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Lax314295.FORelabellingIffAperiodicBimachine

First-order relabellings are exactly the aperiodic bimachines

concepts/Lax314295/FORelabellingIffAperiodicBimachine.lean · lax-314295

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    In the paper

    Theorem

    A string-to-string function is a first-order relabelling if and only if it is computed by an aperiodic bimachine (Theorem C.4.16 of Transducers). The two implications are the separate statements AperiodicBimachineOfFORelabellingAperiodicBimachineOfFORelabelling and FORelabellingOfAperiodicBimachineFORelabellingOfAperiodicBimachine; this statement is their conjunction.

    Lean source view on GitHub

    1import Lax132576.Bimachines
    2import Lax314295.MSORelabellings
    3
    4/-!
    5---
    6title: First-order relabellings are exactly the aperiodic bimachines
    7type: theorem
    8---
    9A string-to-string function is a first-order relabelling if and only if it is
    10computed by an aperiodic bimachine (Theorem C.4.16 of *Transducers*). The two
    11implications are the separate statements `AperiodicBimachineOfFORelabelling`
    12and `FORelabellingOfAperiodicBimachine`; this statement is their
    13conjunction.
    14
    15# Formalization notes
    16
    17Both alphabets are assumed finite.
    18-/
    19
    20namespace Lax314295.FORelabellingIffAperiodicBimachine
    21
    22open Lax132576.Bimachines Lax314295.MSORelabellings
    23
    24/-- A function is a first-order relabelling if and only if an aperiodic bimachine
    25computes it. -/
    26axiom isFORelabelling_iff_isAperiodicBimachine {A B : Type} [Finite A] [Finite B]
    27 (f : List A → List B) : IsFORelabelling f ↔ IsAperiodicBimachine f
    28
    29end Lax314295.FORelabellingIffAperiodicBimachine
    30
    Show Proof

    Formalization notes

    Both alphabets are assumed finite.

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