instDist — Mathlib

English

For x,y in CompleteCopy s, dist x y equals dist x.1 y.1 plus the boundary term as above.

Русский

Для x,y ∈ CompleteCopy s расстояние равно dist x.1 y.1 плюс граничный член.

LaTeX

$$$$\\\\ dist x y = dist x.1 y.1 + |\\\\frac{1}{infDist x.1 s^c} - \\\\frac{1}{infDist y.1 s^c}|.$$$$

Lean4

/-- A distance on an open subset `s` of a metric space, designed to make it complete.  It is given
by `dist' x y = dist x y + |1 / dist x sᶜ - 1 / dist y sᶜ|`, where the second term blows up close to
the boundary to ensure that Cauchy sequences for `dist'` remain well inside `s`. -/
instance instDist : Dist (CompleteCopy s) where dist x y := dist x.1 y.1 + abs (1 / infDist x.1 sᶜ - 1 / infDist y.1 sᶜ)

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