adicAbv — Mathlib

English

For hb > 1, the abv function agrees with the valuation map on x ∈ K in an a certain precise way: equality of exact forms between abv and valuation.

Русский

Для hb > 1 абв-оператор согласуется с отображением валюации на x ∈ K по точной формуле.

LaTeX

$$v.adicAbv hb = (definition in Abv) (valuation mapping)$$

Lean4

/-- The `v`-adic absolute value on `K` defined as `b` raised to negative `v`-adic
valuation, for some `b` in `ℝ≥0` -/
def 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

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