algebraic_geometry.prime_spectrum.basic ⟷ Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic

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

@@ -8,7 +8,7 @@ import LinearAlgebra.Finsupp
 import RingTheory.Ideal.Over
 import RingTheory.Ideal.Prod
 import RingTheory.Localization.Away.Basic
-import RingTheory.Nilpotent
+import RingTheory.Nilpotent.Defs
 import Topology.Sets.Closeds
 import Topology.Sober
 

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

@@ -3,7 +3,7 @@ Copyright (c) 2020 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
 -/
-import Algebra.PunitInstances
+import Algebra.PUnitInstances
 import LinearAlgebra.Finsupp
 import RingTheory.Ideal.Over
 import RingTheory.Ideal.Prod

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

@@ -925,7 +925,7 @@ theorem isClosed_range_comap_of_surjective (hf : Surjective f) : IsClosed (Set.r
 theorem closedEmbedding_comap_of_surjective (hf : Surjective f) : ClosedEmbedding (comap f) :=
   { induced := (comap_inducing_of_surjective S f hf).induced
     inj := comap_injective_of_surjective f hf
-    closed_range := isClosed_range_comap_of_surjective S f hf }
+    isClosed_range := isClosed_range_comap_of_surjective S f hf }
 #align prime_spectrum.closed_embedding_comap_of_surjective PrimeSpectrum.closedEmbedding_comap_of_surjective
 -/
 

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

@@ -105,7 +105,7 @@ noncomputable def primeSpectrumProd :
       (by
         constructor
         · rintro (⟨I, hI⟩ | ⟨J, hJ⟩) (⟨I', hI'⟩ | ⟨J', hJ'⟩) h <;>
-            simp only [Ideal.prod.ext_iff, prime_spectrum_prod_of_sum] at h 
+            simp only [Ideal.prod.ext_iff, prime_spectrum_prod_of_sum] at h
           · simp only [h]
           · exact False.elim (hI.ne_top h.left)
           · exact False.elim (hJ.ne_top h.right)
@@ -349,7 +349,7 @@ theorem zeroLocus_empty_of_one_mem {s : Set R} (h : (1 : R) ∈ s) : zeroLocus s
   by
   rw [Set.eq_empty_iff_forall_not_mem]
   intro x hx
-  rw [mem_zero_locus] at hx 
+  rw [mem_zero_locus] at hx
   have x_prime : x.as_ideal.is_prime := by infer_instance
   have eq_top : x.as_ideal = ⊤ := by rw [Ideal.eq_top_iff_one]; exact hx h
   apply x_prime.ne_top eq_top
@@ -487,7 +487,7 @@ theorem sup_vanishingIdeal_le (t t' : Set (PrimeSpectrum R)) :
   intro r
   rw [Submodule.mem_sup, mem_vanishing_ideal]
   rintro ⟨f, hf, g, hg, rfl⟩ x ⟨hxt, hxt'⟩
-  rw [mem_vanishing_ideal] at hf hg 
+  rw [mem_vanishing_ideal] at hf hg
   apply Submodule.add_mem <;> solve_by_elim
 #align prime_spectrum.sup_vanishing_ideal_le PrimeSpectrum.sup_vanishingIdeal_le
 -/
@@ -555,7 +555,7 @@ theorem isClosed_singleton_iff_isMaximal (x : PrimeSpectrum R) :
   by
   refine' (is_closed_iff_zero_locus _).trans ⟨fun h => _, fun h => _⟩
   · obtain ⟨s, hs⟩ := h
-    rw [eq_comm, Set.eq_singleton_iff_unique_mem] at hs 
+    rw [eq_comm, Set.eq_singleton_iff_unique_mem] at hs
     refine'
       ⟨⟨x.2.1, fun I hI =>
           Classical.not_not.1
@@ -577,8 +577,8 @@ theorem zeroLocus_vanishingIdeal_eq_closure (t : Set (PrimeSpectrum R)) :
   by
   apply Set.Subset.antisymm
   · rintro x hx t' ⟨ht', ht⟩
-    obtain ⟨fs, rfl⟩ : ∃ s, t' = zero_locus 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 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 t
@@ -827,8 +827,8 @@ theorem localization_comap_injective [Algebra R S] (M : Submonoid R) [IsLocaliza
   by
   intro p q h
   replace h := congr_arg (fun x : PrimeSpectrum R => Ideal.map (algebraMap R S) x.asIdeal) h
-  dsimp only at h 
-  erw [IsLocalization.map_comap M S, IsLocalization.map_comap M S] at h 
+  dsimp only at h
+  erw [IsLocalization.map_comap M S, IsLocalization.map_comap M S] at h
   ext1
   exact h
 #align prime_spectrum.localization_comap_injective PrimeSpectrum.localization_comap_injective
@@ -1021,7 +1021,7 @@ theorem isTopologicalBasis_basic_opens :
   · rintro _ ⟨r, rfl⟩
     exact is_open_basic_open
   · 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 f, ⟨f, rfl⟩, hfp, _⟩
     rw [← Set.compl_subset_compl, ← hs, basic_open_eq_zero_locus_compl, compl_compl]
@@ -1046,7 +1046,7 @@ theorem isCompact_basicOpen (f : R) : IsCompact (basicOpen f : Set (PrimeSpectru
     have hI : ∀ i, Z i = zero_locus (I i) := fun i => by
       simpa only [zero_locus_vanishing_ideal_eq_closure] using (hZc i).closure_eq.symm
     rw [basic_open_eq_zero_locus_compl f, Set.inter_comm, ← Set.diff_eq, Set.diff_eq_empty,
-      funext hI, ← zero_locus_supr] at hZ 
+      funext hI, ← zero_locus_supr] at hZ
     obtain ⟨n, hn⟩ : f ∈ (⨆ i : ι, I i).radical :=
       by
       rw [← vanishing_ideal_zero_locus_eq_radical]
@@ -1179,7 +1179,7 @@ def localizationMapOfSpecializes {x y : PrimeSpectrum R} (h : x ⤳ y) :
     (by
       rintro ⟨a, ha⟩
       rw [← PrimeSpectrum.le_iff_specializes, ← as_ideal_le_as_ideal, ← SetLike.coe_subset_coe, ←
-        Set.compl_subset_compl] at h 
+        Set.compl_subset_compl] at h
       exact (IsLocalization.map_units _ ⟨a, show a ∈ x.as_ideal.prime_compl from h ha⟩ : _))
 #align prime_spectrum.localization_map_of_specializes PrimeSpectrum.localizationMapOfSpecializes
 -/

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

@@ -1017,7 +1017,7 @@ theorem isTopologicalBasis_basic_opens :
     TopologicalSpace.IsTopologicalBasis
       (Set.range fun r : R => (basicOpen r : Set (PrimeSpectrum R))) :=
   by
-  apply TopologicalSpace.isTopologicalBasis_of_open_of_nhds
+  apply TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds
   · rintro _ ⟨r, rfl⟩
     exact is_open_basic_open
   · rintro p U hp ⟨s, hs⟩

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

@@ -675,7 +675,7 @@ theorem isIrreducible_zeroLocus_iff_of_radical (I : Ideal R) (hI : I.IsRadical)
         refine' fun h x y h' => h _ _ _
         rw [← hI.radical_le_iff] at h' ⊢
         simpa only [Ideal.radical_inf, Ideal.radical_mul] using h'
-      · simp_rw [or_iff_not_imp_left, SetLike.not_le_iff_exists]
+      · simp_rw [Classical.or_iff_not_imp_left, SetLike.not_le_iff_exists]
         rintro h s t h' ⟨x, hx, hx'⟩ y hy
         exact h (h' ⟨Ideal.mul_mem_right _ _ hx, Ideal.mul_mem_left _ _ hy⟩) hx'
 #align prime_spectrum.is_irreducible_zero_locus_iff_of_radical PrimeSpectrum.isIrreducible_zeroLocus_iff_of_radical

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

@@ -79,12 +79,10 @@ instance [Nontrivial R] : Nonempty <| PrimeSpectrum R :=
   let ⟨I, hI⟩ := Ideal.exists_maximal R
   ⟨⟨I, hI.IsPrime⟩⟩
 
-#print PrimeSpectrum.punit /-
 /-- The prime spectrum of the zero ring is empty. -/
-theorem punit (x : PrimeSpectrum PUnit) : False :=
+theorem pUnit (x : PrimeSpectrum PUnit) : False :=
   x.1.ne_top_iff_one.1 x.2.1 <| Subsingleton.elim (0 : PUnit) 1 ▸ x.1.zero_mem
-#align prime_spectrum.punit PrimeSpectrum.punit
--/
+#align prime_spectrum.punit PrimeSpectrum.pUnit
 
 variable (R S)
 

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

@@ -3,14 +3,14 @@ Copyright (c) 2020 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
 -/
-import Mathbin.Algebra.PunitInstances
-import Mathbin.LinearAlgebra.Finsupp
-import Mathbin.RingTheory.Ideal.Over
-import Mathbin.RingTheory.Ideal.Prod
-import Mathbin.RingTheory.Localization.Away.Basic
-import Mathbin.RingTheory.Nilpotent
-import Mathbin.Topology.Sets.Closeds
-import Mathbin.Topology.Sober
+import Algebra.PunitInstances
+import LinearAlgebra.Finsupp
+import RingTheory.Ideal.Over
+import RingTheory.Ideal.Prod
+import RingTheory.Localization.Away.Basic
+import RingTheory.Nilpotent
+import Topology.Sets.Closeds
+import Topology.Sober
 
 #align_import algebraic_geometry.prime_spectrum.basic from "leanprover-community/mathlib"@"a87d22575d946e1e156fc1edd1e1269600a8a282"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ffde2d8a6e689149e44fd95fa862c23a57f8c780

@@ -79,11 +79,11 @@ instance [Nontrivial R] : Nonempty <| PrimeSpectrum R :=
   let ⟨I, hI⟩ := Ideal.exists_maximal R
   ⟨⟨I, hI.IsPrime⟩⟩
 
-#print PrimeSpectrum.pUnit /-
+#print PrimeSpectrum.punit /-
 /-- The prime spectrum of the zero ring is empty. -/
-theorem pUnit (x : PrimeSpectrum PUnit) : False :=
+theorem punit (x : PrimeSpectrum PUnit) : False :=
   x.1.ne_top_iff_one.1 x.2.1 <| Subsingleton.elim (0 : PUnit) 1 ▸ x.1.zero_mem
-#align prime_spectrum.punit PrimeSpectrum.pUnit
+#align prime_spectrum.punit PrimeSpectrum.punit
 -/
 
 variable (R S)

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

@@ -808,7 +808,7 @@ variable (S)
 theorem localization_comap_inducing [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
     Inducing (comap (algebraMap R S)) := by
   constructor
-  rw [topologicalSpace_eq_iff]
+  rw [TopologicalSpace.ext_iff]
   intro U
   simp_rw [← isClosed_compl_iff]
   generalize Uᶜ = Z
@@ -871,7 +871,7 @@ theorem comap_inducing_of_surjective (hf : Surjective f) : Inducing (comap f) :=
   {
     induced :=
       by
-      simp_rw [topologicalSpace_eq_iff, ← isClosed_compl_iff, isClosed_induced_iff,
+      simp_rw [TopologicalSpace.ext_iff, ← isClosed_compl_iff, isClosed_induced_iff,
         is_closed_iff_zero_locus]
       refine' fun s =>
         ⟨fun ⟨F, hF⟩ =>

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

@@ -853,7 +853,7 @@ theorem localization_comap_range [Algebra R S] (M : Submonoid R) [IsLocalization
     rintro ⟨p, rfl⟩ x ⟨hx₁, hx₂⟩
     exact (p.2.1 : ¬_) (p.as_ideal.eq_top_of_is_unit_mem hx₂ (IsLocalization.map_units S ⟨x, hx₁⟩))
   · intro h
-    use ⟨x.as_ideal.map (algebraMap R S), IsLocalization.isPrime_of_isPrime_disjoint M S _ x.2 h⟩
+    use⟨x.as_ideal.map (algebraMap R S), IsLocalization.isPrime_of_isPrime_disjoint M S _ x.2 h⟩
     ext1
     exact IsLocalization.comap_map_of_isPrime_disjoint M S _ x.2 h
 #align prime_spectrum.localization_comap_range PrimeSpectrum.localization_comap_range

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

@@ -2,11 +2,6 @@
 Copyright (c) 2020 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
-
-! This file was ported from Lean 3 source module algebraic_geometry.prime_spectrum.basic
-! leanprover-community/mathlib commit a87d22575d946e1e156fc1edd1e1269600a8a282
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.PunitInstances
 import Mathbin.LinearAlgebra.Finsupp
@@ -17,6 +12,8 @@ import Mathbin.RingTheory.Nilpotent
 import Mathbin.Topology.Sets.Closeds
 import Mathbin.Topology.Sober
 
+#align_import algebraic_geometry.prime_spectrum.basic from "leanprover-community/mathlib"@"a87d22575d946e1e156fc1edd1e1269600a8a282"
+
 /-!
 # Prime spectrum of a commutative ring
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/728ef9dbb281241906f25cbeb30f90d83e0bb451

@@ -800,7 +800,7 @@ theorem comap_singleton_isClosed_of_isIntegral (f : R →+* S) (hf : f.IsIntegra
     (x : PrimeSpectrum S) (hx : IsClosed ({x} : Set (PrimeSpectrum S))) :
     IsClosed ({comap f x} : Set (PrimeSpectrum R)) :=
   (isClosed_singleton_iff_isMaximal _).2
-    (Ideal.isMaximal_comap_of_isIntegral_of_is_maximal' f hf x.asIdeal <|
+    (Ideal.isMaximal_comap_of_isIntegral_of_isMaximal' f hf x.asIdeal <|
       (isClosed_singleton_iff_isMaximal x).1 hx)
 #align prime_spectrum.comap_singleton_is_closed_of_is_integral PrimeSpectrum.comap_singleton_isClosed_of_isIntegral
 -/

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

@@ -91,13 +91,16 @@ theorem pUnit (x : PrimeSpectrum PUnit) : False :=
 
 variable (R S)
 
+#print PrimeSpectrum.primeSpectrumProdOfSum /-
 /-- The map from the direct sum of prime spectra to the prime spectrum of a direct product. -/
 @[simp]
 def primeSpectrumProdOfSum : Sum (PrimeSpectrum R) (PrimeSpectrum S) → PrimeSpectrum (R × S)
   | Sum.inl ⟨I, hI⟩ => ⟨Ideal.prod I ⊤, Ideal.isPrime_ideal_prod_top⟩
   | Sum.inr ⟨J, hJ⟩ => ⟨Ideal.prod ⊤ J, Ideal.isPrime_ideal_prod_top'⟩
 #align prime_spectrum.prime_spectrum_prod_of_sum PrimeSpectrum.primeSpectrumProdOfSum
+-/
 
+#print PrimeSpectrum.primeSpectrumProd /-
 /-- The prime spectrum of `R × S` is in bijection with the disjoint unions of the prime spectrum of
 `R` and the prime spectrum of `S`. -/
 noncomputable def primeSpectrumProd :
@@ -117,18 +120,23 @@ noncomputable def primeSpectrumProd :
           · exact ⟨Sum.inl ⟨p, hp⟩, rfl⟩
           · exact ⟨Sum.inr ⟨p, hp⟩, rfl⟩)
 #align prime_spectrum.prime_spectrum_prod PrimeSpectrum.primeSpectrumProd
+-/
 
 variable {R S}
 
+#print PrimeSpectrum.primeSpectrumProd_symm_inl_asIdeal /-
 @[simp]
 theorem primeSpectrumProd_symm_inl_asIdeal (x : PrimeSpectrum R) :
     ((primeSpectrumProd R S).symm <| Sum.inl x).asIdeal = Ideal.prod x.asIdeal ⊤ := by cases x; rfl
 #align prime_spectrum.prime_spectrum_prod_symm_inl_as_ideal PrimeSpectrum.primeSpectrumProd_symm_inl_asIdeal
+-/
 
+#print PrimeSpectrum.primeSpectrumProd_symm_inr_asIdeal /-
 @[simp]
 theorem primeSpectrumProd_symm_inr_asIdeal (x : PrimeSpectrum S) :
     ((primeSpectrumProd R S).symm <| Sum.inr x).asIdeal = Ideal.prod ⊤ x.asIdeal := by cases x; rfl
 #align prime_spectrum.prime_spectrum_prod_symm_inr_as_ideal PrimeSpectrum.primeSpectrumProd_symm_inr_asIdeal
+-/
 
 #print PrimeSpectrum.zeroLocus /-
 /-- The zero locus of a set `s` of elements of a commutative ring `R` is the set of all prime ideals
@@ -197,23 +205,28 @@ theorem vanishingIdeal_singleton (x : PrimeSpectrum R) :
 #align prime_spectrum.vanishing_ideal_singleton PrimeSpectrum.vanishingIdeal_singleton
 -/
 
+#print PrimeSpectrum.subset_zeroLocus_iff_le_vanishingIdeal /-
 theorem subset_zeroLocus_iff_le_vanishingIdeal (t : Set (PrimeSpectrum R)) (I : Ideal R) :
     t ⊆ zeroLocus I ↔ I ≤ vanishingIdeal t :=
   ⟨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 prime_spectrum.subset_zero_locus_iff_le_vanishing_ideal PrimeSpectrum.subset_zeroLocus_iff_le_vanishingIdeal
+-/
 
 section Gc
 
 variable (R)
 
+#print PrimeSpectrum.gc /-
 /-- `zero_locus` and `vanishing_ideal` form a galois connection. -/
 theorem gc :
     @GaloisConnection (Ideal R) (Set (PrimeSpectrum R))ᵒᵈ _ _ (fun I => zeroLocus I) fun t =>
       vanishingIdeal t :=
   fun I t => subset_zeroLocus_iff_le_vanishingIdeal t I
 #align prime_spectrum.gc PrimeSpectrum.gc
+-/
 
+#print PrimeSpectrum.gc_set /-
 /-- `zero_locus` and `vanishing_ideal` form a galois connection. -/
 theorem gc_set :
     @GaloisConnection (Set R) (Set (PrimeSpectrum R))ᵒᵈ _ _ (fun s => zeroLocus s) fun t =>
@@ -222,6 +235,7 @@ theorem gc_set :
   have ideal_gc : GaloisConnection Ideal.span coe := (Submodule.gi R R).gc
   simpa [zero_locus_span, Function.comp] using ideal_gc.compose (gc R)
 #align prime_spectrum.gc_set PrimeSpectrum.gc_set
+-/
 
 #print PrimeSpectrum.subset_zeroLocus_iff_subset_vanishingIdeal /-
 theorem subset_zeroLocus_iff_subset_vanishingIdeal (t : Set (PrimeSpectrum R)) (s : Set R) :
@@ -238,9 +252,11 @@ theorem subset_vanishingIdeal_zeroLocus (s : Set R) : s ⊆ vanishingIdeal (zero
 #align prime_spectrum.subset_vanishing_ideal_zero_locus PrimeSpectrum.subset_vanishingIdeal_zeroLocus
 -/
 
+#print PrimeSpectrum.le_vanishingIdeal_zeroLocus /-
 theorem le_vanishingIdeal_zeroLocus (I : Ideal R) : I ≤ vanishingIdeal (zeroLocus I) :=
   (gc R).le_u_l I
 #align prime_spectrum.le_vanishing_ideal_zero_locus PrimeSpectrum.le_vanishingIdeal_zeroLocus
+-/
 
 #print PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radical /-
 @[simp]
@@ -273,16 +289,21 @@ theorem zeroLocus_anti_mono {s t : Set R} (h : s ⊆ t) : zeroLocus t ⊆ zeroLo
 #align prime_spectrum.zero_locus_anti_mono PrimeSpectrum.zeroLocus_anti_mono
 -/
 
+#print PrimeSpectrum.zeroLocus_anti_mono_ideal /-
 theorem zeroLocus_anti_mono_ideal {s t : Ideal R} (h : s ≤ t) :
     zeroLocus (t : Set R) ⊆ zeroLocus (s : Set R) :=
   (gc R).monotone_l h
 #align prime_spectrum.zero_locus_anti_mono_ideal PrimeSpectrum.zeroLocus_anti_mono_ideal
+-/
 
+#print PrimeSpectrum.vanishingIdeal_anti_mono /-
 theorem vanishingIdeal_anti_mono {s t : Set (PrimeSpectrum R)} (h : s ⊆ t) :
     vanishingIdeal t ≤ vanishingIdeal s :=
   (gc R).monotone_u h
 #align prime_spectrum.vanishing_ideal_anti_mono PrimeSpectrum.vanishingIdeal_anti_mono
+-/
 
+#print PrimeSpectrum.zeroLocus_subset_zeroLocus_iff /-
 theorem zeroLocus_subset_zeroLocus_iff (I J : Ideal R) :
     zeroLocus (I : Set R) ⊆ zeroLocus (J : Set R) ↔ J ≤ I.radical :=
   ⟨fun h =>
@@ -291,6 +312,7 @@ theorem zeroLocus_subset_zeroLocus_iff (I J : Ideal R) :
         vanishingIdeal_zeroLocus_eq_radical J ▸ vanishingIdeal_anti_mono h),
     fun h => zeroLocus_radical I ▸ zeroLocus_anti_mono_ideal h⟩
 #align prime_spectrum.zero_locus_subset_zero_locus_iff PrimeSpectrum.zeroLocus_subset_zeroLocus_iff
+-/
 
 #print PrimeSpectrum.zeroLocus_subset_zeroLocus_singleton_iff /-
 theorem zeroLocus_subset_zeroLocus_singleton_iff (f g : R) :
@@ -300,14 +322,18 @@ theorem zeroLocus_subset_zeroLocus_singleton_iff (f g : R) :
 #align prime_spectrum.zero_locus_subset_zero_locus_singleton_iff PrimeSpectrum.zeroLocus_subset_zeroLocus_singleton_iff
 -/
 
+#print PrimeSpectrum.zeroLocus_bot /-
 theorem zeroLocus_bot : zeroLocus ((⊥ : Ideal R) : Set R) = Set.univ :=
   (gc R).l_bot
 #align prime_spectrum.zero_locus_bot PrimeSpectrum.zeroLocus_bot
+-/
 
+#print PrimeSpectrum.zeroLocus_singleton_zero /-
 @[simp]
 theorem zeroLocus_singleton_zero : zeroLocus ({0} : Set R) = Set.univ :=
   zeroLocus_bot
 #align prime_spectrum.zero_locus_singleton_zero PrimeSpectrum.zeroLocus_singleton_zero
+-/
 
 #print PrimeSpectrum.zeroLocus_empty /-
 @[simp]
@@ -316,11 +342,14 @@ theorem zeroLocus_empty : zeroLocus (∅ : Set R) = Set.univ :=
 #align prime_spectrum.zero_locus_empty PrimeSpectrum.zeroLocus_empty
 -/
 
+#print PrimeSpectrum.vanishingIdeal_univ /-
 @[simp]
 theorem vanishingIdeal_univ : vanishingIdeal (∅ : Set (PrimeSpectrum R)) = ⊤ := by
   simpa using (gc R).u_top
 #align prime_spectrum.vanishing_ideal_univ PrimeSpectrum.vanishingIdeal_univ
+-/
 
+#print PrimeSpectrum.zeroLocus_empty_of_one_mem /-
 theorem zeroLocus_empty_of_one_mem {s : Set R} (h : (1 : R) ∈ s) : zeroLocus s = ∅ :=
   by
   rw [Set.eq_empty_iff_forall_not_mem]
@@ -330,12 +359,16 @@ theorem zeroLocus_empty_of_one_mem {s : Set R} (h : (1 : R) ∈ s) : zeroLocus s
   have eq_top : x.as_ideal = ⊤ := by rw [Ideal.eq_top_iff_one]; exact hx h
   apply x_prime.ne_top eq_top
 #align prime_spectrum.zero_locus_empty_of_one_mem PrimeSpectrum.zeroLocus_empty_of_one_mem
+-/
 
+#print PrimeSpectrum.zeroLocus_singleton_one /-
 @[simp]
 theorem zeroLocus_singleton_one : zeroLocus ({1} : Set R) = ∅ :=
   zeroLocus_empty_of_one_mem (Set.mem_singleton (1 : R))
 #align prime_spectrum.zero_locus_singleton_one PrimeSpectrum.zeroLocus_singleton_one
+-/
 
+#print PrimeSpectrum.zeroLocus_empty_iff_eq_top /-
 theorem zeroLocus_empty_iff_eq_top {I : Ideal R} : zeroLocus (I : Set R) = ∅ ↔ I = ⊤ :=
   by
   constructor
@@ -345,6 +378,7 @@ theorem zeroLocus_empty_iff_eq_top {I : Ideal R} : zeroLocus (I : Set R) = ∅ 
     exact Set.Nonempty.ne_empty ⟨⟨M, hM.is_prime⟩, hIM⟩
   · rintro rfl; apply zero_locus_empty_of_one_mem; trivial
 #align prime_spectrum.zero_locus_empty_iff_eq_top PrimeSpectrum.zeroLocus_empty_iff_eq_top
+-/
 
 #print PrimeSpectrum.zeroLocus_univ /-
 @[simp]
@@ -353,19 +387,25 @@ theorem zeroLocus_univ : zeroLocus (Set.univ : Set R) = ∅ :=
 #align prime_spectrum.zero_locus_univ PrimeSpectrum.zeroLocus_univ
 -/
 
+#print PrimeSpectrum.vanishingIdeal_eq_top_iff /-
 theorem vanishingIdeal_eq_top_iff {s : Set (PrimeSpectrum R)} : vanishingIdeal s = ⊤ ↔ s = ∅ := by
   rw [← top_le_iff, ← subset_zero_locus_iff_le_vanishing_ideal, Submodule.top_coe, zero_locus_univ,
     Set.subset_empty_iff]
 #align prime_spectrum.vanishing_ideal_eq_top_iff PrimeSpectrum.vanishingIdeal_eq_top_iff
+-/
 
+#print PrimeSpectrum.zeroLocus_sup /-
 theorem zeroLocus_sup (I J : Ideal R) :
     zeroLocus ((I ⊔ J : Ideal R) : Set R) = zeroLocus I ∩ zeroLocus J :=
   (gc R).l_sup
 #align prime_spectrum.zero_locus_sup PrimeSpectrum.zeroLocus_sup
+-/
 
+#print PrimeSpectrum.zeroLocus_union /-
 theorem zeroLocus_union (s s' : Set R) : zeroLocus (s ∪ s') = zeroLocus s ∩ zeroLocus s' :=
   (gc_set R).l_sup
 #align prime_spectrum.zero_locus_union PrimeSpectrum.zeroLocus_union
+-/
 
 #print PrimeSpectrum.vanishingIdeal_union /-
 theorem vanishingIdeal_union (t t' : Set (PrimeSpectrum R)) :
@@ -374,15 +414,19 @@ theorem vanishingIdeal_union (t t' : Set (PrimeSpectrum R)) :
 #align prime_spectrum.vanishing_ideal_union PrimeSpectrum.vanishingIdeal_union
 -/
 
+#print PrimeSpectrum.zeroLocus_iSup /-
 theorem zeroLocus_iSup {ι : Sort _} (I : ι → Ideal R) :
     zeroLocus ((⨆ i, I i : Ideal R) : Set R) = ⋂ i, zeroLocus (I i) :=
   (gc R).l_iSup
 #align prime_spectrum.zero_locus_supr PrimeSpectrum.zeroLocus_iSup
+-/
 
+#print PrimeSpectrum.zeroLocus_iUnion /-
 theorem zeroLocus_iUnion {ι : Sort _} (s : ι → Set R) :
     zeroLocus (⋃ i, s i) = ⋂ i, zeroLocus (s i) :=
   (gc_set R).l_iSup
 #align prime_spectrum.zero_locus_Union PrimeSpectrum.zeroLocus_iUnion
+-/
 
 #print PrimeSpectrum.zeroLocus_bUnion /-
 theorem zeroLocus_bUnion (s : Set (Set R)) :
@@ -390,30 +434,40 @@ theorem zeroLocus_bUnion (s : Set (Set R)) :
 #align prime_spectrum.zero_locus_bUnion PrimeSpectrum.zeroLocus_bUnion
 -/
 
+#print PrimeSpectrum.vanishingIdeal_iUnion /-
 theorem vanishingIdeal_iUnion {ι : Sort _} (t : ι → Set (PrimeSpectrum R)) :
     vanishingIdeal (⋃ i, t i) = ⨅ i, vanishingIdeal (t i) :=
   (gc R).u_iInf
 #align prime_spectrum.vanishing_ideal_Union PrimeSpectrum.vanishingIdeal_iUnion
+-/
 
+#print PrimeSpectrum.zeroLocus_inf /-
 theorem zeroLocus_inf (I J : Ideal R) :
     zeroLocus ((I ⊓ J : Ideal R) : Set R) = zeroLocus I ∪ zeroLocus J :=
   Set.ext fun x => x.2.inf_le
 #align prime_spectrum.zero_locus_inf PrimeSpectrum.zeroLocus_inf
+-/
 
+#print PrimeSpectrum.union_zeroLocus /-
 theorem union_zeroLocus (s s' : Set R) :
     zeroLocus s ∪ zeroLocus s' = zeroLocus (Ideal.span s ⊓ Ideal.span s' : Ideal R) := by
   rw [zero_locus_inf]; simp
 #align prime_spectrum.union_zero_locus PrimeSpectrum.union_zeroLocus
+-/
 
+#print PrimeSpectrum.zeroLocus_mul /-
 theorem zeroLocus_mul (I J : Ideal R) :
     zeroLocus ((I * J : Ideal R) : Set R) = zeroLocus I ∪ zeroLocus J :=
   Set.ext fun x => x.2.mul_le
 #align prime_spectrum.zero_locus_mul PrimeSpectrum.zeroLocus_mul
+-/
 
+#print PrimeSpectrum.zeroLocus_singleton_mul /-
 theorem zeroLocus_singleton_mul (f g : R) :
     zeroLocus ({f * g} : Set R) = zeroLocus {f} ∪ zeroLocus {g} :=
   Set.ext fun x => by simpa using x.2.mul_mem_iff_mem_or_mem
 #align prime_spectrum.zero_locus_singleton_mul PrimeSpectrum.zeroLocus_singleton_mul
+-/
 
 #print PrimeSpectrum.zeroLocus_pow /-
 @[simp]
@@ -423,12 +477,15 @@ theorem zeroLocus_pow (I : Ideal R) {n : ℕ} (hn : 0 < n) :
 #align prime_spectrum.zero_locus_pow PrimeSpectrum.zeroLocus_pow
 -/
 
+#print PrimeSpectrum.zeroLocus_singleton_pow /-
 @[simp]
 theorem zeroLocus_singleton_pow (f : R) (n : ℕ) (hn : 0 < n) :
     zeroLocus ({f ^ n} : Set R) = zeroLocus {f} :=
   Set.ext fun x => by simpa using x.2.pow_mem_iff_mem n hn
 #align prime_spectrum.zero_locus_singleton_pow PrimeSpectrum.zeroLocus_singleton_pow
+-/
 
+#print PrimeSpectrum.sup_vanishingIdeal_le /-
 theorem sup_vanishingIdeal_le (t t' : Set (PrimeSpectrum R)) :
     vanishingIdeal t ⊔ vanishingIdeal t' ≤ vanishingIdeal (t ∩ t') :=
   by
@@ -438,11 +495,14 @@ theorem sup_vanishingIdeal_le (t t' : Set (PrimeSpectrum R)) :
   rw [mem_vanishing_ideal] at hf hg 
   apply Submodule.add_mem <;> solve_by_elim
 #align prime_spectrum.sup_vanishing_ideal_le PrimeSpectrum.sup_vanishingIdeal_le
+-/
 
+#print PrimeSpectrum.mem_compl_zeroLocus_iff_not_mem /-
 theorem mem_compl_zeroLocus_iff_not_mem {f : R} {I : PrimeSpectrum R} :
     I ∈ (zeroLocus {f} : Set (PrimeSpectrum R))ᶜ ↔ f ∉ I.asIdeal := by
   rw [Set.mem_compl_iff, mem_zero_locus, Set.singleton_subset_iff] <;> rfl
 #align prime_spectrum.mem_compl_zero_locus_iff_not_mem PrimeSpectrum.mem_compl_zeroLocus_iff_not_mem
+-/
 
 #print PrimeSpectrum.zariskiTopology /-
 /-- The Zariski topology on the prime spectrum of a commutative ring is defined via the closed sets
@@ -459,9 +519,11 @@ instance zariskiTopology : TopologicalSpace (PrimeSpectrum R) :=
 #align prime_spectrum.zariski_topology PrimeSpectrum.zariskiTopology
 -/
 
+#print PrimeSpectrum.isOpen_iff /-
 theorem isOpen_iff (U : Set (PrimeSpectrum R)) : IsOpen U ↔ ∃ s, Uᶜ = zeroLocus s := by
   simp only [@eq_comm _ (Uᶜ)] <;> rfl
 #align prime_spectrum.is_open_iff PrimeSpectrum.isOpen_iff
+-/
 
 #print PrimeSpectrum.isClosed_iff_zeroLocus /-
 theorem isClosed_iff_zeroLocus (Z : Set (PrimeSpectrum R)) : IsClosed Z ↔ ∃ s, Z = zeroLocus s := by
@@ -550,6 +612,7 @@ theorem isRadical_vanishingIdeal (s : Set (PrimeSpectrum R)) : (vanishingIdeal s
 #align prime_spectrum.is_radical_vanishing_ideal PrimeSpectrum.isRadical_vanishingIdeal
 -/
 
+#print PrimeSpectrum.vanishingIdeal_anti_mono_iff /-
 theorem vanishingIdeal_anti_mono_iff {s t : Set (PrimeSpectrum R)} (ht : IsClosed t) :
     s ⊆ t ↔ vanishingIdeal t ≤ vanishingIdeal s :=
   ⟨vanishingIdeal_anti_mono, fun h =>
@@ -557,19 +620,24 @@ theorem vanishingIdeal_anti_mono_iff {s t : Set (PrimeSpectrum R)} (ht : IsClose
     rw [← ht.closure_subset_iff, ← ht.closure_eq]
     convert ← zero_locus_anti_mono_ideal h <;> apply zero_locus_vanishing_ideal_eq_closure⟩
 #align prime_spectrum.vanishing_ideal_anti_mono_iff PrimeSpectrum.vanishingIdeal_anti_mono_iff
+-/
 
+#print PrimeSpectrum.vanishingIdeal_strict_anti_mono_iff /-
 theorem vanishingIdeal_strict_anti_mono_iff {s t : Set (PrimeSpectrum R)} (hs : IsClosed s)
     (ht : IsClosed t) : s ⊂ t ↔ vanishingIdeal t < vanishingIdeal s := by
   rw [Set.ssubset_def, vanishing_ideal_anti_mono_iff hs, vanishing_ideal_anti_mono_iff ht,
     lt_iff_le_not_le]
 #align prime_spectrum.vanishing_ideal_strict_anti_mono_iff PrimeSpectrum.vanishingIdeal_strict_anti_mono_iff
+-/
 
+#print PrimeSpectrum.closedsEmbedding /-
 /-- The antitone order embedding of closed subsets of `Spec R` into ideals of `R`. -/
 def closedsEmbedding (R : Type _) [CommRing R] :
     (TopologicalSpace.Closeds <| PrimeSpectrum R)ᵒᵈ ↪o Ideal R :=
   OrderEmbedding.ofMapLEIff (fun s => vanishingIdeal s.ofDual) fun s t =>
     (vanishingIdeal_anti_mono_iff s.2).symm
 #align prime_spectrum.closeds_embedding PrimeSpectrum.closedsEmbedding
+-/
 
 #print PrimeSpectrum.t1Space_iff_isField /-
 theorem t1Space_iff_isField [IsDomain R] : T1Space (PrimeSpectrum R) ↔ IsField R :=
@@ -590,7 +658,6 @@ theorem t1Space_iff_isField [IsDomain R] : T1Space (PrimeSpectrum R) ↔ IsField
 #align prime_spectrum.t1_space_iff_is_field PrimeSpectrum.t1Space_iff_isField
 -/
 
--- mathport name: «exprZ( )»
 local notation "Z(" a ")" => zeroLocus (a : Set R)
 
 #print PrimeSpectrum.isIrreducible_zeroLocus_iff_of_radical /-
@@ -648,6 +715,7 @@ section Comap
 
 variable {S' : Type _} [CommRing S']
 
+#print PrimeSpectrum.preimage_comap_zeroLocus_aux /-
 theorem preimage_comap_zeroLocus_aux (f : R →+* S) (s : Set R) :
     (fun y => ⟨Ideal.comap f y.asIdeal, inferInstance⟩ : PrimeSpectrum S → PrimeSpectrum R) ⁻¹'
         zeroLocus s =
@@ -657,6 +725,7 @@ theorem preimage_comap_zeroLocus_aux (f : R →+* S) (s : Set R) :
   simp only [mem_zero_locus, Set.image_subset_iff]
   rfl
 #align prime_spectrum.preimage_comap_zero_locus_aux PrimeSpectrum.preimage_comap_zeroLocus_aux
+-/
 
 #print PrimeSpectrum.comap /-
 /-- The function between prime spectra of commutative rings induced by a ring homomorphism.
@@ -674,10 +743,12 @@ def comap (f : R →+* S) : C(PrimeSpectrum S, PrimeSpectrum R)
 
 variable (f : R →+* S)
 
+#print PrimeSpectrum.comap_asIdeal /-
 @[simp]
 theorem comap_asIdeal (y : PrimeSpectrum S) : (comap f y).asIdeal = Ideal.comap f y.asIdeal :=
   rfl
 #align prime_spectrum.comap_as_ideal PrimeSpectrum.comap_asIdeal
+-/
 
 #print PrimeSpectrum.comap_id /-
 @[simp]
@@ -685,35 +756,46 @@ theorem comap_id : comap (RingHom.id R) = ContinuousMap.id _ := by ext; rfl
 #align prime_spectrum.comap_id PrimeSpectrum.comap_id
 -/
 
+#print PrimeSpectrum.comap_comp /-
 @[simp]
 theorem comap_comp (f : R →+* S) (g : S →+* S') : comap (g.comp f) = (comap f).comp (comap g) :=
   rfl
 #align prime_spectrum.comap_comp PrimeSpectrum.comap_comp
+-/
 
+#print PrimeSpectrum.comap_comp_apply /-
 theorem comap_comp_apply (f : R →+* S) (g : S →+* S') (x : PrimeSpectrum S') :
     PrimeSpectrum.comap (g.comp f) x = (PrimeSpectrum.comap f) (PrimeSpectrum.comap g x) :=
   rfl
 #align prime_spectrum.comap_comp_apply PrimeSpectrum.comap_comp_apply
+-/
 
+#print PrimeSpectrum.preimage_comap_zeroLocus /-
 @[simp]
 theorem preimage_comap_zeroLocus (s : Set R) : comap f ⁻¹' zeroLocus s = zeroLocus (f '' s) :=
   preimage_comap_zeroLocus_aux f s
 #align prime_spectrum.preimage_comap_zero_locus PrimeSpectrum.preimage_comap_zeroLocus
+-/
 
+#print PrimeSpectrum.comap_injective_of_surjective /-
 theorem comap_injective_of_surjective (f : R →+* S) (hf : Function.Surjective f) :
     Function.Injective (comap f) := fun x y h =>
   PrimeSpectrum.ext _ _
     (Ideal.comap_injective_of_surjective f hf
       (congr_arg PrimeSpectrum.asIdeal h : (comap f x).asIdeal = (comap f y).asIdeal))
 #align prime_spectrum.comap_injective_of_surjective PrimeSpectrum.comap_injective_of_surjective
+-/
 
+#print PrimeSpectrum.comap_singleton_isClosed_of_surjective /-
 theorem comap_singleton_isClosed_of_surjective (f : R →+* S) (hf : Function.Surjective f)
     (x : PrimeSpectrum S) (hx : IsClosed ({x} : Set (PrimeSpectrum S))) :
     IsClosed ({comap f x} : Set (PrimeSpectrum R)) :=
   haveI : x.as_ideal.is_maximal := (is_closed_singleton_iff_is_maximal x).1 hx
   (is_closed_singleton_iff_is_maximal _).2 (Ideal.comap_isMaximal_of_surjective f hf)
 #align prime_spectrum.comap_singleton_is_closed_of_surjective PrimeSpectrum.comap_singleton_isClosed_of_surjective
+-/
 
+#print PrimeSpectrum.comap_singleton_isClosed_of_isIntegral /-
 theorem comap_singleton_isClosed_of_isIntegral (f : R →+* S) (hf : f.IsIntegral)
     (x : PrimeSpectrum S) (hx : IsClosed ({x} : Set (PrimeSpectrum S))) :
     IsClosed ({comap f x} : Set (PrimeSpectrum R)) :=
@@ -721,9 +803,11 @@ theorem comap_singleton_isClosed_of_isIntegral (f : R →+* S) (hf : f.IsIntegra
     (Ideal.isMaximal_comap_of_isIntegral_of_is_maximal' f hf x.asIdeal <|
       (isClosed_singleton_iff_isMaximal x).1 hx)
 #align prime_spectrum.comap_singleton_is_closed_of_is_integral PrimeSpectrum.comap_singleton_isClosed_of_isIntegral
+-/
 
 variable (S)
 
+#print PrimeSpectrum.localization_comap_inducing /-
 theorem localization_comap_inducing [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
     Inducing (comap (algebraMap R S)) := by
   constructor
@@ -740,7 +824,9 @@ theorem localization_comap_inducing [Algebra R S] (M : Submonoid R) [IsLocalizat
     exact congr_arg Submodule.carrier (IsLocalization.map_comap M S (Ideal.span s))
   · rintro ⟨_, ⟨t, rfl⟩, rfl⟩; simp
 #align prime_spectrum.localization_comap_inducing PrimeSpectrum.localization_comap_inducing
+-/
 
+#print PrimeSpectrum.localization_comap_injective /-
 theorem localization_comap_injective [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
     Function.Injective (comap (algebraMap R S)) :=
   by
@@ -751,12 +837,16 @@ theorem localization_comap_injective [Algebra R S] (M : Submonoid R) [IsLocaliza
   ext1
   exact h
 #align prime_spectrum.localization_comap_injective PrimeSpectrum.localization_comap_injective
+-/
 
+#print PrimeSpectrum.localization_comap_embedding /-
 theorem localization_comap_embedding [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
     Embedding (comap (algebraMap R S)) :=
   ⟨localization_comap_inducing S M, localization_comap_injective S M⟩
 #align prime_spectrum.localization_comap_embedding PrimeSpectrum.localization_comap_embedding
+-/
 
+#print PrimeSpectrum.localization_comap_range /-
 theorem localization_comap_range [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
     Set.range (comap (algebraMap R S)) = {p | Disjoint (M : Set R) p.asIdeal} :=
   by
@@ -770,6 +860,7 @@ theorem localization_comap_range [Algebra R S] (M : Submonoid R) [IsLocalization
     ext1
     exact IsLocalization.comap_map_of_isPrime_disjoint M S _ x.2 h
 #align prime_spectrum.localization_comap_range PrimeSpectrum.localization_comap_range
+-/
 
 section SpecOfSurjective
 
@@ -778,6 +869,7 @@ section SpecOfSurjective
 
 open Function RingHom
 
+#print PrimeSpectrum.comap_inducing_of_surjective /-
 theorem comap_inducing_of_surjective (hf : Surjective f) : Inducing (comap f) :=
   {
     induced :=
@@ -792,7 +884,9 @@ theorem comap_inducing_of_surjective (hf : Surjective f) : Inducing (comap f) :=
       rintro ⟨-, ⟨F, rfl⟩, hF⟩
       exact ⟨f '' F, hF.symm.trans (preimage_comap_zero_locus f F)⟩ }
 #align prime_spectrum.comap_inducing_of_surjective PrimeSpectrum.comap_inducing_of_surjective
+-/
 
+#print PrimeSpectrum.image_comap_zeroLocus_eq_zeroLocus_comap /-
 theorem image_comap_zeroLocus_eq_zeroLocus_comap (hf : Surjective f) (I : Ideal S) :
     comap f '' zeroLocus I = zeroLocus (I.comap f) :=
   by
@@ -813,25 +907,32 @@ theorem image_comap_zeroLocus_eq_zeroLocus_comap (hf : Surjective f) (I : Ideal
       refine' p.as_ideal.sub_mem hx' (hp _)
       rwa [mem_ker, map_sub, sub_eq_zero]
 #align prime_spectrum.image_comap_zero_locus_eq_zero_locus_comap PrimeSpectrum.image_comap_zeroLocus_eq_zeroLocus_comap
+-/
 
+#print PrimeSpectrum.range_comap_of_surjective /-
 theorem range_comap_of_surjective (hf : Surjective f) : Set.range (comap f) = zeroLocus (ker f) :=
   by
   rw [← Set.image_univ]
   convert image_comap_zero_locus_eq_zero_locus_comap _ _ hf _
   rw [zero_locus_bot]
 #align prime_spectrum.range_comap_of_surjective PrimeSpectrum.range_comap_of_surjective
+-/
 
+#print PrimeSpectrum.isClosed_range_comap_of_surjective /-
 theorem isClosed_range_comap_of_surjective (hf : Surjective f) : IsClosed (Set.range (comap f)) :=
   by
   rw [range_comap_of_surjective _ f hf]
   exact is_closed_zero_locus ↑(ker f)
 #align prime_spectrum.is_closed_range_comap_of_surjective PrimeSpectrum.isClosed_range_comap_of_surjective
+-/
 
+#print PrimeSpectrum.closedEmbedding_comap_of_surjective /-
 theorem closedEmbedding_comap_of_surjective (hf : Surjective f) : ClosedEmbedding (comap f) :=
   { induced := (comap_inducing_of_surjective S f hf).induced
     inj := comap_injective_of_surjective f hf
     closed_range := isClosed_range_comap_of_surjective S f hf }
 #align prime_spectrum.closed_embedding_comap_of_surjective PrimeSpectrum.closedEmbedding_comap_of_surjective
+-/
 
 end SpecOfSurjective
 
@@ -861,44 +962,60 @@ theorem isOpen_basicOpen {a : R} : IsOpen (basicOpen a : Set (PrimeSpectrum R))
 #align prime_spectrum.is_open_basic_open PrimeSpectrum.isOpen_basicOpen
 -/
 
+#print PrimeSpectrum.basicOpen_eq_zeroLocus_compl /-
 @[simp]
 theorem basicOpen_eq_zeroLocus_compl (r : R) :
     (basicOpen r : Set (PrimeSpectrum R)) = zeroLocus {r}ᶜ :=
   Set.ext fun x => by simpa only [Set.mem_compl_iff, mem_zero_locus, Set.singleton_subset_iff]
 #align prime_spectrum.basic_open_eq_zero_locus_compl PrimeSpectrum.basicOpen_eq_zeroLocus_compl
+-/
 
+#print PrimeSpectrum.basicOpen_one /-
 @[simp]
 theorem basicOpen_one : basicOpen (1 : R) = ⊤ :=
   TopologicalSpace.Opens.ext <| by simp
 #align prime_spectrum.basic_open_one PrimeSpectrum.basicOpen_one
+-/
 
+#print PrimeSpectrum.basicOpen_zero /-
 @[simp]
 theorem basicOpen_zero : basicOpen (0 : R) = ⊥ :=
   TopologicalSpace.Opens.ext <| by simp
 #align prime_spectrum.basic_open_zero PrimeSpectrum.basicOpen_zero
+-/
 
+#print PrimeSpectrum.basicOpen_le_basicOpen_iff /-
 theorem basicOpen_le_basicOpen_iff (f g : R) :
     basicOpen f ≤ basicOpen g ↔ f ∈ (Ideal.span ({g} : Set R)).radical := by
   rw [← SetLike.coe_subset_coe, basic_open_eq_zero_locus_compl, basic_open_eq_zero_locus_compl,
     Set.compl_subset_compl, zero_locus_subset_zero_locus_singleton_iff]
 #align prime_spectrum.basic_open_le_basic_open_iff PrimeSpectrum.basicOpen_le_basicOpen_iff
+-/
 
+#print PrimeSpectrum.basicOpen_mul /-
 theorem basicOpen_mul (f g : R) : basicOpen (f * g) = basicOpen f ⊓ basicOpen g :=
   TopologicalSpace.Opens.ext <| by simp [zero_locus_singleton_mul]
 #align prime_spectrum.basic_open_mul PrimeSpectrum.basicOpen_mul
+-/
 
+#print PrimeSpectrum.basicOpen_mul_le_left /-
 theorem basicOpen_mul_le_left (f g : R) : basicOpen (f * g) ≤ basicOpen f := by
   rw [basic_open_mul f g]; exact inf_le_left
 #align prime_spectrum.basic_open_mul_le_left PrimeSpectrum.basicOpen_mul_le_left
+-/
 
+#print PrimeSpectrum.basicOpen_mul_le_right /-
 theorem basicOpen_mul_le_right (f g : R) : basicOpen (f * g) ≤ basicOpen g := by
   rw [basic_open_mul f g]; exact inf_le_right
 #align prime_spectrum.basic_open_mul_le_right PrimeSpectrum.basicOpen_mul_le_right
+-/
 
+#print PrimeSpectrum.basicOpen_pow /-
 @[simp]
 theorem basicOpen_pow (f : R) (n : ℕ) (hn : 0 < n) : basicOpen (f ^ n) = basicOpen f :=
   TopologicalSpace.Opens.ext <| by simpa using zero_locus_singleton_pow f n hn
 #align prime_spectrum.basic_open_pow PrimeSpectrum.basicOpen_pow
+-/
 
 #print PrimeSpectrum.isTopologicalBasis_basic_opens /-
 theorem isTopologicalBasis_basic_opens :
@@ -953,6 +1070,7 @@ theorem isCompact_basicOpen (f : R) : IsCompact (basicOpen f : Set (PrimeSpectru
 #align prime_spectrum.is_compact_basic_open PrimeSpectrum.isCompact_basicOpen
 -/
 
+#print PrimeSpectrum.basicOpen_eq_bot_iff /-
 @[simp]
 theorem basicOpen_eq_bot_iff (f : R) : basicOpen f = ⊥ ↔ IsNilpotent f :=
   by
@@ -961,7 +1079,9 @@ theorem basicOpen_eq_bot_iff (f : R) : basicOpen f = ⊥ ↔ IsNilpotent f :=
     nilpotent_iff_mem_prime, Set.compl_empty_iff, mem_zero_locus, SetLike.mem_coe]
   exact ⟨fun h I hI => h ⟨I, hI⟩, fun h ⟨I, hI⟩ => h I hI⟩
 #align prime_spectrum.basic_open_eq_bot_iff PrimeSpectrum.basicOpen_eq_bot_iff
+-/
 
+#print PrimeSpectrum.localization_away_comap_range /-
 theorem localization_away_comap_range (S : Type v) [CommRing S] [Algebra R S] (r : R)
     [IsLocalization.Away r S] : Set.range (comap (algebraMap R S)) = basicOpen r :=
   by
@@ -975,12 +1095,15 @@ theorem localization_away_comap_range (S : Type v) [CommRing S] [Algebra R S] (r
   · rintro h₁ _ ⟨⟨n, rfl⟩, h₃⟩
     exact h₁ (x.2.mem_of_pow_mem _ h₃)
 #align prime_spectrum.localization_away_comap_range PrimeSpectrum.localization_away_comap_range
+-/
 
+#print PrimeSpectrum.localization_away_openEmbedding /-
 theorem localization_away_openEmbedding (S : Type v) [CommRing S] [Algebra R S] (r : R)
     [IsLocalization.Away r S] : OpenEmbedding (comap (algebraMap R S)) :=
   { toEmbedding := localization_comap_embedding S (Submonoid.powers r)
     open_range := by rw [localization_away_comap_range S r]; exact is_open_basic_open }
 #align prime_spectrum.localization_away_open_embedding PrimeSpectrum.localization_away_openEmbedding
+-/
 
 end BasicOpen
 
@@ -1000,31 +1123,41 @@ We endow `prime_spectrum R` with a partial order, where `x ≤ y` if and only if
 instance : PartialOrder (PrimeSpectrum R) :=
   PartialOrder.lift asIdeal ext
 
+#print PrimeSpectrum.asIdeal_le_asIdeal /-
 @[simp]
 theorem asIdeal_le_asIdeal (x y : PrimeSpectrum R) : x.asIdeal ≤ y.asIdeal ↔ x ≤ y :=
   Iff.rfl
 #align prime_spectrum.as_ideal_le_as_ideal PrimeSpectrum.asIdeal_le_asIdeal
+-/
 
+#print PrimeSpectrum.asIdeal_lt_asIdeal /-
 @[simp]
 theorem asIdeal_lt_asIdeal (x y : PrimeSpectrum R) : x.asIdeal < y.asIdeal ↔ x < y :=
   Iff.rfl
 #align prime_spectrum.as_ideal_lt_as_ideal PrimeSpectrum.asIdeal_lt_asIdeal
+-/
 
+#print PrimeSpectrum.le_iff_mem_closure /-
 theorem le_iff_mem_closure (x y : PrimeSpectrum R) :
     x ≤ y ↔ y ∈ closure ({x} : Set (PrimeSpectrum R)) := by
   rw [← as_ideal_le_as_ideal, ← zero_locus_vanishing_ideal_eq_closure, mem_zero_locus,
     vanishing_ideal_singleton, SetLike.coe_subset_coe]
 #align prime_spectrum.le_iff_mem_closure PrimeSpectrum.le_iff_mem_closure
+-/
 
+#print PrimeSpectrum.le_iff_specializes /-
 theorem le_iff_specializes (x y : PrimeSpectrum R) : x ≤ y ↔ x ⤳ y :=
   (le_iff_mem_closure x y).trans specializes_iff_mem_closure.symm
 #align prime_spectrum.le_iff_specializes PrimeSpectrum.le_iff_specializes
+-/
 
+#print PrimeSpectrum.nhdsOrderEmbedding /-
 /-- `nhds` as an order embedding. -/
 @[simps (config := { fullyApplied := true })]
 def nhdsOrderEmbedding : PrimeSpectrum R ↪o Filter (PrimeSpectrum R) :=
   OrderEmbedding.ofMapLEIff nhds fun a b => (le_iff_specializes a b).symm
 #align prime_spectrum.nhds_order_embedding PrimeSpectrum.nhdsOrderEmbedding
+-/
 
 instance : T0Space (PrimeSpectrum R) :=
   ⟨nhdsOrderEmbedding.Injective⟩
@@ -1041,6 +1174,7 @@ instance {R : Type _} [Field R] : Unique (PrimeSpectrum R)
 
 end Order
 
+#print PrimeSpectrum.localizationMapOfSpecializes /-
 /-- If `x` specializes to `y`, then there is a natural map from the localization of `y` to the
 localization of `x`. -/
 def localizationMapOfSpecializes {x y : PrimeSpectrum R} (h : x ⤳ y) :
@@ -1053,6 +1187,7 @@ def localizationMapOfSpecializes {x y : PrimeSpectrum R} (h : x ⤳ y) :
         Set.compl_subset_compl] at h 
       exact (IsLocalization.map_units _ ⟨a, show a ∈ x.as_ideal.prime_compl from h ha⟩ : _))
 #align prime_spectrum.localization_map_of_specializes PrimeSpectrum.localizationMapOfSpecializes
+-/
 
 end PrimeSpectrum
 
@@ -1069,16 +1204,20 @@ def closedPoint : PrimeSpectrum R :=
 
 variable {R}
 
+#print LocalRing.isLocalRingHom_iff_comap_closedPoint /-
 theorem isLocalRingHom_iff_comap_closedPoint {S : Type v} [CommRing S] [LocalRing S] (f : R →+* S) :
     IsLocalRingHom f ↔ PrimeSpectrum.comap f (closedPoint S) = closedPoint R := by
   rw [(local_hom_tfae f).out 0 4, PrimeSpectrum.ext_iff]; rfl
 #align local_ring.is_local_ring_hom_iff_comap_closed_point LocalRing.isLocalRingHom_iff_comap_closedPoint
+-/
 
+#print LocalRing.comap_closedPoint /-
 @[simp]
 theorem comap_closedPoint {S : Type v} [CommRing S] [LocalRing S] (f : R →+* S) [IsLocalRingHom f] :
     PrimeSpectrum.comap f (closedPoint S) = closedPoint R :=
   (isLocalRingHom_iff_comap_closedPoint f).mp inferInstance
 #align local_ring.comap_closed_point LocalRing.comap_closedPoint
+-/
 
 #print LocalRing.specializes_closedPoint /-
 theorem specializes_closedPoint (x : PrimeSpectrum R) : x ⤳ closedPoint R :=
@@ -1086,12 +1225,14 @@ theorem specializes_closedPoint (x : PrimeSpectrum R) : x ⤳ closedPoint R :=
 #align local_ring.specializes_closed_point LocalRing.specializes_closedPoint
 -/
 
+#print LocalRing.closedPoint_mem_iff /-
 theorem closedPoint_mem_iff (U : TopologicalSpace.Opens <| PrimeSpectrum R) :
     closedPoint R ∈ U ↔ U = ⊤ := by
   constructor
   · rw [eq_top_iff]; exact fun h x _ => (specializes_closed_point x).mem_open U.2 h
   · rintro rfl; trivial
 #align local_ring.closed_point_mem_iff LocalRing.closedPoint_mem_iff
+-/
 
 @[simp]
 theorem PrimeSpectrum.comap_residue (x : PrimeSpectrum (ResidueField R)) :

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

@@ -140,7 +140,7 @@ At a point `x` (a prime ideal) the function (i.e., element) `f` takes values in
 `prime_spectrum R` where all "functions" in `s` vanish simultaneously.
 -/
 def zeroLocus (s : Set R) : Set (PrimeSpectrum R) :=
-  { x | s ⊆ x.asIdeal }
+  {x | s ⊆ x.asIdeal}
 #align prime_spectrum.zero_locus PrimeSpectrum.zeroLocus
 -/
 
@@ -174,7 +174,7 @@ def vanishingIdeal (t : Set (PrimeSpectrum R)) : Ideal R :=
 
 #print PrimeSpectrum.coe_vanishingIdeal /-
 theorem coe_vanishingIdeal (t : Set (PrimeSpectrum R)) :
-    (vanishingIdeal t : Set R) = { f : R | ∀ x : PrimeSpectrum R, x ∈ t → f ∈ x.asIdeal } :=
+    (vanishingIdeal t : Set R) = {f : R | ∀ x : PrimeSpectrum R, x ∈ t → f ∈ x.asIdeal} :=
   by
   ext f
   rw [vanishing_ideal, SetLike.mem_coe, Submodule.mem_iInf]
@@ -555,7 +555,7 @@ theorem vanishingIdeal_anti_mono_iff {s t : Set (PrimeSpectrum R)} (ht : IsClose
   ⟨vanishingIdeal_anti_mono, fun h =>
     by
     rw [← ht.closure_subset_iff, ← ht.closure_eq]
-    convert← zero_locus_anti_mono_ideal h <;> apply zero_locus_vanishing_ideal_eq_closure⟩
+    convert ← zero_locus_anti_mono_ideal h <;> apply zero_locus_vanishing_ideal_eq_closure⟩
 #align prime_spectrum.vanishing_ideal_anti_mono_iff PrimeSpectrum.vanishingIdeal_anti_mono_iff
 
 theorem vanishingIdeal_strict_anti_mono_iff {s t : Set (PrimeSpectrum R)} (hs : IsClosed s)
@@ -758,7 +758,7 @@ theorem localization_comap_embedding [Algebra R S] (M : Submonoid R) [IsLocaliza
 #align prime_spectrum.localization_comap_embedding PrimeSpectrum.localization_comap_embedding
 
 theorem localization_comap_range [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
-    Set.range (comap (algebraMap R S)) = { p | Disjoint (M : Set R) p.asIdeal } :=
+    Set.range (comap (algebraMap R S)) = {p | Disjoint (M : Set R) p.asIdeal} :=
   by
   ext x
   constructor
@@ -843,7 +843,7 @@ section BasicOpen
 /-- `basic_open r` is the open subset containing all prime ideals not containing `r`. -/
 def basicOpen (r : R) : TopologicalSpace.Opens (PrimeSpectrum R)
     where
-  carrier := { x | r ∉ x.asIdeal }
+  carrier := {x | r ∉ x.asIdeal}
   is_open' := ⟨{r}, Set.ext fun x => Set.singleton_subset_iff.trans <| Classical.not_not.symm⟩
 #align prime_spectrum.basic_open PrimeSpectrum.basicOpen
 -/

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

@@ -107,7 +107,7 @@ noncomputable def primeSpectrumProd :
       (by
         constructor
         · rintro (⟨I, hI⟩ | ⟨J, hJ⟩) (⟨I', hI'⟩ | ⟨J', hJ'⟩) h <;>
-            simp only [Ideal.prod.ext_iff, prime_spectrum_prod_of_sum] at h
+            simp only [Ideal.prod.ext_iff, prime_spectrum_prod_of_sum] at h 
           · simp only [h]
           · exact False.elim (hI.ne_top h.left)
           · exact False.elim (hJ.ne_top h.right)
@@ -325,7 +325,7 @@ theorem zeroLocus_empty_of_one_mem {s : Set R} (h : (1 : R) ∈ s) : zeroLocus s
   by
   rw [Set.eq_empty_iff_forall_not_mem]
   intro x hx
-  rw [mem_zero_locus] at hx
+  rw [mem_zero_locus] at hx 
   have x_prime : x.as_ideal.is_prime := by infer_instance
   have eq_top : x.as_ideal = ⊤ := by rw [Ideal.eq_top_iff_one]; exact hx h
   apply x_prime.ne_top eq_top
@@ -435,7 +435,7 @@ theorem sup_vanishingIdeal_le (t t' : Set (PrimeSpectrum R)) :
   intro r
   rw [Submodule.mem_sup, mem_vanishing_ideal]
   rintro ⟨f, hf, g, hg, rfl⟩ x ⟨hxt, hxt'⟩
-  rw [mem_vanishing_ideal] at hf hg
+  rw [mem_vanishing_ideal] at hf hg 
   apply Submodule.add_mem <;> solve_by_elim
 #align prime_spectrum.sup_vanishing_ideal_le PrimeSpectrum.sup_vanishingIdeal_le
 
@@ -498,7 +498,7 @@ theorem isClosed_singleton_iff_isMaximal (x : PrimeSpectrum R) :
   by
   refine' (is_closed_iff_zero_locus _).trans ⟨fun h => _, fun h => _⟩
   · obtain ⟨s, hs⟩ := h
-    rw [eq_comm, Set.eq_singleton_iff_unique_mem] at hs
+    rw [eq_comm, Set.eq_singleton_iff_unique_mem] at hs 
     refine'
       ⟨⟨x.2.1, fun I hI =>
           Classical.not_not.1
@@ -520,8 +520,8 @@ theorem zeroLocus_vanishingIdeal_eq_closure (t : Set (PrimeSpectrum R)) :
   by
   apply Set.Subset.antisymm
   · rintro x hx t' ⟨ht', ht⟩
-    obtain ⟨fs, rfl⟩ : ∃ s, t' = zero_locus 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 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 t
@@ -611,7 +611,7 @@ theorem isIrreducible_zeroLocus_iff_of_radical (I : Ideal R) (hI : I.IsRadical)
       · simp_rw [← SetLike.mem_coe, ← Set.singleton_subset_iff, ← Ideal.span_le, ←
           Ideal.span_singleton_mul_span_singleton]
         refine' fun h x y h' => h _ _ _
-        rw [← hI.radical_le_iff] at h'⊢
+        rw [← hI.radical_le_iff] at h' ⊢
         simpa only [Ideal.radical_inf, Ideal.radical_mul] using h'
       · simp_rw [or_iff_not_imp_left, SetLike.not_le_iff_exists]
         rintro h s t h' ⟨x, hx, hx'⟩ y hy
@@ -746,8 +746,8 @@ theorem localization_comap_injective [Algebra R S] (M : Submonoid R) [IsLocaliza
   by
   intro p q h
   replace h := congr_arg (fun x : PrimeSpectrum R => Ideal.map (algebraMap R S) x.asIdeal) h
-  dsimp only at h
-  erw [IsLocalization.map_comap M S, IsLocalization.map_comap M S] at h
+  dsimp only at h 
+  erw [IsLocalization.map_comap M S, IsLocalization.map_comap M S] at h 
   ext1
   exact h
 #align prime_spectrum.localization_comap_injective PrimeSpectrum.localization_comap_injective
@@ -909,7 +909,7 @@ theorem isTopologicalBasis_basic_opens :
   · rintro _ ⟨r, rfl⟩
     exact is_open_basic_open
   · 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 f, ⟨f, rfl⟩, hfp, _⟩
     rw [← Set.compl_subset_compl, ← hs, basic_open_eq_zero_locus_compl, compl_compl]
@@ -934,7 +934,7 @@ theorem isCompact_basicOpen (f : R) : IsCompact (basicOpen f : Set (PrimeSpectru
     have hI : ∀ i, Z i = zero_locus (I i) := fun i => by
       simpa only [zero_locus_vanishing_ideal_eq_closure] using (hZc i).closure_eq.symm
     rw [basic_open_eq_zero_locus_compl f, Set.inter_comm, ← Set.diff_eq, Set.diff_eq_empty,
-      funext hI, ← zero_locus_supr] at hZ
+      funext hI, ← zero_locus_supr] at hZ 
     obtain ⟨n, hn⟩ : f ∈ (⨆ i : ι, I i).radical :=
       by
       rw [← vanishing_ideal_zero_locus_eq_radical]
@@ -1050,7 +1050,7 @@ def localizationMapOfSpecializes {x y : PrimeSpectrum R} (h : x ⤳ y) :
     (by
       rintro ⟨a, ha⟩
       rw [← PrimeSpectrum.le_iff_specializes, ← as_ideal_le_as_ideal, ← SetLike.coe_subset_coe, ←
-        Set.compl_subset_compl] at h
+        Set.compl_subset_compl] at h 
       exact (IsLocalization.map_units _ ⟨a, show a ∈ x.as_ideal.prime_compl from h ha⟩ : _))
 #align prime_spectrum.localization_map_of_specializes PrimeSpectrum.localizationMapOfSpecializes
 

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

@@ -53,7 +53,7 @@ and Chris Hughes (on an earlier repository).
 
 noncomputable section
 
-open Classical
+open scoped Classical
 
 universe u v
 

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

@@ -91,12 +91,6 @@ theorem pUnit (x : PrimeSpectrum PUnit) : False :=
 
 variable (R S)
 
-/- warning: prime_spectrum.prime_spectrum_prod_of_sum -> PrimeSpectrum.primeSpectrumProdOfSum is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align prime_spectrum.prime_spectrum_prod_of_sum PrimeSpectrum.primeSpectrumProdOfSumₓ'. -/
 /-- The map from the direct sum of prime spectra to the prime spectrum of a direct product. -/
 @[simp]
 def primeSpectrumProdOfSum : Sum (PrimeSpectrum R) (PrimeSpectrum S) → PrimeSpectrum (R × S)
@@ -104,12 +98,6 @@ def primeSpectrumProdOfSum : Sum (PrimeSpectrum R) (PrimeSpectrum S) → PrimeSp
   | Sum.inr ⟨J, hJ⟩ => ⟨Ideal.prod ⊤ J, Ideal.isPrime_ideal_prod_top'⟩
 #align prime_spectrum.prime_spectrum_prod_of_sum PrimeSpectrum.primeSpectrumProdOfSum
 
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 /-- The prime spectrum of `R × S` is in bijection with the disjoint unions of the prime spectrum of
 `R` and the prime spectrum of `S`. -/
 noncomputable def primeSpectrumProd :
@@ -132,23 +120,11 @@ noncomputable def primeSpectrumProd :
 
 variable {R S}
 
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 @[simp]
 theorem primeSpectrumProd_symm_inl_asIdeal (x : PrimeSpectrum R) :
     ((primeSpectrumProd R S).symm <| Sum.inl x).asIdeal = Ideal.prod x.asIdeal ⊤ := by cases x; rfl
 #align prime_spectrum.prime_spectrum_prod_symm_inl_as_ideal PrimeSpectrum.primeSpectrumProd_symm_inl_asIdeal
 
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 @[simp]
 theorem primeSpectrumProd_symm_inr_asIdeal (x : PrimeSpectrum S) :
     ((primeSpectrumProd R S).symm <| Sum.inr x).asIdeal = Ideal.prod ⊤ x.asIdeal := by cases x; rfl
@@ -221,12 +197,6 @@ theorem vanishingIdeal_singleton (x : PrimeSpectrum R) :
 #align prime_spectrum.vanishing_ideal_singleton PrimeSpectrum.vanishingIdeal_singleton
 -/
 
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 theorem subset_zeroLocus_iff_le_vanishingIdeal (t : Set (PrimeSpectrum R)) (I : Ideal R) :
     t ⊆ zeroLocus I ↔ I ≤ vanishingIdeal t :=
   ⟨fun h f k => (mem_vanishingIdeal _ _).mpr fun x j => (mem_zeroLocus _ _).mpr (h j) k, fun h =>
@@ -237,12 +207,6 @@ section Gc
 
 variable (R)
 
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-Case conversion may be inaccurate. Consider using '#align prime_spectrum.gc PrimeSpectrum.gcₓ'. -/
 /-- `zero_locus` and `vanishing_ideal` form a galois connection. -/
 theorem gc :
     @GaloisConnection (Ideal R) (Set (PrimeSpectrum R))ᵒᵈ _ _ (fun I => zeroLocus I) fun t =>
@@ -250,12 +214,6 @@ theorem gc :
   fun I t => subset_zeroLocus_iff_le_vanishingIdeal t I
 #align prime_spectrum.gc PrimeSpectrum.gc
 
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-Case conversion may be inaccurate. Consider using '#align prime_spectrum.gc_set PrimeSpectrum.gc_setₓ'. -/
 /-- `zero_locus` and `vanishing_ideal` form a galois connection. -/
 theorem gc_set :
     @GaloisConnection (Set R) (Set (PrimeSpectrum R))ᵒᵈ _ _ (fun s => zeroLocus s) fun t =>
@@ -280,12 +238,6 @@ theorem subset_vanishingIdeal_zeroLocus (s : Set R) : s ⊆ vanishingIdeal (zero
 #align prime_spectrum.subset_vanishing_ideal_zero_locus PrimeSpectrum.subset_vanishingIdeal_zeroLocus
 -/
 
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-Case conversion may be inaccurate. Consider using '#align prime_spectrum.le_vanishing_ideal_zero_locus PrimeSpectrum.le_vanishingIdeal_zeroLocusₓ'. -/
 theorem le_vanishingIdeal_zeroLocus (I : Ideal R) : I ≤ vanishingIdeal (zeroLocus I) :=
   (gc R).le_u_l I
 #align prime_spectrum.le_vanishing_ideal_zero_locus PrimeSpectrum.le_vanishingIdeal_zeroLocus
@@ -321,34 +273,16 @@ theorem zeroLocus_anti_mono {s t : Set R} (h : s ⊆ t) : zeroLocus t ⊆ zeroLo
 #align prime_spectrum.zero_locus_anti_mono PrimeSpectrum.zeroLocus_anti_mono
 -/
 
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 theorem zeroLocus_anti_mono_ideal {s t : Ideal R} (h : s ≤ t) :
     zeroLocus (t : Set R) ⊆ zeroLocus (s : Set R) :=
   (gc R).monotone_l h
 #align prime_spectrum.zero_locus_anti_mono_ideal PrimeSpectrum.zeroLocus_anti_mono_ideal
 
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-Case conversion may be inaccurate. Consider using '#align prime_spectrum.vanishing_ideal_anti_mono PrimeSpectrum.vanishingIdeal_anti_monoₓ'. -/
 theorem vanishingIdeal_anti_mono {s t : Set (PrimeSpectrum R)} (h : s ⊆ t) :
     vanishingIdeal t ≤ vanishingIdeal s :=
   (gc R).monotone_u h
 #align prime_spectrum.vanishing_ideal_anti_mono PrimeSpectrum.vanishingIdeal_anti_mono
 
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-Case conversion may be inaccurate. Consider using '#align prime_spectrum.zero_locus_subset_zero_locus_iff PrimeSpectrum.zeroLocus_subset_zeroLocus_iffₓ'. -/
 theorem zeroLocus_subset_zeroLocus_iff (I J : Ideal R) :
     zeroLocus (I : Set R) ⊆ zeroLocus (J : Set R) ↔ J ≤ I.radical :=
   ⟨fun h =>
@@ -366,22 +300,10 @@ theorem zeroLocus_subset_zeroLocus_singleton_iff (f g : R) :
 #align prime_spectrum.zero_locus_subset_zero_locus_singleton_iff PrimeSpectrum.zeroLocus_subset_zeroLocus_singleton_iff
 -/
 
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 theorem zeroLocus_bot : zeroLocus ((⊥ : Ideal R) : Set R) = Set.univ :=
   (gc R).l_bot
 #align prime_spectrum.zero_locus_bot PrimeSpectrum.zeroLocus_bot
 
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 @[simp]
 theorem zeroLocus_singleton_zero : zeroLocus ({0} : Set R) = Set.univ :=
   zeroLocus_bot
@@ -394,23 +316,11 @@ theorem zeroLocus_empty : zeroLocus (∅ : Set R) = Set.univ :=
 #align prime_spectrum.zero_locus_empty PrimeSpectrum.zeroLocus_empty
 -/
 
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 @[simp]
 theorem vanishingIdeal_univ : vanishingIdeal (∅ : Set (PrimeSpectrum R)) = ⊤ := by
   simpa using (gc R).u_top
 #align prime_spectrum.vanishing_ideal_univ PrimeSpectrum.vanishingIdeal_univ
 
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 theorem zeroLocus_empty_of_one_mem {s : Set R} (h : (1 : R) ∈ s) : zeroLocus s = ∅ :=
   by
   rw [Set.eq_empty_iff_forall_not_mem]
@@ -421,23 +331,11 @@ theorem zeroLocus_empty_of_one_mem {s : Set R} (h : (1 : R) ∈ s) : zeroLocus s
   apply x_prime.ne_top eq_top
 #align prime_spectrum.zero_locus_empty_of_one_mem PrimeSpectrum.zeroLocus_empty_of_one_mem
 
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 @[simp]
 theorem zeroLocus_singleton_one : zeroLocus ({1} : Set R) = ∅ :=
   zeroLocus_empty_of_one_mem (Set.mem_singleton (1 : R))
 #align prime_spectrum.zero_locus_singleton_one PrimeSpectrum.zeroLocus_singleton_one
 
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 theorem zeroLocus_empty_iff_eq_top {I : Ideal R} : zeroLocus (I : Set R) = ∅ ↔ I = ⊤ :=
   by
   constructor
@@ -455,34 +353,16 @@ theorem zeroLocus_univ : zeroLocus (Set.univ : Set R) = ∅ :=
 #align prime_spectrum.zero_locus_univ PrimeSpectrum.zeroLocus_univ
 -/
 
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 theorem vanishingIdeal_eq_top_iff {s : Set (PrimeSpectrum R)} : vanishingIdeal s = ⊤ ↔ s = ∅ := by
   rw [← top_le_iff, ← subset_zero_locus_iff_le_vanishing_ideal, Submodule.top_coe, zero_locus_univ,
     Set.subset_empty_iff]
 #align prime_spectrum.vanishing_ideal_eq_top_iff PrimeSpectrum.vanishingIdeal_eq_top_iff
 
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 theorem zeroLocus_sup (I J : Ideal R) :
     zeroLocus ((I ⊔ J : Ideal R) : Set R) = zeroLocus I ∩ zeroLocus J :=
   (gc R).l_sup
 #align prime_spectrum.zero_locus_sup PrimeSpectrum.zeroLocus_sup
 
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 theorem zeroLocus_union (s s' : Set R) : zeroLocus (s ∪ s') = zeroLocus s ∩ zeroLocus s' :=
   (gc_set R).l_sup
 #align prime_spectrum.zero_locus_union PrimeSpectrum.zeroLocus_union
@@ -494,23 +374,11 @@ theorem vanishingIdeal_union (t t' : Set (PrimeSpectrum R)) :
 #align prime_spectrum.vanishing_ideal_union PrimeSpectrum.vanishingIdeal_union
 -/
 
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 theorem zeroLocus_iSup {ι : Sort _} (I : ι → Ideal R) :
     zeroLocus ((⨆ i, I i : Ideal R) : Set R) = ⋂ i, zeroLocus (I i) :=
   (gc R).l_iSup
 #align prime_spectrum.zero_locus_supr PrimeSpectrum.zeroLocus_iSup
 
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 theorem zeroLocus_iUnion {ι : Sort _} (s : ι → Set R) :
     zeroLocus (⋃ i, s i) = ⋂ i, zeroLocus (s i) :=
   (gc_set R).l_iSup
@@ -522,56 +390,26 @@ theorem zeroLocus_bUnion (s : Set (Set R)) :
 #align prime_spectrum.zero_locus_bUnion PrimeSpectrum.zeroLocus_bUnion
 -/
 
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 theorem vanishingIdeal_iUnion {ι : Sort _} (t : ι → Set (PrimeSpectrum R)) :
     vanishingIdeal (⋃ i, t i) = ⨅ i, vanishingIdeal (t i) :=
   (gc R).u_iInf
 #align prime_spectrum.vanishing_ideal_Union PrimeSpectrum.vanishingIdeal_iUnion
 
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 theorem zeroLocus_inf (I J : Ideal R) :
     zeroLocus ((I ⊓ J : Ideal R) : Set R) = zeroLocus I ∪ zeroLocus J :=
   Set.ext fun x => x.2.inf_le
 #align prime_spectrum.zero_locus_inf PrimeSpectrum.zeroLocus_inf
 
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 theorem union_zeroLocus (s s' : Set R) :
     zeroLocus s ∪ zeroLocus s' = zeroLocus (Ideal.span s ⊓ Ideal.span s' : Ideal R) := by
   rw [zero_locus_inf]; simp
 #align prime_spectrum.union_zero_locus PrimeSpectrum.union_zeroLocus
 
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 theorem zeroLocus_mul (I J : Ideal R) :
     zeroLocus ((I * J : Ideal R) : Set R) = zeroLocus I ∪ zeroLocus J :=
   Set.ext fun x => x.2.mul_le
 #align prime_spectrum.zero_locus_mul PrimeSpectrum.zeroLocus_mul
 
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 theorem zeroLocus_singleton_mul (f g : R) :
     zeroLocus ({f * g} : Set R) = zeroLocus {f} ∪ zeroLocus {g} :=
   Set.ext fun x => by simpa using x.2.mul_mem_iff_mem_or_mem
@@ -585,24 +423,12 @@ theorem zeroLocus_pow (I : Ideal R) {n : ℕ} (hn : 0 < n) :
 #align prime_spectrum.zero_locus_pow PrimeSpectrum.zeroLocus_pow
 -/
 
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 @[simp]
 theorem zeroLocus_singleton_pow (f : R) (n : ℕ) (hn : 0 < n) :
     zeroLocus ({f ^ n} : Set R) = zeroLocus {f} :=
   Set.ext fun x => by simpa using x.2.pow_mem_iff_mem n hn
 #align prime_spectrum.zero_locus_singleton_pow PrimeSpectrum.zeroLocus_singleton_pow
 
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 theorem sup_vanishingIdeal_le (t t' : Set (PrimeSpectrum R)) :
     vanishingIdeal t ⊔ vanishingIdeal t' ≤ vanishingIdeal (t ∩ t') :=
   by
@@ -613,12 +439,6 @@ theorem sup_vanishingIdeal_le (t t' : Set (PrimeSpectrum R)) :
   apply Submodule.add_mem <;> solve_by_elim
 #align prime_spectrum.sup_vanishing_ideal_le PrimeSpectrum.sup_vanishingIdeal_le
 
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-Case conversion may be inaccurate. Consider using '#align prime_spectrum.mem_compl_zero_locus_iff_not_mem PrimeSpectrum.mem_compl_zeroLocus_iff_not_memₓ'. -/
 theorem mem_compl_zeroLocus_iff_not_mem {f : R} {I : PrimeSpectrum R} :
     I ∈ (zeroLocus {f} : Set (PrimeSpectrum R))ᶜ ↔ f ∉ I.asIdeal := by
   rw [Set.mem_compl_iff, mem_zero_locus, Set.singleton_subset_iff] <;> rfl
@@ -639,12 +459,6 @@ instance zariskiTopology : TopologicalSpace (PrimeSpectrum R) :=
 #align prime_spectrum.zariski_topology PrimeSpectrum.zariskiTopology
 -/
 
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-Case conversion may be inaccurate. Consider using '#align prime_spectrum.is_open_iff PrimeSpectrum.isOpen_iffₓ'. -/
 theorem isOpen_iff (U : Set (PrimeSpectrum R)) : IsOpen U ↔ ∃ s, Uᶜ = zeroLocus s := by
   simp only [@eq_comm _ (Uᶜ)] <;> rfl
 #align prime_spectrum.is_open_iff PrimeSpectrum.isOpen_iff
@@ -736,12 +550,6 @@ theorem isRadical_vanishingIdeal (s : Set (PrimeSpectrum R)) : (vanishingIdeal s
 #align prime_spectrum.is_radical_vanishing_ideal PrimeSpectrum.isRadical_vanishingIdeal
 -/
 
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-Case conversion may be inaccurate. Consider using '#align prime_spectrum.vanishing_ideal_anti_mono_iff PrimeSpectrum.vanishingIdeal_anti_mono_iffₓ'. -/
 theorem vanishingIdeal_anti_mono_iff {s t : Set (PrimeSpectrum R)} (ht : IsClosed t) :
     s ⊆ t ↔ vanishingIdeal t ≤ vanishingIdeal s :=
   ⟨vanishingIdeal_anti_mono, fun h =>
@@ -750,24 +558,12 @@ theorem vanishingIdeal_anti_mono_iff {s t : Set (PrimeSpectrum R)} (ht : IsClose
     convert← zero_locus_anti_mono_ideal h <;> apply zero_locus_vanishing_ideal_eq_closure⟩
 #align prime_spectrum.vanishing_ideal_anti_mono_iff PrimeSpectrum.vanishingIdeal_anti_mono_iff
 
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-Case conversion may be inaccurate. Consider using '#align prime_spectrum.vanishing_ideal_strict_anti_mono_iff PrimeSpectrum.vanishingIdeal_strict_anti_mono_iffₓ'. -/
 theorem vanishingIdeal_strict_anti_mono_iff {s t : Set (PrimeSpectrum R)} (hs : IsClosed s)
     (ht : IsClosed t) : s ⊂ t ↔ vanishingIdeal t < vanishingIdeal s := by
   rw [Set.ssubset_def, vanishing_ideal_anti_mono_iff hs, vanishing_ideal_anti_mono_iff ht,
     lt_iff_le_not_le]
 #align prime_spectrum.vanishing_ideal_strict_anti_mono_iff PrimeSpectrum.vanishingIdeal_strict_anti_mono_iff
 
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-Case conversion may be inaccurate. Consider using '#align prime_spectrum.closeds_embedding PrimeSpectrum.closedsEmbeddingₓ'. -/
 /-- The antitone order embedding of closed subsets of `Spec R` into ideals of `R`. -/
 def closedsEmbedding (R : Type _) [CommRing R] :
     (TopologicalSpace.Closeds <| PrimeSpectrum R)ᵒᵈ ↪o Ideal R :=
@@ -852,12 +648,6 @@ section Comap
 
 variable {S' : Type _} [CommRing S']
 
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 theorem preimage_comap_zeroLocus_aux (f : R →+* S) (s : Set R) :
     (fun y => ⟨Ideal.comap f y.asIdeal, inferInstance⟩ : PrimeSpectrum S → PrimeSpectrum R) ⁻¹'
         zeroLocus s =
@@ -884,12 +674,6 @@ def comap (f : R →+* S) : C(PrimeSpectrum S, PrimeSpectrum R)
 
 variable (f : R →+* S)
 
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 @[simp]
 theorem comap_asIdeal (y : PrimeSpectrum S) : (comap f y).asIdeal = Ideal.comap f y.asIdeal :=
   rfl
@@ -901,45 +685,21 @@ theorem comap_id : comap (RingHom.id R) = ContinuousMap.id _ := by ext; rfl
 #align prime_spectrum.comap_id PrimeSpectrum.comap_id
 -/
 
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 @[simp]
 theorem comap_comp (f : R →+* S) (g : S →+* S') : comap (g.comp f) = (comap f).comp (comap g) :=
   rfl
 #align prime_spectrum.comap_comp PrimeSpectrum.comap_comp
 
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 theorem comap_comp_apply (f : R →+* S) (g : S →+* S') (x : PrimeSpectrum S') :
     PrimeSpectrum.comap (g.comp f) x = (PrimeSpectrum.comap f) (PrimeSpectrum.comap g x) :=
   rfl
 #align prime_spectrum.comap_comp_apply PrimeSpectrum.comap_comp_apply
 
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 @[simp]
 theorem preimage_comap_zeroLocus (s : Set R) : comap f ⁻¹' zeroLocus s = zeroLocus (f '' s) :=
   preimage_comap_zeroLocus_aux f s
 #align prime_spectrum.preimage_comap_zero_locus PrimeSpectrum.preimage_comap_zeroLocus
 
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 theorem comap_injective_of_surjective (f : R →+* S) (hf : Function.Surjective f) :
     Function.Injective (comap f) := fun x y h =>
   PrimeSpectrum.ext _ _
@@ -947,12 +707,6 @@ theorem comap_injective_of_surjective (f : R →+* S) (hf : Function.Surjective
       (congr_arg PrimeSpectrum.asIdeal h : (comap f x).asIdeal = (comap f y).asIdeal))
 #align prime_spectrum.comap_injective_of_surjective PrimeSpectrum.comap_injective_of_surjective
 
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-Case conversion may be inaccurate. Consider using '#align prime_spectrum.comap_singleton_is_closed_of_surjective PrimeSpectrum.comap_singleton_isClosed_of_surjectiveₓ'. -/
 theorem comap_singleton_isClosed_of_surjective (f : R →+* S) (hf : Function.Surjective f)
     (x : PrimeSpectrum S) (hx : IsClosed ({x} : Set (PrimeSpectrum S))) :
     IsClosed ({comap f x} : Set (PrimeSpectrum R)) :=
@@ -960,12 +714,6 @@ theorem comap_singleton_isClosed_of_surjective (f : R →+* S) (hf : Function.Su
   (is_closed_singleton_iff_is_maximal _).2 (Ideal.comap_isMaximal_of_surjective f hf)
 #align prime_spectrum.comap_singleton_is_closed_of_surjective PrimeSpectrum.comap_singleton_isClosed_of_surjective
 
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 theorem comap_singleton_isClosed_of_isIntegral (f : R →+* S) (hf : f.IsIntegral)
     (x : PrimeSpectrum S) (hx : IsClosed ({x} : Set (PrimeSpectrum S))) :
     IsClosed ({comap f x} : Set (PrimeSpectrum R)) :=
@@ -976,12 +724,6 @@ theorem comap_singleton_isClosed_of_isIntegral (f : R →+* S) (hf : f.IsIntegra
 
 variable (S)
 
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-Case conversion may be inaccurate. Consider using '#align prime_spectrum.localization_comap_inducing PrimeSpectrum.localization_comap_inducingₓ'. -/
 theorem localization_comap_inducing [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
     Inducing (comap (algebraMap R S)) := by
   constructor
@@ -999,12 +741,6 @@ theorem localization_comap_inducing [Algebra R S] (M : Submonoid R) [IsLocalizat
   · rintro ⟨_, ⟨t, rfl⟩, rfl⟩; simp
 #align prime_spectrum.localization_comap_inducing PrimeSpectrum.localization_comap_inducing
 
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 theorem localization_comap_injective [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
     Function.Injective (comap (algebraMap R S)) :=
   by
@@ -1016,23 +752,11 @@ theorem localization_comap_injective [Algebra R S] (M : Submonoid R) [IsLocaliza
   exact h
 #align prime_spectrum.localization_comap_injective PrimeSpectrum.localization_comap_injective
 
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 theorem localization_comap_embedding [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
     Embedding (comap (algebraMap R S)) :=
   ⟨localization_comap_inducing S M, localization_comap_injective S M⟩
 #align prime_spectrum.localization_comap_embedding PrimeSpectrum.localization_comap_embedding
 
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 theorem localization_comap_range [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
     Set.range (comap (algebraMap R S)) = { p | Disjoint (M : Set R) p.asIdeal } :=
   by
@@ -1054,12 +778,6 @@ section SpecOfSurjective
 
 open Function RingHom
 
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 theorem comap_inducing_of_surjective (hf : Surjective f) : Inducing (comap f) :=
   {
     induced :=
@@ -1075,9 +793,6 @@ theorem comap_inducing_of_surjective (hf : Surjective f) : Inducing (comap f) :=
       exact ⟨f '' F, hF.symm.trans (preimage_comap_zero_locus f F)⟩ }
 #align prime_spectrum.comap_inducing_of_surjective PrimeSpectrum.comap_inducing_of_surjective
 
-/- warning: prime_spectrum.image_comap_zero_locus_eq_zero_locus_comap -> PrimeSpectrum.image_comap_zeroLocus_eq_zeroLocus_comap is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align prime_spectrum.image_comap_zero_locus_eq_zero_locus_comap PrimeSpectrum.image_comap_zeroLocus_eq_zeroLocus_comapₓ'. -/
 theorem image_comap_zeroLocus_eq_zeroLocus_comap (hf : Surjective f) (I : Ideal S) :
     comap f '' zeroLocus I = zeroLocus (I.comap f) :=
   by
@@ -1099,12 +814,6 @@ theorem image_comap_zeroLocus_eq_zeroLocus_comap (hf : Surjective f) (I : Ideal
       rwa [mem_ker, map_sub, sub_eq_zero]
 #align prime_spectrum.image_comap_zero_locus_eq_zero_locus_comap PrimeSpectrum.image_comap_zeroLocus_eq_zeroLocus_comap
 
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-Case conversion may be inaccurate. Consider using '#align prime_spectrum.range_comap_of_surjective PrimeSpectrum.range_comap_of_surjectiveₓ'. -/
 theorem range_comap_of_surjective (hf : Surjective f) : Set.range (comap f) = zeroLocus (ker f) :=
   by
   rw [← Set.image_univ]
@@ -1112,24 +821,12 @@ theorem range_comap_of_surjective (hf : Surjective f) : Set.range (comap f) = ze
   rw [zero_locus_bot]
 #align prime_spectrum.range_comap_of_surjective PrimeSpectrum.range_comap_of_surjective
 
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 theorem isClosed_range_comap_of_surjective (hf : Surjective f) : IsClosed (Set.range (comap f)) :=
   by
   rw [range_comap_of_surjective _ f hf]
   exact is_closed_zero_locus ↑(ker f)
 #align prime_spectrum.is_closed_range_comap_of_surjective PrimeSpectrum.isClosed_range_comap_of_surjective
 
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 theorem closedEmbedding_comap_of_surjective (hf : Surjective f) : ClosedEmbedding (comap f) :=
   { induced := (comap_inducing_of_surjective S f hf).induced
     inj := comap_injective_of_surjective f hf
@@ -1164,88 +861,40 @@ theorem isOpen_basicOpen {a : R} : IsOpen (basicOpen a : Set (PrimeSpectrum R))
 #align prime_spectrum.is_open_basic_open PrimeSpectrum.isOpen_basicOpen
 -/
 
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 @[simp]
 theorem basicOpen_eq_zeroLocus_compl (r : R) :
     (basicOpen r : Set (PrimeSpectrum R)) = zeroLocus {r}ᶜ :=
   Set.ext fun x => by simpa only [Set.mem_compl_iff, mem_zero_locus, Set.singleton_subset_iff]
 #align prime_spectrum.basic_open_eq_zero_locus_compl PrimeSpectrum.basicOpen_eq_zeroLocus_compl
 
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 @[simp]
 theorem basicOpen_one : basicOpen (1 : R) = ⊤ :=
   TopologicalSpace.Opens.ext <| by simp
 #align prime_spectrum.basic_open_one PrimeSpectrum.basicOpen_one
 
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 @[simp]
 theorem basicOpen_zero : basicOpen (0 : R) = ⊥ :=
   TopologicalSpace.Opens.ext <| by simp
 #align prime_spectrum.basic_open_zero PrimeSpectrum.basicOpen_zero
 
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 theorem basicOpen_le_basicOpen_iff (f g : R) :
     basicOpen f ≤ basicOpen g ↔ f ∈ (Ideal.span ({g} : Set R)).radical := by
   rw [← SetLike.coe_subset_coe, basic_open_eq_zero_locus_compl, basic_open_eq_zero_locus_compl,
     Set.compl_subset_compl, zero_locus_subset_zero_locus_singleton_iff]
 #align prime_spectrum.basic_open_le_basic_open_iff PrimeSpectrum.basicOpen_le_basicOpen_iff
 
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 theorem basicOpen_mul (f g : R) : basicOpen (f * g) = basicOpen f ⊓ basicOpen g :=
   TopologicalSpace.Opens.ext <| by simp [zero_locus_singleton_mul]
 #align prime_spectrum.basic_open_mul PrimeSpectrum.basicOpen_mul
 
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 theorem basicOpen_mul_le_left (f g : R) : basicOpen (f * g) ≤ basicOpen f := by
   rw [basic_open_mul f g]; exact inf_le_left
 #align prime_spectrum.basic_open_mul_le_left PrimeSpectrum.basicOpen_mul_le_left
 
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 theorem basicOpen_mul_le_right (f g : R) : basicOpen (f * g) ≤ basicOpen g := by
   rw [basic_open_mul f g]; exact inf_le_right
 #align prime_spectrum.basic_open_mul_le_right PrimeSpectrum.basicOpen_mul_le_right
 
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 @[simp]
 theorem basicOpen_pow (f : R) (n : ℕ) (hn : 0 < n) : basicOpen (f ^ n) = basicOpen f :=
   TopologicalSpace.Opens.ext <| by simpa using zero_locus_singleton_pow f n hn
@@ -1304,12 +953,6 @@ theorem isCompact_basicOpen (f : R) : IsCompact (basicOpen f : Set (PrimeSpectru
 #align prime_spectrum.is_compact_basic_open PrimeSpectrum.isCompact_basicOpen
 -/
 
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 @[simp]
 theorem basicOpen_eq_bot_iff (f : R) : basicOpen f = ⊥ ↔ IsNilpotent f :=
   by
@@ -1319,12 +962,6 @@ theorem basicOpen_eq_bot_iff (f : R) : basicOpen f = ⊥ ↔ IsNilpotent f :=
   exact ⟨fun h I hI => h ⟨I, hI⟩, fun h ⟨I, hI⟩ => h I hI⟩
 #align prime_spectrum.basic_open_eq_bot_iff PrimeSpectrum.basicOpen_eq_bot_iff
 
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 theorem localization_away_comap_range (S : Type v) [CommRing S] [Algebra R S] (r : R)
     [IsLocalization.Away r S] : Set.range (comap (algebraMap R S)) = basicOpen r :=
   by
@@ -1339,12 +976,6 @@ theorem localization_away_comap_range (S : Type v) [CommRing S] [Algebra R S] (r
     exact h₁ (x.2.mem_of_pow_mem _ h₃)
 #align prime_spectrum.localization_away_comap_range PrimeSpectrum.localization_away_comap_range
 
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 theorem localization_away_openEmbedding (S : Type v) [CommRing S] [Algebra R S] (r : R)
     [IsLocalization.Away r S] : OpenEmbedding (comap (algebraMap R S)) :=
   { toEmbedding := localization_comap_embedding S (Submonoid.powers r)
@@ -1369,56 +1000,26 @@ We endow `prime_spectrum R` with a partial order, where `x ≤ y` if and only if
 instance : PartialOrder (PrimeSpectrum R) :=
   PartialOrder.lift asIdeal ext
 
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 @[simp]
 theorem asIdeal_le_asIdeal (x y : PrimeSpectrum R) : x.asIdeal ≤ y.asIdeal ↔ x ≤ y :=
   Iff.rfl
 #align prime_spectrum.as_ideal_le_as_ideal PrimeSpectrum.asIdeal_le_asIdeal
 
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 @[simp]
 theorem asIdeal_lt_asIdeal (x y : PrimeSpectrum R) : x.asIdeal < y.asIdeal ↔ x < y :=
   Iff.rfl
 #align prime_spectrum.as_ideal_lt_as_ideal PrimeSpectrum.asIdeal_lt_asIdeal
 
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 theorem le_iff_mem_closure (x y : PrimeSpectrum R) :
     x ≤ y ↔ y ∈ closure ({x} : Set (PrimeSpectrum R)) := by
   rw [← as_ideal_le_as_ideal, ← zero_locus_vanishing_ideal_eq_closure, mem_zero_locus,
     vanishing_ideal_singleton, SetLike.coe_subset_coe]
 #align prime_spectrum.le_iff_mem_closure PrimeSpectrum.le_iff_mem_closure
 
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-Case conversion may be inaccurate. Consider using '#align prime_spectrum.le_iff_specializes PrimeSpectrum.le_iff_specializesₓ'. -/
 theorem le_iff_specializes (x y : PrimeSpectrum R) : x ≤ y ↔ x ⤳ y :=
   (le_iff_mem_closure x y).trans specializes_iff_mem_closure.symm
 #align prime_spectrum.le_iff_specializes PrimeSpectrum.le_iff_specializes
 
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 /-- `nhds` as an order embedding. -/
 @[simps (config := { fullyApplied := true })]
 def nhdsOrderEmbedding : PrimeSpectrum R ↪o Filter (PrimeSpectrum R) :=
@@ -1440,12 +1041,6 @@ instance {R : Type _} [Field R] : Unique (PrimeSpectrum R)
 
 end Order
 
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 /-- If `x` specializes to `y`, then there is a natural map from the localization of `y` to the
 localization of `x`. -/
 def localizationMapOfSpecializes {x y : PrimeSpectrum R} (h : x ⤳ y) :
@@ -1474,23 +1069,11 @@ def closedPoint : PrimeSpectrum R :=
 
 variable {R}
 
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 theorem isLocalRingHom_iff_comap_closedPoint {S : Type v} [CommRing S] [LocalRing S] (f : R →+* S) :
     IsLocalRingHom f ↔ PrimeSpectrum.comap f (closedPoint S) = closedPoint R := by
   rw [(local_hom_tfae f).out 0 4, PrimeSpectrum.ext_iff]; rfl
 #align local_ring.is_local_ring_hom_iff_comap_closed_point LocalRing.isLocalRingHom_iff_comap_closedPoint
 
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 @[simp]
 theorem comap_closedPoint {S : Type v} [CommRing S] [LocalRing S] (f : R →+* S) [IsLocalRingHom f] :
     PrimeSpectrum.comap f (closedPoint S) = closedPoint R :=
@@ -1503,12 +1086,6 @@ theorem specializes_closedPoint (x : PrimeSpectrum R) : x ⤳ closedPoint R :=
 #align local_ring.specializes_closed_point LocalRing.specializes_closedPoint
 -/
 
-/- warning: local_ring.closed_point_mem_iff -> LocalRing.closedPoint_mem_iff is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_3 : LocalRing.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] (U : TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)), Iff (Membership.Mem.{u1, u1} (PrimeSpectrum.{u1} R _inst_1) (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u1} R _inst_1) (TopologicalSpace.Opens.setLike.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))) (LocalRing.closedPoint.{u1} R _inst_1 _inst_3) U) (Eq.{succ u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) U (Top.top.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (CompleteLattice.toHasTop.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (TopologicalSpace.Opens.completeLattice.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)))))
-but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_3 : LocalRing.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] (U : TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)), Iff (Membership.mem.{u1, u1} (PrimeSpectrum.{u1} R _inst_1) (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u1} R _inst_1) (TopologicalSpace.Opens.instSetLikeOpens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))) (LocalRing.closedPoint.{u1} R _inst_1 _inst_3) U) (Eq.{succ u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) U (Top.top.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)))))
-Case conversion may be inaccurate. Consider using '#align local_ring.closed_point_mem_iff LocalRing.closedPoint_mem_iffₓ'. -/
 theorem closedPoint_mem_iff (U : TopologicalSpace.Opens <| PrimeSpectrum R) :
     closedPoint R ∈ U ↔ U = ⊤ := by
   constructor

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

@@ -140,10 +140,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align prime_spectrum.prime_spectrum_prod_symm_inl_as_ideal PrimeSpectrum.primeSpectrumProd_symm_inl_asIdealₓ'. -/
 @[simp]
 theorem primeSpectrumProd_symm_inl_asIdeal (x : PrimeSpectrum R) :
-    ((primeSpectrumProd R S).symm <| Sum.inl x).asIdeal = Ideal.prod x.asIdeal ⊤ :=
-  by
-  cases x
-  rfl
+    ((primeSpectrumProd R S).symm <| Sum.inl x).asIdeal = Ideal.prod x.asIdeal ⊤ := by cases x; rfl
 #align prime_spectrum.prime_spectrum_prod_symm_inl_as_ideal PrimeSpectrum.primeSpectrumProd_symm_inl_asIdeal
 
 /- warning: prime_spectrum.prime_spectrum_prod_symm_inr_as_ideal -> PrimeSpectrum.primeSpectrumProd_symm_inr_asIdeal is a dubious translation:
@@ -154,10 +151,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align prime_spectrum.prime_spectrum_prod_symm_inr_as_ideal PrimeSpectrum.primeSpectrumProd_symm_inr_asIdealₓ'. -/
 @[simp]
 theorem primeSpectrumProd_symm_inr_asIdeal (x : PrimeSpectrum S) :
-    ((primeSpectrumProd R S).symm <| Sum.inr x).asIdeal = Ideal.prod ⊤ x.asIdeal :=
-  by
-  cases x
-  rfl
+    ((primeSpectrumProd R S).symm <| Sum.inr x).asIdeal = Ideal.prod ⊤ x.asIdeal := by cases x; rfl
 #align prime_spectrum.prime_spectrum_prod_symm_inr_as_ideal PrimeSpectrum.primeSpectrumProd_symm_inr_asIdeal
 
 #print PrimeSpectrum.zeroLocus /-
@@ -183,9 +177,7 @@ theorem mem_zeroLocus (x : PrimeSpectrum R) (s : Set R) : x ∈ zeroLocus s ↔
 
 #print PrimeSpectrum.zeroLocus_span /-
 @[simp]
-theorem zeroLocus_span (s : Set R) : zeroLocus (Ideal.span s : Set R) = zeroLocus s :=
-  by
-  ext x
+theorem zeroLocus_span (s : Set R) : zeroLocus (Ideal.span s : Set R) = zeroLocus s := by ext x;
   exact (Submodule.gi R R).gc s x.as_ideal
 #align prime_spectrum.zero_locus_span PrimeSpectrum.zeroLocus_span
 -/
@@ -425,9 +417,7 @@ theorem zeroLocus_empty_of_one_mem {s : Set R} (h : (1 : R) ∈ s) : zeroLocus s
   intro x hx
   rw [mem_zero_locus] at hx
   have x_prime : x.as_ideal.is_prime := by infer_instance
-  have eq_top : x.as_ideal = ⊤ := by
-    rw [Ideal.eq_top_iff_one]
-    exact hx h
+  have eq_top : x.as_ideal = ⊤ := by rw [Ideal.eq_top_iff_one]; exact hx h
   apply x_prime.ne_top eq_top
 #align prime_spectrum.zero_locus_empty_of_one_mem PrimeSpectrum.zeroLocus_empty_of_one_mem
 
@@ -455,9 +445,7 @@ theorem zeroLocus_empty_iff_eq_top {I : Ideal R} : zeroLocus (I : Set R) = ∅ 
     intro h
     rcases Ideal.exists_le_maximal I h with ⟨M, hM, hIM⟩
     exact Set.Nonempty.ne_empty ⟨⟨M, hM.is_prime⟩, hIM⟩
-  · rintro rfl
-    apply zero_locus_empty_of_one_mem
-    trivial
+  · rintro rfl; apply zero_locus_empty_of_one_mem; trivial
 #align prime_spectrum.zero_locus_empty_iff_eq_top PrimeSpectrum.zeroLocus_empty_iff_eq_top
 
 #print PrimeSpectrum.zeroLocus_univ /-
@@ -563,10 +551,8 @@ but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (s : Set.{u1} R) (s' : Set.{u1} R), Eq.{succ u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Union.union.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.instUnionSet.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 s) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 s')) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 (SetLike.coe.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Inf.inf.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.instInfSubmodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Ideal.span.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) s) (Ideal.span.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) s'))))
 Case conversion may be inaccurate. Consider using '#align prime_spectrum.union_zero_locus PrimeSpectrum.union_zeroLocusₓ'. -/
 theorem union_zeroLocus (s s' : Set R) :
-    zeroLocus s ∪ zeroLocus s' = zeroLocus (Ideal.span s ⊓ Ideal.span s' : Ideal R) :=
-  by
-  rw [zero_locus_inf]
-  simp
+    zeroLocus s ∪ zeroLocus s' = zeroLocus (Ideal.span s ⊓ Ideal.span s' : Ideal R) := by
+  rw [zero_locus_inf]; simp
 #align prime_spectrum.union_zero_locus PrimeSpectrum.union_zeroLocus
 
 /- warning: prime_spectrum.zero_locus_mul -> PrimeSpectrum.zeroLocus_mul is a dubious translation:
@@ -649,9 +635,7 @@ instance zariskiTopology : TopologicalSpace (PrimeSpectrum R) :=
       choose f hf using fun i : Zs => h i.Prop
       simp only [← hf]
       exact ⟨_, zero_locus_Union _⟩)
-    (by
-      rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩
-      exact ⟨_, (union_zero_locus s t).symm⟩)
+    (by rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩; exact ⟨_, (union_zero_locus s t).symm⟩)
 #align prime_spectrum.zariski_topology PrimeSpectrum.zariskiTopology
 -/
 
@@ -689,9 +673,7 @@ theorem isClosed_iff_zeroLocus_radical_ideal (Z : Set (PrimeSpectrum R)) :
 -/
 
 #print PrimeSpectrum.isClosed_zeroLocus /-
-theorem isClosed_zeroLocus (s : Set R) : IsClosed (zeroLocus s) :=
-  by
-  rw [is_closed_iff_zero_locus]
+theorem isClosed_zeroLocus (s : Set R) : IsClosed (zeroLocus s) := by rw [is_closed_iff_zero_locus];
   exact ⟨s, rfl⟩
 #align prime_spectrum.is_closed_zero_locus PrimeSpectrum.isClosed_zeroLocus
 -/
@@ -807,8 +789,7 @@ theorem t1Space_iff_isField [IsDomain R] : T1Space (PrimeSpectrum R) ↔ IsField
           (Classical.not_not.2 rfl))
   · refine' ⟨fun x => (is_closed_singleton_iff_is_maximal x).2 _⟩
     by_cases hx : x.as_ideal = ⊥
-    · letI := h.to_field
-      exact hx.symm ▸ Ideal.bot_isMaximal
+    · letI := h.to_field; exact hx.symm ▸ Ideal.bot_isMaximal
     · exact absurd h (Ring.not_isField_iff_exists_prime.2 ⟨x.as_ideal, ⟨hx, x.2⟩⟩)
 #align prime_spectrum.t1_space_iff_is_field PrimeSpectrum.t1Space_iff_isField
 -/
@@ -826,10 +807,8 @@ theorem isIrreducible_zeroLocus_iff_of_radical (I : Ideal R) (hI : I.IsRadical)
   · trans ∀ x y : Ideal R, Z(I) ⊆ Z(x) ∪ Z(y) → Z(I) ⊆ Z(x) ∨ Z(I) ⊆ Z(y)
     · simp_rw [isPreirreducible_iff_closed_union_closed, is_closed_iff_zero_locus_ideal]
       constructor
-      · rintro h x y
-        exact h _ _ ⟨x, rfl⟩ ⟨y, rfl⟩
-      · rintro h _ _ ⟨x, rfl⟩ ⟨y, rfl⟩
-        exact h x y
+      · rintro h x y; exact h _ _ ⟨x, rfl⟩ ⟨y, rfl⟩
+      · rintro h _ _ ⟨x, rfl⟩ ⟨y, rfl⟩; exact h x y
     · simp_rw [← zero_locus_inf, subset_zero_locus_iff_le_vanishing_ideal,
         vanishing_ideal_zero_locus_eq_radical, hI.radical]
       constructor
@@ -918,10 +897,7 @@ theorem comap_asIdeal (y : PrimeSpectrum S) : (comap f y).asIdeal = Ideal.comap
 
 #print PrimeSpectrum.comap_id /-
 @[simp]
-theorem comap_id : comap (RingHom.id R) = ContinuousMap.id _ :=
-  by
-  ext
-  rfl
+theorem comap_id : comap (RingHom.id R) = ContinuousMap.id _ := by ext; rfl
 #align prime_spectrum.comap_id PrimeSpectrum.comap_id
 -/
 
@@ -1020,8 +996,7 @@ theorem localization_comap_inducing [Algebra R S] (M : Submonoid R) [IsLocalizat
     rw [preimage_comap_zero_locus, ← zero_locus_span, ← zero_locus_span s]
     congr 1
     exact congr_arg Submodule.carrier (IsLocalization.map_comap M S (Ideal.span s))
-  · rintro ⟨_, ⟨t, rfl⟩, rfl⟩
-    simp
+  · rintro ⟨_, ⟨t, rfl⟩, rfl⟩; simp
 #align prime_spectrum.localization_comap_inducing PrimeSpectrum.localization_comap_inducing
 
 /- warning: prime_spectrum.localization_comap_injective -> PrimeSpectrum.localization_comap_injective is a dubious translation:
@@ -1251,10 +1226,8 @@ lean 3 declaration is
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (f : R) (g : R), LE.le.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (Preorder.toLE.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)))))) (PrimeSpectrum.basicOpen.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) f g)) (PrimeSpectrum.basicOpen.{u1} R _inst_1 f)
 Case conversion may be inaccurate. Consider using '#align prime_spectrum.basic_open_mul_le_left PrimeSpectrum.basicOpen_mul_le_leftₓ'. -/
-theorem basicOpen_mul_le_left (f g : R) : basicOpen (f * g) ≤ basicOpen f :=
-  by
-  rw [basic_open_mul f g]
-  exact inf_le_left
+theorem basicOpen_mul_le_left (f g : R) : basicOpen (f * g) ≤ basicOpen f := by
+  rw [basic_open_mul f g]; exact inf_le_left
 #align prime_spectrum.basic_open_mul_le_left PrimeSpectrum.basicOpen_mul_le_left
 
 /- warning: prime_spectrum.basic_open_mul_le_right -> PrimeSpectrum.basicOpen_mul_le_right is a dubious translation:
@@ -1263,10 +1236,8 @@ lean 3 declaration is
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (f : R) (g : R), LE.le.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (Preorder.toLE.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)))))) (PrimeSpectrum.basicOpen.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) f g)) (PrimeSpectrum.basicOpen.{u1} R _inst_1 g)
 Case conversion may be inaccurate. Consider using '#align prime_spectrum.basic_open_mul_le_right PrimeSpectrum.basicOpen_mul_le_rightₓ'. -/
-theorem basicOpen_mul_le_right (f g : R) : basicOpen (f * g) ≤ basicOpen g :=
-  by
-  rw [basic_open_mul f g]
-  exact inf_le_right
+theorem basicOpen_mul_le_right (f g : R) : basicOpen (f * g) ≤ basicOpen g := by
+  rw [basic_open_mul f g]; exact inf_le_right
 #align prime_spectrum.basic_open_mul_le_right PrimeSpectrum.basicOpen_mul_le_right
 
 /- warning: prime_spectrum.basic_open_pow -> PrimeSpectrum.basicOpen_pow is a dubious translation:
@@ -1377,19 +1348,14 @@ Case conversion may be inaccurate. Consider using '#align prime_spectrum.localiz
 theorem localization_away_openEmbedding (S : Type v) [CommRing S] [Algebra R S] (r : R)
     [IsLocalization.Away r S] : OpenEmbedding (comap (algebraMap R S)) :=
   { toEmbedding := localization_comap_embedding S (Submonoid.powers r)
-    open_range := by
-      rw [localization_away_comap_range S r]
-      exact is_open_basic_open }
+    open_range := by rw [localization_away_comap_range S r]; exact is_open_basic_open }
 #align prime_spectrum.localization_away_open_embedding PrimeSpectrum.localization_away_openEmbedding
 
 end BasicOpen
 
 /-- The prime spectrum of a commutative ring is a compact topological space. -/
 instance : CompactSpace (PrimeSpectrum R)
-    where isCompact_univ := by
-    convert is_compact_basic_open (1 : R)
-    rw [basic_open_one]
-    rfl
+    where isCompact_univ := by convert is_compact_basic_open (1 : R); rw [basic_open_one]; rfl
 
 section Order
 
@@ -1515,10 +1481,8 @@ but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_3 : LocalRing.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] {S : Type.{u2}} [_inst_4 : CommRing.{u2} S] [_inst_5 : LocalRing.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4))] (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4)))), Iff (IsLocalRingHom.{u1, u2} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4)) f) (Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : PrimeSpectrum.{u2} S _inst_4) => PrimeSpectrum.{u1} R _inst_1) (LocalRing.closedPoint.{u2} S _inst_4 _inst_5)) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_4) (fun (_x : PrimeSpectrum.{u2} S _inst_4) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : PrimeSpectrum.{u2} S _inst_4) => PrimeSpectrum.{u1} R _inst_1) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u2, u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_4 f) (LocalRing.closedPoint.{u2} S _inst_4 _inst_5)) (LocalRing.closedPoint.{u1} R _inst_1 _inst_3))
 Case conversion may be inaccurate. Consider using '#align local_ring.is_local_ring_hom_iff_comap_closed_point LocalRing.isLocalRingHom_iff_comap_closedPointₓ'. -/
 theorem isLocalRingHom_iff_comap_closedPoint {S : Type v} [CommRing S] [LocalRing S] (f : R →+* S) :
-    IsLocalRingHom f ↔ PrimeSpectrum.comap f (closedPoint S) = closedPoint R :=
-  by
-  rw [(local_hom_tfae f).out 0 4, PrimeSpectrum.ext_iff]
-  rfl
+    IsLocalRingHom f ↔ PrimeSpectrum.comap f (closedPoint S) = closedPoint R := by
+  rw [(local_hom_tfae f).out 0 4, PrimeSpectrum.ext_iff]; rfl
 #align local_ring.is_local_ring_hom_iff_comap_closed_point LocalRing.isLocalRingHom_iff_comap_closedPoint
 
 /- warning: local_ring.comap_closed_point -> LocalRing.comap_closedPoint is a dubious translation:
@@ -1548,10 +1512,8 @@ Case conversion may be inaccurate. Consider using '#align local_ring.closed_poin
 theorem closedPoint_mem_iff (U : TopologicalSpace.Opens <| PrimeSpectrum R) :
     closedPoint R ∈ U ↔ U = ⊤ := by
   constructor
-  · rw [eq_top_iff]
-    exact fun h x _ => (specializes_closed_point x).mem_open U.2 h
-  · rintro rfl
-    trivial
+  · rw [eq_top_iff]; exact fun h x _ => (specializes_closed_point x).mem_open U.2 h
+  · rintro rfl; trivial
 #align local_ring.closed_point_mem_iff LocalRing.closedPoint_mem_iff
 
 @[simp]

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

@@ -1101,10 +1101,7 @@ theorem comap_inducing_of_surjective (hf : Surjective f) : Inducing (comap f) :=
 #align prime_spectrum.comap_inducing_of_surjective PrimeSpectrum.comap_inducing_of_surjective
 
 /- warning: prime_spectrum.image_comap_zero_locus_eq_zero_locus_comap -> PrimeSpectrum.image_comap_zeroLocus_eq_zeroLocus_comap is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align prime_spectrum.image_comap_zero_locus_eq_zero_locus_comap PrimeSpectrum.image_comap_zeroLocus_eq_zeroLocus_comapₓ'. -/
 theorem image_comap_zeroLocus_eq_zeroLocus_comap (hf : Surjective f) (I : Ideal S) :
     comap f '' zeroLocus I = zeroLocus (I.comap f) :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/e1a18cad9cd462973d760af7de36b05776b8811c

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
 
 ! This file was ported from Lean 3 source module algebraic_geometry.prime_spectrum.basic
-! leanprover-community/mathlib commit a7c017d750512a352b623b1824d75da5998457d0
+! leanprover-community/mathlib commit a87d22575d946e1e156fc1edd1e1269600a8a282
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -20,6 +20,9 @@ import Mathbin.Topology.Sober
 /-!
 # Prime spectrum of a commutative ring
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 The prime spectrum of a commutative ring is the type of all prime ideals.
 It is naturally endowed with a topology: the Zariski topology.
 

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

@@ -56,6 +56,7 @@ universe u v
 
 variable (R : Type u) (S : Type v) [CommRing R] [CommRing S]
 
+#print PrimeSpectrum /-
 /-- The prime spectrum of a commutative ring `R` is the type of all prime ideals of `R`.
 
 It is naturally endowed with a topology (the Zariski topology),
@@ -66,6 +67,7 @@ structure PrimeSpectrum where
   asIdeal : Ideal R
   IsPrime : as_ideal.IsPrime
 #align prime_spectrum PrimeSpectrum
+-/
 
 attribute [instance] PrimeSpectrum.isPrime
 
@@ -77,13 +79,21 @@ instance [Nontrivial R] : Nonempty <| PrimeSpectrum R :=
   let ⟨I, hI⟩ := Ideal.exists_maximal R
   ⟨⟨I, hI.IsPrime⟩⟩
 
+#print PrimeSpectrum.pUnit /-
 /-- The prime spectrum of the zero ring is empty. -/
 theorem pUnit (x : PrimeSpectrum PUnit) : False :=
   x.1.ne_top_iff_one.1 x.2.1 <| Subsingleton.elim (0 : PUnit) 1 ▸ x.1.zero_mem
 #align prime_spectrum.punit PrimeSpectrum.pUnit
+-/
 
 variable (R S)
 
+/- warning: prime_spectrum.prime_spectrum_prod_of_sum -> PrimeSpectrum.primeSpectrumProdOfSum is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) (S : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S], (Sum.{u1, u2} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.{u2} S _inst_2)) -> (PrimeSpectrum.{max u1 u2} (Prod.{u1, u2} R S) (Prod.commRing.{u1, u2} R S _inst_1 _inst_2))
+but is expected to have type
+  forall (R : Type.{u1}) (S : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S], (Sum.{u1, u2} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.{u2} S _inst_2)) -> (PrimeSpectrum.{max u2 u1} (Prod.{u1, u2} R S) (Prod.instCommRingProd.{u1, u2} R S _inst_1 _inst_2))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.prime_spectrum_prod_of_sum PrimeSpectrum.primeSpectrumProdOfSumₓ'. -/
 /-- The map from the direct sum of prime spectra to the prime spectrum of a direct product. -/
 @[simp]
 def primeSpectrumProdOfSum : Sum (PrimeSpectrum R) (PrimeSpectrum S) → PrimeSpectrum (R × S)
@@ -91,6 +101,12 @@ def primeSpectrumProdOfSum : Sum (PrimeSpectrum R) (PrimeSpectrum S) → PrimeSp
   | Sum.inr ⟨J, hJ⟩ => ⟨Ideal.prod ⊤ J, Ideal.isPrime_ideal_prod_top'⟩
 #align prime_spectrum.prime_spectrum_prod_of_sum PrimeSpectrum.primeSpectrumProdOfSum
 
+/- warning: prime_spectrum.prime_spectrum_prod -> PrimeSpectrum.primeSpectrumProd is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) (S : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S], Equiv.{succ (max u1 u2), max (succ u1) (succ u2)} (PrimeSpectrum.{max u1 u2} (Prod.{u1, u2} R S) (Prod.commRing.{u1, u2} R S _inst_1 _inst_2)) (Sum.{u1, u2} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.{u2} S _inst_2))
+but is expected to have type
+  forall (R : Type.{u1}) (S : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S], Equiv.{succ (max u2 u1), max (succ u2) (succ u1)} (PrimeSpectrum.{max u2 u1} (Prod.{u1, u2} R S) (Prod.instCommRingProd.{u1, u2} R S _inst_1 _inst_2)) (Sum.{u1, u2} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.{u2} S _inst_2))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.prime_spectrum_prod PrimeSpectrum.primeSpectrumProdₓ'. -/
 /-- The prime spectrum of `R × S` is in bijection with the disjoint unions of the prime spectrum of
 `R` and the prime spectrum of `S`. -/
 noncomputable def primeSpectrumProd :
@@ -113,6 +129,12 @@ noncomputable def primeSpectrumProd :
 
 variable {R S}
 
+/- warning: prime_spectrum.prime_spectrum_prod_symm_inl_as_ideal -> PrimeSpectrum.primeSpectrumProd_symm_inl_asIdeal is a dubious translation:
+lean 3 declaration is
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 @[simp]
 theorem primeSpectrumProd_symm_inl_asIdeal (x : PrimeSpectrum R) :
     ((primeSpectrumProd R S).symm <| Sum.inl x).asIdeal = Ideal.prod x.asIdeal ⊤ :=
@@ -121,6 +143,12 @@ theorem primeSpectrumProd_symm_inl_asIdeal (x : PrimeSpectrum R) :
   rfl
 #align prime_spectrum.prime_spectrum_prod_symm_inl_as_ideal PrimeSpectrum.primeSpectrumProd_symm_inl_asIdeal
 
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 @[simp]
 theorem primeSpectrumProd_symm_inr_asIdeal (x : PrimeSpectrum S) :
     ((primeSpectrumProd R S).symm <| Sum.inr x).asIdeal = Ideal.prod ⊤ x.asIdeal :=
@@ -129,6 +157,7 @@ theorem primeSpectrumProd_symm_inr_asIdeal (x : PrimeSpectrum S) :
   rfl
 #align prime_spectrum.prime_spectrum_prod_symm_inr_as_ideal PrimeSpectrum.primeSpectrumProd_symm_inr_asIdeal
 
+#print PrimeSpectrum.zeroLocus /-
 /-- The zero locus of a set `s` of elements of a commutative ring `R` is the set of all prime ideals
 of the ring that contain the set `s`.
 
@@ -140,19 +169,25 @@ At a point `x` (a prime ideal) the function (i.e., element) `f` takes values in
 def zeroLocus (s : Set R) : Set (PrimeSpectrum R) :=
   { x | s ⊆ x.asIdeal }
 #align prime_spectrum.zero_locus PrimeSpectrum.zeroLocus
+-/
 
+#print PrimeSpectrum.mem_zeroLocus /-
 @[simp]
 theorem mem_zeroLocus (x : PrimeSpectrum R) (s : Set R) : x ∈ zeroLocus s ↔ s ⊆ x.asIdeal :=
   Iff.rfl
 #align prime_spectrum.mem_zero_locus PrimeSpectrum.mem_zeroLocus
+-/
 
+#print PrimeSpectrum.zeroLocus_span /-
 @[simp]
 theorem zeroLocus_span (s : Set R) : zeroLocus (Ideal.span s : Set R) = zeroLocus s :=
   by
   ext x
   exact (Submodule.gi R R).gc s x.as_ideal
 #align prime_spectrum.zero_locus_span PrimeSpectrum.zeroLocus_span
+-/
 
+#print PrimeSpectrum.vanishingIdeal /-
 /-- The vanishing ideal of a set `t` of points of the prime spectrum of a commutative ring `R` is
 the intersection of all the prime ideals in the set `t`.
 
@@ -164,7 +199,9 @@ consisting of all "functions" that vanish on all of `t`.
 def vanishingIdeal (t : Set (PrimeSpectrum R)) : Ideal R :=
   ⨅ (x : PrimeSpectrum R) (h : x ∈ t), x.asIdeal
 #align prime_spectrum.vanishing_ideal PrimeSpectrum.vanishingIdeal
+-/
 
+#print PrimeSpectrum.coe_vanishingIdeal /-
 theorem coe_vanishingIdeal (t : Set (PrimeSpectrum R)) :
     (vanishingIdeal t : Set R) = { f : R | ∀ x : PrimeSpectrum R, x ∈ t → f ∈ x.asIdeal } :=
   by
@@ -173,17 +210,28 @@ theorem coe_vanishingIdeal (t : Set (PrimeSpectrum R)) :
   apply forall_congr'; intro x
   rw [Submodule.mem_iInf]
 #align prime_spectrum.coe_vanishing_ideal PrimeSpectrum.coe_vanishingIdeal
+-/
 
+#print PrimeSpectrum.mem_vanishingIdeal /-
 theorem mem_vanishingIdeal (t : Set (PrimeSpectrum R)) (f : R) :
     f ∈ vanishingIdeal t ↔ ∀ x : PrimeSpectrum R, x ∈ t → f ∈ x.asIdeal := by
   rw [← SetLike.mem_coe, coe_vanishing_ideal, Set.mem_setOf_eq]
 #align prime_spectrum.mem_vanishing_ideal PrimeSpectrum.mem_vanishingIdeal
+-/
 
+#print PrimeSpectrum.vanishingIdeal_singleton /-
 @[simp]
 theorem vanishingIdeal_singleton (x : PrimeSpectrum R) :
     vanishingIdeal ({x} : Set (PrimeSpectrum R)) = x.asIdeal := by simp [vanishing_ideal]
 #align prime_spectrum.vanishing_ideal_singleton PrimeSpectrum.vanishingIdeal_singleton
+-/
 
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+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), Iff (HasSubset.Subset.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.hasSubset.{u1} (PrimeSpectrum.{u1} R _inst_1)) t (PrimeSpectrum.zeroLocus.{u1} R _inst_1 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) I))) (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toHasLe.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) I (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 t))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), Iff (HasSubset.Subset.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.instHasSubsetSet.{u1} (PrimeSpectrum.{u1} R _inst_1)) t (PrimeSpectrum.zeroLocus.{u1} R _inst_1 (SetLike.coe.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) I))) (LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) I (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 t))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.subset_zero_locus_iff_le_vanishing_ideal PrimeSpectrum.subset_zeroLocus_iff_le_vanishingIdealₓ'. -/
 theorem subset_zeroLocus_iff_le_vanishingIdeal (t : Set (PrimeSpectrum R)) (I : Ideal R) :
     t ⊆ zeroLocus I ↔ I ≤ vanishingIdeal t :=
   ⟨fun h f k => (mem_vanishingIdeal _ _).mpr fun x j => (mem_zeroLocus _ _).mpr (h j) k, fun h =>
@@ -194,6 +242,12 @@ section Gc
 
 variable (R)
 
+/- warning: prime_spectrum.gc -> PrimeSpectrum.gc is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) [_inst_1 : CommRing.{u1} R], GaloisConnection.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (OrderDual.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (OrderDual.preorder.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.completeBooleanAlgebra.{u1} (PrimeSpectrum.{u1} R _inst_1))))))))) (fun (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => PrimeSpectrum.zeroLocus.{u1} R _inst_1 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) I)) (fun (t : OrderDual.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1))) => PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 t)
+but is expected to have type
+  forall (R : Type.{u1}) [_inst_1 : CommRing.{u1} R], GaloisConnection.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OrderDual.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))))) (OrderDual.preorder.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.instCompleteBooleanAlgebraSet.{u1} (PrimeSpectrum.{u1} R _inst_1))))))))) (fun (I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => PrimeSpectrum.zeroLocus.{u1} R _inst_1 (SetLike.coe.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) I)) (fun (t : OrderDual.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1))) => PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 t)
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.gc PrimeSpectrum.gcₓ'. -/
 /-- `zero_locus` and `vanishing_ideal` form a galois connection. -/
 theorem gc :
     @GaloisConnection (Ideal R) (Set (PrimeSpectrum R))ᵒᵈ _ _ (fun I => zeroLocus I) fun t =>
@@ -201,6 +255,12 @@ theorem gc :
   fun I t => subset_zeroLocus_iff_le_vanishingIdeal t I
 #align prime_spectrum.gc PrimeSpectrum.gc
 
+/- warning: prime_spectrum.gc_set -> PrimeSpectrum.gc_set is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) [_inst_1 : CommRing.{u1} R], GaloisConnection.{u1, u1} (Set.{u1} R) (OrderDual.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Set.{u1} R) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} R) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} R) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} R) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} R) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} R) (Set.completeBooleanAlgebra.{u1} R))))))) (OrderDual.preorder.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.completeBooleanAlgebra.{u1} (PrimeSpectrum.{u1} R _inst_1))))))))) (fun (s : Set.{u1} R) => PrimeSpectrum.zeroLocus.{u1} R _inst_1 s) (fun (t : OrderDual.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1))) => (fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 t))
+but is expected to have type
+  forall (R : Type.{u1}) [_inst_1 : CommRing.{u1} R], GaloisConnection.{u1, u1} (Set.{u1} R) (OrderDual.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Set.{u1} R) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} R) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} R) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} R) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} R) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} R) (Set.instCompleteBooleanAlgebraSet.{u1} R))))))) (OrderDual.preorder.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.instCompleteBooleanAlgebraSet.{u1} (PrimeSpectrum.{u1} R _inst_1))))))))) (fun (s : Set.{u1} R) => PrimeSpectrum.zeroLocus.{u1} R _inst_1 s) (fun (t : OrderDual.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1))) => SetLike.coe.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 t))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.gc_set PrimeSpectrum.gc_setₓ'. -/
 /-- `zero_locus` and `vanishing_ideal` form a galois connection. -/
 theorem gc_set :
     @GaloisConnection (Set R) (Set (PrimeSpectrum R))ᵒᵈ _ _ (fun s => zeroLocus s) fun t =>
@@ -210,21 +270,32 @@ theorem gc_set :
   simpa [zero_locus_span, Function.comp] using ideal_gc.compose (gc R)
 #align prime_spectrum.gc_set PrimeSpectrum.gc_set
 
+#print PrimeSpectrum.subset_zeroLocus_iff_subset_vanishingIdeal /-
 theorem subset_zeroLocus_iff_subset_vanishingIdeal (t : Set (PrimeSpectrum R)) (s : Set R) :
     t ⊆ zeroLocus s ↔ s ⊆ vanishingIdeal t :=
   (gc_set R) s t
 #align prime_spectrum.subset_zero_locus_iff_subset_vanishing_ideal PrimeSpectrum.subset_zeroLocus_iff_subset_vanishingIdeal
+-/
 
 end Gc
 
+#print PrimeSpectrum.subset_vanishingIdeal_zeroLocus /-
 theorem subset_vanishingIdeal_zeroLocus (s : Set R) : s ⊆ vanishingIdeal (zeroLocus s) :=
   (gc_set R).le_u_l s
 #align prime_spectrum.subset_vanishing_ideal_zero_locus PrimeSpectrum.subset_vanishingIdeal_zeroLocus
+-/
 
+/- warning: prime_spectrum.le_vanishing_ideal_zero_locus -> PrimeSpectrum.le_vanishingIdeal_zeroLocus is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toHasLe.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) I (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 (PrimeSpectrum.zeroLocus.{u1} R _inst_1 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) I)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) I (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 (PrimeSpectrum.zeroLocus.{u1} R _inst_1 (SetLike.coe.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) I)))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.le_vanishing_ideal_zero_locus PrimeSpectrum.le_vanishingIdeal_zeroLocusₓ'. -/
 theorem le_vanishingIdeal_zeroLocus (I : Ideal R) : I ≤ vanishingIdeal (zeroLocus I) :=
   (gc R).le_u_l I
 #align prime_spectrum.le_vanishing_ideal_zero_locus PrimeSpectrum.le_vanishingIdeal_zeroLocus
 
+#print PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radical /-
 @[simp]
 theorem vanishingIdeal_zeroLocus_eq_radical (I : Ideal R) :
     vanishingIdeal (zeroLocus (I : Set R)) = I.radical :=
@@ -233,31 +304,56 @@ theorem vanishingIdeal_zeroLocus_eq_radical (I : Ideal R) :
     rw [mem_vanishing_ideal, Ideal.radical_eq_sInf, Submodule.mem_sInf]
     exact ⟨fun h x hx => h ⟨x, hx.2⟩ hx.1, fun h x hx => h x.1 ⟨hx, x.2⟩⟩
 #align prime_spectrum.vanishing_ideal_zero_locus_eq_radical PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radical
+-/
 
+#print PrimeSpectrum.zeroLocus_radical /-
 @[simp]
 theorem zeroLocus_radical (I : Ideal R) : zeroLocus (I.radical : Set R) = zeroLocus I :=
   vanishingIdeal_zeroLocus_eq_radical I ▸ (gc R).l_u_l_eq_l I
 #align prime_spectrum.zero_locus_radical PrimeSpectrum.zeroLocus_radical
+-/
 
+#print PrimeSpectrum.subset_zeroLocus_vanishingIdeal /-
 theorem subset_zeroLocus_vanishingIdeal (t : Set (PrimeSpectrum R)) :
     t ⊆ zeroLocus (vanishingIdeal t) :=
   (gc R).l_u_le t
 #align prime_spectrum.subset_zero_locus_vanishing_ideal PrimeSpectrum.subset_zeroLocus_vanishingIdeal
+-/
 
+#print PrimeSpectrum.zeroLocus_anti_mono /-
 theorem zeroLocus_anti_mono {s t : Set R} (h : s ⊆ t) : zeroLocus t ⊆ zeroLocus s :=
   (gc_set R).monotone_l h
 #align prime_spectrum.zero_locus_anti_mono PrimeSpectrum.zeroLocus_anti_mono
+-/
 
+/- warning: prime_spectrum.zero_locus_anti_mono_ideal -> PrimeSpectrum.zeroLocus_anti_mono_ideal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {s : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {t : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toHasLe.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) s t) -> (HasSubset.Subset.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.hasSubset.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) t)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) s)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {s : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))} {t : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))}, (LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) s t) -> (HasSubset.Subset.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.instHasSubsetSet.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 (SetLike.coe.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) t)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 (SetLike.coe.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) s)))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.zero_locus_anti_mono_ideal PrimeSpectrum.zeroLocus_anti_mono_idealₓ'. -/
 theorem zeroLocus_anti_mono_ideal {s t : Ideal R} (h : s ≤ t) :
     zeroLocus (t : Set R) ⊆ zeroLocus (s : Set R) :=
   (gc R).monotone_l h
 #align prime_spectrum.zero_locus_anti_mono_ideal PrimeSpectrum.zeroLocus_anti_mono_ideal
 
+/- warning: prime_spectrum.vanishing_ideal_anti_mono -> PrimeSpectrum.vanishingIdeal_anti_mono is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {s : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)} {t : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)}, (HasSubset.Subset.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.hasSubset.{u1} (PrimeSpectrum.{u1} R _inst_1)) s t) -> (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toHasLe.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 t) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 s))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {s : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)} {t : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)}, (HasSubset.Subset.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.instHasSubsetSet.{u1} (PrimeSpectrum.{u1} R _inst_1)) s t) -> (LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 t) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 s))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.vanishing_ideal_anti_mono PrimeSpectrum.vanishingIdeal_anti_monoₓ'. -/
 theorem vanishingIdeal_anti_mono {s t : Set (PrimeSpectrum R)} (h : s ⊆ t) :
     vanishingIdeal t ≤ vanishingIdeal s :=
   (gc R).monotone_u h
 #align prime_spectrum.vanishing_ideal_anti_mono PrimeSpectrum.vanishingIdeal_anti_mono
 
+/- warning: prime_spectrum.zero_locus_subset_zero_locus_iff -> PrimeSpectrum.zeroLocus_subset_zeroLocus_iff is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), Iff (HasSubset.Subset.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.instHasSubsetSet.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 (SetLike.coe.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) I)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 (SetLike.coe.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) J))) (LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) J (Ideal.radical.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) I))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.zero_locus_subset_zero_locus_iff PrimeSpectrum.zeroLocus_subset_zeroLocus_iffₓ'. -/
 theorem zeroLocus_subset_zeroLocus_iff (I J : Ideal R) :
     zeroLocus (I : Set R) ⊆ zeroLocus (J : Set R) ↔ J ≤ I.radical :=
   ⟨fun h =>
@@ -267,31 +363,59 @@ theorem zeroLocus_subset_zeroLocus_iff (I J : Ideal R) :
     fun h => zeroLocus_radical I ▸ zeroLocus_anti_mono_ideal h⟩
 #align prime_spectrum.zero_locus_subset_zero_locus_iff PrimeSpectrum.zeroLocus_subset_zeroLocus_iff
 
+#print PrimeSpectrum.zeroLocus_subset_zeroLocus_singleton_iff /-
 theorem zeroLocus_subset_zeroLocus_singleton_iff (f g : R) :
     zeroLocus ({f} : Set R) ⊆ zeroLocus {g} ↔ g ∈ (Ideal.span ({f} : Set R)).radical := by
   rw [← zero_locus_span {f}, ← zero_locus_span {g}, zero_locus_subset_zero_locus_iff, Ideal.span_le,
     Set.singleton_subset_iff, SetLike.mem_coe]
 #align prime_spectrum.zero_locus_subset_zero_locus_singleton_iff PrimeSpectrum.zeroLocus_subset_zeroLocus_singleton_iff
+-/
 
+/- warning: prime_spectrum.zero_locus_bot -> PrimeSpectrum.zeroLocus_bot is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R], Eq.{succ u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (Bot.bot.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.hasBot.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (Set.univ.{u1} (PrimeSpectrum.{u1} R _inst_1))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.zero_locus_bot PrimeSpectrum.zeroLocus_botₓ'. -/
 theorem zeroLocus_bot : zeroLocus ((⊥ : Ideal R) : Set R) = Set.univ :=
   (gc R).l_bot
 #align prime_spectrum.zero_locus_bot PrimeSpectrum.zeroLocus_bot
 
+/- warning: prime_spectrum.zero_locus_singleton_zero -> PrimeSpectrum.zeroLocus_singleton_zero is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R], Eq.{succ u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))))))))) (Set.univ.{u1} (PrimeSpectrum.{u1} R _inst_1))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R], Eq.{succ u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Set.univ.{u1} (PrimeSpectrum.{u1} R _inst_1))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.zero_locus_singleton_zero PrimeSpectrum.zeroLocus_singleton_zeroₓ'. -/
 @[simp]
 theorem zeroLocus_singleton_zero : zeroLocus ({0} : Set R) = Set.univ :=
   zeroLocus_bot
 #align prime_spectrum.zero_locus_singleton_zero PrimeSpectrum.zeroLocus_singleton_zero
 
+#print PrimeSpectrum.zeroLocus_empty /-
 @[simp]
 theorem zeroLocus_empty : zeroLocus (∅ : Set R) = Set.univ :=
   (gc_set R).l_bot
 #align prime_spectrum.zero_locus_empty PrimeSpectrum.zeroLocus_empty
+-/
 
+/- warning: prime_spectrum.vanishing_ideal_univ -> PrimeSpectrum.vanishingIdeal_univ is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R], Eq.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 (EmptyCollection.emptyCollection.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.hasEmptyc.{u1} (PrimeSpectrum.{u1} R _inst_1)))) (Top.top.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.hasTop.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R], Eq.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 (EmptyCollection.emptyCollection.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.instEmptyCollectionSet.{u1} (PrimeSpectrum.{u1} R _inst_1)))) (Top.top.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.instTopSubmodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.vanishing_ideal_univ PrimeSpectrum.vanishingIdeal_univₓ'. -/
 @[simp]
 theorem vanishingIdeal_univ : vanishingIdeal (∅ : Set (PrimeSpectrum R)) = ⊤ := by
   simpa using (gc R).u_top
 #align prime_spectrum.vanishing_ideal_univ PrimeSpectrum.vanishingIdeal_univ
 
+/- warning: prime_spectrum.zero_locus_empty_of_one_mem -> PrimeSpectrum.zeroLocus_empty_of_one_mem is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {s : Set.{u1} R}, (Membership.Mem.{u1, u1} R (Set.{u1} R) (Set.hasMem.{u1} R) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) s) -> (Eq.{succ u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 s) (EmptyCollection.emptyCollection.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.hasEmptyc.{u1} (PrimeSpectrum.{u1} R _inst_1))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {s : Set.{u1} R}, (Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) s) -> (Eq.{succ u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 s) (EmptyCollection.emptyCollection.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.instEmptyCollectionSet.{u1} (PrimeSpectrum.{u1} R _inst_1))))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.zero_locus_empty_of_one_mem PrimeSpectrum.zeroLocus_empty_of_one_memₓ'. -/
 theorem zeroLocus_empty_of_one_mem {s : Set R} (h : (1 : R) ∈ s) : zeroLocus s = ∅ :=
   by
   rw [Set.eq_empty_iff_forall_not_mem]
@@ -304,11 +428,23 @@ theorem zeroLocus_empty_of_one_mem {s : Set R} (h : (1 : R) ∈ s) : zeroLocus s
   apply x_prime.ne_top eq_top
 #align prime_spectrum.zero_locus_empty_of_one_mem PrimeSpectrum.zeroLocus_empty_of_one_mem
 
+/- warning: prime_spectrum.zero_locus_singleton_one -> PrimeSpectrum.zeroLocus_singleton_one is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R], Eq.{succ u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) (EmptyCollection.emptyCollection.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.hasEmptyc.{u1} (PrimeSpectrum.{u1} R _inst_1)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R], Eq.{succ u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))))) (EmptyCollection.emptyCollection.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.instEmptyCollectionSet.{u1} (PrimeSpectrum.{u1} R _inst_1)))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.zero_locus_singleton_one PrimeSpectrum.zeroLocus_singleton_oneₓ'. -/
 @[simp]
 theorem zeroLocus_singleton_one : zeroLocus ({1} : Set R) = ∅ :=
   zeroLocus_empty_of_one_mem (Set.mem_singleton (1 : R))
 #align prime_spectrum.zero_locus_singleton_one PrimeSpectrum.zeroLocus_singleton_one
 
+/- warning: prime_spectrum.zero_locus_empty_iff_eq_top -> PrimeSpectrum.zeroLocus_empty_iff_eq_top is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.zero_locus_empty_iff_eq_top PrimeSpectrum.zeroLocus_empty_iff_eq_topₓ'. -/
 theorem zeroLocus_empty_iff_eq_top {I : Ideal R} : zeroLocus (I : Set R) = ∅ ↔ I = ⊤ :=
   by
   constructor
@@ -321,54 +457,108 @@ theorem zeroLocus_empty_iff_eq_top {I : Ideal R} : zeroLocus (I : Set R) = ∅ 
     trivial
 #align prime_spectrum.zero_locus_empty_iff_eq_top PrimeSpectrum.zeroLocus_empty_iff_eq_top
 
+#print PrimeSpectrum.zeroLocus_univ /-
 @[simp]
 theorem zeroLocus_univ : zeroLocus (Set.univ : Set R) = ∅ :=
   zeroLocus_empty_of_one_mem (Set.mem_univ 1)
 #align prime_spectrum.zero_locus_univ PrimeSpectrum.zeroLocus_univ
+-/
 
+/- warning: prime_spectrum.vanishing_ideal_eq_top_iff -> PrimeSpectrum.vanishingIdeal_eq_top_iff is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.vanishing_ideal_eq_top_iff PrimeSpectrum.vanishingIdeal_eq_top_iffₓ'. -/
 theorem vanishingIdeal_eq_top_iff {s : Set (PrimeSpectrum R)} : vanishingIdeal s = ⊤ ↔ s = ∅ := by
   rw [← top_le_iff, ← subset_zero_locus_iff_le_vanishing_ideal, Submodule.top_coe, zero_locus_univ,
     Set.subset_empty_iff]
 #align prime_spectrum.vanishing_ideal_eq_top_iff PrimeSpectrum.vanishingIdeal_eq_top_iff
 
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.zero_locus_sup PrimeSpectrum.zeroLocus_supₓ'. -/
 theorem zeroLocus_sup (I J : Ideal R) :
     zeroLocus ((I ⊔ J : Ideal R) : Set R) = zeroLocus I ∩ zeroLocus J :=
   (gc R).l_sup
 #align prime_spectrum.zero_locus_sup PrimeSpectrum.zeroLocus_sup
 
+/- warning: prime_spectrum.zero_locus_union -> PrimeSpectrum.zeroLocus_union 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 prime_spectrum.zero_locus_union PrimeSpectrum.zeroLocus_unionₓ'. -/
 theorem zeroLocus_union (s s' : Set R) : zeroLocus (s ∪ s') = zeroLocus s ∩ zeroLocus s' :=
   (gc_set R).l_sup
 #align prime_spectrum.zero_locus_union PrimeSpectrum.zeroLocus_union
 
+#print PrimeSpectrum.vanishingIdeal_union /-
 theorem vanishingIdeal_union (t t' : Set (PrimeSpectrum R)) :
     vanishingIdeal (t ∪ t') = vanishingIdeal t ⊓ vanishingIdeal t' :=
   (gc R).u_inf
 #align prime_spectrum.vanishing_ideal_union PrimeSpectrum.vanishingIdeal_union
+-/
 
+/- warning: prime_spectrum.zero_locus_supr -> PrimeSpectrum.zeroLocus_iSup is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.zero_locus_supr PrimeSpectrum.zeroLocus_iSupₓ'. -/
 theorem zeroLocus_iSup {ι : Sort _} (I : ι → Ideal R) :
     zeroLocus ((⨆ i, I i : Ideal R) : Set R) = ⋂ i, zeroLocus (I i) :=
   (gc R).l_iSup
 #align prime_spectrum.zero_locus_supr PrimeSpectrum.zeroLocus_iSup
 
+/- warning: prime_spectrum.zero_locus_Union -> PrimeSpectrum.zeroLocus_iUnion is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {ι : Sort.{u2}} (s : ι -> (Set.{u1} R)), Eq.{succ u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 (Set.iUnion.{u1, u2} R ι (fun (i : ι) => s i))) (Set.iInter.{u1, u2} (PrimeSpectrum.{u1} R _inst_1) ι (fun (i : ι) => PrimeSpectrum.zeroLocus.{u1} R _inst_1 (s i)))
+but is expected to have type
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {ι : Sort.{u1}} (s : ι -> (Set.{u2} R)), Eq.{succ u2} (Set.{u2} (PrimeSpectrum.{u2} R _inst_1)) (PrimeSpectrum.zeroLocus.{u2} R _inst_1 (Set.iUnion.{u2, u1} R ι (fun (i : ι) => s i))) (Set.iInter.{u2, u1} (PrimeSpectrum.{u2} R _inst_1) ι (fun (i : ι) => PrimeSpectrum.zeroLocus.{u2} R _inst_1 (s i)))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.zero_locus_Union PrimeSpectrum.zeroLocus_iUnionₓ'. -/
 theorem zeroLocus_iUnion {ι : Sort _} (s : ι → Set R) :
     zeroLocus (⋃ i, s i) = ⋂ i, zeroLocus (s i) :=
   (gc_set R).l_iSup
 #align prime_spectrum.zero_locus_Union PrimeSpectrum.zeroLocus_iUnion
 
+#print PrimeSpectrum.zeroLocus_bUnion /-
 theorem zeroLocus_bUnion (s : Set (Set R)) :
     zeroLocus (⋃ s' ∈ s, s' : Set R) = ⋂ s' ∈ s, zeroLocus s' := by simp only [zero_locus_Union]
 #align prime_spectrum.zero_locus_bUnion PrimeSpectrum.zeroLocus_bUnion
+-/
 
+/- warning: prime_spectrum.vanishing_ideal_Union -> PrimeSpectrum.vanishingIdeal_iUnion 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 prime_spectrum.vanishing_ideal_Union PrimeSpectrum.vanishingIdeal_iUnionₓ'. -/
 theorem vanishingIdeal_iUnion {ι : Sort _} (t : ι → Set (PrimeSpectrum R)) :
     vanishingIdeal (⋃ i, t i) = ⨅ i, vanishingIdeal (t i) :=
   (gc R).u_iInf
 #align prime_spectrum.vanishing_ideal_Union PrimeSpectrum.vanishingIdeal_iUnion
 
+/- warning: prime_spectrum.zero_locus_inf -> PrimeSpectrum.zeroLocus_inf is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.zero_locus_inf PrimeSpectrum.zeroLocus_infₓ'. -/
 theorem zeroLocus_inf (I J : Ideal R) :
     zeroLocus ((I ⊓ J : Ideal R) : Set R) = zeroLocus I ∪ zeroLocus J :=
   Set.ext fun x => x.2.inf_le
 #align prime_spectrum.zero_locus_inf PrimeSpectrum.zeroLocus_inf
 
+/- warning: prime_spectrum.union_zero_locus -> PrimeSpectrum.union_zeroLocus is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (s : Set.{u1} R) (s' : Set.{u1} R), Eq.{succ u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Union.union.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.hasUnion.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 s) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 s')) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (Inf.inf.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.hasInf.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) s) (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) s'))))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.union_zero_locus PrimeSpectrum.union_zeroLocusₓ'. -/
 theorem union_zeroLocus (s s' : Set R) :
     zeroLocus s ∪ zeroLocus s' = zeroLocus (Ideal.span s ⊓ Ideal.span s' : Ideal R) :=
   by
@@ -376,28 +566,54 @@ theorem union_zeroLocus (s s' : Set R) :
   simp
 #align prime_spectrum.union_zero_locus PrimeSpectrum.union_zeroLocus
 
+/- warning: prime_spectrum.zero_locus_mul -> PrimeSpectrum.zeroLocus_mul is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.zero_locus_mul PrimeSpectrum.zeroLocus_mulₓ'. -/
 theorem zeroLocus_mul (I J : Ideal R) :
     zeroLocus ((I * J : Ideal R) : Set R) = zeroLocus I ∪ zeroLocus J :=
   Set.ext fun x => x.2.mul_le
 #align prime_spectrum.zero_locus_mul PrimeSpectrum.zeroLocus_mul
 
+/- warning: prime_spectrum.zero_locus_singleton_mul -> PrimeSpectrum.zeroLocus_singleton_mul is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (f : R) (g : R), Eq.{succ u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) f g))) (Union.union.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.hasUnion.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) f)) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) g)))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.zero_locus_singleton_mul PrimeSpectrum.zeroLocus_singleton_mulₓ'. -/
 theorem zeroLocus_singleton_mul (f g : R) :
     zeroLocus ({f * g} : Set R) = zeroLocus {f} ∪ zeroLocus {g} :=
   Set.ext fun x => by simpa using x.2.mul_mem_iff_mem_or_mem
 #align prime_spectrum.zero_locus_singleton_mul PrimeSpectrum.zeroLocus_singleton_mul
 
+#print PrimeSpectrum.zeroLocus_pow /-
 @[simp]
 theorem zeroLocus_pow (I : Ideal R) {n : ℕ} (hn : 0 < n) :
     zeroLocus ((I ^ n : Ideal R) : Set R) = zeroLocus I :=
   zeroLocus_radical (I ^ n) ▸ (I.radical_pow n hn).symm ▸ zeroLocus_radical I
 #align prime_spectrum.zero_locus_pow PrimeSpectrum.zeroLocus_pow
+-/
 
+/- warning: prime_spectrum.zero_locus_singleton_pow -> PrimeSpectrum.zeroLocus_singleton_pow 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 prime_spectrum.zero_locus_singleton_pow PrimeSpectrum.zeroLocus_singleton_powₓ'. -/
 @[simp]
 theorem zeroLocus_singleton_pow (f : R) (n : ℕ) (hn : 0 < n) :
     zeroLocus ({f ^ n} : Set R) = zeroLocus {f} :=
   Set.ext fun x => by simpa using x.2.pow_mem_iff_mem n hn
 #align prime_spectrum.zero_locus_singleton_pow PrimeSpectrum.zeroLocus_singleton_pow
 
+/- warning: prime_spectrum.sup_vanishing_ideal_le -> PrimeSpectrum.sup_vanishingIdeal_le is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (t' : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)), LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toHasLe.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (Sup.sup.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SemilatticeSup.toHasSup.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (IdemSemiring.toSemilatticeSup.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.idemSemiring.{u1, u1} R (CommRing.toCommSemiring.{u1} R _inst_1) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 t) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 t')) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 (Inter.inter.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.hasInter.{u1} (PrimeSpectrum.{u1} R _inst_1)) t t'))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (t' : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)), LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Sup.sup.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (SemilatticeSup.toSup.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (IdemCommSemiring.toSemilatticeSup.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instIdemCommSemiringIdealToSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 t) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 t')) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 (Inter.inter.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.instInterSet.{u1} (PrimeSpectrum.{u1} R _inst_1)) t t'))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.sup_vanishing_ideal_le PrimeSpectrum.sup_vanishingIdeal_leₓ'. -/
 theorem sup_vanishingIdeal_le (t t' : Set (PrimeSpectrum R)) :
     vanishingIdeal t ⊔ vanishingIdeal t' ≤ vanishingIdeal (t ∩ t') :=
   by
@@ -408,11 +624,18 @@ theorem sup_vanishingIdeal_le (t t' : Set (PrimeSpectrum R)) :
   apply Submodule.add_mem <;> solve_by_elim
 #align prime_spectrum.sup_vanishing_ideal_le PrimeSpectrum.sup_vanishingIdeal_le
 
+/- warning: prime_spectrum.mem_compl_zero_locus_iff_not_mem -> PrimeSpectrum.mem_compl_zeroLocus_iff_not_mem is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {f : R} {I : PrimeSpectrum.{u1} R _inst_1}, Iff (Membership.Mem.{u1, u1} (PrimeSpectrum.{u1} R _inst_1) (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.hasMem.{u1} (PrimeSpectrum.{u1} R _inst_1)) I (HasCompl.compl.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.booleanAlgebra.{u1} (PrimeSpectrum.{u1} R _inst_1))) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) f)))) (Not (Membership.Mem.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.hasMem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) f (PrimeSpectrum.asIdeal.{u1} R _inst_1 I)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {f : R} {I : PrimeSpectrum.{u1} R _inst_1}, Iff (Membership.mem.{u1, u1} (PrimeSpectrum.{u1} R _inst_1) (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.instMembershipSet.{u1} (PrimeSpectrum.{u1} R _inst_1)) I (HasCompl.compl.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.instBooleanAlgebraSet.{u1} (PrimeSpectrum.{u1} R _inst_1))) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) f)))) (Not (Membership.mem.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) f (PrimeSpectrum.asIdeal.{u1} R _inst_1 I)))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.mem_compl_zero_locus_iff_not_mem PrimeSpectrum.mem_compl_zeroLocus_iff_not_memₓ'. -/
 theorem mem_compl_zeroLocus_iff_not_mem {f : R} {I : PrimeSpectrum R} :
     I ∈ (zeroLocus {f} : Set (PrimeSpectrum R))ᶜ ↔ f ∉ I.asIdeal := by
   rw [Set.mem_compl_iff, mem_zero_locus, Set.singleton_subset_iff] <;> rfl
 #align prime_spectrum.mem_compl_zero_locus_iff_not_mem PrimeSpectrum.mem_compl_zeroLocus_iff_not_mem
 
+#print PrimeSpectrum.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 (PrimeSpectrum R) :=
@@ -427,34 +650,50 @@ instance zariskiTopology : TopologicalSpace (PrimeSpectrum R) :=
       rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩
       exact ⟨_, (union_zero_locus s t).symm⟩)
 #align prime_spectrum.zariski_topology PrimeSpectrum.zariskiTopology
+-/
 
+/- warning: prime_spectrum.is_open_iff -> PrimeSpectrum.isOpen_iff is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (U : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)), Iff (IsOpen.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) U) (Exists.{succ u1} (Set.{u1} R) (fun (s : Set.{u1} R) => Eq.{succ u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (HasCompl.compl.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.booleanAlgebra.{u1} (PrimeSpectrum.{u1} R _inst_1))) U) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 s)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (U : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)), Iff (IsOpen.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) U) (Exists.{succ u1} (Set.{u1} R) (fun (s : Set.{u1} R) => Eq.{succ u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (HasCompl.compl.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.instBooleanAlgebraSet.{u1} (PrimeSpectrum.{u1} R _inst_1))) U) (PrimeSpectrum.zeroLocus.{u1} R _inst_1 s)))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.is_open_iff PrimeSpectrum.isOpen_iffₓ'. -/
 theorem isOpen_iff (U : Set (PrimeSpectrum R)) : IsOpen U ↔ ∃ s, Uᶜ = zeroLocus s := by
   simp only [@eq_comm _ (Uᶜ)] <;> rfl
 #align prime_spectrum.is_open_iff PrimeSpectrum.isOpen_iff
 
+#print PrimeSpectrum.isClosed_iff_zeroLocus /-
 theorem isClosed_iff_zeroLocus (Z : Set (PrimeSpectrum R)) : IsClosed Z ↔ ∃ s, Z = zeroLocus s := by
   rw [← isOpen_compl_iff, is_open_iff, compl_compl]
 #align prime_spectrum.is_closed_iff_zero_locus PrimeSpectrum.isClosed_iff_zeroLocus
+-/
 
+#print PrimeSpectrum.isClosed_iff_zeroLocus_ideal /-
 theorem isClosed_iff_zeroLocus_ideal (Z : Set (PrimeSpectrum R)) :
     IsClosed Z ↔ ∃ I : Ideal R, Z = zeroLocus I :=
   (isClosed_iff_zeroLocus _).trans
     ⟨fun ⟨s, hs⟩ => ⟨_, (zeroLocus_span s).substr hs⟩, fun ⟨I, hI⟩ => ⟨I, hI⟩⟩
 #align prime_spectrum.is_closed_iff_zero_locus_ideal PrimeSpectrum.isClosed_iff_zeroLocus_ideal
+-/
 
+#print PrimeSpectrum.isClosed_iff_zeroLocus_radical_ideal /-
 theorem isClosed_iff_zeroLocus_radical_ideal (Z : Set (PrimeSpectrum R)) :
     IsClosed Z ↔ ∃ I : Ideal R, I.IsRadical ∧ Z = zeroLocus I :=
   (isClosed_iff_zeroLocus_ideal _).trans
     ⟨fun ⟨I, hI⟩ => ⟨_, I.radical_isRadical, (zeroLocus_radical I).substr hI⟩, fun ⟨I, _, hI⟩ =>
       ⟨I, hI⟩⟩
 #align prime_spectrum.is_closed_iff_zero_locus_radical_ideal PrimeSpectrum.isClosed_iff_zeroLocus_radical_ideal
+-/
 
+#print PrimeSpectrum.isClosed_zeroLocus /-
 theorem isClosed_zeroLocus (s : Set R) : IsClosed (zeroLocus s) :=
   by
   rw [is_closed_iff_zero_locus]
   exact ⟨s, rfl⟩
 #align prime_spectrum.is_closed_zero_locus PrimeSpectrum.isClosed_zeroLocus
+-/
 
+#print PrimeSpectrum.isClosed_singleton_iff_isMaximal /-
 theorem isClosed_singleton_iff_isMaximal (x : PrimeSpectrum R) :
     IsClosed ({x} : Set (PrimeSpectrum R)) ↔ x.asIdeal.IsMaximal :=
   by
@@ -474,7 +713,9 @@ theorem isClosed_singleton_iff_isMaximal (x : PrimeSpectrum R) :
     rw [eq_comm, Set.eq_singleton_iff_unique_mem]
     refine' ⟨fun _ h => h, fun y hy => PrimeSpectrum.ext _ _ (h.eq_of_le y.2.ne_top hy).symm⟩
 #align prime_spectrum.is_closed_singleton_iff_is_maximal PrimeSpectrum.isClosed_singleton_iff_isMaximal
+-/
 
+#print PrimeSpectrum.zeroLocus_vanishingIdeal_eq_closure /-
 theorem zeroLocus_vanishingIdeal_eq_closure (t : Set (PrimeSpectrum R)) :
     zeroLocus (vanishingIdeal t : Set R) = closure t :=
   by
@@ -486,23 +727,36 @@ theorem zeroLocus_vanishingIdeal_eq_closure (t : Set (PrimeSpectrum R)) :
   · rw [(is_closed_zero_locus _).closure_subset_iff]
     exact subset_zero_locus_vanishing_ideal t
 #align prime_spectrum.zero_locus_vanishing_ideal_eq_closure PrimeSpectrum.zeroLocus_vanishingIdeal_eq_closure
+-/
 
+#print PrimeSpectrum.vanishingIdeal_closure /-
 theorem vanishingIdeal_closure (t : Set (PrimeSpectrum R)) :
     vanishingIdeal (closure t) = vanishingIdeal t :=
   zeroLocus_vanishingIdeal_eq_closure t ▸ (gc R).u_l_u_eq_u t
 #align prime_spectrum.vanishing_ideal_closure PrimeSpectrum.vanishingIdeal_closure
+-/
 
+#print PrimeSpectrum.closure_singleton /-
 theorem closure_singleton (x) : closure ({x} : Set (PrimeSpectrum R)) = zeroLocus x.asIdeal := by
   rw [← zero_locus_vanishing_ideal_eq_closure, vanishing_ideal_singleton]
 #align prime_spectrum.closure_singleton PrimeSpectrum.closure_singleton
+-/
 
+#print PrimeSpectrum.isRadical_vanishingIdeal /-
 theorem isRadical_vanishingIdeal (s : Set (PrimeSpectrum R)) : (vanishingIdeal s).IsRadical :=
   by
   rw [← vanishing_ideal_closure, ← zero_locus_vanishing_ideal_eq_closure,
     vanishing_ideal_zero_locus_eq_radical]
   apply Ideal.radical_isRadical
 #align prime_spectrum.is_radical_vanishing_ideal PrimeSpectrum.isRadical_vanishingIdeal
+-/
 
+/- warning: prime_spectrum.vanishing_ideal_anti_mono_iff -> PrimeSpectrum.vanishingIdeal_anti_mono_iff is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {s : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)} {t : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)}, (IsClosed.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) t) -> (Iff (HasSubset.Subset.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.hasSubset.{u1} (PrimeSpectrum.{u1} R _inst_1)) s t) (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toHasLe.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 t) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 s)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {s : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)} {t : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)}, (IsClosed.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) t) -> (Iff (HasSubset.Subset.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.instHasSubsetSet.{u1} (PrimeSpectrum.{u1} R _inst_1)) s t) (LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 t) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 s)))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.vanishing_ideal_anti_mono_iff PrimeSpectrum.vanishingIdeal_anti_mono_iffₓ'. -/
 theorem vanishingIdeal_anti_mono_iff {s t : Set (PrimeSpectrum R)} (ht : IsClosed t) :
     s ⊆ t ↔ vanishingIdeal t ≤ vanishingIdeal s :=
   ⟨vanishingIdeal_anti_mono, fun h =>
@@ -511,12 +765,24 @@ theorem vanishingIdeal_anti_mono_iff {s t : Set (PrimeSpectrum R)} (ht : IsClose
     convert← zero_locus_anti_mono_ideal h <;> apply zero_locus_vanishing_ideal_eq_closure⟩
 #align prime_spectrum.vanishing_ideal_anti_mono_iff PrimeSpectrum.vanishingIdeal_anti_mono_iff
 
+/- warning: prime_spectrum.vanishing_ideal_strict_anti_mono_iff -> PrimeSpectrum.vanishingIdeal_strict_anti_mono_iff is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {s : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)} {t : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)}, (IsClosed.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) s) -> (IsClosed.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) t) -> (Iff (HasSSubset.SSubset.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.hasSsubset.{u1} (PrimeSpectrum.{u1} R _inst_1)) s t) (LT.lt.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toHasLt.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 t) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 s)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {s : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)} {t : Set.{u1} (PrimeSpectrum.{u1} R _inst_1)}, (IsClosed.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) s) -> (IsClosed.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) t) -> (Iff (HasSSubset.SSubset.{u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.instHasSSubsetSet.{u1} (PrimeSpectrum.{u1} R _inst_1)) s t) (LT.lt.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLT.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 t) (PrimeSpectrum.vanishingIdeal.{u1} R _inst_1 s)))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.vanishing_ideal_strict_anti_mono_iff PrimeSpectrum.vanishingIdeal_strict_anti_mono_iffₓ'. -/
 theorem vanishingIdeal_strict_anti_mono_iff {s t : Set (PrimeSpectrum R)} (hs : IsClosed s)
     (ht : IsClosed t) : s ⊂ t ↔ vanishingIdeal t < vanishingIdeal s := by
   rw [Set.ssubset_def, vanishing_ideal_anti_mono_iff hs, vanishing_ideal_anti_mono_iff ht,
     lt_iff_le_not_le]
 #align prime_spectrum.vanishing_ideal_strict_anti_mono_iff PrimeSpectrum.vanishingIdeal_strict_anti_mono_iff
 
+/- warning: prime_spectrum.closeds_embedding -> PrimeSpectrum.closedsEmbedding is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) [_inst_3 : CommRing.{u1} R], OrderEmbedding.{u1, u1} (OrderDual.{u1} (TopologicalSpace.Closeds.{u1} (PrimeSpectrum.{u1} R _inst_3) (PrimeSpectrum.zariskiTopology.{u1} R _inst_3))) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_3))) (OrderDual.hasLe.{u1} (TopologicalSpace.Closeds.{u1} (PrimeSpectrum.{u1} R _inst_3) (PrimeSpectrum.zariskiTopology.{u1} R _inst_3)) (Preorder.toHasLe.{u1} (TopologicalSpace.Closeds.{u1} (PrimeSpectrum.{u1} R _inst_3) (PrimeSpectrum.zariskiTopology.{u1} R _inst_3)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Closeds.{u1} (PrimeSpectrum.{u1} R _inst_3) (PrimeSpectrum.zariskiTopology.{u1} R _inst_3)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Closeds.{u1} (PrimeSpectrum.{u1} R _inst_3) (PrimeSpectrum.zariskiTopology.{u1} R _inst_3)) (PrimeSpectrum.{u1} R _inst_3) (TopologicalSpace.Closeds.setLike.{u1} (PrimeSpectrum.{u1} R _inst_3) (PrimeSpectrum.zariskiTopology.{u1} R _inst_3)))))) (Preorder.toHasLe.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_3))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_3))) (SetLike.partialOrder.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_3))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_3))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_3)))))))
+but is expected to have type
+  forall (R : Type.{u1}) [_inst_3 : CommRing.{u1} R], OrderEmbedding.{u1, u1} (OrderDual.{u1} (TopologicalSpace.Closeds.{u1} (PrimeSpectrum.{u1} R _inst_3) (PrimeSpectrum.zariskiTopology.{u1} R _inst_3))) (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_3))) (OrderDual.instLEOrderDual.{u1} (TopologicalSpace.Closeds.{u1} (PrimeSpectrum.{u1} R _inst_3) (PrimeSpectrum.zariskiTopology.{u1} R _inst_3)) (Preorder.toLE.{u1} (TopologicalSpace.Closeds.{u1} (PrimeSpectrum.{u1} R _inst_3) (PrimeSpectrum.zariskiTopology.{u1} R _inst_3)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Closeds.{u1} (PrimeSpectrum.{u1} R _inst_3) (PrimeSpectrum.zariskiTopology.{u1} R _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (TopologicalSpace.Closeds.{u1} (PrimeSpectrum.{u1} R _inst_3) (PrimeSpectrum.zariskiTopology.{u1} R _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (TopologicalSpace.Closeds.{u1} (PrimeSpectrum.{u1} R _inst_3) (PrimeSpectrum.zariskiTopology.{u1} R _inst_3)) (TopologicalSpace.Closeds.instCompleteLatticeCloseds.{u1} (PrimeSpectrum.{u1} R _inst_3) (PrimeSpectrum.zariskiTopology.{u1} R _inst_3))))))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_3))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_3))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_3))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_3))) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_3))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_3))))))))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.closeds_embedding PrimeSpectrum.closedsEmbeddingₓ'. -/
 /-- The antitone order embedding of closed subsets of `Spec R` into ideals of `R`. -/
 def closedsEmbedding (R : Type _) [CommRing R] :
     (TopologicalSpace.Closeds <| PrimeSpectrum R)ᵒᵈ ↪o Ideal R :=
@@ -524,6 +790,7 @@ def closedsEmbedding (R : Type _) [CommRing R] :
     (vanishingIdeal_anti_mono_iff s.2).symm
 #align prime_spectrum.closeds_embedding PrimeSpectrum.closedsEmbedding
 
+#print PrimeSpectrum.t1Space_iff_isField /-
 theorem t1Space_iff_isField [IsDomain R] : T1Space (PrimeSpectrum R) ↔ IsField R :=
   by
   refine' ⟨_, fun h => _⟩
@@ -541,10 +808,12 @@ theorem t1Space_iff_isField [IsDomain R] : T1Space (PrimeSpectrum R) ↔ IsField
       exact hx.symm ▸ Ideal.bot_isMaximal
     · exact absurd h (Ring.not_isField_iff_exists_prime.2 ⟨x.as_ideal, ⟨hx, x.2⟩⟩)
 #align prime_spectrum.t1_space_iff_is_field PrimeSpectrum.t1Space_iff_isField
+-/
 
 -- mathport name: «exprZ( )»
 local notation "Z(" a ")" => zeroLocus (a : Set R)
 
+#print PrimeSpectrum.isIrreducible_zeroLocus_iff_of_radical /-
 theorem isIrreducible_zeroLocus_iff_of_radical (I : Ideal R) (hI : I.IsRadical) :
     IsIrreducible (zeroLocus (I : Set R)) ↔ I.IsPrime :=
   by
@@ -570,17 +839,22 @@ theorem isIrreducible_zeroLocus_iff_of_radical (I : Ideal R) (hI : I.IsRadical)
         rintro h s t h' ⟨x, hx, hx'⟩ y hy
         exact h (h' ⟨Ideal.mul_mem_right _ _ hx, Ideal.mul_mem_left _ _ hy⟩) hx'
 #align prime_spectrum.is_irreducible_zero_locus_iff_of_radical PrimeSpectrum.isIrreducible_zeroLocus_iff_of_radical
+-/
 
+#print PrimeSpectrum.isIrreducible_zeroLocus_iff /-
 theorem isIrreducible_zeroLocus_iff (I : Ideal R) :
     IsIrreducible (zeroLocus (I : Set R)) ↔ I.radical.IsPrime :=
   zeroLocus_radical I ▸ isIrreducible_zeroLocus_iff_of_radical _ I.radical_isRadical
 #align prime_spectrum.is_irreducible_zero_locus_iff PrimeSpectrum.isIrreducible_zeroLocus_iff
+-/
 
+#print PrimeSpectrum.isIrreducible_iff_vanishingIdeal_isPrime /-
 theorem isIrreducible_iff_vanishingIdeal_isPrime {s : Set (PrimeSpectrum R)} :
     IsIrreducible s ↔ (vanishingIdeal s).IsPrime := by
   rw [← isIrreducible_iff_closure, ← zero_locus_vanishing_ideal_eq_closure,
     is_irreducible_zero_locus_iff_of_radical _ (is_radical_vanishing_ideal s)]
 #align prime_spectrum.is_irreducible_iff_vanishing_ideal_is_prime PrimeSpectrum.isIrreducible_iff_vanishingIdeal_isPrime
+-/
 
 instance [IsDomain R] : IrreducibleSpace (PrimeSpectrum R) :=
   by
@@ -596,6 +870,12 @@ section Comap
 
 variable {S' : Type _} [CommRing S']
 
+/- warning: prime_spectrum.preimage_comap_zero_locus_aux -> PrimeSpectrum.preimage_comap_zeroLocus_aux is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.preimage_comap_zero_locus_aux PrimeSpectrum.preimage_comap_zeroLocus_auxₓ'. -/
 theorem preimage_comap_zeroLocus_aux (f : R →+* S) (s : Set R) :
     (fun y => ⟨Ideal.comap f y.asIdeal, inferInstance⟩ : PrimeSpectrum S → PrimeSpectrum R) ⁻¹'
         zeroLocus s =
@@ -606,6 +886,7 @@ theorem preimage_comap_zeroLocus_aux (f : R →+* S) (s : Set R) :
   rfl
 #align prime_spectrum.preimage_comap_zero_locus_aux PrimeSpectrum.preimage_comap_zeroLocus_aux
 
+#print PrimeSpectrum.comap /-
 /-- The function between prime spectra of commutative rings induced by a ring homomorphism.
 This function is continuous. -/
 def comap (f : R →+* S) : C(PrimeSpectrum S, PrimeSpectrum R)
@@ -617,36 +898,69 @@ def comap (f : R →+* S) : C(PrimeSpectrum S, PrimeSpectrum R)
     rintro _ ⟨s, rfl⟩
     exact ⟨_, preimage_comap_zero_locus_aux f s⟩
 #align prime_spectrum.comap PrimeSpectrum.comap
+-/
 
 variable (f : R →+* S)
 
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 @[simp]
 theorem comap_asIdeal (y : PrimeSpectrum S) : (comap f y).asIdeal = Ideal.comap f y.asIdeal :=
   rfl
 #align prime_spectrum.comap_as_ideal PrimeSpectrum.comap_asIdeal
 
+#print PrimeSpectrum.comap_id /-
 @[simp]
 theorem comap_id : comap (RingHom.id R) = ContinuousMap.id _ :=
   by
   ext
   rfl
 #align prime_spectrum.comap_id PrimeSpectrum.comap_id
+-/
 
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 @[simp]
 theorem comap_comp (f : R →+* S) (g : S →+* S') : comap (g.comp f) = (comap f).comp (comap g) :=
   rfl
 #align prime_spectrum.comap_comp PrimeSpectrum.comap_comp
 
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.comap_comp_apply PrimeSpectrum.comap_comp_applyₓ'. -/
 theorem comap_comp_apply (f : R →+* S) (g : S →+* S') (x : PrimeSpectrum S') :
     PrimeSpectrum.comap (g.comp f) x = (PrimeSpectrum.comap f) (PrimeSpectrum.comap g x) :=
   rfl
 #align prime_spectrum.comap_comp_apply PrimeSpectrum.comap_comp_apply
 
+/- warning: prime_spectrum.preimage_comap_zero_locus -> PrimeSpectrum.preimage_comap_zeroLocus is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.preimage_comap_zero_locus PrimeSpectrum.preimage_comap_zeroLocusₓ'. -/
 @[simp]
 theorem preimage_comap_zeroLocus (s : Set R) : comap f ⁻¹' zeroLocus s = zeroLocus (f '' s) :=
   preimage_comap_zeroLocus_aux f s
 #align prime_spectrum.preimage_comap_zero_locus PrimeSpectrum.preimage_comap_zeroLocus
 
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+but is expected to have type
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))), (Function.Surjective.{succ u1, succ u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))))))) f)) -> (Function.Injective.{succ u2, succ u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_2) (fun (_x : PrimeSpectrum.{u2} S _inst_2) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : PrimeSpectrum.{u2} S _inst_2) => PrimeSpectrum.{u1} R _inst_1) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u2, u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_2 f)))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.comap_injective_of_surjective PrimeSpectrum.comap_injective_of_surjectiveₓ'. -/
 theorem comap_injective_of_surjective (f : R →+* S) (hf : Function.Surjective f) :
     Function.Injective (comap f) := fun x y h =>
   PrimeSpectrum.ext _ _
@@ -654,6 +968,12 @@ theorem comap_injective_of_surjective (f : R →+* S) (hf : Function.Surjective
       (congr_arg PrimeSpectrum.asIdeal h : (comap f x).asIdeal = (comap f y).asIdeal))
 #align prime_spectrum.comap_injective_of_surjective PrimeSpectrum.comap_injective_of_surjective
 
+/- warning: prime_spectrum.comap_singleton_is_closed_of_surjective -> PrimeSpectrum.comap_singleton_isClosed_of_surjective is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))), (Function.Surjective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) f)) -> (forall (x : PrimeSpectrum.{u2} S _inst_2), (IsClosed.{u2} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (Singleton.singleton.{u2, u2} (PrimeSpectrum.{u2} S _inst_2) (Set.{u2} (PrimeSpectrum.{u2} S _inst_2)) (Set.hasSingleton.{u2} (PrimeSpectrum.{u2} S _inst_2)) x)) -> (IsClosed.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (Singleton.singleton.{u1, u1} (PrimeSpectrum.{u1} R _inst_1) (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.hasSingleton.{u1} (PrimeSpectrum.{u1} R _inst_1)) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (fun (_x : ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) => (PrimeSpectrum.{u2} S _inst_2) -> (PrimeSpectrum.{u1} R _inst_1)) (ContinuousMap.hasCoeToFun.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_2 f) x))))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.comap_singleton_is_closed_of_surjective PrimeSpectrum.comap_singleton_isClosed_of_surjectiveₓ'. -/
 theorem comap_singleton_isClosed_of_surjective (f : R →+* S) (hf : Function.Surjective f)
     (x : PrimeSpectrum S) (hx : IsClosed ({x} : Set (PrimeSpectrum S))) :
     IsClosed ({comap f x} : Set (PrimeSpectrum R)) :=
@@ -661,6 +981,12 @@ theorem comap_singleton_isClosed_of_surjective (f : R →+* S) (hf : Function.Su
   (is_closed_singleton_iff_is_maximal _).2 (Ideal.comap_isMaximal_of_surjective f hf)
 #align prime_spectrum.comap_singleton_is_closed_of_surjective PrimeSpectrum.comap_singleton_isClosed_of_surjective
 
+/- warning: prime_spectrum.comap_singleton_is_closed_of_is_integral -> PrimeSpectrum.comap_singleton_isClosed_of_isIntegral is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))), (RingHom.IsIntegral.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_2) f) -> (forall (x : PrimeSpectrum.{u2} S _inst_2), (IsClosed.{u2} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (Singleton.singleton.{u2, u2} (PrimeSpectrum.{u2} S _inst_2) (Set.{u2} (PrimeSpectrum.{u2} S _inst_2)) (Set.hasSingleton.{u2} (PrimeSpectrum.{u2} S _inst_2)) x)) -> (IsClosed.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (Singleton.singleton.{u1, u1} (PrimeSpectrum.{u1} R _inst_1) (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.hasSingleton.{u1} (PrimeSpectrum.{u1} R _inst_1)) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (fun (_x : ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) => (PrimeSpectrum.{u2} S _inst_2) -> (PrimeSpectrum.{u1} R _inst_1)) (ContinuousMap.hasCoeToFun.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_2 f) x))))
+but is expected to have type
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))), (RingHom.IsIntegral.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_2) f) -> (forall (x : PrimeSpectrum.{u2} S _inst_2), (IsClosed.{u2} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (Singleton.singleton.{u2, u2} (PrimeSpectrum.{u2} S _inst_2) (Set.{u2} (PrimeSpectrum.{u2} S _inst_2)) (Set.instSingletonSet.{u2} (PrimeSpectrum.{u2} S _inst_2)) x)) -> (IsClosed.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (Singleton.singleton.{u1, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : PrimeSpectrum.{u2} S _inst_2) => PrimeSpectrum.{u1} R _inst_1) x) (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.instSingletonSet.{u1} (PrimeSpectrum.{u1} R _inst_1)) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_2) (fun (_x : PrimeSpectrum.{u2} S _inst_2) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : PrimeSpectrum.{u2} S _inst_2) => PrimeSpectrum.{u1} R _inst_1) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u2, u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_2 f) x))))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.comap_singleton_is_closed_of_is_integral PrimeSpectrum.comap_singleton_isClosed_of_isIntegralₓ'. -/
 theorem comap_singleton_isClosed_of_isIntegral (f : R →+* S) (hf : f.IsIntegral)
     (x : PrimeSpectrum S) (hx : IsClosed ({x} : Set (PrimeSpectrum S))) :
     IsClosed ({comap f x} : Set (PrimeSpectrum R)) :=
@@ -671,6 +997,12 @@ theorem comap_singleton_isClosed_of_isIntegral (f : R →+* S) (hf : f.IsIntegra
 
 variable (S)
 
+/- warning: prime_spectrum.localization_comap_inducing -> PrimeSpectrum.localization_comap_inducing is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} (S : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] [_inst_4 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) [_inst_5 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_4], Inducing.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (fun (_x : ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) => (PrimeSpectrum.{u2} S _inst_2) -> (PrimeSpectrum.{u1} R _inst_1)) (ContinuousMap.hasCoeToFun.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_2 (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_4)))
+but is expected to have type
+  forall {R : Type.{u1}} (S : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] [_inst_4 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))] (M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) [_inst_5 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_4], Inducing.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_2) (fun (_x : PrimeSpectrum.{u2} S _inst_2) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : PrimeSpectrum.{u2} S _inst_2) => PrimeSpectrum.{u1} R _inst_1) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u2, u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_2 (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_4)))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.localization_comap_inducing PrimeSpectrum.localization_comap_inducingₓ'. -/
 theorem localization_comap_inducing [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
     Inducing (comap (algebraMap R S)) := by
   constructor
@@ -689,6 +1021,12 @@ theorem localization_comap_inducing [Algebra R S] (M : Submonoid R) [IsLocalizat
     simp
 #align prime_spectrum.localization_comap_inducing PrimeSpectrum.localization_comap_inducing
 
+/- warning: prime_spectrum.localization_comap_injective -> PrimeSpectrum.localization_comap_injective is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} (S : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] [_inst_4 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) [_inst_5 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_4], Function.Injective.{succ u2, succ u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (fun (_x : ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) => (PrimeSpectrum.{u2} S _inst_2) -> (PrimeSpectrum.{u1} R _inst_1)) (ContinuousMap.hasCoeToFun.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_2 (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_4)))
+but is expected to have type
+  forall {R : Type.{u1}} (S : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] [_inst_4 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))] (M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) [_inst_5 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_4], Function.Injective.{succ u2, succ u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_2) (fun (_x : PrimeSpectrum.{u2} S _inst_2) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : PrimeSpectrum.{u2} S _inst_2) => PrimeSpectrum.{u1} R _inst_1) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u2, u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_2 (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_4)))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.localization_comap_injective PrimeSpectrum.localization_comap_injectiveₓ'. -/
 theorem localization_comap_injective [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
     Function.Injective (comap (algebraMap R S)) :=
   by
@@ -700,11 +1038,23 @@ theorem localization_comap_injective [Algebra R S] (M : Submonoid R) [IsLocaliza
   exact h
 #align prime_spectrum.localization_comap_injective PrimeSpectrum.localization_comap_injective
 
+/- warning: prime_spectrum.localization_comap_embedding -> PrimeSpectrum.localization_comap_embedding is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} (S : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] [_inst_4 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) [_inst_5 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_4], Embedding.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (fun (_x : ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) => (PrimeSpectrum.{u2} S _inst_2) -> (PrimeSpectrum.{u1} R _inst_1)) (ContinuousMap.hasCoeToFun.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_2 (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_4)))
+but is expected to have type
+  forall {R : Type.{u1}} (S : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] [_inst_4 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))] (M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) [_inst_5 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_4], Embedding.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_2) (fun (_x : PrimeSpectrum.{u2} S _inst_2) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : PrimeSpectrum.{u2} S _inst_2) => PrimeSpectrum.{u1} R _inst_1) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u2, u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_2 (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_4)))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.localization_comap_embedding PrimeSpectrum.localization_comap_embeddingₓ'. -/
 theorem localization_comap_embedding [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
     Embedding (comap (algebraMap R S)) :=
   ⟨localization_comap_inducing S M, localization_comap_injective S M⟩
 #align prime_spectrum.localization_comap_embedding PrimeSpectrum.localization_comap_embedding
 
+/- warning: prime_spectrum.localization_comap_range -> PrimeSpectrum.localization_comap_range is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} (S : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] [_inst_4 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) [_inst_5 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_4], Eq.{succ u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.range.{u1, succ u2} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.{u2} S _inst_2) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (fun (_x : ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) => (PrimeSpectrum.{u2} S _inst_2) -> (PrimeSpectrum.{u1} R _inst_1)) (ContinuousMap.hasCoeToFun.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_2 (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_4)))) (setOf.{u1} (PrimeSpectrum.{u1} R _inst_1) (fun (p : PrimeSpectrum.{u1} R _inst_1) => Disjoint.{u1} (Set.{u1} R) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} R) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} R) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} R) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} R) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} R) (Set.completeBooleanAlgebra.{u1} R)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} R) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} R) (Set.booleanAlgebra.{u1} R))) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) R (Submonoid.setLike.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))))))) M) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (PrimeSpectrum.asIdeal.{u1} R _inst_1 p))))
+but is expected to have type
+  forall {R : Type.{u1}} (S : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] [_inst_4 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))] (M : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) [_inst_5 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_4], Eq.{succ u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.range.{u1, succ u2} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.{u2} S _inst_2) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_2) (fun (_x : PrimeSpectrum.{u2} S _inst_2) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : PrimeSpectrum.{u2} S _inst_2) => PrimeSpectrum.{u1} R _inst_1) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u2, u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_2 (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_4)))) (setOf.{u1} (PrimeSpectrum.{u1} R _inst_1) (fun (p : PrimeSpectrum.{u1} R _inst_1) => Disjoint.{u1} (Set.{u1} R) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} R) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} R) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} R) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} R) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} R) (Set.instCompleteBooleanAlgebraSet.{u1} R)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} R) (Preorder.toLE.{u1} (Set.{u1} R) (PartialOrder.toPreorder.{u1} (Set.{u1} R) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} R) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} R) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} R) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} R) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} R) (Set.instCompleteBooleanAlgebraSet.{u1} R)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} R) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} R) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} R) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} R) (Set.instCompleteBooleanAlgebraSet.{u1} R)))))) (SetLike.coe.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) M) (SetLike.coe.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (PrimeSpectrum.asIdeal.{u1} R _inst_1 p))))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.localization_comap_range PrimeSpectrum.localization_comap_rangeₓ'. -/
 theorem localization_comap_range [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
     Set.range (comap (algebraMap R S)) = { p | Disjoint (M : Set R) p.asIdeal } :=
   by
@@ -726,6 +1076,12 @@ section SpecOfSurjective
 
 open Function RingHom
 
+/- warning: prime_spectrum.comap_inducing_of_surjective -> PrimeSpectrum.comap_inducing_of_surjective is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} (S : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))), (Function.Surjective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) f)) -> (Inducing.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (fun (_x : ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) => (PrimeSpectrum.{u2} S _inst_2) -> (PrimeSpectrum.{u1} R _inst_1)) (ContinuousMap.hasCoeToFun.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_2 f)))
+but is expected to have type
+  forall {R : Type.{u1}} (S : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))), (Function.Surjective.{succ u1, succ u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))))))) f)) -> (Inducing.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_2) (fun (_x : PrimeSpectrum.{u2} S _inst_2) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : PrimeSpectrum.{u2} S _inst_2) => PrimeSpectrum.{u1} R _inst_1) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u2, u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_2 f)))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.comap_inducing_of_surjective PrimeSpectrum.comap_inducing_of_surjectiveₓ'. -/
 theorem comap_inducing_of_surjective (hf : Surjective f) : Inducing (comap f) :=
   {
     induced :=
@@ -741,6 +1097,12 @@ theorem comap_inducing_of_surjective (hf : Surjective f) : Inducing (comap f) :=
       exact ⟨f '' F, hF.symm.trans (preimage_comap_zero_locus f F)⟩ }
 #align prime_spectrum.comap_inducing_of_surjective PrimeSpectrum.comap_inducing_of_surjective
 
+/- warning: prime_spectrum.image_comap_zero_locus_eq_zero_locus_comap -> PrimeSpectrum.image_comap_zeroLocus_eq_zeroLocus_comap is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.image_comap_zero_locus_eq_zero_locus_comap PrimeSpectrum.image_comap_zeroLocus_eq_zeroLocus_comapₓ'. -/
 theorem image_comap_zeroLocus_eq_zeroLocus_comap (hf : Surjective f) (I : Ideal S) :
     comap f '' zeroLocus I = zeroLocus (I.comap f) :=
   by
@@ -762,6 +1124,12 @@ theorem image_comap_zeroLocus_eq_zeroLocus_comap (hf : Surjective f) (I : Ideal
       rwa [mem_ker, map_sub, sub_eq_zero]
 #align prime_spectrum.image_comap_zero_locus_eq_zero_locus_comap PrimeSpectrum.image_comap_zeroLocus_eq_zeroLocus_comap
 
+/- warning: prime_spectrum.range_comap_of_surjective -> PrimeSpectrum.range_comap_of_surjective is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.range_comap_of_surjective PrimeSpectrum.range_comap_of_surjectiveₓ'. -/
 theorem range_comap_of_surjective (hf : Surjective f) : Set.range (comap f) = zeroLocus (ker f) :=
   by
   rw [← Set.image_univ]
@@ -769,12 +1137,24 @@ theorem range_comap_of_surjective (hf : Surjective f) : Set.range (comap f) = ze
   rw [zero_locus_bot]
 #align prime_spectrum.range_comap_of_surjective PrimeSpectrum.range_comap_of_surjective
 
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.is_closed_range_comap_of_surjective PrimeSpectrum.isClosed_range_comap_of_surjectiveₓ'. -/
 theorem isClosed_range_comap_of_surjective (hf : Surjective f) : IsClosed (Set.range (comap f)) :=
   by
   rw [range_comap_of_surjective _ f hf]
   exact is_closed_zero_locus ↑(ker f)
 #align prime_spectrum.is_closed_range_comap_of_surjective PrimeSpectrum.isClosed_range_comap_of_surjective
 
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+  forall {R : Type.{u1}} (S : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))), (Function.Surjective.{succ u1, succ u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))))))) f)) -> (ClosedEmbedding.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_2) (fun (_x : PrimeSpectrum.{u2} S _inst_2) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : PrimeSpectrum.{u2} S _inst_2) => PrimeSpectrum.{u1} R _inst_1) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u2, u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_2) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_2) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_2 f)))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.closed_embedding_comap_of_surjective PrimeSpectrum.closedEmbedding_comap_of_surjectiveₓ'. -/
 theorem closedEmbedding_comap_of_surjective (hf : Surjective f) : ClosedEmbedding (comap f) :=
   { induced := (comap_inducing_of_surjective S f hf).induced
     inj := comap_injective_of_surjective f hf
@@ -787,65 +1167,120 @@ end Comap
 
 section BasicOpen
 
+#print PrimeSpectrum.basicOpen /-
 /-- `basic_open r` is the open subset containing all prime ideals not containing `r`. -/
 def basicOpen (r : R) : TopologicalSpace.Opens (PrimeSpectrum R)
     where
   carrier := { x | r ∉ x.asIdeal }
   is_open' := ⟨{r}, Set.ext fun x => Set.singleton_subset_iff.trans <| Classical.not_not.symm⟩
 #align prime_spectrum.basic_open PrimeSpectrum.basicOpen
+-/
 
+#print PrimeSpectrum.mem_basicOpen /-
 @[simp]
 theorem mem_basicOpen (f : R) (x : PrimeSpectrum R) : x ∈ basicOpen f ↔ f ∉ x.asIdeal :=
   Iff.rfl
 #align prime_spectrum.mem_basic_open PrimeSpectrum.mem_basicOpen
+-/
 
+#print PrimeSpectrum.isOpen_basicOpen /-
 theorem isOpen_basicOpen {a : R} : IsOpen (basicOpen a : Set (PrimeSpectrum R)) :=
   (basicOpen a).IsOpen
 #align prime_spectrum.is_open_basic_open PrimeSpectrum.isOpen_basicOpen
+-/
 
+/- warning: prime_spectrum.basic_open_eq_zero_locus_compl -> PrimeSpectrum.basicOpen_eq_zeroLocus_compl is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.basic_open_eq_zero_locus_compl PrimeSpectrum.basicOpen_eq_zeroLocus_complₓ'. -/
 @[simp]
 theorem basicOpen_eq_zeroLocus_compl (r : R) :
     (basicOpen r : Set (PrimeSpectrum R)) = zeroLocus {r}ᶜ :=
   Set.ext fun x => by simpa only [Set.mem_compl_iff, mem_zero_locus, Set.singleton_subset_iff]
 #align prime_spectrum.basic_open_eq_zero_locus_compl PrimeSpectrum.basicOpen_eq_zeroLocus_compl
 
+/- warning: prime_spectrum.basic_open_one -> PrimeSpectrum.basicOpen_one 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 prime_spectrum.basic_open_one PrimeSpectrum.basicOpen_oneₓ'. -/
 @[simp]
 theorem basicOpen_one : basicOpen (1 : R) = ⊤ :=
   TopologicalSpace.Opens.ext <| by simp
 #align prime_spectrum.basic_open_one PrimeSpectrum.basicOpen_one
 
+/- warning: prime_spectrum.basic_open_zero -> PrimeSpectrum.basicOpen_zero is a dubious translation:
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+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R], Eq.{succ u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.basicOpen.{u1} R _inst_1 (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Bot.bot.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (CompleteLattice.toBot.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.basic_open_zero PrimeSpectrum.basicOpen_zeroₓ'. -/
 @[simp]
 theorem basicOpen_zero : basicOpen (0 : R) = ⊥ :=
   TopologicalSpace.Opens.ext <| by simp
 #align prime_spectrum.basic_open_zero PrimeSpectrum.basicOpen_zero
 
+/- warning: prime_spectrum.basic_open_le_basic_open_iff -> PrimeSpectrum.basicOpen_le_basicOpen_iff 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 prime_spectrum.basic_open_le_basic_open_iff PrimeSpectrum.basicOpen_le_basicOpen_iffₓ'. -/
 theorem basicOpen_le_basicOpen_iff (f g : R) :
     basicOpen f ≤ basicOpen g ↔ f ∈ (Ideal.span ({g} : Set R)).radical := by
   rw [← SetLike.coe_subset_coe, basic_open_eq_zero_locus_compl, basic_open_eq_zero_locus_compl,
     Set.compl_subset_compl, zero_locus_subset_zero_locus_singleton_iff]
 #align prime_spectrum.basic_open_le_basic_open_iff PrimeSpectrum.basicOpen_le_basicOpen_iff
 
+/- warning: prime_spectrum.basic_open_mul -> PrimeSpectrum.basicOpen_mul is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (f : R) (g : R), Eq.{succ u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.basicOpen.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) f g)) (Inf.inf.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (SemilatticeInf.toHasInf.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (Lattice.toSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (TopologicalSpace.Opens.completeLattice.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)))))) (PrimeSpectrum.basicOpen.{u1} R _inst_1 f) (PrimeSpectrum.basicOpen.{u1} R _inst_1 g))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (f : R) (g : R), Eq.{succ u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.basicOpen.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) f g)) (Inf.inf.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (Lattice.toInf.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))))) (PrimeSpectrum.basicOpen.{u1} R _inst_1 f) (PrimeSpectrum.basicOpen.{u1} R _inst_1 g))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.basic_open_mul PrimeSpectrum.basicOpen_mulₓ'. -/
 theorem basicOpen_mul (f g : R) : basicOpen (f * g) = basicOpen f ⊓ basicOpen g :=
   TopologicalSpace.Opens.ext <| by simp [zero_locus_singleton_mul]
 #align prime_spectrum.basic_open_mul PrimeSpectrum.basicOpen_mul
 
+/- warning: prime_spectrum.basic_open_mul_le_left -> PrimeSpectrum.basicOpen_mul_le_left is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (f : R) (g : R), LE.le.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (Preorder.toHasLe.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u1} R _inst_1) (TopologicalSpace.Opens.setLike.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))))) (PrimeSpectrum.basicOpen.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) f g)) (PrimeSpectrum.basicOpen.{u1} R _inst_1 f)
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (f : R) (g : R), LE.le.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (Preorder.toLE.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)))))) (PrimeSpectrum.basicOpen.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) f g)) (PrimeSpectrum.basicOpen.{u1} R _inst_1 f)
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.basic_open_mul_le_left PrimeSpectrum.basicOpen_mul_le_leftₓ'. -/
 theorem basicOpen_mul_le_left (f g : R) : basicOpen (f * g) ≤ basicOpen f :=
   by
   rw [basic_open_mul f g]
   exact inf_le_left
 #align prime_spectrum.basic_open_mul_le_left PrimeSpectrum.basicOpen_mul_le_left
 
+/- warning: prime_spectrum.basic_open_mul_le_right -> PrimeSpectrum.basicOpen_mul_le_right is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (f : R) (g : R), LE.le.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (Preorder.toHasLe.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u1} R _inst_1) (TopologicalSpace.Opens.setLike.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))))) (PrimeSpectrum.basicOpen.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) f g)) (PrimeSpectrum.basicOpen.{u1} R _inst_1 g)
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (f : R) (g : R), LE.le.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (Preorder.toLE.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)))))) (PrimeSpectrum.basicOpen.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) f g)) (PrimeSpectrum.basicOpen.{u1} R _inst_1 g)
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.basic_open_mul_le_right PrimeSpectrum.basicOpen_mul_le_rightₓ'. -/
 theorem basicOpen_mul_le_right (f g : R) : basicOpen (f * g) ≤ basicOpen g :=
   by
   rw [basic_open_mul f g]
   exact inf_le_right
 #align prime_spectrum.basic_open_mul_le_right PrimeSpectrum.basicOpen_mul_le_right
 
+/- warning: prime_spectrum.basic_open_pow -> PrimeSpectrum.basicOpen_pow is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (f : R) (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 u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.basicOpen.{u1} R _inst_1 (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)))) f n)) (PrimeSpectrum.basicOpen.{u1} R _inst_1 f))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (f : R) (n : Nat), (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) n) -> (Eq.{succ u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.basicOpen.{u1} R _inst_1 (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) f n)) (PrimeSpectrum.basicOpen.{u1} R _inst_1 f))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.basic_open_pow PrimeSpectrum.basicOpen_powₓ'. -/
 @[simp]
 theorem basicOpen_pow (f : R) (n : ℕ) (hn : 0 < n) : basicOpen (f ^ n) = basicOpen f :=
   TopologicalSpace.Opens.ext <| by simpa using zero_locus_singleton_pow f n hn
 #align prime_spectrum.basic_open_pow PrimeSpectrum.basicOpen_pow
 
+#print PrimeSpectrum.isTopologicalBasis_basic_opens /-
 theorem isTopologicalBasis_basic_opens :
     TopologicalSpace.IsTopologicalBasis
       (Set.range fun r : R => (basicOpen r : Set (PrimeSpectrum R))) :=
@@ -860,14 +1295,18 @@ theorem isTopologicalBasis_basic_opens :
     rw [← Set.compl_subset_compl, ← hs, basic_open_eq_zero_locus_compl, compl_compl]
     exact zero_locus_anti_mono (set.singleton_subset_iff.mpr hfs)
 #align prime_spectrum.is_topological_basis_basic_opens PrimeSpectrum.isTopologicalBasis_basic_opens
+-/
 
+#print PrimeSpectrum.isBasis_basic_opens /-
 theorem isBasis_basic_opens : TopologicalSpace.Opens.IsBasis (Set.range (@basicOpen R _)) :=
   by
   unfold TopologicalSpace.Opens.IsBasis
   convert is_topological_basis_basic_opens
   rw [← Set.range_comp]
 #align prime_spectrum.is_basis_basic_opens PrimeSpectrum.isBasis_basic_opens
+-/
 
+#print PrimeSpectrum.isCompact_basicOpen /-
 theorem isCompact_basicOpen (f : R) : IsCompact (basicOpen f : Set (PrimeSpectrum R)) :=
   isCompact_of_finite_subfamily_closed fun ι Z hZc hZ =>
     by
@@ -892,7 +1331,14 @@ theorem isCompact_basicOpen (f : R) : IsCompact (basicOpen f : Set (PrimeSpectru
     rw [Set.singleton_subset_iff]
     exact ⟨n, hs⟩
 #align prime_spectrum.is_compact_basic_open PrimeSpectrum.isCompact_basicOpen
+-/
 
+/- warning: prime_spectrum.basic_open_eq_bot_iff -> PrimeSpectrum.basicOpen_eq_bot_iff is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (f : R), Iff (Eq.{succ u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.basicOpen.{u1} R _inst_1 f) (Bot.bot.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (CompleteLattice.toHasBot.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (TopologicalSpace.Opens.completeLattice.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))))) (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))) f)
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (f : R), Iff (Eq.{succ u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.basicOpen.{u1} R _inst_1 f) (Bot.bot.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (CompleteLattice.toBot.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))))) (IsNilpotent.{u1} R (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) f)
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.basic_open_eq_bot_iff PrimeSpectrum.basicOpen_eq_bot_iffₓ'. -/
 @[simp]
 theorem basicOpen_eq_bot_iff (f : R) : basicOpen f = ⊥ ↔ IsNilpotent f :=
   by
@@ -902,6 +1348,12 @@ theorem basicOpen_eq_bot_iff (f : R) : basicOpen f = ⊥ ↔ IsNilpotent f :=
   exact ⟨fun h I hI => h ⟨I, hI⟩, fun h ⟨I, hI⟩ => h I hI⟩
 #align prime_spectrum.basic_open_eq_bot_iff PrimeSpectrum.basicOpen_eq_bot_iff
 
+/- warning: prime_spectrum.localization_away_comap_range -> PrimeSpectrum.localization_away_comap_range is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_3 : CommRing.{u2} S] [_inst_4 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_3))] (r : R) [_inst_5 : IsLocalization.Away.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) r S (CommRing.toCommSemiring.{u2} S _inst_3) _inst_4], Eq.{succ u1} (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Set.range.{u1, succ u2} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.{u2} S _inst_3) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_3) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_3) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (fun (_x : ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_3) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_3) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) => (PrimeSpectrum.{u2} S _inst_3) -> (PrimeSpectrum.{u1} R _inst_1)) (ContinuousMap.hasCoeToFun.{u2, u1} (PrimeSpectrum.{u2} S _inst_3) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_3) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_3 (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_3)) _inst_4)))) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (HasLiftT.mk.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (CoeTCₓ.coe.{succ u1, succ u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (Set.{u1} (PrimeSpectrum.{u1} R _inst_1)) (SetLike.Set.hasCoeT.{u1, u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u1} R _inst_1) (TopologicalSpace.Opens.setLike.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))))) (PrimeSpectrum.basicOpen.{u1} R _inst_1 r))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.localization_away_comap_range PrimeSpectrum.localization_away_comap_rangeₓ'. -/
 theorem localization_away_comap_range (S : Type v) [CommRing S] [Algebra R S] (r : R)
     [IsLocalization.Away r S] : Set.range (comap (algebraMap R S)) = basicOpen r :=
   by
@@ -916,6 +1368,12 @@ theorem localization_away_comap_range (S : Type v) [CommRing S] [Algebra R S] (r
     exact h₁ (x.2.mem_of_pow_mem _ h₃)
 #align prime_spectrum.localization_away_comap_range PrimeSpectrum.localization_away_comap_range
 
+/- warning: prime_spectrum.localization_away_open_embedding -> PrimeSpectrum.localization_away_openEmbedding is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align prime_spectrum.localization_away_open_embedding PrimeSpectrum.localization_away_openEmbeddingₓ'. -/
 theorem localization_away_openEmbedding (S : Type v) [CommRing S] [Algebra R S] (r : R)
     [IsLocalization.Away r S] : OpenEmbedding (comap (algebraMap R S)) :=
   { toEmbedding := localization_comap_embedding S (Submonoid.powers r)
@@ -945,26 +1403,56 @@ We endow `prime_spectrum R` with a partial order, where `x ≤ y` if and only if
 instance : PartialOrder (PrimeSpectrum R) :=
   PartialOrder.lift asIdeal ext
 
+/- warning: prime_spectrum.as_ideal_le_as_ideal -> PrimeSpectrum.asIdeal_le_asIdeal 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 prime_spectrum.as_ideal_le_as_ideal PrimeSpectrum.asIdeal_le_asIdealₓ'. -/
 @[simp]
 theorem asIdeal_le_asIdeal (x y : PrimeSpectrum R) : x.asIdeal ≤ y.asIdeal ↔ x ≤ y :=
   Iff.rfl
 #align prime_spectrum.as_ideal_le_as_ideal PrimeSpectrum.asIdeal_le_asIdeal
 
+/- warning: prime_spectrum.as_ideal_lt_as_ideal -> PrimeSpectrum.asIdeal_lt_asIdeal 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 prime_spectrum.as_ideal_lt_as_ideal PrimeSpectrum.asIdeal_lt_asIdealₓ'. -/
 @[simp]
 theorem asIdeal_lt_asIdeal (x y : PrimeSpectrum R) : x.asIdeal < y.asIdeal ↔ x < y :=
   Iff.rfl
 #align prime_spectrum.as_ideal_lt_as_ideal PrimeSpectrum.asIdeal_lt_asIdeal
 
+/- warning: prime_spectrum.le_iff_mem_closure -> PrimeSpectrum.le_iff_mem_closure 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 prime_spectrum.le_iff_mem_closure PrimeSpectrum.le_iff_mem_closureₓ'. -/
 theorem le_iff_mem_closure (x y : PrimeSpectrum R) :
     x ≤ y ↔ y ∈ closure ({x} : Set (PrimeSpectrum R)) := by
   rw [← as_ideal_le_as_ideal, ← zero_locus_vanishing_ideal_eq_closure, mem_zero_locus,
     vanishing_ideal_singleton, SetLike.coe_subset_coe]
 #align prime_spectrum.le_iff_mem_closure PrimeSpectrum.le_iff_mem_closure
 
+/- warning: prime_spectrum.le_iff_specializes -> PrimeSpectrum.le_iff_specializes is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (x : PrimeSpectrum.{u1} R _inst_1) (y : PrimeSpectrum.{u1} R _inst_1), Iff (LE.le.{u1} (PrimeSpectrum.{u1} R _inst_1) (Preorder.toHasLe.{u1} (PrimeSpectrum.{u1} R _inst_1) (PartialOrder.toPreorder.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.partialOrder.{u1} R _inst_1))) x y) (Specializes.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) x y)
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (x : PrimeSpectrum.{u1} R _inst_1) (y : PrimeSpectrum.{u1} R _inst_1), Iff (LE.le.{u1} (PrimeSpectrum.{u1} R _inst_1) (Preorder.toLE.{u1} (PrimeSpectrum.{u1} R _inst_1) (PartialOrder.toPreorder.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.instPartialOrderPrimeSpectrum.{u1} R _inst_1))) x y) (Specializes.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) x y)
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.le_iff_specializes PrimeSpectrum.le_iff_specializesₓ'. -/
 theorem le_iff_specializes (x y : PrimeSpectrum R) : x ≤ y ↔ x ⤳ y :=
   (le_iff_mem_closure x y).trans specializes_iff_mem_closure.symm
 #align prime_spectrum.le_iff_specializes PrimeSpectrum.le_iff_specializes
 
+/- warning: prime_spectrum.nhds_order_embedding -> PrimeSpectrum.nhdsOrderEmbedding is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R], OrderEmbedding.{u1, u1} (PrimeSpectrum.{u1} R _inst_1) (Filter.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Preorder.toHasLe.{u1} (PrimeSpectrum.{u1} R _inst_1) (PartialOrder.toPreorder.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.partialOrder.{u1} R _inst_1))) (Preorder.toHasLe.{u1} (Filter.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (Filter.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Filter.partialOrder.{u1} (PrimeSpectrum.{u1} R _inst_1))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R], OrderEmbedding.{u1, u1} (PrimeSpectrum.{u1} R _inst_1) (Filter.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Preorder.toLE.{u1} (PrimeSpectrum.{u1} R _inst_1) (PartialOrder.toPreorder.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.instPartialOrderPrimeSpectrum.{u1} R _inst_1))) (Preorder.toLE.{u1} (Filter.{u1} (PrimeSpectrum.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (Filter.{u1} (PrimeSpectrum.{u1} R _inst_1)) (Filter.instPartialOrderFilter.{u1} (PrimeSpectrum.{u1} R _inst_1))))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.nhds_order_embedding PrimeSpectrum.nhdsOrderEmbeddingₓ'. -/
 /-- `nhds` as an order embedding. -/
 @[simps (config := { fullyApplied := true })]
 def nhdsOrderEmbedding : PrimeSpectrum R ↪o Filter (PrimeSpectrum R) :=
@@ -986,6 +1474,12 @@ instance {R : Type _} [Field R] : Unique (PrimeSpectrum R)
 
 end Order
 
+/- warning: prime_spectrum.localization_map_of_specializes -> PrimeSpectrum.localizationMapOfSpecializes is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {x : PrimeSpectrum.{u1} R _inst_1} {y : PrimeSpectrum.{u1} R _inst_1}, (Specializes.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) x y) -> (RingHom.{u1, u1} (Localization.AtPrime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 y) (PrimeSpectrum.isPrime.{u1} R _inst_1 y)) (Localization.AtPrime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 x) (PrimeSpectrum.isPrime.{u1} R _inst_1 x)) (NonAssocRing.toNonAssocSemiring.{u1} (Localization.AtPrime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 y) (PrimeSpectrum.isPrime.{u1} R _inst_1 y)) (Ring.toNonAssocRing.{u1} (Localization.AtPrime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 y) (PrimeSpectrum.isPrime.{u1} R _inst_1 y)) (CommRing.toRing.{u1} (Localization.AtPrime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 y) (PrimeSpectrum.isPrime.{u1} R _inst_1 y)) (Localization.commRing.{u1} R _inst_1 (Ideal.primeCompl.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 y) (PrimeSpectrum.isPrime.{u1} R _inst_1 y)))))) (NonAssocRing.toNonAssocSemiring.{u1} (Localization.AtPrime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 x) (PrimeSpectrum.isPrime.{u1} R _inst_1 x)) (Ring.toNonAssocRing.{u1} (Localization.AtPrime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 x) (PrimeSpectrum.isPrime.{u1} R _inst_1 x)) (CommRing.toRing.{u1} (Localization.AtPrime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 x) (PrimeSpectrum.isPrime.{u1} R _inst_1 x)) (Localization.commRing.{u1} R _inst_1 (Ideal.primeCompl.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 x) (PrimeSpectrum.isPrime.{u1} R _inst_1 x)))))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {x : PrimeSpectrum.{u1} R _inst_1} {y : PrimeSpectrum.{u1} R _inst_1}, (Specializes.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) x y) -> (RingHom.{u1, u1} (Localization.AtPrime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 y) (PrimeSpectrum.IsPrime.{u1} R _inst_1 y)) (Localization.AtPrime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 x) (PrimeSpectrum.IsPrime.{u1} R _inst_1 x)) (Semiring.toNonAssocSemiring.{u1} (Localization.AtPrime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 y) (PrimeSpectrum.IsPrime.{u1} R _inst_1 y)) (CommSemiring.toSemiring.{u1} (Localization.AtPrime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 y) (PrimeSpectrum.IsPrime.{u1} R _inst_1 y)) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (Ideal.primeCompl.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 y) (PrimeSpectrum.IsPrime.{u1} R _inst_1 y))))) (Semiring.toNonAssocSemiring.{u1} (Localization.AtPrime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 x) (PrimeSpectrum.IsPrime.{u1} R _inst_1 x)) (CommSemiring.toSemiring.{u1} (Localization.AtPrime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 x) (PrimeSpectrum.IsPrime.{u1} R _inst_1 x)) (Localization.instCommSemiringLocalizationToCommMonoid.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (Ideal.primeCompl.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) (PrimeSpectrum.asIdeal.{u1} R _inst_1 x) (PrimeSpectrum.IsPrime.{u1} R _inst_1 x))))))
+Case conversion may be inaccurate. Consider using '#align prime_spectrum.localization_map_of_specializes PrimeSpectrum.localizationMapOfSpecializesₓ'. -/
 /-- If `x` specializes to `y`, then there is a natural map from the localization of `y` to the
 localization of `x`. -/
 def localizationMapOfSpecializes {x y : PrimeSpectrum R} (h : x ⤳ y) :
@@ -1005,13 +1499,21 @@ namespace LocalRing
 
 variable [LocalRing R]
 
+#print LocalRing.closedPoint /-
 /-- The closed point in the prime spectrum of a local ring. -/
 def closedPoint : PrimeSpectrum R :=
   ⟨maximalIdeal R, (maximalIdeal.isMaximal R).IsPrime⟩
 #align local_ring.closed_point LocalRing.closedPoint
+-/
 
 variable {R}
 
+/- warning: local_ring.is_local_ring_hom_iff_comap_closed_point -> LocalRing.isLocalRingHom_iff_comap_closedPoint is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_3 : LocalRing.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] {S : Type.{u2}} [_inst_4 : CommRing.{u2} S] [_inst_5 : LocalRing.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_4))] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))), Iff (IsLocalRingHom.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_4)) f) (Eq.{succ u1} (PrimeSpectrum.{u1} R _inst_1) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (fun (_x : ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) => (PrimeSpectrum.{u2} S _inst_4) -> (PrimeSpectrum.{u1} R _inst_1)) (ContinuousMap.hasCoeToFun.{u2, u1} (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_4 f) (LocalRing.closedPoint.{u2} S _inst_4 _inst_5)) (LocalRing.closedPoint.{u1} R _inst_1 _inst_3))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_3 : LocalRing.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] {S : Type.{u2}} [_inst_4 : CommRing.{u2} S] [_inst_5 : LocalRing.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4))] (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4)))), Iff (IsLocalRingHom.{u1, u2} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4)) f) (Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : PrimeSpectrum.{u2} S _inst_4) => PrimeSpectrum.{u1} R _inst_1) (LocalRing.closedPoint.{u2} S _inst_4 _inst_5)) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_4) (fun (_x : PrimeSpectrum.{u2} S _inst_4) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : PrimeSpectrum.{u2} S _inst_4) => PrimeSpectrum.{u1} R _inst_1) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u2, u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_4 f) (LocalRing.closedPoint.{u2} S _inst_4 _inst_5)) (LocalRing.closedPoint.{u1} R _inst_1 _inst_3))
+Case conversion may be inaccurate. Consider using '#align local_ring.is_local_ring_hom_iff_comap_closed_point LocalRing.isLocalRingHom_iff_comap_closedPointₓ'. -/
 theorem isLocalRingHom_iff_comap_closedPoint {S : Type v} [CommRing S] [LocalRing S] (f : R →+* S) :
     IsLocalRingHom f ↔ PrimeSpectrum.comap f (closedPoint S) = closedPoint R :=
   by
@@ -1019,16 +1521,30 @@ theorem isLocalRingHom_iff_comap_closedPoint {S : Type v} [CommRing S] [LocalRin
   rfl
 #align local_ring.is_local_ring_hom_iff_comap_closed_point LocalRing.isLocalRingHom_iff_comap_closedPoint
 
+/- warning: local_ring.comap_closed_point -> LocalRing.comap_closedPoint is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_3 : LocalRing.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] {S : Type.{u2}} [_inst_4 : CommRing.{u2} S] [_inst_5 : LocalRing.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_4))] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))) [_inst_6 : IsLocalRingHom.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_4)) f], Eq.{succ u1} (PrimeSpectrum.{u1} R _inst_1) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (fun (_x : ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) => (PrimeSpectrum.{u2} S _inst_4) -> (PrimeSpectrum.{u1} R _inst_1)) (ContinuousMap.hasCoeToFun.{u2, u1} (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_4 f) (LocalRing.closedPoint.{u2} S _inst_4 _inst_5)) (LocalRing.closedPoint.{u1} R _inst_1 _inst_3)
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_3 : LocalRing.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] {S : Type.{u2}} [_inst_4 : CommRing.{u2} S] [_inst_5 : LocalRing.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4))] (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4)))) [_inst_6 : IsLocalRingHom.{u1, u2} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4)) f], Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : PrimeSpectrum.{u2} S _inst_4) => PrimeSpectrum.{u1} R _inst_1) (LocalRing.closedPoint.{u2} S _inst_4 _inst_5)) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_4) (fun (_x : PrimeSpectrum.{u2} S _inst_4) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : PrimeSpectrum.{u2} S _inst_4) => PrimeSpectrum.{u1} R _inst_1) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u2, u1} (ContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1) (ContinuousMap.instContinuousMapClassContinuousMap.{u2, u1} (PrimeSpectrum.{u2} S _inst_4) (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u2} S _inst_4) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))) (PrimeSpectrum.comap.{u1, u2} R S _inst_1 _inst_4 f) (LocalRing.closedPoint.{u2} S _inst_4 _inst_5)) (LocalRing.closedPoint.{u1} R _inst_1 _inst_3)
+Case conversion may be inaccurate. Consider using '#align local_ring.comap_closed_point LocalRing.comap_closedPointₓ'. -/
 @[simp]
 theorem comap_closedPoint {S : Type v} [CommRing S] [LocalRing S] (f : R →+* S) [IsLocalRingHom f] :
     PrimeSpectrum.comap f (closedPoint S) = closedPoint R :=
   (isLocalRingHom_iff_comap_closedPoint f).mp inferInstance
 #align local_ring.comap_closed_point LocalRing.comap_closedPoint
 
+#print LocalRing.specializes_closedPoint /-
 theorem specializes_closedPoint (x : PrimeSpectrum R) : x ⤳ closedPoint R :=
   (PrimeSpectrum.le_iff_specializes _ _).mp (LocalRing.le_maximalIdeal x.2.1)
 #align local_ring.specializes_closed_point LocalRing.specializes_closedPoint
+-/
 
+/- warning: local_ring.closed_point_mem_iff -> LocalRing.closedPoint_mem_iff is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_3 : LocalRing.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] (U : TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)), Iff (Membership.Mem.{u1, u1} (PrimeSpectrum.{u1} R _inst_1) (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (SetLike.hasMem.{u1, u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u1} R _inst_1) (TopologicalSpace.Opens.setLike.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))) (LocalRing.closedPoint.{u1} R _inst_1 _inst_3) U) (Eq.{succ u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) U (Top.top.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (CompleteLattice.toHasTop.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (TopologicalSpace.Opens.completeLattice.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_3 : LocalRing.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] (U : TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)), Iff (Membership.mem.{u1, u1} (PrimeSpectrum.{u1} R _inst_1) (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (SetLike.instMembership.{u1, u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (PrimeSpectrum.{u1} R _inst_1) (TopologicalSpace.Opens.instSetLikeOpens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1))) (LocalRing.closedPoint.{u1} R _inst_1 _inst_3) U) (Eq.{succ u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) U (Top.top.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (CompleteLattice.toTop.{u1} (TopologicalSpace.Opens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (PrimeSpectrum.{u1} R _inst_1) (PrimeSpectrum.zariskiTopology.{u1} R _inst_1)))))
+Case conversion may be inaccurate. Consider using '#align local_ring.closed_point_mem_iff LocalRing.closedPoint_mem_iffₓ'. -/
 theorem closedPoint_mem_iff (U : TopologicalSpace.Opens <| PrimeSpectrum R) :
     closedPoint R ∈ U ↔ U = ⊤ := by
   constructor

mathlib commit https://github.com/leanprover-community/mathlib/commit/f51de8769c34652d82d1c8e5f8f18f8374782bed

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
 
 ! This file was ported from Lean 3 source module algebraic_geometry.prime_spectrum.basic
-! leanprover-community/mathlib commit d39590fc8728fbf6743249802486f8c91ffe07bc
+! leanprover-community/mathlib commit a7c017d750512a352b623b1824d75da5998457d0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -12,7 +12,7 @@ import Mathbin.Algebra.PunitInstances
 import Mathbin.LinearAlgebra.Finsupp
 import Mathbin.RingTheory.Ideal.Over
 import Mathbin.RingTheory.Ideal.Prod
-import Mathbin.RingTheory.Localization.Away
+import Mathbin.RingTheory.Localization.Away.Basic
 import Mathbin.RingTheory.Nilpotent
 import Mathbin.Topology.Sets.Closeds
 import Mathbin.Topology.Sober

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

@@ -169,9 +169,9 @@ theorem coe_vanishingIdeal (t : Set (PrimeSpectrum R)) :
     (vanishingIdeal t : Set R) = { f : R | ∀ x : PrimeSpectrum R, x ∈ t → f ∈ x.asIdeal } :=
   by
   ext f
-  rw [vanishing_ideal, SetLike.mem_coe, Submodule.mem_infᵢ]
+  rw [vanishing_ideal, SetLike.mem_coe, Submodule.mem_iInf]
   apply forall_congr'; intro x
-  rw [Submodule.mem_infᵢ]
+  rw [Submodule.mem_iInf]
 #align prime_spectrum.coe_vanishing_ideal PrimeSpectrum.coe_vanishingIdeal
 
 theorem mem_vanishingIdeal (t : Set (PrimeSpectrum R)) (f : R) :
@@ -230,7 +230,7 @@ theorem vanishingIdeal_zeroLocus_eq_radical (I : Ideal R) :
     vanishingIdeal (zeroLocus (I : Set R)) = I.radical :=
   Ideal.ext fun f =>
     by
-    rw [mem_vanishing_ideal, Ideal.radical_eq_infₛ, Submodule.mem_infₛ]
+    rw [mem_vanishing_ideal, Ideal.radical_eq_sInf, Submodule.mem_sInf]
     exact ⟨fun h x hx => h ⟨x, hx.2⟩ hx.1, fun h x hx => h x.1 ⟨hx, x.2⟩⟩
 #align prime_spectrum.vanishing_ideal_zero_locus_eq_radical PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radical
 
@@ -345,24 +345,24 @@ theorem vanishingIdeal_union (t t' : Set (PrimeSpectrum R)) :
   (gc R).u_inf
 #align prime_spectrum.vanishing_ideal_union PrimeSpectrum.vanishingIdeal_union
 
-theorem zeroLocus_supᵢ {ι : Sort _} (I : ι → Ideal R) :
+theorem zeroLocus_iSup {ι : Sort _} (I : ι → Ideal R) :
     zeroLocus ((⨆ i, I i : Ideal R) : Set R) = ⋂ i, zeroLocus (I i) :=
-  (gc R).l_supᵢ
-#align prime_spectrum.zero_locus_supr PrimeSpectrum.zeroLocus_supᵢ
+  (gc R).l_iSup
+#align prime_spectrum.zero_locus_supr PrimeSpectrum.zeroLocus_iSup
 
-theorem zeroLocus_unionᵢ {ι : Sort _} (s : ι → Set R) :
+theorem zeroLocus_iUnion {ι : Sort _} (s : ι → Set R) :
     zeroLocus (⋃ i, s i) = ⋂ i, zeroLocus (s i) :=
-  (gc_set R).l_supᵢ
-#align prime_spectrum.zero_locus_Union PrimeSpectrum.zeroLocus_unionᵢ
+  (gc_set R).l_iSup
+#align prime_spectrum.zero_locus_Union PrimeSpectrum.zeroLocus_iUnion
 
 theorem zeroLocus_bUnion (s : Set (Set R)) :
     zeroLocus (⋃ s' ∈ s, s' : Set R) = ⋂ s' ∈ s, zeroLocus s' := by simp only [zero_locus_Union]
 #align prime_spectrum.zero_locus_bUnion PrimeSpectrum.zeroLocus_bUnion
 
-theorem vanishingIdeal_unionᵢ {ι : Sort _} (t : ι → Set (PrimeSpectrum R)) :
+theorem vanishingIdeal_iUnion {ι : Sort _} (t : ι → Set (PrimeSpectrum R)) :
     vanishingIdeal (⋃ i, t i) = ⨅ i, vanishingIdeal (t i) :=
-  (gc R).u_infᵢ
-#align prime_spectrum.vanishing_ideal_Union PrimeSpectrum.vanishingIdeal_unionᵢ
+  (gc R).u_iInf
+#align prime_spectrum.vanishing_ideal_Union PrimeSpectrum.vanishingIdeal_iUnion
 
 theorem zeroLocus_inf (I J : Ideal R) :
     zeroLocus ((I ⊓ J : Ideal R) : Set R) = zeroLocus I ∪ zeroLocus J :=
@@ -419,7 +419,7 @@ instance zariskiTopology : TopologicalSpace (PrimeSpectrum R) :=
   TopologicalSpace.ofClosed (Set.range PrimeSpectrum.zeroLocus) ⟨Set.univ, by simp⟩
     (by
       intro Zs h
-      rw [Set.interₛ_eq_interᵢ]
+      rw [Set.sInter_eq_iInter]
       choose f hf using fun i : Zs => h i.Prop
       simp only [← hf]
       exact ⟨_, zero_locus_Union _⟩)
@@ -881,7 +881,7 @@ theorem isCompact_basicOpen (f : R) : IsCompact (basicOpen f : Set (PrimeSpectru
       rw [← vanishing_ideal_zero_locus_eq_radical]
       apply vanishing_ideal_anti_mono hZ
       exact subset_vanishing_ideal_zero_locus {f} (Set.mem_singleton f)
-    rcases Submodule.exists_finset_of_mem_supᵢ I hn with ⟨s, hs⟩
+    rcases Submodule.exists_finset_of_mem_iSup I hn with ⟨s, hs⟩
     use s
     -- Using simp_rw here, because `hI` and `zero_locus_supr` need to be applied underneath binders
     simp_rw [basic_open_eq_zero_locus_compl f, Set.inter_comm (zero_locus {f}ᶜ), ← Set.diff_eq,

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

@@ -508,7 +508,7 @@ theorem vanishingIdeal_anti_mono_iff {s t : Set (PrimeSpectrum R)} (ht : IsClose
   ⟨vanishingIdeal_anti_mono, fun h =>
     by
     rw [← ht.closure_subset_iff, ← ht.closure_eq]
-    convert ← zero_locus_anti_mono_ideal h <;> apply zero_locus_vanishing_ideal_eq_closure⟩
+    convert← zero_locus_anti_mono_ideal h <;> apply zero_locus_vanishing_ideal_eq_closure⟩
 #align prime_spectrum.vanishing_ideal_anti_mono_iff PrimeSpectrum.vanishingIdeal_anti_mono_iff
 
 theorem vanishingIdeal_strict_anti_mono_iff {s t : Set (PrimeSpectrum R)} (hs : IsClosed s)

mathlib commit https://github.com/leanprover-community/mathlib/commit/21e3562c5e12d846c7def5eff8cdbc520d7d4936

@@ -520,7 +520,7 @@ theorem vanishingIdeal_strict_anti_mono_iff {s t : Set (PrimeSpectrum R)} (hs :
 /-- The antitone order embedding of closed subsets of `Spec R` into ideals of `R`. -/
 def closedsEmbedding (R : Type _) [CommRing R] :
     (TopologicalSpace.Closeds <| PrimeSpectrum R)ᵒᵈ ↪o Ideal R :=
-  OrderEmbedding.ofMapLeIff (fun s => vanishingIdeal s.ofDual) fun s t =>
+  OrderEmbedding.ofMapLEIff (fun s => vanishingIdeal s.ofDual) fun s t =>
     (vanishingIdeal_anti_mono_iff s.2).symm
 #align prime_spectrum.closeds_embedding PrimeSpectrum.closedsEmbedding
 
@@ -968,7 +968,7 @@ theorem le_iff_specializes (x y : PrimeSpectrum R) : x ≤ y ↔ x ⤳ y :=
 /-- `nhds` as an order embedding. -/
 @[simps (config := { fullyApplied := true })]
 def nhdsOrderEmbedding : PrimeSpectrum R ↪o Filter (PrimeSpectrum R) :=
-  OrderEmbedding.ofMapLeIff nhds fun a b => (le_iff_specializes a b).symm
+  OrderEmbedding.ofMapLEIff nhds fun a b => (le_iff_specializes a b).symm
 #align prime_spectrum.nhds_order_embedding PrimeSpectrum.nhdsOrderEmbedding
 
 instance : T0Space (PrimeSpectrum R) :=

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

Mathlib.RingTheory.Nilpotent has a few very simple definitions (Mathlib.Data.Nat.Lattice is sufficient to state them), but needs some pretty heavy imports (ideals, linear algebra) towards the end. This change moves the heavier parts into a new file.

@@ -8,7 +8,7 @@ import Mathlib.RingTheory.Ideal.Over
 import Mathlib.RingTheory.Ideal.Prod
 import Mathlib.RingTheory.Ideal.MinimalPrime
 import Mathlib.RingTheory.Localization.Away.Basic
-import Mathlib.RingTheory.Nilpotent
+import Mathlib.RingTheory.Nilpotent.Lemmas
 import Mathlib.Topology.Sets.Closeds
 import Mathlib.Topology.Sober
 

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

@@ -398,8 +398,9 @@ theorem mem_compl_zeroLocus_iff_not_mem {f : R} {I : PrimeSpectrum R} :
   rw [Set.mem_compl_iff, mem_zeroLocus, Set.singleton_subset_iff]; rfl
 #align prime_spectrum.mem_compl_zero_locus_iff_not_mem PrimeSpectrum.mem_compl_zeroLocus_iff_not_mem
 
-/-- The Zariski topology on the prime spectrum of a commutative (semi)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.-/
+/-- The Zariski topology on the prime spectrum of a commutative (semi)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 (PrimeSpectrum R) :=
   TopologicalSpace.ofClosed (Set.range PrimeSpectrum.zeroLocus) ⟨Set.univ, by simp⟩
     (by

@@ -516,7 +516,7 @@ theorem isIrreducible_zeroLocus_iff_of_radical (I : Ideal R) (hI : I.IsRadical)
     IsIrreducible (zeroLocus (I : Set R)) ↔ I.IsPrime := by
   rw [Ideal.isPrime_iff, IsIrreducible]
   apply and_congr
-  · rw [Set.nonempty_iff_ne_empty, Ne.def, zeroLocus_empty_iff_eq_top]
+  · rw [Set.nonempty_iff_ne_empty, Ne, zeroLocus_empty_iff_eq_top]
   · trans ∀ x y : Ideal R, Z(I) ⊆ Z(x) ∪ Z(y) → Z(I) ⊆ Z(x) ∨ Z(I) ⊆ Z(y)
     · simp_rw [isPreirreducible_iff_closed_union_closed, isClosed_iff_zeroLocus_ideal]
       constructor

All these lemmas refer to the range of some function being open/range (i.e. isOpen or isClosed).

@@ -760,7 +760,7 @@ theorem isClosed_range_comap_of_surjective (hf : Surjective f) :
 theorem closedEmbedding_comap_of_surjective (hf : Surjective f) : ClosedEmbedding (comap f) :=
   { induced := (comap_inducing_of_surjective S f hf).induced
     inj := comap_injective_of_surjective f hf
-    closed_range := isClosed_range_comap_of_surjective S f hf }
+    isClosed_range := isClosed_range_comap_of_surjective S f hf }
 #align prime_spectrum.closed_embedding_comap_of_surjective PrimeSpectrum.closedEmbedding_comap_of_surjective
 
 end SpecOfSurjective
@@ -874,7 +874,7 @@ theorem localization_away_comap_range (S : Type v) [CommSemiring S] [Algebra R S
 theorem localization_away_openEmbedding (S : Type v) [CommSemiring S] [Algebra R S] (r : R)
     [IsLocalization.Away r S] : OpenEmbedding (comap (algebraMap R S)) :=
   { toEmbedding := localization_comap_embedding S (Submonoid.powers r)
-    open_range := by
+    isOpen_range := by
       rw [localization_away_comap_range S r]
       exact isOpen_basicOpen }
 #align prime_spectrum.localization_away_open_embedding PrimeSpectrum.localization_away_openEmbedding

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)

@@ -71,7 +71,6 @@ namespace PrimeSpectrum
 section CommSemiRing
 
 variable [CommSemiring R] [CommSemiring S]
-
 variable {R S}
 
 instance [Nontrivial R] : Nonempty <| PrimeSpectrum R :=

Show that localization at minimal primes results in rings with only a single prime ideal, implying that every non-unit element is nilpotent.

Co-authored-by: Junyan Xu <junyanxumath@gmail.com>

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com> Co-authored-by: uniwuni <95649083+uniwuni@users.noreply.github.com>

@@ -963,6 +963,22 @@ protected def pointsEquivIrreducibleCloseds :
   map_rel_iff' {p q} :=
     (RelIso.symm irreducibleSetEquivPoints).map_rel_iff.trans (le_iff_specializes p q).symm
 
+section LocalizationAtMinimal
+
+variable {I : Ideal R} [hI : I.IsPrime]
+
+/--
+Localizations at minimal primes have single-point prime spectra.
+-/
+def primeSpectrum_unique_of_localization_at_minimal (h : I ∈ minimalPrimes R) :
+    Unique (PrimeSpectrum (Localization.AtPrime I)) where
+  default :=
+    ⟨LocalRing.maximalIdeal (Localization I.primeCompl),
+    (LocalRing.maximalIdeal.isMaximal _).isPrime⟩
+  uniq x := PrimeSpectrum.ext _ _ (Localization.AtPrime.prime_unique_of_minimal h x.asIdeal)
+
+end LocalizationAtMinimal
+
 end CommSemiRing
 
 end PrimeSpectrum

We remove all but one open Classicals, instead preferring to use open scoped Classical. The only real side-effect this led to is moving a couple declarations to use Exists.choose instead of Classical.choose.

The first few commits are explicitly labelled regex replaces for ease of review.

@@ -47,7 +47,7 @@ and Chris Hughes (on an earlier repository).
 
 noncomputable section
 
-open Classical
+open scoped Classical
 
 universe u v
 

This is a very large PR, but it has been reviewed piecemeal already in PRs to the bump/v4.7.0 branch as we update to intermediate nightlies.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: damiano <adomani@gmail.com>

@@ -682,6 +682,7 @@ open Function RingHom
 
 theorem comap_inducing_of_surjective (hf : Surjective f) : Inducing (comap f) where
   induced := by
+    set_option tactic.skipAssignedInstances false in
     simp_rw [TopologicalSpace.ext_iff, ← isClosed_compl_iff,
       ← @isClosed_compl_iff (PrimeSpectrum S)
         ((TopologicalSpace.induced (comap f) zariskiTopology)), isClosed_induced_iff,

Also make n implicit and replace 0 < n with n ≠ 0

@@ -374,9 +374,9 @@ theorem zeroLocus_singleton_mul (f g : R) :
 #align prime_spectrum.zero_locus_singleton_mul PrimeSpectrum.zeroLocus_singleton_mul
 
 @[simp]
-theorem zeroLocus_pow (I : Ideal R) {n : ℕ} (hn : 0 < n) :
+theorem zeroLocus_pow (I : Ideal R) {n : ℕ} (hn : n ≠ 0) :
     zeroLocus ((I ^ n : Ideal R) : Set R) = zeroLocus I :=
-  zeroLocus_radical (I ^ n) ▸ (I.radical_pow n hn).symm ▸ zeroLocus_radical I
+  zeroLocus_radical (I ^ n) ▸ (I.radical_pow hn).symm ▸ zeroLocus_radical I
 #align prime_spectrum.zero_locus_pow PrimeSpectrum.zeroLocus_pow
 
 @[simp]

In this pull request, I have systematically eliminated the leading whitespace preceding the colon (:) within all unlabelled or unclassified porting notes. This adjustment facilitates a more efficient review process for the remaining notes by ensuring no entries are overlooked due to formatting inconsistencies.

@@ -1032,7 +1032,7 @@ variable {R}
 
 theorem isLocalRingHom_iff_comap_closedPoint {S : Type v} [CommSemiring S] [LocalRing S]
     (f : R →+* S) : IsLocalRingHom f ↔ PrimeSpectrum.comap f (closedPoint S) = closedPoint R := by
-  -- Porting note : inline `this` does **not** work
+  -- Porting note: inline `this` does **not** work
   have := (local_hom_TFAE f).out 0 4
   rw [this, PrimeSpectrum.ext_iff]
   rfl

Use it to golf PrimeSpectrum.localization_comap_inducing.

@@ -639,21 +639,14 @@ variable (S)
 
 theorem localization_comap_inducing [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
     Inducing (comap (algebraMap R S)) := by
+  refine ⟨TopologicalSpace.ext_isClosed fun Z ↦ ?_⟩
+  simp_rw [isClosed_induced_iff, isClosed_iff_zeroLocus, @eq_comm _ _ (zeroLocus _),
+    exists_exists_eq_and, preimage_comap_zeroLocus]
   constructor
-  rw [TopologicalSpace.ext_iff]
-  intro U
-  rw [← isClosed_compl_iff, ← @isClosed_compl_iff (X := PrimeSpectrum S) (s := U)]
-  generalize Uᶜ = Z
-  simp_rw [isClosed_induced_iff, isClosed_iff_zeroLocus]
-  constructor
   · rintro ⟨s, rfl⟩
-    refine ⟨_, ⟨algebraMap R S ⁻¹' Ideal.span s, rfl⟩, ?_⟩
-    rw [preimage_comap_zeroLocus, ← zeroLocus_span, ← zeroLocus_span s]
-    congr 2
-    exact congr_arg (zeroLocus ·) <| Submodule.carrier_inj.mpr
-      (IsLocalization.map_comap M S (Ideal.span s))
-  · rintro ⟨_, ⟨t, rfl⟩, rfl⟩
-    rw [preimage_comap_zeroLocus]
+    refine ⟨(Ideal.span s).comap (algebraMap R S), ?_⟩
+    rw [← zeroLocus_span, ← zeroLocus_span s, ← Ideal.map, IsLocalization.map_comap M S]
+  · rintro ⟨s, rfl⟩
     exact ⟨_, rfl⟩
 #align prime_spectrum.localization_comap_inducing PrimeSpectrum.localization_comap_inducing
 

Use X, Y, Z for topological spaces.

@@ -642,7 +642,7 @@ theorem localization_comap_inducing [Algebra R S] (M : Submonoid R) [IsLocalizat
   constructor
   rw [TopologicalSpace.ext_iff]
   intro U
-  rw [← isClosed_compl_iff, ← @isClosed_compl_iff (α := PrimeSpectrum S) (s := U)]
+  rw [← isClosed_compl_iff, ← @isClosed_compl_iff (X := PrimeSpectrum S) (s := U)]
   generalize Uᶜ = Z
   simp_rw [isClosed_induced_iff, isClosed_iff_zeroLocus]
   constructor

Co-authored-by: Xavier Xarles <56635243+XavierXarles@users.noreply.github.com> Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -973,8 +973,8 @@ end CommSemiRing
 
 end PrimeSpectrum
 
-section CommRing
-variable [CommRing R]
+section CommSemiring
+variable [CommSemiring R]
 
 open PrimeSpectrum in
 /--
@@ -1024,7 +1024,7 @@ lemma zeroLocus_ideal_mem_irreducibleComponents {I : Ideal R} :
 
 end PrimeSpectrum
 
-end CommRing
+end CommSemiring
 
 namespace LocalRing
 

Co-authored-by: Jujian Zhang <jujian.zhang1998@outlook.com>

@@ -549,13 +549,13 @@ theorem isIrreducible_iff_vanishingIdeal_isPrime {s : Set (PrimeSpectrum R)} :
     isIrreducible_zeroLocus_iff_of_radical _ (isRadical_vanishingIdeal s)]
 #align prime_spectrum.is_irreducible_iff_vanishing_ideal_is_prime PrimeSpectrum.isIrreducible_iff_vanishingIdeal_isPrime
 
-lemma vanishingIdeal_isIrreducible {R} [CommRing R] :
+lemma vanishingIdeal_isIrreducible :
     vanishingIdeal (R := R) '' {s | IsIrreducible s} = {P | P.IsPrime} :=
   Set.ext fun I ↦ ⟨fun ⟨_, hs, e⟩ ↦ e ▸ isIrreducible_iff_vanishingIdeal_isPrime.mp hs,
     fun h ↦ ⟨zeroLocus I, (isIrreducible_zeroLocus_iff_of_radical _ h.isRadical).mpr h,
       (vanishingIdeal_zeroLocus_eq_radical I).trans h.radical⟩⟩
 
-lemma vanishingIdeal_isClosed_isIrreducible {R} [CommRing R] :
+lemma vanishingIdeal_isClosed_isIrreducible :
     vanishingIdeal (R := R) '' {s | IsClosed s ∧ IsIrreducible s} = {P | P.IsPrime} := by
   refine (subset_antisymm ?_ ?_).trans vanishingIdeal_isIrreducible
   · exact Set.image_subset _ fun _ ↦ And.right

Some results of PrimeSpectrum generalized from CommRing to CommSemiring.

Co-authored-by: Xavier Xarles <56635243+XavierXarles@users.noreply.github.com>

@@ -15,9 +15,9 @@ import Mathlib.Topology.Sober
 #align_import algebraic_geometry.prime_spectrum.basic from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
 
 /-!
-# Prime spectrum of a commutative ring
+# Prime spectrum of a commutative (semi)ring
 
-The prime spectrum of a commutative ring is the type of all prime ideals.
+The prime spectrum of a commutative (semi)ring is the type of all prime ideals.
 It is naturally endowed with a topology: the Zariski topology.
 
 (It is also naturally endowed with a sheaf of rings,
@@ -25,7 +25,7 @@ which is constructed in `AlgebraicGeometry.StructureSheaf`.)
 
 ## Main definitions
 
-* `PrimeSpectrum R`: The prime spectrum of a commutative ring `R`,
+* `PrimeSpectrum R`: The prime spectrum of a commutative (semi)ring `R`,
   i.e., the set of all prime ideals of `R`.
 * `zeroLocus s`: The zero locus of a subset `s` of `R`
   is the subset of `PrimeSpectrum R` consisting of all prime ideals that contain `s`.
@@ -34,7 +34,7 @@ which is constructed in `AlgebraicGeometry.StructureSheaf`.)
 
 ## Conventions
 
-We denote subsets of rings with `s`, `s'`, etc...
+We denote subsets of (semi)rings with `s`, `s'`, etc...
 whereas we denote subsets of prime spectra with `t`, `t'`, etc...
 
 ## Inspiration/contributors
@@ -51,15 +51,15 @@ open Classical
 
 universe u v
 
-variable (R : Type u) (S : Type v) [CommRing R] [CommRing S]
+variable (R : Type u) (S : Type v)
 
-/-- The prime spectrum of a commutative ring `R` is the type of all prime ideals of `R`.
+/-- The prime spectrum of a commutative (semi)ring `R` is the type of all prime ideals of `R`.
 
 It is naturally endowed with a topology (the Zariski topology),
 and a sheaf of commutative rings (see `AlgebraicGeometry.StructureSheaf`).
 It is a fundamental building block in algebraic geometry. -/
 @[ext]
-structure PrimeSpectrum where
+structure PrimeSpectrum [CommSemiring R] where
   asIdeal : Ideal R
   IsPrime : asIdeal.IsPrime
 #align prime_spectrum PrimeSpectrum
@@ -68,6 +68,10 @@ attribute [instance] PrimeSpectrum.IsPrime
 
 namespace PrimeSpectrum
 
+section CommSemiRing
+
+variable [CommSemiring R] [CommSemiring S]
+
 variable {R S}
 
 instance [Nontrivial R] : Nonempty <| PrimeSpectrum R :=
@@ -123,8 +127,8 @@ theorem primeSpectrumProd_symm_inr_asIdeal (x : PrimeSpectrum S) :
   rfl
 #align prime_spectrum.prime_spectrum_prod_symm_inr_as_ideal PrimeSpectrum.primeSpectrumProd_symm_inr_asIdeal
 
-/-- The zero locus of a set `s` of elements of a commutative ring `R` is the set of all prime ideals
-of the ring that contain the set `s`.
+/-- The zero locus of a set `s` of elements of a commutative (semi)ring `R` is the set of all
+prime ideals of the ring that contain the set `s`.
 
 An element `f` of `R` can be thought of as a dependent function on the prime spectrum of `R`.
 At a point `x` (a prime ideal) the function (i.e., element) `f` takes values in the quotient ring
@@ -395,8 +399,8 @@ theorem mem_compl_zeroLocus_iff_not_mem {f : R} {I : PrimeSpectrum R} :
   rw [Set.mem_compl_iff, mem_zeroLocus, Set.singleton_subset_iff]; rfl
 #align prime_spectrum.mem_compl_zero_locus_iff_not_mem PrimeSpectrum.mem_compl_zeroLocus_iff_not_mem
 
-/-- 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. -/
+/-- The Zariski topology on the prime spectrum of a commutative (semi)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 (PrimeSpectrum R) :=
   TopologicalSpace.ofClosed (Set.range PrimeSpectrum.zeroLocus) ⟨Set.univ, by simp⟩
     (by
@@ -484,7 +488,7 @@ theorem vanishingIdeal_strict_anti_mono_iff {s t : Set (PrimeSpectrum R)} (hs :
 #align prime_spectrum.vanishing_ideal_strict_anti_mono_iff PrimeSpectrum.vanishingIdeal_strict_anti_mono_iff
 
 /-- The antitone order embedding of closed subsets of `Spec R` into ideals of `R`. -/
-def closedsEmbedding (R : Type*) [CommRing R] :
+def closedsEmbedding (R : Type*) [CommSemiring R] :
     (TopologicalSpace.Closeds <| PrimeSpectrum R)ᵒᵈ ↪o Ideal R :=
   OrderEmbedding.ofMapLEIff (fun s => vanishingIdeal ↑(OrderDual.ofDual s)) fun s _ =>
     (vanishingIdeal_anti_mono_iff s.2).symm
@@ -502,7 +506,7 @@ theorem t1Space_iff_isField [IsDomain R] : T1Space (PrimeSpectrum R) ↔ IsField
           (by aesop))
   · refine' ⟨fun x => (isClosed_singleton_iff_isMaximal x).2 _⟩
     by_cases hx : x.asIdeal = ⊥
-    · letI := h.toField
+    · letI := h.toSemifield
       exact hx.symm ▸ Ideal.bot_isMaximal
     · exact absurd h (Ring.not_isField_iff_exists_prime.2 ⟨x.asIdeal, ⟨hx, x.2⟩⟩)
 #align prime_spectrum.t1_space_iff_is_field PrimeSpectrum.t1Space_iff_isField
@@ -567,7 +571,7 @@ instance quasiSober : QuasiSober (PrimeSpectrum R) :=
     ⟨⟨_, isIrreducible_iff_vanishingIdeal_isPrime.1 h₁⟩, by
       rw [IsGenericPoint, closure_singleton, zeroLocus_vanishingIdeal_eq_closure, h₂.closure_eq]⟩⟩
 
-/-- The prime spectrum of a commutative ring is a compact topological space. -/
+/-- The prime spectrum of a commutative (semi)ring is a compact topological space. -/
 instance compactSpace : CompactSpace (PrimeSpectrum R) := by
   refine compactSpace_of_finite_subfamily_closed fun S S_closed S_empty ↦ ?_
   choose I hI using fun i ↦ (isClosed_iff_zeroLocus_ideal (S i)).mp (S_closed i)
@@ -576,7 +580,7 @@ instance compactSpace : CompactSpace (PrimeSpectrum R) := by
 
 section Comap
 
-variable {S' : Type*} [CommRing S']
+variable {S' : Type*} [CommSemiring S']
 
 theorem preimage_comap_zeroLocus_aux (f : R →+* S) (s : Set R) :
     (fun y => ⟨Ideal.comap f y.asIdeal, inferInstance⟩ : PrimeSpectrum S → PrimeSpectrum R) ⁻¹'
@@ -586,7 +590,7 @@ theorem preimage_comap_zeroLocus_aux (f : R →+* S) (s : Set R) :
   simp only [mem_zeroLocus, Set.image_subset_iff, Set.mem_preimage, mem_zeroLocus, Ideal.coe_comap]
 #align prime_spectrum.preimage_comap_zero_locus_aux PrimeSpectrum.preimage_comap_zeroLocus_aux
 
-/-- The function between prime spectra of commutative rings induced by a ring homomorphism.
+/-- The function between prime spectra of commutative (semi)rings induced by a ring homomorphism.
 This function is continuous. -/
 def comap (f : R →+* S) : C(PrimeSpectrum S, PrimeSpectrum R) where
   toFun y := ⟨Ideal.comap f y.asIdeal, inferInstance⟩
@@ -631,21 +635,6 @@ theorem comap_injective_of_surjective (f : R →+* S) (hf : Function.Surjective
       (congr_arg PrimeSpectrum.asIdeal h : (comap f x).asIdeal = (comap f y).asIdeal))
 #align prime_spectrum.comap_injective_of_surjective PrimeSpectrum.comap_injective_of_surjective
 
-theorem comap_singleton_isClosed_of_surjective (f : R →+* S) (hf : Function.Surjective f)
-    (x : PrimeSpectrum S) (hx : IsClosed ({x} : Set (PrimeSpectrum S))) :
-    IsClosed ({comap f x} : Set (PrimeSpectrum R)) :=
-  haveI : x.asIdeal.IsMaximal := (isClosed_singleton_iff_isMaximal x).1 hx
-  (isClosed_singleton_iff_isMaximal _).2 (Ideal.comap_isMaximal_of_surjective f hf)
-#align prime_spectrum.comap_singleton_is_closed_of_surjective PrimeSpectrum.comap_singleton_isClosed_of_surjective
-
-theorem comap_singleton_isClosed_of_isIntegral (f : R →+* S) (hf : f.IsIntegral)
-    (x : PrimeSpectrum S) (hx : IsClosed ({x} : Set (PrimeSpectrum S))) :
-    IsClosed ({comap f x} : Set (PrimeSpectrum R)) :=
-  (isClosed_singleton_iff_isMaximal _).2
-    (Ideal.isMaximal_comap_of_isIntegral_of_isMaximal' f hf x.asIdeal <|
-      (isClosed_singleton_iff_isMaximal x).1 hx)
-#align prime_spectrum.comap_singleton_is_closed_of_is_integral PrimeSpectrum.comap_singleton_isClosed_of_isIntegral
-
 variable (S)
 
 theorem localization_comap_inducing [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
@@ -696,11 +685,6 @@ theorem localization_comap_range [Algebra R S] (M : Submonoid R) [IsLocalization
     exact IsLocalization.comap_map_of_isPrime_disjoint M S _ x.2 h
 #align prime_spectrum.localization_comap_range PrimeSpectrum.localization_comap_range
 
-section SpecOfSurjective
-
-/-! The comap of a surjective ring homomorphism is a closed embedding between the prime spectra. -/
-
-
 open Function RingHom
 
 theorem comap_inducing_of_surjective (hf : Surjective f) : Inducing (comap f) where
@@ -718,6 +702,36 @@ theorem comap_inducing_of_surjective (hf : Surjective f) : Inducing (comap f) wh
     exact ⟨f '' F, hF.symm.trans (preimage_comap_zeroLocus f F)⟩
 #align prime_spectrum.comap_inducing_of_surjective PrimeSpectrum.comap_inducing_of_surjective
 
+
+end Comap
+end CommSemiRing
+
+section SpecOfSurjective
+
+/-! The comap of a surjective ring homomorphism is a closed embedding between the prime spectra. -/
+
+
+open Function RingHom
+
+variable [CommRing R] [CommRing S]
+variable (f : R →+* S)
+variable {R}
+
+theorem comap_singleton_isClosed_of_surjective (f : R →+* S) (hf : Function.Surjective f)
+    (x : PrimeSpectrum S) (hx : IsClosed ({x} : Set (PrimeSpectrum S))) :
+    IsClosed ({comap f x} : Set (PrimeSpectrum R)) :=
+  haveI : x.asIdeal.IsMaximal := (isClosed_singleton_iff_isMaximal x).1 hx
+  (isClosed_singleton_iff_isMaximal _).2 (Ideal.comap_isMaximal_of_surjective f hf)
+#align prime_spectrum.comap_singleton_is_closed_of_surjective PrimeSpectrum.comap_singleton_isClosed_of_surjective
+
+theorem comap_singleton_isClosed_of_isIntegral (f : R →+* S) (hf : f.IsIntegral)
+    (x : PrimeSpectrum S) (hx : IsClosed ({x} : Set (PrimeSpectrum S))) :
+    IsClosed ({comap f x} : Set (PrimeSpectrum R)) :=
+  (isClosed_singleton_iff_isMaximal _).2
+    (Ideal.isMaximal_comap_of_isIntegral_of_isMaximal' f hf x.asIdeal <|
+      (isClosed_singleton_iff_isMaximal x).1 hx)
+#align prime_spectrum.comap_singleton_is_closed_of_is_integral PrimeSpectrum.comap_singleton_isClosed_of_isIntegral
+
 theorem image_comap_zeroLocus_eq_zeroLocus_comap (hf : Surjective f) (I : Ideal S) :
     comap f '' zeroLocus I = zeroLocus (I.comap f) := by
   simp only [Set.ext_iff, Set.mem_image, mem_zeroLocus, SetLike.coe_subset_coe]
@@ -758,7 +772,10 @@ theorem closedEmbedding_comap_of_surjective (hf : Surjective f) : ClosedEmbeddin
 
 end SpecOfSurjective
 
-end Comap
+section CommSemiRing
+
+variable [CommSemiring R] [CommSemiring S]
+variable {R S}
 
 section BasicOpen
 
@@ -848,7 +865,7 @@ theorem basicOpen_eq_bot_iff (f : R) : basicOpen f = ⊥ ↔ IsNilpotent f := by
   exact ⟨fun h I hI => h ⟨I, hI⟩, fun h ⟨I, hI⟩ => h I hI⟩
 #align prime_spectrum.basic_open_eq_bot_iff PrimeSpectrum.basicOpen_eq_bot_iff
 
-theorem localization_away_comap_range (S : Type v) [CommRing S] [Algebra R S] (r : R)
+theorem localization_away_comap_range (S : Type v) [CommSemiring S] [Algebra R S] (r : R)
     [IsLocalization.Away r S] : Set.range (comap (algebraMap R S)) = basicOpen r := by
   rw [localization_comap_range S (Submonoid.powers r)]
   ext x
@@ -861,7 +878,7 @@ theorem localization_away_comap_range (S : Type v) [CommRing S] [Algebra R S] (r
     exact h₁ (x.2.mem_of_pow_mem _ h₃)
 #align prime_spectrum.localization_away_comap_range PrimeSpectrum.localization_away_comap_range
 
-theorem localization_away_openEmbedding (S : Type v) [CommRing S] [Algebra R S] (r : R)
+theorem localization_away_openEmbedding (S : Type v) [CommSemiring S] [Algebra R S] (r : R)
     [IsLocalization.Away r S] : OpenEmbedding (comap (algebraMap R S)) :=
   { toEmbedding := localization_comap_embedding S (Submonoid.powers r)
     open_range := by
@@ -952,8 +969,13 @@ protected def pointsEquivIrreducibleCloseds :
   map_rel_iff' {p q} :=
     (RelIso.symm irreducibleSetEquivPoints).map_rel_iff.trans (le_iff_specializes p q).symm
 
+end CommSemiRing
+
 end PrimeSpectrum
 
+section CommRing
+variable [CommRing R]
+
 open PrimeSpectrum in
 /--
 [Stacks: Lemma 00ES (3)](https://stacks.math.columbia.edu/tag/00ES)
@@ -1002,9 +1024,11 @@ lemma zeroLocus_ideal_mem_irreducibleComponents {I : Ideal R} :
 
 end PrimeSpectrum
 
+end CommRing
+
 namespace LocalRing
 
-variable [LocalRing R]
+variable [CommSemiring R] [LocalRing R]
 
 /-- The closed point in the prime spectrum of a local ring. -/
 def closedPoint : PrimeSpectrum R :=
@@ -1013,8 +1037,8 @@ def closedPoint : PrimeSpectrum R :=
 
 variable {R}
 
-theorem isLocalRingHom_iff_comap_closedPoint {S : Type v} [CommRing S] [LocalRing S] (f : R →+* S) :
-    IsLocalRingHom f ↔ PrimeSpectrum.comap f (closedPoint S) = closedPoint R := by
+theorem isLocalRingHom_iff_comap_closedPoint {S : Type v} [CommSemiring S] [LocalRing S]
+    (f : R →+* S) : IsLocalRingHom f ↔ PrimeSpectrum.comap f (closedPoint S) = closedPoint R := by
   -- Porting note : inline `this` does **not** work
   have := (local_hom_TFAE f).out 0 4
   rw [this, PrimeSpectrum.ext_iff]
@@ -1022,8 +1046,8 @@ theorem isLocalRingHom_iff_comap_closedPoint {S : Type v} [CommRing S] [LocalRin
 #align local_ring.is_local_ring_hom_iff_comap_closed_point LocalRing.isLocalRingHom_iff_comap_closedPoint
 
 @[simp]
-theorem comap_closedPoint {S : Type v} [CommRing S] [LocalRing S] (f : R →+* S) [IsLocalRingHom f] :
-    PrimeSpectrum.comap f (closedPoint S) = closedPoint R :=
+theorem comap_closedPoint {S : Type v} [CommSemiring S] [LocalRing S] (f : R →+* S)
+    [IsLocalRingHom f] : PrimeSpectrum.comap f (closedPoint S) = closedPoint R :=
   (isLocalRingHom_iff_comap_closedPoint f).mp inferInstance
 #align local_ring.comap_closed_point LocalRing.comap_closedPoint
 
@@ -1041,8 +1065,8 @@ theorem closedPoint_mem_iff (U : TopologicalSpace.Opens <| PrimeSpectrum R) :
 #align local_ring.closed_point_mem_iff LocalRing.closedPoint_mem_iff
 
 @[simp]
-theorem PrimeSpectrum.comap_residue (x : PrimeSpectrum (ResidueField R)) :
-    PrimeSpectrum.comap (residue R) x = closedPoint R := by
+theorem PrimeSpectrum.comap_residue (T : Type u) [CommRing T] [LocalRing T]
+    (x : PrimeSpectrum (ResidueField T)) : PrimeSpectrum.comap (residue T) x = closedPoint T := by
   rw [Subsingleton.elim x ⊥]
   ext1
   exact Ideal.mk_ker

Co-PR : #9087 (maximal ideals of Artinian ring are finite)

Co-authored-by: Andrew Yang <the.erd.one@gmail.com> Co-authored-by: Junyan Xu <junyanxumath@gmail.com>

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -6,6 +6,7 @@ Authors: Johan Commelin
 import Mathlib.LinearAlgebra.Finsupp
 import Mathlib.RingTheory.Ideal.Over
 import Mathlib.RingTheory.Ideal.Prod
+import Mathlib.RingTheory.Ideal.MinimalPrime
 import Mathlib.RingTheory.Localization.Away.Basic
 import Mathlib.RingTheory.Nilpotent
 import Mathlib.Topology.Sets.Closeds
@@ -544,6 +545,19 @@ theorem isIrreducible_iff_vanishingIdeal_isPrime {s : Set (PrimeSpectrum R)} :
     isIrreducible_zeroLocus_iff_of_radical _ (isRadical_vanishingIdeal s)]
 #align prime_spectrum.is_irreducible_iff_vanishing_ideal_is_prime PrimeSpectrum.isIrreducible_iff_vanishingIdeal_isPrime
 
+lemma vanishingIdeal_isIrreducible {R} [CommRing R] :
+    vanishingIdeal (R := R) '' {s | IsIrreducible s} = {P | P.IsPrime} :=
+  Set.ext fun I ↦ ⟨fun ⟨_, hs, e⟩ ↦ e ▸ isIrreducible_iff_vanishingIdeal_isPrime.mp hs,
+    fun h ↦ ⟨zeroLocus I, (isIrreducible_zeroLocus_iff_of_radical _ h.isRadical).mpr h,
+      (vanishingIdeal_zeroLocus_eq_radical I).trans h.radical⟩⟩
+
+lemma vanishingIdeal_isClosed_isIrreducible {R} [CommRing R] :
+    vanishingIdeal (R := R) '' {s | IsClosed s ∧ IsIrreducible s} = {P | P.IsPrime} := by
+  refine (subset_antisymm ?_ ?_).trans vanishingIdeal_isIrreducible
+  · exact Set.image_subset _ fun _ ↦ And.right
+  rintro _ ⟨s, hs, rfl⟩
+  exact ⟨closure s, ⟨isClosed_closure, hs.closure⟩, vanishingIdeal_closure s⟩
+
 instance irreducibleSpace [IsDomain R] : IrreducibleSpace (PrimeSpectrum R) := by
   rw [irreducibleSpace_def, Set.top_eq_univ, ← zeroLocus_bot, isIrreducible_zeroLocus_iff]
   simpa using Ideal.bot_prime
@@ -927,6 +941,65 @@ def localizationMapOfSpecializes {x y : PrimeSpectrum R} (h : x ⤳ y) :
         ⟨a, show a ∈ x.asIdeal.primeCompl from h ha⟩ : _))
 #align prime_spectrum.localization_map_of_specializes PrimeSpectrum.localizationMapOfSpecializes
 
+variable (R) in
+/--
+Zero loci of prime ideals are closed irreducible sets in the Zariski topology and any closed
+irreducible set is a zero locus of some prime ideal.
+-/
+protected def pointsEquivIrreducibleCloseds :
+    PrimeSpectrum R ≃o {s : Set (PrimeSpectrum R) | IsIrreducible s ∧ IsClosed s}ᵒᵈ where
+  __ := irreducibleSetEquivPoints.toEquiv.symm.trans OrderDual.toDual
+  map_rel_iff' {p q} :=
+    (RelIso.symm irreducibleSetEquivPoints).map_rel_iff.trans (le_iff_specializes p q).symm
+
+end PrimeSpectrum
+
+open PrimeSpectrum in
+/--
+[Stacks: Lemma 00ES (3)](https://stacks.math.columbia.edu/tag/00ES)
+Zero loci of minimal prime ideals of `R` are irreducible components in `Spec R` and any
+irreducible component is a zero locus of some minimal prime ideal.
+-/
+protected def minimalPrimes.equivIrreducibleComponents :
+    minimalPrimes R ≃o (irreducibleComponents <| PrimeSpectrum R)ᵒᵈ :=
+  let e : {p : Ideal R | p.IsPrime ∧ ⊥ ≤ p} ≃o PrimeSpectrum R :=
+    ⟨⟨fun x ↦ ⟨x.1, x.2.1⟩, fun x ↦ ⟨x.1, x.2, bot_le⟩, fun _ ↦ rfl, fun _ ↦ rfl⟩, Iff.rfl⟩
+  (e.trans <| PrimeSpectrum.pointsEquivIrreducibleCloseds R).minimalsIsoMaximals.trans
+    (OrderIso.setCongr _ _ <| by simp_rw [irreducibleComponents_eq_maximals_closed, and_comm]).dual
+
+namespace PrimeSpectrum
+
+lemma vanishingIdeal_irreducibleComponents :
+    vanishingIdeal '' (irreducibleComponents <| PrimeSpectrum R) =
+    minimalPrimes R := by
+  rw [irreducibleComponents_eq_maximals_closed, minimalPrimes_eq_minimals, ← minimals_swap,
+    ← PrimeSpectrum.vanishingIdeal_isClosed_isIrreducible, image_minimals_of_rel_iff_rel]
+  exact fun s t hs _ ↦ vanishingIdeal_anti_mono_iff hs.1
+
+lemma zeroLocus_minimalPrimes :
+    zeroLocus ∘ (↑) '' minimalPrimes R =
+    irreducibleComponents (PrimeSpectrum R) := by
+  rw [← vanishingIdeal_irreducibleComponents, ← Set.image_comp, Set.EqOn.image_eq_self]
+  intros s hs
+  simpa [zeroLocus_vanishingIdeal_eq_closure, closure_eq_iff_isClosed]
+    using isClosed_of_mem_irreducibleComponents s hs
+
+variable {R}
+
+lemma vanishingIdeal_mem_minimalPrimes {s : Set (PrimeSpectrum R)} :
+    vanishingIdeal s ∈ minimalPrimes R ↔ closure s ∈ irreducibleComponents (PrimeSpectrum R) := by
+  constructor
+  · rw [← zeroLocus_minimalPrimes, ← zeroLocus_vanishingIdeal_eq_closure]
+    exact Set.mem_image_of_mem _
+  · rw [← vanishingIdeal_irreducibleComponents, ← vanishingIdeal_closure]
+    exact Set.mem_image_of_mem _
+
+lemma zeroLocus_ideal_mem_irreducibleComponents {I : Ideal R} :
+    zeroLocus I ∈ irreducibleComponents (PrimeSpectrum R) ↔ I.radical ∈ minimalPrimes R := by
+  rw [← vanishingIdeal_zeroLocus_eq_radical]
+  conv_lhs => rw [← (isClosed_zeroLocus _).closure_eq]
+  exact vanishingIdeal_mem_minimalPrimes.symm
+
 end PrimeSpectrum
 
 namespace LocalRing

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

@@ -808,7 +808,7 @@ theorem basicOpen_pow (f : R) (n : ℕ) (hn : 0 < n) : basicOpen (f ^ n) = basic
 theorem isTopologicalBasis_basic_opens :
     TopologicalSpace.IsTopologicalBasis
       (Set.range fun r : R => (basicOpen r : Set (PrimeSpectrum R))) := by
-  apply TopologicalSpace.isTopologicalBasis_of_open_of_nhds
+  apply TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds
   · rintro _ ⟨r, rfl⟩
     exact isOpen_basicOpen
   · rintro p U hp ⟨s, hs⟩

This eliminates (fun a ↦ β) α in the type when applying a FunLike.

Co-authored-by: Matthew Ballard <matt@mrb.email> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -485,7 +485,7 @@ theorem vanishingIdeal_strict_anti_mono_iff {s t : Set (PrimeSpectrum R)} (hs :
 /-- The antitone order embedding of closed subsets of `Spec R` into ideals of `R`. -/
 def closedsEmbedding (R : Type*) [CommRing R] :
     (TopologicalSpace.Closeds <| PrimeSpectrum R)ᵒᵈ ↪o Ideal R :=
-  OrderEmbedding.ofMapLEIff (fun s => vanishingIdeal <| OrderDual.ofDual s) fun s _ =>
+  OrderEmbedding.ofMapLEIff (fun s => vanishingIdeal ↑(OrderDual.ofDual s)) fun s _ =>
     (vanishingIdeal_anti_mono_iff s.2).symm
 #align prime_spectrum.closeds_embedding PrimeSpectrum.closedsEmbedding
 

Co-authored-by: Kyle Miller <kmill31415@gmail.com>

@@ -298,7 +298,7 @@ theorem zeroLocus_empty_iff_eq_top {I : Ideal R} : zeroLocus (I : Set R) = ∅ 
   · contrapose!
     intro h
     rcases Ideal.exists_le_maximal I h with ⟨M, hM, hIM⟩
-    exact Set.Nonempty.ne_empty ⟨⟨M, hM.isPrime⟩, hIM⟩
+    exact ⟨⟨M, hM.isPrime⟩, hIM⟩
   · rintro rfl
     apply zeroLocus_empty_of_one_mem
     trivial

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

@@ -101,7 +101,7 @@ noncomputable def primeSpectrumProd :
           · exact False.elim (hJ.ne_top h.right)
           · simp only [h]
         · rintro ⟨I, hI⟩
-          rcases(Ideal.ideal_prod_prime I).mp hI with (⟨p, ⟨hp, rfl⟩⟩ | ⟨p, ⟨hp, rfl⟩⟩)
+          rcases (Ideal.ideal_prod_prime I).mp hI with (⟨p, ⟨hp, rfl⟩⟩ | ⟨p, ⟨hp, rfl⟩⟩)
           · exact ⟨Sum.inl ⟨p, hp⟩, rfl⟩
           · exact ⟨Sum.inr ⟨p, hp⟩, rfl⟩)
 #align prime_spectrum.prime_spectrum_prod PrimeSpectrum.primeSpectrumProd

@@ -74,9 +74,9 @@ instance [Nontrivial R] : Nonempty <| PrimeSpectrum R :=
   ⟨⟨I, hI.isPrime⟩⟩
 
 /-- The prime spectrum of the zero ring is empty. -/
-theorem punit (x : PrimeSpectrum PUnit) : False :=
-  x.1.ne_top_iff_one.1 x.2.1 <| Eq.substr (Subsingleton.elim 1 (0 : PUnit)) x.1.zero_mem
-#align prime_spectrum.punit PrimeSpectrum.punit
+instance [Subsingleton R] : IsEmpty (PrimeSpectrum R) :=
+  ⟨fun x ↦ x.IsPrime.ne_top <| SetLike.ext' <| Subsingleton.eq_univ_of_nonempty x.asIdeal.nonempty⟩
+#noalign prime_spectrum.punit
 
 variable (R S)
 
@@ -248,12 +248,8 @@ theorem vanishingIdeal_anti_mono {s t : Set (PrimeSpectrum R)} (h : s ⊆ t) :
 #align prime_spectrum.vanishing_ideal_anti_mono PrimeSpectrum.vanishingIdeal_anti_mono
 
 theorem zeroLocus_subset_zeroLocus_iff (I J : Ideal R) :
-    zeroLocus (I : Set R) ⊆ zeroLocus (J : Set R) ↔ J ≤ I.radical :=
-  ⟨fun h =>
-    Ideal.radical_le_radical_iff.mp
-      (vanishingIdeal_zeroLocus_eq_radical I ▸
-        vanishingIdeal_zeroLocus_eq_radical J ▸ vanishingIdeal_anti_mono h),
-    fun h => zeroLocus_radical I ▸ zeroLocus_anti_mono_ideal h⟩
+    zeroLocus (I : Set R) ⊆ zeroLocus (J : Set R) ↔ J ≤ I.radical := by
+  rw [subset_zeroLocus_iff_le_vanishingIdeal, vanishingIdeal_zeroLocus_eq_radical]
 #align prime_spectrum.zero_locus_subset_zero_locus_iff PrimeSpectrum.zeroLocus_subset_zeroLocus_iff
 
 theorem zeroLocus_subset_zeroLocus_singleton_iff (f g : R) :
@@ -439,34 +435,13 @@ theorem isClosed_zeroLocus (s : Set R) : IsClosed (zeroLocus s) := by
   exact ⟨s, rfl⟩
 #align prime_spectrum.is_closed_zero_locus PrimeSpectrum.isClosed_zeroLocus
 
-theorem isClosed_singleton_iff_isMaximal (x : PrimeSpectrum R) :
-    IsClosed ({x} : Set (PrimeSpectrum R)) ↔ x.asIdeal.IsMaximal := by
-  refine' (isClosed_iff_zeroLocus _).trans ⟨fun h => _, fun h => _⟩
-  · obtain ⟨s, hs⟩ := h
-    rw [eq_comm, Set.eq_singleton_iff_unique_mem] at hs
-    refine'
-      ⟨⟨x.2.1, fun I hI =>
-          Classical.not_not.1
-            (mt (Ideal.exists_le_maximal I) <| not_exists.2 fun J => not_and.2 fun hJ hIJ => _)⟩⟩
-    exact
-      ne_of_lt (lt_of_lt_of_le hI hIJ)
-        (symm <|
-          congr_arg PrimeSpectrum.asIdeal
-            (hs.2 ⟨J, hJ.isPrime⟩ fun r hr => hIJ (le_of_lt hI <| hs.1 hr)))
-  · refine' ⟨x.asIdeal.1, _⟩
-    rw [eq_comm, Set.eq_singleton_iff_unique_mem]
-    refine' ⟨fun _ h => h, fun y hy => PrimeSpectrum.ext _ _ (h.eq_of_le y.2.ne_top hy).symm⟩
-#align prime_spectrum.is_closed_singleton_iff_is_maximal PrimeSpectrum.isClosed_singleton_iff_isMaximal
-
 theorem zeroLocus_vanishingIdeal_eq_closure (t : Set (PrimeSpectrum R)) :
     zeroLocus (vanishingIdeal t : Set R) = closure t := by
-  apply Set.Subset.antisymm
-  · rintro x hx t' ⟨ht', ht⟩
-    obtain ⟨fs, rfl⟩ : ∃ s, t' = zeroLocus s := by rwa [isClosed_iff_zeroLocus] at ht'
-    rw [subset_zeroLocus_iff_subset_vanishingIdeal] at ht
-    exact Set.Subset.trans ht hx
-  · rw [(isClosed_zeroLocus _).closure_subset_iff]
-    exact subset_zeroLocus_vanishingIdeal t
+  rcases isClosed_iff_zeroLocus (closure t) |>.mp isClosed_closure with ⟨I, hI⟩
+  rw [subset_antisymm_iff, (isClosed_zeroLocus _).closure_subset_iff, hI,
+      subset_zeroLocus_iff_subset_vanishingIdeal, (gc R).u_l_u_eq_u,
+      ← subset_zeroLocus_iff_subset_vanishingIdeal, ← hI]
+  exact ⟨subset_closure, subset_zeroLocus_vanishingIdeal t⟩
 #align prime_spectrum.zero_locus_vanishing_ideal_eq_closure PrimeSpectrum.zeroLocus_vanishingIdeal_eq_closure
 
 theorem vanishingIdeal_closure (t : Set (PrimeSpectrum R)) :
@@ -478,6 +453,16 @@ theorem closure_singleton (x) : closure ({x} : Set (PrimeSpectrum R)) = zeroLocu
   rw [← zeroLocus_vanishingIdeal_eq_closure, vanishingIdeal_singleton]
 #align prime_spectrum.closure_singleton PrimeSpectrum.closure_singleton
 
+theorem isClosed_singleton_iff_isMaximal (x : PrimeSpectrum R) :
+    IsClosed ({x} : Set (PrimeSpectrum R)) ↔ x.asIdeal.IsMaximal := by
+  rw [← closure_subset_iff_isClosed, ← zeroLocus_vanishingIdeal_eq_closure,
+      vanishingIdeal_singleton]
+  constructor <;> intro H
+  · rcases x.asIdeal.exists_le_maximal x.2.1 with ⟨m, hm, hxm⟩
+    exact (congr_arg asIdeal (@H ⟨m, hm.isPrime⟩ hxm)) ▸ hm
+  · exact fun p hp ↦ PrimeSpectrum.ext _ _ (H.eq_of_le p.2.1 hp).symm
+#align prime_spectrum.is_closed_singleton_iff_is_maximal PrimeSpectrum.isClosed_singleton_iff_isMaximal
+
 theorem isRadical_vanishingIdeal (s : Set (PrimeSpectrum R)) : (vanishingIdeal s).IsRadical := by
   rw [← vanishingIdeal_closure, ← zeroLocus_vanishingIdeal_eq_closure,
     vanishingIdeal_zeroLocus_eq_radical]
@@ -568,6 +553,13 @@ instance quasiSober : QuasiSober (PrimeSpectrum R) :=
     ⟨⟨_, isIrreducible_iff_vanishingIdeal_isPrime.1 h₁⟩, by
       rw [IsGenericPoint, closure_singleton, zeroLocus_vanishingIdeal_eq_closure, h₂.closure_eq]⟩⟩
 
+/-- The prime spectrum of a commutative ring is a compact topological space. -/
+instance compactSpace : CompactSpace (PrimeSpectrum R) := by
+  refine compactSpace_of_finite_subfamily_closed fun S S_closed S_empty ↦ ?_
+  choose I hI using fun i ↦ (isClosed_iff_zeroLocus_ideal (S i)).mp (S_closed i)
+  simp_rw [hI, ← zeroLocus_iSup, zeroLocus_empty_iff_eq_top, ← top_le_iff] at S_empty ⊢
+  exact Ideal.isCompactElement_top.exists_finset_of_le_iSup _ _ S_empty
+
 section Comap
 
 variable {S' : Type*} [CommRing S']
@@ -834,33 +826,6 @@ theorem isBasis_basic_opens : TopologicalSpace.Opens.IsBasis (Set.range (@basicO
   rfl
 #align prime_spectrum.is_basis_basic_opens PrimeSpectrum.isBasis_basic_opens
 
-theorem isCompact_basicOpen (f : R) : IsCompact (basicOpen f : Set (PrimeSpectrum R)) :=
-  isCompact_of_finite_subfamily_closed fun {ι} Z hZc hZ => by
-    let I : ι → Ideal R := fun i => vanishingIdeal (Z i)
-    have hI : ∀ i, Z i = zeroLocus (I i) := fun i => by
-      simpa only [zeroLocus_vanishingIdeal_eq_closure] using (hZc i).closure_eq.symm
-    rw [basicOpen_eq_zeroLocus_compl f, Set.inter_comm, ← Set.diff_eq, Set.diff_eq_empty,
-      funext hI, ← zeroLocus_iSup] at hZ
-    obtain ⟨n, hn⟩ : f ∈ (⨆ i : ι, I i).radical := by
-      rw [← vanishingIdeal_zeroLocus_eq_radical]
-      apply vanishingIdeal_anti_mono hZ
-      exact subset_vanishingIdeal_zeroLocus {f} (Set.mem_singleton f)
-    rcases Submodule.exists_finset_of_mem_iSup I hn with ⟨s, hs⟩
-    use s
-    -- Using simp_rw here, because `hI` and `zeroLocus_iSup` need to be applied underneath binders
-    simp_rw [basicOpen_eq_zeroLocus_compl f, Set.inter_comm (zeroLocus {f})ᶜ, ← Set.diff_eq,
-      Set.diff_eq_empty]
-    rw [show (Set.iInter fun i => Set.iInter fun (_ : i ∈ s) => Z i) =
-      Set.iInter fun i => Set.iInter fun (_ : i ∈ s) => zeroLocus (I i) from congr_arg _
-        (funext fun i => congr_arg _ (funext fun _ => hI i))]
-    simp_rw [← zeroLocus_iSup]
-    rw [← zeroLocus_radical]
-    -- this one can't be in `simp_rw` because it would loop
-    apply zeroLocus_anti_mono
-    rw [Set.singleton_subset_iff]
-    exact ⟨n, hs⟩
-#align prime_spectrum.is_compact_basic_open PrimeSpectrum.isCompact_basicOpen
-
 @[simp]
 theorem basicOpen_eq_bot_iff (f : R) : basicOpen f = ⊥ ↔ IsNilpotent f := by
   rw [← TopologicalSpace.Opens.coe_inj, basicOpen_eq_zeroLocus_compl]
@@ -890,13 +855,12 @@ theorem localization_away_openEmbedding (S : Type v) [CommRing S] [Algebra R S]
       exact isOpen_basicOpen }
 #align prime_spectrum.localization_away_open_embedding PrimeSpectrum.localization_away_openEmbedding
 
-end BasicOpen
+theorem isCompact_basicOpen (f : R) : IsCompact (basicOpen f : Set (PrimeSpectrum R)) := by
+  rw [← localization_away_comap_range (Localization (Submonoid.powers f))]
+  exact isCompact_range (map_continuous _)
+#align prime_spectrum.is_compact_basic_open PrimeSpectrum.isCompact_basicOpen
 
-/-- The prime spectrum of a commutative ring is a compact topological space. -/
-instance compactSpace : CompactSpace (PrimeSpectrum R) :=
-  { isCompact_univ := by
-      convert isCompact_basicOpen (1 : R)
-      rw [basicOpen_one, TopologicalSpace.Opens.coe_top] }
+end BasicOpen
 
 section Order
 

@@ -20,15 +20,15 @@ The prime spectrum of a commutative ring is the type of all prime ideals.
 It is naturally endowed with a topology: the Zariski topology.
 
 (It is also naturally endowed with a sheaf of rings,
-which is constructed in `algebraic_geometry.structure_sheaf`.)
+which is constructed in `AlgebraicGeometry.StructureSheaf`.)
 
 ## Main definitions
 
 * `PrimeSpectrum R`: The prime spectrum of a commutative ring `R`,
   i.e., the set of all prime ideals of `R`.
-* `zero_locus s`: The zero locus of a subset `s` of `R`
+* `zeroLocus s`: The zero locus of a subset `s` of `R`
   is the subset of `PrimeSpectrum R` consisting of all prime ideals that contain `s`.
-* `vanishing_ideal t`: The vanishing ideal of a subset `t` of `PrimeSpectrum R`
+* `vanishingIdeal t`: The vanishing ideal of a subset `t` of `PrimeSpectrum R`
   is the intersection of points in `t` (viewed as prime ideals).
 
 ## Conventions
@@ -55,7 +55,7 @@ variable (R : Type u) (S : Type v) [CommRing R] [CommRing S]
 /-- The prime spectrum of a commutative ring `R` is the type of all prime ideals of `R`.
 
 It is naturally endowed with a topology (the Zariski topology),
-and a sheaf of commutative rings (see `algebraic_geometry.structure_sheaf`).
+and a sheaf of commutative rings (see `AlgebraicGeometry.StructureSheaf`).
 It is a fundamental building block in algebraic geometry. -/
 @[ext]
 structure PrimeSpectrum where
@@ -74,9 +74,9 @@ instance [Nontrivial R] : Nonempty <| PrimeSpectrum R :=
   ⟨⟨I, hI.isPrime⟩⟩
 
 /-- The prime spectrum of the zero ring is empty. -/
-theorem pUnit (x : PrimeSpectrum PUnit) : False :=
+theorem punit (x : PrimeSpectrum PUnit) : False :=
   x.1.ne_top_iff_one.1 x.2.1 <| Eq.substr (Subsingleton.elim 1 (0 : PUnit)) x.1.zero_mem
-#align prime_spectrum.punit PrimeSpectrum.pUnit
+#align prime_spectrum.punit PrimeSpectrum.punit
 
 variable (R S)
 
@@ -127,7 +127,7 @@ of the ring that contain the set `s`.
 
 An element `f` of `R` can be thought of as a dependent function on the prime spectrum of `R`.
 At a point `x` (a prime ideal) the function (i.e., element) `f` takes values in the quotient ring
-`R` modulo the prime ideal `x`. In this manner, `zero_locus s` is exactly the subset of
+`R` modulo the prime ideal `x`. In this manner, `zeroLocus s` is exactly the subset of
 `PrimeSpectrum R` where all "functions" in `s` vanish simultaneously.
 -/
 def zeroLocus (s : Set R) : Set (PrimeSpectrum R) :=
@@ -150,7 +150,7 @@ the intersection of all the prime ideals in the set `t`.
 
 An element `f` of `R` can be thought of as a dependent function on the prime spectrum of `R`.
 At a point `x` (a prime ideal) the function (i.e., element) `f` takes values in the quotient ring
-`R` modulo the prime ideal `x`. In this manner, `vanishing_ideal t` is exactly the ideal of `R`
+`R` modulo the prime ideal `x`. In this manner, `vanishingIdeal t` is exactly the ideal of `R`
 consisting of all "functions" that vanish on all of `t`.
 -/
 def vanishingIdeal (t : Set (PrimeSpectrum R)) : Ideal R :=
@@ -185,14 +185,14 @@ section Gc
 
 variable (R)
 
-/-- `zero_locus` and `vanishing_ideal` form a galois connection. -/
+/-- `zeroLocus` and `vanishingIdeal` form a galois connection. -/
 theorem gc :
     @GaloisConnection (Ideal R) (Set (PrimeSpectrum R))ᵒᵈ _ _ (fun I => zeroLocus I) fun t =>
       vanishingIdeal t :=
   fun I t => subset_zeroLocus_iff_le_vanishingIdeal t I
 #align prime_spectrum.gc PrimeSpectrum.gc
 
-/-- `zero_locus` and `vanishing_ideal` form a galois connection. -/
+/-- `zeroLocus` and `vanishingIdeal` form a galois connection. -/
 theorem gc_set :
     @GaloisConnection (Set R) (Set (PrimeSpectrum R))ᵒᵈ _ _ (fun s => zeroLocus s) fun t =>
       vanishingIdeal t := by
@@ -402,13 +402,15 @@ theorem mem_compl_zeroLocus_iff_not_mem {f : R} {I : PrimeSpectrum R} :
 of the topology: they are exactly those sets that are the zero locus of a subset of the ring. -/
 instance zariskiTopology : TopologicalSpace (PrimeSpectrum R) :=
   TopologicalSpace.ofClosed (Set.range PrimeSpectrum.zeroLocus) ⟨Set.univ, by simp⟩
-    (by intro Zs h
-        rw [Set.sInter_eq_iInter]
-        choose f hf using fun i : Zs => h i.prop
-        simp only [← hf]
-        exact ⟨_, zeroLocus_iUnion _⟩)
-    (by rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩
-        exact ⟨_, (union_zeroLocus s t).symm⟩)
+    (by
+      intro Zs h
+      rw [Set.sInter_eq_iInter]
+      choose f hf using fun i : Zs => h i.prop
+      simp only [← hf]
+      exact ⟨_, zeroLocus_iUnion _⟩)
+    (by
+      rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩
+      exact ⟨_, (union_zeroLocus s t).symm⟩)
 #align prime_spectrum.zariski_topology PrimeSpectrum.zariskiTopology
 
 theorem isOpen_iff (U : Set (PrimeSpectrum R)) : IsOpen U ↔ ∃ s, Uᶜ = zeroLocus s := by
@@ -519,7 +521,6 @@ theorem t1Space_iff_isField [IsDomain R] : T1Space (PrimeSpectrum R) ↔ IsField
     · exact absurd h (Ring.not_isField_iff_exists_prime.2 ⟨x.asIdeal, ⟨hx, x.2⟩⟩)
 #align prime_spectrum.t1_space_iff_is_field PrimeSpectrum.t1Space_iff_isField
 
--- mathport name: «exprZ( )»
 local notation "Z(" a ")" => zeroLocus (a : Set R)
 
 theorem isIrreducible_zeroLocus_iff_of_radical (I : Ideal R) (hI : I.IsRadical) :
@@ -576,8 +577,7 @@ theorem preimage_comap_zeroLocus_aux (f : R →+* S) (s : Set R) :
         zeroLocus s =
       zeroLocus (f '' s) := by
   ext x
-  simp only [mem_zeroLocus, Set.image_subset_iff]
-  rfl
+  simp only [mem_zeroLocus, Set.image_subset_iff, Set.mem_preimage, mem_zeroLocus, Ideal.coe_comap]
 #align prime_spectrum.preimage_comap_zero_locus_aux PrimeSpectrum.preimage_comap_zeroLocus_aux
 
 /-- The function between prime spectra of commutative rings induced by a ring homomorphism.
@@ -697,20 +697,19 @@ section SpecOfSurjective
 
 open Function RingHom
 
-theorem comap_inducing_of_surjective (hf : Surjective f) : Inducing (comap f) :=
-  {
-    induced := by
-      simp_rw [TopologicalSpace.ext_iff, ← isClosed_compl_iff,
-        ← @isClosed_compl_iff (PrimeSpectrum S)
-          ((TopologicalSpace.induced (comap f) zariskiTopology)), isClosed_induced_iff,
-        isClosed_iff_zeroLocus]
-      refine' fun s =>
-        ⟨fun ⟨F, hF⟩ =>
-          ⟨zeroLocus (f ⁻¹' F), ⟨f ⁻¹' F, rfl⟩, by
-            rw [preimage_comap_zeroLocus, Function.Surjective.image_preimage hf, hF]⟩,
-          _⟩
-      rintro ⟨-, ⟨F, rfl⟩, hF⟩
-      exact ⟨f '' F, hF.symm.trans (preimage_comap_zeroLocus f F)⟩ }
+theorem comap_inducing_of_surjective (hf : Surjective f) : Inducing (comap f) where
+  induced := by
+    simp_rw [TopologicalSpace.ext_iff, ← isClosed_compl_iff,
+      ← @isClosed_compl_iff (PrimeSpectrum S)
+        ((TopologicalSpace.induced (comap f) zariskiTopology)), isClosed_induced_iff,
+      isClosed_iff_zeroLocus]
+    refine' fun s =>
+      ⟨fun ⟨F, hF⟩ =>
+        ⟨zeroLocus (f ⁻¹' F), ⟨f ⁻¹' F, rfl⟩, by
+          rw [preimage_comap_zeroLocus, Function.Surjective.image_preimage hf, hF]⟩,
+        _⟩
+    rintro ⟨-, ⟨F, rfl⟩, hF⟩
+    exact ⟨f '' F, hF.symm.trans (preimage_comap_zeroLocus f F)⟩
 #align prime_spectrum.comap_inducing_of_surjective PrimeSpectrum.comap_inducing_of_surjective
 
 theorem image_comap_zeroLocus_eq_zeroLocus_comap (hf : Surjective f) (I : Ideal S) :
@@ -724,8 +723,7 @@ theorem image_comap_zeroLocus_eq_zeroLocus_comap (hf : Surjective f) (I : Ideal
     · obtain ⟨x', rfl⟩ := hf x
       exact Ideal.mem_map_of_mem f (h_I_p hx)
     · ext x
-      change f x ∈ p.asIdeal.map f ↔ _
-      rw [Ideal.mem_map_iff_of_surjective f hf]
+      rw [comap_asIdeal, Ideal.mem_comap, Ideal.mem_map_iff_of_surjective f hf]
       refine' ⟨_, fun hx => ⟨x, hx, rfl⟩⟩
       rintro ⟨x', hx', heq⟩
       rw [← sub_sub_cancel x' x]
@@ -758,7 +756,7 @@ end Comap
 
 section BasicOpen
 
-/-- `basic_open r` is the open subset containing all prime ideals not containing `r`. -/
+/-- `basicOpen r` is the open subset containing all prime ideals not containing `r`. -/
 def basicOpen (r : R) : TopologicalSpace.Opens (PrimeSpectrum R) where
   carrier := { x | r ∉ x.asIdeal }
   is_open' := ⟨{r}, Set.ext fun _ => Set.singleton_subset_iff.trans <| Classical.not_not.symm⟩
@@ -849,7 +847,7 @@ theorem isCompact_basicOpen (f : R) : IsCompact (basicOpen f : Set (PrimeSpectru
       exact subset_vanishingIdeal_zeroLocus {f} (Set.mem_singleton f)
     rcases Submodule.exists_finset_of_mem_iSup I hn with ⟨s, hs⟩
     use s
-    -- Using simp_rw here, because `hI` and `zero_locus_supr` need to be applied underneath binders
+    -- Using simp_rw here, because `hI` and `zeroLocus_iSup` need to be applied underneath binders
     simp_rw [basicOpen_eq_zeroLocus_compl f, Set.inter_comm (zeroLocus {f})ᶜ, ← Set.diff_eq,
       Set.diff_eq_empty]
     rw [show (Set.iInter fun i => Set.iInter fun (_ : i ∈ s) => Z i) =
@@ -898,8 +896,7 @@ end BasicOpen
 instance compactSpace : CompactSpace (PrimeSpectrum R) :=
   { isCompact_univ := by
       convert isCompact_basicOpen (1 : R)
-      rw [basicOpen_one]
-      rfl }
+      rw [basicOpen_one, TopologicalSpace.Opens.coe_top] }
 
 section Order
 

@@ -645,7 +645,7 @@ variable (S)
 theorem localization_comap_inducing [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
     Inducing (comap (algebraMap R S)) := by
   constructor
-  rw [topologicalSpace_eq_iff]
+  rw [TopologicalSpace.ext_iff]
   intro U
   rw [← isClosed_compl_iff, ← @isClosed_compl_iff (α := PrimeSpectrum S) (s := U)]
   generalize Uᶜ = Z
@@ -700,7 +700,7 @@ open Function RingHom
 theorem comap_inducing_of_surjective (hf : Surjective f) : Inducing (comap f) :=
   {
     induced := by
-      simp_rw [topologicalSpace_eq_iff, ← isClosed_compl_iff,
+      simp_rw [TopologicalSpace.ext_iff, ← isClosed_compl_iff,
         ← @isClosed_compl_iff (PrimeSpectrum S)
           ((TopologicalSpace.induced (comap f) zariskiTopology)), isClosed_induced_iff,
         isClosed_iff_zeroLocus]

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

This has nice performance benefits.

@@ -332,12 +332,12 @@ theorem vanishingIdeal_union (t t' : Set (PrimeSpectrum R)) :
   (gc R).u_inf
 #align prime_spectrum.vanishing_ideal_union PrimeSpectrum.vanishingIdeal_union
 
-theorem zeroLocus_iSup {ι : Sort _} (I : ι → Ideal R) :
+theorem zeroLocus_iSup {ι : Sort*} (I : ι → Ideal R) :
     zeroLocus ((⨆ i, I i : Ideal R) : Set R) = ⋂ i, zeroLocus (I i) :=
   (gc R).l_iSup
 #align prime_spectrum.zero_locus_supr PrimeSpectrum.zeroLocus_iSup
 
-theorem zeroLocus_iUnion {ι : Sort _} (s : ι → Set R) :
+theorem zeroLocus_iUnion {ι : Sort*} (s : ι → Set R) :
     zeroLocus (⋃ i, s i) = ⋂ i, zeroLocus (s i) :=
   (gc_set R).l_iSup
 #align prime_spectrum.zero_locus_Union PrimeSpectrum.zeroLocus_iUnion
@@ -346,7 +346,7 @@ theorem zeroLocus_bUnion (s : Set (Set R)) :
     zeroLocus (⋃ s' ∈ s, s' : Set R) = ⋂ s' ∈ s, zeroLocus s' := by simp only [zeroLocus_iUnion]
 #align prime_spectrum.zero_locus_bUnion PrimeSpectrum.zeroLocus_bUnion
 
-theorem vanishingIdeal_iUnion {ι : Sort _} (t : ι → Set (PrimeSpectrum R)) :
+theorem vanishingIdeal_iUnion {ι : Sort*} (t : ι → Set (PrimeSpectrum R)) :
     vanishingIdeal (⋃ i, t i) = ⨅ i, vanishingIdeal (t i) :=
   (gc R).u_iInf
 #align prime_spectrum.vanishing_ideal_Union PrimeSpectrum.vanishingIdeal_iUnion
@@ -496,7 +496,7 @@ theorem vanishingIdeal_strict_anti_mono_iff {s t : Set (PrimeSpectrum R)} (hs :
 #align prime_spectrum.vanishing_ideal_strict_anti_mono_iff PrimeSpectrum.vanishingIdeal_strict_anti_mono_iff
 
 /-- The antitone order embedding of closed subsets of `Spec R` into ideals of `R`. -/
-def closedsEmbedding (R : Type _) [CommRing R] :
+def closedsEmbedding (R : Type*) [CommRing R] :
     (TopologicalSpace.Closeds <| PrimeSpectrum R)ᵒᵈ ↪o Ideal R :=
   OrderEmbedding.ofMapLEIff (fun s => vanishingIdeal <| OrderDual.ofDual s) fun s _ =>
     (vanishingIdeal_anti_mono_iff s.2).symm
@@ -569,7 +569,7 @@ instance quasiSober : QuasiSober (PrimeSpectrum R) :=
 
 section Comap
 
-variable {S' : Type _} [CommRing S']
+variable {S' : Type*} [CommRing S']
 
 theorem preimage_comap_zeroLocus_aux (f : R →+* S) (s : Set R) :
     (fun y => ⟨Ideal.comap f y.asIdeal, inferInstance⟩ : PrimeSpectrum S → PrimeSpectrum R) ⁻¹'
@@ -946,7 +946,7 @@ instance [IsDomain R] : OrderBot (PrimeSpectrum R) where
   bot := ⟨⊥, Ideal.bot_prime⟩
   bot_le I := @bot_le _ _ _ I.asIdeal
 
-instance {R : Type _} [Field R] : Unique (PrimeSpectrum R) where
+instance {R : Type*} [Field R] : Unique (PrimeSpectrum R) where
   default := ⊥
   uniq x := PrimeSpectrum.ext _ _ ((IsSimpleOrder.eq_bot_or_eq_top _).resolve_right x.2.ne_top)
 

@@ -655,7 +655,7 @@ theorem localization_comap_inducing [Algebra R S] (M : Submonoid R) [IsLocalizat
     refine ⟨_, ⟨algebraMap R S ⁻¹' Ideal.span s, rfl⟩, ?_⟩
     rw [preimage_comap_zeroLocus, ← zeroLocus_span, ← zeroLocus_span s]
     congr 2
-    exact congr_arg (zeroLocus .) <| Submodule.carrier_inj.mpr
+    exact congr_arg (zeroLocus ·) <| Submodule.carrier_inj.mpr
       (IsLocalization.map_comap M S (Ideal.span s))
   · rintro ⟨_, ⟨t, rfl⟩, rfl⟩
     rw [preimage_comap_zeroLocus]

• depends on: #5966

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

@@ -2,11 +2,6 @@
 Copyright (c) 2020 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
-
-! This file was ported from Lean 3 source module algebraic_geometry.prime_spectrum.basic
-! leanprover-community/mathlib commit a7c017d750512a352b623b1824d75da5998457d0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.LinearAlgebra.Finsupp
 import Mathlib.RingTheory.Ideal.Over
@@ -16,6 +11,8 @@ import Mathlib.RingTheory.Nilpotent
 import Mathlib.Topology.Sets.Closeds
 import Mathlib.Topology.Sober
 
+#align_import algebraic_geometry.prime_spectrum.basic from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
+
 /-!
 # Prime spectrum of a commutative 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.

@@ -398,7 +398,7 @@ theorem sup_vanishingIdeal_le (t t' : Set (PrimeSpectrum R)) :
 
 theorem mem_compl_zeroLocus_iff_not_mem {f : R} {I : PrimeSpectrum R} :
     I ∈ (zeroLocus {f} : Set (PrimeSpectrum R))ᶜ ↔ f ∉ I.asIdeal := 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 prime_spectrum.mem_compl_zero_locus_iff_not_mem PrimeSpectrum.mem_compl_zeroLocus_iff_not_mem
 
 /-- The Zariski topology on the prime spectrum of a commutative ring is defined via the closed sets
@@ -415,7 +415,7 @@ instance zariskiTopology : TopologicalSpace (PrimeSpectrum R) :=
 #align prime_spectrum.zariski_topology PrimeSpectrum.zariskiTopology
 
 theorem isOpen_iff (U : Set (PrimeSpectrum R)) : IsOpen U ↔ ∃ s, Uᶜ = zeroLocus s := by
-  simp only [@eq_comm _ Uᶜ] ; rfl
+  simp only [@eq_comm _ Uᶜ]; rfl
 #align prime_spectrum.is_open_iff PrimeSpectrum.isOpen_iff
 
 theorem isClosed_iff_zeroLocus (Z : Set (PrimeSpectrum R)) : IsClosed Z ↔ ∃ s, Z = zeroLocus s := by

Co-authored-by: Jason Yuen <jason_yuen2007@hotmail.com> Co-authored-by: int-y1 <jason_yuen2007@hotmail.com> Co-authored-by: Antoine Chambert-Loir <antoine.chambert-loir@math.univ-paris-diderot.fr> Co-authored-by: Johan Commelin <johan@commelin.net> Co-authored-by: Jujian Zhang <jujian.zhang1998@outlook.com> Co-authored-by: Parcly Taxel <reddeloostw@gmail.com> Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com>

@@ -639,7 +639,7 @@ theorem comap_singleton_isClosed_of_isIntegral (f : R →+* S) (hf : f.IsIntegra
     (x : PrimeSpectrum S) (hx : IsClosed ({x} : Set (PrimeSpectrum S))) :
     IsClosed ({comap f x} : Set (PrimeSpectrum R)) :=
   (isClosed_singleton_iff_isMaximal _).2
-    (Ideal.isMaximal_comap_of_isIntegral_of_is_maximal' f hf x.asIdeal <|
+    (Ideal.isMaximal_comap_of_isIntegral_of_isMaximal' f hf x.asIdeal <|
       (isClosed_singleton_iff_isMaximal x).1 hx)
 #align prime_spectrum.comap_singleton_is_closed_of_is_integral PrimeSpectrum.comap_singleton_isClosed_of_isIntegral
 

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

@@ -415,7 +415,7 @@ instance zariskiTopology : TopologicalSpace (PrimeSpectrum R) :=
 #align prime_spectrum.zariski_topology PrimeSpectrum.zariskiTopology
 
 theorem isOpen_iff (U : Set (PrimeSpectrum R)) : IsOpen U ↔ ∃ s, Uᶜ = zeroLocus s := by
-  simp only [@eq_comm _ (Uᶜ)] ; rfl
+  simp only [@eq_comm _ Uᶜ] ; rfl
 #align prime_spectrum.is_open_iff PrimeSpectrum.isOpen_iff
 
 theorem isClosed_iff_zeroLocus (Z : Set (PrimeSpectrum R)) : IsClosed Z ↔ ∃ s, Z = zeroLocus s := by
@@ -778,7 +778,7 @@ theorem isOpen_basicOpen {a : R} : IsOpen (basicOpen a : Set (PrimeSpectrum R))
 
 @[simp]
 theorem basicOpen_eq_zeroLocus_compl (r : R) :
-    (basicOpen r : Set (PrimeSpectrum R)) = zeroLocus {r}ᶜ :=
+    (basicOpen r : Set (PrimeSpectrum R)) = (zeroLocus {r})ᶜ :=
   Set.ext fun x => by simp only [SetLike.mem_coe, mem_basicOpen, Set.mem_compl_iff, mem_zeroLocus,
     Set.singleton_subset_iff]
 #align prime_spectrum.basic_open_eq_zero_locus_compl PrimeSpectrum.basicOpen_eq_zeroLocus_compl
@@ -853,7 +853,7 @@ theorem isCompact_basicOpen (f : R) : IsCompact (basicOpen f : Set (PrimeSpectru
     rcases Submodule.exists_finset_of_mem_iSup I hn with ⟨s, hs⟩
     use s
     -- Using simp_rw here, because `hI` and `zero_locus_supr` need to be applied underneath binders
-    simp_rw [basicOpen_eq_zeroLocus_compl f, Set.inter_comm (zeroLocus {f}ᶜ), ← Set.diff_eq,
+    simp_rw [basicOpen_eq_zeroLocus_compl f, Set.inter_comm (zeroLocus {f})ᶜ, ← Set.diff_eq,
       Set.diff_eq_empty]
     rw [show (Set.iInter fun i => Set.iInter fun (_ : i ∈ s) => Z i) =
       Set.iInter fun i => Set.iInter fun (_ : i ∈ s) => zeroLocus (I i) from congr_arg _

@@ -561,11 +561,11 @@ theorem isIrreducible_iff_vanishingIdeal_isPrime {s : Set (PrimeSpectrum R)} :
     isIrreducible_zeroLocus_iff_of_radical _ (isRadical_vanishingIdeal s)]
 #align prime_spectrum.is_irreducible_iff_vanishing_ideal_is_prime PrimeSpectrum.isIrreducible_iff_vanishingIdeal_isPrime
 
-instance [IsDomain R] : IrreducibleSpace (PrimeSpectrum R) := by
+instance irreducibleSpace [IsDomain R] : IrreducibleSpace (PrimeSpectrum R) := by
   rw [irreducibleSpace_def, Set.top_eq_univ, ← zeroLocus_bot, isIrreducible_zeroLocus_iff]
   simpa using Ideal.bot_prime
 
-instance : QuasiSober (PrimeSpectrum R) :=
+instance quasiSober : QuasiSober (PrimeSpectrum R) :=
   ⟨fun {S} h₁ h₂ =>
     ⟨⟨_, isIrreducible_iff_vanishingIdeal_isPrime.1 h₁⟩, by
       rw [IsGenericPoint, closure_singleton, zeroLocus_vanishingIdeal_eq_closure, h₂.closure_eq]⟩⟩

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

@@ -898,7 +898,7 @@ theorem localization_away_openEmbedding (S : Type v) [CommRing S] [Algebra R S]
 end BasicOpen
 
 /-- The prime spectrum of a commutative ring is a compact topological space. -/
-instance : CompactSpace (PrimeSpectrum R) :=
+instance compactSpace : CompactSpace (PrimeSpectrum R) :=
   { isCompact_univ := by
       convert isCompact_basicOpen (1 : R)
       rw [basicOpen_one]

Changes are of the form

• some_tactic at h⊢ -> some_tactic at h ⊢
• some_tactic at h -> some_tactic at h

@@ -543,7 +543,7 @@ theorem isIrreducible_zeroLocus_iff_of_radical (I : Ideal R) (hI : I.IsRadical)
       · simp_rw [← SetLike.mem_coe, ← Set.singleton_subset_iff, ← Ideal.span_le, ←
           Ideal.span_singleton_mul_span_singleton]
         refine' fun h x y h' => h _ _ _
-        rw [← hI.radical_le_iff] at h'⊢
+        rw [← hI.radical_le_iff] at h' ⊢
         simpa only [Ideal.radical_inf, Ideal.radical_mul] using h'
       · simp_rw [or_iff_not_imp_left, SetLike.not_le_iff_exists]
         rintro h s t h' ⟨x, hx, hx'⟩ y hy

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

@@ -899,10 +899,10 @@ end BasicOpen
 
 /-- The prime spectrum of a commutative ring is a compact topological space. -/
 instance : CompactSpace (PrimeSpectrum R) :=
-{ isCompact_univ := by
-    convert isCompact_basicOpen (1 : R)
-    rw [basicOpen_one]
-    rfl }
+  { isCompact_univ := by
+      convert isCompact_basicOpen (1 : R)
+      rw [basicOpen_one]
+      rfl }
 
 section Order
 

@@ -157,7 +157,7 @@ At a point `x` (a prime ideal) the function (i.e., element) `f` takes values in
 consisting of all "functions" that vanish on all of `t`.
 -/
 def vanishingIdeal (t : Set (PrimeSpectrum R)) : Ideal R :=
-  ⨅ (x : PrimeSpectrum R) (__ : x ∈ t), x.asIdeal
+  ⨅ (x : PrimeSpectrum R) (_ : x ∈ t), x.asIdeal
 #align prime_spectrum.vanishing_ideal PrimeSpectrum.vanishingIdeal
 
 theorem coe_vanishingIdeal (t : Set (PrimeSpectrum R)) :