nonarchimedean — Mathlib

English

The p-adic norm is non-archimedean: padicNorm p (q + r) ≤ max(padicNorm p q, padicNorm p r).

Русский

Nон-архи-метрическая норма: padicNorm p (q + r) ≤ max(padicNorm p q, padicNorm p r).

LaTeX

$$$ padicNorm\ p\ (q + r) \leq \max( padicNorm\ p\ q, padicNorm\ p\ r ) $$$

Lean4

/-- The `p`-adic norm is nonarchimedean: the norm of `p + q` is at most the max of the norm of `p`
and the norm of `q`. -/
protected theorem nonarchimedean {q r : ℚ} : padicNorm p (q + r) ≤ max (padicNorm p q) (padicNorm p r) :=
  by
  wlog hle : padicValRat p q ≤ padicValRat p r generalizing q r
  · rw [add_comm, max_comm]
    exact this (le_of_not_ge hle)
  exact nonarchimedean_aux hle

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