ring_theory.dedekind_domain.ideal ⟷ Mathlib.RingTheory.DedekindDomain.Ideal

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

@@ -8,7 +8,7 @@ import AlgebraicGeometry.PrimeSpectrum.Maximal
 import AlgebraicGeometry.PrimeSpectrum.Noetherian
 import Order.Hom.Basic
 import RingTheory.DedekindDomain.Basic
-import RingTheory.FractionalIdeal
+import RingTheory.FractionalIdeal.Basic
 import RingTheory.PrincipalIdealDomain
 import RingTheory.ChainOfDivisors
 
@@ -295,7 +295,7 @@ noncomputable instance : InvOneClass (FractionalIdeal R₁⁰ K) :=
 
 end FractionalIdeal
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) (fraction_ring[fraction_ring] A))) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) (fraction_ring[fraction_ring] A))) -/
 #print IsDedekindDomainInv /-
 /-- A Dedekind domain is an integral domain such that every fractional ideal has an inverse.
 
@@ -313,7 +313,7 @@ open FractionalIdeal
 
 variable {R A K}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) K)) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) K)) -/
 #print isDedekindDomainInv_iff /-
 theorem isDedekindDomainInv_iff [Algebra A K] [IsFractionRing A K] :
     IsDedekindDomainInv A ↔ ∀ (I) (_ : I ≠ (⊥ : FractionalIdeal A⁰ K)), I * I⁻¹ = 1 :=
@@ -541,7 +541,6 @@ theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I
 #align fractional_ideal.exists_not_mem_one_of_ne_bot FractionalIdeal.exists_not_mem_one_of_ne_bot
 -/
 
-#print FractionalIdeal.one_mem_inv_coe_ideal /-
 theorem one_mem_inv_coe_ideal {I : Ideal A} (hI : I ≠ ⊥) : (1 : K) ∈ (I : FractionalIdeal A⁰ K)⁻¹ :=
   by
   rw [mem_inv_iff (coe_ideal_ne_zero.mpr hI)]
@@ -550,7 +549,6 @@ theorem one_mem_inv_coe_ideal {I : Ideal A} (hI : I ≠ ⊥) : (1 : K) ∈ (I :
   exact coe_ideal_le_one hy
   assumption
 #align fractional_ideal.one_mem_inv_coe_ideal FractionalIdeal.one_mem_inv_coe_ideal
--/
 
 #print FractionalIdeal.mul_inv_cancel_of_le_one /-
 theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
@@ -625,7 +623,7 @@ theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ 
   induction' i with i ih
   · rw [pow_zero]; exact one_mem_inv_coe_ideal hI0
   · show x ^ i.succ ∈ (I⁻¹ : FractionalIdeal A⁰ K)
-    rw [pow_succ]; exact x_mul_mem _ ih
+    rw [pow_succ']; exact x_mul_mem _ ih
 #align fractional_ideal.coe_ideal_mul_inv FractionalIdeal.coe_ideal_mul_inv
 -/
 
@@ -882,7 +880,7 @@ theorem Ideal.isPrime_iff_bot_or_prime {P : Ideal A} : IsPrime P ↔ P = ⊥ ∨
 theorem Ideal.pow_right_strictAnti (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
     StrictAnti ((· ^ ·) I : ℕ → Ideal A) :=
   strictAnti_nat_of_succ_lt fun e =>
-    Ideal.dvdNotUnit_iff_lt.mp ⟨pow_ne_zero _ hI0, I, mt isUnit_iff.mp hI1, pow_succ' I e⟩
+    Ideal.dvdNotUnit_iff_lt.mp ⟨pow_ne_zero _ hI0, I, mt isUnit_iff.mp hI1, pow_succ I e⟩
 #align ideal.strict_anti_pow Ideal.pow_right_strictAnti
 -/
 
@@ -1455,7 +1453,7 @@ theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n :
     (h : a * b ∈ I ^ n) : a ∈ I ∨ b ∈ I ^ n :=
   by
   cases n; · simp
-  by_cases hI0 : I = ⊥; · simpa [pow_succ, hI0] using h
+  by_cases hI0 : I = ⊥; · simpa [pow_succ', hI0] using h
   simp only [← Submodule.span_singleton_le_iff_mem, Ideal.submodule_span_eq, ← Ideal.dvd_iff_le, ←
     Ideal.span_singleton_mul_span_singleton] at h ⊢
   by_cases ha : I ∣ span {a}
@@ -1528,7 +1526,7 @@ theorem Ideal.prod_le_prime {ι : Type _} {s : Finset ι} {f : ι → Ideal R} {
 #align ideal.prod_le_prime Ideal.prod_le_prime
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 #print IsDedekindDomain.inf_prime_pow_eq_prod /-
 /-- The intersection of distinct prime powers in a Dedekind domain is the product of these
 prime powers. -/
@@ -1628,7 +1626,7 @@ noncomputable def Ideal.quotientMulEquivQuotientProd (I J : Ideal R) (coprime :
 #align ideal.quotient_mul_equiv_quotient_prod Ideal.quotientMulEquivQuotientProd
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 #print IsDedekindDomain.quotientEquivPiOfFinsetProdEq /-
 /-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
 `∏ i in s, P i ^ e i`, then `R ⧸ I` factors as `Π (i : s), R ⧸ (P i ^ e i)`.
@@ -1646,7 +1644,7 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfFinsetProdEq {ι : Type _} {
 #align is_dedekind_domain.quotient_equiv_pi_of_finset_prod_eq IsDedekindDomain.quotientEquivPiOfFinsetProdEq
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 #print IsDedekindDomain.exists_representative_mod_finset /-
 /-- Corollary of the Chinese remainder theorem: given elements `x i : R / P i ^ e i`,
 we can choose a representative `y : R` such that `y ≡ x i (mod P i ^ e i)`.-/
@@ -1662,7 +1660,7 @@ theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type _} {s : Fin
 #align is_dedekind_domain.exists_representative_mod_finset IsDedekindDomain.exists_representative_mod_finset
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 #print IsDedekindDomain.exists_forall_sub_mem_ideal /-
 /-- Corollary of the Chinese remainder theorem: given elements `x i : R`,
 we can choose a representative `y : R` such that `y - x i ∈ P i ^ e i`.-/

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

@@ -273,7 +273,7 @@ theorem invertible_iff_generator_nonzero (I : FractionalIdeal R₁⁰ K)
     rw [eq_span_singleton_of_principal I]
     intro hI
     have := mem_span_singleton_self _ (generator (I : Submodule R₁ K))
-    rw [hI, mem_zero_iff] at this 
+    rw [hI, mem_zero_iff] at this
     contradiction
 #align fractional_ideal.invertible_iff_generator_nonzero FractionalIdeal.invertible_iff_generator_nonzero
 -/
@@ -418,9 +418,9 @@ theorem dimensionLEOne : DimensionLEOne A :=
   obtain ⟨z, hzM, hzp⟩ := SetLike.exists_of_lt hM
   -- We have `z * y ∈ M * (M⁻¹ * P) = P`.
   have zy_mem := mul_mem_mul (mem_coe_ideal_of_mem A⁰ hzM) hx
-  rw [← RingHom.map_mul, ← mul_assoc, h.mul_inv_eq_one M'_ne, one_mul] at zy_mem 
+  rw [← RingHom.map_mul, ← mul_assoc, h.mul_inv_eq_one M'_ne, one_mul] at zy_mem
   obtain ⟨zy, hzy, zy_eq⟩ := (mem_coe_ideal A⁰).mp zy_mem
-  rw [IsFractionRing.injective A (FractionRing A) zy_eq] at hzy 
+  rw [IsFractionRing.injective A (FractionRing A) zy_eq] at hzy
   -- But `P` is a prime ideal, so `z ∉ P` implies `y ∈ P`, as desired.
   exact mem_coe_ideal_of_mem A⁰ (Or.resolve_left (hP.mem_or_mem hzy) hzp)
 #align is_dedekind_domain_inv.dimension_le_one IsDedekindDomainInv.dimensionLEOne
@@ -466,7 +466,7 @@ theorem exists_multiset_prod_cons_le_and_prod_not_le [IsDedekindDomain A] (hNF :
   have := Multiset.map_erase PrimeSpectrum.asIdeal PrimeSpectrum.ext P Z
   obtain ⟨hP0, hZP0⟩ : P.as_ideal ≠ ⊥ ∧ ((Z.erase P).map PrimeSpectrum.asIdeal).Prod ≠ ⊥ := by
     rwa [Ne.def, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
-      this] at hprodZ 
+      this] at hprodZ
   -- By maximality of `P` and `M`, we have that `P ≤ M` implies `P = M`.
   have hPM' := (IsDedekindDomain.dimensionLEOne _ hP0 P.is_prime).eq_of_le hM.ne_top hPM
   subst hPM'
@@ -529,13 +529,13 @@ theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I
       apply hle
       rw [Multiset.prod_cons]
       exact Submodule.smul_mem_smul h_Iy hbZ
-    rw [Ideal.mem_span_singleton'] at h_yb 
+    rw [Ideal.mem_span_singleton'] at h_yb
     rcases h_yb with ⟨c, hc⟩
     rw [← hc, RingHom.map_mul, mul_assoc, mul_inv_cancel hnz_fa, mul_one]
     apply coe_mem_one
   · refine' mt (mem_one_iff _).mp _
     rintro ⟨x', h₂_abs⟩
-    rw [← div_eq_mul_inv, eq_div_iff_mul_eq hnz_fa, ← RingHom.map_mul] at h₂_abs 
+    rw [← div_eq_mul_inv, eq_div_iff_mul_eq hnz_fa, ← RingHom.map_mul] at h₂_abs
     have := ideal.mem_span_singleton'.mpr ⟨x', IsFractionRing.injective A K h₂_abs⟩
     contradiction
 #align fractional_ideal.exists_not_mem_one_of_ne_bot FractionalIdeal.exists_not_mem_one_of_ne_bot
@@ -577,7 +577,7 @@ theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I
     ∃ x : K, x ∈ (J : FractionalIdeal A⁰ K)⁻¹ ∧ x ∉ (1 : FractionalIdeal A⁰ K) :=
     exists_not_mem_one_of_ne_bot hNF hJ0 hJ1
   contrapose! hx1 with h_abs
-  rw [hJ] at hx 
+  rw [hJ] at hx
   exact hI hx
 #align fractional_ideal.mul_inv_cancel_of_le_one FractionalIdeal.mul_inv_cancel_of_le_one
 -/
@@ -599,7 +599,7 @@ theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ 
   -- In particular, we'll show all `x ∈ J⁻¹` are integral.
   suffices x ∈ integralClosure A K by
     rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range, ←
-        mem_one_iff] at this  <;>
+        mem_one_iff] at this <;>
       assumption
   -- For that, we'll find a subalgebra that is f.g. as a module and contains `x`.
   -- `A` is a noetherian ring, so we just need to find a subalgebra between `{x}` and `I⁻¹`.
@@ -694,7 +694,7 @@ protected theorem div_eq_mul_inv [IsDedekindDomain A] (I J : FractionalIdeal A
   refine' le_antisymm ((mul_right_le_iff hJ).mp _) ((le_div_iff_mul_le hJ).mpr _)
   · rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one, mul_le]
     intro x hx y hy
-    rw [mem_div_iff_of_nonzero hJ] at hx 
+    rw [mem_div_iff_of_nonzero hJ] at hx
     exact hx y hy
   rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one]
   exact le_refl I
@@ -767,7 +767,7 @@ instance Ideal.isDomain : IsDomain (Ideal A) :=
 theorem Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
   ⟨Ideal.le_of_dvd, fun h => by
     by_cases hI : I = ⊥
-    · have hJ : J = ⊥ := by rwa [hI, ← eq_bot_iff] at h 
+    · have hJ : J = ⊥ := by rwa [hI, ← eq_bot_iff] at h
       rw [hI, hJ]
     have hI' : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coe_ideal_ne_zero.mpr hI
     have : (I : FractionalIdeal A⁰ (FractionRing A))⁻¹ * J ≤ 1 :=
@@ -846,7 +846,7 @@ theorem Ideal.isPrime_of_prime {P : Ideal A} (h : Prime P) : IsPrime P :=
   by
   refine' ⟨_, fun x y hxy => _⟩
   · rintro rfl
-    rw [← Ideal.one_eq_top] at h 
+    rw [← Ideal.one_eq_top] at h
     exact h.not_unit isUnit_one
   · simp only [← Ideal.dvd_span_singleton, ← Ideal.span_singleton_mul_span_singleton] at hxy ⊢
     exact h.dvd_or_dvd hxy
@@ -913,7 +913,7 @@ theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime
   have := pow_ne_zero (i + 1) hP
   rw [← Ideal.dvdNotUnit_iff_lt, dvd_not_unit_iff_normalized_factors_lt_normalized_factors,
     normalized_factors_pow, normalized_factors_irreducible P_prime'.irreducible,
-    Multiset.nsmul_singleton, Multiset.lt_replicate_succ] at hlt 
+    Multiset.nsmul_singleton, Multiset.lt_replicate_succ] at hlt
   rw [← Ideal.dvd_iff_le, dvd_iff_normalized_factors_le_normalized_factors, normalized_factors_pow,
     normalized_factors_irreducible P_prime'.irreducible, Multiset.nsmul_singleton]
   all_goals assumption
@@ -957,7 +957,7 @@ theorem Ideal.exist_integer_multiples_not_mem {J : Ideal A} (hJ : J ≠ ⊤) {ι
   suffices ↑J / I < I⁻¹
     by
     obtain ⟨_, a, hI, hpI⟩ := set_like.lt_iff_le_and_exists.mp this
-    rw [mem_inv_iff hI0] at hI 
+    rw [mem_inv_iff hI0] at hI
     refine' ⟨a, fun i hi => _, _⟩
     -- By definition, `a ∈ I⁻¹` multiplies elements of `I` into elements of `1`,
     -- in other words, `a * f i` is an integer.
@@ -1096,7 +1096,7 @@ theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
     by
     apply normalized_factors_prod_of_prime
     intro p hp
-    rw [mem_inter] at hp 
+    rw [mem_inter] at hp
     exact prime_of_normalized_factor p hp.left
   have :=
     Multiset.prod_ne_zero_of_prime (normalized_factors I ∩ normalized_factors J) fun _ h =>
@@ -1114,7 +1114,7 @@ theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
       rw [Multiset.count_inter]
       exact le_min (count_le_of_ideal_ge le_sup_left hI a) (count_le_of_ideal_ge le_sup_right hJ a)
     · intro p hp
-      rw [mem_inter] at hp 
+      rw [mem_inter] at hp
       exact prime_of_normalized_factor p hp.left
     · exact ne_bot_of_le_ne_bot hI le_sup_left
     · exact this
@@ -1135,7 +1135,7 @@ theorem irreducible_pow_sup_of_le (hJ : Irreducible J) (n : ℕ) (hn : ↑n ≤
   by_cases hI : I = ⊥
   · simp_all
   rw [irreducible_pow_sup hI hJ, min_eq_right]
-  rwa [multiplicity_eq_count_normalized_factors hJ hI, PartENat.coe_le_coe, normalize_eq J] at hn 
+  rwa [multiplicity_eq_count_normalized_factors hJ hI, PartENat.coe_le_coe, normalize_eq J] at hn
 #align irreducible_pow_sup_of_le irreducible_pow_sup_of_le
 -/
 
@@ -1149,7 +1149,7 @@ theorem irreducible_pow_sup_of_ge (hI : I ≠ ⊥) (hJ : Irreducible J) (n : ℕ
   ·
     rw [← PartENat.natCast_inj, PartENat.natCast_get,
       multiplicity_eq_count_normalized_factors hJ hI, normalize_eq J]
-  · rwa [multiplicity_eq_count_normalized_factors hJ hI, PartENat.coe_le_coe, normalize_eq J] at hn 
+  · rwa [multiplicity_eq_count_normalized_factors hJ hI, PartENat.coe_le_coe, normalize_eq J] at hn
 #align irreducible_pow_sup_of_ge irreducible_pow_sup_of_ge
 -/
 
@@ -1268,7 +1268,7 @@ def idealFactorsFunOfQuotHom {f : R ⧸ I →+* A ⧸ J} (hf : Function.Surjecti
       by
       have : J.Quotient.mk.ker ≤ comap J.Quotient.mk (map f (map I.Quotient.mk X)) :=
         ker_le_comap J.Quotient.mk
-      rw [mk_ker] at this 
+      rw [mk_ker] at this
       exact dvd_iff_le.mpr this⟩
   monotone' := by
     rintro ⟨X, hX⟩ ⟨Y, hY⟩ h
@@ -1363,7 +1363,7 @@ theorem idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFact
   by
   by_cases hI : I = ⊥
   · exfalso
-    rw [hI, bot_eq_zero, normalized_factors_zero, ← Multiset.empty_eq_zero] at hL 
+    rw [hI, bot_eq_zero, normalized_factors_zero, ← Multiset.empty_eq_zero] at hL
     exact hL
   · apply mem_normalizedFactors_factor_dvd_iso_of_mem_normalizedFactors hI hJ hL _
     rintro ⟨l, hl⟩ ⟨l', hl'⟩
@@ -1460,7 +1460,7 @@ theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n :
     Ideal.span_singleton_mul_span_singleton] at h ⊢
   by_cases ha : I ∣ span {a}
   · exact Or.inl ha
-  rw [mul_comm] at h 
+  rw [mul_comm] at h
   exact Or.inr (Prime.pow_dvd_of_dvd_mul_right ((Ideal.prime_iff_isPrime hI0).mpr hI) _ ha h)
 #align ideal.is_prime.mul_mem_pow Ideal.IsPrime.mul_mem_pow
 -/
@@ -1703,7 +1703,7 @@ theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [Normaliza
   by
   by_cases hb : b = 0
   · rw [ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalized_factors_zero]
-    rw [hb, normalized_factors_zero] at ha 
+    rw [hb, normalized_factors_zero] at ha
     simpa only [Multiset.not_mem_zero]
   · suffices Prime (Ideal.span ({a} : Set R))
       by
@@ -1740,7 +1740,7 @@ theorem multiplicity_eq_multiplicity_span [DecidableRel ((· ∣ ·) : R → R 
           ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _)))
   · suffices ¬Finite (Ideal.span ({a} : Set R)) (Ideal.span ({b} : Set R))
       by
-      rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this 
+      rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this
       rw [h, this]
     refine'
       not_finite_iff_forall.mpr fun n =>
@@ -1766,7 +1766,7 @@ noncomputable def normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r 
       constructor
       · rintro ⟨a, ha⟩ ⟨b, hb⟩ h
         rw [Subtype.mk_eq_mk, Ideal.span_singleton_eq_span_singleton, Subtype.coe_mk,
-          Subtype.coe_mk] at h 
+          Subtype.coe_mk] at h
         exact subtype.mk_eq_mk.mpr (mem_normalized_factors_eq_of_associated ha hb h)
       · rintro ⟨i, hi⟩
         letI : i.is_principal := inferInstance

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

@@ -463,11 +463,22 @@ theorem exists_multiset_prod_cons_le_and_prod_not_le [IsDedekindDomain A] (hNF :
   -- Then in fact there is a `P ∈ Z` with `P ≤ M`.
   obtain ⟨P, hPZ, rfl⟩ := multiset.mem_map.mp hPZ'
   classical
+  have := Multiset.map_erase PrimeSpectrum.asIdeal PrimeSpectrum.ext P Z
+  obtain ⟨hP0, hZP0⟩ : P.as_ideal ≠ ⊥ ∧ ((Z.erase P).map PrimeSpectrum.asIdeal).Prod ≠ ⊥ := by
+    rwa [Ne.def, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
+      this] at hprodZ 
+  -- By maximality of `P` and `M`, we have that `P ≤ M` implies `P = M`.
+  have hPM' := (IsDedekindDomain.dimensionLEOne _ hP0 P.is_prime).eq_of_le hM.ne_top hPM
+  subst hPM'
+  -- By minimality of `Z`, erasing `P` from `Z` is exactly what we need.
+  refine' ⟨Z.erase P, _, _⟩
+  · convert hZI
+    rw [this, Multiset.cons_erase hPZ']
+  · refine' fun h => h_eraseZ (Z.erase P) ⟨h, _⟩ (multiset.erase_lt.mpr hPZ)
+    exact hZP0
 #align exists_multiset_prod_cons_le_and_prod_not_le exists_multiset_prod_cons_le_and_prod_not_le
 -/
 
--- By maximality of `P` and `M`, we have that `P ≤ M` implies `P = M`.
--- By minimality of `Z`, erasing `P` from `Z` is exactly what we need.
 namespace FractionalIdeal
 
 open Ideal

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

@@ -463,22 +463,11 @@ theorem exists_multiset_prod_cons_le_and_prod_not_le [IsDedekindDomain A] (hNF :
   -- Then in fact there is a `P ∈ Z` with `P ≤ M`.
   obtain ⟨P, hPZ, rfl⟩ := multiset.mem_map.mp hPZ'
   classical
-  have := Multiset.map_erase PrimeSpectrum.asIdeal PrimeSpectrum.ext P Z
-  obtain ⟨hP0, hZP0⟩ : P.as_ideal ≠ ⊥ ∧ ((Z.erase P).map PrimeSpectrum.asIdeal).Prod ≠ ⊥ := by
-    rwa [Ne.def, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
-      this] at hprodZ 
-  -- By maximality of `P` and `M`, we have that `P ≤ M` implies `P = M`.
-  have hPM' := (IsDedekindDomain.dimensionLEOne _ hP0 P.is_prime).eq_of_le hM.ne_top hPM
-  subst hPM'
-  -- By minimality of `Z`, erasing `P` from `Z` is exactly what we need.
-  refine' ⟨Z.erase P, _, _⟩
-  · convert hZI
-    rw [this, Multiset.cons_erase hPZ']
-  · refine' fun h => h_eraseZ (Z.erase P) ⟨h, _⟩ (multiset.erase_lt.mpr hPZ)
-    exact hZP0
 #align exists_multiset_prod_cons_le_and_prod_not_le exists_multiset_prod_cons_le_and_prod_not_le
 -/
 
+-- By maximality of `P` and `M`, we have that `P ≤ M` implies `P = M`.
+-- By minimality of `Z`, erasing `P` from `Z` is exactly what we need.
 namespace FractionalIdeal
 
 open Ideal

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

@@ -878,12 +878,12 @@ theorem Ideal.isPrime_iff_bot_or_prime {P : Ideal A} : IsPrime P ↔ P = ⊥ ∨
 #align ideal.is_prime_iff_bot_or_prime Ideal.isPrime_iff_bot_or_prime
 -/
 
-#print Ideal.strictAnti_pow /-
-theorem Ideal.strictAnti_pow (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
+#print Ideal.pow_right_strictAnti /-
+theorem Ideal.pow_right_strictAnti (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
     StrictAnti ((· ^ ·) I : ℕ → Ideal A) :=
   strictAnti_nat_of_succ_lt fun e =>
     Ideal.dvdNotUnit_iff_lt.mp ⟨pow_ne_zero _ hI0, I, mt isUnit_iff.mp hI1, pow_succ' I e⟩
-#align ideal.strict_anti_pow Ideal.strictAnti_pow
+#align ideal.strict_anti_pow Ideal.pow_right_strictAnti
 -/
 
 #print Ideal.pow_lt_self /-
@@ -895,7 +895,7 @@ theorem Ideal.pow_lt_self (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) (e :
 #print Ideal.exists_mem_pow_not_mem_pow_succ /-
 theorem Ideal.exists_mem_pow_not_mem_pow_succ (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) (e : ℕ) :
     ∃ x ∈ I ^ e, x ∉ I ^ (e + 1) :=
-  SetLike.exists_of_lt (I.strictAnti_pow hI0 hI1 e.lt_succ_self)
+  SetLike.exists_of_lt (I.pow_right_strictAnti hI0 hI1 e.lt_succ_self)
 #align ideal.exists_mem_pow_not_mem_pow_succ Ideal.exists_mem_pow_not_mem_pow_succ
 -/
 

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

@@ -603,7 +603,7 @@ theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ 
       assumption
   -- For that, we'll find a subalgebra that is f.g. as a module and contains `x`.
   -- `A` is a noetherian ring, so we just need to find a subalgebra between `{x}` and `I⁻¹`.
-  rw [mem_integralClosure_iff_mem_FG]
+  rw [mem_integralClosure_iff_mem_fg]
   have x_mul_mem : ∀ b ∈ (I⁻¹ : FractionalIdeal A⁰ K), x * b ∈ (I⁻¹ : FractionalIdeal A⁰ K) :=
     by
     intro b hb

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

@@ -3,14 +3,14 @@ Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
 -/
-import Mathbin.Algebra.Algebra.Subalgebra.Pointwise
-import Mathbin.AlgebraicGeometry.PrimeSpectrum.Maximal
-import Mathbin.AlgebraicGeometry.PrimeSpectrum.Noetherian
-import Mathbin.Order.Hom.Basic
-import Mathbin.RingTheory.DedekindDomain.Basic
-import Mathbin.RingTheory.FractionalIdeal
-import Mathbin.RingTheory.PrincipalIdealDomain
-import Mathbin.RingTheory.ChainOfDivisors
+import Algebra.Algebra.Subalgebra.Pointwise
+import AlgebraicGeometry.PrimeSpectrum.Maximal
+import AlgebraicGeometry.PrimeSpectrum.Noetherian
+import Order.Hom.Basic
+import RingTheory.DedekindDomain.Basic
+import RingTheory.FractionalIdeal
+import RingTheory.PrincipalIdealDomain
+import RingTheory.ChainOfDivisors
 
 #align_import ring_theory.dedekind_domain.ideal from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"
 
@@ -295,7 +295,7 @@ noncomputable instance : InvOneClass (FractionalIdeal R₁⁰ K) :=
 
 end FractionalIdeal
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) (fraction_ring[fraction_ring] A))) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) (fraction_ring[fraction_ring] A))) -/
 #print IsDedekindDomainInv /-
 /-- A Dedekind domain is an integral domain such that every fractional ideal has an inverse.
 
@@ -313,7 +313,7 @@ open FractionalIdeal
 
 variable {R A K}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) K)) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) K)) -/
 #print isDedekindDomainInv_iff /-
 theorem isDedekindDomainInv_iff [Algebra A K] [IsFractionRing A K] :
     IsDedekindDomainInv A ↔ ∀ (I) (_ : I ≠ (⊥ : FractionalIdeal A⁰ K)), I * I⁻¹ = 1 :=
@@ -1528,7 +1528,7 @@ theorem Ideal.prod_le_prime {ι : Type _} {s : Finset ι} {f : ι → Ideal R} {
 #align ideal.prod_le_prime Ideal.prod_le_prime
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 #print IsDedekindDomain.inf_prime_pow_eq_prod /-
 /-- The intersection of distinct prime powers in a Dedekind domain is the product of these
 prime powers. -/
@@ -1628,7 +1628,7 @@ noncomputable def Ideal.quotientMulEquivQuotientProd (I J : Ideal R) (coprime :
 #align ideal.quotient_mul_equiv_quotient_prod Ideal.quotientMulEquivQuotientProd
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 #print IsDedekindDomain.quotientEquivPiOfFinsetProdEq /-
 /-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
 `∏ i in s, P i ^ e i`, then `R ⧸ I` factors as `Π (i : s), R ⧸ (P i ^ e i)`.
@@ -1646,7 +1646,7 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfFinsetProdEq {ι : Type _} {
 #align is_dedekind_domain.quotient_equiv_pi_of_finset_prod_eq IsDedekindDomain.quotientEquivPiOfFinsetProdEq
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 #print IsDedekindDomain.exists_representative_mod_finset /-
 /-- Corollary of the Chinese remainder theorem: given elements `x i : R / P i ^ e i`,
 we can choose a representative `y : R` such that `y ≡ x i (mod P i ^ e i)`.-/
@@ -1662,7 +1662,7 @@ theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type _} {s : Fin
 #align is_dedekind_domain.exists_representative_mod_finset IsDedekindDomain.exists_representative_mod_finset
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 #print IsDedekindDomain.exists_forall_sub_mem_ideal /-
 /-- Corollary of the Chinese remainder theorem: given elements `x i : R`,
 we can choose a representative `y : R` such that `y - x i ∈ P i ^ e i`.-/

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

@@ -728,7 +728,7 @@ noncomputable instance FractionalIdeal.semifield : Semifield (FractionalIdeal A
     inv := fun I => I⁻¹
     inv_zero := inv_zero' _
     div := (· / ·)
-    div_eq_mul_inv := FractionalIdeal.div_eq_mul_inv
+    div_eq_hMul_inv := FractionalIdeal.div_eq_mul_inv
     mul_inv_cancel := fun I => FractionalIdeal.mul_inv_cancel }
 #align fractional_ideal.semifield FractionalIdeal.semifield
 -/

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

@@ -283,7 +283,7 @@ theorem isPrincipal_inv (I : FractionalIdeal R₁⁰ K) [Submodule.IsPrincipal (
     (h : I ≠ 0) : Submodule.IsPrincipal I⁻¹.1 :=
   by
   rw [val_eq_coe, is_principal_iff]
-  use (generator (I : Submodule R₁ K))⁻¹
+  use(generator (I : Submodule R₁ K))⁻¹
   have hI : I * span_singleton _ (generator (I : Submodule R₁ K))⁻¹ = 1
   apply mul_generator_self_inv _ I h
   exact (right_inverse_eq _ I (span_singleton _ (generator (I : Submodule R₁ K))⁻¹) hI).symm
@@ -1287,7 +1287,7 @@ theorem idealFactorsFunOfQuotHom_id :
     idealFactorsFunOfQuotHom (RingHom.id (A ⧸ J)).is_surjective = OrderHom.id :=
   OrderHom.ext _ _
     (funext fun X => by
-      simp only [idealFactorsFunOfQuotHom, map_id, OrderHom.coe_fun_mk, OrderHom.id_coe, id.def,
+      simp only [idealFactorsFunOfQuotHom, map_id, OrderHom.coe_mk, OrderHom.id_coe, id.def,
         comap_map_of_surjective J.Quotient.mk quotient.mk_surjective, ←
         RingHom.ker_eq_comap_bot J.Quotient.mk, mk_ker, sup_eq_left.mpr (dvd_iff_le.mp X.prop),
         Subtype.coe_eta])
@@ -1303,8 +1303,8 @@ theorem idealFactorsFunOfQuotHom_comp {f : R ⧸ I →+* A ⧸ J} {g : A ⧸ J 
       idealFactorsFunOfQuotHom (show Function.Surjective (g.comp f) from hg.comp hf) :=
   by
   refine' OrderHom.ext _ _ (funext fun x => _)
-  rw [idealFactorsFunOfQuotHom, idealFactorsFunOfQuotHom, OrderHom.comp_coe, OrderHom.coe_fun_mk,
-    OrderHom.coe_fun_mk, Function.comp_apply, idealFactorsFunOfQuotHom, OrderHom.coe_fun_mk,
+  rw [idealFactorsFunOfQuotHom, idealFactorsFunOfQuotHom, OrderHom.comp_coe, OrderHom.coe_mk,
+    OrderHom.coe_mk, Function.comp_apply, idealFactorsFunOfQuotHom, OrderHom.coe_mk,
     Subtype.mk_eq_mk, Subtype.coe_mk, map_comap_of_surjective J.Quotient.mk quotient.mk_surjective,
     map_map]
 #align ideal_factors_fun_of_quot_hom_comp idealFactorsFunOfQuotHom_comp
@@ -1776,7 +1776,7 @@ noncomputable def normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r 
             (Submodule.IsPrincipal.prime_generator_of_isPrime i
                 (prime_of_normalized_factor i hi).NeZero).Irreducible
             _
-        · use ⟨a, ha⟩
+        · use⟨a, ha⟩
           simp only [Subtype.coe_mk, Subtype.mk_eq_mk, ← span_singleton_eq_span_singleton.mpr ha',
             Ideal.span_singleton_generator]
         ·

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

@@ -2,11 +2,6 @@
 Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-
-! This file was ported from Lean 3 source module ring_theory.dedekind_domain.ideal
-! leanprover-community/mathlib commit 36938f775671ff28bea1c0310f1608e4afbb22e0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.Algebra.Subalgebra.Pointwise
 import Mathbin.AlgebraicGeometry.PrimeSpectrum.Maximal
@@ -17,6 +12,8 @@ import Mathbin.RingTheory.FractionalIdeal
 import Mathbin.RingTheory.PrincipalIdealDomain
 import Mathbin.RingTheory.ChainOfDivisors
 
+#align_import ring_theory.dedekind_domain.ideal from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"
+
 /-!
 # Dedekind domains and ideals
 
@@ -298,7 +295,7 @@ noncomputable instance : InvOneClass (FractionalIdeal R₁⁰ K) :=
 
 end FractionalIdeal
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) (fraction_ring[fraction_ring] A))) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) (fraction_ring[fraction_ring] A))) -/
 #print IsDedekindDomainInv /-
 /-- A Dedekind domain is an integral domain such that every fractional ideal has an inverse.
 
@@ -316,7 +313,7 @@ open FractionalIdeal
 
 variable {R A K}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) K)) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) K)) -/
 #print isDedekindDomainInv_iff /-
 theorem isDedekindDomainInv_iff [Algebra A K] [IsFractionRing A K] :
     IsDedekindDomainInv A ↔ ∀ (I) (_ : I ≠ (⊥ : FractionalIdeal A⁰ K)), I * I⁻¹ = 1 :=
@@ -1531,7 +1528,7 @@ theorem Ideal.prod_le_prime {ι : Type _} {s : Finset ι} {f : ι → Ideal R} {
 #align ideal.prod_le_prime Ideal.prod_le_prime
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 #print IsDedekindDomain.inf_prime_pow_eq_prod /-
 /-- The intersection of distinct prime powers in a Dedekind domain is the product of these
 prime powers. -/
@@ -1631,7 +1628,7 @@ noncomputable def Ideal.quotientMulEquivQuotientProd (I J : Ideal R) (coprime :
 #align ideal.quotient_mul_equiv_quotient_prod Ideal.quotientMulEquivQuotientProd
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 #print IsDedekindDomain.quotientEquivPiOfFinsetProdEq /-
 /-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
 `∏ i in s, P i ^ e i`, then `R ⧸ I` factors as `Π (i : s), R ⧸ (P i ^ e i)`.
@@ -1649,7 +1646,7 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfFinsetProdEq {ι : Type _} {
 #align is_dedekind_domain.quotient_equiv_pi_of_finset_prod_eq IsDedekindDomain.quotientEquivPiOfFinsetProdEq
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 #print IsDedekindDomain.exists_representative_mod_finset /-
 /-- Corollary of the Chinese remainder theorem: given elements `x i : R / P i ^ e i`,
 we can choose a representative `y : R` such that `y ≡ x i (mod P i ^ e i)`.-/
@@ -1665,7 +1662,7 @@ theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type _} {s : Fin
 #align is_dedekind_domain.exists_representative_mod_finset IsDedekindDomain.exists_representative_mod_finset
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 #print IsDedekindDomain.exists_forall_sub_mem_ideal /-
 /-- Corollary of the Chinese remainder theorem: given elements `x i : R`,
 we can choose a representative `y : R` such that `y - x i ∈ P i ^ e i`.-/

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

@@ -75,45 +75,62 @@ variable {I J : FractionalIdeal R₁⁰ K}
 noncomputable instance : Inv (FractionalIdeal R₁⁰ K) :=
   ⟨fun I => 1 / I⟩
 
+#print FractionalIdeal.inv_eq /-
 theorem inv_eq : I⁻¹ = 1 / I :=
   rfl
 #align fractional_ideal.inv_eq FractionalIdeal.inv_eq
+-/
 
+#print FractionalIdeal.inv_zero' /-
 theorem inv_zero' : (0 : FractionalIdeal R₁⁰ K)⁻¹ = 0 :=
   div_zero
 #align fractional_ideal.inv_zero' FractionalIdeal.inv_zero'
+-/
 
+#print FractionalIdeal.inv_nonzero /-
 theorem inv_nonzero {J : FractionalIdeal R₁⁰ K} (h : J ≠ 0) :
     J⁻¹ = ⟨(1 : FractionalIdeal R₁⁰ K) / J, fractional_div_of_nonzero h⟩ :=
   div_nonzero _
 #align fractional_ideal.inv_nonzero FractionalIdeal.inv_nonzero
+-/
 
+#print FractionalIdeal.coe_inv_of_nonzero /-
 theorem coe_inv_of_nonzero {J : FractionalIdeal R₁⁰ K} (h : J ≠ 0) :
     (↑J⁻¹ : Submodule R₁ K) = IsLocalization.coeSubmodule K ⊤ / J := by rwa [inv_nonzero _]; rfl;
   assumption
 #align fractional_ideal.coe_inv_of_nonzero FractionalIdeal.coe_inv_of_nonzero
+-/
 
 variable {K}
 
+#print FractionalIdeal.mem_inv_iff /-
 theorem mem_inv_iff (hI : I ≠ 0) {x : K} : x ∈ I⁻¹ ↔ ∀ y ∈ I, x * y ∈ (1 : FractionalIdeal R₁⁰ K) :=
   mem_div_iff_of_nonzero hI
 #align fractional_ideal.mem_inv_iff FractionalIdeal.mem_inv_iff
+-/
 
+#print FractionalIdeal.inv_anti_mono /-
 theorem inv_anti_mono (hI : I ≠ 0) (hJ : J ≠ 0) (hIJ : I ≤ J) : J⁻¹ ≤ I⁻¹ := fun x => by
   simp only [mem_inv_iff hI, mem_inv_iff hJ]; exact fun h y hy => h y (hIJ hy)
 #align fractional_ideal.inv_anti_mono FractionalIdeal.inv_anti_mono
+-/
 
+#print FractionalIdeal.le_self_mul_inv /-
 theorem le_self_mul_inv {I : FractionalIdeal R₁⁰ K} (hI : I ≤ (1 : FractionalIdeal R₁⁰ K)) :
     I ≤ I * I⁻¹ :=
   le_self_mul_one_div hI
 #align fractional_ideal.le_self_mul_inv FractionalIdeal.le_self_mul_inv
+-/
 
 variable (K)
 
+#print FractionalIdeal.coe_ideal_le_self_mul_inv /-
 theorem coe_ideal_le_self_mul_inv (I : Ideal R₁) : (I : FractionalIdeal R₁⁰ K) ≤ I * I⁻¹ :=
   le_self_mul_inv coeIdeal_le_one
 #align fractional_ideal.coe_ideal_le_self_mul_inv FractionalIdeal.coe_ideal_le_self_mul_inv
+-/
 
+#print FractionalIdeal.right_inverse_eq /-
 /-- `I⁻¹` is the inverse of `I` if `I` has an inverse. -/
 theorem right_inverse_eq (I J : FractionalIdeal R₁⁰ K) (h : I * J = 1) : J = I⁻¹ :=
   by
@@ -135,69 +152,93 @@ theorem right_inverse_eq (I J : FractionalIdeal R₁⁰ K) (h : I * J = 1) : J =
   rw [mul_comm]
   exact mul_mem_mul hx hy
 #align fractional_ideal.right_inverse_eq FractionalIdeal.right_inverse_eq
+-/
 
+#print FractionalIdeal.mul_inv_cancel_iff /-
 theorem mul_inv_cancel_iff {I : FractionalIdeal R₁⁰ K} : I * I⁻¹ = 1 ↔ ∃ J, I * J = 1 :=
   ⟨fun h => ⟨I⁻¹, h⟩, fun ⟨J, hJ⟩ => by rwa [← right_inverse_eq K I J hJ]⟩
 #align fractional_ideal.mul_inv_cancel_iff FractionalIdeal.mul_inv_cancel_iff
+-/
 
+#print FractionalIdeal.mul_inv_cancel_iff_isUnit /-
 theorem mul_inv_cancel_iff_isUnit {I : FractionalIdeal R₁⁰ K} : I * I⁻¹ = 1 ↔ IsUnit I :=
   (mul_inv_cancel_iff K).trans isUnit_iff_exists_inv.symm
 #align fractional_ideal.mul_inv_cancel_iff_is_unit FractionalIdeal.mul_inv_cancel_iff_isUnit
+-/
 
 variable {K' : Type _} [Field K'] [Algebra R₁ K'] [IsFractionRing R₁ K']
 
+#print FractionalIdeal.map_inv /-
 @[simp]
 theorem map_inv (I : FractionalIdeal R₁⁰ K) (h : K ≃ₐ[R₁] K') :
     I⁻¹.map (h : K →ₐ[R₁] K') = (I.map h)⁻¹ := by rw [inv_eq, map_div, map_one, inv_eq]
 #align fractional_ideal.map_inv FractionalIdeal.map_inv
+-/
 
 open Submodule Submodule.IsPrincipal
 
+#print FractionalIdeal.spanSingleton_inv /-
 @[simp]
 theorem spanSingleton_inv (x : K) : (spanSingleton R₁⁰ x)⁻¹ = spanSingleton _ x⁻¹ :=
   one_div_spanSingleton x
 #align fractional_ideal.span_singleton_inv FractionalIdeal.spanSingleton_inv
+-/
 
+#print FractionalIdeal.spanSingleton_div_spanSingleton /-
 @[simp]
 theorem spanSingleton_div_spanSingleton (x y : K) :
     spanSingleton R₁⁰ x / spanSingleton R₁⁰ y = spanSingleton R₁⁰ (x / y) := by
   rw [div_span_singleton, mul_comm, span_singleton_mul_span_singleton, div_eq_mul_inv]
 #align fractional_ideal.span_singleton_div_span_singleton FractionalIdeal.spanSingleton_div_spanSingleton
+-/
 
+#print FractionalIdeal.spanSingleton_div_self /-
 theorem spanSingleton_div_self {x : K} (hx : x ≠ 0) :
     spanSingleton R₁⁰ x / spanSingleton R₁⁰ x = 1 := by
   rw [span_singleton_div_span_singleton, div_self hx, span_singleton_one]
 #align fractional_ideal.span_singleton_div_self FractionalIdeal.spanSingleton_div_self
+-/
 
+#print FractionalIdeal.coe_ideal_span_singleton_div_self /-
 theorem coe_ideal_span_singleton_div_self {x : R₁} (hx : x ≠ 0) :
     (Ideal.span ({x} : Set R₁) : FractionalIdeal R₁⁰ K) / Ideal.span ({x} : Set R₁) = 1 := by
   rw [coe_ideal_span_singleton,
     span_singleton_div_self K <|
       (map_ne_zero_iff _ <| NoZeroSMulDivisors.algebraMap_injective R₁ K).mpr hx]
 #align fractional_ideal.coe_ideal_span_singleton_div_self FractionalIdeal.coe_ideal_span_singleton_div_self
+-/
 
+#print FractionalIdeal.spanSingleton_mul_inv /-
 theorem spanSingleton_mul_inv {x : K} (hx : x ≠ 0) :
     spanSingleton R₁⁰ x * (spanSingleton R₁⁰ x)⁻¹ = 1 := by
   rw [span_singleton_inv, span_singleton_mul_span_singleton, mul_inv_cancel hx, span_singleton_one]
 #align fractional_ideal.span_singleton_mul_inv FractionalIdeal.spanSingleton_mul_inv
+-/
 
+#print FractionalIdeal.coe_ideal_span_singleton_mul_inv /-
 theorem coe_ideal_span_singleton_mul_inv {x : R₁} (hx : x ≠ 0) :
     (Ideal.span ({x} : Set R₁) : FractionalIdeal R₁⁰ K) * (Ideal.span ({x} : Set R₁))⁻¹ = 1 := by
   rw [coe_ideal_span_singleton,
     span_singleton_mul_inv K <|
       (map_ne_zero_iff _ <| NoZeroSMulDivisors.algebraMap_injective R₁ K).mpr hx]
 #align fractional_ideal.coe_ideal_span_singleton_mul_inv FractionalIdeal.coe_ideal_span_singleton_mul_inv
+-/
 
+#print FractionalIdeal.spanSingleton_inv_mul /-
 theorem spanSingleton_inv_mul {x : K} (hx : x ≠ 0) :
     (spanSingleton R₁⁰ x)⁻¹ * spanSingleton R₁⁰ x = 1 := by
   rw [mul_comm, span_singleton_mul_inv K hx]
 #align fractional_ideal.span_singleton_inv_mul FractionalIdeal.spanSingleton_inv_mul
+-/
 
+#print FractionalIdeal.coe_ideal_span_singleton_inv_mul /-
 theorem coe_ideal_span_singleton_inv_mul {x : R₁} (hx : x ≠ 0) :
     (Ideal.span ({x} : Set R₁) : FractionalIdeal R₁⁰ K)⁻¹ * Ideal.span ({x} : Set R₁) = 1 := by
   rw [mul_comm, coe_ideal_span_singleton_mul_inv K hx]
 #align fractional_ideal.coe_ideal_span_singleton_inv_mul FractionalIdeal.coe_ideal_span_singleton_inv_mul
+-/
 
+#print FractionalIdeal.mul_generator_self_inv /-
 theorem mul_generator_self_inv {R₁ : Type _} [CommRing R₁] [Algebra R₁ K] [IsLocalization R₁⁰ K]
     (I : FractionalIdeal R₁⁰ K) [Submodule.IsPrincipal (I : Submodule R₁ K)] (h : I ≠ 0) :
     I * spanSingleton _ (generator (I : Submodule R₁ K))⁻¹ = 1 :=
@@ -211,13 +252,17 @@ theorem mul_generator_self_inv {R₁ : Type _} [CommRing R₁] [Algebra R₁ K]
   apply h
   rw [eq_span_singleton_of_principal I, generator_I_eq_zero, span_singleton_zero]
 #align fractional_ideal.mul_generator_self_inv FractionalIdeal.mul_generator_self_inv
+-/
 
+#print FractionalIdeal.invertible_of_principal /-
 theorem invertible_of_principal (I : FractionalIdeal R₁⁰ K)
     [Submodule.IsPrincipal (I : Submodule R₁ K)] (h : I ≠ 0) : I * I⁻¹ = 1 :=
   mul_div_self_cancel_iff.mpr
     ⟨spanSingleton _ (generator (I : Submodule R₁ K))⁻¹, mul_generator_self_inv _ I h⟩
 #align fractional_ideal.invertible_of_principal FractionalIdeal.invertible_of_principal
+-/
 
+#print FractionalIdeal.invertible_iff_generator_nonzero /-
 theorem invertible_iff_generator_nonzero (I : FractionalIdeal R₁⁰ K)
     [Submodule.IsPrincipal (I : Submodule R₁ K)] :
     I * I⁻¹ = 1 ↔ generator (I : Submodule R₁ K) ≠ 0 :=
@@ -234,7 +279,9 @@ theorem invertible_iff_generator_nonzero (I : FractionalIdeal R₁⁰ K)
     rw [hI, mem_zero_iff] at this 
     contradiction
 #align fractional_ideal.invertible_iff_generator_nonzero FractionalIdeal.invertible_iff_generator_nonzero
+-/
 
+#print FractionalIdeal.isPrincipal_inv /-
 theorem isPrincipal_inv (I : FractionalIdeal R₁⁰ K) [Submodule.IsPrincipal (I : Submodule R₁ K)]
     (h : I ≠ 0) : Submodule.IsPrincipal I⁻¹.1 :=
   by
@@ -244,6 +291,7 @@ theorem isPrincipal_inv (I : FractionalIdeal R₁⁰ K) [Submodule.IsPrincipal (
   apply mul_generator_self_inv _ I h
   exact (right_inverse_eq _ I (span_singleton _ (generator (I : Submodule R₁ K))⁻¹) hI).symm
 #align fractional_ideal.is_principal_inv FractionalIdeal.isPrincipal_inv
+-/
 
 noncomputable instance : InvOneClass (FractionalIdeal R₁⁰ K) :=
   { FractionalIdeal.hasOne, FractionalIdeal.hasInv K with inv_one := div_one }
@@ -269,6 +317,7 @@ open FractionalIdeal
 variable {R A K}
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) K)) -/
+#print isDedekindDomainInv_iff /-
 theorem isDedekindDomainInv_iff [Algebra A K] [IsFractionRing A K] :
     IsDedekindDomainInv A ↔ ∀ (I) (_ : I ≠ (⊥ : FractionalIdeal A⁰ K)), I * I⁻¹ = 1 :=
   by
@@ -277,7 +326,9 @@ theorem isDedekindDomainInv_iff [Algebra A K] [IsFractionRing A K] :
   rw [← h.to_equiv.apply_eq_iff_eq]
   simp [IsDedekindDomainInv, show ⇑h.to_equiv = h from rfl]
 #align is_dedekind_domain_inv_iff isDedekindDomainInv_iff
+-/
 
+#print FractionalIdeal.adjoinIntegral_eq_one_of_isUnit /-
 theorem FractionalIdeal.adjoinIntegral_eq_one_of_isUnit [Algebra A K] [IsFractionRing A K] (x : K)
     (hx : IsIntegral A x) (hI : IsUnit (adjoinIntegral A⁰ x hx)) : adjoinIntegral A⁰ x hx = 1 :=
   by
@@ -286,24 +337,29 @@ theorem FractionalIdeal.adjoinIntegral_eq_one_of_isUnit [Algebra A K] [IsFractio
   convert congr_arg (· * I⁻¹) mul_self <;>
     simp only [(mul_inv_cancel_iff_is_unit K).mpr hI, mul_assoc, mul_one]
 #align fractional_ideal.adjoin_integral_eq_one_of_is_unit FractionalIdeal.adjoinIntegral_eq_one_of_isUnit
+-/
 
 namespace IsDedekindDomainInv
 
 variable [Algebra A K] [IsFractionRing A K] (h : IsDedekindDomainInv A)
 
-include h
-
+#print IsDedekindDomainInv.mul_inv_eq_one /-
 theorem mul_inv_eq_one {I : FractionalIdeal A⁰ K} (hI : I ≠ 0) : I * I⁻¹ = 1 :=
   isDedekindDomainInv_iff.mp h I hI
 #align is_dedekind_domain_inv.mul_inv_eq_one IsDedekindDomainInv.mul_inv_eq_one
+-/
 
+#print IsDedekindDomainInv.inv_mul_eq_one /-
 theorem inv_mul_eq_one {I : FractionalIdeal A⁰ K} (hI : I ≠ 0) : I⁻¹ * I = 1 :=
   (mul_comm _ _).trans (h.mul_inv_eq_one hI)
 #align is_dedekind_domain_inv.inv_mul_eq_one IsDedekindDomainInv.inv_mul_eq_one
+-/
 
+#print IsDedekindDomainInv.isUnit /-
 protected theorem isUnit {I : FractionalIdeal A⁰ K} (hI : I ≠ 0) : IsUnit I :=
   isUnit_of_mul_eq_one _ _ (h.mul_inv_eq_one hI)
 #align is_dedekind_domain_inv.is_unit IsDedekindDomainInv.isUnit
+-/
 
 #print IsDedekindDomainInv.isNoetherianRing /-
 theorem isNoetherianRing : IsNoetherianRing A :=
@@ -385,6 +441,7 @@ end IsDedekindDomainInv
 
 variable [Algebra A K] [IsFractionRing A K]
 
+#print exists_multiset_prod_cons_le_and_prod_not_le /-
 /-- Specialization of `exists_prime_spectrum_prod_le_and_ne_bot_of_domain` to Dedekind domains:
 Let `I : ideal A` be a nonzero ideal, where `A` is a Dedekind domain that is not a field.
 Then `exists_prime_spectrum_prod_le_and_ne_bot_of_domain` states we can find a product of prime
@@ -423,11 +480,13 @@ theorem exists_multiset_prod_cons_le_and_prod_not_le [IsDedekindDomain A] (hNF :
   · refine' fun h => h_eraseZ (Z.erase P) ⟨h, _⟩ (multiset.erase_lt.mpr hPZ)
     exact hZP0
 #align exists_multiset_prod_cons_le_and_prod_not_le exists_multiset_prod_cons_le_and_prod_not_le
+-/
 
 namespace FractionalIdeal
 
 open Ideal
 
+#print FractionalIdeal.exists_not_mem_one_of_ne_bot /-
 theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
     (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
     ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) :=
@@ -483,7 +542,9 @@ theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I
     have := ideal.mem_span_singleton'.mpr ⟨x', IsFractionRing.injective A K h₂_abs⟩
     contradiction
 #align fractional_ideal.exists_not_mem_one_of_ne_bot FractionalIdeal.exists_not_mem_one_of_ne_bot
+-/
 
+#print FractionalIdeal.one_mem_inv_coe_ideal /-
 theorem one_mem_inv_coe_ideal {I : Ideal A} (hI : I ≠ ⊥) : (1 : K) ∈ (I : FractionalIdeal A⁰ K)⁻¹ :=
   by
   rw [mem_inv_iff (coe_ideal_ne_zero.mpr hI)]
@@ -492,7 +553,9 @@ theorem one_mem_inv_coe_ideal {I : Ideal A} (hI : I ≠ ⊥) : (1 : K) ∈ (I :
   exact coe_ideal_le_one hy
   assumption
 #align fractional_ideal.one_mem_inv_coe_ideal FractionalIdeal.one_mem_inv_coe_ideal
+-/
 
+#print FractionalIdeal.mul_inv_cancel_of_le_one /-
 theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
     (hI : ((I * I⁻¹)⁻¹ : FractionalIdeal A⁰ K) ≤ 1) : (I * I⁻¹ : FractionalIdeal A⁰ K) = 1 :=
   by
@@ -520,7 +583,9 @@ theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I
   rw [hJ] at hx 
   exact hI hx
 #align fractional_ideal.mul_inv_cancel_of_le_one FractionalIdeal.mul_inv_cancel_of_le_one
+-/
 
+#print FractionalIdeal.coe_ideal_mul_inv /-
 /-- Nonzero integral ideals in a Dedekind domain are invertible.
 
 We will use this to show that nonzero fractional ideals are invertible,
@@ -565,7 +630,9 @@ theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ 
   · show x ^ i.succ ∈ (I⁻¹ : FractionalIdeal A⁰ K)
     rw [pow_succ]; exact x_mul_mem _ ih
 #align fractional_ideal.coe_ideal_mul_inv FractionalIdeal.coe_ideal_mul_inv
+-/
 
+#print FractionalIdeal.mul_inv_cancel /-
 /-- Nonzero fractional ideals in a Dedekind domain are units.
 
 This is also available as `_root_.mul_inv_cancel`, using the
@@ -586,7 +653,9 @@ protected theorem mul_inv_cancel [IsDedekindDomain A] {I : FractionalIdeal A⁰
   · exact mt ((injective_iff_map_eq_zero (algebraMap A K)).mp (IsFractionRing.injective A K) _) ha
   · exact coe_ideal_ne_zero.mp (right_ne_zero_of_mul hne)
 #align fractional_ideal.mul_inv_cancel FractionalIdeal.mul_inv_cancel
+-/
 
+#print FractionalIdeal.mul_right_le_iff /-
 theorem mul_right_le_iff [IsDedekindDomain A] {J : FractionalIdeal A⁰ K} (hJ : J ≠ 0) :
     ∀ {I I'}, I * J ≤ I' * J ↔ I ≤ I' := by
   intro I I'
@@ -595,21 +664,29 @@ theorem mul_right_le_iff [IsDedekindDomain A] {J : FractionalIdeal A⁰ K} (hJ :
     convert mul_right_mono J⁻¹ h <;> rw [mul_assoc, FractionalIdeal.mul_inv_cancel hJ, mul_one]
   · exact fun h => mul_right_mono J h
 #align fractional_ideal.mul_right_le_iff FractionalIdeal.mul_right_le_iff
+-/
 
+#print FractionalIdeal.mul_left_le_iff /-
 theorem mul_left_le_iff [IsDedekindDomain A] {J : FractionalIdeal A⁰ K} (hJ : J ≠ 0) {I I'} :
     J * I ≤ J * I' ↔ I ≤ I' := by convert mul_right_le_iff hJ using 1 <;> simp only [mul_comm]
 #align fractional_ideal.mul_left_le_iff FractionalIdeal.mul_left_le_iff
+-/
 
+#print FractionalIdeal.mul_right_strictMono /-
 theorem mul_right_strictMono [IsDedekindDomain A] {I : FractionalIdeal A⁰ K} (hI : I ≠ 0) :
     StrictMono (· * I) :=
   strictMono_of_le_iff_le fun _ _ => (mul_right_le_iff hI).symm
 #align fractional_ideal.mul_right_strict_mono FractionalIdeal.mul_right_strictMono
+-/
 
+#print FractionalIdeal.mul_left_strictMono /-
 theorem mul_left_strictMono [IsDedekindDomain A] {I : FractionalIdeal A⁰ K} (hI : I ≠ 0) :
     StrictMono ((· * ·) I) :=
   strictMono_of_le_iff_le fun _ _ => (mul_left_le_iff hI).symm
 #align fractional_ideal.mul_left_strict_mono FractionalIdeal.mul_left_strictMono
+-/
 
+#print FractionalIdeal.div_eq_mul_inv /-
 /-- This is also available as `_root_.div_eq_mul_inv`, using the
 `comm_group_with_zero` instance defined below.
 -/
@@ -625,6 +702,7 @@ protected theorem div_eq_mul_inv [IsDedekindDomain A] (I J : FractionalIdeal A
   rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one]
   exact le_refl I
 #align fractional_ideal.div_eq_mul_inv FractionalIdeal.div_eq_mul_inv
+-/
 
 end FractionalIdeal
 
@@ -687,6 +765,7 @@ instance Ideal.isDomain : IsDomain (Ideal A) :=
 #align ideal.is_domain Ideal.isDomain
 -/
 
+#print Ideal.dvd_iff_le /-
 /-- For ideals in a Dedekind domain, to divide is to contain. -/
 theorem Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
   ⟨Ideal.le_of_dvd, fun h => by
@@ -702,7 +781,9 @@ theorem Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
     refine' coe_ideal_injective (show (J : FractionalIdeal A⁰ (FractionRing A)) = ↑(I * H) from _)
     rw [coe_ideal_mul, hH, ← mul_assoc, mul_inv_cancel hI', one_mul]⟩
 #align ideal.dvd_iff_le Ideal.dvd_iff_le
+-/
 
+#print Ideal.dvdNotUnit_iff_lt /-
 theorem Ideal.dvdNotUnit_iff_lt {I J : Ideal A} : DvdNotUnit I J ↔ J < I :=
   ⟨fun ⟨hI, H, hunit, hmul⟩ =>
     lt_of_le_of_ne (Ideal.dvd_iff_le.mp ⟨H, hmul⟩)
@@ -715,6 +796,7 @@ theorem Ideal.dvdNotUnit_iff_lt {I J : Ideal A} : DvdNotUnit I J ↔ J < I :=
     dvdNotUnit_of_dvd_of_not_dvd (Ideal.dvd_iff_le.mpr (le_of_lt h))
       (mt Ideal.dvd_iff_le.mp (not_le_of_lt h))⟩
 #align ideal.dvd_not_unit_iff_lt Ideal.dvdNotUnit_iff_lt
+-/
 
 instance : WfDvdMonoid (Ideal A)
     where wellFounded_dvdNotUnit :=
@@ -755,11 +837,14 @@ instance Ideal.normalizationMonoid : NormalizationMonoid (Ideal A) :=
 #align ideal.normalization_monoid Ideal.normalizationMonoid
 -/
 
+#print Ideal.dvd_span_singleton /-
 @[simp]
 theorem Ideal.dvd_span_singleton {I : Ideal A} {x : A} : I ∣ Ideal.span {x} ↔ x ∈ I :=
   Ideal.dvd_iff_le.trans (Ideal.span_le.trans Set.singleton_subset_iff)
 #align ideal.dvd_span_singleton Ideal.dvd_span_singleton
+-/
 
+#print Ideal.isPrime_of_prime /-
 theorem Ideal.isPrime_of_prime {P : Ideal A} (h : Prime P) : IsPrime P :=
   by
   refine' ⟨_, fun x y hxy => _⟩
@@ -769,43 +854,57 @@ theorem Ideal.isPrime_of_prime {P : Ideal A} (h : Prime P) : IsPrime P :=
   · simp only [← Ideal.dvd_span_singleton, ← Ideal.span_singleton_mul_span_singleton] at hxy ⊢
     exact h.dvd_or_dvd hxy
 #align ideal.is_prime_of_prime Ideal.isPrime_of_prime
+-/
 
+#print Ideal.prime_of_isPrime /-
 theorem Ideal.prime_of_isPrime {P : Ideal A} (hP : P ≠ ⊥) (h : IsPrime P) : Prime P :=
   by
   refine' ⟨hP, mt ideal.is_unit_iff.mp h.ne_top, fun I J hIJ => _⟩
   simpa only [Ideal.dvd_iff_le] using h.mul_le.mp (Ideal.le_of_dvd hIJ)
 #align ideal.prime_of_is_prime Ideal.prime_of_isPrime
+-/
 
+#print Ideal.prime_iff_isPrime /-
 /-- In a Dedekind domain, the (nonzero) prime elements of the monoid with zero `ideal A`
 are exactly the prime ideals. -/
 theorem Ideal.prime_iff_isPrime {P : Ideal A} (hP : P ≠ ⊥) : Prime P ↔ IsPrime P :=
   ⟨Ideal.isPrime_of_prime, Ideal.prime_of_isPrime hP⟩
 #align ideal.prime_iff_is_prime Ideal.prime_iff_isPrime
+-/
 
+#print Ideal.isPrime_iff_bot_or_prime /-
 /-- In a Dedekind domain, the the prime ideals are the zero ideal together with the prime elements
 of the monoid with zero `ideal A`. -/
 theorem Ideal.isPrime_iff_bot_or_prime {P : Ideal A} : IsPrime P ↔ P = ⊥ ∨ Prime P :=
   ⟨fun hp => (eq_or_ne P ⊥).imp_right fun hp0 => Ideal.prime_of_isPrime hp0 hp, fun hp =>
     hp.elim (fun h => h.symm ▸ Ideal.bot_prime) Ideal.isPrime_of_prime⟩
 #align ideal.is_prime_iff_bot_or_prime Ideal.isPrime_iff_bot_or_prime
+-/
 
+#print Ideal.strictAnti_pow /-
 theorem Ideal.strictAnti_pow (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
     StrictAnti ((· ^ ·) I : ℕ → Ideal A) :=
   strictAnti_nat_of_succ_lt fun e =>
     Ideal.dvdNotUnit_iff_lt.mp ⟨pow_ne_zero _ hI0, I, mt isUnit_iff.mp hI1, pow_succ' I e⟩
 #align ideal.strict_anti_pow Ideal.strictAnti_pow
+-/
 
+#print Ideal.pow_lt_self /-
 theorem Ideal.pow_lt_self (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) (e : ℕ) (he : 2 ≤ e) :
     I ^ e < I := by convert I.strict_anti_pow hI0 hI1 he <;> rw [pow_one]
 #align ideal.pow_lt_self Ideal.pow_lt_self
+-/
 
+#print Ideal.exists_mem_pow_not_mem_pow_succ /-
 theorem Ideal.exists_mem_pow_not_mem_pow_succ (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) (e : ℕ) :
     ∃ x ∈ I ^ e, x ∉ I ^ (e + 1) :=
   SetLike.exists_of_lt (I.strictAnti_pow hI0 hI1 e.lt_succ_self)
 #align ideal.exists_mem_pow_not_mem_pow_succ Ideal.exists_mem_pow_not_mem_pow_succ
+-/
 
 open UniqueFactorizationMonoid
 
+#print Ideal.eq_prime_pow_of_succ_lt_of_le /-
 theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥)
     {i : ℕ} (hlt : P ^ (i + 1) < I) (hle : I ≤ P ^ i) : I = P ^ i :=
   by
@@ -822,22 +921,28 @@ theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime
     normalized_factors_irreducible P_prime'.irreducible, Multiset.nsmul_singleton]
   all_goals assumption
 #align ideal.eq_prime_pow_of_succ_lt_of_le Ideal.eq_prime_pow_of_succ_lt_of_le
+-/
 
+#print Ideal.pow_succ_lt_pow /-
 theorem Ideal.pow_succ_lt_pow {P : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥) (i : ℕ) :
     P ^ (i + 1) < P ^ i :=
   lt_of_le_of_ne (Ideal.pow_le_pow (Nat.le_succ _))
     (mt (pow_eq_pow_iff hP (mt Ideal.isUnit_iff.mp P_prime.ne_top)).mp i.succ_ne_self)
 #align ideal.pow_succ_lt_pow Ideal.pow_succ_lt_pow
+-/
 
+#print Associates.le_singleton_iff /-
 theorem Associates.le_singleton_iff (x : A) (n : ℕ) (I : Ideal A) :
     Associates.mk I ^ n ≤ Associates.mk (Ideal.span {x}) ↔ x ∈ I ^ n := by
   rw [← Associates.dvd_eq_le, ← Associates.mk_pow, Associates.mk_dvd_mk, Ideal.dvd_span_singleton]
 #align associates.le_singleton_iff Associates.le_singleton_iff
+-/
 
 open FractionalIdeal
 
 variable {A K}
 
+#print Ideal.exist_integer_multiples_not_mem /-
 /-- Strengthening of `is_localization.exist_integer_multiples`:
 Let `J ≠ ⊤` be an ideal in a Dedekind domain `A`, and `f ≠ 0` a finite collection
 of elements of `K = Frac(A)`, then we can multiply the elements of `f` by some `a : K`
@@ -879,6 +984,7 @@ theorem Ideal.exist_integer_multiples_not_mem {J : Ideal A} (hJ : J ≠ ⊤) {ι
       strictMono_of_le_iff_le (fun _ _ => (coe_ideal_le_coe_ideal K).symm)
         (lt_top_iff_ne_top.mpr hJ)
 #align ideal.exist_integer_multiples_not_mem Ideal.exist_integer_multiples_not_mem
+-/
 
 section Gcd
 
@@ -932,20 +1038,26 @@ instance : NormalizedGCDMonoid (Ideal A) :=
     normalize_gcd := fun _ _ => normalize_eq _
     normalize_lcm := fun _ _ => normalize_eq _ }
 
+#print Ideal.gcd_eq_sup /-
 -- In fact, any lawful gcd and lcm would equal sup and inf respectively.
 @[simp]
 theorem gcd_eq_sup (I J : Ideal A) : gcd I J = I ⊔ J :=
   rfl
 #align ideal.gcd_eq_sup Ideal.gcd_eq_sup
+-/
 
+#print Ideal.lcm_eq_inf /-
 @[simp]
 theorem lcm_eq_inf (I J : Ideal A) : lcm I J = I ⊓ J :=
   rfl
 #align ideal.lcm_eq_inf Ideal.lcm_eq_inf
+-/
 
+#print Ideal.inf_eq_mul_of_coprime /-
 theorem inf_eq_mul_of_coprime {I J : Ideal A} (coprime : I ⊔ J = ⊤) : I ⊓ J = I * J := by
   rw [← associated_iff_eq.mp (gcd_mul_lcm I J), lcm_eq_inf I J, gcd_eq_sup, coprime, top_mul]
 #align ideal.inf_eq_mul_of_coprime Ideal.inf_eq_mul_of_coprime
+-/
 
 end Ideal
 
@@ -961,10 +1073,13 @@ open scoped Classical
 
 open Multiset UniqueFactorizationMonoid Ideal
 
+#print prod_normalizedFactors_eq_self /-
 theorem prod_normalizedFactors_eq_self (hI : I ≠ ⊥) : (normalizedFactors I).Prod = I :=
   associated_iff_eq.1 (normalizedFactors_prod hI)
 #align prod_normalized_factors_eq_self prod_normalizedFactors_eq_self
+-/
 
+#print count_le_of_ideal_ge /-
 theorem count_le_of_ideal_ge {I J : Ideal T} (h : I ≤ J) (hI : I ≠ ⊥) (K : Ideal T) :
     count K (normalizedFactors J) ≤ count K (normalizedFactors I) :=
   le_iff_count.1
@@ -972,7 +1087,9 @@ theorem count_le_of_ideal_ge {I J : Ideal T} (h : I ≤ J) (hI : I ≠ ⊥) (K :
       (dvd_iff_le.2 h))
     _
 #align count_le_of_ideal_ge count_le_of_ideal_ge
+-/
 
+#print sup_eq_prod_inf_factors /-
 theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
     I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).Prod :=
   by
@@ -1005,13 +1122,17 @@ theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
     · exact ne_bot_of_le_ne_bot hI le_sup_left
     · exact this
 #align sup_eq_prod_inf_factors sup_eq_prod_inf_factors
+-/
 
+#print irreducible_pow_sup /-
 theorem irreducible_pow_sup (hI : I ≠ ⊥) (hJ : Irreducible J) (n : ℕ) :
     J ^ n ⊔ I = J ^ min ((normalizedFactors I).count J) n := by
   rw [sup_eq_prod_inf_factors (pow_ne_zero n hJ.ne_zero) hI, min_comm,
     normalized_factors_of_irreducible_pow hJ, normalize_eq J, replicate_inter, prod_replicate]
 #align irreducible_pow_sup irreducible_pow_sup
+-/
 
+#print irreducible_pow_sup_of_le /-
 theorem irreducible_pow_sup_of_le (hJ : Irreducible J) (n : ℕ) (hn : ↑n ≤ multiplicity J I) :
     J ^ n ⊔ I = J ^ n := by
   by_cases hI : I = ⊥
@@ -1019,7 +1140,9 @@ theorem irreducible_pow_sup_of_le (hJ : Irreducible J) (n : ℕ) (hn : ↑n ≤
   rw [irreducible_pow_sup hI hJ, min_eq_right]
   rwa [multiplicity_eq_count_normalized_factors hJ hI, PartENat.coe_le_coe, normalize_eq J] at hn 
 #align irreducible_pow_sup_of_le irreducible_pow_sup_of_le
+-/
 
+#print irreducible_pow_sup_of_ge /-
 theorem irreducible_pow_sup_of_ge (hI : I ≠ ⊥) (hJ : Irreducible J) (n : ℕ)
     (hn : multiplicity J I ≤ n) :
     J ^ n ⊔ I = J ^ (multiplicity J I).get (PartENat.dom_of_le_natCast hn) :=
@@ -1031,6 +1154,7 @@ theorem irreducible_pow_sup_of_ge (hI : I ≠ ⊥) (hJ : Irreducible J) (n : ℕ
       multiplicity_eq_count_normalized_factors hJ hI, normalize_eq J]
   · rwa [multiplicity_eq_count_normalized_factors hJ hI, PartENat.coe_le_coe, normalize_eq J] at hn 
 #align irreducible_pow_sup_of_ge irreducible_pow_sup_of_ge
+-/
 
 end IsDedekindDomain
 
@@ -1047,6 +1171,7 @@ namespace IsDedekindDomain
 
 variable [IsDomain R] [IsDedekindDomain R]
 
+#print IsDedekindDomain.HeightOneSpectrum /-
 /-- The height one prime spectrum of a Dedekind domain `R` is the type of nonzero prime ideals of
 `R`. Note that this equals the maximal spectrum if `R` has Krull dimension 1. -/
 @[ext, nolint has_nonempty_instance unused_arguments]
@@ -1055,6 +1180,7 @@ structure HeightOneSpectrum where
   IsPrime : as_ideal.IsPrime
   ne_bot : as_ideal ≠ ⊥
 #align is_dedekind_domain.height_one_spectrum IsDedekindDomain.HeightOneSpectrum
+-/
 
 attribute [instance] height_one_spectrum.is_prime
 
@@ -1100,6 +1226,7 @@ def equivMaximalSpectrum (hR : ¬IsField R) : HeightOneSpectrum R ≃ MaximalSpe
 
 variable (R K)
 
+#print IsDedekindDomain.HeightOneSpectrum.iInf_localization_eq_bot /-
 /-- A Dedekind domain is equal to the intersection of its localizations at all its height one
 non-zero prime ideals viewed as subalgebras of its field of fractions. -/
 theorem iInf_localization_eq_bot [Algebra R K] [hK : IsFractionRing R K] :
@@ -1120,6 +1247,7 @@ theorem iInf_localization_eq_bot [Algebra R K] [hK : IsFractionRing R K] :
   · exact fun hx ⟨v, hv⟩ => hx ((equiv_maximal_spectrum hR).symm ⟨v, hv⟩)
   · exact fun hx ⟨v, hv, hbot⟩ => hx ⟨v, dimension_le_one v hbot hv⟩
 #align is_dedekind_domain.height_one_spectrum.infi_localization_eq_bot IsDedekindDomain.HeightOneSpectrum.iInf_localization_eq_bot
+-/
 
 end HeightOneSpectrum
 
@@ -1131,6 +1259,7 @@ open Ideal
 
 variable {R} {A} [IsDedekindDomain A] {I : Ideal R} {J : Ideal A}
 
+#print idealFactorsFunOfQuotHom /-
 /-- The map from ideals of `R` dividing `I` to the ideals of `A` dividing `J` induced by
   a homomorphism `f : R/I →+* A/J` -/
 @[simps]
@@ -1153,7 +1282,9 @@ def idealFactorsFunOfQuotHom {f : R ⧸ I →+* A ⧸ J} (hf : Function.Surjecti
     rwa [map_le_iff_le_comap, comap_map_of_surjective I.Quotient.mk quotient.mk_surjective, ←
       RingHom.ker_eq_comap_bot, mk_ker, sup_eq_left.mpr <| le_of_dvd hY]
 #align ideal_factors_fun_of_quot_hom idealFactorsFunOfQuotHom
+-/
 
+#print idealFactorsFunOfQuotHom_id /-
 @[simp]
 theorem idealFactorsFunOfQuotHom_id :
     idealFactorsFunOfQuotHom (RingHom.id (A ⧸ J)).is_surjective = OrderHom.id :=
@@ -1164,9 +1295,11 @@ theorem idealFactorsFunOfQuotHom_id :
         RingHom.ker_eq_comap_bot J.Quotient.mk, mk_ker, sup_eq_left.mpr (dvd_iff_le.mp X.prop),
         Subtype.coe_eta])
 #align ideal_factors_fun_of_quot_hom_id idealFactorsFunOfQuotHom_id
+-/
 
 variable {B : Type _} [CommRing B] [IsDomain B] [IsDedekindDomain B] {L : Ideal B}
 
+#print idealFactorsFunOfQuotHom_comp /-
 theorem idealFactorsFunOfQuotHom_comp {f : R ⧸ I →+* A ⧸ J} {g : A ⧸ J →+* B ⧸ L}
     (hf : Function.Surjective f) (hg : Function.Surjective g) :
     (idealFactorsFunOfQuotHom hg).comp (idealFactorsFunOfQuotHom hf) =
@@ -1178,9 +1311,11 @@ theorem idealFactorsFunOfQuotHom_comp {f : R ⧸ I →+* A ⧸ J} {g : A ⧸ J 
     Subtype.mk_eq_mk, Subtype.coe_mk, map_comap_of_surjective J.Quotient.mk quotient.mk_surjective,
     map_map]
 #align ideal_factors_fun_of_quot_hom_comp idealFactorsFunOfQuotHom_comp
+-/
 
 variable [IsDomain R] [IsDedekindDomain R] (f : R ⧸ I ≃+* A ⧸ J)
 
+#print idealFactorsEquivOfQuotEquiv /-
 /-- The bijection between ideals of `R` dividing `I` and the ideals of `A` dividing `J` induced by
   an isomorphism `f : R/I ≅ A/J`. -/
 @[simps]
@@ -1198,12 +1333,16 @@ def idealFactorsEquivOfQuotEquiv : {p : Ideal R | p ∣ I} ≃o {p : Ideal A | p
         idealFactorsFunOfQuotHom_comp, ← RingEquiv.toRingHom_eq_coe, ← RingEquiv.toRingHom_eq_coe, ←
         RingEquiv.toRingHom_trans, RingEquiv.self_trans_symm, RingEquiv.toRingHom_refl])
 #align ideal_factors_equiv_of_quot_equiv idealFactorsEquivOfQuotEquiv
+-/
 
+#print idealFactorsEquivOfQuotEquiv_symm /-
 theorem idealFactorsEquivOfQuotEquiv_symm :
     (idealFactorsEquivOfQuotEquiv f).symm = idealFactorsEquivOfQuotEquiv f.symm :=
   rfl
 #align ideal_factors_equiv_of_quot_equiv_symm idealFactorsEquivOfQuotEquiv_symm
+-/
 
+#print idealFactorsEquivOfQuotEquiv_is_dvd_iso /-
 theorem idealFactorsEquivOfQuotEquiv_is_dvd_iso {L M : Ideal R} (hL : L ∣ I) (hM : M ∣ I) :
     (idealFactorsEquivOfQuotEquiv f ⟨L, hL⟩ : Ideal A) ∣ idealFactorsEquivOfQuotEquiv f ⟨M, hM⟩ ↔
       L ∣ M :=
@@ -1214,11 +1353,13 @@ theorem idealFactorsEquivOfQuotEquiv_is_dvd_iso {L M : Ideal R} (hL : L ∣ I) (
     by rw [dvd_iff_le, dvd_iff_le, Subtype.coe_le_coe, this, Subtype.mk_le_mk]
   exact (idealFactorsEquivOfQuotEquiv f).le_iff_le
 #align ideal_factors_equiv_of_quot_equiv_is_dvd_iso idealFactorsEquivOfQuotEquiv_is_dvd_iso
+-/
 
 open UniqueFactorizationMonoid
 
 variable [DecidableEq (Ideal R)] [DecidableEq (Ideal A)]
 
+#print idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors /-
 theorem idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors (hJ : J ≠ ⊥)
     {L : Ideal R} (hL : L ∈ normalizedFactors I) :
     ↑(idealFactorsEquivOfQuotEquiv f ⟨L, dvd_of_mem_normalizedFactors hL⟩) ∈ normalizedFactors J :=
@@ -1232,7 +1373,9 @@ theorem idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFact
     rw [Subtype.coe_mk, Subtype.coe_mk]
     apply idealFactorsEquivOfQuotEquiv_is_dvd_iso f
 #align ideal_factors_equiv_of_quot_equiv_mem_normalized_factors_of_mem_normalized_factors idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors
+-/
 
+#print normalizedFactorsEquivOfQuotEquiv /-
 /-- The bijection between the sets of normalized factors of I and J induced by a ring
     isomorphism `f : R/I ≅ A/J`. -/
 @[simps apply]
@@ -1252,18 +1395,22 @@ def normalizedFactorsEquivOfQuotEquiv (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
   left_inv := fun ⟨j, hj⟩ => by simp
   right_inv := fun ⟨j, hj⟩ => by simp
 #align normalized_factors_equiv_of_quot_equiv normalizedFactorsEquivOfQuotEquiv
+-/
 
+#print normalizedFactorsEquivOfQuotEquiv_symm /-
 @[simp]
 theorem normalizedFactorsEquivOfQuotEquiv_symm (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
     (normalizedFactorsEquivOfQuotEquiv f hI hJ).symm =
       normalizedFactorsEquivOfQuotEquiv f.symm hJ hI :=
   rfl
 #align normalized_factors_equiv_of_quot_equiv_symm normalizedFactorsEquivOfQuotEquiv_symm
+-/
 
 variable [DecidableRel ((· ∣ ·) : Ideal R → Ideal R → Prop)]
 
 variable [DecidableRel ((· ∣ ·) : Ideal A → Ideal A → Prop)]
 
+#print normalizedFactorsEquivOfQuotEquiv_multiplicity_eq_multiplicity /-
 /-- The map `normalized_factors_equiv_of_quot_equiv` preserves multiplicities. -/
 theorem normalizedFactorsEquivOfQuotEquiv_multiplicity_eq_multiplicity (hI : I ≠ ⊥) (hJ : J ≠ ⊥)
     (L : Ideal R) (hL : L ∈ normalizedFactors I) :
@@ -1274,6 +1421,7 @@ theorem normalizedFactorsEquivOfQuotEquiv_multiplicity_eq_multiplicity (hI : I 
     multiplicity_factor_dvd_iso_eq_multiplicity_of_mem_normalizedFactors hI hJ hL
       fun ⟨l, hl⟩ ⟨l', hl'⟩ => idealFactorsEquivOfQuotEquiv_is_dvd_iso f hl hl'
 #align normalized_factors_equiv_of_quot_equiv_multiplicity_eq_multiplicity normalizedFactorsEquivOfQuotEquiv_multiplicity_eq_multiplicity
+-/
 
 end
 
@@ -1285,22 +1433,27 @@ open scoped BigOperators
 
 variable {R}
 
+#print Ring.DimensionLeOne.prime_le_prime_iff_eq /-
 theorem Ring.DimensionLeOne.prime_le_prime_iff_eq (h : Ring.DimensionLEOne R) {P Q : Ideal R}
     [hP : P.IsPrime] [hQ : Q.IsPrime] (hP0 : P ≠ ⊥) : P ≤ Q ↔ P = Q :=
   ⟨(h P hP0 hP).eq_of_le hQ.ne_top, Eq.le⟩
 #align ring.dimension_le_one.prime_le_prime_iff_eq Ring.DimensionLeOne.prime_le_prime_iff_eq
+-/
 
+#print Ideal.coprime_of_no_prime_ge /-
 theorem Ideal.coprime_of_no_prime_ge {I J : Ideal R} (h : ∀ P, I ≤ P → J ≤ P → ¬IsPrime P) :
     I ⊔ J = ⊤ := by
   by_contra hIJ
   obtain ⟨P, hP, hIJ⟩ := Ideal.exists_le_maximal _ hIJ
   exact h P (le_trans le_sup_left hIJ) (le_trans le_sup_right hIJ) hP.is_prime
 #align ideal.coprime_of_no_prime_ge Ideal.coprime_of_no_prime_ge
+-/
 
 section DedekindDomain
 
 variable {R} [IsDomain R] [IsDedekindDomain R]
 
+#print Ideal.IsPrime.mul_mem_pow /-
 theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n : ℕ}
     (h : a * b ∈ I ^ n) : a ∈ I ∨ b ∈ I ^ n :=
   by
@@ -1313,6 +1466,7 @@ theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n :
   rw [mul_comm] at h 
   exact Or.inr (Prime.pow_dvd_of_dvd_mul_right ((Ideal.prime_iff_isPrime hI0).mpr hI) _ ha h)
 #align ideal.is_prime.mul_mem_pow Ideal.IsPrime.mul_mem_pow
+-/
 
 section
 
@@ -1365,6 +1519,7 @@ theorem Ideal.pow_le_prime_iff {I P : Ideal R} [hP : P.IsPrime] {n : ℕ} (hn :
 #align ideal.pow_le_prime_iff Ideal.pow_le_prime_iff
 -/
 
+#print Ideal.prod_le_prime /-
 theorem Ideal.prod_le_prime {ι : Type _} {s : Finset ι} {f : ι → Ideal R} {P : Ideal R}
     [hP : P.IsPrime] : ∏ i in s, f i ≤ P ↔ ∃ i ∈ s, f i ≤ P :=
   by
@@ -1374,8 +1529,10 @@ theorem Ideal.prod_le_prime {ι : Type _} {s : Finset ι} {f : ι → Ideal R} {
   simp only [← Ideal.dvd_iff_le]
   exact ((Ideal.prime_iff_isPrime hP0).mpr hP).dvd_finset_prod_iff _
 #align ideal.prod_le_prime Ideal.prod_le_prime
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+#print IsDedekindDomain.inf_prime_pow_eq_prod /-
 /-- The intersection of distinct prime powers in a Dedekind domain is the product of these
 prime powers. -/
 theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type _} (s : Finset ι) (f : ι → Ideal R)
@@ -1408,7 +1565,9 @@ theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type _} (s : Finset ι) (f
   · exact (Prime a (Finset.mem_insert_self a s)).NeZero
   · exact (Prime b (Finset.mem_insert_of_mem hb)).NeZero
 #align is_dedekind_domain.inf_prime_pow_eq_prod IsDedekindDomain.inf_prime_pow_eq_prod
+-/
 
+#print IsDedekindDomain.quotientEquivPiOfProdEq /-
 /-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
 `∏ i, P i ^ e i`, then `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`. -/
 noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type _} [Fintype ι] (I : Ideal R)
@@ -1432,9 +1591,11 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type _} [Fintyp
                 ((is_dedekind_domain.dimension_le_one.prime_le_prime_iff_eq (Prime j).NeZero).mp
                     (Ideal.le_of_pow_le_prime hPj)).symm))
 #align is_dedekind_domain.quotient_equiv_pi_of_prod_eq IsDedekindDomain.quotientEquivPiOfProdEq
+-/
 
 open scoped Classical
 
+#print IsDedekindDomain.quotientEquivPiFactors /-
 /-- **Chinese remainder theorem** for a Dedekind domain: `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`,
 where `P i` ranges over the prime factors of `I` and `e i` over the multiplicities. -/
 noncomputable def IsDedekindDomain.quotientEquivPiFactors {I : Ideal R} (hI : I ≠ ⊥) :
@@ -1450,22 +1611,28 @@ noncomputable def IsDedekindDomain.quotientEquivPiFactors {I : Ideal R} (hI : I
       _ = (factors I).Prod := by rw [Multiset.map_id']
       _ = I := (@associated_iff_eq (Ideal R) _ Ideal.uniqueUnits _ _).mp (factors_prod hI))
 #align is_dedekind_domain.quotient_equiv_pi_factors IsDedekindDomain.quotientEquivPiFactors
+-/
 
+#print IsDedekindDomain.quotientEquivPiFactors_mk /-
 @[simp]
 theorem IsDedekindDomain.quotientEquivPiFactors_mk {I : Ideal R} (hI : I ≠ ⊥) (x : R) :
     IsDedekindDomain.quotientEquivPiFactors hI (Ideal.Quotient.mk I x) = fun P =>
       Ideal.Quotient.mk _ x :=
   rfl
 #align is_dedekind_domain.quotient_equiv_pi_factors_mk IsDedekindDomain.quotientEquivPiFactors_mk
+-/
 
+#print Ideal.quotientMulEquivQuotientProd /-
 /-- **Chinese remainder theorem**, specialized to two ideals. -/
 noncomputable def Ideal.quotientMulEquivQuotientProd (I J : Ideal R) (coprime : I ⊔ J = ⊤) :
     R ⧸ I * J ≃+* (R ⧸ I) × R ⧸ J :=
   RingEquiv.trans (Ideal.quotEquivOfEq (inf_eq_mul_of_coprime coprime).symm)
     (Ideal.quotientInfEquivQuotientProd I J coprime)
 #align ideal.quotient_mul_equiv_quotient_prod Ideal.quotientMulEquivQuotientProd
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+#print IsDedekindDomain.quotientEquivPiOfFinsetProdEq /-
 /-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
 `∏ i in s, P i ^ e i`, then `R ⧸ I` factors as `Π (i : s), R ⧸ (P i ^ e i)`.
 
@@ -1480,8 +1647,10 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfFinsetProdEq {ι : Type _} {
     (fun i => Prime i i.2) (fun i j h => coprime i i.2 j j.2 (Subtype.coe_injective.Ne h))
     (trans (Finset.prod_coe_sort s fun i => P i ^ e i) prod_eq)
 #align is_dedekind_domain.quotient_equiv_pi_of_finset_prod_eq IsDedekindDomain.quotientEquivPiOfFinsetProdEq
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+#print IsDedekindDomain.exists_representative_mod_finset /-
 /-- Corollary of the Chinese remainder theorem: given elements `x i : R / P i ^ e i`,
 we can choose a representative `y : R` such that `y ≡ x i (mod P i ^ e i)`.-/
 theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type _} {s : Finset ι}
@@ -1494,8 +1663,10 @@ theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type _} {s : Fin
   obtain ⟨z, rfl⟩ := Ideal.Quotient.mk_surjective y
   exact ⟨z, fun i hi => rfl⟩
 #align is_dedekind_domain.exists_representative_mod_finset IsDedekindDomain.exists_representative_mod_finset
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+#print IsDedekindDomain.exists_forall_sub_mem_ideal /-
 /-- Corollary of the Chinese remainder theorem: given elements `x i : R`,
 we can choose a representative `y : R` such that `y - x i ∈ P i ^ e i`.-/
 theorem IsDedekindDomain.exists_forall_sub_mem_ideal {ι : Type _} {s : Finset ι} (P : ι → Ideal R)
@@ -1508,6 +1679,7 @@ theorem IsDedekindDomain.exists_forall_sub_mem_ideal {ι : Type _} {s : Finset 
       Ideal.Quotient.mk _ (x i)
   exact ⟨y, fun i hi => ideal.quotient.eq.mp (hy i hi)⟩
 #align is_dedekind_domain.exists_forall_sub_mem_ideal IsDedekindDomain.exists_forall_sub_mem_ideal
+-/
 
 end DedekindDomain
 
@@ -1519,11 +1691,13 @@ open multiplicity UniqueFactorizationMonoid Ideal
 
 variable {R} [IsDomain R] [IsPrincipalIdealRing R]
 
+#print span_singleton_dvd_span_singleton_iff_dvd /-
 theorem span_singleton_dvd_span_singleton_iff_dvd {a b : R} :
     Ideal.span {a} ∣ Ideal.span ({b} : Set R) ↔ a ∣ b :=
   ⟨fun h => mem_span_singleton.mp (dvd_iff_le.mp h (mem_span_singleton.mpr (dvd_refl b))), fun h =>
     dvd_iff_le.mpr fun d hd => mem_span_singleton.mpr (dvd_trans h (mem_span_singleton.mp hd))⟩
 #align span_singleton_dvd_span_singleton_iff_dvd span_singleton_dvd_span_singleton_iff_dvd
+-/
 
 #print singleton_span_mem_normalizedFactors_of_mem_normalizedFactors /-
 theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
@@ -1551,6 +1725,7 @@ theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [Normaliza
 #align singleton_span_mem_normalized_factors_of_mem_normalized_factors singleton_span_mem_normalizedFactors_of_mem_normalizedFactors
 -/
 
+#print multiplicity_eq_multiplicity_span /-
 theorem multiplicity_eq_multiplicity_span [DecidableRel ((· ∣ ·) : R → R → Prop)]
     [DecidableRel ((· ∣ ·) : Ideal R → Ideal R → Prop)] {a b : R} :
     multiplicity (Ideal.span {a}) (Ideal.span ({b} : Set R)) = multiplicity a b :=
@@ -1576,9 +1751,11 @@ theorem multiplicity_eq_multiplicity_span [DecidableRel ((· ∣ ·) : R → R 
         rw [Ideal.span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]
         exact not_finite_iff_forall.mp h n
 #align multiplicity_eq_multiplicity_span multiplicity_eq_multiplicity_span
+-/
 
 variable [DecidableEq R] [DecidableEq (Ideal R)] [NormalizationMonoid R]
 
+#print normalizedFactorsEquivSpanNormalizedFactors /-
 /-- The bijection between the (normalized) prime factors of `r` and the (normalized) prime factors
     of `span {r}` -/
 @[simps]
@@ -1611,9 +1788,11 @@ noncomputable def normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r 
               ((show Ideal.span {r} ≤ i from dvd_iff_le.mp (dvd_of_mem_normalized_factors hi))
                 (mem_span_singleton.mpr (dvd_refl r))))
 #align normalized_factors_equiv_span_normalized_factors normalizedFactorsEquivSpanNormalizedFactors
+-/
 
 variable [DecidableRel ((· ∣ ·) : R → R → Prop)] [DecidableRel ((· ∣ ·) : Ideal R → Ideal R → Prop)]
 
+#print multiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_multiplicity /-
 /-- The bijection `normalized_factors_equiv_span_normalized_factors` between the set of prime
     factors of `r` and the set of prime factors of the ideal `⟨r⟩` preserves multiplicities. -/
 theorem multiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_multiplicity {r d : R}
@@ -1625,7 +1804,9 @@ theorem multiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_multiplicity
   simp only [normalizedFactorsEquivSpanNormalizedFactors, multiplicity_eq_multiplicity_span,
     Subtype.coe_mk, Equiv.ofBijective_apply]
 #align multiplicity_normalized_factors_equiv_span_normalized_factors_eq_multiplicity multiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_multiplicity
+-/
 
+#print multiplicity_normalizedFactorsEquivSpanNormalizedFactors_symm_eq_multiplicity /-
 /-- The bijection `normalized_factors_equiv_span_normalized_factors.symm` between the set of prime
     factors of the ideal `⟨r⟩` and the set of prime factors of `r` preserves multiplicities. -/
 theorem multiplicity_normalizedFactorsEquivSpanNormalizedFactors_symm_eq_multiplicity {r : R}
@@ -1638,6 +1819,7 @@ theorem multiplicity_normalizedFactorsEquivSpanNormalizedFactors_symm_eq_multipl
   rw [hx.symm, Equiv.symm_apply_apply, Subtype.coe_mk,
     multiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_multiplicity hr ha, hx]
 #align multiplicity_normalized_factors_equiv_span_normalized_factors_symm_eq_multiplicity multiplicity_normalizedFactorsEquivSpanNormalizedFactors_symm_eq_multiplicity
+-/
 
 end PID
 

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

@@ -1366,7 +1366,7 @@ theorem Ideal.pow_le_prime_iff {I P : Ideal R} [hP : P.IsPrime] {n : ℕ} (hn :
 -/
 
 theorem Ideal.prod_le_prime {ι : Type _} {s : Finset ι} {f : ι → Ideal R} {P : Ideal R}
-    [hP : P.IsPrime] : (∏ i in s, f i) ≤ P ↔ ∃ i ∈ s, f i ≤ P :=
+    [hP : P.IsPrime] : ∏ i in s, f i ≤ P ↔ ∃ i ∈ s, f i ≤ P :=
   by
   by_cases hP0 : P = ⊥
   · simp only [hP0, le_bot_iff]
@@ -1413,7 +1413,7 @@ theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type _} (s : Finset ι) (f
 `∏ i, P i ^ e i`, then `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`. -/
 noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type _} [Fintype ι] (I : Ideal R)
     (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i, Prime (P i)) (coprime : ∀ i j, i ≠ j → P i ≠ P j)
-    (prod_eq : (∏ i, P i ^ e i) = I) : R ⧸ I ≃+* ∀ i, R ⧸ P i ^ e i :=
+    (prod_eq : ∏ i, P i ^ e i = I) : R ⧸ I ≃+* ∀ i, R ⧸ P i ^ e i :=
   (Ideal.quotEquivOfEq
         (by
           simp only [← prod_eq, Finset.inf_eq_iInf, Finset.mem_univ, ciInf_pos, ←
@@ -1443,7 +1443,7 @@ noncomputable def IsDedekindDomain.quotientEquivPiFactors {I : Ideal R} (hI : I
     (fun P : (factors I).toFinset => prime_of_factor _ (Multiset.mem_toFinset.mp P.Prop))
     (fun i j hij => Subtype.coe_injective.Ne hij)
     (calc
-      (∏ P : (factors I).toFinset, (P : Ideal R) ^ (factors I).count (P : Ideal R)) =
+      ∏ P : (factors I).toFinset, (P : Ideal R) ^ (factors I).count (P : Ideal R) =
           ∏ P in (factors I).toFinset, P ^ (factors I).count P :=
         (factors I).toFinset.prod_coe_sort fun P => P ^ (factors I).count P
       _ = ((factors I).map fun P => P).Prod := (Finset.prod_multiset_map_count (factors I) id).symm
@@ -1475,7 +1475,7 @@ the product to a finite subset `s` of a potentially infinite indexing type `ι`.
 noncomputable def IsDedekindDomain.quotientEquivPiOfFinsetProdEq {ι : Type _} {s : Finset ι}
     (I : Ideal R) (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
     (coprime : ∀ (i) (_ : i ∈ s) (j) (_ : j ∈ s), i ≠ j → P i ≠ P j)
-    (prod_eq : (∏ i in s, P i ^ e i) = I) : R ⧸ I ≃+* ∀ i : s, R ⧸ P i ^ e i :=
+    (prod_eq : ∏ i in s, P i ^ e i = I) : R ⧸ I ≃+* ∀ i : s, R ⧸ P i ^ e i :=
   IsDedekindDomain.quotientEquivPiOfProdEq I (fun i : s => P i) (fun i : s => e i)
     (fun i => Prime i i.2) (fun i j h => coprime i i.2 j j.2 (Subtype.coe_injective.Ne h))
     (trans (Finset.prod_coe_sort s fun i => P i ^ e i) prod_eq)

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

@@ -351,7 +351,6 @@ theorem dimensionLEOne : DimensionLEOne A :=
       (1 : FractionalIdeal A⁰ (FractionRing A)) = _ * _ * _ := _
       _ ≤ _ * _ := (mul_right_mono (P⁻¹ * M : FractionalIdeal A⁰ (FractionRing A)) this)
       _ = M := _
-      
     · rw [mul_assoc, ← mul_assoc ↑P, h.mul_inv_eq_one P'_ne, one_mul, h.inv_mul_eq_one M'_ne]
     · rw [← mul_assoc ↑P, h.mul_inv_eq_one P'_ne, one_mul]
     · infer_instance
@@ -874,7 +873,6 @@ theorem Ideal.exist_integer_multiples_not_mem {J : Ideal A} (hJ : J ≠ ⊤) {ι
     ↑J / I = ↑J * I⁻¹ := div_eq_mul_inv (↑J) I
     _ < 1 * I⁻¹ := (mul_right_strict_mono (inv_ne_zero hI0) _)
     _ = I⁻¹ := one_mul _
-    
   · rw [← coe_ideal_top]
     -- And multiplying by `I⁻¹` is indeed strictly monotone.
     exact
@@ -1450,8 +1448,7 @@ noncomputable def IsDedekindDomain.quotientEquivPiFactors {I : Ideal R} (hI : I
         (factors I).toFinset.prod_coe_sort fun P => P ^ (factors I).count P
       _ = ((factors I).map fun P => P).Prod := (Finset.prod_multiset_map_count (factors I) id).symm
       _ = (factors I).Prod := by rw [Multiset.map_id']
-      _ = I := (@associated_iff_eq (Ideal R) _ Ideal.uniqueUnits _ _).mp (factors_prod hI)
-      )
+      _ = I := (@associated_iff_eq (Ideal R) _ Ideal.uniqueUnits _ _).mp (factors_prod hI))
 #align is_dedekind_domain.quotient_equiv_pi_factors IsDedekindDomain.quotientEquivPiFactors
 
 @[simp]

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -250,7 +250,7 @@ noncomputable instance : InvOneClass (FractionalIdeal R₁⁰ K) :=
 
 end FractionalIdeal
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) (fraction_ring[fraction_ring] A))) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) (fraction_ring[fraction_ring] A))) -/
 #print IsDedekindDomainInv /-
 /-- A Dedekind domain is an integral domain such that every fractional ideal has an inverse.
 
@@ -268,7 +268,7 @@ open FractionalIdeal
 
 variable {R A K}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) K)) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) K)) -/
 theorem isDedekindDomainInv_iff [Algebra A K] [IsFractionRing A K] :
     IsDedekindDomainInv A ↔ ∀ (I) (_ : I ≠ (⊥ : FractionalIdeal A⁰ K)), I * I⁻¹ = 1 :=
   by
@@ -1377,7 +1377,7 @@ theorem Ideal.prod_le_prime {ι : Type _} {s : Finset ι} {f : ι → Ideal R} {
   exact ((Ideal.prime_iff_isPrime hP0).mpr hP).dvd_finset_prod_iff _
 #align ideal.prod_le_prime Ideal.prod_le_prime
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 /-- The intersection of distinct prime powers in a Dedekind domain is the product of these
 prime powers. -/
 theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type _} (s : Finset ι) (f : ι → Ideal R)
@@ -1468,7 +1468,7 @@ noncomputable def Ideal.quotientMulEquivQuotientProd (I J : Ideal R) (coprime :
     (Ideal.quotientInfEquivQuotientProd I J coprime)
 #align ideal.quotient_mul_equiv_quotient_prod Ideal.quotientMulEquivQuotientProd
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 /-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
 `∏ i in s, P i ^ e i`, then `R ⧸ I` factors as `Π (i : s), R ⧸ (P i ^ e i)`.
 
@@ -1484,7 +1484,7 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfFinsetProdEq {ι : Type _} {
     (trans (Finset.prod_coe_sort s fun i => P i ^ e i) prod_eq)
 #align is_dedekind_domain.quotient_equiv_pi_of_finset_prod_eq IsDedekindDomain.quotientEquivPiOfFinsetProdEq
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 /-- Corollary of the Chinese remainder theorem: given elements `x i : R / P i ^ e i`,
 we can choose a representative `y : R` such that `y ≡ x i (mod P i ^ e i)`.-/
 theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type _} {s : Finset ι}
@@ -1498,7 +1498,7 @@ theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type _} {s : Fin
   exact ⟨z, fun i hi => rfl⟩
 #align is_dedekind_domain.exists_representative_mod_finset IsDedekindDomain.exists_representative_mod_finset
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 /-- Corollary of the Chinese remainder theorem: given elements `x i : R`,
 we can choose a representative `y : R` such that `y - x i ∈ P i ^ e i`.-/
 theorem IsDedekindDomain.exists_forall_sub_mem_ideal {ι : Type _} {s : Finset ι} (P : ι → Ideal R)

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -332,8 +332,8 @@ theorem integrallyClosed : IsIntegrallyClosed A :=
 
 open Ring
 
-#print IsDedekindDomainInv.dimensionLeOne /-
-theorem dimensionLeOne : DimensionLEOne A :=
+#print IsDedekindDomainInv.dimensionLEOne /-
+theorem dimensionLEOne : DimensionLEOne A :=
   by
   -- We're going to show that `P` is maximal because any (maximal) ideal `M`
   -- that is strictly larger would be `⊤`.
@@ -371,7 +371,7 @@ theorem dimensionLeOne : DimensionLEOne A :=
   rw [IsFractionRing.injective A (FractionRing A) zy_eq] at hzy 
   -- But `P` is a prime ideal, so `z ∉ P` implies `y ∈ P`, as desired.
   exact mem_coe_ideal_of_mem A⁰ (Or.resolve_left (hP.mem_or_mem hzy) hzp)
-#align is_dedekind_domain_inv.dimension_le_one IsDedekindDomainInv.dimensionLeOne
+#align is_dedekind_domain_inv.dimension_le_one IsDedekindDomainInv.dimensionLEOne
 -/
 
 #print IsDedekindDomainInv.isDedekindDomain /-

mathlib commit https://github.com/leanprover-community/mathlib/commit/58a272265b5e05f258161260dd2c5d247213cbd3

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
 
 ! This file was ported from Lean 3 source module ring_theory.dedekind_domain.ideal
-! leanprover-community/mathlib commit 2bbc7e3884ba234309d2a43b19144105a753292e
+! leanprover-community/mathlib commit 36938f775671ff28bea1c0310f1608e4afbb22e0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -20,6 +20,9 @@ import Mathbin.RingTheory.ChainOfDivisors
 /-!
 # Dedekind domains and ideals
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file, we show a ring is a Dedekind domain iff all fractional ideals are invertible.
 Then we prove some results on the unique factorization monoid structure of the ideals.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/13361559d66b84f80b6d5a1c4a26aa5054766725

@@ -248,6 +248,7 @@ noncomputable instance : InvOneClass (FractionalIdeal R₁⁰ K) :=
 end FractionalIdeal
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) (fraction_ring[fraction_ring] A))) -/
+#print IsDedekindDomainInv /-
 /-- A Dedekind domain is an integral domain such that every fractional ideal has an inverse.
 
 This is equivalent to `is_dedekind_domain`.
@@ -258,6 +259,7 @@ assuming `is_dedekind_domain A`, which implies `is_dedekind_domain_inv`. For **i
 def IsDedekindDomainInv : Prop :=
   ∀ (I) (_ : I ≠ (⊥ : FractionalIdeal A⁰ (FractionRing A))), I * I⁻¹ = 1
 #align is_dedekind_domain_inv IsDedekindDomainInv
+-/
 
 open FractionalIdeal
 
@@ -300,6 +302,7 @@ protected theorem isUnit {I : FractionalIdeal A⁰ K} (hI : I ≠ 0) : IsUnit I
   isUnit_of_mul_eq_one _ _ (h.mul_inv_eq_one hI)
 #align is_dedekind_domain_inv.is_unit IsDedekindDomainInv.isUnit
 
+#print IsDedekindDomainInv.isNoetherianRing /-
 theorem isNoetherianRing : IsNoetherianRing A :=
   by
   refine' is_noetherian_ring_iff.mpr ⟨fun I : Ideal A => _⟩
@@ -308,8 +311,10 @@ theorem isNoetherianRing : IsNoetherianRing A :=
   have hI : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coe_ideal_ne_zero.mpr hI
   exact I.fg_of_is_unit (IsFractionRing.injective A (FractionRing A)) (h.is_unit hI)
 #align is_dedekind_domain_inv.is_noetherian_ring IsDedekindDomainInv.isNoetherianRing
+-/
 
-theorem integrally_closed : IsIntegrallyClosed A :=
+#print IsDedekindDomainInv.integrallyClosed /-
+theorem integrallyClosed : IsIntegrallyClosed A :=
   by
   -- It suffices to show that for integral `x`,
   -- `A[x]` (which is a fractional ideal) is in fact equal to `A`.
@@ -319,11 +324,13 @@ theorem integrally_closed : IsIntegrallyClosed A :=
     FractionalIdeal.adjoinIntegral_eq_one_of_isUnit x hx (h.is_unit _)]
   · exact mem_adjoin_integral_self A⁰ x hx
   · exact fun h => one_ne_zero (eq_zero_iff.mp h 1 (Subalgebra.one_mem _))
-#align is_dedekind_domain_inv.integrally_closed IsDedekindDomainInv.integrally_closed
+#align is_dedekind_domain_inv.integrally_closed IsDedekindDomainInv.integrallyClosed
+-/
 
 open Ring
 
-theorem dimensionLEOne : DimensionLEOne A :=
+#print IsDedekindDomainInv.dimensionLeOne /-
+theorem dimensionLeOne : DimensionLEOne A :=
   by
   -- We're going to show that `P` is maximal because any (maximal) ideal `M`
   -- that is strictly larger would be `⊤`.
@@ -361,13 +368,16 @@ theorem dimensionLEOne : DimensionLEOne A :=
   rw [IsFractionRing.injective A (FractionRing A) zy_eq] at hzy 
   -- But `P` is a prime ideal, so `z ∉ P` implies `y ∈ P`, as desired.
   exact mem_coe_ideal_of_mem A⁰ (Or.resolve_left (hP.mem_or_mem hzy) hzp)
-#align is_dedekind_domain_inv.dimension_le_one IsDedekindDomainInv.dimensionLEOne
+#align is_dedekind_domain_inv.dimension_le_one IsDedekindDomainInv.dimensionLeOne
+-/
 
+#print IsDedekindDomainInv.isDedekindDomain /-
 /-- Showing one side of the equivalence between the definitions
 `is_dedekind_domain_inv` and `is_dedekind_domain` of Dedekind domains. -/
 theorem isDedekindDomain : IsDedekindDomain A :=
-  ⟨h.IsNoetherianRing, h.DimensionLEOne, h.integrally_closed⟩
+  ⟨h.IsNoetherianRing, h.DimensionLEOne, h.integrallyClosed⟩
 #align is_dedekind_domain_inv.is_dedekind_domain IsDedekindDomainInv.isDedekindDomain
+-/
 
 end IsDedekindDomainInv
 
@@ -616,11 +626,13 @@ protected theorem div_eq_mul_inv [IsDedekindDomain A] (I J : FractionalIdeal A
 
 end FractionalIdeal
 
+#print isDedekindDomain_iff_isDedekindDomainInv /-
 /-- `is_dedekind_domain` and `is_dedekind_domain_inv` are equivalent ways
 to express that an integral domain is a Dedekind domain. -/
 theorem isDedekindDomain_iff_isDedekindDomainInv : IsDedekindDomain A ↔ IsDedekindDomainInv A :=
   ⟨fun h I hI => FractionalIdeal.mul_inv_cancel hI, fun h => h.IsDedekindDomain⟩
 #align is_dedekind_domain_iff_is_dedekind_domain_inv isDedekindDomain_iff_isDedekindDomainInv
+-/
 
 end Inverse
 
@@ -632,6 +644,7 @@ open FractionalIdeal
 
 open Ideal
 
+#print FractionalIdeal.semifield /-
 noncomputable instance FractionalIdeal.semifield : Semifield (FractionalIdeal A⁰ K) :=
   { FractionalIdeal.commSemiring,
     coeIdeal_injective.Nontrivial with
@@ -641,7 +654,9 @@ noncomputable instance FractionalIdeal.semifield : Semifield (FractionalIdeal A
     div_eq_mul_inv := FractionalIdeal.div_eq_mul_inv
     mul_inv_cancel := fun I => FractionalIdeal.mul_inv_cancel }
 #align fractional_ideal.semifield FractionalIdeal.semifield
+-/
 
+#print FractionalIdeal.cancelCommMonoidWithZero /-
 /-- Fractional ideals have cancellative multiplication in a Dedekind domain.
 
 Although this instance is a direct consequence of the instance
@@ -654,16 +669,21 @@ instance FractionalIdeal.cancelCommMonoidWithZero :
     FractionalIdeal.commSemiring,-- Project out the computable fields first.
     (by infer_instance : CancelCommMonoidWithZero (FractionalIdeal A⁰ K)) with }
 #align fractional_ideal.cancel_comm_monoid_with_zero FractionalIdeal.cancelCommMonoidWithZero
+-/
 
+#print Ideal.cancelCommMonoidWithZero /-
 instance Ideal.cancelCommMonoidWithZero : CancelCommMonoidWithZero (Ideal A) :=
   { Ideal.idemCommSemiring,
     Function.Injective.cancelCommMonoidWithZero (coeIdealHom A⁰ (FractionRing A)) coeIdeal_injective
       (RingHom.map_zero _) (RingHom.map_one _) (RingHom.map_mul _) (RingHom.map_pow _) with }
 #align ideal.cancel_comm_monoid_with_zero Ideal.cancelCommMonoidWithZero
+-/
 
+#print Ideal.isDomain /-
 instance Ideal.isDomain : IsDomain (Ideal A) :=
   { (inferInstance : IsCancelMulZero _), Ideal.nontrivial with }
 #align ideal.is_domain Ideal.isDomain
+-/
 
 /-- For ideals in a Dedekind domain, to divide is to contain. -/
 theorem Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
@@ -703,6 +723,7 @@ instance : WfDvdMonoid (Ideal A)
     ext
     rw [Ideal.dvdNotUnit_iff_lt]
 
+#print Ideal.uniqueFactorizationMonoid /-
 instance Ideal.uniqueFactorizationMonoid : UniqueFactorizationMonoid (Ideal A) :=
   { Ideal.wfDvdMonoid with
     irreducible_iff_prime := fun P =>
@@ -724,10 +745,13 @@ instance Ideal.uniqueFactorizationMonoid : UniqueFactorizationMonoid (Ideal A) :
               mt this.is_prime.mem_or_mem (not_or_of_not x_not_mem y_not_mem)⟩⟩,
         Prime.irreducible⟩ }
 #align ideal.unique_factorization_monoid Ideal.uniqueFactorizationMonoid
+-/
 
+#print Ideal.normalizationMonoid /-
 instance Ideal.normalizationMonoid : NormalizationMonoid (Ideal A) :=
   normalizationMonoidOfUniqueUnits
 #align ideal.normalization_monoid Ideal.normalizationMonoid
+-/
 
 @[simp]
 theorem Ideal.dvd_span_singleton {I : Ideal A} {x : A} : I ∣ Ideal.span {x} ↔ x ∈ I :=
@@ -866,6 +890,7 @@ and the lcm is their infimum, and use this to instantiate `normalized_gcd_monoid
 -/
 
 
+#print Ideal.sup_mul_inf /-
 @[simp]
 theorem sup_mul_inf (I J : Ideal A) : (I ⊔ J) * (I ⊓ J) = I * J :=
   by
@@ -889,6 +914,7 @@ theorem sup_mul_inf (I J : Ideal A) : (I ⊔ J) * (I ⊓ J) = I * J :=
   rw [← hgcd, ← hlcm, associated_iff_eq.mp (gcd_mul_lcm _ _)]
   infer_instance
 #align ideal.sup_mul_inf Ideal.sup_mul_inf
+-/
 
 /-- Ideals in a Dedekind domain have gcd and lcm operators that (trivially) are compatible with
 the normalization operator. -/
@@ -1035,22 +1061,31 @@ variable (v : HeightOneSpectrum R) {R}
 
 namespace HeightOneSpectrum
 
+#print IsDedekindDomain.HeightOneSpectrum.isMaximal /-
 instance isMaximal : v.asIdeal.IsMaximal :=
   dimensionLEOne v.asIdeal v.ne_bot v.IsPrime
 #align is_dedekind_domain.height_one_spectrum.is_maximal IsDedekindDomain.HeightOneSpectrum.isMaximal
+-/
 
+#print IsDedekindDomain.HeightOneSpectrum.prime /-
 theorem prime : Prime v.asIdeal :=
   Ideal.prime_of_isPrime v.ne_bot v.IsPrime
 #align is_dedekind_domain.height_one_spectrum.prime IsDedekindDomain.HeightOneSpectrum.prime
+-/
 
+#print IsDedekindDomain.HeightOneSpectrum.irreducible /-
 theorem irreducible : Irreducible v.asIdeal :=
   UniqueFactorizationMonoid.irreducible_iff_prime.mpr v.Prime
 #align is_dedekind_domain.height_one_spectrum.irreducible IsDedekindDomain.HeightOneSpectrum.irreducible
+-/
 
+#print IsDedekindDomain.HeightOneSpectrum.associates_irreducible /-
 theorem associates_irreducible : Irreducible <| Associates.mk v.asIdeal :=
   (Associates.irreducible_mk _).mpr v.Irreducible
 #align is_dedekind_domain.height_one_spectrum.associates_irreducible IsDedekindDomain.HeightOneSpectrum.associates_irreducible
+-/
 
+#print IsDedekindDomain.HeightOneSpectrum.equivMaximalSpectrum /-
 /-- An equivalence between the height one and maximal spectra for rings of Krull dimension 1. -/
 def equivMaximalSpectrum (hR : ¬IsField R) : HeightOneSpectrum R ≃ MaximalSpectrum R
     where
@@ -1060,6 +1095,7 @@ def equivMaximalSpectrum (hR : ¬IsField R) : HeightOneSpectrum R ≃ MaximalSpe
   left_inv := fun ⟨_, _, _⟩ => rfl
   right_inv := fun ⟨_, _⟩ => rfl
 #align is_dedekind_domain.height_one_spectrum.equiv_maximal_spectrum IsDedekindDomain.HeightOneSpectrum.equivMaximalSpectrum
+-/
 
 variable (R K)
 
@@ -1248,10 +1284,10 @@ open scoped BigOperators
 
 variable {R}
 
-theorem Ring.DimensionLEOne.prime_le_prime_iff_eq (h : Ring.DimensionLEOne R) {P Q : Ideal R}
+theorem Ring.DimensionLeOne.prime_le_prime_iff_eq (h : Ring.DimensionLEOne R) {P Q : Ideal R}
     [hP : P.IsPrime] [hQ : Q.IsPrime] (hP0 : P ≠ ⊥) : P ≤ Q ↔ P = Q :=
   ⟨(h P hP0 hP).eq_of_le hQ.ne_top, Eq.le⟩
-#align ring.dimension_le_one.prime_le_prime_iff_eq Ring.DimensionLEOne.prime_le_prime_iff_eq
+#align ring.dimension_le_one.prime_le_prime_iff_eq Ring.DimensionLeOne.prime_le_prime_iff_eq
 
 theorem Ideal.coprime_of_no_prime_ge {I J : Ideal R} (h : ∀ P, I ≤ P → J ≤ P → ¬IsPrime P) :
     I ⊔ J = ⊤ := by
@@ -1281,17 +1317,20 @@ section
 
 open scoped Classical
 
+#print Ideal.count_normalizedFactors_eq /-
 theorem Ideal.count_normalizedFactors_eq {p x : Ideal R} [hp : p.IsPrime] {n : ℕ} (hle : x ≤ p ^ n)
     (hlt : ¬x ≤ p ^ (n + 1)) : (normalizedFactors x).count p = n :=
   count_normalizedFactors_eq' ((Ideal.isPrime_iff_bot_or_prime.mp hp).imp_right Prime.irreducible)
     (by haveI : Unique (Ideal R)ˣ := Ideal.uniqueUnits; apply normalize_eq)
     (by convert ideal.dvd_iff_le.mpr hle) (by convert mt Ideal.le_of_dvd hlt)
 #align ideal.count_normalized_factors_eq Ideal.count_normalizedFactors_eq
+-/
 
 /- Warning: even though a pure term-mode proof typechecks (the `by convert` can simply be
   removed), it's slower to the point of a possible timeout. -/
 end
 
+#print Ideal.le_mul_of_no_prime_factors /-
 theorem Ideal.le_mul_of_no_prime_factors {I J K : Ideal R}
     (coprime : ∀ P, J ≤ P → K ≤ P → ¬IsPrime P) (hJ : I ≤ J) (hK : I ≤ K) : I ≤ J * K :=
   by
@@ -1305,7 +1344,9 @@ theorem Ideal.le_mul_of_no_prime_factors {I J K : Ideal R}
       (UniqueFactorizationMonoid.dvd_of_dvd_mul_right_of_no_prime_factors hJ0
         (fun P hPJ hPK => mt Ideal.isPrime_of_prime (coprime P hPJ hPK)) hJ)
 #align ideal.le_mul_of_no_prime_factors Ideal.le_mul_of_no_prime_factors
+-/
 
+#print Ideal.le_of_pow_le_prime /-
 theorem Ideal.le_of_pow_le_prime {I P : Ideal R} [hP : P.IsPrime] {n : ℕ} (h : I ^ n ≤ P) : I ≤ P :=
   by
   by_cases hP0 : P = ⊥
@@ -1314,11 +1355,14 @@ theorem Ideal.le_of_pow_le_prime {I P : Ideal R} [hP : P.IsPrime] {n : ℕ} (h :
   rw [← Ideal.dvd_iff_le] at h ⊢
   exact ((Ideal.prime_iff_isPrime hP0).mpr hP).dvd_of_dvd_pow h
 #align ideal.le_of_pow_le_prime Ideal.le_of_pow_le_prime
+-/
 
+#print Ideal.pow_le_prime_iff /-
 theorem Ideal.pow_le_prime_iff {I P : Ideal R} [hP : P.IsPrime] {n : ℕ} (hn : n ≠ 0) :
     I ^ n ≤ P ↔ I ≤ P :=
   ⟨Ideal.le_of_pow_le_prime, fun h => trans (Ideal.pow_le_self hn) h⟩
 #align ideal.pow_le_prime_iff Ideal.pow_le_prime_iff
+-/
 
 theorem Ideal.prod_le_prime {ι : Type _} {s : Finset ι} {f : ι → Ideal R} {P : Ideal R}
     [hP : P.IsPrime] : (∏ i in s, f i) ≤ P ↔ ∃ i ∈ s, f i ≤ P :=
@@ -1481,6 +1525,7 @@ theorem span_singleton_dvd_span_singleton_iff_dvd {a b : R} :
     dvd_iff_le.mpr fun d hd => mem_span_singleton.mpr (dvd_trans h (mem_span_singleton.mp hd))⟩
 #align span_singleton_dvd_span_singleton_iff_dvd span_singleton_dvd_span_singleton_iff_dvd
 
+#print singleton_span_mem_normalizedFactors_of_mem_normalizedFactors /-
 theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
     [DecidableEq R] [DecidableEq (Ideal R)] {a b : R} (ha : a ∈ normalizedFactors b) :
     Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : Set R)) :=
@@ -1504,6 +1549,7 @@ theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [Normaliza
     by_contra
     exact (prime_of_normalized_factor a ha).NeZero (span_singleton_eq_bot.mp h)
 #align singleton_span_mem_normalized_factors_of_mem_normalized_factors singleton_span_mem_normalizedFactors_of_mem_normalizedFactors
+-/
 
 theorem multiplicity_eq_multiplicity_span [DecidableRel ((· ∣ ·) : R → R → Prop)]
     [DecidableRel ((· ∣ ·) : Ideal R → Ideal R → Prop)] {a b : R} :

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

@@ -309,7 +309,7 @@ theorem isNoetherianRing : IsNoetherianRing A :=
   exact I.fg_of_is_unit (IsFractionRing.injective A (FractionRing A)) (h.is_unit hI)
 #align is_dedekind_domain_inv.is_noetherian_ring IsDedekindDomainInv.isNoetherianRing
 
-theorem integrallyClosed : IsIntegrallyClosed A :=
+theorem integrally_closed : IsIntegrallyClosed A :=
   by
   -- It suffices to show that for integral `x`,
   -- `A[x]` (which is a fractional ideal) is in fact equal to `A`.
@@ -319,11 +319,11 @@ theorem integrallyClosed : IsIntegrallyClosed A :=
     FractionalIdeal.adjoinIntegral_eq_one_of_isUnit x hx (h.is_unit _)]
   · exact mem_adjoin_integral_self A⁰ x hx
   · exact fun h => one_ne_zero (eq_zero_iff.mp h 1 (Subalgebra.one_mem _))
-#align is_dedekind_domain_inv.integrally_closed IsDedekindDomainInv.integrallyClosed
+#align is_dedekind_domain_inv.integrally_closed IsDedekindDomainInv.integrally_closed
 
 open Ring
 
-theorem dimensionLeOne : DimensionLeOne A :=
+theorem dimensionLEOne : DimensionLEOne A :=
   by
   -- We're going to show that `P` is maximal because any (maximal) ideal `M`
   -- that is strictly larger would be `⊤`.
@@ -361,12 +361,12 @@ theorem dimensionLeOne : DimensionLeOne A :=
   rw [IsFractionRing.injective A (FractionRing A) zy_eq] at hzy 
   -- But `P` is a prime ideal, so `z ∉ P` implies `y ∈ P`, as desired.
   exact mem_coe_ideal_of_mem A⁰ (Or.resolve_left (hP.mem_or_mem hzy) hzp)
-#align is_dedekind_domain_inv.dimension_le_one IsDedekindDomainInv.dimensionLeOne
+#align is_dedekind_domain_inv.dimension_le_one IsDedekindDomainInv.dimensionLEOne
 
 /-- Showing one side of the equivalence between the definitions
 `is_dedekind_domain_inv` and `is_dedekind_domain` of Dedekind domains. -/
 theorem isDedekindDomain : IsDedekindDomain A :=
-  ⟨h.IsNoetherianRing, h.DimensionLeOne, h.integrallyClosed⟩
+  ⟨h.IsNoetherianRing, h.DimensionLEOne, h.integrally_closed⟩
 #align is_dedekind_domain_inv.is_dedekind_domain IsDedekindDomainInv.isDedekindDomain
 
 end IsDedekindDomainInv
@@ -397,19 +397,19 @@ theorem exists_multiset_prod_cons_le_and_prod_not_le [IsDedekindDomain A] (hNF :
   -- Then in fact there is a `P ∈ Z` with `P ≤ M`.
   obtain ⟨P, hPZ, rfl⟩ := multiset.mem_map.mp hPZ'
   classical
-    have := Multiset.map_erase PrimeSpectrum.asIdeal PrimeSpectrum.ext P Z
-    obtain ⟨hP0, hZP0⟩ : P.as_ideal ≠ ⊥ ∧ ((Z.erase P).map PrimeSpectrum.asIdeal).Prod ≠ ⊥ := by
-      rwa [Ne.def, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
-        this] at hprodZ 
-    -- By maximality of `P` and `M`, we have that `P ≤ M` implies `P = M`.
-    have hPM' := (IsDedekindDomain.dimensionLeOne _ hP0 P.is_prime).eq_of_le hM.ne_top hPM
-    subst hPM'
-    -- By minimality of `Z`, erasing `P` from `Z` is exactly what we need.
-    refine' ⟨Z.erase P, _, _⟩
-    · convert hZI
-      rw [this, Multiset.cons_erase hPZ']
-    · refine' fun h => h_eraseZ (Z.erase P) ⟨h, _⟩ (multiset.erase_lt.mpr hPZ)
-      exact hZP0
+  have := Multiset.map_erase PrimeSpectrum.asIdeal PrimeSpectrum.ext P Z
+  obtain ⟨hP0, hZP0⟩ : P.as_ideal ≠ ⊥ ∧ ((Z.erase P).map PrimeSpectrum.asIdeal).Prod ≠ ⊥ := by
+    rwa [Ne.def, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
+      this] at hprodZ 
+  -- By maximality of `P` and `M`, we have that `P ≤ M` implies `P = M`.
+  have hPM' := (IsDedekindDomain.dimensionLEOne _ hP0 P.is_prime).eq_of_le hM.ne_top hPM
+  subst hPM'
+  -- By minimality of `Z`, erasing `P` from `Z` is exactly what we need.
+  refine' ⟨Z.erase P, _, _⟩
+  · convert hZI
+    rw [this, Multiset.cons_erase hPZ']
+  · refine' fun h => h_eraseZ (Z.erase P) ⟨h, _⟩ (multiset.erase_lt.mpr hPZ)
+    exact hZP0
 #align exists_multiset_prod_cons_le_and_prod_not_le exists_multiset_prod_cons_le_and_prod_not_le
 
 namespace FractionalIdeal
@@ -1036,7 +1036,7 @@ variable (v : HeightOneSpectrum R) {R}
 namespace HeightOneSpectrum
 
 instance isMaximal : v.asIdeal.IsMaximal :=
-  dimensionLeOne v.asIdeal v.ne_bot v.IsPrime
+  dimensionLEOne v.asIdeal v.ne_bot v.IsPrime
 #align is_dedekind_domain.height_one_spectrum.is_maximal IsDedekindDomain.HeightOneSpectrum.isMaximal
 
 theorem prime : Prime v.asIdeal :=
@@ -1054,7 +1054,7 @@ theorem associates_irreducible : Irreducible <| Associates.mk v.asIdeal :=
 /-- An equivalence between the height one and maximal spectra for rings of Krull dimension 1. -/
 def equivMaximalSpectrum (hR : ¬IsField R) : HeightOneSpectrum R ≃ MaximalSpectrum R
     where
-  toFun v := ⟨v.asIdeal, dimensionLeOne v.asIdeal v.ne_bot v.IsPrime⟩
+  toFun v := ⟨v.asIdeal, dimensionLEOne v.asIdeal v.ne_bot v.IsPrime⟩
   invFun v :=
     ⟨v.asIdeal, v.IsMaximal.IsPrime, Ring.ne_bot_of_isMaximal_of_not_isField v.IsMaximal hR⟩
   left_inv := fun ⟨_, _, _⟩ => rfl
@@ -1098,7 +1098,7 @@ variable {R} {A} [IsDedekindDomain A] {I : Ideal R} {J : Ideal A}
   a homomorphism `f : R/I →+* A/J` -/
 @[simps]
 def idealFactorsFunOfQuotHom {f : R ⧸ I →+* A ⧸ J} (hf : Function.Surjective f) :
-    { p : Ideal R | p ∣ I } →o { p : Ideal A | p ∣ J }
+    {p : Ideal R | p ∣ I} →o {p : Ideal A | p ∣ J}
     where
   toFun X :=
     ⟨comap J.Quotient.mk (map f (map I.Quotient.mk X)),
@@ -1147,7 +1147,7 @@ variable [IsDomain R] [IsDedekindDomain R] (f : R ⧸ I ≃+* A ⧸ J)
 /-- The bijection between ideals of `R` dividing `I` and the ideals of `A` dividing `J` induced by
   an isomorphism `f : R/I ≅ A/J`. -/
 @[simps]
-def idealFactorsEquivOfQuotEquiv : { p : Ideal R | p ∣ I } ≃o { p : Ideal A | p ∣ J } :=
+def idealFactorsEquivOfQuotEquiv : {p : Ideal R | p ∣ I} ≃o {p : Ideal A | p ∣ J} :=
   OrderIso.ofHomInv
     (idealFactorsFunOfQuotHom (show Function.Surjective (f : R ⧸ I →+* A ⧸ J) from f.Surjective))
     (idealFactorsFunOfQuotHom
@@ -1173,7 +1173,7 @@ theorem idealFactorsEquivOfQuotEquiv_is_dvd_iso {L M : Ideal R} (hL : L ∣ I) (
   by
   suffices
     idealFactorsEquivOfQuotEquiv f ⟨M, hM⟩ ≤ idealFactorsEquivOfQuotEquiv f ⟨L, hL⟩ ↔
-      (⟨M, hM⟩ : { p : Ideal R | p ∣ I }) ≤ ⟨L, hL⟩
+      (⟨M, hM⟩ : {p : Ideal R | p ∣ I}) ≤ ⟨L, hL⟩
     by rw [dvd_iff_le, dvd_iff_le, Subtype.coe_le_coe, this, Subtype.mk_le_mk]
   exact (idealFactorsEquivOfQuotEquiv f).le_iff_le
 #align ideal_factors_equiv_of_quot_equiv_is_dvd_iso idealFactorsEquivOfQuotEquiv_is_dvd_iso
@@ -1200,7 +1200,7 @@ theorem idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFact
     isomorphism `f : R/I ≅ A/J`. -/
 @[simps apply]
 def normalizedFactorsEquivOfQuotEquiv (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
-    { L : Ideal R | L ∈ normalizedFactors I } ≃ { M : Ideal A | M ∈ normalizedFactors J }
+    {L : Ideal R | L ∈ normalizedFactors I} ≃ {M : Ideal A | M ∈ normalizedFactors J}
     where
   toFun j :=
     ⟨idealFactorsEquivOfQuotEquiv f ⟨↑j, dvd_of_mem_normalizedFactors j.Prop⟩,
@@ -1248,10 +1248,10 @@ open scoped BigOperators
 
 variable {R}
 
-theorem Ring.DimensionLeOne.prime_le_prime_iff_eq (h : Ring.DimensionLeOne R) {P Q : Ideal R}
+theorem Ring.DimensionLEOne.prime_le_prime_iff_eq (h : Ring.DimensionLEOne R) {P Q : Ideal R}
     [hP : P.IsPrime] [hQ : Q.IsPrime] (hP0 : P ≠ ⊥) : P ≤ Q ↔ P = Q :=
   ⟨(h P hP0 hP).eq_of_le hQ.ne_top, Eq.le⟩
-#align ring.dimension_le_one.prime_le_prime_iff_eq Ring.DimensionLeOne.prime_le_prime_iff_eq
+#align ring.dimension_le_one.prime_le_prime_iff_eq Ring.DimensionLEOne.prime_le_prime_iff_eq
 
 theorem Ideal.coprime_of_no_prime_ge {I J : Ideal R} (h : ∀ P, I ≤ P → J ≤ P → ¬IsPrime P) :
     I ⊔ J = ⊤ := by
@@ -1537,8 +1537,8 @@ variable [DecidableEq R] [DecidableEq (Ideal R)] [NormalizationMonoid R]
     of `span {r}` -/
 @[simps]
 noncomputable def normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r ≠ 0) :
-    { d : R | d ∈ normalizedFactors r } ≃
-      { I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R)) } :=
+    {d : R | d ∈ normalizedFactors r} ≃
+      {I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R))} :=
   Equiv.ofBijective
     (fun d =>
       ⟨Ideal.span {↑d}, singleton_span_mem_normalizedFactors_of_mem_normalizedFactors d.Prop⟩)
@@ -1583,7 +1583,7 @@ theorem multiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_multiplicity
 /-- The bijection `normalized_factors_equiv_span_normalized_factors.symm` between the set of prime
     factors of the ideal `⟨r⟩` and the set of prime factors of `r` preserves multiplicities. -/
 theorem multiplicity_normalizedFactorsEquivSpanNormalizedFactors_symm_eq_multiplicity {r : R}
-    (hr : r ≠ 0) (I : { I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R)) }) :
+    (hr : r ≠ 0) (I : {I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R))}) :
     multiplicity ((normalizedFactorsEquivSpanNormalizedFactors hr).symm I : R) r =
       multiplicity (I : Ideal R) (Ideal.span {r}) :=
   by

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

@@ -228,7 +228,7 @@ theorem invertible_iff_generator_nonzero (I : FractionalIdeal R₁⁰ K)
     rw [eq_span_singleton_of_principal I]
     intro hI
     have := mem_span_singleton_self _ (generator (I : Submodule R₁ K))
-    rw [hI, mem_zero_iff] at this
+    rw [hI, mem_zero_iff] at this 
     contradiction
 #align fractional_ideal.invertible_iff_generator_nonzero FractionalIdeal.invertible_iff_generator_nonzero
 
@@ -356,9 +356,9 @@ theorem dimensionLeOne : DimensionLeOne A :=
   obtain ⟨z, hzM, hzp⟩ := SetLike.exists_of_lt hM
   -- We have `z * y ∈ M * (M⁻¹ * P) = P`.
   have zy_mem := mul_mem_mul (mem_coe_ideal_of_mem A⁰ hzM) hx
-  rw [← RingHom.map_mul, ← mul_assoc, h.mul_inv_eq_one M'_ne, one_mul] at zy_mem
+  rw [← RingHom.map_mul, ← mul_assoc, h.mul_inv_eq_one M'_ne, one_mul] at zy_mem 
   obtain ⟨zy, hzy, zy_eq⟩ := (mem_coe_ideal A⁰).mp zy_mem
-  rw [IsFractionRing.injective A (FractionRing A) zy_eq] at hzy
+  rw [IsFractionRing.injective A (FractionRing A) zy_eq] at hzy 
   -- But `P` is a prime ideal, so `z ∉ P` implies `y ∈ P`, as desired.
   exact mem_coe_ideal_of_mem A⁰ (Or.resolve_left (hP.mem_or_mem hzy) hzp)
 #align is_dedekind_domain_inv.dimension_le_one IsDedekindDomainInv.dimensionLeOne
@@ -400,7 +400,7 @@ theorem exists_multiset_prod_cons_le_and_prod_not_le [IsDedekindDomain A] (hNF :
     have := Multiset.map_erase PrimeSpectrum.asIdeal PrimeSpectrum.ext P Z
     obtain ⟨hP0, hZP0⟩ : P.as_ideal ≠ ⊥ ∧ ((Z.erase P).map PrimeSpectrum.asIdeal).Prod ≠ ⊥ := by
       rwa [Ne.def, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
-        this] at hprodZ
+        this] at hprodZ 
     -- By maximality of `P` and `M`, we have that `P ≤ M` implies `P = M`.
     have hPM' := (IsDedekindDomain.dimensionLeOne _ hP0 P.is_prime).eq_of_le hM.ne_top hPM
     subst hPM'
@@ -461,13 +461,13 @@ theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I
       apply hle
       rw [Multiset.prod_cons]
       exact Submodule.smul_mem_smul h_Iy hbZ
-    rw [Ideal.mem_span_singleton'] at h_yb
+    rw [Ideal.mem_span_singleton'] at h_yb 
     rcases h_yb with ⟨c, hc⟩
     rw [← hc, RingHom.map_mul, mul_assoc, mul_inv_cancel hnz_fa, mul_one]
     apply coe_mem_one
   · refine' mt (mem_one_iff _).mp _
     rintro ⟨x', h₂_abs⟩
-    rw [← div_eq_mul_inv, eq_div_iff_mul_eq hnz_fa, ← RingHom.map_mul] at h₂_abs
+    rw [← div_eq_mul_inv, eq_div_iff_mul_eq hnz_fa, ← RingHom.map_mul] at h₂_abs 
     have := ideal.mem_span_singleton'.mpr ⟨x', IsFractionRing.injective A K h₂_abs⟩
     contradiction
 #align fractional_ideal.exists_not_mem_one_of_ne_bot FractionalIdeal.exists_not_mem_one_of_ne_bot
@@ -505,7 +505,7 @@ theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I
     ∃ x : K, x ∈ (J : FractionalIdeal A⁰ K)⁻¹ ∧ x ∉ (1 : FractionalIdeal A⁰ K) :=
     exists_not_mem_one_of_ne_bot hNF hJ0 hJ1
   contrapose! hx1 with h_abs
-  rw [hJ] at hx
+  rw [hJ] at hx 
   exact hI hx
 #align fractional_ideal.mul_inv_cancel_of_le_one FractionalIdeal.mul_inv_cancel_of_le_one
 
@@ -525,7 +525,7 @@ theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ 
   -- In particular, we'll show all `x ∈ J⁻¹` are integral.
   suffices x ∈ integralClosure A K by
     rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range, ←
-        mem_one_iff] at this <;>
+        mem_one_iff] at this  <;>
       assumption
   -- For that, we'll find a subalgebra that is f.g. as a module and contains `x`.
   -- `A` is a noetherian ring, so we just need to find a subalgebra between `{x}` and `I⁻¹`.
@@ -533,10 +533,10 @@ theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ 
   have x_mul_mem : ∀ b ∈ (I⁻¹ : FractionalIdeal A⁰ K), x * b ∈ (I⁻¹ : FractionalIdeal A⁰ K) :=
     by
     intro b hb
-    rw [mem_inv_iff] at hx⊢
+    rw [mem_inv_iff] at hx ⊢
     swap; · exact coe_ideal_ne_zero.mpr hI0
     swap; · exact hJ0
-    simp only [mul_assoc, mul_comm b] at hx⊢
+    simp only [mul_assoc, mul_comm b] at hx ⊢
     intro y hy
     exact hx _ (mul_mem_mul hy hb)
   -- It turns out the subalgebra consisting of all `p(x)` for `p : A[X]` works.
@@ -563,7 +563,7 @@ protected theorem mul_inv_cancel [IsDedekindDomain A] {I : FractionalIdeal A⁰
     I * I⁻¹ = 1 :=
   by
   obtain ⟨a, J, ha, hJ⟩ :
-    ∃ (a : A)(aI : Ideal A), a ≠ 0 ∧ I = span_singleton A⁰ (algebraMap _ _ a)⁻¹ * aI :=
+    ∃ (a : A) (aI : Ideal A), a ≠ 0 ∧ I = span_singleton A⁰ (algebraMap _ _ a)⁻¹ * aI :=
     exists_eq_span_singleton_mul I
   suffices h₂ : I * (span_singleton A⁰ (algebraMap _ _ a) * J⁻¹) = 1
   · rw [mul_inv_cancel_iff]
@@ -608,7 +608,7 @@ protected theorem div_eq_mul_inv [IsDedekindDomain A] (I J : FractionalIdeal A
   refine' le_antisymm ((mul_right_le_iff hJ).mp _) ((le_div_iff_mul_le hJ).mpr _)
   · rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one, mul_le]
     intro x hx y hy
-    rw [mem_div_iff_of_nonzero hJ] at hx
+    rw [mem_div_iff_of_nonzero hJ] at hx 
     exact hx y hy
   rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one]
   exact le_refl I
@@ -669,7 +669,7 @@ instance Ideal.isDomain : IsDomain (Ideal A) :=
 theorem Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
   ⟨Ideal.le_of_dvd, fun h => by
     by_cases hI : I = ⊥
-    · have hJ : J = ⊥ := by rwa [hI, ← eq_bot_iff] at h
+    · have hJ : J = ⊥ := by rwa [hI, ← eq_bot_iff] at h 
       rw [hI, hJ]
     have hI' : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coe_ideal_ne_zero.mpr hI
     have : (I : FractionalIdeal A⁰ (FractionRing A))⁻¹ * J ≤ 1 :=
@@ -738,9 +738,9 @@ theorem Ideal.isPrime_of_prime {P : Ideal A} (h : Prime P) : IsPrime P :=
   by
   refine' ⟨_, fun x y hxy => _⟩
   · rintro rfl
-    rw [← Ideal.one_eq_top] at h
+    rw [← Ideal.one_eq_top] at h 
     exact h.not_unit isUnit_one
-  · simp only [← Ideal.dvd_span_singleton, ← Ideal.span_singleton_mul_span_singleton] at hxy⊢
+  · simp only [← Ideal.dvd_span_singleton, ← Ideal.span_singleton_mul_span_singleton] at hxy ⊢
     exact h.dvd_or_dvd hxy
 #align ideal.is_prime_of_prime Ideal.isPrime_of_prime
 
@@ -791,7 +791,7 @@ theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime
   have := pow_ne_zero (i + 1) hP
   rw [← Ideal.dvdNotUnit_iff_lt, dvd_not_unit_iff_normalized_factors_lt_normalized_factors,
     normalized_factors_pow, normalized_factors_irreducible P_prime'.irreducible,
-    Multiset.nsmul_singleton, Multiset.lt_replicate_succ] at hlt
+    Multiset.nsmul_singleton, Multiset.lt_replicate_succ] at hlt 
   rw [← Ideal.dvd_iff_le, dvd_iff_normalized_factors_le_normalized_factors, normalized_factors_pow,
     normalized_factors_irreducible P_prime'.irreducible, Multiset.nsmul_singleton]
   all_goals assumption
@@ -829,7 +829,7 @@ theorem Ideal.exist_integer_multiples_not_mem {J : Ideal A} (hJ : J ≠ ⊤) {ι
   suffices ↑J / I < I⁻¹
     by
     obtain ⟨_, a, hI, hpI⟩ := set_like.lt_iff_le_and_exists.mp this
-    rw [mem_inv_iff hI0] at hI
+    rw [mem_inv_iff hI0] at hI 
     refine' ⟨a, fun i hi => _, _⟩
     -- By definition, `a ∈ I⁻¹` multiplies elements of `I` into elements of `1`,
     -- in other words, `a * f i` is an integer.
@@ -955,7 +955,7 @@ theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
     by
     apply normalized_factors_prod_of_prime
     intro p hp
-    rw [mem_inter] at hp
+    rw [mem_inter] at hp 
     exact prime_of_normalized_factor p hp.left
   have :=
     Multiset.prod_ne_zero_of_prime (normalized_factors I ∩ normalized_factors J) fun _ h =>
@@ -973,7 +973,7 @@ theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
       rw [Multiset.count_inter]
       exact le_min (count_le_of_ideal_ge le_sup_left hI a) (count_le_of_ideal_ge le_sup_right hJ a)
     · intro p hp
-      rw [mem_inter] at hp
+      rw [mem_inter] at hp 
       exact prime_of_normalized_factor p hp.left
     · exact ne_bot_of_le_ne_bot hI le_sup_left
     · exact this
@@ -990,7 +990,7 @@ theorem irreducible_pow_sup_of_le (hJ : Irreducible J) (n : ℕ) (hn : ↑n ≤
   by_cases hI : I = ⊥
   · simp_all
   rw [irreducible_pow_sup hI hJ, min_eq_right]
-  rwa [multiplicity_eq_count_normalized_factors hJ hI, PartENat.coe_le_coe, normalize_eq J] at hn
+  rwa [multiplicity_eq_count_normalized_factors hJ hI, PartENat.coe_le_coe, normalize_eq J] at hn 
 #align irreducible_pow_sup_of_le irreducible_pow_sup_of_le
 
 theorem irreducible_pow_sup_of_ge (hI : I ≠ ⊥) (hJ : Irreducible J) (n : ℕ)
@@ -1002,7 +1002,7 @@ theorem irreducible_pow_sup_of_ge (hI : I ≠ ⊥) (hJ : Irreducible J) (n : ℕ
   ·
     rw [← PartENat.natCast_inj, PartENat.natCast_get,
       multiplicity_eq_count_normalized_factors hJ hI, normalize_eq J]
-  · rwa [multiplicity_eq_count_normalized_factors hJ hI, PartENat.coe_le_coe, normalize_eq J] at hn
+  · rwa [multiplicity_eq_count_normalized_factors hJ hI, PartENat.coe_le_coe, normalize_eq J] at hn 
 #align irreducible_pow_sup_of_ge irreducible_pow_sup_of_ge
 
 end IsDedekindDomain
@@ -1105,11 +1105,11 @@ def idealFactorsFunOfQuotHom {f : R ⧸ I →+* A ⧸ J} (hf : Function.Surjecti
       by
       have : J.Quotient.mk.ker ≤ comap J.Quotient.mk (map f (map I.Quotient.mk X)) :=
         ker_le_comap J.Quotient.mk
-      rw [mk_ker] at this
+      rw [mk_ker] at this 
       exact dvd_iff_le.mpr this⟩
   monotone' := by
     rintro ⟨X, hX⟩ ⟨Y, hY⟩ h
-    rw [← Subtype.coe_le_coe, Subtype.coe_mk, Subtype.coe_mk] at h⊢
+    rw [← Subtype.coe_le_coe, Subtype.coe_mk, Subtype.coe_mk] at h ⊢
     rw [Subtype.coe_mk, comap_le_comap_iff_of_surjective J.Quotient.mk quotient.mk_surjective,
       map_le_iff_le_comap, Subtype.coe_mk, comap_map_of_surjective _ hf (map I.Quotient.mk Y)]
     suffices map I.Quotient.mk X ≤ map I.Quotient.mk Y by exact le_sup_of_le_left this
@@ -1188,7 +1188,7 @@ theorem idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFact
   by
   by_cases hI : I = ⊥
   · exfalso
-    rw [hI, bot_eq_zero, normalized_factors_zero, ← Multiset.empty_eq_zero] at hL
+    rw [hI, bot_eq_zero, normalized_factors_zero, ← Multiset.empty_eq_zero] at hL 
     exact hL
   · apply mem_normalizedFactors_factor_dvd_iso_of_mem_normalizedFactors hI hJ hL _
     rintro ⟨l, hl⟩ ⟨l', hl'⟩
@@ -1270,10 +1270,10 @@ theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n :
   cases n; · simp
   by_cases hI0 : I = ⊥; · simpa [pow_succ, hI0] using h
   simp only [← Submodule.span_singleton_le_iff_mem, Ideal.submodule_span_eq, ← Ideal.dvd_iff_le, ←
-    Ideal.span_singleton_mul_span_singleton] at h⊢
+    Ideal.span_singleton_mul_span_singleton] at h ⊢
   by_cases ha : I ∣ span {a}
   · exact Or.inl ha
-  rw [mul_comm] at h
+  rw [mul_comm] at h 
   exact Or.inr (Prime.pow_dvd_of_dvd_mul_right ((Ideal.prime_iff_isPrime hI0).mpr hI) _ ha h)
 #align ideal.is_prime.mul_mem_pow Ideal.IsPrime.mul_mem_pow
 
@@ -1295,7 +1295,7 @@ end
 theorem Ideal.le_mul_of_no_prime_factors {I J K : Ideal R}
     (coprime : ∀ P, J ≤ P → K ≤ P → ¬IsPrime P) (hJ : I ≤ J) (hK : I ≤ K) : I ≤ J * K :=
   by
-  simp only [← Ideal.dvd_iff_le] at coprime hJ hK⊢
+  simp only [← Ideal.dvd_iff_le] at coprime hJ hK ⊢
   by_cases hJ0 : J = 0
   · simpa only [hJ0, MulZeroClass.zero_mul] using hJ
   obtain ⟨I', rfl⟩ := hK
@@ -1309,9 +1309,9 @@ theorem Ideal.le_mul_of_no_prime_factors {I J K : Ideal R}
 theorem Ideal.le_of_pow_le_prime {I P : Ideal R} [hP : P.IsPrime] {n : ℕ} (h : I ^ n ≤ P) : I ≤ P :=
   by
   by_cases hP0 : P = ⊥
-  · simp only [hP0, le_bot_iff] at h⊢
+  · simp only [hP0, le_bot_iff] at h ⊢
     exact pow_eq_zero h
-  rw [← Ideal.dvd_iff_le] at h⊢
+  rw [← Ideal.dvd_iff_le] at h ⊢
   exact ((Ideal.prime_iff_isPrime hP0).mpr hP).dvd_of_dvd_pow h
 #align ideal.le_of_pow_le_prime Ideal.le_of_pow_le_prime
 
@@ -1487,7 +1487,7 @@ theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [Normaliza
   by
   by_cases hb : b = 0
   · rw [ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalized_factors_zero]
-    rw [hb, normalized_factors_zero] at ha
+    rw [hb, normalized_factors_zero] at ha 
     simpa only [Multiset.not_mem_zero]
   · suffices Prime (Ideal.span ({a} : Set R))
       by
@@ -1522,7 +1522,7 @@ theorem multiplicity_eq_multiplicity_span [DecidableRel ((· ∣ ·) : R → R 
           ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _)))
   · suffices ¬Finite (Ideal.span ({a} : Set R)) (Ideal.span ({b} : Set R))
       by
-      rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this
+      rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this 
       rw [h, this]
     refine'
       not_finite_iff_forall.mpr fun n =>
@@ -1546,7 +1546,7 @@ noncomputable def normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r 
       constructor
       · rintro ⟨a, ha⟩ ⟨b, hb⟩ h
         rw [Subtype.mk_eq_mk, Ideal.span_singleton_eq_span_singleton, Subtype.coe_mk,
-          Subtype.coe_mk] at h
+          Subtype.coe_mk] at h 
         exact subtype.mk_eq_mk.mpr (mem_normalized_factors_eq_of_associated ha hb h)
       · rintro ⟨i, hi⟩
         letI : i.is_principal := inferInstance

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

@@ -1153,11 +1153,11 @@ def idealFactorsEquivOfQuotEquiv : { p : Ideal R | p ∣ I } ≃o { p : Ideal A
     (idealFactorsFunOfQuotHom
       (show Function.Surjective (f.symm : A ⧸ J →+* R ⧸ I) from f.symm.Surjective))
     (by
-      simp only [← idealFactorsFunOfQuotHom_id, [anonymous], [anonymous],
+      simp only [← idealFactorsFunOfQuotHom_id, OrderHom.coe_eq, OrderHom.coe_eq,
         idealFactorsFunOfQuotHom_comp, ← RingEquiv.toRingHom_eq_coe, ← RingEquiv.toRingHom_eq_coe, ←
         RingEquiv.toRingHom_trans, RingEquiv.symm_trans_self, RingEquiv.toRingHom_refl])
     (by
-      simp only [← idealFactorsFunOfQuotHom_id, [anonymous], [anonymous],
+      simp only [← idealFactorsFunOfQuotHom_id, OrderHom.coe_eq, OrderHom.coe_eq,
         idealFactorsFunOfQuotHom_comp, ← RingEquiv.toRingHom_eq_coe, ← RingEquiv.toRingHom_eq_coe, ←
         RingEquiv.toRingHom_trans, RingEquiv.self_trans_symm, RingEquiv.toRingHom_refl])
 #align ideal_factors_equiv_of_quot_equiv idealFactorsEquivOfQuotEquiv

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

@@ -57,7 +57,7 @@ dedekind domain, dedekind ring
 
 variable (R A K : Type _) [CommRing R] [CommRing A] [Field K]
 
-open nonZeroDivisors Polynomial
+open scoped nonZeroDivisors Polynomial
 
 variable [IsDomain A]
 
@@ -930,7 +930,7 @@ section IsDedekindDomain
 
 variable {T : Type _} [CommRing T] [IsDomain T] [IsDedekindDomain T] {I J : Ideal T}
 
-open Classical
+open scoped Classical
 
 open Multiset UniqueFactorizationMonoid Ideal
 
@@ -1244,7 +1244,7 @@ section ChineseRemainder
 
 open Ideal UniqueFactorizationMonoid
 
-open BigOperators
+open scoped BigOperators
 
 variable {R}
 
@@ -1279,7 +1279,7 @@ theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n :
 
 section
 
-open Classical
+open scoped Classical
 
 theorem Ideal.count_normalizedFactors_eq {p x : Ideal R} [hp : p.IsPrime] {n : ℕ} (hle : x ≤ p ^ n)
     (hlt : ¬x ≤ p ^ (n + 1)) : (normalizedFactors x).count p = n :=
@@ -1388,7 +1388,7 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type _} [Fintyp
                     (Ideal.le_of_pow_le_prime hPj)).symm))
 #align is_dedekind_domain.quotient_equiv_pi_of_prod_eq IsDedekindDomain.quotientEquivPiOfProdEq
 
-open Classical
+open scoped Classical
 
 /-- **Chinese remainder theorem** for a Dedekind domain: `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`,
 where `P i` ranges over the prime factors of `I` and `e i` over the multiplicities. -/

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

@@ -86,10 +86,7 @@ theorem inv_nonzero {J : FractionalIdeal R₁⁰ K} (h : J ≠ 0) :
 #align fractional_ideal.inv_nonzero FractionalIdeal.inv_nonzero
 
 theorem coe_inv_of_nonzero {J : FractionalIdeal R₁⁰ K} (h : J ≠ 0) :
-    (↑J⁻¹ : Submodule R₁ K) = IsLocalization.coeSubmodule K ⊤ / J :=
-  by
-  rwa [inv_nonzero _]
-  rfl
+    (↑J⁻¹ : Submodule R₁ K) = IsLocalization.coeSubmodule K ⊤ / J := by rwa [inv_nonzero _]; rfl;
   assumption
 #align fractional_ideal.coe_inv_of_nonzero FractionalIdeal.coe_inv_of_nonzero
 
@@ -99,10 +96,8 @@ theorem mem_inv_iff (hI : I ≠ 0) {x : K} : x ∈ I⁻¹ ↔ ∀ y ∈ I, x * y
   mem_div_iff_of_nonzero hI
 #align fractional_ideal.mem_inv_iff FractionalIdeal.mem_inv_iff
 
-theorem inv_anti_mono (hI : I ≠ 0) (hJ : J ≠ 0) (hIJ : I ≤ J) : J⁻¹ ≤ I⁻¹ := fun x =>
-  by
-  simp only [mem_inv_iff hI, mem_inv_iff hJ]
-  exact fun h y hy => h y (hIJ hy)
+theorem inv_anti_mono (hI : I ≠ 0) (hJ : J ≠ 0) (hIJ : I ≤ J) : J⁻¹ ≤ I⁻¹ := fun x => by
+  simp only [mem_inv_iff hI, mem_inv_iff hJ]; exact fun h y hy => h y (hIJ hy)
 #align fractional_ideal.inv_anti_mono FractionalIdeal.inv_anti_mono
 
 theorem le_self_mul_inv {I : FractionalIdeal R₁⁰ K} (hI : I ≤ (1 : FractionalIdeal R₁⁰ K)) :
@@ -282,9 +277,7 @@ theorem FractionalIdeal.adjoinIntegral_eq_one_of_isUnit [Algebra A K] [IsFractio
     (hx : IsIntegral A x) (hI : IsUnit (adjoinIntegral A⁰ x hx)) : adjoinIntegral A⁰ x hx = 1 :=
   by
   set I := adjoin_integral A⁰ x hx
-  have mul_self : I * I = I := by
-    apply coe_to_submodule_injective
-    simp
+  have mul_self : I * I = I := by apply coe_to_submodule_injective; simp
   convert congr_arg (· * I⁻¹) mul_self <;>
     simp only [(mul_inv_cancel_iff_is_unit K).mpr hI, mul_assoc, mul_one]
 #align fractional_ideal.adjoin_integral_eq_one_of_is_unit FractionalIdeal.adjoinIntegral_eq_one_of_isUnit
@@ -311,8 +304,7 @@ theorem isNoetherianRing : IsNoetherianRing A :=
   by
   refine' is_noetherian_ring_iff.mpr ⟨fun I : Ideal A => _⟩
   by_cases hI : I = ⊥
-  · rw [hI]
-    apply Submodule.fg_bot
+  · rw [hI]; apply Submodule.fg_bot
   have hI : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coe_ideal_ne_zero.mpr hI
   exact I.fg_of_is_unit (IsFractionRing.injective A (FractionRing A)) (h.is_unit hI)
 #align is_dedekind_domain_inv.is_noetherian_ring IsDedekindDomainInv.isNoetherianRing
@@ -400,9 +392,7 @@ theorem exists_multiset_prod_cons_le_and_prod_not_le [IsDedekindDomain A] (hNF :
       (fun Z => (Z.map PrimeSpectrum.asIdeal).Prod ≤ I ∧ (Z.map PrimeSpectrum.asIdeal).Prod ≠ ⊥)
       ⟨Z₀, hZ₀⟩
   have hZM : Multiset.prod (Z.map PrimeSpectrum.asIdeal) ≤ M := le_trans hZI hIM
-  have hZ0 : Z ≠ 0 := by
-    rintro rfl
-    simpa [hM.ne_top] using hZM
+  have hZ0 : Z ≠ 0 := by rintro rfl; simpa [hM.ne_top] using hZM
   obtain ⟨_, hPZ', hPM⟩ := (hM.is_prime.multiset_prod_le (mt multiset.map_eq_zero.mp hZ0)).mp hZM
   -- Then in fact there is a `P ∈ Z` with `P ≤ M`.
   obtain ⟨P, hPZ, rfl⟩ := multiset.mem_map.mp hPZ'
@@ -498,8 +488,7 @@ theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I
   by_cases hI1 : I = ⊤
   · rw [hI1, coe_ideal_top, one_mul, inv_one]
   by_cases hNF : IsField A
-  · letI := hNF.to_field
-    rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩
+  · letI := hNF.to_field; rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩
   -- We'll show a contradiction with `exists_not_mem_one_of_ne_bot`:
   -- `J⁻¹ = (I * I⁻¹)⁻¹` cannot have an element `x ∉ 1`, so it must equal `1`.
   obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * I⁻¹ :=
@@ -531,8 +520,7 @@ theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ 
   -- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.
   apply mul_inv_cancel_of_le_one hI0
   by_cases hJ0 : (I * I⁻¹ : FractionalIdeal A⁰ K) = 0
-  · rw [hJ0, inv_zero']
-    exact zero_le _
+  · rw [hJ0, inv_zero']; exact zero_le _
   intro x hx
   -- In particular, we'll show all `x ∈ J⁻¹` are integral.
   suffices x ∈ integralClosure A K by
@@ -546,10 +534,8 @@ theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ 
     by
     intro b hb
     rw [mem_inv_iff] at hx⊢
-    swap
-    · exact coe_ideal_ne_zero.mpr hI0
-    swap
-    · exact hJ0
+    swap; · exact coe_ideal_ne_zero.mpr hI0
+    swap; · exact hJ0
     simp only [mul_assoc, mul_comm b] at hx⊢
     intro y hy
     exact hx _ (mul_mem_mul hy hb)
@@ -563,11 +549,9 @@ theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ 
   refine' Submodule.sum_mem _ fun i hi => Submodule.smul_mem _ _ _
   clear hi
   induction' i with i ih
-  · rw [pow_zero]
-    exact one_mem_inv_coe_ideal hI0
+  · rw [pow_zero]; exact one_mem_inv_coe_ideal hI0
   · show x ^ i.succ ∈ (I⁻¹ : FractionalIdeal A⁰ K)
-    rw [pow_succ]
-    exact x_mul_mem _ ih
+    rw [pow_succ]; exact x_mul_mem _ ih
 #align fractional_ideal.coe_ideal_mul_inv FractionalIdeal.coe_ideal_mul_inv
 
 /-- Nonzero fractional ideals in a Dedekind domain are units.
@@ -595,7 +579,7 @@ theorem mul_right_le_iff [IsDedekindDomain A] {J : FractionalIdeal A⁰ K} (hJ :
     ∀ {I I'}, I * J ≤ I' * J ↔ I ≤ I' := by
   intro I I'
   constructor
-  · intro h
+  · intro h;
     convert mul_right_mono J⁻¹ h <;> rw [mul_assoc, FractionalIdeal.mul_inv_cancel hJ, mul_one]
   · exact fun h => mul_right_mono J h
 #align fractional_ideal.mul_right_le_iff FractionalIdeal.mul_right_le_iff
@@ -854,16 +838,10 @@ theorem Ideal.exist_integer_multiples_not_mem {J : Ideal A} (hJ : J ≠ ⊤) {ι
       -- And if all `a`-multiples of `I` are an element of `J`,
       -- then `a` is actually an element of `J / I`, contradiction.
       refine' (mem_div_iff_of_nonzero hI0).mpr fun y hy => Submodule.span_induction hy _ _ _ _
-      · rintro _ ⟨i, hi, rfl⟩
-        exact hpI i hi
-      · rw [MulZeroClass.mul_zero]
-        exact Submodule.zero_mem _
-      · intro x y hx hy
-        rw [mul_add]
-        exact Submodule.add_mem _ hx hy
-      · intro b x hx
-        rw [mul_smul_comm]
-        exact Submodule.smul_mem _ b hx
+      · rintro _ ⟨i, hi, rfl⟩; exact hpI i hi
+      · rw [MulZeroClass.mul_zero]; exact Submodule.zero_mem _
+      · intro x y hx hy; rw [mul_add]; exact Submodule.add_mem _ hx hy
+      · intro b x hx; rw [mul_smul_comm]; exact Submodule.smul_mem _ b hx
   -- To show the inclusion of `J / I` into `I⁻¹ = 1 / I`, note that `J < I`.
   calc
     ↑J / I = ↑J * I⁻¹ := div_eq_mul_inv (↑J) I
@@ -1306,9 +1284,7 @@ open Classical
 theorem Ideal.count_normalizedFactors_eq {p x : Ideal R} [hp : p.IsPrime] {n : ℕ} (hle : x ≤ p ^ n)
     (hlt : ¬x ≤ p ^ (n + 1)) : (normalizedFactors x).count p = n :=
   count_normalizedFactors_eq' ((Ideal.isPrime_iff_bot_or_prime.mp hp).imp_right Prime.irreducible)
-    (by
-      haveI : Unique (Ideal R)ˣ := Ideal.uniqueUnits
-      apply normalize_eq)
+    (by haveI : Unique (Ideal R)ˣ := Ideal.uniqueUnits; apply normalize_eq)
     (by convert ideal.dvd_iff_le.mpr hle) (by convert mt Ideal.le_of_dvd hlt)
 #align ideal.count_normalized_factors_eq Ideal.count_normalizedFactors_eq
 
@@ -1383,8 +1359,7 @@ theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type _} (s : Finset ι) (f
             (Ideal.le_of_pow_le_prime hPa)).trans
         ((is_dedekind_domain.dimension_le_one.prime_le_prime_iff_eq _).mp
             (Ideal.le_of_pow_le_prime hPb)).symm)
-  · rintro rfl
-    contradiction
+  · rintro rfl; contradiction
   · exact (Prime a (Finset.mem_insert_self a s)).NeZero
   · exact (Prime b (Finset.mem_insert_of_mem hb)).NeZero
 #align is_dedekind_domain.inf_prime_pow_eq_prod IsDedekindDomain.inf_prime_pow_eq_prod

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

@@ -113,7 +113,7 @@ theorem le_self_mul_inv {I : FractionalIdeal R₁⁰ K} (hI : I ≤ (1 : Fractio
 variable (K)
 
 theorem coe_ideal_le_self_mul_inv (I : Ideal R₁) : (I : FractionalIdeal R₁⁰ K) ≤ I * I⁻¹ :=
-  le_self_mul_inv coe_ideal_le_one
+  le_self_mul_inv coeIdeal_le_one
 #align fractional_ideal.coe_ideal_le_self_mul_inv FractionalIdeal.coe_ideal_le_self_mul_inv
 
 /-- `I⁻¹` is the inverse of `I` if `I` has an inverse. -/
@@ -650,7 +650,7 @@ open Ideal
 
 noncomputable instance FractionalIdeal.semifield : Semifield (FractionalIdeal A⁰ K) :=
   { FractionalIdeal.commSemiring,
-    coe_ideal_injective.Nontrivial with
+    coeIdeal_injective.Nontrivial with
     inv := fun I => I⁻¹
     inv_zero := inv_zero' _
     div := (· / ·)
@@ -673,9 +673,8 @@ instance FractionalIdeal.cancelCommMonoidWithZero :
 
 instance Ideal.cancelCommMonoidWithZero : CancelCommMonoidWithZero (Ideal A) :=
   { Ideal.idemCommSemiring,
-    Function.Injective.cancelCommMonoidWithZero (coeIdealHom A⁰ (FractionRing A))
-      coe_ideal_injective (RingHom.map_zero _) (RingHom.map_one _) (RingHom.map_mul _)
-      (RingHom.map_pow _) with }
+    Function.Injective.cancelCommMonoidWithZero (coeIdealHom A⁰ (FractionRing A)) coeIdeal_injective
+      (RingHom.map_zero _) (RingHom.map_one _) (RingHom.map_mul _) (RingHom.map_pow _) with }
 #align ideal.cancel_comm_monoid_with_zero Ideal.cancelCommMonoidWithZero
 
 instance Ideal.isDomain : IsDomain (Ideal A) :=

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

@@ -541,7 +541,7 @@ theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ 
       assumption
   -- For that, we'll find a subalgebra that is f.g. as a module and contains `x`.
   -- `A` is a noetherian ring, so we just need to find a subalgebra between `{x}` and `I⁻¹`.
-  rw [mem_integralClosure_iff_mem_fG]
+  rw [mem_integralClosure_iff_mem_FG]
   have x_mul_mem : ∀ b ∈ (I⁻¹ : FractionalIdeal A⁰ K), x * b ∈ (I⁻¹ : FractionalIdeal A⁰ K) :=
     by
     intro b hb

mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8

@@ -541,7 +541,7 @@ theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ 
       assumption
   -- For that, we'll find a subalgebra that is f.g. as a module and contains `x`.
   -- `A` is a noetherian ring, so we just need to find a subalgebra between `{x}` and `I⁻¹`.
-  rw [mem_integralClosure_iff_mem_fg]
+  rw [mem_integralClosure_iff_mem_fG]
   have x_mul_mem : ∀ b ∈ (I⁻¹ : FractionalIdeal A⁰ K), x * b ∈ (I⁻¹ : FractionalIdeal A⁰ K) :=
     by
     intro b hb

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

@@ -1088,13 +1088,13 @@ variable (R K)
 
 /-- A Dedekind domain is equal to the intersection of its localizations at all its height one
 non-zero prime ideals viewed as subalgebras of its field of fractions. -/
-theorem infᵢ_localization_eq_bot [Algebra R K] [hK : IsFractionRing R K] :
+theorem iInf_localization_eq_bot [Algebra R K] [hK : IsFractionRing R K] :
     (⨅ v : HeightOneSpectrum R,
         Localization.subalgebra.ofField K _ v.asIdeal.primeCompl_le_nonZeroDivisors) =
       ⊥ :=
   by
   ext x
-  rw [Algebra.mem_infᵢ]
+  rw [Algebra.mem_iInf]
   constructor
   by_cases hR : IsField R
   · rcases function.bijective_iff_has_inverse.mp
@@ -1102,10 +1102,10 @@ theorem infᵢ_localization_eq_bot [Algebra R K] [hK : IsFractionRing R K] :
       ⟨algebra_map_inv, _, algebra_map_right_inv⟩
     exact fun _ => algebra.mem_bot.mpr ⟨algebra_map_inv x, algebra_map_right_inv x⟩
     exact hK
-  all_goals rw [← MaximalSpectrum.infᵢ_localization_eq_bot, Algebra.mem_infᵢ]
+  all_goals rw [← MaximalSpectrum.iInf_localization_eq_bot, Algebra.mem_iInf]
   · exact fun hx ⟨v, hv⟩ => hx ((equiv_maximal_spectrum hR).symm ⟨v, hv⟩)
   · exact fun hx ⟨v, hv, hbot⟩ => hx ⟨v, dimension_le_one v hbot hv⟩
-#align is_dedekind_domain.height_one_spectrum.infi_localization_eq_bot IsDedekindDomain.HeightOneSpectrum.infᵢ_localization_eq_bot
+#align is_dedekind_domain.height_one_spectrum.infi_localization_eq_bot IsDedekindDomain.HeightOneSpectrum.iInf_localization_eq_bot
 
 end HeightOneSpectrum
 
@@ -1397,7 +1397,7 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type _} [Fintyp
     (prod_eq : (∏ i, P i ^ e i) = I) : R ⧸ I ≃+* ∀ i, R ⧸ P i ^ e i :=
   (Ideal.quotEquivOfEq
         (by
-          simp only [← prod_eq, Finset.inf_eq_infᵢ, Finset.mem_univ, cinfᵢ_pos, ←
+          simp only [← prod_eq, Finset.inf_eq_iInf, Finset.mem_univ, ciInf_pos, ←
             IsDedekindDomain.inf_prime_pow_eq_prod _ _ _ (fun i _ => Prime i) fun i _ j _ =>
               coprime i j])).trans <|
     Ideal.quotientInfRingEquivPiQuotient _ fun i j hij =>

mathlib commit https://github.com/leanprover-community/mathlib/commit/730c6d4cab72b9d84fcfb9e95e8796e9cd8f40ba

@@ -1257,7 +1257,7 @@ theorem normalizedFactorsEquivOfQuotEquiv_multiplicity_eq_multiplicity (hI : I 
   by
   rw [normalizedFactorsEquivOfQuotEquiv, Equiv.coe_fn_mk, Subtype.coe_mk]
   exact
-    multiplicity_factor_dvd_iso_eq_multiplicity_of_mem_normalized_factor hI hJ hL
+    multiplicity_factor_dvd_iso_eq_multiplicity_of_mem_normalizedFactors hI hJ hL
       fun ⟨l, hl⟩ ⟨l', hl'⟩ => idealFactorsEquivOfQuotEquiv_is_dvd_iso f hl hl'
 #align normalized_factors_equiv_of_quot_equiv_multiplicity_eq_multiplicity normalizedFactorsEquivOfQuotEquiv_multiplicity_eq_multiplicity
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/290a7ba01fbcab1b64757bdaa270d28f4dcede35

@@ -894,7 +894,7 @@ theorem sup_mul_inf (I J : Ideal A) : (I ⊔ J) * (I ⊓ J) = I * J :=
   by
   letI := Classical.decEq (Ideal A)
   letI := Classical.decEq (Associates (Ideal A))
-  letI := UniqueFactorizationMonoid.toNormalizedGcdMonoid (Ideal A)
+  letI := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
   have hgcd : gcd I J = I ⊔ J :=
     by
     rw [gcd_eq_normalize _ _, normalize_eq]

mathlib commit https://github.com/leanprover-community/mathlib/commit/2af0836443b4cfb5feda0df0051acdb398304931

@@ -557,7 +557,7 @@ theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ 
   refine'
     ⟨AlgHom.range (Polynomial.aeval x : A[X] →ₐ[A] K),
       is_noetherian_submodule.mp (IsNoetherian I⁻¹) _ fun y hy => _,
-      ⟨Polynomial.X, Polynomial.aeval_x x⟩⟩
+      ⟨Polynomial.X, Polynomial.aeval_X x⟩⟩
   obtain ⟨p, rfl⟩ := (AlgHom.mem_range _).mp hy
   rw [Polynomial.aeval_eq_sum_range]
   refine' Submodule.sum_mem _ fun i hi => Submodule.smul_mem _ _ _

mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212

@@ -620,7 +620,7 @@ theorem mul_left_strictMono [IsDedekindDomain A] {I : FractionalIdeal A⁰ K} (h
 protected theorem div_eq_mul_inv [IsDedekindDomain A] (I J : FractionalIdeal A⁰ K) :
     I / J = I * J⁻¹ := by
   by_cases hJ : J = 0
-  · rw [hJ, div_zero, inv_zero', mul_zero]
+  · rw [hJ, div_zero, inv_zero', MulZeroClass.mul_zero]
   refine' le_antisymm ((mul_right_le_iff hJ).mp _) ((le_div_iff_mul_le hJ).mpr _)
   · rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one, mul_le]
     intro x hx y hy
@@ -857,7 +857,7 @@ theorem Ideal.exist_integer_multiples_not_mem {J : Ideal A} (hJ : J ≠ ⊤) {ι
       refine' (mem_div_iff_of_nonzero hI0).mpr fun y hy => Submodule.span_induction hy _ _ _ _
       · rintro _ ⟨i, hi, rfl⟩
         exact hpI i hi
-      · rw [mul_zero]
+      · rw [MulZeroClass.mul_zero]
         exact Submodule.zero_mem _
       · intro x y hx hy
         rw [mul_add]
@@ -1322,7 +1322,7 @@ theorem Ideal.le_mul_of_no_prime_factors {I J K : Ideal R}
   by
   simp only [← Ideal.dvd_iff_le] at coprime hJ hK⊢
   by_cases hJ0 : J = 0
-  · simpa only [hJ0, zero_mul] using hJ
+  · simpa only [hJ0, MulZeroClass.zero_mul] using hJ
   obtain ⟨I', rfl⟩ := hK
   rw [mul_comm]
   exact

mathlib commit https://github.com/leanprover-community/mathlib/commit/38f16f960f5006c6c0c2bac7b0aba5273188f4e5

@@ -557,7 +557,7 @@ theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ 
   refine'
     ⟨AlgHom.range (Polynomial.aeval x : A[X] →ₐ[A] K),
       is_noetherian_submodule.mp (IsNoetherian I⁻¹) _ fun y hy => _,
-      ⟨Polynomial.x, Polynomial.aeval_x x⟩⟩
+      ⟨Polynomial.X, Polynomial.aeval_x x⟩⟩
   obtain ⟨p, rfl⟩ := (AlgHom.mem_range _).mp hy
   rw [Polynomial.aeval_eq_sum_range]
   refine' Submodule.sum_mem _ fun i hi => Submodule.smul_mem _ _ _

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

@@ -252,7 +252,7 @@ noncomputable instance : InvOneClass (FractionalIdeal R₁⁰ K) :=
 
 end FractionalIdeal
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) (fraction_ring[fraction_ring] A))) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) (fraction_ring[fraction_ring] A))) -/
 /-- A Dedekind domain is an integral domain such that every fractional ideal has an inverse.
 
 This is equivalent to `is_dedekind_domain`.
@@ -268,7 +268,7 @@ open FractionalIdeal
 
 variable {R A K}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) K)) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (I «expr ≠ » («expr⊥»() : fractional_ideal[fractional_ideal] non_zero_divisors(A) K)) -/
 theorem isDedekindDomainInv_iff [Algebra A K] [IsFractionRing A K] :
     IsDedekindDomainInv A ↔ ∀ (I) (_ : I ≠ (⊥ : FractionalIdeal A⁰ K)), I * I⁻¹ = 1 :=
   by
@@ -347,7 +347,7 @@ theorem dimensionLeOne : DimensionLeOne A :=
     rw [eq_top_iff, ← coe_ideal_le_coe_ideal (FractionRing A), coe_ideal_top]
     calc
       (1 : FractionalIdeal A⁰ (FractionRing A)) = _ * _ * _ := _
-      _ ≤ _ * _ := mul_right_mono (P⁻¹ * M : FractionalIdeal A⁰ (FractionRing A)) this
+      _ ≤ _ * _ := (mul_right_mono (P⁻¹ * M : FractionalIdeal A⁰ (FractionRing A)) this)
       _ = M := _
       
     · rw [mul_assoc, ← mul_assoc ↑P, h.mul_inv_eq_one P'_ne, one_mul, h.inv_mul_eq_one M'_ne]
@@ -868,7 +868,7 @@ theorem Ideal.exist_integer_multiples_not_mem {J : Ideal A} (hJ : J ≠ ⊤) {ι
   -- To show the inclusion of `J / I` into `I⁻¹ = 1 / I`, note that `J < I`.
   calc
     ↑J / I = ↑J * I⁻¹ := div_eq_mul_inv (↑J) I
-    _ < 1 * I⁻¹ := mul_right_strict_mono (inv_ne_zero hI0) _
+    _ < 1 * I⁻¹ := (mul_right_strict_mono (inv_ne_zero hI0) _)
     _ = I⁻¹ := one_mul _
     
   · rw [← coe_ideal_top]
@@ -1355,7 +1355,7 @@ theorem Ideal.prod_le_prime {ι : Type _} {s : Finset ι} {f : ι → Ideal R} {
   exact ((Ideal.prime_iff_isPrime hP0).mpr hP).dvd_finset_prod_iff _
 #align ideal.prod_le_prime Ideal.prod_le_prime
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 /-- The intersection of distinct prime powers in a Dedekind domain is the product of these
 prime powers. -/
 theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type _} (s : Finset ι) (f : ι → Ideal R)
@@ -1447,7 +1447,7 @@ noncomputable def Ideal.quotientMulEquivQuotientProd (I J : Ideal R) (coprime :
     (Ideal.quotientInfEquivQuotientProd I J coprime)
 #align ideal.quotient_mul_equiv_quotient_prod Ideal.quotientMulEquivQuotientProd
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 /-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
 `∏ i in s, P i ^ e i`, then `R ⧸ I` factors as `Π (i : s), R ⧸ (P i ^ e i)`.
 
@@ -1463,7 +1463,7 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfFinsetProdEq {ι : Type _} {
     (trans (Finset.prod_coe_sort s fun i => P i ^ e i) prod_eq)
 #align is_dedekind_domain.quotient_equiv_pi_of_finset_prod_eq IsDedekindDomain.quotientEquivPiOfFinsetProdEq
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 /-- Corollary of the Chinese remainder theorem: given elements `x i : R / P i ^ e i`,
 we can choose a representative `y : R` such that `y ≡ x i (mod P i ^ e i)`.-/
 theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type _} {s : Finset ι}
@@ -1477,7 +1477,7 @@ theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type _} {s : Fin
   exact ⟨z, fun i hi => rfl⟩
 #align is_dedekind_domain.exists_representative_mod_finset IsDedekindDomain.exists_representative_mod_finset
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (i j «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i j «expr ∈ » s) -/
 /-- Corollary of the Chinese remainder theorem: given elements `x i : R`,
 we can choose a representative `y : R` such that `y - x i ∈ P i ^ e i`.-/
 theorem IsDedekindDomain.exists_forall_sub_mem_ideal {ι : Type _} {s : Finset ι} (P : ι → Ideal R)

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

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

@@ -75,7 +75,9 @@ theorem inv_nonzero {J : FractionalIdeal R₁⁰ K} (h : J ≠ 0) :
 
 theorem coe_inv_of_nonzero {J : FractionalIdeal R₁⁰ K} (h : J ≠ 0) :
     (↑J⁻¹ : Submodule R₁ K) = IsLocalization.coeSubmodule K ⊤ / (J : Submodule R₁ K) := by
-  rw [inv_nonzero]; rfl; assumption
+  rw [inv_nonzero]
+  · rfl
+  · assumption
 #align fractional_ideal.coe_inv_of_nonzero FractionalIdeal.coe_inv_of_nonzero
 
 variable {K}

Define the canonical coercion from the nonnegative rationals to any division semiring.

From LeanAPAP

@@ -588,6 +588,7 @@ noncomputable instance FractionalIdeal.semifield : Semifield (FractionalIdeal A
   inv_zero := inv_zero' _
   div_eq_mul_inv := FractionalIdeal.div_eq_mul_inv
   mul_inv_cancel _ := FractionalIdeal.mul_inv_cancel
+  nnqsmul := _
 #align fractional_ideal.semifield FractionalIdeal.semifield
 
 /-- Fractional ideals have cancellative multiplication in a Dedekind domain.

This is the parts of the diff of #11203 which don't mention NNRat.cast.

• Use more where notation.
• Write qsmul := _ instead of qsmul := qsmulRec _ to make the instances more robust to definition changes.
• Delete qsmulRec.
• Move qsmul before ratCast_def in instance declarations.
• Name more instances.
• Rename rat_smul to qsmul.

@@ -583,13 +583,11 @@ open FractionalIdeal
 
 open Ideal
 
-noncomputable instance FractionalIdeal.semifield : Semifield (FractionalIdeal A⁰ K) :=
-  { coeIdeal_injective.nontrivial with
-    inv := fun I => I⁻¹
-    inv_zero := inv_zero' _
-    div := (· / ·)
-    div_eq_mul_inv := FractionalIdeal.div_eq_mul_inv
-    mul_inv_cancel := fun I => FractionalIdeal.mul_inv_cancel }
+noncomputable instance FractionalIdeal.semifield : Semifield (FractionalIdeal A⁰ K) where
+  __ := coeIdeal_injective.nontrivial
+  inv_zero := inv_zero' _
+  div_eq_mul_inv := FractionalIdeal.div_eq_mul_inv
+  mul_inv_cancel _ := FractionalIdeal.mul_inv_cancel
 #align fractional_ideal.semifield FractionalIdeal.semifield
 
 /-- Fractional ideals have cancellative multiplication in a Dedekind domain.

New simp tags or simp lemmas

associated_one_iff_isUnit, associated_zero_iff_eq_zero, Associates.mk_eq_one, Associates.mk_dvd_mk, Associates.mk_isRelPrime_iff, Associates.mk_zero, Associates.quot_out_zero, Associates.le_zero, Associates.prime_mk, Associates.irreducible_mk, Associates.mk_dvdNotUnit_mk_iff, Associates.factors_le, Associates.prod_factors

New gcongr tags

Associates.factors_mono, Associates.prod_mono

Change explicit args to implicit

Associates.prime_mk, Associates.irreducible_mk, Associates.one_or_eq_of_le_of_prime

Change typeclass assumptions

• drop [Nontrivial _] here and there, mostly in cases when a lemma has _ ≠ _ assumption
• drop all decidability assumptions in Associates.FactorSetMem
• drop decidability assumptions when they aren't needed to formulate a theorem

Use WithTop API

Use WithTop.some and ⊤ instead of Option.some and none in UniqueFactorizationDomain.

Renames

• Associates.isPrimal_iff → Associates.isPrimal_mk;
• Associates.mk_le_mk_iff_dvd_iff → Associates.mk_le_mk_iff_dvd;
• Associates.factors_0 → Associates.factors_zero;
• Associates.factors_eq_none_iff_zero → Associates.factors_eq_top_iff_zero

@@ -1019,7 +1019,7 @@ theorem irreducible : Irreducible v.asIdeal :=
 #align is_dedekind_domain.height_one_spectrum.irreducible IsDedekindDomain.HeightOneSpectrum.irreducible
 
 theorem associates_irreducible : Irreducible <| Associates.mk v.asIdeal :=
-  (Associates.irreducible_mk _).mpr v.irreducible
+  Associates.irreducible_mk.mpr v.irreducible
 #align is_dedekind_domain.height_one_spectrum.associates_irreducible IsDedekindDomain.HeightOneSpectrum.associates_irreducible
 
 /-- An equivalence between the height one and maximal spectra for rings of Krull dimension 1. -/

Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>

@@ -1091,7 +1091,7 @@ theorem idealFactorsFunOfQuotHom_id :
     idealFactorsFunOfQuotHom (RingHom.id (A ⧸ J)).surjective = OrderHom.id :=
   OrderHom.ext _ _
     (funext fun X => by
-      simp only [idealFactorsFunOfQuotHom, map_id, OrderHom.coe_mk, OrderHom.id_coe, id.def,
+      simp only [idealFactorsFunOfQuotHom, map_id, OrderHom.coe_mk, OrderHom.id_coe, id,
         comap_map_of_surjective (Ideal.Quotient.mk J) Quotient.mk_surjective, ←
         RingHom.ker_eq_comap_bot (Ideal.Quotient.mk J), mk_ker,
         sup_eq_left.mpr (dvd_iff_le.mp X.prop), Subtype.coe_eta])

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -818,7 +818,7 @@ theorem Ideal.exist_integer_multiples_not_mem {J : Ideal A} (hJ : J ≠ ⊤) {ι
   -- To show the inclusion of `J / I` into `I⁻¹ = 1 / I`, note that `J < I`.
   calc
     ↑J / I = ↑J * I⁻¹ := div_eq_mul_inv (↑J) I
-    _ < 1 * I⁻¹ := (mul_right_strictMono (inv_ne_zero hI0) ?_)
+    _ < 1 * I⁻¹ := mul_right_strictMono (inv_ne_zero hI0) ?_
     _ = I⁻¹ := one_mul _
   · rw [← coeIdeal_top]
     -- And multiplying by `I⁻¹` is indeed strictly monotone.

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

@@ -994,7 +994,7 @@ variable [IsDedekindDomain R]
 
 /-- The height one prime spectrum of a Dedekind domain `R` is the type of nonzero prime ideals of
 `R`. Note that this equals the maximal spectrum if `R` has Krull dimension 1. -/
--- Porting note: removed `has_nonempty_instance`, linter doesn't exist yet
+-- Porting note(#5171): removed `has_nonempty_instance`, linter doesn't exist yet
 @[ext, nolint unusedArguments]
 structure HeightOneSpectrum where
   asIdeal : Ideal R

This refactor adds a new definition IsIntegrallyClosedIn R A equal to IsIntegralClosure R A A, and redefines IsIntegrallyClosed R to equal IsIntegrallyClosed R (FractionRing A). This should make it possible and convenient to generalize away from the fraction fields.

This also more closely approximates the conventions of the Stacks project.

This is a second attempt at the refactor, after #7116 which was much more messy.

@@ -294,7 +294,7 @@ theorem isNoetherianRing : IsNoetherianRing A := by
 theorem integrallyClosed : IsIntegrallyClosed A := by
   -- It suffices to show that for integral `x`,
   -- `A[x]` (which is a fractional ideal) is in fact equal to `A`.
-  refine ⟨fun {x hx} => ?_⟩
+  refine (isIntegrallyClosed_iff (FractionRing A)).mpr (fun {x hx} => ?_)
   rw [← Set.mem_range, ← Algebra.mem_bot, ← Subalgebra.mem_toSubmodule, Algebra.toSubmodule_bot,
     Submodule.one_eq_span, ← coe_spanSingleton A⁰ (1 : FractionRing A), spanSingleton_one, ←
     FractionalIdeal.adjoinIntegral_eq_one_of_isUnit x hx (h.isUnit _)]

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

@@ -1395,7 +1395,7 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfFinsetProdEq {ι : Type*} {s
 #align is_dedekind_domain.quotient_equiv_pi_of_finset_prod_eq IsDedekindDomain.quotientEquivPiOfFinsetProdEq
 
 /-- Corollary of the Chinese remainder theorem: given elements `x i : R / P i ^ e i`,
-we can choose a representative `y : R` such that `y ≡ x i (mod P i ^ e i)`.-/
+we can choose a representative `y : R` such that `y ≡ x i (mod P i ^ e i)`. -/
 theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type*} {s : Finset ι}
     (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
     (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → P i ≠ P j) (x : ∀ i : s, R ⧸ P i ^ e i) :
@@ -1407,7 +1407,7 @@ theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type*} {s : Fins
 #align is_dedekind_domain.exists_representative_mod_finset IsDedekindDomain.exists_representative_mod_finset
 
 /-- Corollary of the Chinese remainder theorem: given elements `x i : R`,
-we can choose a representative `y : R` such that `y - x i ∈ P i ^ e i`.-/
+we can choose a representative `y : R` such that `y - x i ∈ P i ^ e i`. -/
 theorem IsDedekindDomain.exists_forall_sub_mem_ideal {ι : Type*} {s : Finset ι} (P : ι → Ideal R)
     (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
     (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → P i ≠ P j) (x : s → R) :

@@ -385,7 +385,7 @@ theorem exists_multiset_prod_cons_le_and_prod_not_le [IsDedekindDomain A] (hNF :
   classical
     have := Multiset.map_erase PrimeSpectrum.asIdeal PrimeSpectrum.ext P Z
     obtain ⟨hP0, hZP0⟩ : P.asIdeal ≠ ⊥ ∧ ((Z.erase P).map PrimeSpectrum.asIdeal).prod ≠ ⊥ := by
-      rwa [Ne.def, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
+      rwa [Ne, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
         this] at hprodZ
     -- By maximality of `P` and `M`, we have that `P ≤ M` implies `P = M`.
     have hPM' := (P.IsPrime.isMaximal hP0).eq_of_le hM.ne_top hPM

We change the following field in the definition of an additive commutative monoid:

 nsmul_succ : ∀ (n : ℕ) (x : G),
-  AddMonoid.nsmul (n + 1) x = x + AddMonoid.nsmul n x
+  AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x

where the latter is more natural

We adjust the definitions of ^ in monoids, groups, etc. Originally there was a warning comment about why this natural order was preferred

use x * npowRec n x and not npowRec n x * x in the definition to make sure that definitional unfolding of npowRec is blocked, to avoid deep recursion issues.

but it seems to no longer apply.

Remarks on the PR :

• pow_succ and pow_succ' have switched their meanings.
• Most of the time, the proofs were adjusted by priming/unpriming one lemma, or exchanging left and right; a few proofs were more complicated to adjust.
• In particular, [Mathlib/NumberTheory/RamificationInertia.lean] used Ideal.IsPrime.mul_mem_pow which is defined in [Mathlib/RingTheory/DedekindDomain/Ideal.lean]. Changing the order of operation forced me to add the symmetric lemma Ideal.IsPrime.mem_pow_mul.
• the docstring for Cauchy condensation test in [Mathlib/Analysis/PSeries.lean] was mathematically incorrect, I added the mention that the function is antitone.

@@ -501,7 +501,7 @@ theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ 
   induction' i with i ih
   · rw [pow_zero]; exact one_mem_inv_coe_ideal hI0
   · show x ^ i.succ ∈ (I⁻¹ : FractionalIdeal A⁰ K)
-    rw [pow_succ]; exact x_mul_mem _ ih
+    rw [pow_succ']; exact x_mul_mem _ ih
 #align fractional_ideal.coe_ideal_mul_inv FractionalIdeal.coe_ideal_mul_inv
 
 /-- Nonzero fractional ideals in a Dedekind domain are units.
@@ -730,7 +730,7 @@ nonrec theorem Ideal.mem_normalizedFactors_iff {p I : Ideal A} (hI : I ≠ ⊥)
 theorem Ideal.pow_right_strictAnti (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
     StrictAnti (I ^ · : ℕ → Ideal A) :=
   strictAnti_nat_of_succ_lt fun e =>
-    Ideal.dvdNotUnit_iff_lt.mp ⟨pow_ne_zero _ hI0, I, mt isUnit_iff.mp hI1, pow_succ' I e⟩
+    Ideal.dvdNotUnit_iff_lt.mp ⟨pow_ne_zero _ hI0, I, mt isUnit_iff.mp hI1, pow_succ I e⟩
 #align ideal.strict_anti_pow Ideal.pow_right_strictAnti
 
 theorem Ideal.pow_lt_self (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) (e : ℕ) (he : 2 ≤ e) :
@@ -1240,6 +1240,12 @@ theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n :
   exact Or.inr (Prime.pow_dvd_of_dvd_mul_right ((Ideal.prime_iff_isPrime hI0).mpr hI) _ ha h)
 #align ideal.is_prime.mul_mem_pow Ideal.IsPrime.mul_mem_pow
 
+theorem Ideal.IsPrime.mem_pow_mul (I : Ideal R) [hI : I.IsPrime] {a b : R} {n : ℕ}
+    (h : a * b ∈ I ^ n) : a ∈ I ^ n ∨ b ∈ I := by
+  rw [mul_comm] at h
+  rw [or_comm]
+  exact Ideal.IsPrime.mul_mem_pow _ h
+
 section
 
 open scoped Classical

Probably because in mathlib4 the definition IsDedekindDomain extends Domain, and this was not the case in mathlib3, there are unused hypothesis of the form

variable [IsDomain R] [IsDedekindDomain R]

and this PR removes the first one, that can be inferred by the second, both in variable declarations and in theorem/definition assumptions. A regex search has been performed on the library to search for all occurrences and none is left.

@@ -54,8 +54,6 @@ variable (R A K : Type*) [CommRing R] [CommRing A] [Field K]
 
 open scoped nonZeroDivisors Polynomial
 
-variable [IsDomain A]
-
 section Inverse
 
 namespace FractionalIdeal
@@ -229,10 +227,14 @@ theorem isPrincipal_inv (I : FractionalIdeal R₁⁰ K) [Submodule.IsPrincipal (
   exact (right_inverse_eq _ I (spanSingleton _ (generator (I : Submodule R₁ K))⁻¹) hI).symm
 #align fractional_ideal.is_principal_inv FractionalIdeal.isPrincipal_inv
 
+
 noncomputable instance : InvOneClass (FractionalIdeal R₁⁰ K) := { inv_one := div_one }
 
 end FractionalIdeal
 
+section IsDedekindDomainInv
+
+variable [IsDomain A]
 /-- A Dedekind domain is an integral domain such that every fractional ideal has an inverse.
 
 This is equivalent to `IsDedekindDomain`.
@@ -346,8 +348,20 @@ theorem isDedekindDomain : IsDedekindDomain A :=
 
 end IsDedekindDomainInv
 
+end IsDedekindDomainInv
+
 variable [Algebra A K] [IsFractionRing A K]
 
+variable {A K}
+
+theorem one_mem_inv_coe_ideal [IsDomain A] {I : Ideal A} (hI : I ≠ ⊥) :
+    (1 : K) ∈ (I : FractionalIdeal A⁰ K)⁻¹ := by
+  rw [FractionalIdeal.mem_inv_iff (FractionalIdeal.coeIdeal_ne_zero.mpr hI)]
+  intro y hy
+  rw [one_mul]
+  exact FractionalIdeal.coeIdeal_le_one hy
+-- #align fractional_ideal.one_mem_inv_coe_ideal FractionalIdeal.one_mem_inv_coe_ideal
+
 /-- Specialization of `exists_primeSpectrum_prod_le_and_ne_bot_of_domain` to Dedekind domains:
 Let `I : Ideal A` be a nonzero ideal, where `A` is a Dedekind domain that is not a field.
 Then `exists_primeSpectrum_prod_le_and_ne_bot_of_domain` states we can find a product of prime
@@ -383,7 +397,6 @@ theorem exists_multiset_prod_cons_le_and_prod_not_le [IsDedekindDomain A] (hNF :
     · refine fun h => h_eraseZ (Z.erase P) ⟨h, ?_⟩ (Multiset.erase_lt.mpr hPZ)
       exact hZP0
 #align exists_multiset_prod_cons_le_and_prod_not_le exists_multiset_prod_cons_le_and_prod_not_le
-
 namespace FractionalIdeal
 
 open Ideal
@@ -434,14 +447,6 @@ theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] {I : Ideal A} (hI0 : I
   Set.not_subset.1 <| not_inv_le_one_of_ne_bot hI0 hI1
 #align fractional_ideal.exists_not_mem_one_of_ne_bot FractionalIdeal.exists_not_mem_one_of_ne_bot
 
-theorem one_mem_inv_coe_ideal {I : Ideal A} (hI : I ≠ ⊥) :
-    (1 : K) ∈ (I : FractionalIdeal A⁰ K)⁻¹ := by
-  rw [mem_inv_iff (coeIdeal_ne_zero.mpr hI)]
-  intro y hy
-  rw [one_mul]
-  exact coeIdeal_le_one hy
-#align fractional_ideal.one_mem_inv_coe_ideal FractionalIdeal.one_mem_inv_coe_ideal
-
 theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
     (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
   -- We'll show a contradiction with `exists_not_mem_one_of_ne_bot`:
@@ -562,7 +567,8 @@ end FractionalIdeal
 
 /-- `IsDedekindDomain` and `IsDedekindDomainInv` are equivalent ways
 to express that an integral domain is a Dedekind domain. -/
-theorem isDedekindDomain_iff_isDedekindDomainInv : IsDedekindDomain A ↔ IsDedekindDomainInv A :=
+theorem isDedekindDomain_iff_isDedekindDomainInv [IsDomain A] :
+    IsDedekindDomain A ↔ IsDedekindDomainInv A :=
   ⟨fun _h _I hI => FractionalIdeal.mul_inv_cancel hI, fun h => h.isDedekindDomain⟩
 #align is_dedekind_domain_iff_is_dedekind_domain_inv isDedekindDomain_iff_isDedekindDomainInv
 
@@ -901,7 +907,7 @@ end IsDedekindDomain
 
 section IsDedekindDomain
 
-variable {T : Type*} [CommRing T] [IsDomain T] [IsDedekindDomain T] {I J : Ideal T}
+variable {T : Type*} [CommRing T] [IsDedekindDomain T] {I J : Ideal T}
 
 open scoped Classical
 
@@ -984,7 +990,7 @@ one are prime and irreducible.
 
 namespace IsDedekindDomain
 
-variable [IsDomain R] [IsDedekindDomain R]
+variable [IsDedekindDomain R]
 
 /-- The height one prime spectrum of a Dedekind domain `R` is the type of nonzero prime ideals of
 `R`. Note that this equals the maximal spectrum if `R` has Krull dimension 1. -/
@@ -1091,7 +1097,7 @@ theorem idealFactorsFunOfQuotHom_id :
         sup_eq_left.mpr (dvd_iff_le.mp X.prop), Subtype.coe_eta])
 #align ideal_factors_fun_of_quot_hom_id idealFactorsFunOfQuotHom_id
 
-variable {B : Type*} [CommRing B] [IsDomain B] [IsDedekindDomain B] {L : Ideal B}
+variable {B : Type*} [CommRing B] [IsDedekindDomain B] {L : Ideal B}
 
 theorem idealFactorsFunOfQuotHom_comp {f : R ⧸ I →+* A ⧸ J} {g : A ⧸ J →+* B ⧸ L}
     (hf : Function.Surjective f) (hg : Function.Surjective g) :
@@ -1104,7 +1110,7 @@ theorem idealFactorsFunOfQuotHom_comp {f : R ⧸ I →+* A ⧸ J} {g : A ⧸ J 
     Quotient.mk_surjective, map_map]
 #align ideal_factors_fun_of_quot_hom_comp idealFactorsFunOfQuotHom_comp
 
-variable [IsDomain R] [IsDedekindDomain R] (f : R ⧸ I ≃+* A ⧸ J)
+variable [IsDedekindDomain R] (f : R ⧸ I ≃+* A ⧸ J)
 
 /-- The bijection between ideals of `R` dividing `I` and the ideals of `A` dividing `J` induced by
   an isomorphism `f : R/I ≅ A/J`. -/
@@ -1220,7 +1226,7 @@ theorem Ideal.coprime_of_no_prime_ge {I J : Ideal R} (h : ∀ P, I ≤ P → J 
 
 section DedekindDomain
 
-variable [IsDomain R] [IsDedekindDomain R]
+variable [IsDedekindDomain R]
 
 theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n : ℕ}
     (h : a * b ∈ I ^ n) : a ∈ I ∨ b ∈ I ^ n := by

@@ -712,8 +712,7 @@ theorem Ideal.prime_generator_of_prime {P : Ideal A} (h : Prime P) [P.IsPrincipa
   prime_generator_of_isPrime _ h.ne_zero
 
 open UniqueFactorizationMonoid in
-nonrec theorem Ideal.mem_normalizedFactors_iff [DecidableEq (Ideal A)]
-    {p I : Ideal A} (hI : I ≠ ⊥) :
+nonrec theorem Ideal.mem_normalizedFactors_iff {p I : Ideal A} (hI : I ≠ ⊥) :
     p ∈ normalizedFactors I ↔ p.IsPrime ∧ I ≤ p := by
   rw [← Ideal.dvd_iff_le]
   by_cases hp : p = 0

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)

@@ -61,7 +61,6 @@ section Inverse
 namespace FractionalIdeal
 
 variable {R₁ : Type*} [CommRing R₁] [IsDomain R₁] [Algebra R₁ K] [IsFractionRing R₁ K]
-
 variable {I J : FractionalIdeal R₁⁰ K}
 
 noncomputable instance : Inv (FractionalIdeal R₁⁰ K) := ⟨fun I => 1 / I⟩
@@ -572,7 +571,6 @@ end Inverse
 section IsDedekindDomain
 
 variable {R A}
-
 variable [IsDedekindDomain A] [Algebra A K] [IsFractionRing A K]
 
 open FractionalIdeal
@@ -1057,7 +1055,6 @@ section
 open Ideal
 
 variable {R A}
-
 variable [IsDedekindDomain A] {I : Ideal R} {J : Ideal A}
 
 /-- The map from ideals of `R` dividing `I` to the ideals of `A` dividing `J` induced by
@@ -1187,7 +1184,6 @@ theorem normalizedFactorsEquivOfQuotEquiv_symm (hI : I ≠ ⊥) (hJ : J ≠ ⊥)
 #align normalized_factors_equiv_of_quot_equiv_symm normalizedFactorsEquivOfQuotEquiv_symm
 
 variable [DecidableRel ((· ∣ ·) : Ideal R → Ideal R → Prop)]
-
 variable [DecidableRel ((· ∣ ·) : Ideal A → Ideal A → Prop)]
 
 /-- The map `normalizedFactorsEquivOfQuotEquiv` preserves multiplicities. -/
@@ -1420,7 +1416,6 @@ section PID
 open multiplicity UniqueFactorizationMonoid Ideal
 
 variable {R}
-
 variable [IsDomain R] [IsPrincipalIdealRing R]
 
 theorem span_singleton_dvd_span_singleton_iff_dvd {a b : R} :

It doesn't make a lot of sense for factors to require a DecidableEq assumption since it's not used in the statement, and the definition is already noncomputable. This PR removes that assumption and updates some lemmas later in the file accordingly.

@@ -746,7 +746,6 @@ open UniqueFactorizationMonoid
 
 theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥)
     {i : ℕ} (hlt : P ^ (i + 1) < I) (hle : I ≤ P ^ i) : I = P ^ i := by
-  have := Classical.decEq (Ideal A)
   refine le_antisymm hle ?_
   have P_prime' := Ideal.prime_of_isPrime hP P_prime
   have h1 : I ≠ ⊥ := (lt_of_le_of_lt bot_le hlt).ne'
@@ -889,8 +888,7 @@ theorem inf_eq_mul_of_coprime {I J : Ideal A} (coprime : IsCoprime I J) : I ⊓
 theorem isCoprime_iff_gcd {I J : Ideal A} : IsCoprime I J ↔ gcd I J = 1 := by
   rw [Ideal.isCoprime_iff_codisjoint, codisjoint_iff, one_eq_top, gcd_eq_sup]
 
-theorem factors_span_eq [DecidableEq K[X]] [DecidableEq (Ideal K[X])] {p : K[X]} :
-    factors (span {p}) = (factors p).map (fun q ↦ span {q}) := by
+theorem factors_span_eq {p : K[X]} : factors (span {p}) = (factors p).map (fun q ↦ span {q}) := by
   rcases eq_or_ne p 0 with rfl | hp; · simpa [Set.singleton_zero] using normalizedFactors_zero
   have : ∀ q ∈ (factors p).map (fun q ↦ span {q}), Prime q := fun q hq ↦ by
     obtain ⟨r, hr, rfl⟩ := Multiset.mem_map.mp hq
@@ -1442,7 +1440,7 @@ theorem Ideal.squarefree_span_singleton {a : R} :
     exact isUnit_iff.mpr <| eq_top_of_isUnit_mem _ (Submodule.IsPrincipal.generator_mem I) (h _ hI)
 
 theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
-    [DecidableEq R] [DecidableEq (Ideal R)] {a b : R} (ha : a ∈ normalizedFactors b) :
+    {a b : R} (ha : a ∈ normalizedFactors b) :
     Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : Set R)) := by
   by_cases hb : b = 0
   · rw [Ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalizedFactors_zero]

Co-authored-by: Patrick Massot <patrickmassot@free.fr> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -255,13 +255,13 @@ theorem isDedekindDomainInv_iff [Algebra A K] [IsFractionRing A K] :
     FractionalIdeal.mapEquiv (FractionRing.algEquiv A K)
   refine h.toEquiv.forall_congr (fun {x} => ?_)
   rw [← h.toEquiv.apply_eq_iff_eq]
-  simp [IsDedekindDomainInv]
+  simp [h, IsDedekindDomainInv]
 #align is_dedekind_domain_inv_iff isDedekindDomainInv_iff
 
 theorem FractionalIdeal.adjoinIntegral_eq_one_of_isUnit [Algebra A K] [IsFractionRing A K] (x : K)
     (hx : IsIntegral A x) (hI : IsUnit (adjoinIntegral A⁰ x hx)) : adjoinIntegral A⁰ x hx = 1 := by
   set I := adjoinIntegral A⁰ x hx
-  have mul_self : I * I = I := by apply coeToSubmodule_injective; simp
+  have mul_self : I * I = I := by apply coeToSubmodule_injective; simp [I]
   convert congr_arg (· * I⁻¹) mul_self <;>
     simp only [(mul_inv_cancel_iff_isUnit K).mpr hI, mul_assoc, mul_one]
 #align fractional_ideal.adjoin_integral_eq_one_of_is_unit FractionalIdeal.adjoinIntegral_eq_one_of_isUnit

Remove unnecessary "rw"s.

@@ -1532,7 +1532,7 @@ theorem multiplicity_normalizedFactorsEquivSpanNormalizedFactors_symm_eq_multipl
   obtain ⟨x, hx⟩ := (normalizedFactorsEquivSpanNormalizedFactors hr).surjective I
   obtain ⟨a, ha⟩ := x
   rw [hx.symm, Equiv.symm_apply_apply, Subtype.coe_mk,
-    multiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_multiplicity hr ha, hx]
+    multiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_multiplicity hr ha]
 #align multiplicity_normalized_factors_equiv_span_normalized_factors_symm_eq_multiplicity multiplicity_normalizedFactorsEquivSpanNormalizedFactors_symm_eq_multiplicity
 
 end PID

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

This follows on from #6964.

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

@@ -110,9 +110,8 @@ theorem coe_ideal_le_self_mul_inv (I : Ideal R₁) :
 /-- `I⁻¹` is the inverse of `I` if `I` has an inverse. -/
 theorem right_inverse_eq (I J : FractionalIdeal R₁⁰ K) (h : I * J = 1) : J = I⁻¹ := by
   have hI : I ≠ 0 := ne_zero_of_mul_eq_one I J h
-  suffices h' : I * (1 / I) = 1
-  · exact congr_arg Units.inv <|
-      @Units.ext _ _ (Units.mkOfMulEqOne _ _ h) (Units.mkOfMulEqOne _ _ h') rfl
+  suffices h' : I * (1 / I) = 1 from
+    congr_arg Units.inv <| @Units.ext _ _ (Units.mkOfMulEqOne _ _ h) (Units.mkOfMulEqOne _ _ h') rfl
   apply le_antisymm
   · apply mul_le.mpr _
     intro x hx y hy
@@ -226,8 +225,8 @@ theorem isPrincipal_inv (I : FractionalIdeal R₁⁰ K) [Submodule.IsPrincipal (
     (h : I ≠ 0) : Submodule.IsPrincipal I⁻¹.1 := by
   rw [val_eq_coe, isPrincipal_iff]
   use (generator (I : Submodule R₁ K))⁻¹
-  have hI : I * spanSingleton _ (generator (I : Submodule R₁ K))⁻¹ = 1
-  apply mul_generator_self_inv _ I h
+  have hI : I * spanSingleton _ (generator (I : Submodule R₁ K))⁻¹ = 1 :=
+    mul_generator_self_inv _ I h
   exact (right_inverse_eq _ I (spanSingleton _ (generator (I : Submodule R₁ K))⁻¹) hI).symm
 #align fractional_ideal.is_principal_inv FractionalIdeal.isPrincipal_inv
 
@@ -511,8 +510,8 @@ protected theorem mul_inv_cancel [IsDedekindDomain A] {I : FractionalIdeal A⁰
   obtain ⟨a, J, ha, hJ⟩ :
     ∃ (a : A) (aI : Ideal A), a ≠ 0 ∧ I = spanSingleton A⁰ (algebraMap A K a)⁻¹ * aI :=
     exists_eq_spanSingleton_mul I
-  suffices h₂ : I * (spanSingleton A⁰ (algebraMap _ _ a) * (J : FractionalIdeal A⁰ K)⁻¹) = 1
-  · rw [mul_inv_cancel_iff]
+  suffices h₂ : I * (spanSingleton A⁰ (algebraMap _ _ a) * (J : FractionalIdeal A⁰ K)⁻¹) = 1 by
+    rw [mul_inv_cancel_iff]
     exact ⟨spanSingleton A⁰ (algebraMap _ _ a) * (J : FractionalIdeal A⁰ K)⁻¹, h₂⟩
   subst hJ
   rw [mul_assoc, mul_left_comm (J : FractionalIdeal A⁰ K), coe_ideal_mul_inv, mul_one,
@@ -1151,8 +1150,8 @@ theorem idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFact
     {L : Ideal R} (hL : L ∈ normalizedFactors I) :
     ↑(idealFactorsEquivOfQuotEquiv f ⟨L, dvd_of_mem_normalizedFactors hL⟩)
       ∈ normalizedFactors J := by
-  have hI : I ≠ ⊥
-  · intro hI
+  have hI : I ≠ ⊥ := by
+    intro hI
     rw [hI, bot_eq_zero, normalizedFactors_zero, ← Multiset.empty_eq_zero] at hL
     exact Finset.not_mem_empty _ hL
   refine mem_normalizedFactors_factor_dvd_iso_of_mem_normalizedFactors hI hJ hL

@@ -595,12 +595,9 @@ Although this instance is a direct consequence of the instance
 `FractionalIdeal.semifield`, we define this instance to provide
 a computable alternative.
 -/
--- Porting note: added noncomputable because otherwise it fails, so it seems that the goal
--- is not achieved... Previous attempt was
--- { @FractionalIdeal.commSemiring A _ A⁰ K _ _,  -- Project out the computable fields first.
---   (by infer_instance : CancelCommMonoidWithZero (FractionalIdeal A⁰ K)) with }
-noncomputable instance FractionalIdeal.cancelCommMonoidWithZero :
-    CancelCommMonoidWithZero (FractionalIdeal A⁰ K) := inferInstance
+instance FractionalIdeal.cancelCommMonoidWithZero :
+    CancelCommMonoidWithZero (FractionalIdeal A⁰ K) where
+  __ : CommSemiring (FractionalIdeal A⁰ K) := inferInstance
 #align fractional_ideal.cancel_comm_monoid_with_zero FractionalIdeal.cancelCommMonoidWithZero
 
 instance Ideal.cancelCommMonoidWithZero : CancelCommMonoidWithZero (Ideal A) :=

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Joachim Breitner <mail@joachim-breitner.de>

@@ -596,11 +596,11 @@ Although this instance is a direct consequence of the instance
 a computable alternative.
 -/
 -- Porting note: added noncomputable because otherwise it fails, so it seems that the goal
--- is not achieved...
+-- is not achieved... Previous attempt was
+-- { @FractionalIdeal.commSemiring A _ A⁰ K _ _,  -- Project out the computable fields first.
+--   (by infer_instance : CancelCommMonoidWithZero (FractionalIdeal A⁰ K)) with }
 noncomputable instance FractionalIdeal.cancelCommMonoidWithZero :
-    CancelCommMonoidWithZero (FractionalIdeal A⁰ K) :=
-  { @FractionalIdeal.commSemiring A _ A⁰ K _ _,  -- Project out the computable fields first.
-    (by infer_instance : CancelCommMonoidWithZero (FractionalIdeal A⁰ K)) with }
+    CancelCommMonoidWithZero (FractionalIdeal A⁰ K) := inferInstance
 #align fractional_ideal.cancel_comm_monoid_with_zero FractionalIdeal.cancelCommMonoidWithZero
 
 instance Ideal.cancelCommMonoidWithZero : CancelCommMonoidWithZero (Ideal A) :=

The main result is Module.End.isSemisimple_of_squarefree_aeval_eq_zero

@@ -703,6 +703,20 @@ theorem Ideal.isPrime_iff_bot_or_prime {P : Ideal A} : IsPrime P ↔ P = ⊥ ∨
     hp.elim (fun h => h.symm ▸ Ideal.bot_prime) Ideal.isPrime_of_prime⟩
 #align ideal.is_prime_iff_bot_or_prime Ideal.isPrime_iff_bot_or_prime
 
+@[simp]
+theorem Ideal.prime_span_singleton_iff {a : A} : Prime (Ideal.span {a}) ↔ Prime a := by
+  rcases eq_or_ne a 0 with rfl | ha
+  · rw [Set.singleton_zero, span_zero, ← Ideal.zero_eq_bot, ← not_iff_not]
+    simp only [not_prime_zero, not_false_eq_true]
+  · have ha' : span {a} ≠ ⊥ := by simpa only [ne_eq, span_singleton_eq_bot] using ha
+    rw [Ideal.prime_iff_isPrime ha', Ideal.span_singleton_prime ha]
+
+open Submodule.IsPrincipal in
+theorem Ideal.prime_generator_of_prime {P : Ideal A} (h : Prime P) [P.IsPrincipal] :
+    Prime (generator P) :=
+  have : Ideal.IsPrime P := Ideal.isPrime_of_prime h
+  prime_generator_of_isPrime _ h.ne_zero
+
 open UniqueFactorizationMonoid in
 nonrec theorem Ideal.mem_normalizedFactors_iff [DecidableEq (Ideal A)]
     {p I : Ideal A} (hI : I ≠ ⊥) :
@@ -879,6 +893,15 @@ theorem inf_eq_mul_of_coprime {I J : Ideal A} (coprime : IsCoprime I J) : I ⊓
 theorem isCoprime_iff_gcd {I J : Ideal A} : IsCoprime I J ↔ gcd I J = 1 := by
   rw [Ideal.isCoprime_iff_codisjoint, codisjoint_iff, one_eq_top, gcd_eq_sup]
 
+theorem factors_span_eq [DecidableEq K[X]] [DecidableEq (Ideal K[X])] {p : K[X]} :
+    factors (span {p}) = (factors p).map (fun q ↦ span {q}) := by
+  rcases eq_or_ne p 0 with rfl | hp; · simpa [Set.singleton_zero] using normalizedFactors_zero
+  have : ∀ q ∈ (factors p).map (fun q ↦ span {q}), Prime q := fun q hq ↦ by
+    obtain ⟨r, hr, rfl⟩ := Multiset.mem_map.mp hq
+    exact prime_span_singleton_iff.mpr <| prime_of_factor r hr
+  rw [← span_singleton_eq_span_singleton.mpr (factors_prod hp), ← multiset_prod_span_singleton,
+    factors_eq_normalizedFactors, normalizedFactors_prod_of_prime this]
+
 end Ideal
 
 end Gcd
@@ -1412,6 +1435,16 @@ theorem span_singleton_dvd_span_singleton_iff_dvd {a b : R} :
     dvd_iff_le.mpr fun _d hd => mem_span_singleton.mpr (dvd_trans h (mem_span_singleton.mp hd))⟩
 #align span_singleton_dvd_span_singleton_iff_dvd span_singleton_dvd_span_singleton_iff_dvd
 
+@[simp]
+theorem Ideal.squarefree_span_singleton {a : R} :
+    Squarefree (span {a}) ↔ Squarefree a := by
+  refine ⟨fun h x hx ↦ ?_, fun h I hI ↦ ?_⟩
+  · rw [← span_singleton_dvd_span_singleton_iff_dvd, ← span_singleton_mul_span_singleton] at hx
+    simpa using h _ hx
+  · rw [← span_singleton_generator I, span_singleton_mul_span_singleton,
+      span_singleton_dvd_span_singleton_iff_dvd] at hI
+    exact isUnit_iff.mpr <| eq_top_of_isUnit_mem _ (Submodule.IsPrincipal.generator_mem I) (h _ hI)
+
 theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
     [DecidableEq R] [DecidableEq (Ideal R)] {a b : R} (ha : a ∈ normalizedFactors b) :
     Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : Set R)) := by

From flt-regular

@@ -703,6 +703,17 @@ theorem Ideal.isPrime_iff_bot_or_prime {P : Ideal A} : IsPrime P ↔ P = ⊥ ∨
     hp.elim (fun h => h.symm ▸ Ideal.bot_prime) Ideal.isPrime_of_prime⟩
 #align ideal.is_prime_iff_bot_or_prime Ideal.isPrime_iff_bot_or_prime
 
+open UniqueFactorizationMonoid in
+nonrec theorem Ideal.mem_normalizedFactors_iff [DecidableEq (Ideal A)]
+    {p I : Ideal A} (hI : I ≠ ⊥) :
+    p ∈ normalizedFactors I ↔ p.IsPrime ∧ I ≤ p := by
+  rw [← Ideal.dvd_iff_le]
+  by_cases hp : p = 0
+  · rw [← zero_eq_bot] at hI
+    simp only [hp, zero_not_mem_normalizedFactors, zero_dvd_iff, hI, false_iff, not_and,
+      not_false_eq_true, implies_true]
+  · rwa [mem_normalizedFactors_iff hI, prime_iff_isPrime]
+
 theorem Ideal.pow_right_strictAnti (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
     StrictAnti (I ^ · : ℕ → Ideal A) :=
   strictAnti_nat_of_succ_lt fun e =>

The file RingTheory.FractionalIdeal.Basic is more than 1600 lines long. This PR splits it into two files: Basic and Operations following the model of RingTheory.Ideal.Basic and RingTheory.Ideal.Operations

Co-authored-by: Mario Carneiro <mcarneir@andrew.cmu.edu> Co-authored-by: Jz Pan <acme_pjz@hotmail.com>

@@ -6,11 +6,9 @@ Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
 import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
 import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
 import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
-import Mathlib.Order.Hom.Basic
-import Mathlib.RingTheory.DedekindDomain.Basic
-import Mathlib.RingTheory.FractionalIdeal.Basic
-import Mathlib.RingTheory.PrincipalIdealDomain
 import Mathlib.RingTheory.ChainOfDivisors
+import Mathlib.RingTheory.DedekindDomain.Basic
+import Mathlib.RingTheory.FractionalIdeal.Operations
 
 #align_import ring_theory.dedekind_domain.ideal from "leanprover-community/mathlib"@"2bbc7e3884ba234309d2a43b19144105a753292e"
 

This PR defines the absolute ideal norm of a fractional ideal I : FractionalIdeal R⁰ K where K is a fraction field of R as a zero-preserving group homomorphism with values in ℚ and proves that it generalises the norm on (integral) ideals (and some other classical result).

Also in this PR:

• Add the directory Mathlib/RingTheory/FractionalIdeal and move the file Mathlib/RingTheory/FractionalIdeal.lean to Mathlib/RingTheory/FractionalIdeal/Basic.lean. The new results are in Mathlib/RingTheory/FractionalIdeal/Norm.lean
• Define the numerator and denominator of a fractional ideal. These are used to define the norm. Also define a linear equiv between a fractional ideal and its numerator.
• Several technical lemmas.

@@ -8,7 +8,7 @@ import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
 import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
 import Mathlib.Order.Hom.Basic
 import Mathlib.RingTheory.DedekindDomain.Basic
-import Mathlib.RingTheory.FractionalIdeal
+import Mathlib.RingTheory.FractionalIdeal.Basic
 import Mathlib.RingTheory.PrincipalIdealDomain
 import Mathlib.RingTheory.ChainOfDivisors
 

Changes in this PR shouldn't change the public API. The only changes about ∃ x ∈ s, _ is inside a proof.

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

@@ -245,7 +245,7 @@ assuming `IsDedekindDomain A`, which implies `IsDedekindDomainInv`. For **integr
 `IsDedekindDomain`(`_inv`) implies only `Ideal.cancelCommMonoidWithZero`.
 -/
 def IsDedekindDomainInv : Prop :=
-  ∀ (I) (_ : I ≠ (⊥ : FractionalIdeal A⁰ (FractionRing A))), I * I⁻¹ = 1
+  ∀ I ≠ (⊥ : FractionalIdeal A⁰ (FractionRing A)), I * I⁻¹ = 1
 #align is_dedekind_domain_inv IsDedekindDomainInv
 
 open FractionalIdeal
@@ -253,7 +253,7 @@ open FractionalIdeal
 variable {R A K}
 
 theorem isDedekindDomainInv_iff [Algebra A K] [IsFractionRing A K] :
-    IsDedekindDomainInv A ↔ ∀ (I) (_ : I ≠ (⊥ : FractionalIdeal A⁰ K)), I * I⁻¹ = 1 := by
+    IsDedekindDomainInv A ↔ ∀ I ≠ (⊥ : FractionalIdeal A⁰ K), I * I⁻¹ = 1 := by
   let h : FractionalIdeal A⁰ (FractionRing A) ≃+* FractionalIdeal A⁰ K :=
     FractionalIdeal.mapEquiv (FractionRing.algEquiv A K)
   refine h.toEquiv.forall_congr (fun {x} => ?_)

• Use wlog tactic instead of an explicit suffices.
• Restate a lemma in terms of an inequality between fractional ideals.
• Drop an unneeded assumption, thanks to @erdOne

Co-authored-by: @erdOne

Co-authored-by: erd1 <the.erd.one@gmail.com>

@@ -392,39 +392,30 @@ namespace FractionalIdeal
 
 open Ideal
 
-theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
-    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
-    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
-  -- WLOG, let `I` be maximal.
-  suffices
-    ∀ {M : Ideal A} (_hM : M.IsMaximal),
-      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
-    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
-    have hM0 := (M.bot_lt_of_maximal hNF).ne'
-    obtain ⟨x, hxM, hx1⟩ := this hM
-    refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
-        rw [coeIdeal_ne_zero] <;> assumption
-  -- Let `a` be a nonzero element of `M` and `J` the ideal generated by `a`.
-  intro M hM
-  obtain ⟨⟨a, haM⟩, ha0⟩ := Submodule.nonzero_mem_of_bot_lt (M.bot_lt_of_maximal hNF)
+lemma not_inv_le_one_of_ne_bot [IsDedekindDomain A] {I : Ideal A}
+    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) : ¬(I⁻¹ : FractionalIdeal A⁰ K) ≤ 1 := by
+  have hNF : ¬IsField A := fun h ↦ letI := h.toField; (eq_bot_or_eq_top I).elim hI0 hI1
+  wlog hM : I.IsMaximal generalizing I
+  · rcases I.exists_le_maximal hI1 with ⟨M, hmax, hIM⟩
+    have hMbot : M ≠ ⊥ := (M.bot_lt_of_maximal hNF).ne'
+    refine mt (le_trans <| inv_anti_mono ?_ ?_ ?_) (this hMbot hmax.ne_top hmax) <;>
+      simpa only [coeIdeal_ne_zero, coeIdeal_le_coeIdeal]
+  have hI0 : ⊥ < I := I.bot_lt_of_maximal hNF
+  obtain ⟨⟨a, haI⟩, ha0⟩ := Submodule.nonzero_mem_of_bot_lt hI0
   replace ha0 : a ≠ 0 := Subtype.coe_injective.ne ha0
   let J : Ideal A := Ideal.span {a}
   have hJ0 : J ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp ha0
-  have hJM : J ≤ M := Ideal.span_le.mpr (Set.singleton_subset_iff.mpr haM)
-  have hM0 : ⊥ < M := M.bot_lt_of_maximal hNF
+  have hJI : J ≤ I := I.span_singleton_le_iff_mem.2 haI
   -- Then we can find a product of prime (hence maximal) ideals contained in `J`,
   -- such that removing element `M` from the product is not contained in `J`.
-  obtain ⟨Z, hle, hnle⟩ := exists_multiset_prod_cons_le_and_prod_not_le hNF hJ0 hJM
+  obtain ⟨Z, hle, hnle⟩ := exists_multiset_prod_cons_le_and_prod_not_le hNF hJ0 hJI
   -- Choose an element `b` of the product that is not in `J`.
   obtain ⟨b, hbZ, hbJ⟩ := SetLike.not_le_iff_exists.mp hnle
   have hnz_fa : algebraMap A K a ≠ 0 :=
     mt ((injective_iff_map_eq_zero _).mp (IsFractionRing.injective A K) a) ha0
-  have _hb0 : algebraMap A K b ≠ 0 :=
-    mt ((injective_iff_map_eq_zero _).mp (IsFractionRing.injective A K) b) fun h =>
-      hbJ <| h.symm ▸ J.zero_mem
   -- Then `b a⁻¹ : K` is in `M⁻¹` but not in `1`.
-  refine' ⟨algebraMap A K b * (algebraMap A K a)⁻¹, (mem_inv_iff _).mpr _, _⟩
-  · exact coeIdeal_ne_zero.mpr hM0.ne'
+  refine Set.not_subset.2 ⟨algebraMap A K b * (algebraMap A K a)⁻¹, (mem_inv_iff ?_).mpr ?_, ?_⟩
+  · exact coeIdeal_ne_zero.mpr hI0.ne'
   · rintro y₀ hy₀
     obtain ⟨y, h_Iy, rfl⟩ := (mem_coeIdeal _).mp hy₀
     rw [mul_comm, ← mul_assoc, ← RingHom.map_mul]
@@ -441,6 +432,10 @@ theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I
     rw [← div_eq_mul_inv, eq_div_iff_mul_eq hnz_fa, ← RingHom.map_mul] at h₂_abs
     have := Ideal.mem_span_singleton'.mpr ⟨x', IsFractionRing.injective A K h₂_abs⟩
     contradiction
+
+theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
+    (hI1 : I ≠ ⊤) : ∃ x ∈ (I⁻¹ : FractionalIdeal A⁰ K), x ∉ (1 : FractionalIdeal A⁰ K) :=
+  Set.not_subset.1 <| not_inv_le_one_of_ne_bot hI0 hI1
 #align fractional_ideal.exists_not_mem_one_of_ne_bot FractionalIdeal.exists_not_mem_one_of_ne_bot
 
 theorem one_mem_inv_coe_ideal {I : Ideal A} (hI : I ≠ ⊥) :
@@ -453,11 +448,6 @@ theorem one_mem_inv_coe_ideal {I : Ideal A} (hI : I ≠ ⊥) :
 
 theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
     (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
-  -- Handle a few trivial cases.
-  by_cases hI1 : I = ⊤
-  · rw [hI1, coeIdeal_top, one_mul, inv_one]
-  by_cases hNF : IsField A
-  · letI := hNF.toField; rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩
   -- We'll show a contradiction with `exists_not_mem_one_of_ne_bot`:
   -- `J⁻¹ = (I * I⁻¹)⁻¹` cannot have an element `x ∉ 1`, so it must equal `1`.
   obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * (I : FractionalIdeal A⁰ K)⁻¹ :=
@@ -469,12 +459,7 @@ theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I
     exact coe_ideal_le_self_mul_inv K I
   by_cases hJ1 : J = ⊤
   · rw [← hJ, hJ1, coeIdeal_top]
-  obtain ⟨x, hx, hx1⟩ :
-    ∃ x : K, x ∈ (J : FractionalIdeal A⁰ K)⁻¹ ∧ x ∉ (1 : FractionalIdeal A⁰ K) :=
-    exists_not_mem_one_of_ne_bot hNF hJ0 hJ1
-  contrapose! hx1 with h_abs
-  rw [hJ] at hx
-  exact hI hx
+  exact (not_inv_le_one_of_ne_bot (K := K) hJ0 hJ1 (hJ ▸ hI)).elim
 #align fractional_ideal.mul_inv_cancel_of_le_one FractionalIdeal.mul_inv_cancel_of_le_one
 
 /-- Nonzero integral ideals in a Dedekind domain are invertible.

Performed with a regex search for ∀ (.) (.), \1 ≠ \2 →, and a few variants to catch implicit binders and explicit types.

I have deliberately avoided trying to make the analogous Set.Pairwise transformation (or any Pairwise (foo on bar) transformations) in this PR, to keep the diff small.

Co-authored-by: Yaël Dillies <yael.dillies@gmail.com>

@@ -1308,18 +1308,19 @@ theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f :
 /-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
 `∏ i, P i ^ e i`, then `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`. -/
 noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype ι] (I : Ideal R)
-    (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i, Prime (P i)) (coprime : ∀ i j, i ≠ j → P i ≠ P j)
+    (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i, Prime (P i))
+    (coprime : Pairwise fun i j => P i ≠ P j)
     (prod_eq : ∏ i, P i ^ e i = I) : R ⧸ I ≃+* ∀ i, R ⧸ P i ^ e i :=
   (Ideal.quotEquivOfEq
     (by
       simp only [← prod_eq, Finset.inf_eq_iInf, Finset.mem_univ, ciInf_pos,
-        ← IsDedekindDomain.inf_prime_pow_eq_prod _ _ _ (fun i _ => prime i) fun i _ j _ =>
-        coprime i j])).trans <|
+        ← IsDedekindDomain.inf_prime_pow_eq_prod _ _ _ (fun i _ => prime i)
+        (coprime.set_pairwise _)])).trans <|
     Ideal.quotientInfRingEquivPiQuotient _ fun i j hij => Ideal.coprime_of_no_prime_ge (by
       intro P hPi hPj hPp
       haveI := Ideal.isPrime_of_prime (prime i)
       haveI := Ideal.isPrime_of_prime (prime j)
-      refine coprime i j hij ?_
+      refine coprime hij ?_
       refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
         (Ideal.le_of_pow_le_prime hPi)).trans
         ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp

Search for [∀∃].*(_ and manually replace some occurrences with more readable versions. In case of ∀, the new expressions are defeq to the old ones. In case of ∃, they differ by exists_prop.

In some rare cases, golf proofs that needed fixing.

@@ -1279,7 +1279,7 @@ theorem Ideal.prod_le_prime {ι : Type*} {s : Finset ι} {f : ι → Ideal R} {P
 prime powers. -/
 theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f : ι → Ideal R)
     (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (f i))
-    (coprime : ∀ (i) (_ : i ∈ s) (j) (_ : j ∈ s), i ≠ j → f i ≠ f j) :
+    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → f i ≠ f j) :
     (s.inf fun i => f i ^ e i) = ∏ i in s, f i ^ e i := by
   letI := Classical.decEq ι
   revert prime coprime
@@ -1368,7 +1368,7 @@ the product to a finite subset `s` of a potentially infinite indexing type `ι`.
 -/
 noncomputable def IsDedekindDomain.quotientEquivPiOfFinsetProdEq {ι : Type*} {s : Finset ι}
     (I : Ideal R) (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
-    (coprime : ∀ (i) (_ : i ∈ s) (j) (_ : j ∈ s), i ≠ j → P i ≠ P j)
+    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → P i ≠ P j)
     (prod_eq : ∏ i in s, P i ^ e i = I) : R ⧸ I ≃+* ∀ i : s, R ⧸ P i ^ e i :=
   IsDedekindDomain.quotientEquivPiOfProdEq I (fun i : s => P i) (fun i : s => e i)
     (fun i => prime i i.2) (fun i j h => coprime i i.2 j j.2 (Subtype.coe_injective.ne h))
@@ -1379,7 +1379,7 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfFinsetProdEq {ι : Type*} {s
 we can choose a representative `y : R` such that `y ≡ x i (mod P i ^ e i)`.-/
 theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type*} {s : Finset ι}
     (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
-    (coprime : ∀ (i) (_ : i ∈ s) (j) (_ : j ∈ s), i ≠ j → P i ≠ P j) (x : ∀ i : s, R ⧸ P i ^ e i) :
+    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → P i ≠ P j) (x : ∀ i : s, R ⧸ P i ^ e i) :
     ∃ y, ∀ (i) (hi : i ∈ s), Ideal.Quotient.mk (P i ^ e i) y = x ⟨i, hi⟩ := by
   let f := IsDedekindDomain.quotientEquivPiOfFinsetProdEq _ P e prime coprime rfl
   obtain ⟨y, rfl⟩ := f.surjective x
@@ -1391,7 +1391,7 @@ theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type*} {s : Fins
 we can choose a representative `y : R` such that `y - x i ∈ P i ^ e i`.-/
 theorem IsDedekindDomain.exists_forall_sub_mem_ideal {ι : Type*} {s : Finset ι} (P : ι → Ideal R)
     (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
-    (coprime : ∀ (i) (_ : i ∈ s) (j) (_ : j ∈ s), i ≠ j → P i ≠ P j) (x : s → R) :
+    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → P i ≠ P j) (x : s → R) :
     ∃ y, ∀ (i) (hi : i ∈ s), y - x ⟨i, hi⟩ ∈ P i ^ e i := by
   obtain ⟨y, hy⟩ :=
     IsDedekindDomain.exists_representative_mod_finset P e prime coprime fun i =>

@@ -768,9 +768,18 @@ theorem Associates.le_singleton_iff (x : A) (n : ℕ) (I : Ideal A) :
     Ideal.dvd_span_singleton]
 #align associates.le_singleton_iff Associates.le_singleton_iff
 
+variable {K}
+
+lemma FractionalIdeal.le_inv_comm {I J : FractionalIdeal A⁰ K} (hI : I ≠ 0) (hJ : J ≠ 0) :
+    I ≤ J⁻¹ ↔ J ≤ I⁻¹ := by
+  rw [inv_eq, inv_eq, le_div_iff_mul_le hI, le_div_iff_mul_le hJ, mul_comm]
+
+lemma FractionalIdeal.inv_le_comm {I J : FractionalIdeal A⁰ K} (hI : I ≠ 0) (hJ : J ≠ 0) :
+    I⁻¹ ≤ J ↔ J⁻¹ ≤ I := by
+  simpa using le_inv_comm (A := A) (K := K) (inv_ne_zero hI) (inv_ne_zero hJ)
+
 open FractionalIdeal
 
-variable {K}
 
 /-- Strengthening of `IsLocalization.exist_integer_multiples`:
 Let `J ≠ ⊤` be an ideal in a Dedekind domain `A`, and `f ≠ 0` a finite collection

The names for lemmas about monotonicity of (a ^ ·) and (· ^ n) were a mess. This PR tidies up everything related by following the naming convention for (a * ·) and (· * b). Namely, (a ^ ·) is pow_right and (· ^ n) is pow_left in lemma names. All lemma renames follow the corresponding multiplication lemma names closely.

Renames

Algebra.GroupPower.Order

• pow_mono → pow_right_mono
• pow_le_pow → pow_le_pow_right
• pow_le_pow_of_le_left → pow_le_pow_left
• pow_lt_pow_of_lt_left → pow_lt_pow_left
• strictMonoOn_pow → pow_left_strictMonoOn
• pow_strictMono_right → pow_right_strictMono
• pow_lt_pow → pow_lt_pow_right
• pow_lt_pow_iff → pow_lt_pow_iff_right
• pow_le_pow_iff → pow_le_pow_iff_right
• self_lt_pow → lt_self_pow
• strictAnti_pow → pow_right_strictAnti
• pow_lt_pow_iff_of_lt_one → pow_lt_pow_iff_right_of_lt_one
• pow_lt_pow_of_lt_one → pow_lt_pow_right_of_lt_one
• lt_of_pow_lt_pow → lt_of_pow_lt_pow_left
• le_of_pow_le_pow → le_of_pow_le_pow_left
• pow_lt_pow₀ → pow_lt_pow_right₀

Algebra.GroupPower.CovariantClass

• pow_le_pow_of_le_left' → pow_le_pow_left'
• nsmul_le_nsmul_of_le_right → nsmul_le_nsmul_right
• pow_lt_pow' → pow_lt_pow_right'
• nsmul_lt_nsmul → nsmul_lt_nsmul_left
• pow_strictMono_left → pow_right_strictMono'
• nsmul_strictMono_right → nsmul_left_strictMono
• StrictMono.pow_right' → StrictMono.pow_const
• StrictMono.nsmul_left → StrictMono.const_nsmul
• pow_strictMono_right' → pow_left_strictMono
• nsmul_strictMono_left → nsmul_right_strictMono
• Monotone.pow_right → Monotone.pow_const
• Monotone.nsmul_left → Monotone.const_nsmul
• lt_of_pow_lt_pow' → lt_of_pow_lt_pow_left'
• lt_of_nsmul_lt_nsmul → lt_of_nsmul_lt_nsmul_right
• pow_le_pow' → pow_le_pow_right'
• nsmul_le_nsmul → nsmul_le_nsmul_left
• pow_le_pow_of_le_one' → pow_le_pow_right_of_le_one'
• nsmul_le_nsmul_of_nonpos → nsmul_le_nsmul_left_of_nonpos
• le_of_pow_le_pow' → le_of_pow_le_pow_left'
• le_of_nsmul_le_nsmul' → le_of_nsmul_le_nsmul_right'
• pow_le_pow_iff' → pow_le_pow_iff_right'
• nsmul_le_nsmul_iff → nsmul_le_nsmul_iff_left
• pow_lt_pow_iff' → pow_lt_pow_iff_right'
• nsmul_lt_nsmul_iff → nsmul_lt_nsmul_iff_left

Data.Nat.Pow

• Nat.pow_lt_pow_of_lt_left → Nat.pow_lt_pow_left
• Nat.pow_le_iff_le_left → Nat.pow_le_pow_iff_left
• Nat.pow_lt_iff_lt_left → Nat.pow_lt_pow_iff_left

Lemmas added

• pow_le_pow_iff_left
• pow_lt_pow_iff_left
• pow_right_injective
• pow_right_inj
• Nat.pow_le_pow_left to have the correct name since Nat.pow_le_pow_of_le_left is in Std.
• Nat.pow_le_pow_right to have the correct name since Nat.pow_le_pow_of_le_right is in Std.

Lemmas removed

• self_le_pow was a duplicate of le_self_pow.
• Nat.pow_lt_pow_of_lt_right is defeq to pow_lt_pow_right.
• Nat.pow_right_strictMono is defeq to pow_right_strictMono.
• Nat.pow_le_iff_le_right is defeq to pow_le_pow_iff_right.
• Nat.pow_lt_iff_lt_right is defeq to pow_lt_pow_iff_right.

Other changes

• A bunch of proofs have been golfed.
• Some lemma assumptions have been turned from 0 < n or 1 ≤ n to n ≠ 0.
• A few Nat lemmas have been protected.
• One docstring has been fixed.

@@ -720,22 +720,22 @@ theorem Ideal.isPrime_iff_bot_or_prime {P : Ideal A} : IsPrime P ↔ P = ⊥ ∨
     hp.elim (fun h => h.symm ▸ Ideal.bot_prime) Ideal.isPrime_of_prime⟩
 #align ideal.is_prime_iff_bot_or_prime Ideal.isPrime_iff_bot_or_prime
 
-theorem Ideal.strictAnti_pow (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
+theorem Ideal.pow_right_strictAnti (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
     StrictAnti (I ^ · : ℕ → Ideal A) :=
   strictAnti_nat_of_succ_lt fun e =>
     Ideal.dvdNotUnit_iff_lt.mp ⟨pow_ne_zero _ hI0, I, mt isUnit_iff.mp hI1, pow_succ' I e⟩
-#align ideal.strict_anti_pow Ideal.strictAnti_pow
+#align ideal.strict_anti_pow Ideal.pow_right_strictAnti
 
 theorem Ideal.pow_lt_self (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) (e : ℕ) (he : 2 ≤ e) :
     I ^ e < I := by
-  convert I.strictAnti_pow hI0 hI1 he
+  convert I.pow_right_strictAnti hI0 hI1 he
   dsimp only
   rw [pow_one]
 #align ideal.pow_lt_self Ideal.pow_lt_self
 
 theorem Ideal.exists_mem_pow_not_mem_pow_succ (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) (e : ℕ) :
     ∃ x ∈ I ^ e, x ∉ I ^ (e + 1) :=
-  SetLike.exists_of_lt (I.strictAnti_pow hI0 hI1 e.lt_succ_self)
+  SetLike.exists_of_lt (I.pow_right_strictAnti hI0 hI1 e.lt_succ_self)
 #align ideal.exists_mem_pow_not_mem_pow_succ Ideal.exists_mem_pow_not_mem_pow_succ
 
 open UniqueFactorizationMonoid
@@ -758,7 +758,7 @@ theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime
 
 theorem Ideal.pow_succ_lt_pow {P : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥) (i : ℕ) :
     P ^ (i + 1) < P ^ i :=
-  lt_of_le_of_ne (Ideal.pow_le_pow (Nat.le_succ _))
+  lt_of_le_of_ne (Ideal.pow_le_pow_right (Nat.le_succ _))
     (mt (pow_eq_pow_iff hP (mt Ideal.isUnit_iff.mp P_prime.ne_top)).mp i.succ_ne_self)
 #align ideal.pow_succ_lt_pow Ideal.pow_succ_lt_pow
 

Also generalizes Ideal.subset_union to Subgroup.

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

@@ -369,9 +369,7 @@ theorem exists_multiset_prod_cons_le_and_prod_not_le [IsDedekindDomain A] (hNF :
     wellFounded_lt.has_min
       {Z | (Z.map PrimeSpectrum.asIdeal).prod ≤ I ∧ (Z.map PrimeSpectrum.asIdeal).prod ≠ ⊥}
       ⟨Z₀, hZ₀.1, hZ₀.2⟩
-  have hZM : Multiset.prod (Z.map PrimeSpectrum.asIdeal) ≤ M := le_trans hZI hIM
-  have hZ0 : Z ≠ 0 := by rintro rfl; simp [hM.ne_top] at hZM
-  obtain ⟨_, hPZ', hPM⟩ := (hM.isPrime.multiset_prod_le (mt Multiset.map_eq_zero.mp hZ0)).mp hZM
+  obtain ⟨_, hPZ', hPM⟩ := hM.isPrime.multiset_prod_le.mp (hZI.trans hIM)
   -- Then in fact there is a `P ∈ Z` with `P ≤ M`.
   obtain ⟨P, hPZ, rfl⟩ := Multiset.mem_map.mp hPZ'
   classical

I used the regex \(\(· (.) ·\) (.)\), replacing with ($2 $1 ·).

@@ -560,7 +560,7 @@ theorem mul_right_strictMono [IsDedekindDomain A] {I : FractionalIdeal A⁰ K} (
 #align fractional_ideal.mul_right_strict_mono FractionalIdeal.mul_right_strictMono
 
 theorem mul_left_strictMono [IsDedekindDomain A] {I : FractionalIdeal A⁰ K} (hI : I ≠ 0) :
-    StrictMono ((· * ·) I) :=
+    StrictMono (I * ·) :=
   strictMono_of_le_iff_le fun _ _ => (mul_left_le_iff hI).symm
 #align fractional_ideal.mul_left_strict_mono FractionalIdeal.mul_left_strictMono
 
@@ -723,7 +723,7 @@ theorem Ideal.isPrime_iff_bot_or_prime {P : Ideal A} : IsPrime P ↔ P = ⊥ ∨
 #align ideal.is_prime_iff_bot_or_prime Ideal.isPrime_iff_bot_or_prime
 
 theorem Ideal.strictAnti_pow (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
-    StrictAnti ((· ^ ·) I : ℕ → Ideal A) :=
+    StrictAnti (I ^ · : ℕ → Ideal A) :=
   strictAnti_nat_of_succ_lt fun e =>
     Ideal.dvdNotUnit_iff_lt.mp ⟨pow_ne_zero _ hI0, I, mt isUnit_iff.mp hI1, pow_succ' I e⟩
 #align ideal.strict_anti_pow Ideal.strictAnti_pow

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -298,7 +298,7 @@ theorem integrallyClosed : IsIntegrallyClosed A := by
   -- `A[x]` (which is a fractional ideal) is in fact equal to `A`.
   refine ⟨fun {x hx} => ?_⟩
   rw [← Set.mem_range, ← Algebra.mem_bot, ← Subalgebra.mem_toSubmodule, Algebra.toSubmodule_bot,
-    Submodule.one_eq_span, ←coe_spanSingleton A⁰ (1 : FractionRing A), spanSingleton_one, ←
+    Submodule.one_eq_span, ← coe_spanSingleton A⁰ (1 : FractionRing A), spanSingleton_one, ←
     FractionalIdeal.adjoinIntegral_eq_one_of_isUnit x hx (h.isUnit _)]
   · exact mem_adjoinIntegral_self A⁰ x hx
   · exact fun h => one_ne_zero (eq_zero_iff.mp h 1 (Algebra.adjoin A {x}).one_mem)

This PR tests a string-based tool for renaming declarations.

Inspired by this Zulip thread, I am trying to reduce the diff of #8406.

This PR makes the following renames:

| From | To |

@@ -497,7 +497,7 @@ theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ 
       ← mem_one_iff] at this
   -- For that, we'll find a subalgebra that is f.g. as a module and contains `x`.
   -- `A` is a noetherian ring, so we just need to find a subalgebra between `{x}` and `I⁻¹`.
-  rw [mem_integralClosure_iff_mem_FG]
+  rw [mem_integralClosure_iff_mem_fg]
   have x_mul_mem : ∀ b ∈ (I⁻¹ : FractionalIdeal A⁰ K), x * b ∈ (I⁻¹ : FractionalIdeal A⁰ K) := by
     intro b hb
     rw [mem_inv_iff (coeIdeal_ne_zero.mpr hI0)]

We already have WellFoundedLT/WellFoundedGT as wrappers around IsWellFounded, but we didn't have the corresponding wrapper lemmas.

@@ -366,9 +366,9 @@ theorem exists_multiset_prod_cons_le_and_prod_not_le [IsDedekindDomain A] (hNF :
   -- Let `Z` be a minimal set of prime ideals such that their product is contained in `J`.
   obtain ⟨Z₀, hZ₀⟩ := PrimeSpectrum.exists_primeSpectrum_prod_le_and_ne_bot_of_domain hNF hI0
   obtain ⟨Z, ⟨hZI, hprodZ⟩, h_eraseZ⟩ :=
-    Multiset.wellFounded_lt.has_min
-      (fun Z => (Z.map PrimeSpectrum.asIdeal).prod ≤ I ∧ (Z.map PrimeSpectrum.asIdeal).prod ≠ ⊥)
-      ⟨Z₀, hZ₀⟩
+    wellFounded_lt.has_min
+      {Z | (Z.map PrimeSpectrum.asIdeal).prod ≤ I ∧ (Z.map PrimeSpectrum.asIdeal).prod ≠ ⊥}
+      ⟨Z₀, hZ₀.1, hZ₀.2⟩
   have hZM : Multiset.prod (Z.map PrimeSpectrum.asIdeal) ≤ M := le_trans hZI hIM
   have hZ0 : Z ≠ 0 := by rintro rfl; simp [hM.ne_top] at hZM
   obtain ⟨_, hPZ', hPM⟩ := (hM.isPrime.multiset_prod_le (mt Multiset.map_eq_zero.mp hZ0)).mp hZM

And the same thing for StarSubalgebra R A. IntermediateField was already handled in #7957.

As a result, nine (obvious) lemmas are now true by definition.

This slightly adjusts the statement of Algebra.toSubmodule_bot to make it simpler and true by definition; the original statement can be recovered by rewriting by Submodule.one_eq_span, which I've had to add in some downstream proofs.

@@ -297,8 +297,8 @@ theorem integrallyClosed : IsIntegrallyClosed A := by
   -- It suffices to show that for integral `x`,
   -- `A[x]` (which is a fractional ideal) is in fact equal to `A`.
   refine ⟨fun {x hx} => ?_⟩
-  rw [← Set.mem_range, ← Algebra.mem_bot, ← Subalgebra.mem_toSubmodule, Algebra.toSubmodule_bot, ←
-    coe_spanSingleton A⁰ (1 : FractionRing A), spanSingleton_one, ←
+  rw [← Set.mem_range, ← Algebra.mem_bot, ← Subalgebra.mem_toSubmodule, Algebra.toSubmodule_bot,
+    Submodule.one_eq_span, ←coe_spanSingleton A⁰ (1 : FractionRing A), spanSingleton_one, ←
     FractionalIdeal.adjoinIntegral_eq_one_of_isUnit x hx (h.isUnit _)]
   · exact mem_adjoinIntegral_self A⁰ x hx
   · exact fun h => one_ne_zero (eq_zero_iff.mp h 1 (Algebra.adjoin A {x}).one_mem)

This reverts commit f3695eb2.

@@ -1156,7 +1156,10 @@ def normalizedFactorsEquivOfQuotEquiv (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
         idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors f.symm hI
           j.prop⟩
   left_inv := fun ⟨j, hj⟩ => by simp
-  right_inv := fun ⟨j, hj⟩ => by simp
+  right_inv := fun ⟨j, hj⟩ => by
+    simp
+    -- This used to be the end of the proof before leanprover/lean4#2644
+    erw [OrderIso.apply_symm_apply]
 #align normalized_factors_equiv_of_quot_equiv normalizedFactorsEquivOfQuotEquiv
 
 @[simp]

This reverts commit 26eb2b0a.

@@ -1156,10 +1156,7 @@ def normalizedFactorsEquivOfQuotEquiv (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
         idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors f.symm hI
           j.prop⟩
   left_inv := fun ⟨j, hj⟩ => by simp
-  right_inv := fun ⟨j, hj⟩ => by
-    simp
-    -- This used to be the end of the proof before leanprover/lean4#2644
-    erw [OrderIso.apply_symm_apply]
+  right_inv := fun ⟨j, hj⟩ => by simp
 #align normalized_factors_equiv_of_quot_equiv normalizedFactorsEquivOfQuotEquiv
 
 @[simp]

This includes all the changes from #7606.

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

@@ -1156,7 +1156,10 @@ def normalizedFactorsEquivOfQuotEquiv (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
         idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors f.symm hI
           j.prop⟩
   left_inv := fun ⟨j, hj⟩ => by simp
-  right_inv := fun ⟨j, hj⟩ => by simp
+  right_inv := fun ⟨j, hj⟩ => by
+    simp
+    -- This used to be the end of the proof before leanprover/lean4#2644
+    erw [OrderIso.apply_symm_apply]
 #align normalized_factors_equiv_of_quot_equiv normalizedFactorsEquivOfQuotEquiv
 
 @[simp]

@@ -1305,7 +1305,7 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype
       simp only [← prod_eq, Finset.inf_eq_iInf, Finset.mem_univ, ciInf_pos,
         ← IsDedekindDomain.inf_prime_pow_eq_prod _ _ _ (fun i _ => prime i) fun i _ j _ =>
         coprime i j])).trans <|
-    Ideal.quotientInfRingEquivPiQuotient _ fun i j hij => (Ideal.coprime_of_no_prime_ge (by
+    Ideal.quotientInfRingEquivPiQuotient _ fun i j hij => Ideal.coprime_of_no_prime_ge (by
       intro P hPi hPj hPp
       haveI := Ideal.isPrime_of_prime (prime i)
       haveI := Ideal.isPrime_of_prime (prime j)
@@ -1315,7 +1315,7 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype
         ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
           (Ideal.le_of_pow_le_prime hPj)).symm
       · exact (prime i).ne_zero
-      · exact (prime j).ne_zero)).sup_eq
+      · exact (prime j).ne_zero)
 #align is_dedekind_domain.quotient_equiv_pi_of_prod_eq IsDedekindDomain.quotientEquivPiOfProdEq
 
 open scoped Classical
@@ -1346,8 +1346,8 @@ theorem IsDedekindDomain.quotientEquivPiFactors_mk {I : Ideal R} (hI : I ≠ ⊥
 /-- **Chinese remainder theorem**, specialized to two ideals. -/
 noncomputable def Ideal.quotientMulEquivQuotientProd (I J : Ideal R) (coprime : IsCoprime I J) :
     R ⧸ I * J ≃+* (R ⧸ I) × R ⧸ J :=
-  RingEquiv.trans (Ideal.quotEquivOfEq (inf_eq_mul_of_coprime coprime).symm)
-    (Ideal.quotientInfEquivQuotientProd I J coprime.sup_eq)
+  Ideal.quotEquivOfEq (inf_eq_mul_of_coprime coprime).symm |>.trans <|
+    Ideal.quotientInfEquivQuotientProd I J coprime
 #align ideal.quotient_mul_equiv_quotient_prod Ideal.quotientMulEquivQuotientProd
 
 /-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as

Make IsCoprime I J the preferred way to say that two ideals are coprime, provide lemmas translating to other formulations.

@@ -871,8 +871,8 @@ theorem gcd_eq_sup (I J : Ideal A) : gcd I J = I ⊔ J := rfl
 theorem lcm_eq_inf (I J : Ideal A) : lcm I J = I ⊓ J := rfl
 #align ideal.lcm_eq_inf Ideal.lcm_eq_inf
 
-theorem inf_eq_mul_of_coprime {I J : Ideal A} (coprime : I ⊔ J = ⊤) : I ⊓ J = I * J := by
-  rw [← associated_iff_eq.mp (gcd_mul_lcm I J), lcm_eq_inf I J, gcd_eq_sup, coprime, top_mul]
+theorem inf_eq_mul_of_coprime {I J : Ideal A} (coprime : IsCoprime I J) : I ⊓ J = I * J := by
+  rw [← associated_iff_eq.mp (gcd_mul_lcm I J), lcm_eq_inf I J, gcd_eq_sup, coprime.sup_eq, top_mul]
 #align ideal.inf_eq_mul_of_coprime Ideal.inf_eq_mul_of_coprime
 
 theorem isCoprime_iff_gcd {I J : Ideal A} : IsCoprime I J ↔ gcd I J = 1 := by
@@ -1195,7 +1195,8 @@ theorem Ring.DimensionLeOne.prime_le_prime_iff_eq [Ring.DimensionLEOne R] {P Q :
 #align ring.dimension_le_one.prime_le_prime_iff_eq Ring.DimensionLeOne.prime_le_prime_iff_eq
 
 theorem Ideal.coprime_of_no_prime_ge {I J : Ideal R} (h : ∀ P, I ≤ P → J ≤ P → ¬IsPrime P) :
-    I ⊔ J = ⊤ := by
+    IsCoprime I J := by
+  rw [isCoprime_iff_sup_eq]
   by_contra hIJ
   obtain ⟨P, hP, hIJ⟩ := Ideal.exists_le_maximal _ hIJ
   exact h P (le_trans le_sup_left hIJ) (le_trans le_sup_right hIJ) hP.isPrime
@@ -1304,7 +1305,7 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype
       simp only [← prod_eq, Finset.inf_eq_iInf, Finset.mem_univ, ciInf_pos,
         ← IsDedekindDomain.inf_prime_pow_eq_prod _ _ _ (fun i _ => prime i) fun i _ j _ =>
         coprime i j])).trans <|
-    Ideal.quotientInfRingEquivPiQuotient _ fun i j hij => Ideal.coprime_of_no_prime_ge (by
+    Ideal.quotientInfRingEquivPiQuotient _ fun i j hij => (Ideal.coprime_of_no_prime_ge (by
       intro P hPi hPj hPp
       haveI := Ideal.isPrime_of_prime (prime i)
       haveI := Ideal.isPrime_of_prime (prime j)
@@ -1314,7 +1315,7 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype
         ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
           (Ideal.le_of_pow_le_prime hPj)).symm
       · exact (prime i).ne_zero
-      · exact (prime j).ne_zero)
+      · exact (prime j).ne_zero)).sup_eq
 #align is_dedekind_domain.quotient_equiv_pi_of_prod_eq IsDedekindDomain.quotientEquivPiOfProdEq
 
 open scoped Classical
@@ -1343,10 +1344,10 @@ theorem IsDedekindDomain.quotientEquivPiFactors_mk {I : Ideal R} (hI : I ≠ ⊥
 #align is_dedekind_domain.quotient_equiv_pi_factors_mk IsDedekindDomain.quotientEquivPiFactors_mk
 
 /-- **Chinese remainder theorem**, specialized to two ideals. -/
-noncomputable def Ideal.quotientMulEquivQuotientProd (I J : Ideal R) (coprime : I ⊔ J = ⊤) :
+noncomputable def Ideal.quotientMulEquivQuotientProd (I J : Ideal R) (coprime : IsCoprime I J) :
     R ⧸ I * J ≃+* (R ⧸ I) × R ⧸ J :=
   RingEquiv.trans (Ideal.quotEquivOfEq (inf_eq_mul_of_coprime coprime).symm)
-    (Ideal.quotientInfEquivQuotientProd I J coprime)
+    (Ideal.quotientInfEquivQuotientProd I J coprime.sup_eq)
 #align ideal.quotient_mul_equiv_quotient_prod Ideal.quotientMulEquivQuotientProd
 
 /-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as

From flt-regular.

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

@@ -875,6 +875,9 @@ theorem inf_eq_mul_of_coprime {I J : Ideal A} (coprime : I ⊔ J = ⊤) : I ⊓
   rw [← associated_iff_eq.mp (gcd_mul_lcm I J), lcm_eq_inf I J, gcd_eq_sup, coprime, top_mul]
 #align ideal.inf_eq_mul_of_coprime Ideal.inf_eq_mul_of_coprime
 
+theorem isCoprime_iff_gcd {I J : Ideal A} : IsCoprime I J ↔ gcd I J = 1 := by
+  rw [Ideal.isCoprime_iff_codisjoint, codisjoint_iff, one_eq_top, gcd_eq_sup]
+
 end Ideal
 
 end Gcd

This PR defines IsDedekindRing as IsDedekindDomain without the bundled IsDomain hypothesis.

Zulip thread: https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/Should.20.60IsDedekindDomain.60.20extend.20.60IsDomain.60.3F

Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>

@@ -663,7 +663,7 @@ theorem Ideal.dvdNotUnit_iff_lt {I J : Ideal A} : DvdNotUnit I J ↔ J < I :=
 instance : WfDvdMonoid (Ideal A) where
   wellFounded_dvdNotUnit := by
     have : WellFounded ((· > ·) : Ideal A → Ideal A → Prop) :=
-      isNoetherian_iff_wellFounded.mp (isNoetherianRing_iff.mp IsDedekindDomain.toIsNoetherian)
+      isNoetherian_iff_wellFounded.mp (isNoetherianRing_iff.mp IsDedekindRing.toIsNoetherian)
     convert this
     ext
     rw [Ideal.dvdNotUnit_iff_lt]

@@ -313,8 +313,7 @@ theorem dimensionLEOne : DimensionLEOne A := ⟨by
   refine Ideal.isMaximal_def.mpr ⟨hP.ne_top, fun M hM => ?_⟩
   -- We may assume `P` and `M` (as fractional ideals) are nonzero.
   have P'_ne : (P : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr P_ne
-  have M'_ne : (M : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 :=
-    coeIdeal_ne_zero.mpr (lt_of_le_of_lt bot_le hM).ne'
+  have M'_ne : (M : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hM.ne_bot
   -- In particular, we'll show `M⁻¹ * P ≤ P`
   suffices (M⁻¹ : FractionalIdeal A⁰ (FractionRing A)) * P ≤ P by
     rw [eq_top_iff, ← coeIdeal_le_coeIdeal (FractionRing A), coeIdeal_top]
@@ -353,9 +352,9 @@ end IsDedekindDomainInv
 
 variable [Algebra A K] [IsFractionRing A K]
 
-/-- Specialization of `exists_prime_spectrum_prod_le_and_ne_bot_of_domain` to Dedekind domains:
+/-- Specialization of `exists_primeSpectrum_prod_le_and_ne_bot_of_domain` to Dedekind domains:
 Let `I : Ideal A` be a nonzero ideal, where `A` is a Dedekind domain that is not a field.
-Then `exists_prime_spectrum_prod_le_and_ne_bot_of_domain` states we can find a product of prime
+Then `exists_primeSpectrum_prod_le_and_ne_bot_of_domain` states we can find a product of prime
 ideals that is contained within `I`. This lemma extends that result by making the product minimal:
 let `M` be a maximal ideal that contains `I`, then the product including `M` is contained within `I`
 and the product excluding `M` is not contained within `I`. -/
@@ -403,14 +402,12 @@ theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I
     ∀ {M : Ideal A} (_hM : M.IsMaximal),
       ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
     obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
-    skip
     have hM0 := (M.bot_lt_of_maximal hNF).ne'
     obtain ⟨x, hxM, hx1⟩ := this hM
     refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
         rw [coeIdeal_ne_zero] <;> assumption
   -- Let `a` be a nonzero element of `M` and `J` the ideal generated by `a`.
   intro M hM
-  skip
   obtain ⟨⟨a, haM⟩, ha0⟩ := Submodule.nonzero_mem_of_bot_lt (M.bot_lt_of_maximal hNF)
   replace ha0 : a ≠ 0 := Subtype.coe_injective.ne ha0
   let J : Ideal A := Ideal.span {a}
@@ -503,11 +500,9 @@ theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ 
   rw [mem_integralClosure_iff_mem_FG]
   have x_mul_mem : ∀ b ∈ (I⁻¹ : FractionalIdeal A⁰ K), x * b ∈ (I⁻¹ : FractionalIdeal A⁰ K) := by
     intro b hb
-    rw [mem_inv_iff]
+    rw [mem_inv_iff (coeIdeal_ne_zero.mpr hI0)]
     dsimp only at hx
-    rw [val_eq_coe, mem_coe, mem_inv_iff] at hx
-    swap; · exact hJ0
-    swap; · exact coeIdeal_ne_zero.mpr hI0
+    rw [val_eq_coe, mem_coe, mem_inv_iff hJ0] at hx
     simp only [mul_assoc, mul_comm b] at hx ⊢
     intro y hy
     exact hx _ (mul_mem_mul hy hb)
@@ -528,7 +523,7 @@ theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ 
 /-- Nonzero fractional ideals in a Dedekind domain are units.
 
 This is also available as `_root_.mul_inv_cancel`, using the
-`CommGroupWithZero` instance defined below.
+`Semifield` instance defined below.
 -/
 protected theorem mul_inv_cancel [IsDedekindDomain A] {I : FractionalIdeal A⁰ K} (hne : I ≠ 0) :
     I * I⁻¹ = 1 := by
@@ -549,7 +544,7 @@ theorem mul_right_le_iff [IsDedekindDomain A] {J : FractionalIdeal A⁰ K} (hJ :
     ∀ {I I'}, I * J ≤ I' * J ↔ I ≤ I' := by
   intro I I'
   constructor
-  · intro h;
+  · intro h
     convert mul_right_mono J⁻¹ h <;> dsimp only <;>
     rw [mul_assoc, FractionalIdeal.mul_inv_cancel hJ, mul_one]
   · exact fun h => mul_right_mono J h
@@ -570,7 +565,7 @@ theorem mul_left_strictMono [IsDedekindDomain A] {I : FractionalIdeal A⁰ K} (h
 #align fractional_ideal.mul_left_strict_mono FractionalIdeal.mul_left_strictMono
 
 /-- This is also available as `_root_.div_eq_mul_inv`, using the
-`CommGroupWithZero` instance defined below.
+`Semifield` instance defined below.
 -/
 protected theorem div_eq_mul_inv [IsDedekindDomain A] (I J : FractionalIdeal A⁰ K) :
     I / J = I * J⁻¹ := by
@@ -616,7 +611,7 @@ noncomputable instance FractionalIdeal.semifield : Semifield (FractionalIdeal A
 /-- Fractional ideals have cancellative multiplication in a Dedekind domain.
 
 Although this instance is a direct consequence of the instance
-`fractional_ideal.comm_group_with_zero`, we define this instance to provide
+`FractionalIdeal.semifield`, we define this instance to provide
 a computable alternative.
 -/
 -- Porting note: added noncomputable because otherwise it fails, so it seems that the goal
@@ -964,7 +959,7 @@ end IsDedekindDomain
 ### Height one spectrum of a Dedekind domain
 If `R` is a Dedekind domain of Krull dimension 1, the maximal ideals of `R` are exactly its nonzero
 prime ideals.
-We define `height_one_spectrum` and provide lemmas to recover the facts that prime ideals of height
+We define `HeightOneSpectrum` and provide lemmas to recover the facts that prime ideals of height
 one are prime and irreducible.
 -/
 
@@ -1104,12 +1099,10 @@ def idealFactorsEquivOfQuotEquiv : { p : Ideal R | p ∣ I } ≃o { p : Ideal A
     ?_ ?_
   · have := idealFactorsFunOfQuotHom_comp fsym_surj f_surj
     simp only [RingEquiv.comp_symm, idealFactorsFunOfQuotHom_id] at this
-    rw [← this]
-    congr
+    rw [← this, OrderHom.coe_eq, OrderHom.coe_eq]
   · have := idealFactorsFunOfQuotHom_comp f_surj fsym_surj
     simp only [RingEquiv.symm_comp, idealFactorsFunOfQuotHom_id] at this
-    rw [← this]
-    congr
+    rw [← this, OrderHom.coe_eq, OrderHom.coe_eq]
 #align ideal_factors_equiv_of_quot_equiv idealFactorsEquivOfQuotEquiv
 
 theorem idealFactorsEquivOfQuotEquiv_symm :
@@ -1134,15 +1127,15 @@ theorem idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFact
     {L : Ideal R} (hL : L ∈ normalizedFactors I) :
     ↑(idealFactorsEquivOfQuotEquiv f ⟨L, dvd_of_mem_normalizedFactors hL⟩)
       ∈ normalizedFactors J := by
-  by_cases hI : I = ⊥
-  · exfalso
+  have hI : I ≠ ⊥
+  · intro hI
     rw [hI, bot_eq_zero, normalizedFactors_zero, ← Multiset.empty_eq_zero] at hL
     exact Finset.not_mem_empty _ hL
-  · refine mem_normalizedFactors_factor_dvd_iso_of_mem_normalizedFactors hI hJ hL
-      (d := (idealFactorsEquivOfQuotEquiv f).toEquiv) ?_
-    rintro ⟨l, hl⟩ ⟨l', hl'⟩
-    rw [Subtype.coe_mk, Subtype.coe_mk]
-    apply idealFactorsEquivOfQuotEquiv_is_dvd_iso f
+  refine mem_normalizedFactors_factor_dvd_iso_of_mem_normalizedFactors hI hJ hL
+    (d := (idealFactorsEquivOfQuotEquiv f).toEquiv) ?_
+  rintro ⟨l, hl⟩ ⟨l', hl'⟩
+  rw [Subtype.coe_mk, Subtype.coe_mk]
+  apply idealFactorsEquivOfQuotEquiv_is_dvd_iso f
 #align ideal_factors_equiv_of_quot_equiv_mem_normalized_factors_of_mem_normalized_factors idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors
 
 /-- The bijection between the sets of normalized factors of I and J induced by a ring
@@ -1285,18 +1278,17 @@ theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f :
   rw [Finset.inf_insert, Finset.prod_insert ha, ih]
   refine' le_antisymm (Ideal.le_mul_of_no_prime_factors _ inf_le_left inf_le_right) Ideal.mul_le_inf
   intro P hPa hPs hPp
-  haveI := hPp
   obtain ⟨b, hb, hPb⟩ := Ideal.prod_le_prime.mp hPs
   haveI := Ideal.isPrime_of_prime (prime a (Finset.mem_insert_self a s))
   haveI := Ideal.isPrime_of_prime (prime b (Finset.mem_insert_of_mem hb))
   refine coprime a (Finset.mem_insert_self a s) b (Finset.mem_insert_of_mem hb) ?_ ?_
-  · rintro rfl; contradiction
+  · exact (ne_of_mem_of_not_mem hb ha).symm
   · refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
       (Ideal.le_of_pow_le_prime hPa)).trans
       ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
       (Ideal.le_of_pow_le_prime hPb)).symm
-    exact (prime a (Finset.mem_insert_self a s)).ne_zero
-    exact (prime b (Finset.mem_insert_of_mem hb)).ne_zero
+    · exact (prime a (Finset.mem_insert_self a s)).ne_zero
+    · exact (prime b (Finset.mem_insert_of_mem hb)).ne_zero
 #align is_dedekind_domain.inf_prime_pow_eq_prod IsDedekindDomain.inf_prime_pow_eq_prod
 
 /-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
@@ -1311,7 +1303,6 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype
         coprime i j])).trans <|
     Ideal.quotientInfRingEquivPiQuotient _ fun i j hij => Ideal.coprime_of_no_prime_ge (by
       intro P hPi hPj hPp
-      haveI := hPp
       haveI := Ideal.isPrime_of_prime (prime i)
       haveI := Ideal.isPrime_of_prime (prime j)
       refine coprime i j hij ?_
@@ -1319,8 +1310,8 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype
         (Ideal.le_of_pow_le_prime hPi)).trans
         ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
           (Ideal.le_of_pow_le_prime hPj)).symm
-      exact (prime i).ne_zero
-      exact (prime j).ne_zero)
+      · exact (prime i).ne_zero
+      · exact (prime j).ne_zero)
 #align is_dedekind_domain.quotient_equiv_pi_of_prod_eq IsDedekindDomain.quotientEquivPiOfProdEq
 
 open scoped Classical

Search&replace MulZeroClass.mul_zero -> mul_zero, MulZeroClass.zero_mul -> zero_mul.

These were introduced by Mathport, as the full name of mul_zero is actually MulZeroClass.mul_zero (it's exported with the short name).

@@ -575,7 +575,7 @@ theorem mul_left_strictMono [IsDedekindDomain A] {I : FractionalIdeal A⁰ K} (h
 protected theorem div_eq_mul_inv [IsDedekindDomain A] (I J : FractionalIdeal A⁰ K) :
     I / J = I * J⁻¹ := by
   by_cases hJ : J = 0
-  · rw [hJ, div_zero, inv_zero', MulZeroClass.mul_zero]
+  · rw [hJ, div_zero, inv_zero', mul_zero]
   refine' le_antisymm ((mul_right_le_iff hJ).mp _) ((le_div_iff_mul_le hJ).mpr _)
   · rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one, mul_le]
     intro x hx y hy
@@ -804,7 +804,7 @@ theorem Ideal.exist_integer_multiples_not_mem {J : Ideal A} (hJ : J ≠ ⊤) {ι
       -- then `a` is actually an element of `J / I`, contradiction.
       refine' (mem_div_iff_of_nonzero hI0).mpr fun y hy => Submodule.span_induction hy _ _ _ _
       · rintro _ ⟨i, hi, rfl⟩; exact hpI i hi
-      · rw [MulZeroClass.mul_zero]; exact Submodule.zero_mem _
+      · rw [mul_zero]; exact Submodule.zero_mem _
       · intro x y hx hy; rw [mul_add]; exact Submodule.add_mem _ hx hy
       · intro b x hx; rw [mul_smul_comm]; exact Submodule.smul_mem _ b hx
   -- To show the inclusion of `J / I` into `I⁻¹ = 1 / I`, note that `J < I`.
@@ -1237,7 +1237,7 @@ theorem Ideal.le_mul_of_no_prime_factors {I J K : Ideal R}
     (coprime : ∀ P, J ≤ P → K ≤ P → ¬IsPrime P) (hJ : I ≤ J) (hK : I ≤ K) : I ≤ J * K := by
   simp only [← Ideal.dvd_iff_le] at coprime hJ hK ⊢
   by_cases hJ0 : J = 0
-  · simpa only [hJ0, MulZeroClass.zero_mul] using hJ
+  · simpa only [hJ0, zero_mul] using hJ
   obtain ⟨I', rfl⟩ := hK
   rw [mul_comm]
   refine mul_dvd_mul_left K

This redefines the IsDedekindDomain class to be the conjunction through extends of the classes IsDomain, IsNoetherianRing, DimensionLEOne and IsIntegrallyClosed.

Zulip thread: https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/Should.20.60IsDedekindDomain.60.20extend.20.60IsDomain.60.3F

@@ -346,7 +346,7 @@ theorem dimensionLEOne : DimensionLEOne A := ⟨by
 /-- Showing one side of the equivalence between the definitions
 `IsDedekindDomainInv` and `IsDedekindDomain` of Dedekind domains. -/
 theorem isDedekindDomain : IsDedekindDomain A :=
-  ⟨h.isNoetherianRing, h.dimensionLEOne, h.integrallyClosed⟩
+  { h.isNoetherianRing, h.dimensionLEOne, h.integrallyClosed with }
 #align is_dedekind_domain_inv.is_dedekind_domain IsDedekindDomainInv.isDedekindDomain
 
 end IsDedekindDomainInv
@@ -668,7 +668,7 @@ theorem Ideal.dvdNotUnit_iff_lt {I J : Ideal A} : DvdNotUnit I J ↔ J < I :=
 instance : WfDvdMonoid (Ideal A) where
   wellFounded_dvdNotUnit := by
     have : WellFounded ((· > ·) : Ideal A → Ideal A → Prop) :=
-      isNoetherian_iff_wellFounded.mp (isNoetherianRing_iff.mp IsDedekindDomain.isNoetherianRing)
+      isNoetherian_iff_wellFounded.mp (isNoetherianRing_iff.mp IsDedekindDomain.toIsNoetherian)
     convert this
     ext
     rw [Ideal.dvdNotUnit_iff_lt]

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

This has nice performance benefits.

@@ -52,7 +52,7 @@ dedekind domain, dedekind ring
 -/
 
 
-variable (R A K : Type _) [CommRing R] [CommRing A] [Field K]
+variable (R A K : Type*) [CommRing R] [CommRing A] [Field K]
 
 open scoped nonZeroDivisors Polynomial
 
@@ -62,7 +62,7 @@ section Inverse
 
 namespace FractionalIdeal
 
-variable {R₁ : Type _} [CommRing R₁] [IsDomain R₁] [Algebra R₁ K] [IsFractionRing R₁ K]
+variable {R₁ : Type*} [CommRing R₁] [IsDomain R₁] [Algebra R₁ K] [IsFractionRing R₁ K]
 
 variable {I J : FractionalIdeal R₁⁰ K}
 
@@ -136,7 +136,7 @@ theorem mul_inv_cancel_iff_isUnit {I : FractionalIdeal R₁⁰ K} : I * I⁻¹ =
   (mul_inv_cancel_iff K).trans isUnit_iff_exists_inv.symm
 #align fractional_ideal.mul_inv_cancel_iff_is_unit FractionalIdeal.mul_inv_cancel_iff_isUnit
 
-variable {K' : Type _} [Field K'] [Algebra R₁ K'] [IsFractionRing R₁ K']
+variable {K' : Type*} [Field K'] [Algebra R₁ K'] [IsFractionRing R₁ K']
 
 @[simp]
 theorem map_inv (I : FractionalIdeal R₁⁰ K) (h : K ≃ₐ[R₁] K') :
@@ -191,7 +191,7 @@ theorem coe_ideal_span_singleton_inv_mul {x : R₁} (hx : x ≠ 0) :
   rw [mul_comm, coe_ideal_span_singleton_mul_inv K hx]
 #align fractional_ideal.coe_ideal_span_singleton_inv_mul FractionalIdeal.coe_ideal_span_singleton_inv_mul
 
-theorem mul_generator_self_inv {R₁ : Type _} [CommRing R₁] [Algebra R₁ K] [IsLocalization R₁⁰ K]
+theorem mul_generator_self_inv {R₁ : Type*} [CommRing R₁] [Algebra R₁ K] [IsLocalization R₁⁰ K]
     (I : FractionalIdeal R₁⁰ K) [Submodule.IsPrincipal (I : Submodule R₁ K)] (h : I ≠ 0) :
     I * spanSingleton _ (generator (I : Submodule R₁ K))⁻¹ = 1 := by
   -- Rewrite only the `I` that appears alone.
@@ -783,7 +783,7 @@ variable {K}
 Let `J ≠ ⊤` be an ideal in a Dedekind domain `A`, and `f ≠ 0` a finite collection
 of elements of `K = Frac(A)`, then we can multiply the elements of `f` by some `a : K`
 to find a collection of elements of `A` that is not completely contained in `J`. -/
-theorem Ideal.exist_integer_multiples_not_mem {J : Ideal A} (hJ : J ≠ ⊤) {ι : Type _} (s : Finset ι)
+theorem Ideal.exist_integer_multiples_not_mem {J : Ideal A} (hJ : J ≠ ⊤) {ι : Type*} (s : Finset ι)
     (f : ι → K) {j} (hjs : j ∈ s) (hjf : f j ≠ 0) :
     ∃ a : K,
       (∀ i ∈ s, IsLocalization.IsInteger A (a * f i)) ∧
@@ -888,7 +888,7 @@ end IsDedekindDomain
 
 section IsDedekindDomain
 
-variable {T : Type _} [CommRing T] [IsDomain T] [IsDedekindDomain T] {I J : Ideal T}
+variable {T : Type*} [CommRing T] [IsDomain T] [IsDedekindDomain T] {I J : Ideal T}
 
 open scoped Classical
 
@@ -1079,7 +1079,7 @@ theorem idealFactorsFunOfQuotHom_id :
         sup_eq_left.mpr (dvd_iff_le.mp X.prop), Subtype.coe_eta])
 #align ideal_factors_fun_of_quot_hom_id idealFactorsFunOfQuotHom_id
 
-variable {B : Type _} [CommRing B] [IsDomain B] [IsDedekindDomain B] {L : Ideal B}
+variable {B : Type*} [CommRing B] [IsDomain B] [IsDedekindDomain B] {L : Ideal B}
 
 theorem idealFactorsFunOfQuotHom_comp {f : R ⧸ I →+* A ⧸ J} {g : A ⧸ J →+* B ⧸ L}
     (hf : Function.Surjective f) (hg : Function.Surjective g) :
@@ -1259,7 +1259,7 @@ theorem Ideal.pow_le_prime_iff {I P : Ideal R} [_hP : P.IsPrime] {n : ℕ} (hn :
   ⟨Ideal.le_of_pow_le_prime, fun h => _root_.trans (Ideal.pow_le_self hn) h⟩
 #align ideal.pow_le_prime_iff Ideal.pow_le_prime_iff
 
-theorem Ideal.prod_le_prime {ι : Type _} {s : Finset ι} {f : ι → Ideal R} {P : Ideal R}
+theorem Ideal.prod_le_prime {ι : Type*} {s : Finset ι} {f : ι → Ideal R} {P : Ideal R}
     [hP : P.IsPrime] : ∏ i in s, f i ≤ P ↔ ∃ i ∈ s, f i ≤ P := by
   by_cases hP0 : P = ⊥
   · simp only [hP0, le_bot_iff]
@@ -1270,7 +1270,7 @@ theorem Ideal.prod_le_prime {ι : Type _} {s : Finset ι} {f : ι → Ideal R} {
 
 /-- The intersection of distinct prime powers in a Dedekind domain is the product of these
 prime powers. -/
-theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type _} (s : Finset ι) (f : ι → Ideal R)
+theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f : ι → Ideal R)
     (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (f i))
     (coprime : ∀ (i) (_ : i ∈ s) (j) (_ : j ∈ s), i ≠ j → f i ≠ f j) :
     (s.inf fun i => f i ^ e i) = ∏ i in s, f i ^ e i := by
@@ -1301,7 +1301,7 @@ theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type _} (s : Finset ι) (f
 
 /-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
 `∏ i, P i ^ e i`, then `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`. -/
-noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type _} [Fintype ι] (I : Ideal R)
+noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype ι] (I : Ideal R)
     (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i, Prime (P i)) (coprime : ∀ i j, i ≠ j → P i ≠ P j)
     (prod_eq : ∏ i, P i ^ e i = I) : R ⧸ I ≃+* ∀ i, R ⧸ P i ^ e i :=
   (Ideal.quotEquivOfEq
@@ -1361,7 +1361,7 @@ noncomputable def Ideal.quotientMulEquivQuotientProd (I J : Ideal R) (coprime :
 This is a version of `IsDedekindDomain.quotientEquivPiOfProdEq` where we restrict
 the product to a finite subset `s` of a potentially infinite indexing type `ι`.
 -/
-noncomputable def IsDedekindDomain.quotientEquivPiOfFinsetProdEq {ι : Type _} {s : Finset ι}
+noncomputable def IsDedekindDomain.quotientEquivPiOfFinsetProdEq {ι : Type*} {s : Finset ι}
     (I : Ideal R) (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
     (coprime : ∀ (i) (_ : i ∈ s) (j) (_ : j ∈ s), i ≠ j → P i ≠ P j)
     (prod_eq : ∏ i in s, P i ^ e i = I) : R ⧸ I ≃+* ∀ i : s, R ⧸ P i ^ e i :=
@@ -1372,7 +1372,7 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfFinsetProdEq {ι : Type _} {
 
 /-- Corollary of the Chinese remainder theorem: given elements `x i : R / P i ^ e i`,
 we can choose a representative `y : R` such that `y ≡ x i (mod P i ^ e i)`.-/
-theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type _} {s : Finset ι}
+theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type*} {s : Finset ι}
     (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
     (coprime : ∀ (i) (_ : i ∈ s) (j) (_ : j ∈ s), i ≠ j → P i ≠ P j) (x : ∀ i : s, R ⧸ P i ^ e i) :
     ∃ y, ∀ (i) (hi : i ∈ s), Ideal.Quotient.mk (P i ^ e i) y = x ⟨i, hi⟩ := by
@@ -1384,7 +1384,7 @@ theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type _} {s : Fin
 
 /-- Corollary of the Chinese remainder theorem: given elements `x i : R`,
 we can choose a representative `y : R` such that `y - x i ∈ P i ^ e i`.-/
-theorem IsDedekindDomain.exists_forall_sub_mem_ideal {ι : Type _} {s : Finset ι} (P : ι → Ideal R)
+theorem IsDedekindDomain.exists_forall_sub_mem_ideal {ι : Type*} {s : Finset ι} (P : ι → Ideal R)
     (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
     (coprime : ∀ (i) (_ : i ∈ s) (j) (_ : j ∈ s), i ≠ j → P i ≠ P j) (x : s → R) :
     ∃ y, ∀ (i) (hi : i ∈ s), y - x ⟨i, hi⟩ ∈ P i ^ e i := by

Co-authored-by: Jujian Zhang <jujian.zhang1998@outlook.com> Co-authored-by: Oliver Nash <github@olivernash.org>

@@ -1046,9 +1046,9 @@ variable [IsDedekindDomain A] {I : Ideal R} {J : Ideal A}
 
 /-- The map from ideals of `R` dividing `I` to the ideals of `A` dividing `J` induced by
   a homomorphism `f : R/I →+* A/J` -/
-@[simps]
+@[simps] -- Porting note: use `Subtype` instead of `Set` to make linter happy
 def idealFactorsFunOfQuotHom {f : R ⧸ I →+* A ⧸ J} (hf : Function.Surjective f) :
-    { p : Ideal R | p ∣ I } →o { p : Ideal A | p ∣ J } where
+    {p : Ideal R // p ∣ I} →o {p : Ideal A // p ∣ J} where
   toFun X := ⟨comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) X)), by
     have : RingHom.ker (Ideal.Quotient.mk J) ≤
         comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) X)) :=
@@ -1066,13 +1066,14 @@ def idealFactorsFunOfQuotHom {f : R ⧸ I →+* A ⧸ J} (hf : Function.Surjecti
     rwa [map_le_iff_le_comap, comap_map_of_surjective (Ideal.Quotient.mk I)
       Quotient.mk_surjective, ← RingHom.ker_eq_comap_bot, mk_ker, sup_eq_left.mpr <| le_of_dvd hY]
 #align ideal_factors_fun_of_quot_hom idealFactorsFunOfQuotHom
+#align ideal_factors_fun_of_quot_hom_coe_coe idealFactorsFunOfQuotHom_coe_coe
 
 @[simp]
 theorem idealFactorsFunOfQuotHom_id :
     idealFactorsFunOfQuotHom (RingHom.id (A ⧸ J)).surjective = OrderHom.id :=
   OrderHom.ext _ _
     (funext fun X => by
-      simp only [idealFactorsFunOfQuotHom, map_id, OrderHom.coe_fun_mk, OrderHom.id_coe, id.def,
+      simp only [idealFactorsFunOfQuotHom, map_id, OrderHom.coe_mk, OrderHom.id_coe, id.def,
         comap_map_of_surjective (Ideal.Quotient.mk J) Quotient.mk_surjective, ←
         RingHom.ker_eq_comap_bot (Ideal.Quotient.mk J), mk_ker,
         sup_eq_left.mpr (dvd_iff_le.mp X.prop), Subtype.coe_eta])
@@ -1085,8 +1086,8 @@ theorem idealFactorsFunOfQuotHom_comp {f : R ⧸ I →+* A ⧸ J} {g : A ⧸ J 
     (idealFactorsFunOfQuotHom hg).comp (idealFactorsFunOfQuotHom hf) =
       idealFactorsFunOfQuotHom (show Function.Surjective (g.comp f) from hg.comp hf) := by
   refine OrderHom.ext _ _ (funext fun x => ?_)
-  rw [idealFactorsFunOfQuotHom, idealFactorsFunOfQuotHom, OrderHom.comp_coe, OrderHom.coe_fun_mk,
-    OrderHom.coe_fun_mk, Function.comp_apply, idealFactorsFunOfQuotHom, OrderHom.coe_fun_mk,
+  rw [idealFactorsFunOfQuotHom, idealFactorsFunOfQuotHom, OrderHom.comp_coe, OrderHom.coe_mk,
+    OrderHom.coe_mk, Function.comp_apply, idealFactorsFunOfQuotHom, OrderHom.coe_mk,
     Subtype.mk_eq_mk, Subtype.coe_mk, map_comap_of_surjective (Ideal.Quotient.mk J)
     Quotient.mk_surjective, map_map]
 #align ideal_factors_fun_of_quot_hom_comp idealFactorsFunOfQuotHom_comp

The predicate that a ring has Krull dimension at most one was a regular def. I believe we should turn it into a class because:

• The property follows from the ring structure, e.g. because it is a PID or because it is an integral closure.
• We pass it around as a whole hypothesis, something instance synthesis can deal well with.
• It makes the definition of IsDedekindDomain the conjunction of a number of classes, so we could switch to extends for all its fields.

The main change in API is the addition of Ideal.IsPrime.isMaximal which is a restatement of the Krull dimension property with convenient dot notation: turn a prime ideal into a maximal ideal given the hypothesis that it's not zero.

Zulip thread: https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/Should.20.60IsDedekindDomain.60.20extend.20.60IsDomain.60.3F/near/374515392

@@ -306,7 +306,7 @@ theorem integrallyClosed : IsIntegrallyClosed A := by
 
 open Ring
 
-theorem dimensionLEOne : DimensionLEOne A := by
+theorem dimensionLEOne : DimensionLEOne A := ⟨by
   -- We're going to show that `P` is maximal because any (maximal) ideal `M`
   -- that is strictly larger would be `⊤`.
   rintro P P_ne hP
@@ -340,7 +340,7 @@ theorem dimensionLEOne : DimensionLEOne A := by
   obtain ⟨zy, hzy, zy_eq⟩ := (mem_coeIdeal A⁰).mp zy_mem
   rw [IsFractionRing.injective A (FractionRing A) zy_eq] at hzy
   -- But `P` is a prime ideal, so `z ∉ P` implies `y ∈ P`, as desired.
-  exact mem_coeIdeal_of_mem A⁰ (Or.resolve_left (hP.mem_or_mem hzy) hzp)
+  exact mem_coeIdeal_of_mem A⁰ (Or.resolve_left (hP.mem_or_mem hzy) hzp)⟩
 #align is_dedekind_domain_inv.dimension_le_one IsDedekindDomainInv.dimensionLEOne
 
 /-- Showing one side of the equivalence between the definitions
@@ -381,7 +381,7 @@ theorem exists_multiset_prod_cons_le_and_prod_not_le [IsDedekindDomain A] (hNF :
       rwa [Ne.def, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
         this] at hprodZ
     -- By maximality of `P` and `M`, we have that `P ≤ M` implies `P = M`.
-    have hPM' := (IsDedekindDomain.dimensionLEOne _ hP0 P.IsPrime).eq_of_le hM.ne_top hPM
+    have hPM' := (P.IsPrime.isMaximal hP0).eq_of_le hM.ne_top hPM
     subst hPM'
     -- By minimality of `Z`, erasing `P` from `Z` is exactly what we need.
     refine ⟨Z.erase P, ?_, ?_⟩
@@ -989,7 +989,7 @@ variable (v : HeightOneSpectrum R) {R}
 
 namespace HeightOneSpectrum
 
-instance isMaximal : v.asIdeal.IsMaximal := dimensionLEOne v.asIdeal v.ne_bot v.isPrime
+instance isMaximal : v.asIdeal.IsMaximal := v.isPrime.isMaximal v.ne_bot
 #align is_dedekind_domain.height_one_spectrum.is_maximal IsDedekindDomain.HeightOneSpectrum.isMaximal
 
 theorem prime : Prime v.asIdeal := Ideal.prime_of_isPrime v.ne_bot v.isPrime
@@ -1005,7 +1005,7 @@ theorem associates_irreducible : Irreducible <| Associates.mk v.asIdeal :=
 
 /-- An equivalence between the height one and maximal spectra for rings of Krull dimension 1. -/
 def equivMaximalSpectrum (hR : ¬IsField R) : HeightOneSpectrum R ≃ MaximalSpectrum R where
-  toFun v := ⟨v.asIdeal, dimensionLEOne v.asIdeal v.ne_bot v.isPrime⟩
+  toFun v := ⟨v.asIdeal, v.isPrime.isMaximal v.ne_bot⟩
   invFun v :=
     ⟨v.asIdeal, v.IsMaximal.isPrime, Ring.ne_bot_of_isMaximal_of_not_isField v.IsMaximal hR⟩
   left_inv := fun ⟨_, _, _⟩ => rfl
@@ -1029,7 +1029,7 @@ theorem iInf_localization_eq_bot [Algebra R K] [hK : IsFractionRing R K] :
     exact fun _ => Algebra.mem_bot.mpr ⟨algebra_map_inv x, algebra_map_right_inv x⟩
   all_goals rw [← MaximalSpectrum.iInf_localization_eq_bot, Algebra.mem_iInf]
   · exact fun hx ⟨v, hv⟩ => hx ((equivMaximalSpectrum hR).symm ⟨v, hv⟩)
-  · exact fun hx ⟨v, hv, hbot⟩ => hx ⟨v, dimensionLEOne v hbot hv⟩
+  · exact fun hx ⟨v, hv, hbot⟩ => hx ⟨v, hv.isMaximal hbot⟩
 #align is_dedekind_domain.height_one_spectrum.infi_localization_eq_bot IsDedekindDomain.HeightOneSpectrum.iInf_localization_eq_bot
 
 end HeightOneSpectrum
@@ -1192,9 +1192,9 @@ open scoped BigOperators
 
 variable {R}
 
-theorem Ring.DimensionLeOne.prime_le_prime_iff_eq (h : Ring.DimensionLEOne R) {P Q : Ideal R}
+theorem Ring.DimensionLeOne.prime_le_prime_iff_eq [Ring.DimensionLEOne R] {P Q : Ideal R}
     [hP : P.IsPrime] [hQ : Q.IsPrime] (hP0 : P ≠ ⊥) : P ≤ Q ↔ P = Q :=
-  ⟨(h P hP0 hP).eq_of_le hQ.ne_top, Eq.le⟩
+  ⟨(hP.isMaximal hP0).eq_of_le hQ.ne_top, Eq.le⟩
 #align ring.dimension_le_one.prime_le_prime_iff_eq Ring.DimensionLeOne.prime_le_prime_iff_eq
 
 theorem Ideal.coprime_of_no_prime_ge {I J : Ideal R} (h : ∀ P, I ≤ P → J ≤ P → ¬IsPrime P) :
@@ -1290,9 +1290,9 @@ theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type _} (s : Finset ι) (f
   haveI := Ideal.isPrime_of_prime (prime b (Finset.mem_insert_of_mem hb))
   refine coprime a (Finset.mem_insert_self a s) b (Finset.mem_insert_of_mem hb) ?_ ?_
   · rintro rfl; contradiction
-  · refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq IsDedekindDomain.dimensionLEOne ?_).mp
+  · refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
       (Ideal.le_of_pow_le_prime hPa)).trans
-      ((Ring.DimensionLeOne.prime_le_prime_iff_eq IsDedekindDomain.dimensionLEOne ?_).mp
+      ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
       (Ideal.le_of_pow_le_prime hPb)).symm
     exact (prime a (Finset.mem_insert_self a s)).ne_zero
     exact (prime b (Finset.mem_insert_of_mem hb)).ne_zero
@@ -1314,9 +1314,9 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type _} [Fintyp
       haveI := Ideal.isPrime_of_prime (prime i)
       haveI := Ideal.isPrime_of_prime (prime j)
       refine coprime i j hij ?_
-      refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq IsDedekindDomain.dimensionLEOne ?_).mp
+      refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
         (Ideal.le_of_pow_le_prime hPi)).trans
-        ((Ring.DimensionLeOne.prime_le_prime_iff_eq IsDedekindDomain.dimensionLEOne ?_).mp
+        ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
           (Ideal.le_of_pow_le_prime hPj)).symm
       exact (prime i).ne_zero
       exact (prime j).ne_zero)

• 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 Kenji Nakagawa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-
-! This file was ported from Lean 3 source module ring_theory.dedekind_domain.ideal
-! leanprover-community/mathlib commit 2bbc7e3884ba234309d2a43b19144105a753292e
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
 import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
@@ -17,6 +12,8 @@ import Mathlib.RingTheory.FractionalIdeal
 import Mathlib.RingTheory.PrincipalIdealDomain
 import Mathlib.RingTheory.ChainOfDivisors
 
+#align_import ring_theory.dedekind_domain.ideal from "leanprover-community/mathlib"@"2bbc7e3884ba234309d2a43b19144105a753292e"
+
 /-!
 # Dedekind domains and ideals
 

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

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

@@ -328,7 +328,7 @@ theorem dimensionLEOne : DimensionLEOne A := by
       _ = M := ?_
     · rw [mul_assoc, ← mul_assoc (P : FractionalIdeal A⁰ (FractionRing A)), h.mul_inv_eq_one P'_ne,
       one_mul, h.inv_mul_eq_one M'_ne]
-    · rw [← mul_assoc  (P : FractionalIdeal A⁰ (FractionRing A)), h.mul_inv_eq_one P'_ne, one_mul]
+    · rw [← mul_assoc (P : FractionalIdeal A⁰ (FractionRing A)), h.mul_inv_eq_one P'_ne, one_mul]
   -- Suppose we have `x ∈ M⁻¹ * P`, then in fact `x = algebraMap _ _ y` for some `y`.
   intro x hx
   have le_one : (M⁻¹ : FractionalIdeal A⁰ (FractionRing A)) * P ≤ 1 := by
@@ -491,10 +491,10 @@ We will use this to show that nonzero fractional ideals are invertible,
 and finally conclude that fractional ideals in a Dedekind domain form a group with zero.
 -/
 theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ ⊥) :
-    I * (I : FractionalIdeal A⁰ K)⁻¹  = 1 := by
+    I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
   -- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.
   apply mul_inv_cancel_of_le_one hI0
-  by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹  = 0
+  by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹ = 0
   · rw [hJ0, inv_zero']; exact zero_le _
   intro x hx
   -- In particular, we'll show all `x ∈ J⁻¹` are integral.
@@ -668,13 +668,13 @@ theorem Ideal.dvdNotUnit_iff_lt {I J : Ideal A} : DvdNotUnit I J ↔ J < I :=
       (mt Ideal.dvd_iff_le.mp (not_le_of_lt h))⟩
 #align ideal.dvd_not_unit_iff_lt Ideal.dvdNotUnit_iff_lt
 
-instance : WfDvdMonoid (Ideal A)
-    where wellFounded_dvdNotUnit := by  {
-  have : WellFounded ((· > ·) : Ideal A → Ideal A → Prop) :=
-    isNoetherian_iff_wellFounded.mp (isNoetherianRing_iff.mp IsDedekindDomain.isNoetherianRing)
-  convert this
-  ext
-  rw [Ideal.dvdNotUnit_iff_lt] }
+instance : WfDvdMonoid (Ideal A) where
+  wellFounded_dvdNotUnit := by
+    have : WellFounded ((· > ·) : Ideal A → Ideal A → Prop) :=
+      isNoetherian_iff_wellFounded.mp (isNoetherianRing_iff.mp IsDedekindDomain.isNoetherianRing)
+    convert this
+    ext
+    rw [Ideal.dvdNotUnit_iff_lt]
 
 instance Ideal.uniqueFactorizationMonoid : UniqueFactorizationMonoid (Ideal A) :=
   { irreducible_iff_prime := by
@@ -867,7 +867,7 @@ instance : NormalizedGCDMonoid (Ideal A) :=
     lcm_zero_left := fun _ => by simp only [zero_eq_bot, bot_inf_eq]
     lcm_zero_right := fun _ => by simp only [zero_eq_bot, inf_bot_eq]
     gcd_mul_lcm := fun _ _ => by rw [associated_iff_eq, sup_mul_inf]
-    normalize_gcd := fun _ _ =>  normalize_eq _
+    normalize_gcd := fun _ _ => normalize_eq _
     normalize_lcm := fun _ _ => normalize_eq _ }
 
 -- In fact, any lawful gcd and lcm would equal sup and inf respectively.

• depends on: https://github.com/leanprover/std4/pull/163
• depends on: #5584
• depends on: #5715
• depends on: #5729
• depends on: https://github.com/leanprover/std4/pull/173

Co-authored-by: Komyyy <pol_tta@outlook.jp> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -1027,8 +1027,8 @@ theorem iInf_localization_eq_bot [Algebra R K] [hK : IsFractionRing R K] :
   constructor
   by_cases hR : IsField R
   · rcases Function.bijective_iff_has_inverse.mp
-      (IsField.localization_map_bijective (flip nonZeroDivisors.ne_zero rfl : 0 ∉ R⁰) hR) with
-      ⟨algebra_map_inv, _, algebra_map_right_inv⟩
+      (IsField.localization_map_bijective (Rₘ := K) (flip nonZeroDivisors.ne_zero rfl : 0 ∉ R⁰) hR)
+      with ⟨algebra_map_inv, _, algebra_map_right_inv⟩
     exact fun _ => Algebra.mem_bot.mpr ⟨algebra_map_inv x, algebra_map_right_inv x⟩
   all_goals rw [← MaximalSpectrum.iInf_localization_eq_bot, Algebra.mem_iInf]
   · exact fun hx ⟨v, hv⟩ => hx ((equivMaximalSpectrum hR).symm ⟨v, hv⟩)

• Also remove most superfluous parentheses around big operators (∑, ∏ and variants).
• roughly the used regex: ([^a-zA-Zα-ωΑ-Ω'𝓝ℳ₀𝕂ₛ)]) \(([∑∏][^()∑∏]*,[^()∑∏:]*)\) ([⊂⊆=<≤]) replaced by $1 $2 $3

@@ -1262,7 +1262,7 @@ theorem Ideal.pow_le_prime_iff {I P : Ideal R} [_hP : P.IsPrime] {n : ℕ} (hn :
 #align ideal.pow_le_prime_iff Ideal.pow_le_prime_iff
 
 theorem Ideal.prod_le_prime {ι : Type _} {s : Finset ι} {f : ι → Ideal R} {P : Ideal R}
-    [hP : P.IsPrime] : (∏ i in s, f i) ≤ P ↔ ∃ i ∈ s, f i ≤ P := by
+    [hP : P.IsPrime] : ∏ i in s, f i ≤ P ↔ ∃ i ∈ s, f i ≤ P := by
   by_cases hP0 : P = ⊥
   · simp only [hP0, le_bot_iff]
     rw [← Ideal.zero_eq_bot, Finset.prod_eq_zero_iff]
@@ -1305,7 +1305,7 @@ theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type _} (s : Finset ι) (f
 `∏ i, P i ^ e i`, then `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`. -/
 noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type _} [Fintype ι] (I : Ideal R)
     (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i, Prime (P i)) (coprime : ∀ i j, i ≠ j → P i ≠ P j)
-    (prod_eq : (∏ i, P i ^ e i) = I) : R ⧸ I ≃+* ∀ i, R ⧸ P i ^ e i :=
+    (prod_eq : ∏ i, P i ^ e i = I) : R ⧸ I ≃+* ∀ i, R ⧸ P i ^ e i :=
   (Ideal.quotEquivOfEq
     (by
       simp only [← prod_eq, Finset.inf_eq_iInf, Finset.mem_univ, ciInf_pos,
@@ -1366,7 +1366,7 @@ the product to a finite subset `s` of a potentially infinite indexing type `ι`.
 noncomputable def IsDedekindDomain.quotientEquivPiOfFinsetProdEq {ι : Type _} {s : Finset ι}
     (I : Ideal R) (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
     (coprime : ∀ (i) (_ : i ∈ s) (j) (_ : j ∈ s), i ≠ j → P i ≠ P j)
-    (prod_eq : (∏ i in s, P i ^ e i) = I) : R ⧸ I ≃+* ∀ i : s, R ⧸ P i ^ e i :=
+    (prod_eq : ∏ i in s, P i ^ e i = I) : R ⧸ I ≃+* ∀ i : s, R ⧸ P i ^ e i :=
   IsDedekindDomain.quotientEquivPiOfProdEq I (fun i : s => P i) (fun i : s => e i)
     (fun i => prime i i.2) (fun i j h => coprime i i.2 j j.2 (Subtype.coe_injective.ne h))
     (_root_.trans (Finset.prod_coe_sort s fun i => P i ^ e i) prod_eq)

Co-authored-by: Chris Hughes <chrishughes24@gmail.com>

@@ -982,20 +982,20 @@ variable [IsDomain R] [IsDedekindDomain R]
 @[ext, nolint unusedArguments]
 structure HeightOneSpectrum where
   asIdeal : Ideal R
-  IsPrime : asIdeal.IsPrime
+  isPrime : asIdeal.IsPrime
   ne_bot : asIdeal ≠ ⊥
 #align is_dedekind_domain.height_one_spectrum IsDedekindDomain.HeightOneSpectrum
 
-attribute [instance] HeightOneSpectrum.IsPrime
+attribute [instance] HeightOneSpectrum.isPrime
 
 variable (v : HeightOneSpectrum R) {R}
 
 namespace HeightOneSpectrum
 
-instance isMaximal : v.asIdeal.IsMaximal := dimensionLEOne v.asIdeal v.ne_bot v.IsPrime
+instance isMaximal : v.asIdeal.IsMaximal := dimensionLEOne v.asIdeal v.ne_bot v.isPrime
 #align is_dedekind_domain.height_one_spectrum.is_maximal IsDedekindDomain.HeightOneSpectrum.isMaximal
 
-theorem prime : Prime v.asIdeal := Ideal.prime_of_isPrime v.ne_bot v.IsPrime
+theorem prime : Prime v.asIdeal := Ideal.prime_of_isPrime v.ne_bot v.isPrime
 #align is_dedekind_domain.height_one_spectrum.prime IsDedekindDomain.HeightOneSpectrum.prime
 
 theorem irreducible : Irreducible v.asIdeal :=
@@ -1008,7 +1008,7 @@ theorem associates_irreducible : Irreducible <| Associates.mk v.asIdeal :=
 
 /-- An equivalence between the height one and maximal spectra for rings of Krull dimension 1. -/
 def equivMaximalSpectrum (hR : ¬IsField R) : HeightOneSpectrum R ≃ MaximalSpectrum R where
-  toFun v := ⟨v.asIdeal, dimensionLEOne v.asIdeal v.ne_bot v.IsPrime⟩
+  toFun v := ⟨v.asIdeal, dimensionLEOne v.asIdeal v.ne_bot v.isPrime⟩
   invFun v :=
     ⟨v.asIdeal, v.IsMaximal.isPrime, Ring.ne_bot_of_isMaximal_of_not_isField v.IsMaximal hR⟩
   left_inv := fun ⟨_, _, _⟩ => rfl

These are all doc fixes

@@ -723,7 +723,7 @@ theorem Ideal.prime_iff_isPrime {P : Ideal A} (hP : P ≠ ⊥) : Prime P ↔ IsP
   ⟨Ideal.isPrime_of_prime, Ideal.prime_of_isPrime hP⟩
 #align ideal.prime_iff_is_prime Ideal.prime_iff_isPrime
 
-/-- In a Dedekind domain, the the prime ideals are the zero ideal together with the prime elements
+/-- In a Dedekind domain, the prime ideals are the zero ideal together with the prime elements
 of the monoid with zero `Ideal A`. -/
 theorem Ideal.isPrime_iff_bot_or_prime {P : Ideal A} : IsPrime P ↔ P = ⊥ ∨ Prime P :=
   ⟨fun hp => (eq_or_ne P ⊥).imp_right fun hp0 => Ideal.prime_of_isPrime hp0 hp, fun hp =>

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -309,7 +309,7 @@ theorem integrallyClosed : IsIntegrallyClosed A := by
 
 open Ring
 
-theorem dimensionLeOne : DimensionLEOne A := by
+theorem dimensionLEOne : DimensionLEOne A := by
   -- We're going to show that `P` is maximal because any (maximal) ideal `M`
   -- that is strictly larger would be `⊤`.
   rintro P P_ne hP
@@ -344,12 +344,12 @@ theorem dimensionLeOne : DimensionLEOne A := by
   rw [IsFractionRing.injective A (FractionRing A) zy_eq] at hzy
   -- But `P` is a prime ideal, so `z ∉ P` implies `y ∈ P`, as desired.
   exact mem_coeIdeal_of_mem A⁰ (Or.resolve_left (hP.mem_or_mem hzy) hzp)
-#align is_dedekind_domain_inv.dimension_le_one IsDedekindDomainInv.dimensionLeOne
+#align is_dedekind_domain_inv.dimension_le_one IsDedekindDomainInv.dimensionLEOne
 
 /-- Showing one side of the equivalence between the definitions
 `IsDedekindDomainInv` and `IsDedekindDomain` of Dedekind domains. -/
 theorem isDedekindDomain : IsDedekindDomain A :=
-  ⟨h.isNoetherianRing, h.dimensionLeOne, h.integrallyClosed⟩
+  ⟨h.isNoetherianRing, h.dimensionLEOne, h.integrallyClosed⟩
 #align is_dedekind_domain_inv.is_dedekind_domain IsDedekindDomainInv.isDedekindDomain
 
 end IsDedekindDomainInv
@@ -384,7 +384,7 @@ theorem exists_multiset_prod_cons_le_and_prod_not_le [IsDedekindDomain A] (hNF :
       rwa [Ne.def, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
         this] at hprodZ
     -- By maximality of `P` and `M`, we have that `P ≤ M` implies `P = M`.
-    have hPM' := (IsDedekindDomain.dimensionLeOne _ hP0 P.IsPrime).eq_of_le hM.ne_top hPM
+    have hPM' := (IsDedekindDomain.dimensionLEOne _ hP0 P.IsPrime).eq_of_le hM.ne_top hPM
     subst hPM'
     -- By minimality of `Z`, erasing `P` from `Z` is exactly what we need.
     refine ⟨Z.erase P, ?_, ?_⟩
@@ -992,7 +992,7 @@ variable (v : HeightOneSpectrum R) {R}
 
 namespace HeightOneSpectrum
 
-instance isMaximal : v.asIdeal.IsMaximal := dimensionLeOne v.asIdeal v.ne_bot v.IsPrime
+instance isMaximal : v.asIdeal.IsMaximal := dimensionLEOne v.asIdeal v.ne_bot v.IsPrime
 #align is_dedekind_domain.height_one_spectrum.is_maximal IsDedekindDomain.HeightOneSpectrum.isMaximal
 
 theorem prime : Prime v.asIdeal := Ideal.prime_of_isPrime v.ne_bot v.IsPrime
@@ -1008,7 +1008,7 @@ theorem associates_irreducible : Irreducible <| Associates.mk v.asIdeal :=
 
 /-- An equivalence between the height one and maximal spectra for rings of Krull dimension 1. -/
 def equivMaximalSpectrum (hR : ¬IsField R) : HeightOneSpectrum R ≃ MaximalSpectrum R where
-  toFun v := ⟨v.asIdeal, dimensionLeOne v.asIdeal v.ne_bot v.IsPrime⟩
+  toFun v := ⟨v.asIdeal, dimensionLEOne v.asIdeal v.ne_bot v.IsPrime⟩
   invFun v :=
     ⟨v.asIdeal, v.IsMaximal.isPrime, Ring.ne_bot_of_isMaximal_of_not_isField v.IsMaximal hR⟩
   left_inv := fun ⟨_, _, _⟩ => rfl
@@ -1032,7 +1032,7 @@ theorem iInf_localization_eq_bot [Algebra R K] [hK : IsFractionRing R K] :
     exact fun _ => Algebra.mem_bot.mpr ⟨algebra_map_inv x, algebra_map_right_inv x⟩
   all_goals rw [← MaximalSpectrum.iInf_localization_eq_bot, Algebra.mem_iInf]
   · exact fun hx ⟨v, hv⟩ => hx ((equivMaximalSpectrum hR).symm ⟨v, hv⟩)
-  · exact fun hx ⟨v, hv, hbot⟩ => hx ⟨v, dimensionLeOne v hbot hv⟩
+  · exact fun hx ⟨v, hv, hbot⟩ => hx ⟨v, dimensionLEOne v hbot hv⟩
 #align is_dedekind_domain.height_one_spectrum.infi_localization_eq_bot IsDedekindDomain.HeightOneSpectrum.iInf_localization_eq_bot
 
 end HeightOneSpectrum
@@ -1293,9 +1293,9 @@ theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type _} (s : Finset ι) (f
   haveI := Ideal.isPrime_of_prime (prime b (Finset.mem_insert_of_mem hb))
   refine coprime a (Finset.mem_insert_self a s) b (Finset.mem_insert_of_mem hb) ?_ ?_
   · rintro rfl; contradiction
-  · refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq IsDedekindDomain.dimensionLeOne ?_).mp
+  · refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq IsDedekindDomain.dimensionLEOne ?_).mp
       (Ideal.le_of_pow_le_prime hPa)).trans
-      ((Ring.DimensionLeOne.prime_le_prime_iff_eq IsDedekindDomain.dimensionLeOne ?_).mp
+      ((Ring.DimensionLeOne.prime_le_prime_iff_eq IsDedekindDomain.dimensionLEOne ?_).mp
       (Ideal.le_of_pow_le_prime hPb)).symm
     exact (prime a (Finset.mem_insert_self a s)).ne_zero
     exact (prime b (Finset.mem_insert_of_mem hb)).ne_zero
@@ -1317,9 +1317,9 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type _} [Fintyp
       haveI := Ideal.isPrime_of_prime (prime i)
       haveI := Ideal.isPrime_of_prime (prime j)
       refine coprime i j hij ?_
-      refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq IsDedekindDomain.dimensionLeOne ?_).mp
+      refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq IsDedekindDomain.dimensionLEOne ?_).mp
         (Ideal.le_of_pow_le_prime hPi)).trans
-        ((Ring.DimensionLeOne.prime_le_prime_iff_eq IsDedekindDomain.dimensionLeOne ?_).mp
+        ((Ring.DimensionLeOne.prime_le_prime_iff_eq IsDedekindDomain.dimensionLEOne ?_).mp
           (Ideal.le_of_pow_le_prime hPj)).symm
       exact (prime i).ne_zero
       exact (prime j).ne_zero)

• An instance from RingTheory.FractionalIdeal was used explicitly so I had to give it a name
• At three places, an equiv is defined and the lemmas produced by [simps] (or [simps!]) make the simpNF linter very unhappy so I commented out the [simps] and left a porting note