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feat(Padics): add an instance IsValuativeTopology ℚ_[p] #42998

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feat(Padics): add an instance IsValuativeTopology ℚ_[p] #42998

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feat(Padics): `IsValuativeTopology ℚ_[p]`
WenrongZou Aug 21, 2026
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update import
WenrongZou Aug 21, 2026
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13 changes: 13 additions & 0 deletions Mathlib/NumberTheory/Padics/PadicNumbers.lean
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Expand Up @@ -1198,6 +1198,19 @@ lemma norm_eq_zpow_log_mulValuation {x : ℚ_[p]} (hx : x ≠ 0) :
‖x‖ = (p : ℝ) ^ (log (mulValuation x)) := by
simp [norm_eq_zpow_neg_valuation, hx]

lemma norm_lt_zpow_iff_mulValuation_lt_exp {x : ℚ_[p]} {m : ℤ} :
‖x‖ < (p : ℝ) ^ m ↔ mulValuation x < exp m := by
have h1p : (1 : ℝ) < p := mod_cast hp.out.one_lt
by_cases hx : x = 0
· simpa [hx] using zpow_pos (by positivity) m
· rw [norm_eq_zpow_neg_valuation hx, zpow_lt_zpow_iff_right₀ h1p, mulValuation_toFun,
ite_eq_right hx, exp_lt_exp]

lemma norm_lt_norm_iff_mulValuation_lt {x y : ℚ_[p]} (hy : y ≠ 0) :
‖x‖ < ‖y‖ ↔ mulValuation x < mulValuation y := by
rw [norm_eq_zpow_log_mulValuation hy, norm_lt_zpow_iff_mulValuation_lt_exp,
exp_log (mulValuation.ne_zero_iff.mpr hy)]

/-- The additive `p`-adic valuation on `ℚ_[p]`, as an `addValuation`. -/
def addValuation : AddValuation ℚ_[p] (WithTop ℤ) :=
AddValuation.of addValuationDef AddValuation.map_zero AddValuation.map_one AddValuation.map_add
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23 changes: 22 additions & 1 deletion Mathlib/NumberTheory/Padics/ValuativeRel.lean
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Expand Up @@ -7,6 +7,7 @@ module

public import Mathlib.NumberTheory.Padics.PadicNumbers
public import Mathlib.RingTheory.Valuation.RankOne
public import Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

/-!
# p-adic numbers with a valuative relation
Expand All @@ -21,7 +22,7 @@ public section
variable {p : ℕ} [hp : Fact p.Prime] {Γ₀ : Type*} [LinearOrderedCommMonoidWithZero Γ₀]
(v : Valuation ℚ_[p] Γ₀)

open ValuativeRel WithZero
open ValuativeRel WithZero Valuation MonoidWithZeroHom.ValueGroup₀

namespace Padic

Expand All @@ -44,4 +45,24 @@ instance : IsNontrivial ℚ_[p] where

instance : IsRankLeOne ℚ_[p] := .of_compatible_mulArchimedean mulValuation

instance : IsValuativeTopology ℚ_[p] := by
refine IsValuativeTopology.of_mem_nhds_zero_iff_vle (mulValuation (p := p)) fun {s} ↦ ?_
rw [Metric.mem_nhds_iff]
have h1p : (1 : ℝ) < p := mod_cast hp.out.one_lt
constructor
· -- A metric ball `‖·‖ < ε` contains the valuation ball of radius `v (p ^ n)` for large `n`.
intro ⟨ε, hε, hball⟩
obtain ⟨n, hn⟩ := exists_pow_lt_of_lt_one hε (inv_lt_one_of_one_lt₀ h1p)
have ha0 : (p : ℚ_[p]) ^ n ≠ 0 := pow_ne_zero _ (Nat.cast_ne_zero.mpr hp.out.ne_zero)
have hnorm : ‖(p : ℚ_[p]) ^ n‖ < ε := by simpa [norm_pow, norm_p] using hn
refine ⟨Units.mk0 (mulValuation.restrict (p ^ n)) (by simpa using ha0),
fun z hz ↦ hball (mem_ball_zero_iff.mpr ?_)⟩
simp only [Set.mem_ofPred_eq, Units.val_mk0, restrict_lt_iff] at hz
exact ((norm_lt_norm_iff_mulValuation_lt ha0).mpr hz).trans hnorm
· -- Conversely, a valuation ball `v · < γ` is the metric ball of radius `p ^ log γ`.
intro ⟨γ, hγ⟩
refine ⟨p ^ log (embedding γ.val), zpow_pos (by positivity) _, fun _ hz ↦ hγ ?_⟩
rw [mem_ball_zero_iff, norm_lt_zpow_iff_mulValuation_lt_exp, exp_log (by simp)] at hz
simpa [restrict_lt_iff_lt_embedding] using hz

end Padic
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