_iSup
 #align projective_spectrum.zero_locus_supr_homogeneous_ideal ProjectiveSpectrum.zeroLocus_iSup_homogeneousIdeal
 
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 theorem zeroLocus_iUnion {γ : Sort _} (s : γ → Set A) :
     zeroLocus 𝒜 (⋃ i, s i) = ⋂ i, zeroLocus 𝒜 (s i) :=
   (gc_set 𝒜).l_iSup
 #align projective_spectrum.zero_locus_Union ProjectiveSpectrum.zeroLocus_iUnion
 
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 theorem zeroLocus_bUnion (s : Set (Set A)) :
     zeroLocus 𝒜 (⋃ s' ∈ s, s' : Set A) = ⋂ s' ∈ s, zeroLocus 𝒜 s' := by simp only [zero_locus_Union]
 #align projective_spectrum.zero_locus_bUnion ProjectiveSpectrum.zeroLocus_bUnion
 
-/- warning: projective_spectrum.vanishing_ideal_Union -> ProjectiveSpectrum.vanishingIdeal_iUnion is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.vanishing_ideal_Union ProjectiveSpectrum.vanishingIdeal_iUnionₓ'. -/
 theorem vanishingIdeal_iUnion {γ : Sort _} (t : γ → Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (⋃ i, t i) = ⨅ i, vanishingIdeal (t i) :=
   HomogeneousIdeal.toIdeal_injective <| by
     convert(gc_ideal 𝒜).u_iInf <;> exact HomogeneousIdeal.toIdeal_iInf _
 #align projective_spectrum.vanishing_ideal_Union ProjectiveSpectrum.vanishingIdeal_iUnion
 
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 theorem zeroLocus_inf (I J : Ideal A) :
     zeroLocus 𝒜 ((I ⊓ J : Ideal A) : Set A) = zeroLocus 𝒜 I ∪ zeroLocus 𝒜 J :=
   Set.ext fun x => x.IsPrime.inf_le
 #align projective_spectrum.zero_locus_inf ProjectiveSpectrum.zeroLocus_inf
 
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-Case conversion may be inaccurate. Consider using '#align projective_spectrum.union_zero_locus ProjectiveSpectrum.union_zeroLocusₓ'. -/
 theorem union_zeroLocus (s s' : Set A) :
     zeroLocus 𝒜 s ∪ zeroLocus 𝒜 s' = zeroLocus 𝒜 (Ideal.span s ⊓ Ideal.span s' : Ideal A) := by
   rw [zero_locus_inf]; simp
 #align projective_spectrum.union_zero_locus ProjectiveSpectrum.union_zeroLocus
 
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 theorem zeroLocus_mul_ideal (I J : Ideal A) :
     zeroLocus 𝒜 ((I * J : Ideal A) : Set A) = zeroLocus 𝒜 I ∪ zeroLocus 𝒜 J :=
   Set.ext fun x => x.IsPrime.mul_le
 #align projective_spectrum.zero_locus_mul_ideal ProjectiveSpectrum.zeroLocus_mul_ideal
 
-/- warning: projective_spectrum.zero_locus_mul_homogeneous_ideal -> ProjectiveSpectrum.zeroLocus_mul_homogeneousIdeal is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_mul_homogeneous_ideal ProjectiveSpectrum.zeroLocus_mul_homogeneousIdealₓ'. -/
 theorem zeroLocus_mul_homogeneousIdeal (I J : HomogeneousIdeal 𝒜) :
     zeroLocus 𝒜 ((I * J : HomogeneousIdeal 𝒜) : Set A) = zeroLocus 𝒜 I ∪ zeroLocus 𝒜 J :=
   Set.ext fun x => x.IsPrime.mul_le
 #align projective_spectrum.zero_locus_mul_homogeneous_ideal ProjectiveSpectrum.zeroLocus_mul_homogeneousIdeal
 
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-Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_singleton_mul ProjectiveSpectrum.zeroLocus_singleton_mulₓ'. -/
 theorem zeroLocus_singleton_mul (f g : A) :
     zeroLocus 𝒜 ({f * g} : Set A) = zeroLocus 𝒜 {f} ∪ zeroLocus 𝒜 {g} :=
   Set.ext fun x => by simpa using x.is_prime.mul_mem_iff_mem_or_mem
 #align projective_spectrum.zero_locus_singleton_mul ProjectiveSpectrum.zeroLocus_singleton_mul
 
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 @[simp]
 theorem zeroLocus_singleton_pow (f : A) (n : ℕ) (hn : 0 < n) :
     zeroLocus 𝒜 ({f ^ n} : Set A) = zeroLocus 𝒜 {f} :=
   Set.ext fun x => by simpa using x.is_prime.pow_mem_iff_mem n hn
 #align projective_spectrum.zero_locus_singleton_pow ProjectiveSpectrum.zeroLocus_singleton_pow
 
-/- warning: projective_spectrum.sup_vanishing_ideal_le -> ProjectiveSpectrum.sup_vanishingIdeal_le is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.sup_vanishing_ideal_le ProjectiveSpectrum.sup_vanishingIdeal_leₓ'. -/
 theorem sup_vanishingIdeal_le (t t' : Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal t ⊔ vanishingIdeal t' ≤ vanishingIdeal (t ∩ t') :=
   by
@@ -535,20 +322,11 @@ theorem sup_vanishingIdeal_le (t t' : Set (ProjectiveSpectrum 𝒜)) :
   apply Submodule.add_mem <;> solve_by_elim
 #align projective_spectrum.sup_vanishing_ideal_le ProjectiveSpectrum.sup_vanishingIdeal_le
 
-/- warning: projective_spectrum.mem_compl_zero_locus_iff_not_mem -> ProjectiveSpectrum.mem_compl_zeroLocus_iff_not_mem is a dubious translation:
-<too large>
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 theorem mem_compl_zeroLocus_iff_not_mem {f : A} {I : ProjectiveSpectrum 𝒜} :
     I ∈ (zeroLocus 𝒜 {f} : Set (ProjectiveSpectrum 𝒜))ᶜ ↔ f ∉ I.asHomogeneousIdeal := by
   rw [Set.mem_compl_iff, mem_zero_locus, Set.singleton_subset_iff] <;> rfl
 #align projective_spectrum.mem_compl_zero_locus_iff_not_mem ProjectiveSpectrum.mem_compl_zeroLocus_iff_not_mem
 
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-Case conversion may be inaccurate. Consider using '#align projective_spectrum.zariski_topology ProjectiveSpectrum.zariskiTopologyₓ'. -/
 /-- The Zariski topology on the prime spectrum of a commutative ring is defined via the closed sets
 of the topology: they are exactly those sets that are the zero locus of a subset of the ring. -/
 instance zariskiTopology : TopologicalSpace (ProjectiveSpectrum 𝒜) :=
@@ -563,50 +341,23 @@ instance zariskiTopology : TopologicalSpace (ProjectiveSpectrum 𝒜) :=
     (by rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩; exact ⟨_, (union_zero_locus 𝒜 s t).symm⟩)
 #align projective_spectrum.zariski_topology ProjectiveSpectrum.zariskiTopology
 
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 /-- The underlying topology of `Proj` is the projective spectrum of graded ring `A`. -/
 def top : TopCat :=
   TopCat.of (ProjectiveSpectrum 𝒜)
 #align projective_spectrum.Top ProjectiveSpectrum.top
 
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 theorem isOpen_iff (U : Set (ProjectiveSpectrum 𝒜)) : IsOpen U ↔ ∃ s, Uᶜ = zeroLocus 𝒜 s := by
   simp only [@eq_comm _ (Uᶜ)] <;> rfl
 #align projective_spectrum.is_open_iff ProjectiveSpectrum.isOpen_iff
 
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-Case conversion may be inaccurate. Consider using '#align projective_spectrum.is_closed_iff_zero_locus ProjectiveSpectrum.isClosed_iff_zeroLocusₓ'. -/
 theorem isClosed_iff_zeroLocus (Z : Set (ProjectiveSpectrum 𝒜)) :
     IsClosed Z ↔ ∃ s, Z = zeroLocus 𝒜 s := by rw [← isOpen_compl_iff, is_open_iff, compl_compl]
 #align projective_spectrum.is_closed_iff_zero_locus ProjectiveSpectrum.isClosed_iff_zeroLocus
 
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-Case conversion may be inaccurate. Consider using '#align projective_spectrum.is_closed_zero_locus ProjectiveSpectrum.isClosed_zeroLocusₓ'. -/
 theorem isClosed_zeroLocus (s : Set A) : IsClosed (zeroLocus 𝒜 s) := by
   rw [is_closed_iff_zero_locus]; exact ⟨s, rfl⟩
 #align projective_spectrum.is_closed_zero_locus ProjectiveSpectrum.isClosed_zeroLocus
 
-/- warning: projective_spectrum.zero_locus_vanishing_ideal_eq_closure -> ProjectiveSpectrum.zeroLocus_vanishingIdeal_eq_closure is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_vanishing_ideal_eq_closure ProjectiveSpectrum.zeroLocus_vanishingIdeal_eq_closureₓ'. -/
 theorem zeroLocus_vanishingIdeal_eq_closure (t : Set (ProjectiveSpectrum 𝒜)) :
     zeroLocus 𝒜 (vanishingIdeal t : Set A) = closure t :=
   by
@@ -619,12 +370,6 @@ theorem zeroLocus_vanishingIdeal_eq_closure (t : Set (ProjectiveSpectrum 𝒜))
     exact subset_zero_locus_vanishing_ideal 𝒜 t
 #align projective_spectrum.zero_locus_vanishing_ideal_eq_closure ProjectiveSpectrum.zeroLocus_vanishingIdeal_eq_closure
 
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 theorem vanishingIdeal_closure (t : Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (closure t) = vanishingIdeal t :=
   by
@@ -637,12 +382,6 @@ theorem vanishingIdeal_closure (t : Set (ProjectiveSpectrum 𝒜)) :
 
 section BasicOpen
 
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-Case conversion may be inaccurate. Consider using '#align projective_spectrum.basic_open ProjectiveSpectrum.basicOpenₓ'. -/
 /-- `basic_open r` is the open subset containing all prime ideals not containing `r`. -/
 def basicOpen (r : A) : TopologicalSpace.Opens (ProjectiveSpectrum 𝒜)
     where
@@ -650,114 +389,54 @@ def basicOpen (r : A) : TopologicalSpace.Opens (ProjectiveSpectrum 𝒜)
   is_open' := ⟨{r}, Set.ext fun x => Set.singleton_subset_iff.trans <| Classical.not_not.symm⟩
 #align projective_spectrum.basic_open ProjectiveSpectrum.basicOpen
 
-/- warning: projective_spectrum.mem_basic_open -> ProjectiveSpectrum.mem_basicOpen is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.mem_basic_open ProjectiveSpectrum.mem_basicOpenₓ'. -/
 @[simp]
 theorem mem_basicOpen (f : A) (x : ProjectiveSpectrum 𝒜) :
     x ∈ basicOpen 𝒜 f ↔ f ∉ x.asHomogeneousIdeal :=
   Iff.rfl
 #align projective_spectrum.mem_basic_open ProjectiveSpectrum.mem_basicOpen
 
-/- warning: projective_spectrum.mem_coe_basic_open -> ProjectiveSpectrum.mem_coe_basicOpen is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.mem_coe_basic_open ProjectiveSpectrum.mem_coe_basicOpenₓ'. -/
 theorem mem_coe_basicOpen (f : A) (x : ProjectiveSpectrum 𝒜) :
     x ∈ (↑(basicOpen 𝒜 f) : Set (ProjectiveSpectrum 𝒜)) ↔ f ∉ x.asHomogeneousIdeal :=
   Iff.rfl
 #align projective_spectrum.mem_coe_basic_open ProjectiveSpectrum.mem_coe_basicOpen
 
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 theorem isOpen_basicOpen {a : A} : IsOpen (basicOpen 𝒜 a : Set (ProjectiveSpectrum 𝒜)) :=
   (basicOpen 𝒜 a).IsOpen
 #align projective_spectrum.is_open_basic_open ProjectiveSpectrum.isOpen_basicOpen
 
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 @[simp]
 theorem basicOpen_eq_zeroLocus_compl (r : A) :
     (basicOpen 𝒜 r : Set (ProjectiveSpectrum 𝒜)) = zeroLocus 𝒜 {r}ᶜ :=
   Set.ext fun x => by simpa only [Set.mem_compl_iff, mem_zero_locus, Set.singleton_subset_iff]
 #align projective_spectrum.basic_open_eq_zero_locus_compl ProjectiveSpectrum.basicOpen_eq_zeroLocus_compl
 
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 @[simp]
 theorem basicOpen_one : basicOpen 𝒜 (1 : A) = ⊤ :=
   TopologicalSpace.Opens.ext <| by simp
 #align projective_spectrum.basic_open_one ProjectiveSpectrum.basicOpen_one
 
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 @[simp]
 theorem basicOpen_zero : basicOpen 𝒜 (0 : A) = ⊥ :=
   TopologicalSpace.Opens.ext <| by simp
 #align projective_spectrum.basic_open_zero ProjectiveSpectrum.basicOpen_zero
 
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 theorem basicOpen_mul (f g : A) : basicOpen 𝒜 (f * g) = basicOpen 𝒜 f ⊓ basicOpen 𝒜 g :=
   TopologicalSpace.Opens.ext <| by simp [zero_locus_singleton_mul]
 #align projective_spectrum.basic_open_mul ProjectiveSpectrum.basicOpen_mul
 
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 theorem basicOpen_mul_le_left (f g : A) : basicOpen 𝒜 (f * g) ≤ basicOpen 𝒜 f := by
   rw [basic_open_mul 𝒜 f g]; exact inf_le_left
 #align projective_spectrum.basic_open_mul_le_left ProjectiveSpectrum.basicOpen_mul_le_left
 
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 theorem basicOpen_mul_le_right (f g : A) : basicOpen 𝒜 (f * g) ≤ basicOpen 𝒜 g := by
   rw [basic_open_mul 𝒜 f g]; exact inf_le_right
 #align projective_spectrum.basic_open_mul_le_right ProjectiveSpectrum.basicOpen_mul_le_right
 
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 @[simp]
 theorem basicOpen_pow (f : A) (n : ℕ) (hn : 0 < n) : basicOpen 𝒜 (f ^ n) = basicOpen 𝒜 f :=
   TopologicalSpace.Opens.ext <| by simpa using zero_locus_singleton_pow 𝒜 f n hn
 #align projective_spectrum.basic_open_pow ProjectiveSpectrum.basicOpen_pow
 
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-Case conversion may be inaccurate. Consider using '#align projective_spectrum.basic_open_eq_union_of_projection ProjectiveSpectrum.basicOpen_eq_union_of_projectionₓ'. -/
 theorem basicOpen_eq_union_of_projection (f : A) :
     basicOpen 𝒜 f = ⨆ i : ℕ, basicOpen 𝒜 (GradedAlgebra.proj 𝒜 i f) :=
   TopologicalSpace.Opens.ext <|
@@ -776,12 +455,6 @@ theorem basicOpen_eq_union_of_projection (f : A) :
         exact fun rid => hz (z.1.2 i rid)
 #align projective_spectrum.basic_open_eq_union_of_projection ProjectiveSpectrum.basicOpen_eq_union_of_projection
 
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-Case conversion may be inaccurate. Consider using '#align projective_spectrum.is_topological_basis_basic_opens ProjectiveSpectrum.isTopologicalBasis_basic_opensₓ'. -/
 theorem isTopologicalBasis_basic_opens :
     TopologicalSpace.IsTopologicalBasis
       (Set.range fun r : A => (basicOpen 𝒜 r : Set (ProjectiveSpectrum 𝒜))) :=
@@ -812,30 +485,18 @@ where `x ≤ y` if and only if `y ∈ closure {x}`.
 instance : PartialOrder (ProjectiveSpectrum 𝒜) :=
   PartialOrder.lift asHomogeneousIdeal fun ⟨_, _, _⟩ ⟨_, _, _⟩ => mk.inj_eq.mpr
 
-/- warning: projective_spectrum.as_ideal_le_as_ideal -> ProjectiveSpectrum.as_ideal_le_as_ideal is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.as_ideal_le_as_ideal ProjectiveSpectrum.as_ideal_le_as_idealₓ'. -/
 @[simp]
 theorem as_ideal_le_as_ideal (x y : ProjectiveSpectrum 𝒜) :
     x.asHomogeneousIdeal ≤ y.asHomogeneousIdeal ↔ x ≤ y :=
   Iff.rfl
 #align projective_spectrum.as_ideal_le_as_ideal ProjectiveSpectrum.as_ideal_le_as_ideal
 
-/- warning: projective_spectrum.as_ideal_lt_as_ideal -> ProjectiveSpectrum.as_ideal_lt_as_ideal is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.as_ideal_lt_as_ideal ProjectiveSpectrum.as_ideal_lt_as_idealₓ'. -/
 @[simp]
 theorem as_ideal_lt_as_ideal (x y : ProjectiveSpectrum 𝒜) :
     x.asHomogeneousIdeal < y.asHomogeneousIdeal ↔ x < y :=
   Iff.rfl
 #align projective_spectrum.as_ideal_lt_as_ideal ProjectiveSpectrum.as_ideal_lt_as_ideal
 
-/- warning: projective_spectrum.le_iff_mem_closure -> ProjectiveSpectrum.le_iff_mem_closure is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (x : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (y : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), Iff (LE.le.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Preorder.toHasLe.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (PartialOrder.toPreorder.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.partialOrder.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) x y) (Membership.Mem.{u2, u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasMem.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) y (closure.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Singleton.singleton.{u2, u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasSingleton.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x)))
-but is expected to have type
-  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_3 : Algebra.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] (𝒜 : Nat -> (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u2, u1} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 𝒜] (x : ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (y : ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), Iff (LE.le.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Preorder.toLE.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (PartialOrder.toPreorder.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.instPartialOrderProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) x y) (Membership.mem.{u1, u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instMembershipSet.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) y (closure.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Singleton.singleton.{u1, u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instSingletonSet.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x)))
-Case conversion may be inaccurate. Consider using '#align projective_spectrum.le_iff_mem_closure ProjectiveSpectrum.le_iff_mem_closureₓ'. -/
 theorem le_iff_mem_closure (x y : ProjectiveSpectrum 𝒜) :
     x ≤ y ↔ y ∈ closure ({x} : Set (ProjectiveSpectrum 𝒜)) :=
   by

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

@@ -107,9 +107,7 @@ but is expected to have type
   forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (s : Set.{u2} A), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (Ideal.span.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) s))) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s)
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_span ProjectiveSpectrum.zeroLocus_spanₓ'. -/
 @[simp]
-theorem zeroLocus_span (s : Set A) : zeroLocus 𝒜 (Ideal.span s) = zeroLocus 𝒜 s :=
-  by
-  ext x
+theorem zeroLocus_span (s : Set A) : zeroLocus 𝒜 (Ideal.span s) = zeroLocus 𝒜 s := by ext x;
   exact (Submodule.gi _ _).gc s x.as_homogeneous_ideal.to_ideal
 #align projective_spectrum.zero_locus_span ProjectiveSpectrum.zeroLocus_span
 
@@ -477,10 +475,8 @@ but is expected to have type
   forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (s : Set.{u2} A) (s' : Set.{u2} A), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Union.union.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instUnionSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s')) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (Inf.inf.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Submodule.instInfSubmodule.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (Ideal.span.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) s) (Ideal.span.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) s'))))
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.union_zero_locus ProjectiveSpectrum.union_zeroLocusₓ'. -/
 theorem union_zeroLocus (s s' : Set A) :
-    zeroLocus 𝒜 s ∪ zeroLocus 𝒜 s' = zeroLocus 𝒜 (Ideal.span s ⊓ Ideal.span s' : Ideal A) :=
-  by
-  rw [zero_locus_inf]
-  simp
+    zeroLocus 𝒜 s ∪ zeroLocus 𝒜 s' = zeroLocus 𝒜 (Ideal.span s ⊓ Ideal.span s' : Ideal A) := by
+  rw [zero_locus_inf]; simp
 #align projective_spectrum.union_zero_locus ProjectiveSpectrum.union_zeroLocus
 
 /- warning: projective_spectrum.zero_locus_mul_ideal -> ProjectiveSpectrum.zeroLocus_mul_ideal is a dubious translation:
@@ -564,9 +560,7 @@ instance zariskiTopology : TopologicalSpace (ProjectiveSpectrum 𝒜) :=
       have hf : ∀ i : Zs, ↑i = zero_locus 𝒜 (f i) := fun i => (Classical.choose_spec (h i.2)).symm
       simp only [hf]
       exact ⟨_, zero_locus_Union 𝒜 _⟩)
-    (by
-      rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩
-      exact ⟨_, (union_zero_locus 𝒜 s t).symm⟩)
+    (by rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩; exact ⟨_, (union_zero_locus 𝒜 s t).symm⟩)
 #align projective_spectrum.zariski_topology ProjectiveSpectrum.zariskiTopology
 
 /- warning: projective_spectrum.Top -> ProjectiveSpectrum.top is a dubious translation:
@@ -606,10 +600,8 @@ lean 3 declaration is
 but is expected to have type
   forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (s : Set.{u2} A), IsClosed.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s)
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.is_closed_zero_locus ProjectiveSpectrum.isClosed_zeroLocusₓ'. -/
-theorem isClosed_zeroLocus (s : Set A) : IsClosed (zeroLocus 𝒜 s) :=
-  by
-  rw [is_closed_iff_zero_locus]
-  exact ⟨s, rfl⟩
+theorem isClosed_zeroLocus (s : Set A) : IsClosed (zeroLocus 𝒜 s) := by
+  rw [is_closed_iff_zero_locus]; exact ⟨s, rfl⟩
 #align projective_spectrum.is_closed_zero_locus ProjectiveSpectrum.isClosed_zeroLocus
 
 /- warning: projective_spectrum.zero_locus_vanishing_ideal_eq_closure -> ProjectiveSpectrum.zeroLocus_vanishingIdeal_eq_closure is a dubious translation:
@@ -735,10 +727,8 @@ lean 3 declaration is
 but is expected to have type
   forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (f : A) (g : A), LE.le.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Preorder.toLE.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)))))) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (NonUnitalNonAssocRing.toMul.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) f g)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 f)
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.basic_open_mul_le_left ProjectiveSpectrum.basicOpen_mul_le_leftₓ'. -/
-theorem basicOpen_mul_le_left (f g : A) : basicOpen 𝒜 (f * g) ≤ basicOpen 𝒜 f :=
-  by
-  rw [basic_open_mul 𝒜 f g]
-  exact inf_le_left
+theorem basicOpen_mul_le_left (f g : A) : basicOpen 𝒜 (f * g) ≤ basicOpen 𝒜 f := by
+  rw [basic_open_mul 𝒜 f g]; exact inf_le_left
 #align projective_spectrum.basic_open_mul_le_left ProjectiveSpectrum.basicOpen_mul_le_left
 
 /- warning: projective_spectrum.basic_open_mul_le_right -> ProjectiveSpectrum.basicOpen_mul_le_right is a dubious translation:
@@ -747,10 +737,8 @@ lean 3 declaration is
 but is expected to have type
   forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (f : A) (g : A), LE.le.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Preorder.toLE.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)))))) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (NonUnitalNonAssocRing.toMul.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) f g)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 g)
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.basic_open_mul_le_right ProjectiveSpectrum.basicOpen_mul_le_rightₓ'. -/
-theorem basicOpen_mul_le_right (f g : A) : basicOpen 𝒜 (f * g) ≤ basicOpen 𝒜 g :=
-  by
-  rw [basic_open_mul 𝒜 f g]
-  exact inf_le_right
+theorem basicOpen_mul_le_right (f g : A) : basicOpen 𝒜 (f * g) ≤ basicOpen 𝒜 g := by
+  rw [basic_open_mul 𝒜 f g]; exact inf_le_right
 #align projective_spectrum.basic_open_mul_le_right ProjectiveSpectrum.basicOpen_mul_le_right
 
 /- warning: projective_spectrum.basic_open_pow -> ProjectiveSpectrum.basicOpen_pow is a dubious translation:

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

@@ -92,10 +92,7 @@ def zeroLocus (s : Set A) : Set (ProjectiveSpectrum 𝒜) :=
 #align projective_spectrum.zero_locus ProjectiveSpectrum.zeroLocus
 
 /- warning: projective_spectrum.mem_zero_locus -> ProjectiveSpectrum.mem_zeroLocus is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (x : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (s : Set.{u2} A), Iff (Membership.Mem.{u2, u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasMem.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s)) (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) s ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A 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(Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A 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(Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4)))) (ProjectiveSpectrum.asHomogeneousIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x)))
-but is expected to have type
-  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_3 : Algebra.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] (𝒜 : Nat -> (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u2, u1} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 𝒜] (x : ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (s : Set.{u1} A), Iff (Membership.mem.{u1, u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instMembershipSet.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x (ProjectiveSpectrum.zeroLocus.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s)) (HasSubset.Subset.{u1} (Set.{u1} A) (Set.instHasSubsetSet.{u1} A) s (SetLike.coe.{u1, u1} (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (ProjectiveSpectrum.asHomogeneousIdeal.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x)))
+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.mem_zero_locus ProjectiveSpectrum.mem_zeroLocusₓ'. -/
 @[simp]
 theorem mem_zeroLocus (x : ProjectiveSpectrum 𝒜) (s : Set A) :
@@ -136,10 +133,7 @@ def vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) : HomogeneousIdeal 𝒜 :
 #align projective_spectrum.vanishing_ideal ProjectiveSpectrum.vanishingIdeal
 
 /- warning: projective_spectrum.coe_vanishing_ideal -> ProjectiveSpectrum.coe_vanishingIdeal is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))} [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)), Eq.{succ u2} (Set.{u2} A) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t)) (setOf.{u2} A (fun (f : A) => forall (x : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), (Membership.Mem.{u2, u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasMem.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x t) -> (Membership.Mem.{u2, u2} A (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) (SetLike.hasMem.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4)) f (ProjectiveSpectrum.asHomogeneousIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x))))
-but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] {𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))} [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)), Eq.{succ u2} (Set.{u2} A) (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t)) (setOf.{u2} A (fun (f : A) => forall (x : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), (Membership.mem.{u2, u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instMembershipSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x t) -> (Membership.mem.{u2, u2} A (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) 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+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.coe_vanishing_ideal ProjectiveSpectrum.coe_vanishingIdealₓ'. -/
 theorem coe_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
     (vanishingIdeal t : Set A) =
@@ -153,10 +147,7 @@ theorem coe_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
 #align projective_spectrum.coe_vanishing_ideal ProjectiveSpectrum.coe_vanishingIdeal
 
 /- warning: projective_spectrum.mem_vanishing_ideal -> ProjectiveSpectrum.mem_vanishingIdeal is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.mem_vanishing_ideal ProjectiveSpectrum.mem_vanishingIdealₓ'. -/
 theorem mem_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) (f : A) :
     f ∈ vanishingIdeal t ↔ ∀ x : ProjectiveSpectrum 𝒜, x ∈ t → f ∈ x.asHomogeneousIdeal := by
@@ -204,10 +195,7 @@ theorem gc_ideal :
 #align projective_spectrum.gc_ideal ProjectiveSpectrum.gc_ideal
 
 /- warning: projective_spectrum.gc_set -> ProjectiveSpectrum.gc_set is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.gc_set ProjectiveSpectrum.gc_setₓ'. -/
 /-- `zero_locus` and `vanishing_ideal` form a galois connection. -/
 theorem gc_set :
@@ -219,10 +207,7 @@ theorem gc_set :
 #align projective_spectrum.gc_set ProjectiveSpectrum.gc_set
 
 /- warning: projective_spectrum.gc_homogeneous_ideal -> ProjectiveSpectrum.gc_homogeneousIdeal is a dubious translation:
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u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) => ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t)
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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) => ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A 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ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t)
+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.gc_homogeneous_ideal ProjectiveSpectrum.gc_homogeneousIdealₓ'. -/
 theorem gc_homogeneousIdeal :
     @GaloisConnection (HomogeneousIdeal 𝒜) (Set (ProjectiveSpectrum 𝒜))ᵒᵈ _ _
@@ -233,10 +218,7 @@ theorem gc_homogeneousIdeal :
 #align projective_spectrum.gc_homogeneous_ideal ProjectiveSpectrum.gc_homogeneousIdeal
 
 /- warning: projective_spectrum.subset_zero_locus_iff_subset_vanishing_ideal -> ProjectiveSpectrum.subset_zeroLocus_iff_subset_vanishingIdeal is a dubious translation:
-lean 3 declaration is
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+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.subset_zero_locus_iff_subset_vanishing_ideal ProjectiveSpectrum.subset_zeroLocus_iff_subset_vanishingIdealₓ'. -/
 theorem subset_zeroLocus_iff_subset_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) (s : Set A) :
     t ⊆ zeroLocus 𝒜 s ↔ s ⊆ vanishingIdeal t :=
@@ -244,10 +226,7 @@ theorem subset_zeroLocus_iff_subset_vanishingIdeal (t : Set (ProjectiveSpectrum
 #align projective_spectrum.subset_zero_locus_iff_subset_vanishing_ideal ProjectiveSpectrum.subset_zeroLocus_iff_subset_vanishingIdeal
 
 /- warning: projective_spectrum.subset_vanishing_ideal_zero_locus -> ProjectiveSpectrum.subset_vanishingIdeal_zeroLocus is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.subset_vanishing_ideal_zero_locus ProjectiveSpectrum.subset_vanishingIdeal_zeroLocusₓ'. -/
 theorem subset_vanishingIdeal_zeroLocus (s : Set A) : s ⊆ vanishingIdeal (zeroLocus 𝒜 s) :=
   (gc_set _).le_u_l s
@@ -265,10 +244,7 @@ theorem ideal_le_vanishingIdeal_zeroLocus (I : Ideal A) :
 #align projective_spectrum.ideal_le_vanishing_ideal_zero_locus ProjectiveSpectrum.ideal_le_vanishingIdeal_zeroLocus
 
 /- warning: projective_spectrum.homogeneous_ideal_le_vanishing_ideal_zero_locus -> ProjectiveSpectrum.homogeneousIdeal_le_vanishingIdeal_zeroLocus is a dubious translation:
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(CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) I)))
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-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (I : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A 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(NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (Preorder.toLE.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (PartialOrder.toPreorder.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A 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(NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A 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+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.homogeneous_ideal_le_vanishing_ideal_zero_locus ProjectiveSpectrum.homogeneousIdeal_le_vanishingIdeal_zeroLocusₓ'. -/
 theorem homogeneousIdeal_le_vanishingIdeal_zeroLocus (I : HomogeneousIdeal 𝒜) :
     I ≤ vanishingIdeal (zeroLocus 𝒜 I) :=
@@ -276,10 +252,7 @@ theorem homogeneousIdeal_le_vanishingIdeal_zeroLocus (I : HomogeneousIdeal 𝒜)
 #align projective_spectrum.homogeneous_ideal_le_vanishing_ideal_zero_locus ProjectiveSpectrum.homogeneousIdeal_le_vanishingIdeal_zeroLocus
 
 /- warning: projective_spectrum.subset_zero_locus_vanishing_ideal -> ProjectiveSpectrum.subset_zeroLocus_vanishingIdeal is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.subset_zero_locus_vanishing_ideal ProjectiveSpectrum.subset_zeroLocus_vanishingIdealₓ'. -/
 theorem subset_zeroLocus_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
     t ⊆ zeroLocus 𝒜 (vanishingIdeal t) :=
@@ -308,10 +281,7 @@ theorem zeroLocus_anti_mono_ideal {s t : Ideal A} (h : s ≤ t) :
 #align projective_spectrum.zero_locus_anti_mono_ideal ProjectiveSpectrum.zeroLocus_anti_mono_ideal
 
 /- warning: projective_spectrum.zero_locus_anti_mono_homogeneous_ideal -> ProjectiveSpectrum.zeroLocus_anti_mono_homogeneousIdeal is a dubious translation:
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-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] {s : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A 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Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4} {t : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4}, (LE.le.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Preorder.toHasLe.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (PartialOrder.toPreorder.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.partialOrder.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4))) s t) -> (HasSubset.Subset.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasSubset.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) t)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A 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(AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) s)))
-but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] {s : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4} {t : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4}, (LE.le.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (Preorder.toLE.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (PartialOrder.toPreorder.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A 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s)))
+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_anti_mono_homogeneous_ideal ProjectiveSpectrum.zeroLocus_anti_mono_homogeneousIdealₓ'. -/
 theorem zeroLocus_anti_mono_homogeneousIdeal {s t : HomogeneousIdeal 𝒜} (h : s ≤ t) :
     zeroLocus 𝒜 (t : Set A) ⊆ zeroLocus 𝒜 (s : Set A) :=
@@ -319,10 +289,7 @@ theorem zeroLocus_anti_mono_homogeneousIdeal {s t : HomogeneousIdeal 𝒜} (h :
 #align projective_spectrum.zero_locus_anti_mono_homogeneous_ideal ProjectiveSpectrum.zeroLocus_anti_mono_homogeneousIdeal
 
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_inst_4)}, (HasSubset.Subset.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instHasSubsetSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) s t) -> (LE.le.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A 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(Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (PartialOrder.toPreorder.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.instPartialOrderHomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4))) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s))
+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.vanishing_ideal_anti_mono ProjectiveSpectrum.vanishingIdeal_anti_monoₓ'. -/
 theorem vanishingIdeal_anti_mono {s t : Set (ProjectiveSpectrum 𝒜)} (h : s ⊆ t) :
     vanishingIdeal t ≤ vanishingIdeal s :=
@@ -418,10 +385,7 @@ theorem zeroLocus_sup_ideal (I J : Ideal A) :
 #align projective_spectrum.zero_locus_sup_ideal ProjectiveSpectrum.zeroLocus_sup_ideal
 
 /- warning: projective_spectrum.zero_locus_sup_homogeneous_ideal -> ProjectiveSpectrum.zeroLocus_sup_homogeneousIdeal is a dubious translation:
-lean 3 declaration is
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(Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat 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(NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) (Sup.sup.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.hasSup.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4) I J))) (Inter.inter.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasInter.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) I)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) J)))
-but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (I : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (J : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (Sup.sup.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.instSupHomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4) I J))) (Inter.inter.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instInterSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 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+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_sup_homogeneous_ideal ProjectiveSpectrum.zeroLocus_sup_homogeneousIdealₓ'. -/
 theorem zeroLocus_sup_homogeneousIdeal (I J : HomogeneousIdeal 𝒜) :
     zeroLocus 𝒜 ((I ⊔ J : HomogeneousIdeal 𝒜) : Set A) = zeroLocus _ I ∩ zeroLocus _ J :=
@@ -439,10 +403,7 @@ theorem zeroLocus_union (s s' : Set A) : zeroLocus 𝒜 (s ∪ s') = zeroLocus _
 #align projective_spectrum.zero_locus_union ProjectiveSpectrum.zeroLocus_union
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.vanishing_ideal_union ProjectiveSpectrum.vanishingIdeal_unionₓ'. -/
 theorem vanishingIdeal_union (t t' : Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (t ∪ t') = vanishingIdeal t ⊓ vanishingIdeal t' := by
@@ -461,10 +422,7 @@ theorem zeroLocus_iSup_ideal {γ : Sort _} (I : γ → Ideal A) :
 #align projective_spectrum.zero_locus_supr_ideal ProjectiveSpectrum.zeroLocus_iSup_ideal
 
 /- warning: projective_spectrum.zero_locus_supr_homogeneous_ideal -> ProjectiveSpectrum.zeroLocus_iSup_homogeneousIdeal is a dubious translation:
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-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] {γ : Sort.{u3}} (I : γ -> (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A 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(CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) (iSup.{u2, u3} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.hasSup.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4) γ (fun (i : γ) => I i)))) (Set.iInter.{u2, u3} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) γ (fun (i : γ) => ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} 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-but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] {γ : Sort.{u3}} (I : γ -> (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) 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A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A 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(CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (iSup.{u2, u3} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) 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(CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4) γ (fun (i : γ) => I i)))) (Set.iInter.{u2, u3} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) γ (fun (i : γ) => ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (I i))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_supr_homogeneous_ideal ProjectiveSpectrum.zeroLocus_iSup_homogeneousIdealₓ'. -/
 theorem zeroLocus_iSup_homogeneousIdeal {γ : Sort _} (I : γ → HomogeneousIdeal 𝒜) :
     zeroLocus _ ((⨆ i, I i : HomogeneousIdeal 𝒜) : Set A) = ⋂ i, zeroLocus 𝒜 (I i) :=
@@ -493,10 +451,7 @@ theorem zeroLocus_bUnion (s : Set (Set A)) :
 #align projective_spectrum.zero_locus_bUnion ProjectiveSpectrum.zeroLocus_bUnion
 
 /- warning: projective_spectrum.vanishing_ideal_Union -> ProjectiveSpectrum.vanishingIdeal_iUnion is a dubious translation:
-lean 3 declaration is
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u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4) γ (fun (i : γ) => ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (t i)))
+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.vanishing_ideal_Union ProjectiveSpectrum.vanishingIdeal_iUnionₓ'. -/
 theorem vanishingIdeal_iUnion {γ : Sort _} (t : γ → Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (⋃ i, t i) = ⨅ i, vanishingIdeal (t i) :=
@@ -540,10 +495,7 @@ theorem zeroLocus_mul_ideal (I J : Ideal A) :
 #align projective_spectrum.zero_locus_mul_ideal ProjectiveSpectrum.zeroLocus_mul_ideal
 
 /- warning: projective_spectrum.zero_locus_mul_homogeneous_ideal -> ProjectiveSpectrum.zeroLocus_mul_homogeneousIdeal is a dubious translation:
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(NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) (HMul.hMul.{u2, u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (instHMul.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.hasMul.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (CommRing.toCommSemiring.{u2} A _inst_2) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4)) I J))) (Union.union.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasUnion.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A 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(Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) I)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} 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(SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) J)))
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-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (I : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (J : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HMul.hMul.{u2, u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (instHMul.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (instMulHomogeneousIdealToSemiring.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommRing.toCommSemiring.{u2} A _inst_2) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4)) I J))) (Union.union.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instUnionSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) I)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) J)))
+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_mul_homogeneous_ideal ProjectiveSpectrum.zeroLocus_mul_homogeneousIdealₓ'. -/
 theorem zeroLocus_mul_homogeneousIdeal (I J : HomogeneousIdeal 𝒜) :
     zeroLocus 𝒜 ((I * J : HomogeneousIdeal 𝒜) : Set A) = zeroLocus 𝒜 I ∪ zeroLocus 𝒜 J :=
@@ -574,10 +526,7 @@ theorem zeroLocus_singleton_pow (f : A) (n : ℕ) (hn : 0 < n) :
 #align projective_spectrum.zero_locus_singleton_pow ProjectiveSpectrum.zeroLocus_singleton_pow
 
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u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t')) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Inter.inter.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instInterSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) t t'))
+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.sup_vanishing_ideal_le ProjectiveSpectrum.sup_vanishingIdeal_leₓ'. -/
 theorem sup_vanishingIdeal_le (t t' : Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal t ⊔ vanishingIdeal t' ≤ vanishingIdeal (t ∩ t') :=
@@ -591,10 +540,7 @@ theorem sup_vanishingIdeal_le (t t' : Set (ProjectiveSpectrum 𝒜)) :
 #align projective_spectrum.sup_vanishing_ideal_le ProjectiveSpectrum.sup_vanishingIdeal_le
 
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(fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4)) f (ProjectiveSpectrum.asHomogeneousIdeal.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 I)))
+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.mem_compl_zero_locus_iff_not_mem ProjectiveSpectrum.mem_compl_zeroLocus_iff_not_memₓ'. -/
 theorem mem_compl_zeroLocus_iff_not_mem {f : A} {I : ProjectiveSpectrum 𝒜} :
     I ∈ (zeroLocus 𝒜 {f} : Set (ProjectiveSpectrum 𝒜))ᶜ ↔ f ∉ I.asHomogeneousIdeal := by
@@ -667,10 +613,7 @@ theorem isClosed_zeroLocus (s : Set A) : IsClosed (zeroLocus 𝒜 s) :=
 #align projective_spectrum.is_closed_zero_locus ProjectiveSpectrum.isClosed_zeroLocus
 
 /- warning: projective_spectrum.zero_locus_vanishing_ideal_eq_closure -> ProjectiveSpectrum.zeroLocus_vanishingIdeal_eq_closure is a dubious translation:
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_inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 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_inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t))) (closure.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) t)
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+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_vanishing_ideal_eq_closure ProjectiveSpectrum.zeroLocus_vanishingIdeal_eq_closureₓ'. -/
 theorem zeroLocus_vanishingIdeal_eq_closure (t : Set (ProjectiveSpectrum 𝒜)) :
     zeroLocus 𝒜 (vanishingIdeal t : Set A) = closure t :=
@@ -716,10 +659,7 @@ def basicOpen (r : A) : TopologicalSpace.Opens (ProjectiveSpectrum 𝒜)
 #align projective_spectrum.basic_open ProjectiveSpectrum.basicOpen
 
 /- warning: projective_spectrum.mem_basic_open -> ProjectiveSpectrum.mem_basicOpen is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.mem_basic_open ProjectiveSpectrum.mem_basicOpenₓ'. -/
 @[simp]
 theorem mem_basicOpen (f : A) (x : ProjectiveSpectrum 𝒜) :
@@ -728,10 +668,7 @@ theorem mem_basicOpen (f : A) (x : ProjectiveSpectrum 𝒜) :
 #align projective_spectrum.mem_basic_open ProjectiveSpectrum.mem_basicOpen
 
 /- warning: projective_spectrum.mem_coe_basic_open -> ProjectiveSpectrum.mem_coe_basicOpen is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.mem_coe_basic_open ProjectiveSpectrum.mem_coe_basicOpenₓ'. -/
 theorem mem_coe_basicOpen (f : A) (x : ProjectiveSpectrum 𝒜) :
     x ∈ (↑(basicOpen 𝒜 f) : Set (ProjectiveSpectrum 𝒜)) ↔ f ∉ x.asHomogeneousIdeal :=
@@ -888,10 +825,7 @@ instance : PartialOrder (ProjectiveSpectrum 𝒜) :=
   PartialOrder.lift asHomogeneousIdeal fun ⟨_, _, _⟩ ⟨_, _, _⟩ => mk.inj_eq.mpr
 
 /- warning: projective_spectrum.as_ideal_le_as_ideal -> ProjectiveSpectrum.as_ideal_le_as_ideal is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.as_ideal_le_as_ideal ProjectiveSpectrum.as_ideal_le_as_idealₓ'. -/
 @[simp]
 theorem as_ideal_le_as_ideal (x y : ProjectiveSpectrum 𝒜) :
@@ -900,10 +834,7 @@ theorem as_ideal_le_as_ideal (x y : ProjectiveSpectrum 𝒜) :
 #align projective_spectrum.as_ideal_le_as_ideal ProjectiveSpectrum.as_ideal_le_as_ideal
 
 /- warning: projective_spectrum.as_ideal_lt_as_ideal -> ProjectiveSpectrum.as_ideal_lt_as_ideal is a dubious translation:
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-  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_3 : Algebra.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] (𝒜 : Nat -> (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u2, u1} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 𝒜] (x : ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (y : ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), Iff (LT.lt.{u1} (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (Preorder.toLT.{u1} (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (PartialOrder.toPreorder.{u1} (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.instPartialOrderHomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 _inst_4))) (ProjectiveSpectrum.asHomogeneousIdeal.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x) (ProjectiveSpectrum.asHomogeneousIdeal.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 y)) (LT.lt.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Preorder.toLT.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (PartialOrder.toPreorder.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.instPartialOrderProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) x y)
+<too large>
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.as_ideal_lt_as_ideal ProjectiveSpectrum.as_ideal_lt_as_idealₓ'. -/
 @[simp]
 theorem as_ideal_lt_as_ideal (x y : ProjectiveSpectrum 𝒜) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/8d33f09cd7089ecf074b4791907588245aec5d1b

@@ -831,7 +831,7 @@ theorem basicOpen_pow (f : A) (n : ℕ) (hn : 0 < n) : basicOpen 𝒜 (f ^ n) =
 lean 3 declaration is
   forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (f : A), Eq.{succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 f) (iSup.{u2, 1} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (TopologicalSpace.Opens.completeLattice.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)))) Nat (fun (i : Nat) => ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (coeFn.{succ u2, succ u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (fun (_x : LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) => A -> A) (LinearMap.hasCoeToFun.{u1, u1, u2, u2} R R A A (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (GradedAlgebra.proj.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜 _inst_4 i) f)))
 but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (f : A), Eq.{succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 f) (iSup.{u2, 1} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ConditionallyCompleteLattice.toSupSet.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)))) Nat (fun (i : Nat) => ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (FunLike.coe.{succ u2, succ u2, succ u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : A) => A) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u2} R R A A (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (GradedAlgebra.proj.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜 _inst_4 i) f)))
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (f : A), Eq.{succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 f) (iSup.{u2, 1} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ConditionallyCompleteLattice.toSupSet.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)))) Nat (fun (i : Nat) => ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (FunLike.coe.{succ u2, succ u2, succ u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : A) => A) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u2} R R A A (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (GradedAlgebra.proj.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜 _inst_4 i) f)))
 Case conversion may be inaccurate. Consider using '#align projective_spectrum.basic_open_eq_union_of_projection ProjectiveSpectrum.basicOpen_eq_union_of_projectionₓ'. -/
 theorem basicOpen_eq_union_of_projection (f : A) :
     basicOpen 𝒜 f = ⨆ i : ℕ, basicOpen 𝒜 (GradedAlgebra.proj 𝒜 i f) :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jujian Zhang, Johan Commelin
 
 ! This file was ported from Lean 3 source module algebraic_geometry.projective_spectrum.topology
-! leanprover-community/mathlib commit d39590fc8728fbf6743249802486f8c91ffe07bc
+! leanprover-community/mathlib commit 4280f5f32e16755ec7985ce11e189b6cd6ff6735
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.Topology.Sets.Opens
 /-!
 # Projective spectrum of a graded ring
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 The projective spectrum of a graded commutative ring is the subtype of all homogenous ideals that
 are prime and do not contain the irrelevant ideal.
 It is naturally endowed with a topology: the Zariski topology.

mathlib commit https://github.com/leanprover-community/mathlib/commit/33c67ae661dd8988516ff7f247b0be3018cdd952

@@ -52,6 +52,12 @@ variable [CommSemiring R] [CommRing A] [Algebra R A]
 
 variable (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜]
 
+/- warning: projective_spectrum -> ProjectiveSpectrum is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜], Type.{u2}
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜], Type.{u2}
+Case conversion may be inaccurate. Consider using '#align projective_spectrum ProjectiveSpectrumₓ'. -/
 /-- The projective spectrum of a graded commutative ring is the subtype of all homogenous ideals
 that are prime and do not contain the irrelevant ideal. -/
 @[ext, nolint has_nonempty_instance]
@@ -65,6 +71,12 @@ attribute [instance] ProjectiveSpectrum.isPrime
 
 namespace ProjectiveSpectrum
 
+/- warning: projective_spectrum.zero_locus -> ProjectiveSpectrum.zeroLocus is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜], (Set.{u2} A) -> (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜], (Set.{u2} A) -> (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus ProjectiveSpectrum.zeroLocusₓ'. -/
 /-- The zero locus of a set `s` of elements of a commutative ring `A` is the set of all relevant
 homogeneous prime ideals of the ring that contain the set `s`.
 
@@ -76,12 +88,24 @@ def zeroLocus (s : Set A) : Set (ProjectiveSpectrum 𝒜) :=
   { x | s ⊆ x.asHomogeneousIdeal }
 #align projective_spectrum.zero_locus ProjectiveSpectrum.zeroLocus
 
+/- warning: projective_spectrum.mem_zero_locus -> ProjectiveSpectrum.mem_zeroLocus is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (x : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (s : Set.{u2} A), Iff (Membership.Mem.{u2, u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasMem.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s)) (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) s ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A 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(CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4)))) (ProjectiveSpectrum.asHomogeneousIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x)))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_3 : Algebra.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] (𝒜 : Nat -> (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u2, u1} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 𝒜] (x : ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (s : Set.{u1} A), Iff (Membership.mem.{u1, u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instMembershipSet.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x (ProjectiveSpectrum.zeroLocus.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s)) (HasSubset.Subset.{u1} (Set.{u1} A) (Set.instHasSubsetSet.{u1} A) s (SetLike.coe.{u1, u1} (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (ProjectiveSpectrum.asHomogeneousIdeal.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.mem_zero_locus ProjectiveSpectrum.mem_zeroLocusₓ'. -/
 @[simp]
 theorem mem_zeroLocus (x : ProjectiveSpectrum 𝒜) (s : Set A) :
     x ∈ zeroLocus 𝒜 s ↔ s ⊆ x.asHomogeneousIdeal :=
   Iff.rfl
 #align projective_spectrum.mem_zero_locus ProjectiveSpectrum.mem_zeroLocus
 
+/- warning: projective_spectrum.zero_locus_span -> ProjectiveSpectrum.zeroLocus_span is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (s : Set.{u2} A), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))))) (Ideal.span.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) s))) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s)
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (s : Set.{u2} A), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (Ideal.span.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) s))) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s)
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_span ProjectiveSpectrum.zeroLocus_spanₓ'. -/
 @[simp]
 theorem zeroLocus_span (s : Set A) : zeroLocus 𝒜 (Ideal.span s) = zeroLocus 𝒜 s :=
   by
@@ -91,6 +115,12 @@ theorem zeroLocus_span (s : Set A) : zeroLocus 𝒜 (Ideal.span s) = zeroLocus 
 
 variable {𝒜}
 
+/- warning: projective_spectrum.vanishing_ideal -> ProjectiveSpectrum.vanishingIdeal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))} [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜], (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) -> (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] {𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))} [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜], (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) -> (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4)
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.vanishing_ideal ProjectiveSpectrum.vanishingIdealₓ'. -/
 /-- The vanishing ideal of a set `t` of points of the projective spectrum of a commutative ring `R`
 is the intersection of all the relevant homogeneous prime ideals in the set `t`.
 
@@ -102,6 +132,12 @@ def vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) : HomogeneousIdeal 𝒜 :
   ⨅ (x : ProjectiveSpectrum 𝒜) (h : x ∈ t), x.asHomogeneousIdeal
 #align projective_spectrum.vanishing_ideal ProjectiveSpectrum.vanishingIdeal
 
+/- warning: projective_spectrum.coe_vanishing_ideal -> ProjectiveSpectrum.coe_vanishingIdeal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))} [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)), Eq.{succ u2} (Set.{u2} A) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) 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Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t)) (setOf.{u2} A (fun (f : A) => forall (x : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), (Membership.Mem.{u2, u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasMem.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x t) -> (Membership.Mem.{u2, u2} A (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) (SetLike.hasMem.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4)) f (ProjectiveSpectrum.asHomogeneousIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x))))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] {𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))} [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)), Eq.{succ u2} (Set.{u2} A) (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t)) (setOf.{u2} A (fun (f : A) => forall (x : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), (Membership.mem.{u2, u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instMembershipSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x t) -> (Membership.mem.{u2, u2} A (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (SetLike.instMembership.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4)) f (ProjectiveSpectrum.asHomogeneousIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x))))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.coe_vanishing_ideal ProjectiveSpectrum.coe_vanishingIdealₓ'. -/
 theorem coe_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
     (vanishingIdeal t : Set A) =
       { f | ∀ x : ProjectiveSpectrum 𝒜, x ∈ t → f ∈ x.asHomogeneousIdeal } :=
@@ -113,17 +149,35 @@ theorem coe_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
   rw [HomogeneousIdeal.toIdeal_iInf, Submodule.mem_iInf, HomogeneousIdeal.mem_iff]
 #align projective_spectrum.coe_vanishing_ideal ProjectiveSpectrum.coe_vanishingIdeal
 
+/- warning: projective_spectrum.mem_vanishing_ideal -> ProjectiveSpectrum.mem_vanishingIdeal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))} [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (f : A), Iff (Membership.Mem.{u2, u2} A (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (SetLike.hasMem.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} 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(CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4)) f (ProjectiveSpectrum.asHomogeneousIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] {𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))} [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (f : A), Iff (Membership.mem.{u2, u2} A (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (SetLike.instMembership.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4)) f (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t)) (forall (x : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), (Membership.mem.{u2, u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instMembershipSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x t) -> (Membership.mem.{u2, u2} A (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (SetLike.instMembership.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4)) f (ProjectiveSpectrum.asHomogeneousIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.mem_vanishing_ideal ProjectiveSpectrum.mem_vanishingIdealₓ'. -/
 theorem mem_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) (f : A) :
     f ∈ vanishingIdeal t ↔ ∀ x : ProjectiveSpectrum 𝒜, x ∈ t → f ∈ x.asHomogeneousIdeal := by
   rw [← SetLike.mem_coe, coe_vanishing_ideal, Set.mem_setOf_eq]
 #align projective_spectrum.mem_vanishing_ideal ProjectiveSpectrum.mem_vanishingIdeal
 
+/- warning: projective_spectrum.vanishing_ideal_singleton -> ProjectiveSpectrum.vanishingIdeal_singleton is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))} [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (x : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), Eq.{succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u2, u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasSingleton.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x)) (ProjectiveSpectrum.asHomogeneousIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x)
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_3 : Algebra.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {𝒜 : Nat -> (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3))} [_inst_4 : GradedAlgebra.{0, u2, u1} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 𝒜] (x : ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), Eq.{succ u1} (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (ProjectiveSpectrum.vanishingIdeal.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u1, u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instSingletonSet.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x)) (ProjectiveSpectrum.asHomogeneousIdeal.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x)
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.vanishing_ideal_singleton ProjectiveSpectrum.vanishingIdeal_singletonₓ'. -/
 @[simp]
 theorem vanishingIdeal_singleton (x : ProjectiveSpectrum 𝒜) :
     vanishingIdeal ({x} : Set (ProjectiveSpectrum 𝒜)) = x.asHomogeneousIdeal := by
   simp [vanishing_ideal]
 #align projective_spectrum.vanishing_ideal_singleton ProjectiveSpectrum.vanishingIdeal_singleton
 
+/- warning: projective_spectrum.subset_zero_locus_iff_le_vanishing_ideal -> ProjectiveSpectrum.subset_zeroLocus_iff_le_vanishingIdeal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))} [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (I : Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))), Iff (HasSubset.Subset.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasSubset.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) t (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))))) I))) (LE.le.{u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Preorder.toHasLe.{u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (SetLike.partialOrder.{u2, u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))))) I (HomogeneousIdeal.toIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4 (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] {𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))} [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (I : Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))), Iff (HasSubset.Subset.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instHasSubsetSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) t (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) I))) (LE.le.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Preorder.toLE.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Submodule.completeLattice.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))))) I (HomogeneousIdeal.toIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4 (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.subset_zero_locus_iff_le_vanishing_ideal ProjectiveSpectrum.subset_zeroLocus_iff_le_vanishingIdealₓ'. -/
 theorem subset_zeroLocus_iff_le_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) (I : Ideal A) :
     t ⊆ zeroLocus 𝒜 I ↔ I ≤ (vanishingIdeal t).toIdeal :=
   ⟨fun h f k => (mem_vanishingIdeal _ _).mpr fun x j => (mem_zeroLocus _ _ _).mpr (h j) k, fun h =>
@@ -133,6 +187,12 @@ theorem subset_zeroLocus_iff_le_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜
 
 variable (𝒜)
 
+/- warning: projective_spectrum.gc_ideal -> ProjectiveSpectrum.gc_ideal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜], GaloisConnection.{u2, u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (OrderDual.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (SetLike.partialOrder.{u2, u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))) (OrderDual.preorder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (PartialOrder.toPreorder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.completeBooleanAlgebra.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))))))))) (fun (I : Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) => ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))))) I)) (fun (t : OrderDual.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) => HomogeneousIdeal.toIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4 (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜], GaloisConnection.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (OrderDual.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Submodule.completeLattice.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))))) (OrderDual.preorder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (PartialOrder.toPreorder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instCompleteBooleanAlgebraSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))))))))) (fun (I : Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) => ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) I)) (fun (t : OrderDual.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) => HomogeneousIdeal.toIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4 (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.gc_ideal ProjectiveSpectrum.gc_idealₓ'. -/
 /-- `zero_locus` and `vanishing_ideal` form a galois connection. -/
 theorem gc_ideal :
     @GaloisConnection (Ideal A) (Set (ProjectiveSpectrum 𝒜))ᵒᵈ _ _ (fun I => zeroLocus 𝒜 I) fun t =>
@@ -140,6 +200,12 @@ theorem gc_ideal :
   fun I t => subset_zeroLocus_iff_le_vanishingIdeal t I
 #align projective_spectrum.gc_ideal ProjectiveSpectrum.gc_ideal
 
+/- warning: projective_spectrum.gc_set -> ProjectiveSpectrum.gc_set is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜], GaloisConnection.{u2, u2} (Set.{u2} A) (OrderDual.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) (PartialOrder.toPreorder.{u2} (Set.{u2} A) (CompleteSemilatticeInf.toPartialOrder.{u2} (Set.{u2} A) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Set.{u2} A) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} A) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} A) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} A) (Set.completeBooleanAlgebra.{u2} A))))))) (OrderDual.preorder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (PartialOrder.toPreorder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) 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(CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜], GaloisConnection.{u2, u2} (Set.{u2} A) (OrderDual.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) (PartialOrder.toPreorder.{u2} (Set.{u2} A) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} A) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} A) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} A) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} A) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} A) (Set.instCompleteBooleanAlgebraSet.{u2} A))))))) (OrderDual.preorder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (PartialOrder.toPreorder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instCompleteBooleanAlgebraSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))))))))) (fun (s : Set.{u2} A) => ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s) (fun (t : OrderDual.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) => SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.gc_set ProjectiveSpectrum.gc_setₓ'. -/
 /-- `zero_locus` and `vanishing_ideal` form a galois connection. -/
 theorem gc_set :
     @GaloisConnection (Set A) (Set (ProjectiveSpectrum 𝒜))ᵒᵈ _ _ (fun s => zeroLocus 𝒜 s) fun t =>
@@ -149,6 +215,12 @@ theorem gc_set :
   simpa [zero_locus_span, Function.comp] using GaloisConnection.compose ideal_gc (gc_ideal 𝒜)
 #align projective_spectrum.gc_set ProjectiveSpectrum.gc_set
 
+/- warning: projective_spectrum.gc_homogeneous_ideal -> ProjectiveSpectrum.gc_homogeneousIdeal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜], GaloisConnection.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A 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(ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.completeBooleanAlgebra.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))))))))) (fun (I : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) => ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) I)) (fun (t : OrderDual.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) => ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t)
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜], GaloisConnection.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (OrderDual.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) (PartialOrder.toPreorder.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.instPartialOrderHomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4)) (OrderDual.preorder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (PartialOrder.toPreorder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instCompleteBooleanAlgebraSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))))))))) (fun (I : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) => ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) 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(CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) I)) (fun (t : OrderDual.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) => ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t)
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.gc_homogeneous_ideal ProjectiveSpectrum.gc_homogeneousIdealₓ'. -/
 theorem gc_homogeneousIdeal :
     @GaloisConnection (HomogeneousIdeal 𝒜) (Set (ProjectiveSpectrum 𝒜))ᵒᵈ _ _
       (fun I => zeroLocus 𝒜 I) fun t => vanishingIdeal t :=
@@ -157,133 +229,295 @@ theorem gc_homogeneousIdeal :
     subset_zero_locus_iff_le_vanishing_ideal t I.to_ideal
 #align projective_spectrum.gc_homogeneous_ideal ProjectiveSpectrum.gc_homogeneousIdeal
 
+/- warning: projective_spectrum.subset_zero_locus_iff_subset_vanishing_ideal -> ProjectiveSpectrum.subset_zeroLocus_iff_subset_vanishingIdeal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (s : Set.{u2} A), Iff (HasSubset.Subset.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasSubset.{u2} 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+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (s : Set.{u2} A), Iff (HasSubset.Subset.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instHasSubsetSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) t (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s)) (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) s (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.subset_zero_locus_iff_subset_vanishing_ideal ProjectiveSpectrum.subset_zeroLocus_iff_subset_vanishingIdealₓ'. -/
 theorem subset_zeroLocus_iff_subset_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) (s : Set A) :
     t ⊆ zeroLocus 𝒜 s ↔ s ⊆ vanishingIdeal t :=
   (gc_set _) s t
 #align projective_spectrum.subset_zero_locus_iff_subset_vanishing_ideal ProjectiveSpectrum.subset_zeroLocus_iff_subset_vanishingIdeal
 
+/- warning: projective_spectrum.subset_vanishing_ideal_zero_locus -> ProjectiveSpectrum.subset_vanishingIdeal_zeroLocus is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (s : Set.{u2} A), HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) s ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (s : Set.{u2} A), HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) s (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.subset_vanishing_ideal_zero_locus ProjectiveSpectrum.subset_vanishingIdeal_zeroLocusₓ'. -/
 theorem subset_vanishingIdeal_zeroLocus (s : Set A) : s ⊆ vanishingIdeal (zeroLocus 𝒜 s) :=
   (gc_set _).le_u_l s
 #align projective_spectrum.subset_vanishing_ideal_zero_locus ProjectiveSpectrum.subset_vanishingIdeal_zeroLocus
 
+/- warning: projective_spectrum.ideal_le_vanishing_ideal_zero_locus -> ProjectiveSpectrum.ideal_le_vanishingIdeal_zeroLocus is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (I : Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))), LE.le.{u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Preorder.toHasLe.{u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (SetLike.partialOrder.{u2, u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))))) I (HomogeneousIdeal.toIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4 (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))))) I))))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (I : Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))), LE.le.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Preorder.toLE.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Submodule.completeLattice.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))))) I (HomogeneousIdeal.toIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4 (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) I))))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.ideal_le_vanishing_ideal_zero_locus ProjectiveSpectrum.ideal_le_vanishingIdeal_zeroLocusₓ'. -/
 theorem ideal_le_vanishingIdeal_zeroLocus (I : Ideal A) :
     I ≤ (vanishingIdeal (zeroLocus 𝒜 I)).toIdeal :=
   (gc_ideal _).le_u_l I
 #align projective_spectrum.ideal_le_vanishing_ideal_zero_locus ProjectiveSpectrum.ideal_le_vanishingIdeal_zeroLocus
 
+/- warning: projective_spectrum.homogeneous_ideal_le_vanishing_ideal_zero_locus -> ProjectiveSpectrum.homogeneousIdeal_le_vanishingIdeal_zeroLocus is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (I : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4), LE.le.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Preorder.toHasLe.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (PartialOrder.toPreorder.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.partialOrder.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4))) I (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) I)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (I : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4), LE.le.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (Preorder.toLE.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (PartialOrder.toPreorder.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.instPartialOrderHomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4))) I (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) I)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.homogeneous_ideal_le_vanishing_ideal_zero_locus ProjectiveSpectrum.homogeneousIdeal_le_vanishingIdeal_zeroLocusₓ'. -/
 theorem homogeneousIdeal_le_vanishingIdeal_zeroLocus (I : HomogeneousIdeal 𝒜) :
     I ≤ vanishingIdeal (zeroLocus 𝒜 I) :=
   (gc_homogeneousIdeal _).le_u_l I
 #align projective_spectrum.homogeneous_ideal_le_vanishing_ideal_zero_locus ProjectiveSpectrum.homogeneousIdeal_le_vanishingIdeal_zeroLocus
 
+/- warning: projective_spectrum.subset_zero_locus_vanishing_ideal -> ProjectiveSpectrum.subset_zeroLocus_vanishingIdeal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)), HasSubset.Subset.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasSubset.{u2} (ProjectiveSpectrum.{u1, u2} R A 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+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)), HasSubset.Subset.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instHasSubsetSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) t (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.subset_zero_locus_vanishing_ideal ProjectiveSpectrum.subset_zeroLocus_vanishingIdealₓ'. -/
 theorem subset_zeroLocus_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
     t ⊆ zeroLocus 𝒜 (vanishingIdeal t) :=
   (gc_ideal _).l_u_le t
 #align projective_spectrum.subset_zero_locus_vanishing_ideal ProjectiveSpectrum.subset_zeroLocus_vanishingIdeal
 
+/- warning: projective_spectrum.zero_locus_anti_mono -> ProjectiveSpectrum.zeroLocus_anti_mono is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] {s : Set.{u2} A} {t : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.hasSubset.{u2} A) s t) -> (HasSubset.Subset.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasSubset.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] {s : Set.{u2} A} {t : Set.{u2} A}, (HasSubset.Subset.{u2} (Set.{u2} A) (Set.instHasSubsetSet.{u2} A) s t) -> (HasSubset.Subset.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instHasSubsetSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_anti_mono ProjectiveSpectrum.zeroLocus_anti_monoₓ'. -/
 theorem zeroLocus_anti_mono {s t : Set A} (h : s ⊆ t) : zeroLocus 𝒜 t ⊆ zeroLocus 𝒜 s :=
   (gc_set _).monotone_l h
 #align projective_spectrum.zero_locus_anti_mono ProjectiveSpectrum.zeroLocus_anti_mono
 
+/- warning: projective_spectrum.zero_locus_anti_mono_ideal -> ProjectiveSpectrum.zeroLocus_anti_mono_ideal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] {s : Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))} {t : Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))}, (LE.le.{u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Preorder.toHasLe.{u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (SetLike.partialOrder.{u2, u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))))) s t) -> (HasSubset.Subset.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasSubset.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))))) t)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))))) s)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] {s : Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))} {t : Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))}, (LE.le.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Preorder.toLE.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Submodule.completeLattice.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))))) s t) -> (HasSubset.Subset.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instHasSubsetSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) t)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) s)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_anti_mono_ideal ProjectiveSpectrum.zeroLocus_anti_mono_idealₓ'. -/
 theorem zeroLocus_anti_mono_ideal {s t : Ideal A} (h : s ≤ t) :
     zeroLocus 𝒜 (t : Set A) ⊆ zeroLocus 𝒜 (s : Set A) :=
   (gc_ideal _).monotone_l h
 #align projective_spectrum.zero_locus_anti_mono_ideal ProjectiveSpectrum.zeroLocus_anti_mono_ideal
 
+/- warning: projective_spectrum.zero_locus_anti_mono_homogeneous_ideal -> ProjectiveSpectrum.zeroLocus_anti_mono_homogeneousIdeal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] {s : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A 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(NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Preorder.toHasLe.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (PartialOrder.toPreorder.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.partialOrder.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4))) s t) -> (HasSubset.Subset.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasSubset.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) t)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) s)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] {s : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4} {t : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4}, (LE.le.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (Preorder.toLE.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (PartialOrder.toPreorder.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.instPartialOrderHomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4))) s t) -> (HasSubset.Subset.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instHasSubsetSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) t)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) s)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_anti_mono_homogeneous_ideal ProjectiveSpectrum.zeroLocus_anti_mono_homogeneousIdealₓ'. -/
 theorem zeroLocus_anti_mono_homogeneousIdeal {s t : HomogeneousIdeal 𝒜} (h : s ≤ t) :
     zeroLocus 𝒜 (t : Set A) ⊆ zeroLocus 𝒜 (s : Set A) :=
   (gc_homogeneousIdeal _).monotone_l h
 #align projective_spectrum.zero_locus_anti_mono_homogeneous_ideal ProjectiveSpectrum.zeroLocus_anti_mono_homogeneousIdeal
 
+/- warning: projective_spectrum.vanishing_ideal_anti_mono -> ProjectiveSpectrum.vanishingIdeal_anti_mono is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] {s : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)} {t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)}, (HasSubset.Subset.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasSubset.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) s t) -> (LE.le.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Preorder.toHasLe.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (PartialOrder.toPreorder.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.partialOrder.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 _inst_4))) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] {s : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)} {t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)}, (HasSubset.Subset.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instHasSubsetSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) s t) -> (LE.le.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (Preorder.toLE.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (PartialOrder.toPreorder.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.instPartialOrderHomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4))) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.vanishing_ideal_anti_mono ProjectiveSpectrum.vanishingIdeal_anti_monoₓ'. -/
 theorem vanishingIdeal_anti_mono {s t : Set (ProjectiveSpectrum 𝒜)} (h : s ⊆ t) :
     vanishingIdeal t ≤ vanishingIdeal s :=
   (gc_ideal _).monotone_u h
 #align projective_spectrum.vanishing_ideal_anti_mono ProjectiveSpectrum.vanishingIdeal_anti_mono
 
+/- warning: projective_spectrum.zero_locus_bot -> ProjectiveSpectrum.zeroLocus_bot is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_bot ProjectiveSpectrum.zeroLocus_botₓ'. -/
 theorem zeroLocus_bot : zeroLocus 𝒜 ((⊥ : Ideal A) : Set A) = Set.univ :=
   (gc_ideal 𝒜).l_bot
 #align projective_spectrum.zero_locus_bot ProjectiveSpectrum.zeroLocus_bot
 
+/- warning: projective_spectrum.zero_locus_singleton_zero -> ProjectiveSpectrum.zeroLocus_singleton_zero is a dubious translation:
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+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜], Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.instSingletonSet.{u2} A) (OfNat.ofNat.{u2} A 0 (Zero.toOfNat0.{u2} A (CommMonoidWithZero.toZero.{u2} A (CommSemiring.toCommMonoidWithZero.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))))) (Set.univ.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_singleton_zero ProjectiveSpectrum.zeroLocus_singleton_zeroₓ'. -/
 @[simp]
 theorem zeroLocus_singleton_zero : zeroLocus 𝒜 ({0} : Set A) = Set.univ :=
   zeroLocus_bot _
 #align projective_spectrum.zero_locus_singleton_zero ProjectiveSpectrum.zeroLocus_singleton_zero
 
+/- warning: projective_spectrum.zero_locus_empty -> ProjectiveSpectrum.zeroLocus_empty is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_empty ProjectiveSpectrum.zeroLocus_emptyₓ'. -/
 @[simp]
 theorem zeroLocus_empty : zeroLocus 𝒜 (∅ : Set A) = Set.univ :=
   (gc_set 𝒜).l_bot
 #align projective_spectrum.zero_locus_empty ProjectiveSpectrum.zeroLocus_empty
 
+/- warning: projective_spectrum.vanishing_ideal_univ -> ProjectiveSpectrum.vanishingIdeal_univ is a dubious translation:
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(CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.instTopHomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.vanishing_ideal_univ ProjectiveSpectrum.vanishingIdeal_univₓ'. -/
 @[simp]
 theorem vanishingIdeal_univ : vanishingIdeal (∅ : Set (ProjectiveSpectrum 𝒜)) = ⊤ := by
   simpa using (gc_ideal _).u_top
 #align projective_spectrum.vanishing_ideal_univ ProjectiveSpectrum.vanishingIdeal_univ
 
+/- warning: projective_spectrum.zero_locus_empty_of_one_mem -> ProjectiveSpectrum.zeroLocus_empty_of_one_mem is a dubious translation:
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_empty_of_one_mem ProjectiveSpectrum.zeroLocus_empty_of_one_memₓ'. -/
 theorem zeroLocus_empty_of_one_mem {s : Set A} (h : (1 : A) ∈ s) : zeroLocus 𝒜 s = ∅ :=
   Set.eq_empty_iff_forall_not_mem.mpr fun x hx =>
     (inferInstance : x.asHomogeneousIdeal.toIdeal.IsPrime).ne_top <|
       x.asHomogeneousIdeal.toIdeal.eq_top_iff_one.mpr <| hx h
 #align projective_spectrum.zero_locus_empty_of_one_mem ProjectiveSpectrum.zeroLocus_empty_of_one_mem
 
+/- warning: projective_spectrum.zero_locus_singleton_one -> ProjectiveSpectrum.zeroLocus_singleton_one is a dubious translation:
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_singleton_one ProjectiveSpectrum.zeroLocus_singleton_oneₓ'. -/
 @[simp]
 theorem zeroLocus_singleton_one : zeroLocus 𝒜 ({1} : Set A) = ∅ :=
   zeroLocus_empty_of_one_mem 𝒜 (Set.mem_singleton (1 : A))
 #align projective_spectrum.zero_locus_singleton_one ProjectiveSpectrum.zeroLocus_singleton_one
 
+/- warning: projective_spectrum.zero_locus_univ -> ProjectiveSpectrum.zeroLocus_univ is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_univ ProjectiveSpectrum.zeroLocus_univₓ'. -/
 @[simp]
 theorem zeroLocus_univ : zeroLocus 𝒜 (Set.univ : Set A) = ∅ :=
   zeroLocus_empty_of_one_mem _ (Set.mem_univ 1)
 #align projective_spectrum.zero_locus_univ ProjectiveSpectrum.zeroLocus_univ
 
+/- warning: projective_spectrum.zero_locus_sup_ideal -> ProjectiveSpectrum.zeroLocus_sup_ideal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (I : Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (J : Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))))) (Sup.sup.{u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (SemilatticeSup.toHasSup.{u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (IdemSemiring.toSemilatticeSup.{u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Submodule.idemSemiring.{u2, u2} A (CommRing.toCommSemiring.{u2} A _inst_2) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Algebra.id.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) I J))) (Inter.inter.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasInter.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))))) I)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))))) J)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (I : Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (J : Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (Sup.sup.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (SemilatticeSup.toSup.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (IdemCommSemiring.toSemilatticeSup.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Ideal.instIdemCommSemiringIdealToSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) I J))) (Inter.inter.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instInterSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) I)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) J)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_sup_ideal ProjectiveSpectrum.zeroLocus_sup_idealₓ'. -/
 theorem zeroLocus_sup_ideal (I J : Ideal A) :
     zeroLocus 𝒜 ((I ⊔ J : Ideal A) : Set A) = zeroLocus _ I ∩ zeroLocus _ J :=
   (gc_ideal 𝒜).l_sup
 #align projective_spectrum.zero_locus_sup_ideal ProjectiveSpectrum.zeroLocus_sup_ideal
 
+/- warning: projective_spectrum.zero_locus_sup_homogeneous_ideal -> ProjectiveSpectrum.zeroLocus_sup_homogeneousIdeal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (I : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A 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(NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A 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(Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) (Sup.sup.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.hasSup.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4) I J))) (Inter.inter.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasInter.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) I)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) J)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (I : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (J : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (Sup.sup.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.instSupHomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4) I J))) (Inter.inter.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instInterSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 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(fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) J)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_sup_homogeneous_ideal ProjectiveSpectrum.zeroLocus_sup_homogeneousIdealₓ'. -/
 theorem zeroLocus_sup_homogeneousIdeal (I J : HomogeneousIdeal 𝒜) :
     zeroLocus 𝒜 ((I ⊔ J : HomogeneousIdeal 𝒜) : Set A) = zeroLocus _ I ∩ zeroLocus _ J :=
   (gc_homogeneousIdeal 𝒜).l_sup
 #align projective_spectrum.zero_locus_sup_homogeneous_ideal ProjectiveSpectrum.zeroLocus_sup_homogeneousIdeal
 
+/- warning: projective_spectrum.zero_locus_union -> ProjectiveSpectrum.zeroLocus_union is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (s : Set.{u2} A) (s' : Set.{u2} A), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Union.union.{u2} (Set.{u2} A) (Set.instUnionSet.{u2} A) s s')) (Inter.inter.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instInterSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s'))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_union ProjectiveSpectrum.zeroLocus_unionₓ'. -/
 theorem zeroLocus_union (s s' : Set A) : zeroLocus 𝒜 (s ∪ s') = zeroLocus _ s ∩ zeroLocus _ s' :=
   (gc_set 𝒜).l_sup
 #align projective_spectrum.zero_locus_union ProjectiveSpectrum.zeroLocus_union
 
+/- warning: projective_spectrum.vanishing_ideal_union -> ProjectiveSpectrum.vanishingIdeal_union is a dubious translation:
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(NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t'))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.vanishing_ideal_union ProjectiveSpectrum.vanishingIdeal_unionₓ'. -/
 theorem vanishingIdeal_union (t t' : Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (t ∪ t') = vanishingIdeal t ⊓ vanishingIdeal t' := by
   ext1 <;> convert(gc_ideal 𝒜).u_inf
 #align projective_spectrum.vanishing_ideal_union ProjectiveSpectrum.vanishingIdeal_union
 
+/- warning: projective_spectrum.zero_locus_supr_ideal -> ProjectiveSpectrum.zeroLocus_iSup_ideal is a dubious translation:
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+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] {γ : Sort.{u3}} (I : γ -> (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))))) (iSup.{u2, u3} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (ConditionallyCompleteLattice.toHasSup.{u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Submodule.completeLattice.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))) γ (fun (i : γ) => I i)))) (Set.iInter.{u2, u3} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) γ (fun (i : γ) => ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))))) (I i))))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] {γ : Sort.{u3}} (I : γ -> (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (iSup.{u2, u3} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (ConditionallyCompleteLattice.toSupSet.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Submodule.completeLattice.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) γ (fun (i : γ) => I i)))) (Set.iInter.{u2, u3} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) γ (fun (i : γ) => ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (I i))))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_supr_ideal ProjectiveSpectrum.zeroLocus_iSup_idealₓ'. -/
 theorem zeroLocus_iSup_ideal {γ : Sort _} (I : γ → Ideal A) :
     zeroLocus _ ((⨆ i, I i : Ideal A) : Set A) = ⋂ i, zeroLocus 𝒜 (I i) :=
   (gc_ideal 𝒜).l_iSup
 #align projective_spectrum.zero_locus_supr_ideal ProjectiveSpectrum.zeroLocus_iSup_ideal
 
+/- warning: projective_spectrum.zero_locus_supr_homogeneous_ideal -> ProjectiveSpectrum.zeroLocus_iSup_homogeneousIdeal is a dubious translation:
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(CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A 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_inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) (iSup.{u2, u3} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.hasSup.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4) γ (fun (i : γ) => I i)))) (Set.iInter.{u2, u3} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) γ (fun (i : γ) => ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) (I i))))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] {γ : Sort.{u3}} (I : γ -> (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4)), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 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(CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (iSup.{u2, u3} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) 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(fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (I i))))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_supr_homogeneous_ideal ProjectiveSpectrum.zeroLocus_iSup_homogeneousIdealₓ'. -/
 theorem zeroLocus_iSup_homogeneousIdeal {γ : Sort _} (I : γ → HomogeneousIdeal 𝒜) :
     zeroLocus _ ((⨆ i, I i : HomogeneousIdeal 𝒜) : Set A) = ⋂ i, zeroLocus 𝒜 (I i) :=
   (gc_homogeneousIdeal 𝒜).l_iSup
 #align projective_spectrum.zero_locus_supr_homogeneous_ideal ProjectiveSpectrum.zeroLocus_iSup_homogeneousIdeal
 
+/- warning: projective_spectrum.zero_locus_Union -> ProjectiveSpectrum.zeroLocus_iUnion is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] {γ : Sort.{u3}} (s : γ -> (Set.{u2} A)), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Set.iUnion.{u2, u3} A γ (fun (i : γ) => s i))) (Set.iInter.{u2, u3} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) γ (fun (i : γ) => ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (s i)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] {γ : Sort.{u3}} (s : γ -> (Set.{u2} A)), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Set.iUnion.{u2, u3} A γ (fun (i : γ) => s i))) (Set.iInter.{u2, u3} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) γ (fun (i : γ) => ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (s i)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_Union ProjectiveSpectrum.zeroLocus_iUnionₓ'. -/
 theorem zeroLocus_iUnion {γ : Sort _} (s : γ → Set A) :
     zeroLocus 𝒜 (⋃ i, s i) = ⋂ i, zeroLocus 𝒜 (s i) :=
   (gc_set 𝒜).l_iSup
 #align projective_spectrum.zero_locus_Union ProjectiveSpectrum.zeroLocus_iUnion
 
+/- warning: projective_spectrum.zero_locus_bUnion -> ProjectiveSpectrum.zeroLocus_bUnion is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (s : Set.{u2} (Set.{u2} A)), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Set.iUnion.{u2, succ u2} A (Set.{u2} A) (fun (s' : Set.{u2} A) => Set.iUnion.{u2, 0} A (Membership.Mem.{u2, u2} (Set.{u2} A) (Set.{u2} (Set.{u2} A)) (Set.hasMem.{u2} (Set.{u2} A)) s' s) (fun (H : Membership.Mem.{u2, u2} (Set.{u2} A) (Set.{u2} (Set.{u2} A)) (Set.hasMem.{u2} (Set.{u2} A)) s' s) => s')))) (Set.iInter.{u2, succ u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u2} A) (fun (s' : Set.{u2} A) => Set.iInter.{u2, 0} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Membership.Mem.{u2, u2} (Set.{u2} A) (Set.{u2} (Set.{u2} A)) (Set.hasMem.{u2} (Set.{u2} A)) s' s) (fun (H : Membership.Mem.{u2, u2} (Set.{u2} A) (Set.{u2} (Set.{u2} A)) (Set.hasMem.{u2} (Set.{u2} A)) s' s) => ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s')))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (s : Set.{u2} (Set.{u2} A)), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Set.iUnion.{u2, succ u2} A (Set.{u2} A) (fun (s' : Set.{u2} A) => Set.iUnion.{u2, 0} A (Membership.mem.{u2, u2} (Set.{u2} A) (Set.{u2} (Set.{u2} A)) (Set.instMembershipSet.{u2} (Set.{u2} A)) s' s) (fun (H : Membership.mem.{u2, u2} (Set.{u2} A) (Set.{u2} (Set.{u2} A)) (Set.instMembershipSet.{u2} (Set.{u2} A)) s' s) => s')))) (Set.iInter.{u2, succ u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u2} A) (fun (s' : Set.{u2} A) => Set.iInter.{u2, 0} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Membership.mem.{u2, u2} (Set.{u2} A) (Set.{u2} (Set.{u2} A)) (Set.instMembershipSet.{u2} (Set.{u2} A)) s' s) (fun (H : Membership.mem.{u2, u2} (Set.{u2} A) (Set.{u2} (Set.{u2} A)) (Set.instMembershipSet.{u2} (Set.{u2} A)) s' s) => ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s')))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_bUnion ProjectiveSpectrum.zeroLocus_bUnionₓ'. -/
 theorem zeroLocus_bUnion (s : Set (Set A)) :
     zeroLocus 𝒜 (⋃ s' ∈ s, s' : Set A) = ⋂ s' ∈ s, zeroLocus 𝒜 s' := by simp only [zero_locus_Union]
 #align projective_spectrum.zero_locus_bUnion ProjectiveSpectrum.zeroLocus_bUnion
 
+/- warning: projective_spectrum.vanishing_ideal_Union -> ProjectiveSpectrum.vanishingIdeal_iUnion is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] {γ : Sort.{u3}} (t : γ -> (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))), Eq.{succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Set.iUnion.{u2, u3} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) γ (fun (i : γ) => t i))) (iInf.{u2, u3} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.hasInf.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 _inst_4) γ (fun (i : γ) => ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (t i)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] {γ : Sort.{u3}} (t : γ -> (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))), Eq.{succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Set.iUnion.{u2, u3} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) γ (fun (i : γ) => t i))) (iInf.{u2, u3} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.instInfSetHomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4) γ (fun (i : γ) => ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (t i)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.vanishing_ideal_Union ProjectiveSpectrum.vanishingIdeal_iUnionₓ'. -/
 theorem vanishingIdeal_iUnion {γ : Sort _} (t : γ → Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (⋃ i, t i) = ⨅ i, vanishingIdeal (t i) :=
   HomogeneousIdeal.toIdeal_injective <| by
     convert(gc_ideal 𝒜).u_iInf <;> exact HomogeneousIdeal.toIdeal_iInf _
 #align projective_spectrum.vanishing_ideal_Union ProjectiveSpectrum.vanishingIdeal_iUnion
 
+/- warning: projective_spectrum.zero_locus_inf -> ProjectiveSpectrum.zeroLocus_inf is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_inf ProjectiveSpectrum.zeroLocus_infₓ'. -/
 theorem zeroLocus_inf (I J : Ideal A) :
     zeroLocus 𝒜 ((I ⊓ J : Ideal A) : Set A) = zeroLocus 𝒜 I ∪ zeroLocus 𝒜 J :=
   Set.ext fun x => x.IsPrime.inf_le
 #align projective_spectrum.zero_locus_inf ProjectiveSpectrum.zeroLocus_inf
 
+/- warning: projective_spectrum.union_zero_locus -> ProjectiveSpectrum.union_zeroLocus is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (s : Set.{u2} A) (s' : Set.{u2} A), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Union.union.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasUnion.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s')) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))))) (Inf.inf.{u2} (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Submodule.hasInf.{u2, u2} A A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))) (Ideal.span.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) s) (Ideal.span.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) s'))))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (s : Set.{u2} A) (s' : Set.{u2} A), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Union.union.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instUnionSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s')) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (Inf.inf.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Submodule.instInfSubmodule.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (Ideal.span.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) s) (Ideal.span.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) s'))))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.union_zero_locus ProjectiveSpectrum.union_zeroLocusₓ'. -/
 theorem union_zeroLocus (s s' : Set A) :
     zeroLocus 𝒜 s ∪ zeroLocus 𝒜 s' = zeroLocus 𝒜 (Ideal.span s ⊓ Ideal.span s' : Ideal A) :=
   by
@@ -291,27 +525,57 @@ theorem union_zeroLocus (s s' : Set A) :
   simp
 #align projective_spectrum.union_zero_locus ProjectiveSpectrum.union_zeroLocus
 
+/- warning: projective_spectrum.zero_locus_mul_ideal -> ProjectiveSpectrum.zeroLocus_mul_ideal is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (I : Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (J : Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (HMul.hMul.{u2, u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (instHMul.{u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Ideal.instMulIdealToSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) I J))) (Union.union.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instUnionSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) I)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) A (Submodule.setLike.{u2, u2} A A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Semiring.toModule.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) J)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_mul_ideal ProjectiveSpectrum.zeroLocus_mul_idealₓ'. -/
 theorem zeroLocus_mul_ideal (I J : Ideal A) :
     zeroLocus 𝒜 ((I * J : Ideal A) : Set A) = zeroLocus 𝒜 I ∪ zeroLocus 𝒜 J :=
   Set.ext fun x => x.IsPrime.mul_le
 #align projective_spectrum.zero_locus_mul_ideal ProjectiveSpectrum.zeroLocus_mul_ideal
 
+/- warning: projective_spectrum.zero_locus_mul_homogeneous_ideal -> ProjectiveSpectrum.zeroLocus_mul_homogeneousIdeal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (I : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A 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(NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A 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(Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) (HMul.hMul.{u2, u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (instHMul.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.hasMul.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (CommRing.toCommSemiring.{u2} A _inst_2) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4)) I J))) (Union.union.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasUnion.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) I)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) J)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (I : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (J : HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HMul.hMul.{u2, u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A 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(NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4)) I J))) (Union.union.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instUnionSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A 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(Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) J)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_mul_homogeneous_ideal ProjectiveSpectrum.zeroLocus_mul_homogeneousIdealₓ'. -/
 theorem zeroLocus_mul_homogeneousIdeal (I J : HomogeneousIdeal 𝒜) :
     zeroLocus 𝒜 ((I * J : HomogeneousIdeal 𝒜) : Set A) = zeroLocus 𝒜 I ∪ zeroLocus 𝒜 J :=
   Set.ext fun x => x.IsPrime.mul_le
 #align projective_spectrum.zero_locus_mul_homogeneous_ideal ProjectiveSpectrum.zeroLocus_mul_homogeneousIdeal
 
+/- warning: projective_spectrum.zero_locus_singleton_mul -> ProjectiveSpectrum.zeroLocus_singleton_mul is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (f : A) (g : A), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.hasSingleton.{u2} A) (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (Distrib.toHasMul.{u2} A (Ring.toDistrib.{u2} A (CommRing.toRing.{u2} A _inst_2)))) f g))) (Union.union.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasUnion.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.hasSingleton.{u2} A) f)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.hasSingleton.{u2} A) g)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (f : A) (g : A), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.instSingletonSet.{u2} A) (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (NonUnitalNonAssocRing.toMul.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) f g))) (Union.union.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instUnionSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.instSingletonSet.{u2} A) f)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.instSingletonSet.{u2} A) g)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_singleton_mul ProjectiveSpectrum.zeroLocus_singleton_mulₓ'. -/
 theorem zeroLocus_singleton_mul (f g : A) :
     zeroLocus 𝒜 ({f * g} : Set A) = zeroLocus 𝒜 {f} ∪ zeroLocus 𝒜 {g} :=
   Set.ext fun x => by simpa using x.is_prime.mul_mem_iff_mem_or_mem
 #align projective_spectrum.zero_locus_singleton_mul ProjectiveSpectrum.zeroLocus_singleton_mul
 
+/- warning: projective_spectrum.zero_locus_singleton_pow -> ProjectiveSpectrum.zeroLocus_singleton_pow is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (f : A) (n : Nat), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) n) -> (Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.hasSingleton.{u2} A) (HPow.hPow.{u2, 0, u2} A Nat A (instHPow.{u2, 0} A Nat (Monoid.Pow.{u2} A (Ring.toMonoid.{u2} A (CommRing.toRing.{u2} A _inst_2)))) f n))) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.hasSingleton.{u2} A) f)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (f : A) (n : Nat), (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) n) -> (Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.instSingletonSet.{u2} A) (HPow.hPow.{u2, 0, u2} A Nat A (instHPow.{u2, 0} A Nat (Monoid.Pow.{u2} A (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) f n))) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.instSingletonSet.{u2} A) f)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_singleton_pow ProjectiveSpectrum.zeroLocus_singleton_powₓ'. -/
 @[simp]
 theorem zeroLocus_singleton_pow (f : A) (n : ℕ) (hn : 0 < n) :
     zeroLocus 𝒜 ({f ^ n} : Set A) = zeroLocus 𝒜 {f} :=
   Set.ext fun x => by simpa using x.is_prime.pow_mem_iff_mem n hn
 #align projective_spectrum.zero_locus_singleton_pow ProjectiveSpectrum.zeroLocus_singleton_pow
 
+/- warning: projective_spectrum.sup_vanishing_ideal_le -> ProjectiveSpectrum.sup_vanishingIdeal_le is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (t' : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)), LE.le.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (Preorder.toLE.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (PartialOrder.toPreorder.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.instPartialOrderHomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4))) (Sup.sup.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.instSupHomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t')) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Inter.inter.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instInterSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) t t'))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.sup_vanishing_ideal_le ProjectiveSpectrum.sup_vanishingIdeal_leₓ'. -/
 theorem sup_vanishingIdeal_le (t t' : Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal t ⊔ vanishingIdeal t' ≤ vanishingIdeal (t ∩ t') :=
   by
@@ -323,11 +587,23 @@ theorem sup_vanishingIdeal_le (t t' : Set (ProjectiveSpectrum 𝒜)) :
   apply Submodule.add_mem <;> solve_by_elim
 #align projective_spectrum.sup_vanishing_ideal_le ProjectiveSpectrum.sup_vanishingIdeal_le
 
+/- warning: projective_spectrum.mem_compl_zero_locus_iff_not_mem -> ProjectiveSpectrum.mem_compl_zeroLocus_iff_not_mem is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] {f : A} {I : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4}, Iff (Membership.Mem.{u2, u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasMem.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) I (HasCompl.compl.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.booleanAlgebra.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.hasSingleton.{u2} A) f)))) (Not (Membership.Mem.{u2, u2} A (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) (SetLike.hasMem.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4)) f (ProjectiveSpectrum.asHomogeneousIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 I)))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_3 : Algebra.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] (𝒜 : Nat -> (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u2, u1} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 𝒜] {f : A} {I : ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4}, Iff (Membership.mem.{u1, u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instMembershipSet.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) I (HasCompl.compl.{u1} (Set.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instBooleanAlgebraSet.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) (ProjectiveSpectrum.zeroLocus.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u1, u1} A (Set.{u1} A) (Set.instSingletonSet.{u1} A) f)))) (Not (Membership.mem.{u1, u1} A (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (SetLike.instMembership.{u1, u1} (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4)) f (ProjectiveSpectrum.asHomogeneousIdeal.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 I)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.mem_compl_zero_locus_iff_not_mem ProjectiveSpectrum.mem_compl_zeroLocus_iff_not_memₓ'. -/
 theorem mem_compl_zeroLocus_iff_not_mem {f : A} {I : ProjectiveSpectrum 𝒜} :
     I ∈ (zeroLocus 𝒜 {f} : Set (ProjectiveSpectrum 𝒜))ᶜ ↔ f ∉ I.asHomogeneousIdeal := by
   rw [Set.mem_compl_iff, mem_zero_locus, Set.singleton_subset_iff] <;> rfl
 #align projective_spectrum.mem_compl_zero_locus_iff_not_mem ProjectiveSpectrum.mem_compl_zeroLocus_iff_not_mem
 
+/- warning: projective_spectrum.zariski_topology -> ProjectiveSpectrum.zariskiTopology is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜], TopologicalSpace.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜], TopologicalSpace.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zariski_topology ProjectiveSpectrum.zariskiTopologyₓ'. -/
 /-- The Zariski topology on the prime spectrum of a commutative ring is defined via the closed sets
 of the topology: they are exactly those sets that are the zero locus of a subset of the ring. -/
 instance zariskiTopology : TopologicalSpace (ProjectiveSpectrum 𝒜) :=
@@ -344,25 +620,55 @@ instance zariskiTopology : TopologicalSpace (ProjectiveSpectrum 𝒜) :=
       exact ⟨_, (union_zero_locus 𝒜 s t).symm⟩)
 #align projective_spectrum.zariski_topology ProjectiveSpectrum.zariskiTopology
 
+/- warning: projective_spectrum.Top -> ProjectiveSpectrum.top is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜], TopCat.{u2}
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜], TopCat.{u2}
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.Top ProjectiveSpectrum.topₓ'. -/
 /-- The underlying topology of `Proj` is the projective spectrum of graded ring `A`. -/
 def top : TopCat :=
   TopCat.of (ProjectiveSpectrum 𝒜)
 #align projective_spectrum.Top ProjectiveSpectrum.top
 
+/- warning: projective_spectrum.is_open_iff -> ProjectiveSpectrum.isOpen_iff is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (U : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)), Iff (IsOpen.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) U) (Exists.{succ u2} (Set.{u2} A) (fun (s : Set.{u2} A) => Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (HasCompl.compl.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.booleanAlgebra.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) U) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (U : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)), Iff (IsOpen.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) U) (Exists.{succ u2} (Set.{u2} A) (fun (s : Set.{u2} A) => Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (HasCompl.compl.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instBooleanAlgebraSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) U) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.is_open_iff ProjectiveSpectrum.isOpen_iffₓ'. -/
 theorem isOpen_iff (U : Set (ProjectiveSpectrum 𝒜)) : IsOpen U ↔ ∃ s, Uᶜ = zeroLocus 𝒜 s := by
   simp only [@eq_comm _ (Uᶜ)] <;> rfl
 #align projective_spectrum.is_open_iff ProjectiveSpectrum.isOpen_iff
 
+/- warning: projective_spectrum.is_closed_iff_zero_locus -> ProjectiveSpectrum.isClosed_iff_zeroLocus is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (Z : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)), Iff (IsClosed.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) Z) (Exists.{succ u2} (Set.{u2} A) (fun (s : Set.{u2} A) => Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) Z (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (Z : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)), Iff (IsClosed.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) Z) (Exists.{succ u2} (Set.{u2} A) (fun (s : Set.{u2} A) => Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) Z (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.is_closed_iff_zero_locus ProjectiveSpectrum.isClosed_iff_zeroLocusₓ'. -/
 theorem isClosed_iff_zeroLocus (Z : Set (ProjectiveSpectrum 𝒜)) :
     IsClosed Z ↔ ∃ s, Z = zeroLocus 𝒜 s := by rw [← isOpen_compl_iff, is_open_iff, compl_compl]
 #align projective_spectrum.is_closed_iff_zero_locus ProjectiveSpectrum.isClosed_iff_zeroLocus
 
+/- warning: projective_spectrum.is_closed_zero_locus -> ProjectiveSpectrum.isClosed_zeroLocus is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (s : Set.{u2} A), IsClosed.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s)
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (s : Set.{u2} A), IsClosed.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 s)
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.is_closed_zero_locus ProjectiveSpectrum.isClosed_zeroLocusₓ'. -/
 theorem isClosed_zeroLocus (s : Set A) : IsClosed (zeroLocus 𝒜 s) :=
   by
   rw [is_closed_iff_zero_locus]
   exact ⟨s, rfl⟩
 #align projective_spectrum.is_closed_zero_locus ProjectiveSpectrum.isClosed_zeroLocus
 
+/- warning: projective_spectrum.zero_locus_vanishing_ideal_eq_closure -> ProjectiveSpectrum.zeroLocus_vanishingIdeal_eq_closure is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4)))) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t))) (closure.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) t)
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (SetLike.coe.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t))) (closure.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) t)
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.zero_locus_vanishing_ideal_eq_closure ProjectiveSpectrum.zeroLocus_vanishingIdeal_eq_closureₓ'. -/
 theorem zeroLocus_vanishingIdeal_eq_closure (t : Set (ProjectiveSpectrum 𝒜)) :
     zeroLocus 𝒜 (vanishingIdeal t : Set A) = closure t :=
   by
@@ -375,6 +681,12 @@ theorem zeroLocus_vanishingIdeal_eq_closure (t : Set (ProjectiveSpectrum 𝒜))
     exact subset_zero_locus_vanishing_ideal 𝒜 t
 #align projective_spectrum.zero_locus_vanishing_ideal_eq_closure ProjectiveSpectrum.zeroLocus_vanishingIdeal_eq_closure
 
+/- warning: projective_spectrum.vanishing_ideal_closure -> ProjectiveSpectrum.vanishingIdeal_closure is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)), Eq.{succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (ProjectiveSpectrum.vanishingIdeal._proof_1.{u1, u2} R A _inst_1 _inst_2 _inst_3) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_4) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (closure.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) t)) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t)
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (t : Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)), Eq.{succ u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (closure.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) t)) (ProjectiveSpectrum.vanishingIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 t)
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.vanishing_ideal_closure ProjectiveSpectrum.vanishingIdeal_closureₓ'. -/
 theorem vanishingIdeal_closure (t : Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (closure t) = vanishingIdeal t :=
   by
@@ -387,6 +699,12 @@ theorem vanishingIdeal_closure (t : Set (ProjectiveSpectrum 𝒜)) :
 
 section BasicOpen
 
+/- warning: projective_spectrum.basic_open -> ProjectiveSpectrum.basicOpen is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜], A -> (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜], A -> (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.basic_open ProjectiveSpectrum.basicOpenₓ'. -/
 /-- `basic_open r` is the open subset containing all prime ideals not containing `r`. -/
 def basicOpen (r : A) : TopologicalSpace.Opens (ProjectiveSpectrum 𝒜)
     where
@@ -394,58 +712,124 @@ def basicOpen (r : A) : TopologicalSpace.Opens (ProjectiveSpectrum 𝒜)
   is_open' := ⟨{r}, Set.ext fun x => Set.singleton_subset_iff.trans <| Classical.not_not.symm⟩
 #align projective_spectrum.basic_open ProjectiveSpectrum.basicOpen
 
+/- warning: projective_spectrum.mem_basic_open -> ProjectiveSpectrum.mem_basicOpen is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (f : A) (x : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), Iff (Membership.Mem.{u2, u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (SetLike.hasMem.{u2, u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (TopologicalSpace.Opens.setLike.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) x (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 f)) (Not (Membership.Mem.{u2, u2} A (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) (SetLike.hasMem.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} 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(CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4)) f (ProjectiveSpectrum.asHomogeneousIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x)))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_3 : Algebra.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] (𝒜 : Nat -> (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u2, u1} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 𝒜] (f : A) (x : ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), Iff (Membership.mem.{u1, u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (TopologicalSpace.Opens.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (TopologicalSpace.Opens.instSetLikeOpens.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) x (ProjectiveSpectrum.basicOpen.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 f)) (Not (Membership.mem.{u1, u1} A (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (SetLike.instMembership.{u1, u1} (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) A (HomogeneousIdeal.setLike.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4)) f (ProjectiveSpectrum.asHomogeneousIdeal.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.mem_basic_open ProjectiveSpectrum.mem_basicOpenₓ'. -/
 @[simp]
 theorem mem_basicOpen (f : A) (x : ProjectiveSpectrum 𝒜) :
     x ∈ basicOpen 𝒜 f ↔ f ∉ x.asHomogeneousIdeal :=
   Iff.rfl
 #align projective_spectrum.mem_basic_open ProjectiveSpectrum.mem_basicOpen
 
+/- warning: projective_spectrum.mem_coe_basic_open -> ProjectiveSpectrum.mem_coe_basicOpen is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (f : A) (x : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), Iff (Membership.Mem.{u2, u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasMem.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (HasLiftT.mk.{succ u2, succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CoeTCₓ.coe.{succ u2, succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (SetLike.Set.hasCoeT.{u2, u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (TopologicalSpace.Opens.setLike.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))))) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 f))) (Not (Membership.Mem.{u2, u2} A (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) (SetLike.hasMem.{u2, u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) A (HomogeneousIdeal.setLike.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4)) f (ProjectiveSpectrum.asHomogeneousIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x)))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_3 : Algebra.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] (𝒜 : Nat -> (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u2, u1} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 𝒜] (f : A) (x : ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), Iff (Membership.mem.{u1, u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instMembershipSet.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x (SetLike.coe.{u1, u1} (TopologicalSpace.Opens.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (TopologicalSpace.Opens.instSetLikeOpens.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.basicOpen.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 f))) (Not (Membership.mem.{u1, u1} A (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A 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(Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4)) f (ProjectiveSpectrum.asHomogeneousIdeal.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.mem_coe_basic_open ProjectiveSpectrum.mem_coe_basicOpenₓ'. -/
 theorem mem_coe_basicOpen (f : A) (x : ProjectiveSpectrum 𝒜) :
     x ∈ (↑(basicOpen 𝒜 f) : Set (ProjectiveSpectrum 𝒜)) ↔ f ∉ x.asHomogeneousIdeal :=
   Iff.rfl
 #align projective_spectrum.mem_coe_basic_open ProjectiveSpectrum.mem_coe_basicOpen
 
+/- warning: projective_spectrum.is_open_basic_open -> ProjectiveSpectrum.isOpen_basicOpen is a dubious translation:
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+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] {a : A}, IsOpen.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (SetLike.coe.{u2, u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (TopologicalSpace.Opens.instSetLikeOpens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 a))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.is_open_basic_open ProjectiveSpectrum.isOpen_basicOpenₓ'. -/
 theorem isOpen_basicOpen {a : A} : IsOpen (basicOpen 𝒜 a : Set (ProjectiveSpectrum 𝒜)) :=
   (basicOpen 𝒜 a).IsOpen
 #align projective_spectrum.is_open_basic_open ProjectiveSpectrum.isOpen_basicOpen
 
+/- warning: projective_spectrum.basic_open_eq_zero_locus_compl -> ProjectiveSpectrum.basicOpen_eq_zeroLocus_compl is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (r : A), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (HasLiftT.mk.{succ u2, succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CoeTCₓ.coe.{succ u2, succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (SetLike.Set.hasCoeT.{u2, u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (TopologicalSpace.Opens.setLike.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))))) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 r)) (HasCompl.compl.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.booleanAlgebra.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.hasSingleton.{u2} A) r)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (r : A), Eq.{succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (SetLike.coe.{u2, u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (TopologicalSpace.Opens.instSetLikeOpens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 r)) (HasCompl.compl.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instBooleanAlgebraSet.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) (ProjectiveSpectrum.zeroLocus.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.instSingletonSet.{u2} A) r)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.basic_open_eq_zero_locus_compl ProjectiveSpectrum.basicOpen_eq_zeroLocus_complₓ'. -/
 @[simp]
 theorem basicOpen_eq_zeroLocus_compl (r : A) :
     (basicOpen 𝒜 r : Set (ProjectiveSpectrum 𝒜)) = zeroLocus 𝒜 {r}ᶜ :=
   Set.ext fun x => by simpa only [Set.mem_compl_iff, mem_zero_locus, Set.singleton_subset_iff]
 #align projective_spectrum.basic_open_eq_zero_locus_compl ProjectiveSpectrum.basicOpen_eq_zeroLocus_compl
 
+/- warning: projective_spectrum.basic_open_one -> ProjectiveSpectrum.basicOpen_one is a dubious translation:
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+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜], Eq.{succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (OfNat.ofNat.{u2} A 1 (OfNat.mk.{u2} A 1 (One.one.{u2} A (AddMonoidWithOne.toOne.{u2} A (AddGroupWithOne.toAddMonoidWithOne.{u2} A (AddCommGroupWithOne.toAddGroupWithOne.{u2} A (Ring.toAddCommGroupWithOne.{u2} A (CommRing.toRing.{u2} A _inst_2))))))))) (Top.top.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.toHasTop.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (TopologicalSpace.Opens.completeLattice.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜], Eq.{succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (OfNat.ofNat.{u2} A 1 (One.toOfNat1.{u2} A (Semiring.toOne.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (Top.top.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.toTop.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.basic_open_one ProjectiveSpectrum.basicOpen_oneₓ'. -/
 @[simp]
 theorem basicOpen_one : basicOpen 𝒜 (1 : A) = ⊤ :=
   TopologicalSpace.Opens.ext <| by simp
 #align projective_spectrum.basic_open_one ProjectiveSpectrum.basicOpen_one
 
+/- warning: projective_spectrum.basic_open_zero -> ProjectiveSpectrum.basicOpen_zero is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜], Eq.{succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (OfNat.ofNat.{u2} A 0 (OfNat.mk.{u2} A 0 (Zero.zero.{u2} A (MulZeroClass.toHasZero.{u2} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))))) (Bot.bot.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.toHasBot.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (TopologicalSpace.Opens.completeLattice.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜], Eq.{succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (OfNat.ofNat.{u2} A 0 (Zero.toOfNat0.{u2} A (CommMonoidWithZero.toZero.{u2} A (CommSemiring.toCommMonoidWithZero.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (Bot.bot.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.toBot.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.basic_open_zero ProjectiveSpectrum.basicOpen_zeroₓ'. -/
 @[simp]
 theorem basicOpen_zero : basicOpen 𝒜 (0 : A) = ⊥ :=
   TopologicalSpace.Opens.ext <| by simp
 #align projective_spectrum.basic_open_zero ProjectiveSpectrum.basicOpen_zero
 
+/- warning: projective_spectrum.basic_open_mul -> ProjectiveSpectrum.basicOpen_mul is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (f : A) (g : A), Eq.{succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (Distrib.toHasMul.{u2} A (Ring.toDistrib.{u2} A (CommRing.toRing.{u2} A _inst_2)))) f g)) (Inf.inf.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (SemilatticeInf.toHasInf.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Lattice.toSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ConditionallyCompleteLattice.toLattice.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (TopologicalSpace.Opens.completeLattice.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)))))) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 f) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 g))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (f : A) (g : A), Eq.{succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (NonUnitalNonAssocRing.toMul.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) f g)) (Inf.inf.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Lattice.toInf.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ConditionallyCompleteLattice.toLattice.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))))) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 f) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 g))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.basic_open_mul ProjectiveSpectrum.basicOpen_mulₓ'. -/
 theorem basicOpen_mul (f g : A) : basicOpen 𝒜 (f * g) = basicOpen 𝒜 f ⊓ basicOpen 𝒜 g :=
   TopologicalSpace.Opens.ext <| by simp [zero_locus_singleton_mul]
 #align projective_spectrum.basic_open_mul ProjectiveSpectrum.basicOpen_mul
 
+/- warning: projective_spectrum.basic_open_mul_le_left -> ProjectiveSpectrum.basicOpen_mul_le_left is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (f : A) (g : A), LE.le.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Preorder.toHasLe.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (SetLike.partialOrder.{u2, u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (TopologicalSpace.Opens.setLike.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))))) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (Distrib.toHasMul.{u2} A (Ring.toDistrib.{u2} A (CommRing.toRing.{u2} A _inst_2)))) f g)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 f)
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (f : A) (g : A), LE.le.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Preorder.toLE.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)))))) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (NonUnitalNonAssocRing.toMul.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) f g)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 f)
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.basic_open_mul_le_left ProjectiveSpectrum.basicOpen_mul_le_leftₓ'. -/
 theorem basicOpen_mul_le_left (f g : A) : basicOpen 𝒜 (f * g) ≤ basicOpen 𝒜 f :=
   by
   rw [basic_open_mul 𝒜 f g]
   exact inf_le_left
 #align projective_spectrum.basic_open_mul_le_left ProjectiveSpectrum.basicOpen_mul_le_left
 
+/- warning: projective_spectrum.basic_open_mul_le_right -> ProjectiveSpectrum.basicOpen_mul_le_right is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (f : A) (g : A), LE.le.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Preorder.toHasLe.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (SetLike.partialOrder.{u2, u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (TopologicalSpace.Opens.setLike.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))))) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (Distrib.toHasMul.{u2} A (Ring.toDistrib.{u2} A (CommRing.toRing.{u2} A _inst_2)))) f g)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 g)
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (f : A) (g : A), LE.le.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Preorder.toLE.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)))))) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (NonUnitalNonAssocRing.toMul.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) f g)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 g)
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.basic_open_mul_le_right ProjectiveSpectrum.basicOpen_mul_le_rightₓ'. -/
 theorem basicOpen_mul_le_right (f g : A) : basicOpen 𝒜 (f * g) ≤ basicOpen 𝒜 g :=
   by
   rw [basic_open_mul 𝒜 f g]
   exact inf_le_right
 #align projective_spectrum.basic_open_mul_le_right ProjectiveSpectrum.basicOpen_mul_le_right
 
+/- warning: projective_spectrum.basic_open_pow -> ProjectiveSpectrum.basicOpen_pow is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (f : A) (n : Nat), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) n) -> (Eq.{succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (HPow.hPow.{u2, 0, u2} A Nat A (instHPow.{u2, 0} A Nat (Monoid.Pow.{u2} A (Ring.toMonoid.{u2} A (CommRing.toRing.{u2} A _inst_2)))) f n)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 f))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (f : A) (n : Nat), (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) n) -> (Eq.{succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (HPow.hPow.{u2, 0, u2} A Nat A (instHPow.{u2, 0} A Nat (Monoid.Pow.{u2} A (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) f n)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 f))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.basic_open_pow ProjectiveSpectrum.basicOpen_powₓ'. -/
 @[simp]
 theorem basicOpen_pow (f : A) (n : ℕ) (hn : 0 < n) : basicOpen 𝒜 (f ^ n) = basicOpen 𝒜 f :=
   TopologicalSpace.Opens.ext <| by simpa using zero_locus_singleton_pow 𝒜 f n hn
 #align projective_spectrum.basic_open_pow ProjectiveSpectrum.basicOpen_pow
 
+/- warning: projective_spectrum.basic_open_eq_union_of_projection -> ProjectiveSpectrum.basicOpen_eq_union_of_projection is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (f : A), Eq.{succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 f) (iSup.{u2, 1} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ConditionallyCompleteLattice.toHasSup.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (TopologicalSpace.Opens.completeLattice.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)))) Nat (fun (i : Nat) => ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (coeFn.{succ u2, succ u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (fun (_x : LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) => A -> A) (LinearMap.hasCoeToFun.{u1, u1, u2, u2} R R A A (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (GradedAlgebra.proj.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜 _inst_4 i) f)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜] (f : A), Eq.{succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 f) (iSup.{u2, 1} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ConditionallyCompleteLattice.toSupSet.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)))) Nat (fun (i : Nat) => ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 (FunLike.coe.{succ u2, succ u2, succ u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : A) => A) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u2} R R A A (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (GradedAlgebra.proj.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜 _inst_4 i) f)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.basic_open_eq_union_of_projection ProjectiveSpectrum.basicOpen_eq_union_of_projectionₓ'. -/
 theorem basicOpen_eq_union_of_projection (f : A) :
     basicOpen 𝒜 f = ⨆ i : ℕ, basicOpen 𝒜 (GradedAlgebra.proj 𝒜 i f) :=
   TopologicalSpace.Opens.ext <|
@@ -464,6 +848,12 @@ theorem basicOpen_eq_union_of_projection (f : A) :
         exact fun rid => hz (z.1.2 i rid)
 #align projective_spectrum.basic_open_eq_union_of_projection ProjectiveSpectrum.basicOpen_eq_union_of_projection
 
+/- warning: projective_spectrum.is_topological_basis_basic_opens -> ProjectiveSpectrum.isTopologicalBasis_basic_opens is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜], TopologicalSpace.IsTopologicalBasis.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.range.{u2, succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) A (fun (r : A) => (fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (HasLiftT.mk.{succ u2, succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (CoeTCₓ.coe.{succ u2, succ u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (SetLike.Set.hasCoeT.{u2, u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (TopologicalSpace.Opens.setLike.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))))) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 r)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 𝒜], TopologicalSpace.IsTopologicalBasis.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.range.{u2, succ u2} (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) A (fun (r : A) => SetLike.coe.{u2, u2} (TopologicalSpace.Opens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (TopologicalSpace.Opens.instSetLikeOpens.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (ProjectiveSpectrum.basicOpen.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 r)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.is_topological_basis_basic_opens ProjectiveSpectrum.isTopologicalBasis_basic_opensₓ'. -/
 theorem isTopologicalBasis_basic_opens :
     TopologicalSpace.IsTopologicalBasis
       (Set.range fun r : A => (basicOpen 𝒜 r : Set (ProjectiveSpectrum 𝒜))) :=
@@ -494,18 +884,36 @@ where `x ≤ y` if and only if `y ∈ closure {x}`.
 instance : PartialOrder (ProjectiveSpectrum 𝒜) :=
   PartialOrder.lift asHomogeneousIdeal fun ⟨_, _, _⟩ ⟨_, _, _⟩ => mk.inj_eq.mpr
 
+/- warning: projective_spectrum.as_ideal_le_as_ideal -> ProjectiveSpectrum.as_ideal_le_as_ideal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (x : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (y : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), Iff (LE.le.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) (Preorder.toHasLe.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) (PartialOrder.toPreorder.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) (HomogeneousIdeal.partialOrder.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4))) (ProjectiveSpectrum.asHomogeneousIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x) (ProjectiveSpectrum.asHomogeneousIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 y)) (LE.le.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Preorder.toHasLe.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (PartialOrder.toPreorder.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.partialOrder.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) x y)
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_3 : Algebra.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] (𝒜 : Nat -> (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u2, u1} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 𝒜] (x : ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (y : ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), Iff (LE.le.{u1} (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (Preorder.toLE.{u1} (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (PartialOrder.toPreorder.{u1} (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.instPartialOrderHomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 _inst_4))) (ProjectiveSpectrum.asHomogeneousIdeal.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x) (ProjectiveSpectrum.asHomogeneousIdeal.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 y)) (LE.le.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Preorder.toLE.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (PartialOrder.toPreorder.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.instPartialOrderProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) x y)
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.as_ideal_le_as_ideal ProjectiveSpectrum.as_ideal_le_as_idealₓ'. -/
 @[simp]
 theorem as_ideal_le_as_ideal (x y : ProjectiveSpectrum 𝒜) :
     x.asHomogeneousIdeal ≤ y.asHomogeneousIdeal ↔ x ≤ y :=
   Iff.rfl
 #align projective_spectrum.as_ideal_le_as_ideal ProjectiveSpectrum.as_ideal_le_as_ideal
 
+/- warning: projective_spectrum.as_ideal_lt_as_ideal -> ProjectiveSpectrum.as_ideal_lt_as_ideal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (x : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (y : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), Iff (LT.lt.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) (Preorder.toHasLt.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) (PartialOrder.toPreorder.{u2} (HomogeneousIdeal.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) _inst_4) (HomogeneousIdeal.partialOrder.{0, u2, u2} Nat (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (AddCommMonoid.toAddMonoid.{0} Nat (OrderedAddCommMonoid.toAddCommMonoid.{0} Nat (CanonicallyOrderedAddMonoid.toOrderedAddCommMonoid.{0} Nat (CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddMonoid.{0} Nat Nat.canonicallyOrderedCommSemiring)))) (Submodule.setLike.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) 𝒜 _inst_4))) (ProjectiveSpectrum.asHomogeneousIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x) (ProjectiveSpectrum.asHomogeneousIdeal.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 y)) (LT.lt.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Preorder.toHasLt.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (PartialOrder.toPreorder.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.partialOrder.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) x y)
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_3 : Algebra.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] (𝒜 : Nat -> (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u2, u1} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 𝒜] (x : ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (y : ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), Iff (LT.lt.{u1} (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (Preorder.toLT.{u1} (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (PartialOrder.toPreorder.{u1} (HomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_4) (HomogeneousIdeal.instPartialOrderHomogeneousIdeal.{0, u1, u1} Nat (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) (Submodule.addSubmonoidClass.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3)) 𝒜 _inst_4))) (ProjectiveSpectrum.asHomogeneousIdeal.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 x) (ProjectiveSpectrum.asHomogeneousIdeal.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4 y)) (LT.lt.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Preorder.toLT.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (PartialOrder.toPreorder.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.instPartialOrderProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) x y)
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.as_ideal_lt_as_ideal ProjectiveSpectrum.as_ideal_lt_as_idealₓ'. -/
 @[simp]
 theorem as_ideal_lt_as_ideal (x y : ProjectiveSpectrum 𝒜) :
     x.asHomogeneousIdeal < y.asHomogeneousIdeal ↔ x < y :=
   Iff.rfl
 #align projective_spectrum.as_ideal_lt_as_ideal ProjectiveSpectrum.as_ideal_lt_as_ideal
 
+/- warning: projective_spectrum.le_iff_mem_closure -> ProjectiveSpectrum.le_iff_mem_closure is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (𝒜 : Nat -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u1, u2} Nat R A (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) Nat.addMonoid _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 𝒜] (x : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (y : ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), Iff (LE.le.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Preorder.toHasLe.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (PartialOrder.toPreorder.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.partialOrder.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) x y) (Membership.Mem.{u2, u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasMem.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) y (closure.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Singleton.singleton.{u2, u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.hasSingleton.{u2} (ProjectiveSpectrum.{u1, u2} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x)))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_3 : Algebra.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] (𝒜 : Nat -> (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3))) [_inst_4 : GradedAlgebra.{0, u2, u1} Nat R A (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.addMonoid _inst_1 (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 𝒜] (x : ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (y : ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4), Iff (LE.le.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Preorder.toLE.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (PartialOrder.toPreorder.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.instPartialOrderProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4))) x y) (Membership.mem.{u1, u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instMembershipSet.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) y (closure.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (ProjectiveSpectrum.zariskiTopology.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Singleton.singleton.{u1, u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4) (Set.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) (Set.instSingletonSet.{u1} (ProjectiveSpectrum.{u2, u1} R A _inst_1 _inst_2 _inst_3 𝒜 _inst_4)) x)))
+Case conversion may be inaccurate. Consider using '#align projective_spectrum.le_iff_mem_closure ProjectiveSpectrum.le_iff_mem_closureₓ'. -/
 theorem le_iff_mem_closure (x y : ProjectiveSpectrum 𝒜) :
     x ≤ y ↔ y ∈ closure ({x} : Set (ProjectiveSpectrum 𝒜)) :=
   by

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

@@ -107,10 +107,10 @@ theorem coe_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
       { f | ∀ x : ProjectiveSpectrum 𝒜, x ∈ t → f ∈ x.asHomogeneousIdeal } :=
   by
   ext f
-  rw [vanishing_ideal, SetLike.mem_coe, ← HomogeneousIdeal.mem_iff, HomogeneousIdeal.toIdeal_infᵢ,
-    Submodule.mem_infᵢ]
+  rw [vanishing_ideal, SetLike.mem_coe, ← HomogeneousIdeal.mem_iff, HomogeneousIdeal.toIdeal_iInf,
+    Submodule.mem_iInf]
   apply forall_congr' fun x => _
-  rw [HomogeneousIdeal.toIdeal_infᵢ, Submodule.mem_infᵢ, HomogeneousIdeal.mem_iff]
+  rw [HomogeneousIdeal.toIdeal_iInf, Submodule.mem_iInf, HomogeneousIdeal.mem_iff]
 #align projective_spectrum.coe_vanishing_ideal ProjectiveSpectrum.coe_vanishingIdeal
 
 theorem mem_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) (f : A) :
@@ -254,30 +254,30 @@ theorem vanishingIdeal_union (t t' : Set (ProjectiveSpectrum 𝒜)) :
   ext1 <;> convert(gc_ideal 𝒜).u_inf
 #align projective_spectrum.vanishing_ideal_union ProjectiveSpectrum.vanishingIdeal_union
 
-theorem zeroLocus_supᵢ_ideal {γ : Sort _} (I : γ → Ideal A) :
+theorem zeroLocus_iSup_ideal {γ : Sort _} (I : γ → Ideal A) :
     zeroLocus _ ((⨆ i, I i : Ideal A) : Set A) = ⋂ i, zeroLocus 𝒜 (I i) :=
-  (gc_ideal 𝒜).l_supᵢ
-#align projective_spectrum.zero_locus_supr_ideal ProjectiveSpectrum.zeroLocus_supᵢ_ideal
+  (gc_ideal 𝒜).l_iSup
+#align projective_spectrum.zero_locus_supr_ideal ProjectiveSpectrum.zeroLocus_iSup_ideal
 
-theorem zeroLocus_supᵢ_homogeneousIdeal {γ : Sort _} (I : γ → HomogeneousIdeal 𝒜) :
+theorem zeroLocus_iSup_homogeneousIdeal {γ : Sort _} (I : γ → HomogeneousIdeal 𝒜) :
     zeroLocus _ ((⨆ i, I i : HomogeneousIdeal 𝒜) : Set A) = ⋂ i, zeroLocus 𝒜 (I i) :=
-  (gc_homogeneousIdeal 𝒜).l_supᵢ
-#align projective_spectrum.zero_locus_supr_homogeneous_ideal ProjectiveSpectrum.zeroLocus_supᵢ_homogeneousIdeal
+  (gc_homogeneousIdeal 𝒜).l_iSup
+#align projective_spectrum.zero_locus_supr_homogeneous_ideal ProjectiveSpectrum.zeroLocus_iSup_homogeneousIdeal
 
-theorem zeroLocus_unionᵢ {γ : Sort _} (s : γ → Set A) :
+theorem zeroLocus_iUnion {γ : Sort _} (s : γ → Set A) :
     zeroLocus 𝒜 (⋃ i, s i) = ⋂ i, zeroLocus 𝒜 (s i) :=
-  (gc_set 𝒜).l_supᵢ
-#align projective_spectrum.zero_locus_Union ProjectiveSpectrum.zeroLocus_unionᵢ
+  (gc_set 𝒜).l_iSup
+#align projective_spectrum.zero_locus_Union ProjectiveSpectrum.zeroLocus_iUnion
 
 theorem zeroLocus_bUnion (s : Set (Set A)) :
     zeroLocus 𝒜 (⋃ s' ∈ s, s' : Set A) = ⋂ s' ∈ s, zeroLocus 𝒜 s' := by simp only [zero_locus_Union]
 #align projective_spectrum.zero_locus_bUnion ProjectiveSpectrum.zeroLocus_bUnion
 
-theorem vanishingIdeal_unionᵢ {γ : Sort _} (t : γ → Set (ProjectiveSpectrum 𝒜)) :
+theorem vanishingIdeal_iUnion {γ : Sort _} (t : γ → Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (⋃ i, t i) = ⨅ i, vanishingIdeal (t i) :=
   HomogeneousIdeal.toIdeal_injective <| by
-    convert(gc_ideal 𝒜).u_infᵢ <;> exact HomogeneousIdeal.toIdeal_infᵢ _
-#align projective_spectrum.vanishing_ideal_Union ProjectiveSpectrum.vanishingIdeal_unionᵢ
+    convert(gc_ideal 𝒜).u_iInf <;> exact HomogeneousIdeal.toIdeal_iInf _
+#align projective_spectrum.vanishing_ideal_Union ProjectiveSpectrum.vanishingIdeal_iUnion
 
 theorem zeroLocus_inf (I J : Ideal A) :
     zeroLocus 𝒜 ((I ⊓ J : Ideal A) : Set A) = zeroLocus 𝒜 I ∪ zeroLocus 𝒜 J :=
@@ -334,7 +334,7 @@ instance zariskiTopology : TopologicalSpace (ProjectiveSpectrum 𝒜) :=
   TopologicalSpace.ofClosed (Set.range (ProjectiveSpectrum.zeroLocus 𝒜)) ⟨Set.univ, by simp⟩
     (by
       intro Zs h
-      rw [Set.interₛ_eq_interᵢ]
+      rw [Set.sInter_eq_iInter]
       let f : Zs → Set _ := fun i => Classical.choose (h i.2)
       have hf : ∀ i : Zs, ↑i = zero_locus 𝒜 (f i) := fun i => (Classical.choose_spec (h i.2)).symm
       simp only [hf]
@@ -450,7 +450,7 @@ theorem basicOpen_eq_union_of_projection (f : A) :
     basicOpen 𝒜 f = ⨆ i : ℕ, basicOpen 𝒜 (GradedAlgebra.proj 𝒜 i f) :=
   TopologicalSpace.Opens.ext <|
     Set.ext fun z => by
-      erw [mem_coe_basic_open, TopologicalSpace.Opens.mem_supₛ]
+      erw [mem_coe_basic_open, TopologicalSpace.Opens.mem_sSup]
       constructor <;> intro hz
       · rcases show ∃ i, GradedAlgebra.proj 𝒜 i f ∉ z.as_homogeneous_ideal
             by

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

@@ -251,7 +251,7 @@ theorem zeroLocus_union (s s' : Set A) : zeroLocus 𝒜 (s ∪ s') = zeroLocus _
 
 theorem vanishingIdeal_union (t t' : Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (t ∪ t') = vanishingIdeal t ⊓ vanishingIdeal t' := by
-  ext1 <;> convert (gc_ideal 𝒜).u_inf
+  ext1 <;> convert(gc_ideal 𝒜).u_inf
 #align projective_spectrum.vanishing_ideal_union ProjectiveSpectrum.vanishingIdeal_union
 
 theorem zeroLocus_supᵢ_ideal {γ : Sort _} (I : γ → Ideal A) :
@@ -276,7 +276,7 @@ theorem zeroLocus_bUnion (s : Set (Set A)) :
 theorem vanishingIdeal_unionᵢ {γ : Sort _} (t : γ → Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (⋃ i, t i) = ⨅ i, vanishingIdeal (t i) :=
   HomogeneousIdeal.toIdeal_injective <| by
-    convert (gc_ideal 𝒜).u_infᵢ <;> exact HomogeneousIdeal.toIdeal_infᵢ _
+    convert(gc_ideal 𝒜).u_infᵢ <;> exact HomogeneousIdeal.toIdeal_infᵢ _
 #align projective_spectrum.vanishing_ideal_Union ProjectiveSpectrum.vanishingIdeal_unionᵢ
 
 theorem zeroLocus_inf (I J : Ideal A) :

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

Moves definition of and lemmas related to Set.Subsingleton and Set.Nontrivial to a new file, so that Basic can be shorter.

@@ -6,7 +6,7 @@ Authors: Jujian Zhang, Johan Commelin
 import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
 import Mathlib.Topology.Category.TopCat.Basic
 import Mathlib.Topology.Sets.Opens
-import Mathlib.Data.Set.Basic
+import Mathlib.Data.Set.Subsingleton
 
 #align_import algebraic_geometry.projective_spectrum.topology from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
 

Reference the newly created issues #12094 and #12096, as well as the pre-existing #5171. Change all references to #10927 to #5171. Some of these changes were not labelled as "porting note"; change this for good measure.

@@ -46,7 +46,7 @@ variable {R A : Type*}
 variable [CommSemiring R] [CommRing A] [Algebra R A]
 variable (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜]
 
--- porting note (#10927): removed @[nolint has_nonempty_instance]
+-- porting note (#5171): removed @[nolint has_nonempty_instance]
 /-- The projective spectrum of a graded commutative ring is the subtype of all homogenous ideals
 that are prime and do not contain the irrelevant ideal. -/
 @[ext]

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)

@@ -43,9 +43,7 @@ noncomputable section
 open DirectSum BigOperators Pointwise SetLike TopCat TopologicalSpace CategoryTheory Opposite
 
 variable {R A : Type*}
-
 variable [CommSemiring R] [CommRing A] [Algebra R A]
-
 variable (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜]
 
 -- porting note (#10927): removed @[nolint has_nonempty_instance]

Classifies by adding issue number (#10927) to porting notes claiming removed @[nolint has_nonempty_instance].

@@ -48,7 +48,7 @@ variable [CommSemiring R] [CommRing A] [Algebra R A]
 
 variable (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜]
 
--- porting note: removed @[nolint has_nonempty_instance]
+-- porting note (#10927): removed @[nolint has_nonempty_instance]
 /-- The projective spectrum of a graded commutative ring is the subtype of all homogenous ideals
 that are prime and do not contain the irrelevant ideal. -/
 @[ext]

• Remove Data.Set.Basic from scripts/noshake.json.
• Remove an exception that was used by examples only, move these examples to a new test file.
• Drop an exception for Order.Filter.Basic dependency on Control.Traversable.Instances, as the relevant parts were moved to Order.Filter.ListTraverse.
• Run lake exe shake --fix.

@@ -6,6 +6,7 @@ Authors: Jujian Zhang, Johan Commelin
 import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
 import Mathlib.Topology.Category.TopCat.Basic
 import Mathlib.Topology.Sets.Opens
+import Mathlib.Data.Set.Basic
 
 #align_import algebraic_geometry.projective_spectrum.topology from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
 

Mostly, this means replacing "of_open" by "of_isOpen". A few lemmas names were misleading and are corrected differently. Zulip discussion.

@@ -453,7 +453,7 @@ theorem basicOpen_eq_union_of_projection (f : A) :
 theorem isTopologicalBasis_basic_opens :
     TopologicalSpace.IsTopologicalBasis
       (Set.range fun r : A => (basicOpen 𝒜 r : Set (ProjectiveSpectrum 𝒜))) := by
-  apply TopologicalSpace.isTopologicalBasis_of_open_of_nhds
+  apply TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds
   · rintro _ ⟨r, rfl⟩
     exact isOpen_basicOpen 𝒜
   · rintro p U hp ⟨s, hs⟩

Replace rcases( with rcases (. Same thing for convert( and congrm(. No other change.

@@ -270,7 +270,7 @@ theorem zeroLocus_bUnion (s : Set (Set A)) :
 theorem vanishingIdeal_iUnion {γ : Sort*} (t : γ → Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (⋃ i, t i) = ⨅ i, vanishingIdeal (t i) :=
   HomogeneousIdeal.toIdeal_injective <| by
-    convert(gc_ideal 𝒜).u_iInf; exact HomogeneousIdeal.toIdeal_iInf _
+    convert (gc_ideal 𝒜).u_iInf; exact HomogeneousIdeal.toIdeal_iInf _
 #align projective_spectrum.vanishing_ideal_Union ProjectiveSpectrum.vanishingIdeal_iUnion
 
 theorem zeroLocus_inf (I J : Ideal A) :

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

@@ -371,7 +371,6 @@ theorem zeroLocus_vanishingIdeal_eq_closure (t : Set (ProjectiveSpectrum 𝒜))
 theorem vanishingIdeal_closure (t : Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (closure t) = vanishingIdeal t := by
   have := (gc_ideal 𝒜).u_l_u_eq_u t
-  dsimp only at this
   ext1
   erw [zeroLocus_vanishingIdeal_eq_closure 𝒜 t] at this
   exact this

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

This has nice performance benefits.

@@ -41,7 +41,7 @@ noncomputable section
 
 open DirectSum BigOperators Pointwise SetLike TopCat TopologicalSpace CategoryTheory Opposite
 
-variable {R A : Type _}
+variable {R A : Type*}
 
 variable [CommSemiring R] [CommRing A] [Algebra R A]
 
@@ -247,17 +247,17 @@ theorem vanishingIdeal_union (t t' : Set (ProjectiveSpectrum 𝒜)) :
   ext1; exact (gc_ideal 𝒜).u_inf
 #align projective_spectrum.vanishing_ideal_union ProjectiveSpectrum.vanishingIdeal_union
 
-theorem zeroLocus_iSup_ideal {γ : Sort _} (I : γ → Ideal A) :
+theorem zeroLocus_iSup_ideal {γ : Sort*} (I : γ → Ideal A) :
     zeroLocus _ ((⨆ i, I i : Ideal A) : Set A) = ⋂ i, zeroLocus 𝒜 (I i) :=
   (gc_ideal 𝒜).l_iSup
 #align projective_spectrum.zero_locus_supr_ideal ProjectiveSpectrum.zeroLocus_iSup_ideal
 
-theorem zeroLocus_iSup_homogeneousIdeal {γ : Sort _} (I : γ → HomogeneousIdeal 𝒜) :
+theorem zeroLocus_iSup_homogeneousIdeal {γ : Sort*} (I : γ → HomogeneousIdeal 𝒜) :
     zeroLocus _ ((⨆ i, I i : HomogeneousIdeal 𝒜) : Set A) = ⋂ i, zeroLocus 𝒜 (I i) :=
   (gc_homogeneousIdeal 𝒜).l_iSup
 #align projective_spectrum.zero_locus_supr_homogeneous_ideal ProjectiveSpectrum.zeroLocus_iSup_homogeneousIdeal
 
-theorem zeroLocus_iUnion {γ : Sort _} (s : γ → Set A) :
+theorem zeroLocus_iUnion {γ : Sort*} (s : γ → Set A) :
     zeroLocus 𝒜 (⋃ i, s i) = ⋂ i, zeroLocus 𝒜 (s i) :=
   (gc_set 𝒜).l_iSup
 #align projective_spectrum.zero_locus_Union ProjectiveSpectrum.zeroLocus_iUnion
@@ -267,7 +267,7 @@ theorem zeroLocus_bUnion (s : Set (Set A)) :
   by simp only [zeroLocus_iUnion]
 #align projective_spectrum.zero_locus_bUnion ProjectiveSpectrum.zeroLocus_bUnion
 
-theorem vanishingIdeal_iUnion {γ : Sort _} (t : γ → Set (ProjectiveSpectrum 𝒜)) :
+theorem vanishingIdeal_iUnion {γ : Sort*} (t : γ → Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (⋃ i, t i) = ⨅ i, vanishingIdeal (t i) :=
   HomogeneousIdeal.toIdeal_injective <| by
     convert(gc_ideal 𝒜).u_iInf; exact HomogeneousIdeal.toIdeal_iInf _

• 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) 2020 Jujian Zhang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jujian Zhang, Johan Commelin
-
-! This file was ported from Lean 3 source module algebraic_geometry.projective_spectrum.topology
-! 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.RingTheory.GradedAlgebra.HomogeneousIdeal
 import Mathlib.Topology.Category.TopCat.Basic
 import Mathlib.Topology.Sets.Opens
 
+#align_import algebraic_geometry.projective_spectrum.topology from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
+
 /-!
 # Projective spectrum of a graded ring
 

This is the second half of the changes originally in #5699, removing all occurrences of ; after a space and implementing a linter rule to enforce it.

In most cases this 2-character substring has a space after it, so the following command was run first:

find . -type f -name "*.lean" -exec sed -i -E 's/ ; /; /g' {} \;

The remaining cases were few enough in number that they were done manually.

@@ -247,7 +247,7 @@ theorem zeroLocus_union (s s' : Set A) : zeroLocus 𝒜 (s ∪ s') = zeroLocus _
 
 theorem vanishingIdeal_union (t t' : Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (t ∪ t') = vanishingIdeal t ⊓ vanishingIdeal t' := by
-  ext1 ; exact (gc_ideal 𝒜).u_inf
+  ext1; exact (gc_ideal 𝒜).u_inf
 #align projective_spectrum.vanishing_ideal_union ProjectiveSpectrum.vanishingIdeal_union
 
 theorem zeroLocus_iSup_ideal {γ : Sort _} (I : γ → Ideal A) :
@@ -273,7 +273,7 @@ theorem zeroLocus_bUnion (s : Set (Set A)) :
 theorem vanishingIdeal_iUnion {γ : Sort _} (t : γ → Set (ProjectiveSpectrum 𝒜)) :
     vanishingIdeal (⋃ i, t i) = ⨅ i, vanishingIdeal (t i) :=
   HomogeneousIdeal.toIdeal_injective <| by
-    convert(gc_ideal 𝒜).u_iInf ; exact HomogeneousIdeal.toIdeal_iInf _
+    convert(gc_ideal 𝒜).u_iInf; exact HomogeneousIdeal.toIdeal_iInf _
 #align projective_spectrum.vanishing_ideal_Union ProjectiveSpectrum.vanishingIdeal_iUnion
 
 theorem zeroLocus_inf (I J : Ideal A) :
@@ -320,7 +320,7 @@ theorem sup_vanishingIdeal_le (t t' : Set (ProjectiveSpectrum 𝒜)) :
 
 theorem mem_compl_zeroLocus_iff_not_mem {f : A} {I : ProjectiveSpectrum 𝒜} :
     I ∈ (zeroLocus 𝒜 {f} : Set (ProjectiveSpectrum 𝒜))ᶜ ↔ f ∉ I.asHomogeneousIdeal := by
-  rw [Set.mem_compl_iff, mem_zeroLocus, Set.singleton_subset_iff] ; rfl
+  rw [Set.mem_compl_iff, mem_zeroLocus, Set.singleton_subset_iff]; rfl
 #align projective_spectrum.mem_compl_zero_locus_iff_not_mem ProjectiveSpectrum.mem_compl_zeroLocus_iff_not_mem
 
 /-- The Zariski topology on the prime spectrum of a commutative ring is defined via the closed sets
@@ -348,7 +348,7 @@ set_option linter.uppercaseLean3 false in
 #align projective_spectrum.Top ProjectiveSpectrum.top
 
 theorem isOpen_iff (U : Set (ProjectiveSpectrum 𝒜)) : IsOpen U ↔ ∃ s, Uᶜ = zeroLocus 𝒜 s := by
-  simp only [@eq_comm _ Uᶜ] ; rfl
+  simp only [@eq_comm _ Uᶜ]; rfl
 #align projective_spectrum.is_open_iff ProjectiveSpectrum.isOpen_iff
 
 theorem isClosed_iff_zeroLocus (Z : Set (ProjectiveSpectrum 𝒜)) :
@@ -406,7 +406,7 @@ theorem isOpen_basicOpen {a : A} : IsOpen (basicOpen 𝒜 a : Set (ProjectiveSpe
 @[simp]
 theorem basicOpen_eq_zeroLocus_compl (r : A) :
     (basicOpen 𝒜 r : Set (ProjectiveSpectrum 𝒜)) = (zeroLocus 𝒜 {r})ᶜ :=
-  Set.ext fun x => by simp only [Set.mem_compl_iff, mem_zeroLocus, Set.singleton_subset_iff] ; rfl
+  Set.ext fun x => by simp only [Set.mem_compl_iff, mem_zeroLocus, Set.singleton_subset_iff]; rfl
 #align projective_spectrum.basic_open_eq_zero_locus_compl ProjectiveSpectrum.basicOpen_eq_zeroLocus_compl
 
 @[simp]

Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>

@@ -348,7 +348,7 @@ set_option linter.uppercaseLean3 false in
 #align projective_spectrum.Top ProjectiveSpectrum.top
 
 theorem isOpen_iff (U : Set (ProjectiveSpectrum 𝒜)) : IsOpen U ↔ ∃ s, Uᶜ = zeroLocus 𝒜 s := by
-  simp only [@eq_comm _ (Uᶜ)] ; rfl
+  simp only [@eq_comm _ Uᶜ] ; rfl
 #align projective_spectrum.is_open_iff ProjectiveSpectrum.isOpen_iff
 
 theorem isClosed_iff_zeroLocus (Z : Set (ProjectiveSpectrum 𝒜)) :
@@ -405,7 +405,7 @@ theorem isOpen_basicOpen {a : A} : IsOpen (basicOpen 𝒜 a : Set (ProjectiveSpe
 
 @[simp]
 theorem basicOpen_eq_zeroLocus_compl (r : A) :
-    (basicOpen 𝒜 r : Set (ProjectiveSpectrum 𝒜)) = zeroLocus 𝒜 {r}ᶜ :=
+    (basicOpen 𝒜 r : Set (ProjectiveSpectrum 𝒜)) = (zeroLocus 𝒜 {r})ᶜ :=
   Set.ext fun x => by simp only [Set.mem_compl_iff, mem_zeroLocus, Set.singleton_subset_iff] ; rfl
 #align projective_spectrum.basic_open_eq_zero_locus_compl ProjectiveSpectrum.basicOpen_eq_zeroLocus_compl
 

We disable the "relaxed" auto-implicit feature, so only single character identifiers become eligible as auto-implicits. See discussion on zulip and 2.

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

@@ -57,7 +57,7 @@ that are prime and do not contain the irrelevant ideal. -/
 structure ProjectiveSpectrum where
   asHomogeneousIdeal : HomogeneousIdeal 𝒜
   isPrime : asHomogeneousIdeal.toIdeal.IsPrime
-  not_irrelevant_le : ¬HomogeneousIdeal.irrelevant 𝒜 ≤ as_homogeneous_ideal
+  not_irrelevant_le : ¬HomogeneousIdeal.irrelevant 𝒜 ≤ asHomogeneousIdeal
 #align projective_spectrum ProjectiveSpectrum
 
 attribute [instance] ProjectiveSpectrum.isPrime

@@ -97,7 +97,7 @@ At a point `x` (a homogeneous prime ideal) the function (i.e., element) `f` take
 quotient ring `A` modulo the prime ideal `x`. In this manner, `vanishingIdeal t` is exactly the
 ideal of `A` consisting of all "functions" that vanish on all of `t`. -/
 def vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) : HomogeneousIdeal 𝒜 :=
-  ⨅ (x : ProjectiveSpectrum 𝒜) (_h : x ∈ t), x.asHomogeneousIdeal
+  ⨅ (x : ProjectiveSpectrum 𝒜) (_ : x ∈ t), x.asHomogeneousIdeal
 #align projective_spectrum.vanishing_ideal ProjectiveSpectrum.vanishingIdeal
 
 theorem coe_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :