fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
IsCoprime.prod_left_iff : IsCoprime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsCoprime (s i) x := by
classical
refine Finset.induction_on t (iff_of_true isCoprime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsCoprime.mul_left_iff, ih, Finset.forall_mem_insert] | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsCoprime.prod_left_iff | null |
IsCoprime.prod_right_iff : IsCoprime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsCoprime x (s i) := by
simpa only [isCoprime_comm] using IsCoprime.prod_left_iff (R := R) | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsCoprime.prod_right_iff | null |
IsCoprime.of_prod_left (H1 : IsCoprime (∏ i ∈ t, s i) x) (i : I) (hit : i ∈ t) :
IsCoprime (s i) x :=
IsCoprime.prod_left_iff.1 H1 i hit | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsCoprime.of_prod_left | null |
IsCoprime.of_prod_right (H1 : IsCoprime x (∏ i ∈ t, s i)) (i : I) (hit : i ∈ t) :
IsCoprime x (s i) :=
IsCoprime.prod_right_iff.1 H1 i hit | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsCoprime.of_prod_right | null |
Finset.prod_dvd_of_coprime
(Hs : (t : Set I).Pairwise (IsCoprime on s)) (Hs1 : (∀ i ∈ t, s i ∣ z)) :
(∏ x ∈ t, s x) ∣ z := by
classical
induction t using Finset.induction_on with
| empty => simp
| insert a r har ih =>
rw [Finset.prod_insert har]
refine IsCoprime.mul_dvd ?_ ?_ ?_
· refine IsCoprime.prod_right fun i hir ↦ ?_
exact Hs (by simp) (by simp [hir]) (ne_of_mem_of_not_mem hir har).symm
· exact Hs1 a (Finset.mem_insert_self a r)
· refine ih (Hs.mono ?_) fun i hi ↦ Hs1 i <| Finset.mem_insert_of_mem hi
simp only [Finset.coe_insert, Set.subset_insert] | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | Finset.prod_dvd_of_coprime | null |
Fintype.prod_dvd_of_coprime [Fintype I] (Hs : Pairwise (IsCoprime on s))
(Hs1 : ∀ i, s i ∣ z) : (∏ x, s x) ∣ z :=
Finset.prod_dvd_of_coprime (Hs.set_pairwise _) fun i _ ↦ Hs1 i | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | Fintype.prod_dvd_of_coprime | null |
exists_sum_eq_one_iff_pairwise_coprime [DecidableEq I] (h : t.Nonempty) :
(∃ μ : I → R, (∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j) = 1) ↔
Pairwise (IsCoprime on fun i : t ↦ s i) := by
induction h using Finset.Nonempty.cons_induction with
| singleton =>
simp [exists_apply_eq, Pairwise, Function.onFun]
| cons a t hat h ih =>
rw [pairwise_cons']
have mem : ∀ x ∈ t, a ∈ insert a t \ {x} := fun x hx ↦ by
rw [mem_sdiff, mem_singleton]
exact ⟨mem_insert_self _ _, fun ha ↦ hat (ha ▸ hx)⟩
constructor
· rintro ⟨μ, hμ⟩
rw [sum_cons, cons_eq_insert, sdiff_singleton_eq_erase, erase_insert hat] at hμ
refine ⟨ih.mp ⟨Pi.single h.choose (μ a * s h.choose) + μ * fun _ ↦ s a, ?_⟩, fun b hb ↦ ?_⟩
· rw [prod_eq_mul_prod_diff_singleton h.choose_spec, ← mul_assoc, ←
@if_pos _ _ h.choose_spec R (_ * _) 0, ← sum_pi_single', ← sum_add_distrib] at hμ
rw [← hμ, sum_congr rfl]
intro x hx
convert add_mul (R := R) _ _ _ using 2
· by_cases hx : x = h.choose
· rw [hx, Pi.single_eq_same, Pi.single_eq_same]
· rw [Pi.single_eq_of_ne hx, Pi.single_eq_of_ne hx, zero_mul]
· convert (mul_assoc _ _ _).symm
rw [prod_eq_prod_diff_singleton_mul (mem x hx), mul_comm, sdiff_sdiff_comm,
sdiff_singleton_eq_erase a, erase_insert hat]
· have : IsCoprime (s b) (s a) :=
⟨μ a * ∏ i ∈ t \ {b}, s i, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j, ?_⟩
· exact ⟨this.symm, this⟩
rw [mul_assoc, ← prod_eq_prod_diff_singleton_mul hb, sum_mul, ← hμ, sum_congr rfl]
intro x hx
rw [mul_assoc]
congr
rw [prod_eq_prod_diff_singleton_mul (mem x hx) _]
congr 2
rw [sdiff_sdiff_comm, sdiff_singleton_eq_erase a, erase_insert hat]
· rintro ⟨hs, Hb⟩
obtain ⟨μ, hμ⟩ := ih.mpr hs
obtain ⟨u, v, huv⟩ := IsCoprime.prod_left fun b hb ↦ (Hb b hb).right
use fun i ↦ if i = a then u else v * μ i
have hμ' : (∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a)) = v * s a := by
rw [← mul_sum, ← sum_mul, hμ, one_mul]
rw [sum_cons, cons_eq_insert, sdiff_singleton_eq_erase, erase_insert hat]
simp only [↓reduceIte, ite_mul]
rw [← huv, ← hμ', sum_congr rfl]
intro x hx
rw [mul_assoc, if_neg fun ha : x = a ↦ hat (ha.casesOn hx)]
rw [mul_assoc]
congr
rw [prod_eq_prod_diff_singleton_mul (mem x hx) _]
congr 2
... | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | exists_sum_eq_one_iff_pairwise_coprime | null |
exists_sum_eq_one_iff_pairwise_coprime' [Fintype I] [Nonempty I] [DecidableEq I] :
(∃ μ : I → R, (∑ i : I, μ i * ∏ j ∈ {i}ᶜ, s j) = 1) ↔ Pairwise (IsCoprime on s) := by
convert exists_sum_eq_one_iff_pairwise_coprime Finset.univ_nonempty (s := s) using 1
simp only [pairwise_subtype_iff_pairwise_finset', coe_univ, Set.pairwise_univ] | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | exists_sum_eq_one_iff_pairwise_coprime' | null |
pairwise_coprime_iff_coprime_prod [DecidableEq I] :
Pairwise (IsCoprime on fun i : t ↦ s i) ↔ ∀ i ∈ t, IsCoprime (s i) (∏ j ∈ t \ {i}, s j) := by
rw [Finset.pairwise_subtype_iff_pairwise_finset']
refine ⟨fun hp i hi ↦ IsCoprime.prod_right_iff.mpr fun j hj ↦ ?_, fun hp ↦ ?_⟩
· rw [Finset.mem_sdiff, Finset.mem_singleton] at hj
exact (hp hj.1 hi hj.2).symm
· rintro i hi j hj h
apply IsCoprime.prod_right_iff.mp (hp i hi)
exact Finset.mem_sdiff.mpr ⟨hj, fun f ↦ h (Finset.mem_singleton.mp f).symm⟩
variable {m n : ℕ} | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | pairwise_coprime_iff_coprime_prod | null |
IsCoprime.pow_left (H : IsCoprime x y) : IsCoprime (x ^ m) y := by
rw [← Finset.card_range m, ← Finset.prod_const]
exact IsCoprime.prod_left fun _ _ ↦ H | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsCoprime.pow_left | null |
IsCoprime.pow_right (H : IsCoprime x y) : IsCoprime x (y ^ n) := by
rw [← Finset.card_range n, ← Finset.prod_const]
exact IsCoprime.prod_right fun _ _ ↦ H | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsCoprime.pow_right | null |
IsCoprime.pow (H : IsCoprime x y) : IsCoprime (x ^ m) (y ^ n) :=
H.pow_left.pow_right | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsCoprime.pow | null |
IsCoprime.pow_left_iff (hm : 0 < m) : IsCoprime (x ^ m) y ↔ IsCoprime x y := by
refine ⟨fun h ↦ ?_, IsCoprime.pow_left⟩
rw [← Finset.card_range m, ← Finset.prod_const] at h
exact h.of_prod_left 0 (Finset.mem_range.mpr hm) | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsCoprime.pow_left_iff | null |
IsCoprime.pow_right_iff (hm : 0 < m) : IsCoprime x (y ^ m) ↔ IsCoprime x y :=
isCoprime_comm.trans <| (IsCoprime.pow_left_iff hm).trans <| isCoprime_comm | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsCoprime.pow_right_iff | null |
IsCoprime.pow_iff (hm : 0 < m) (hn : 0 < n) : IsCoprime (x ^ m) (y ^ n) ↔ IsCoprime x y :=
(IsCoprime.pow_left_iff hm).trans <| IsCoprime.pow_right_iff hn | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsCoprime.pow_iff | null |
IsRelPrime.prod_left : (∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x := by
classical
refine Finset.induction_on t (fun _ ↦ isRelPrime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2) | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsRelPrime.prod_left | null |
IsRelPrime.prod_right : (∀ i ∈ t, IsRelPrime x (s i)) → IsRelPrime x (∏ i ∈ t, s i) := by
simpa only [isRelPrime_comm] using IsRelPrime.prod_left (α := α) | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsRelPrime.prod_right | null |
IsRelPrime.prod_left_iff : IsRelPrime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsRelPrime (s i) x := by
classical
refine Finset.induction_on t (iff_of_true isRelPrime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsRelPrime.mul_left_iff, ih, Finset.forall_mem_insert] | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsRelPrime.prod_left_iff | null |
IsRelPrime.prod_right_iff : IsRelPrime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsRelPrime x (s i) := by
simpa only [isRelPrime_comm] using IsRelPrime.prod_left_iff (α := α) | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsRelPrime.prod_right_iff | null |
IsRelPrime.of_prod_left (H1 : IsRelPrime (∏ i ∈ t, s i) x) (i : I) (hit : i ∈ t) :
IsRelPrime (s i) x :=
IsRelPrime.prod_left_iff.1 H1 i hit | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsRelPrime.of_prod_left | null |
IsRelPrime.of_prod_right (H1 : IsRelPrime x (∏ i ∈ t, s i)) (i : I) (hit : i ∈ t) :
IsRelPrime x (s i) :=
IsRelPrime.prod_right_iff.1 H1 i hit | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | IsRelPrime.of_prod_right | null |
Finset.prod_dvd_of_isRelPrime :
(t : Set I).Pairwise (IsRelPrime on s) → (∀ i ∈ t, s i ∣ z) → (∏ x ∈ t, s x) ∣ z := by
classical
exact Finset.induction_on t (fun _ _ ↦ one_dvd z)
(by
intro a r har ih Hs Hs1
rw [Finset.prod_insert har]
have aux1 : a ∈ (↑(insert a r) : Set I) := Finset.mem_insert_self a r
refine
(IsRelPrime.prod_right fun i hir ↦
Hs aux1 (Finset.mem_insert_of_mem hir) <| by
rintro rfl
exact har hir).mul_dvd
(Hs1 a aux1) (ih (Hs.mono ?_) fun i hi ↦ Hs1 i <| Finset.mem_insert_of_mem hi)
simp only [Finset.coe_insert, Set.subset_insert]) | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | Finset.prod_dvd_of_isRelPrime | null |
Fintype.prod_dvd_of_isRelPrime [Fintype I] (Hs : Pairwise (IsRelPrime on s))
(Hs1 : ∀ i, s i ∣ z) : (∏ x, s x) ∣ z :=
Finset.prod_dvd_of_isRelPrime (Hs.set_pairwise _) fun i _ ↦ Hs1 i | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | Fintype.prod_dvd_of_isRelPrime | null |
pairwise_isRelPrime_iff_isRelPrime_prod [DecidableEq I] :
Pairwise (IsRelPrime on fun i : t ↦ s i) ↔ ∀ i ∈ t, IsRelPrime (s i) (∏ j ∈ t \ {i}, s j) := by
refine ⟨fun hp i hi ↦ IsRelPrime.prod_right_iff.mpr fun j hj ↦ ?_, fun hp ↦ ?_⟩
· rw [Finset.mem_sdiff, Finset.mem_singleton] at hj
obtain ⟨hj, ji⟩ := hj
exact @hp ⟨i, hi⟩ ⟨j, hj⟩ fun h ↦ ji (congrArg Subtype.val h).symm
· rintro ⟨i, hi⟩ ⟨j, hj⟩ h
apply IsRelPrime.prod_right_iff.mp (hp i hi)
grind | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | pairwise_isRelPrime_iff_isRelPrime_prod | null |
pow_left (H : IsRelPrime x y) : IsRelPrime (x ^ m) y := by
rw [← Finset.card_range m, ← Finset.prod_const]
exact IsRelPrime.prod_left fun _ _ ↦ H | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | pow_left | null |
pow_right (H : IsRelPrime x y) : IsRelPrime x (y ^ n) := by
rw [← Finset.card_range n, ← Finset.prod_const]
exact IsRelPrime.prod_right fun _ _ ↦ H | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | pow_right | null |
pow (H : IsRelPrime x y) : IsRelPrime (x ^ m) (y ^ n) :=
H.pow_left.pow_right | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | pow | null |
pow_left_iff (hm : 0 < m) : IsRelPrime (x ^ m) y ↔ IsRelPrime x y := by
refine ⟨fun h ↦ ?_, IsRelPrime.pow_left⟩
rw [← Finset.card_range m, ← Finset.prod_const] at h
exact h.of_prod_left 0 (Finset.mem_range.mpr hm) | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | pow_left_iff | null |
pow_right_iff (hm : 0 < m) : IsRelPrime x (y ^ m) ↔ IsRelPrime x y :=
isRelPrime_comm.trans <| (IsRelPrime.pow_left_iff hm).trans <| isRelPrime_comm | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | pow_right_iff | null |
pow_iff (hm : 0 < m) (hn : 0 < n) :
IsRelPrime (x ^ m) (y ^ n) ↔ IsRelPrime x y :=
(IsRelPrime.pow_left_iff hm).trans (IsRelPrime.pow_right_iff hn) | theorem | RingTheory | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.RingTheory.Coprime.Basic"
] | Mathlib/RingTheory/Coprime/Lemmas.lean | pow_iff | null |
intValuationDef (r : R) : ℤᵐ⁰ :=
if r = 0 then 0
else
↑(Multiplicative.ofAdd
(-(Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {r} : Ideal R)).factors : ℤ)) | def | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuationDef | The additive `v`-adic valuation of `r ∈ R` is the exponent of `v` in the factorization of the
ideal `(r)`, if `r` is nonzero, or infinity, if `r = 0`. `intValuationDef` is the corresponding
multiplicative valuation. |
intValuationDef_if_pos {r : R} (hr : r = 0) : v.intValuationDef r = 0 :=
if_pos hr
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuationDef_if_pos | null |
intValuationDef_zero : v.intValuationDef 0 = 0 :=
if_pos rfl
open scoped Classical in | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuationDef_zero | null |
intValuationDef_if_neg {r : R} (hr : r ≠ 0) :
v.intValuationDef r =
Multiplicative.ofAdd
(-(Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {r} : Ideal R)).factors : ℤ) :=
if_neg hr | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuationDef_if_neg | null |
intValuation.map_zero' : v.intValuationDef 0 = 0 :=
v.intValuationDef_if_pos (Eq.refl 0) | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation.map_zero' | The `v`-adic valuation of `0 : R` equals 0. |
intValuation.map_one' : v.intValuationDef 1 = 1 := by
classical
rw [v.intValuationDef_if_neg (zero_ne_one.symm : (1 : R) ≠ 0), Ideal.span_singleton_one, ←
Ideal.one_eq_top, Associates.mk_one, Associates.factors_one,
Associates.count_zero (by apply v.associates_irreducible), Int.ofNat_zero, neg_zero, ofAdd_zero,
WithZero.coe_one] | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation.map_one' | The `v`-adic valuation of `1 : R` equals 1. |
intValuation.map_mul' (x y : R) :
v.intValuationDef (x * y) = v.intValuationDef x * v.intValuationDef y := by
classical
simp only [intValuationDef]
by_cases hx : x = 0
· rw [hx, zero_mul, if_pos (Eq.refl _), zero_mul]
· by_cases hy : y = 0
· rw [hy, mul_zero, if_pos (Eq.refl _), mul_zero]
· rw [if_neg hx, if_neg hy, if_neg (mul_ne_zero hx hy), ← WithZero.coe_mul, WithZero.coe_inj, ←
ofAdd_add, ← Ideal.span_singleton_mul_span_singleton, ← Associates.mk_mul_mk, ← neg_add,
Associates.count_mul (by apply Associates.mk_ne_zero'.mpr hx)
(by apply Associates.mk_ne_zero'.mpr hy) (by apply v.associates_irreducible)]
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation.map_mul' | The `v`-adic valuation of a product equals the product of the valuations. |
intValuation.le_max_iff_min_le {a b c : ℕ} :
Multiplicative.ofAdd (-c : ℤ) ≤
max (Multiplicative.ofAdd (-a : ℤ)) (Multiplicative.ofAdd (-b : ℤ)) ↔
min a b ≤ c := by
rw [le_max_iff, ofAdd_le, ofAdd_le, neg_le_neg_iff, neg_le_neg_iff, Int.ofNat_le, Int.ofNat_le, ←
min_le_iff] | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation.le_max_iff_min_le | null |
intValuation.map_add_le_max' (x y : R) :
v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y) := by
classical
by_cases hx : x = 0
· rw [hx, zero_add]
conv_rhs => rw [intValuationDef, if_pos (Eq.refl _)]
rw [max_eq_right (WithZero.zero_le (v.intValuationDef y))]
· by_cases hy : y = 0
· rw [hy, add_zero]
conv_rhs => rw [max_comm, intValuationDef, if_pos (Eq.refl _)]
rw [max_eq_right (WithZero.zero_le (v.intValuationDef x))]
· by_cases hxy : x + y = 0
· rw [intValuationDef, if_pos hxy]; exact zero_le'
· rw [v.intValuationDef_if_neg hxy, v.intValuationDef_if_neg hx, v.intValuationDef_if_neg hy,
WithZero.le_max_iff, intValuation.le_max_iff_min_le]
set nmin :=
min ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span { x })).factors)
((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span { y })).factors)
have h_dvd_x : x ∈ v.asIdeal ^ nmin := by
rw [← Associates.le_singleton_iff x nmin _,
Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hx) _]
· exact min_le_left _ _
apply v.associates_irreducible
have h_dvd_y : y ∈ v.asIdeal ^ nmin := by
rw [← Associates.le_singleton_iff y nmin _,
Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hy) _]
· exact min_le_right _ _
apply v.associates_irreducible
have h_dvd_xy : Associates.mk v.asIdeal ^ nmin ≤ Associates.mk (Ideal.span {x + y}) := by
rw [Associates.le_singleton_iff]
exact Ideal.add_mem (v.asIdeal ^ nmin) h_dvd_x h_dvd_y
rw [Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hxy) _] at h_dvd_xy
· exact h_dvd_xy
apply v.associates_irreducible | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation.map_add_le_max' | The `v`-adic valuation of a sum is bounded above by the maximum of the valuations. |
intValuation : Valuation R ℤᵐ⁰ where
toFun := v.intValuationDef
map_zero' := intValuation.map_zero' v
map_one' := intValuation.map_one' v
map_mul' := intValuation.map_mul' v
map_add_le_max' := intValuation.map_add_le_max' v | def | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation | The `v`-adic valuation on `R`. |
intValuation_apply {r : R} (v : IsDedekindDomain.HeightOneSpectrum R) :
intValuation v r = intValuationDef v r := rfl
open scoped Classical in | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation_apply | null |
intValuation_def {r : R} :
v.intValuation r = if r = 0 then 0 else
↑(Multiplicative.ofAdd
(-(Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {r} : Ideal R)).factors : ℤ)) :=
rfl
@[deprecated intValuation_apply (since := "2025-04-26")] | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation_def | null |
intValuation_toFun (r : R) :
v.intValuation r = v.intValuationDef r := rfl
open scoped Classical in | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation_toFun | null |
intValuation_if_neg {r : R} (hr : r ≠ 0) :
v.intValuation r =
Multiplicative.ofAdd
(-(Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {r} : Ideal R)).factors : ℤ) :=
intValuationDef_if_neg _ hr | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation_if_neg | null |
intValuation_ne_zero (x : R) (hx : x ≠ 0) : v.intValuation x ≠ 0 := by
rw [v.intValuation_if_neg hx]
exact WithZero.coe_ne_zero | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation_ne_zero | Nonzero elements have nonzero adic valuation. |
intValuation_ne_zero' (x : nonZeroDivisors R) : v.intValuation x ≠ 0 :=
v.intValuation_ne_zero x (nonZeroDivisors.coe_ne_zero x) | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation_ne_zero' | Nonzero divisors have nonzero valuation. |
intValuation_zero_lt (x : nonZeroDivisors R) : 0 < v.intValuation x := by
rw [v.intValuation_if_neg (nonZeroDivisors.coe_ne_zero x)]
exact WithZero.zero_lt_coe _
@[deprecated (since := "2025-05-11")]
alias intValuation_zero_le := intValuation_zero_lt | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation_zero_lt | Nonzero divisors have valuation greater than zero. |
intValuation_le_one (x : R) : v.intValuation x ≤ 1 := by
by_cases hx : x = 0
· rw [hx, Valuation.map_zero]; exact WithZero.zero_le 1
· rw [v.intValuation_if_neg hx, ← WithZero.coe_one, ← ofAdd_zero, WithZero.coe_le_coe, ofAdd_le,
Right.neg_nonpos_iff]
exact Int.natCast_nonneg _ | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation_le_one | The `v`-adic valuation on `R` is bounded above by 1. |
intValuation_lt_one_iff_dvd (r : R) :
v.intValuation r < 1 ↔ v.asIdeal ∣ Ideal.span {r} := by
classical
by_cases hr : r = 0
· simp [hr]
· rw [v.intValuation_if_neg hr, ← WithZero.coe_one, ← ofAdd_zero, WithZero.coe_lt_coe, ofAdd_lt,
neg_lt_zero, ← Int.ofNat_zero, Int.ofNat_lt, zero_lt_iff]
have h : (Ideal.span {r} : Ideal R) ≠ 0 := by
rw [Ne, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]
exact hr
apply Associates.count_ne_zero_iff_dvd h (by apply v.irreducible) | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation_lt_one_iff_dvd | The `v`-adic valuation of `r ∈ R` is less than 1 if and only if `v` divides the ideal `(r)`. |
intValuation_lt_one_iff_mem (r : R) :
v.intValuation r < 1 ↔ r ∈ v.asIdeal := by
rw [intValuation_lt_one_iff_dvd, Ideal.dvd_span_singleton] | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation_lt_one_iff_mem | The `v`-adic valuation of `r ∈ R` is less than 1 if and only if `r ∈ v`. |
intValuation_le_pow_iff_dvd (r : R) (n : ℕ) :
v.intValuation r ≤ Multiplicative.ofAdd (-(n : ℤ)) ↔ v.asIdeal ^ n ∣ Ideal.span {r} := by
classical
by_cases hr : r = 0
· simp_rw [hr, Valuation.map_zero, Ideal.dvd_span_singleton, zero_le', Submodule.zero_mem]
· rw [v.intValuation_if_neg hr, WithZero.coe_le_coe, ofAdd_le, neg_le_neg_iff, Int.ofNat_le,
Ideal.dvd_span_singleton, ← Associates.le_singleton_iff,
Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hr)
(by apply v.associates_irreducible)] | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation_le_pow_iff_dvd | The `v`-adic valuation of `r ∈ R` is less than `Multiplicative.ofAdd (-n)` if and only if
`vⁿ` divides the ideal `(r)`. |
intValuation_le_pow_iff_mem (r : R) (n : ℕ) :
v.intValuation r ≤ Multiplicative.ofAdd (-(n : ℤ)) ↔ r ∈ v.asIdeal ^ n := by
rw [intValuation_le_pow_iff_dvd, Ideal.dvd_span_singleton] | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation_le_pow_iff_mem | The `v`-adic valuation of `r ∈ R` is less than `Multiplicative.ofAdd (-n)` if and only if
`r ∈ vⁿ`. |
intValuation_exists_uniformizer :
∃ π : R, v.intValuation π = Multiplicative.ofAdd (-1 : ℤ) := by
classical
have hv : _root_.Irreducible (Associates.mk v.asIdeal) := v.associates_irreducible
have hlt : v.asIdeal ^ 2 < v.asIdeal := by
rw [← Ideal.dvdNotUnit_iff_lt]
exact
⟨v.ne_bot, v.asIdeal, (not_congr Ideal.isUnit_iff).mpr (Ideal.IsPrime.ne_top v.isPrime),
sq v.asIdeal⟩
obtain ⟨π, mem, notMem⟩ := SetLike.exists_of_lt hlt
have hπ : Associates.mk (Ideal.span {π}) ≠ 0 := by
rw [Associates.mk_ne_zero']
intro h
rw [h] at notMem
exact notMem (Submodule.zero_mem (v.asIdeal ^ 2))
use π
rw [intValuation_if_neg _ (Associates.mk_ne_zero'.mp hπ), WithZero.coe_inj]
apply congr_arg
rw [neg_inj, ← Int.ofNat_one, Int.natCast_inj]
rw [← Ideal.dvd_span_singleton, ← Associates.mk_le_mk_iff_dvd] at mem notMem
rw [← pow_one (Associates.mk v.asIdeal), Associates.prime_pow_dvd_iff_le hπ hv] at mem
rw [Associates.mk_pow, Associates.prime_pow_dvd_iff_le hπ hv, not_le] at notMem
exact Nat.eq_of_le_of_lt_succ mem notMem | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation_exists_uniformizer | There exists `π ∈ R` with `v`-adic valuation `Multiplicative.ofAdd (-1)`. |
intValuation_singleton {r : R} (hr : r ≠ 0) (hv : v.asIdeal = Ideal.span {r}) :
v.intValuation r = Multiplicative.ofAdd (-1 : ℤ) := by
classical
rw [v.intValuation_if_neg hr, ← hv, Associates.count_self, Int.ofNat_one,
ofAdd_neg, WithZero.coe_inv]
apply v.associates_irreducible
/-! ### Adic valuations on the field of fractions `K` -/
variable (K) in | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | intValuation_singleton | The `I`-adic valuation of a generator of `I` equals `(-1 : ℤᵐ⁰)` |
valuation (v : HeightOneSpectrum R) : Valuation K ℤᵐ⁰ :=
v.intValuation.extendToLocalization
(fun r hr => Set.mem_compl <| v.intValuation_ne_zero' ⟨r, hr⟩) K | def | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | valuation | The `v`-adic valuation of `x ∈ K` is the valuation of `r` divided by the valuation of `s`,
where `r` and `s` are chosen so that `x = r/s`. |
valuation_def (x : K) :
v.valuation K x =
v.intValuation.extendToLocalization
(fun r hr => Set.mem_compl (v.intValuation_ne_zero' ⟨r, hr⟩)) K x :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | valuation_def | null |
valuation_of_mk' {r : R} {s : nonZeroDivisors R} :
v.valuation K (IsLocalization.mk' K r s) = v.intValuation r / v.intValuation s := by
rw [valuation_def, Valuation.extendToLocalization_mk', div_eq_mul_inv]
open scoped algebraMap in | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | valuation_of_mk' | The `v`-adic valuation of `r/s ∈ K` is the valuation of `r` divided by the valuation of `s`. |
valuation_of_algebraMap (r : R) : v.valuation K r = v.intValuation r := by
rw [valuation_def, Valuation.extendToLocalization_apply_map_apply]
open scoped algebraMap in
@[deprecated valuation_of_algebraMap (since := "2025-05-11")] | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | valuation_of_algebraMap | The `v`-adic valuation on `K` extends the `v`-adic valuation on `R`. |
valuation_eq_intValuationDef (r : R) : v.valuation K r = v.intValuationDef r :=
Valuation.extendToLocalization_apply_map_apply ..
open scoped algebraMap in | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | valuation_eq_intValuationDef | null |
valuation_le_one (r : R) : v.valuation K r ≤ 1 := by
rw [valuation_of_algebraMap]; exact v.intValuation_le_one r
open scoped algebraMap in | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | valuation_le_one | The `v`-adic valuation on `R` is bounded above by 1. |
valuation_lt_one_iff_dvd (r : R) :
v.valuation K r < 1 ↔ v.asIdeal ∣ Ideal.span {r} := by
rw [valuation_of_algebraMap]; exact v.intValuation_lt_one_iff_dvd r
open scoped algebraMap in | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | valuation_lt_one_iff_dvd | The `v`-adic valuation of `r ∈ R` is less than 1 if and only if `v` divides the ideal `(r)`. |
valuation_lt_one_iff_mem (r : R) :
v.valuation K r < 1 ↔ r ∈ v.asIdeal := by
rw [valuation_of_algebraMap]; exact v.intValuation_lt_one_iff_mem r
variable (K) | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | valuation_lt_one_iff_mem | The `v`-adic valuation of `r ∈ R` is less than 1 if and only if `r ∈ v`. |
valuation_exists_uniformizer : ∃ π : K,
v.valuation K π = Multiplicative.ofAdd (-1 : ℤ) := by
obtain ⟨r, hr⟩ := v.intValuation_exists_uniformizer
use algebraMap R K r
rw [valuation_def, Valuation.extendToLocalization_apply_map_apply]
exact hr | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | valuation_exists_uniformizer | There exists `π ∈ K` with `v`-adic valuation `Multiplicative.ofAdd (-1)`. |
valuation_surjective :
Function.Surjective (v.valuation K) := by
intro x
induction x with
| zero => simp
| coe x =>
induction x with | ofAdd x
obtain ⟨π, hπ⟩ := v.valuation_exists_uniformizer K
refine ⟨π ^ (- x), ?_⟩
simp [hπ, ← WithZero.coe_zpow, ← ofAdd_zsmul] | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | valuation_surjective | null |
valuation_uniformizer_ne_zero : Classical.choose (v.valuation_exists_uniformizer K) ≠ 0 :=
haveI hu := Classical.choose_spec (v.valuation_exists_uniformizer K)
(Valuation.ne_zero_iff _).mp (ne_of_eq_of_ne hu WithZero.coe_ne_zero) | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | valuation_uniformizer_ne_zero | Uniformizers are nonzero. |
mem_integers_of_valuation_le_one (x : K)
(h : ∀ v : HeightOneSpectrum R, v.valuation K x ≤ 1) : x ∈ (algebraMap R K).range := by
obtain ⟨⟨n, d, hd⟩, hx⟩ := IsLocalization.surj (nonZeroDivisors R) x
obtain rfl : x = IsLocalization.mk' K n ⟨d, hd⟩ := IsLocalization.eq_mk'_iff_mul_eq.mpr hx
obtain rfl | hn0 := eq_or_ne n 0
· simp
have hd0 := nonZeroDivisors.ne_zero hd
suffices Ideal.span {d} ∣ (Ideal.span {n} : Ideal R) by
obtain ⟨z, rfl⟩ := Ideal.span_singleton_le_span_singleton.1 (Ideal.le_of_dvd this)
use z
rw [map_mul, mul_comm, mul_eq_mul_left_iff] at hx
exact (hx.resolve_right fun h => by simp [hd0] at h).symm
classical
have ine {r : R} : r ≠ 0 → Ideal.span {r} ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp
rw [← Associates.mk_le_mk_iff_dvd, ← Associates.factors_le, Associates.factors_mk _ (ine hn0),
Associates.factors_mk _ (ine hd0), WithTop.coe_le_coe, Multiset.le_iff_count]
rintro ⟨v, hv⟩
obtain ⟨v, rfl⟩ := Associates.mk_surjective v
have hv' := hv
rw [Associates.irreducible_mk, irreducible_iff_prime] at hv
specialize h ⟨v, Ideal.isPrime_of_prime hv, hv.ne_zero⟩
simp_rw [valuation_of_mk', intValuation_if_neg _ hn0, intValuation_if_neg _ hd0,
← WithZero.coe_div, ← WithZero.coe_one, WithZero.coe_le_coe, Associates.factors_mk _ (ine hn0),
Associates.factors_mk _ (ine hd0), Associates.count_some hv'] at h
simpa using h
/-! ### Completions with respect to adic valuations
Given a Dedekind domain `R` with field of fractions `K` and a maximal ideal `v` of `R`, we define
the completion of `K` with respect to its `v`-adic valuation, denoted `v.adicCompletion`, and its
ring of integers, denoted `v.adicCompletionIntegers`. -/
variable {K} | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | mem_integers_of_valuation_le_one | null |
adicValued : Valued K ℤᵐ⁰ :=
Valued.mk' (v.valuation K) | def | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | adicValued | `K` as a valued field with the `v`-adic valuation. |
adicValued_apply {x : K} : v.adicValued.v x = v.valuation K x :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | adicValued_apply | null |
adicValued_apply' (x : WithVal (v.valuation K)) : v.adicValued.v x = v.valuation K x :=
rfl
variable (K) | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | adicValued_apply' | null |
adicCompletion := (v.valuation K).Completion | abbrev | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | adicCompletion | The completion of `K` with respect to its `v`-adic valuation. |
valuedAdicCompletion_def {x : v.adicCompletion K} : Valued.v x = Valued.extension x :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | valuedAdicCompletion_def | null |
valuedAdicCompletion_surjective :
Function.Surjective (Valued.v : (v.adicCompletion K) → ℤᵐ⁰) :=
Valued.valuedCompletion_surjective_iff.mpr (v.valuation_surjective K) | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | valuedAdicCompletion_surjective | null |
adicCompletionIntegers : ValuationSubring (v.adicCompletion K) :=
Valued.v.valuationSubring | def | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | adicCompletionIntegers | The ring of integers of `adicCompletion`. |
mem_adicCompletionIntegers {x : v.adicCompletion K} :
x ∈ v.adicCompletionIntegers K ↔ Valued.v x ≤ 1 :=
Iff.rfl | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | mem_adicCompletionIntegers | null |
notMem_adicCompletionIntegers {x : v.adicCompletion K} :
x ∉ v.adicCompletionIntegers K ↔ 1 < Valued.v x := by
rw [not_congr <| mem_adicCompletionIntegers R K v]
exact not_le
@[deprecated (since := "2025-05-23")]
alias not_mem_adicCompletionIntegers := notMem_adicCompletionIntegers | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | notMem_adicCompletionIntegers | null |
adicValued.uniformContinuousConstSMul :
UniformContinuousConstSMul S (WithVal <| v.valuation K) := by
refine ⟨fun l ↦ ?_⟩
simp_rw [Algebra.smul_def]
exact (Ring.uniformContinuousConstSMul (WithVal <| v.valuation K)).uniformContinuous_const_smul _
open UniformSpace in | instance | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | adicValued.uniformContinuousConstSMul | null |
coe_smul_adicCompletion (r : S) (x : WithVal (v.valuation K)) :
(↑(r • x) : v.adicCompletion K) = r • (↑x : v.adicCompletion K) :=
UniformSpace.Completion.coe_smul r x | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | coe_smul_adicCompletion | null |
algebraMap_adicCompletion : ⇑(algebraMap S <| v.adicCompletion K) = (↑) ∘ algebraMap S K :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | algebraMap_adicCompletion | null |
coe_algebraMap_mem (r : R) : ↑((algebraMap R K) r) ∈ adicCompletionIntegers K v := by
rw [mem_adicCompletionIntegers]
letI : Valued K ℤᵐ⁰ := adicValued v
dsimp only [adicCompletion]
rw [Valued.valuedCompletion_apply]
exact v.valuation_le_one _ | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | coe_algebraMap_mem | null |
@[simp]
algebraMap_adicCompletionIntegers_apply (r : R) :
algebraMap R (v.adicCompletionIntegers K) r = (algebraMap R K r : v.adicCompletion K) :=
rfl | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | algebraMap_adicCompletionIntegers_apply | null |
valuedAdicCompletion_eq_valuation (r : R) :
Valued.v (r : v.adicCompletion K) = v.valuation K r := by
convert Valued.valuedCompletion_apply (r : K)
variable {R K} in | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | valuedAdicCompletion_eq_valuation | The valuation on the completion agrees with the global valuation on elements of the
integer ring. |
valuedAdicCompletion_eq_valuation' (k : K) :
Valued.v (k : v.adicCompletion K) = v.valuation K k := by
convert Valued.valuedCompletion_apply k
variable {R K} in
open scoped algebraMap in -- to make the coercion from `R` fire | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | valuedAdicCompletion_eq_valuation' | The valuation on the completion agrees with the global valuation on elements of the field. |
coe_mem_adicCompletionIntegers (r : R) :
(r : adicCompletion K v) ∈ adicCompletionIntegers K v := by
rw [mem_adicCompletionIntegers, valuedAdicCompletion_eq_valuation]
exact valuation_le_one v r
@[simp] | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | coe_mem_adicCompletionIntegers | A global integer is in the local integers. |
coe_smul_adicCompletionIntegers (r : R) (x : v.adicCompletionIntegers K) :
(↑(r • x) : v.adicCompletion K) = r • (x : v.adicCompletion K) :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | coe_smul_adicCompletionIntegers | null |
adicCompletion.instIsScalarTower' :
IsScalarTower R (v.adicCompletionIntegers K) (v.adicCompletion K) where
smul_assoc x y z := by simp only [Algebra.smul_def]; apply mul_assoc | instance | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | adicCompletion.instIsScalarTower' | null |
adicCompletion.mul_nonZeroDivisor_mem_adicCompletionIntegers (v : HeightOneSpectrum R)
(a : v.adicCompletion K) : ∃ b ∈ R⁰, a * b ∈ v.adicCompletionIntegers K := by
by_cases ha : a ∈ v.adicCompletionIntegers K
· use 1
simp [ha]
· rw [notMem_adicCompletionIntegers] at ha
obtain ⟨ϖ, hϖ⟩ := intValuation_exists_uniformizer v
have : Valued.v (algebraMap R (v.adicCompletion K) ϖ) = (exp (1 : ℤ))⁻¹ := by
simp [valuedAdicCompletion_eq_valuation, valuation_of_algebraMap, hϖ, exp]
have hϖ0 : ϖ ≠ 0 := by rintro rfl; simp at hϖ
refine ⟨ϖ^(log (Valued.v a)).natAbs, pow_mem (mem_nonZeroDivisors_of_ne_zero hϖ0) _, ?_⟩
simp only [map_pow, mem_adicCompletionIntegers, map_mul, this, inv_pow, ← exp_nsmul, nsmul_one,
Int.natCast_natAbs]
exact mul_inv_le_one_of_le₀ (le_exp_log.trans (by simp [le_abs_self])) (zero_le _) | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | adicCompletion.mul_nonZeroDivisor_mem_adicCompletionIntegers | null |
adicAbvDef (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) :=
fun x ↦ toNNReal (ne_zero_of_lt hb) (v.valuation K x)
variable {R K} in | def | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | adicAbvDef | The `v`-adic absolute value function on `K` defined as `b` raised to negative `v`-adic
valuation, for some `b` in `ℝ≥0` |
isNonarchimedean_adicAbvDef {b : NNReal} (hb : 1 < b) :
IsNonarchimedean (α := K) (fun x ↦ v.adicAbvDef hb x) := by
intro x y
simp only [adicAbvDef]
have h_mono := (toNNReal_strictMono hb).monotone
rw [← h_mono.map_max]
exact h_mono ((v.valuation _).map_add x y)
variable {R K} in | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | isNonarchimedean_adicAbvDef | null |
adicAbv (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) : AbsoluteValue K ℝ where
toFun x := v.adicAbvDef hb x
map_mul' _ _ := by simp [adicAbvDef]
nonneg' _ := NNReal.zero_le_coe
eq_zero' _ := by simp [adicAbvDef]
add_le' _ _ := (isNonarchimedean_adicAbvDef v hb).add_le fun _ ↦ bot_le
variable {R K} in | def | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | adicAbv | The `v`-adic absolute value on `K` defined as `b` raised to negative `v`-adic
valuation, for some `b` in `ℝ≥0` |
isNonarchimedean_adicAbv (v : HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) :
IsNonarchimedean (α := K) (v.adicAbv hb) := isNonarchimedean_adicAbvDef v hb
variable {R K} in | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | isNonarchimedean_adicAbv | The `v`-adic absolute value is nonarchimedean |
adicAbv_coe_le_one {b : NNReal} (hb : 1 < b) (r : R) :
v.adicAbv hb (algebraMap R K r) ≤ 1 := by
simpa [adicAbv, adicAbvDef, toNNReal_le_one_iff hb] using valuation_le_one v r
variable {R K} in | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | adicAbv_coe_le_one | null |
adicAbv_coe_lt_one_iff {b : NNReal} (hb : 1 < b) (r : R) :
v.adicAbv hb (algebraMap R K r) < 1 ↔ r ∈ v.asIdeal := by
simpa [adicAbv, adicAbvDef, toNNReal_lt_one_iff hb] using valuation_lt_one_iff_mem v r
variable {R K} in | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | adicAbv_coe_lt_one_iff | null |
adicAbv_coe_eq_one_iff {b : NNReal} (hb : 1 < b) (r : R) :
v.adicAbv hb (algebraMap R K r) = 1 ↔ r ∉ v.asIdeal := by
rw [← not_iff_not, not_not, ← v.adicAbv_coe_lt_one_iff (K := K) hb, ne_iff_lt_iff_le]
exact adicAbv_coe_le_one v hb r | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.IsNonarchimedean",
"Mathlib.Data.Int.WithZero",
"Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas",
"Mathlib.RingTheory.Valuation.ExtendToLocalization",
"Mathlib.Topology.Algebra.Valued.ValuedField",
"Mathlib.Topology.Algebra.Valued.WithVal"
] | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | adicAbv_coe_eq_one_iff | null |
Ring.DimensionLEOne : Prop where
(maximalOfPrime : ∀ {p : Ideal R}, p ≠ ⊥ → p.IsPrime → p.IsMaximal)
open Ideal Ring | class | RingTheory | [
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Polynomial.RationalRoot"
] | Mathlib/RingTheory/DedekindDomain/Basic.lean | Ring.DimensionLEOne | A ring `R` has Krull dimension at most one if all nonzero prime ideals are maximal. |
Ideal.IsPrime.isMaximal {R : Type*} [CommRing R] [DimensionLEOne R]
{p : Ideal R} (h : p.IsPrime) (hp : p ≠ ⊥) : p.IsMaximal :=
DimensionLEOne.maximalOfPrime hp h | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Polynomial.RationalRoot"
] | Mathlib/RingTheory/DedekindDomain/Basic.lean | Ideal.IsPrime.isMaximal | null |
DimensionLEOne.principal_ideal_ring [IsDomain A] [IsPrincipalIdealRing A] :
DimensionLEOne A where
maximalOfPrime := fun nonzero _ =>
IsPrime.to_maximal_ideal nonzero | instance | RingTheory | [
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Polynomial.RationalRoot"
] | Mathlib/RingTheory/DedekindDomain/Basic.lean | DimensionLEOne.principal_ideal_ring | null |
DimensionLEOne.isIntegralClosure (B : Type*) [CommRing B] [IsDomain B] [Nontrivial R]
[Algebra R A] [Algebra R B] [Algebra B A] [IsScalarTower R B A] [IsIntegralClosure B R A]
[DimensionLEOne R] : DimensionLEOne B where
maximalOfPrime := fun {p} ne_bot _ =>
IsIntegralClosure.isMaximal_of_isMaximal_comap (R := R) A p
(Ideal.IsPrime.isMaximal inferInstance (IsIntegralClosure.comap_ne_bot A ne_bot))
nonrec instance DimensionLEOne.integralClosure [Nontrivial R] [IsDomain A] [Algebra R A]
[DimensionLEOne R] : DimensionLEOne (integralClosure R A) :=
DimensionLEOne.isIntegralClosure R A (integralClosure R A)
variable {R} | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Polynomial.RationalRoot"
] | Mathlib/RingTheory/DedekindDomain/Basic.lean | DimensionLEOne.isIntegralClosure | null |
DimensionLEOne.not_lt_lt [Ring.DimensionLEOne R] (p₀ p₁ p₂ : Ideal R) [hp₁ : p₁.IsPrime]
[hp₂ : p₂.IsPrime] : ¬(p₀ < p₁ ∧ p₁ < p₂)
| ⟨h01, h12⟩ => h12.ne ((hp₁.isMaximal (bot_le.trans_lt h01).ne').eq_of_le hp₂.ne_top h12.le) | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Polynomial.RationalRoot"
] | Mathlib/RingTheory/DedekindDomain/Basic.lean | DimensionLEOne.not_lt_lt | null |
DimensionLEOne.eq_bot_of_lt [Ring.DimensionLEOne R] (p P : Ideal R) [p.IsPrime]
[P.IsPrime] (hpP : p < P) : p = ⊥ :=
by_contra fun hp0 => not_lt_lt ⊥ p P ⟨Ne.bot_lt hp0, hpP⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Polynomial.RationalRoot"
] | Mathlib/RingTheory/DedekindDomain/Basic.lean | DimensionLEOne.eq_bot_of_lt | null |
IsDedekindRing : Prop
extends IsNoetherian A A, DimensionLEOne A, IsIntegralClosure A A (FractionRing A) | class | RingTheory | [
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Polynomial.RationalRoot"
] | Mathlib/RingTheory/DedekindDomain/Basic.lean | IsDedekindRing | A Dedekind ring is a commutative ring that is Noetherian, integrally closed, and
has Krull dimension at most one.
This is exactly `IsDedekindDomain` minus the `IsDomain` hypothesis.
The integral closure condition is independent of the choice of field of fractions:
use `isDedekindRing_iff` to prove `IsDedekindRing` for a given `fraction_map`. |
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