instIsPrincipalIdealRingSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegers

English

The adic completion integers form a principal ideal ring.

Русский

Целые приородки адикционной завершенности образуют кольцо главных идеалов.

LaTeX

$$$\text{IsPrincipalIdealRing}((v.adicCompletionIntegers\ K))$$$

Lean4

instance : IsPrincipalIdealRing (v.adicCompletionIntegers K) :=
  by
  unfold HeightOneSpectrum.adicCompletionIntegers
  rw [(Valuation.valuationSubring.integers (Valued.v)).isPrincipalIdealRing_iff_not_denselyOrdered,
    WithZero.denselyOrdered_set_iff_subsingleton]
  simpa using Valued.v.range_nontrivial

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