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Lax916827.TwoWayOfRegular

Every regular function is computed by a two-way transducer

concepts/Lax916827/TwoWayOfRegular.lean · lax-916827

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    Concept map

    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

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    In the paper

    Theorem

    Every regular function is computed by a two-way transducer (Corollary C.2.8 of Transducers). Two-way transducers are closed under composition (Theorem C.2.5) and contain the rational functions (Corollary C.2.7), and map reverse and map duplicate are computed by explicit two-way transducers that sweep each block backwards, respectively twice.

    Lean source view on GitHub

    1import Lax916827.RegularFunctions
    2import Lax916827.TwoWayTransducers
    3
    4/-!
    5---
    6title: Every regular function is computed by a two-way transducer
    7type: theorem
    8---
    9Every regular function is computed by a two-way transducer (Corollary C.2.8 of
    10*Transducers*). Two-way transducers are closed under composition (Theorem
    11C.2.5) and contain the rational functions (Corollary C.2.7), and map reverse
    12and map duplicate are computed by explicit two-way transducers that sweep each
    13block backwards, respectively twice.
    14
    15# Formalization notes
    16
    17Both alphabets are assumed finite; the induction on the composition tree uses
    18the finiteness of the intermediate alphabets built into the closure.
    19-/
    20
    21namespace Lax916827.TwoWayOfRegular
    22
    23open Lax916827.RegularFunctions Lax916827.TwoWayTransducers
    24
    25/-- A regular function is computed by a two-way transducer. -/
    26axiom isTwoWay_of_isRegularFun {A B : Type} [Finite A] [Finite B] {f : List A → List B}
    27 (hf : IsRegularFun f) : IsTwoWay f
    28
    29end Lax916827.TwoWayOfRegular
    30
    Show Proof

    Formalization notes

    Both alphabets are assumed finite; the induction on the composition tree uses the finiteness of the intermediate alphabets built into the closure.

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