refactor(Probability): define statistical distance on arbitrary types - #1065
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Defines statistical distance as `(1/2) * ∑' a, |p a - q a|` instead of a finite sum. Absolute summability follows from unit mass, so the `MetricSpace (PMF α)` instance and the `StatisticallyClose` API now apply to PMFs on any type, such as `List Bool`. On finite types, `dist_eq` still gives the finite-sum formula; the general formula is `dist_eq_tsum`. This removes `[Fintype α]` from the metric instance, `dist_le_one`, `dist_eq_one_of_disjoint_support`, `dist_bind_le`, `dist_map_le`, and the `StatisticallyClose` lemmas, and `[Fintype Commitment]` from `StatisticallyHiding` and its theorems. Callers that pass these instances explicitly need to drop them, and `dist_eq` no longer holds by `rfl`. The private `sum_toReal` and `bind_apply_toReal` copies in this file are replaced by the public lemmas from #1062. The perfect hiding/binding impossibility for commitments is now a corollary of the statistical version, and the test instantiates it with commitments in `ℕ`.
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Extends statistical distance and its postprocessing bounds from finite types to arbitrary PMFs using an absolutely summable infinite sum. The scoped
MetricSpace (PMF α)instance andStatisticallyCloseAPI now apply to outcomes such asList Bool.dist_eqretains the finite-sum formula, whiledist_eq_tsumgives the general formula.This also removes the finite-commitment requirement from
StatisticallyHiding. The perfect hiding/binding impossibility theorem becomes a corollary of the statistical version, with a test using commitments inℕ. Callers passing the removed finiteness instances explicitly need to drop them, anddist_eqno longer holds byrfl.Depends on #1062.
Composed with Claude Code; reviewed and rebased with Codex.