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Lax132576.SubsequentialCharacterisation

Machine-independent characterisation of subsequential functions

concepts/Lax132576/SubsequentialCharacterisation.lean · lax-132576

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    In the paper

    Theorem

    A partial function f:A∗→B∗f : A^* \to B^* is subsequential if and only if it is continuous and has bounded variation: for all w1,w2w_1, w_2,

    sup⁡w∥f(ww1),f(ww2)∥<∞,\sup_w \|f(w w_1), f(w w_2)\| < \infty,

    ww ranging over the strings for which both values are defined (Theorem B.4.8 of Transducers, Choffrut). The construction of the transducer splits the output after a prefix into a branching part, which depends on the future, and a non-branching part, both of which are shown regular; the book's Claims B.4.9–B.4.12 are the steps of that construction.

    Lean source view on GitHub

    1import Lax765601.Continuity
    2import Lax132576.SubsequentialTransducers
    3import Lax132576.LeftDistance
    4
    5/-!
    6---
    7title: Machine-independent characterisation of subsequential functions
    8type: theorem
    9---
    10A partial function f:A∗→B∗f : A^* \to B^* is subsequential if and only if it is
    11continuous and has bounded variation: for all w1,w2w_1, w_2,
    12sup⁡w∥f(ww1),f(ww2)∥<∞,\sup_w \|f(w w_1), f(w w_2)\| < \infty,
    13ww ranging over the strings for which both values are defined (Theorem B.4.8
    14of *Transducers*, Choffrut). The construction of the transducer splits the
    15output after a prefix into a *branching* part, which depends on the future,
    16and a *non-branching* part, both of which are shown regular; the book's
    17Claims B.4.9–B.4.12 are the steps of that construction.
    18
    19# Formalization notes
    20
    21Continuity of a partial function is `PartialContinuous` of
    22`Lax765601.Continuity`, and bounded variation is `BoundedVariation` of
    23`LeftDistance`. Both alphabets are assumed finite.
    24-/
    25
    26namespace Lax132576.SubsequentialCharacterisation
    27
    28open Lax765601.Continuity Lax132576.SubsequentialTransducers Lax132576.LeftDistance
    29
    30/-- A partial function is subsequential if and only if it is continuous and has
    31bounded variation. -/
    32axiom isSubsequential_iff {A B : Type} [Finite A] [Finite B] (f : List A → Option (List B)) :
    33 IsSubsequential f ↔ PartialContinuous f ∧ BoundedVariation f
    34
    35end Lax132576.SubsequentialCharacterisation
    36
    Show Proof

    Formalization notes

    Continuity of a partial function is PartialContinuousPartialContinuous of Lax765601.ContinuityLax765601.Continuity, and bounded variation is BoundedVariationBoundedVariation of LeftDistanceLeftDistance. Both alphabets are assumed finite.

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