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Lax314295.AperiodicBimachineOfFORelabelling

First-order relabellings are computed by aperiodic bimachines

concepts/Lax314295/AperiodicBimachineOfFORelabelling.lean · lax-314295

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    In the paper

    Theorem

    Every first-order relabelling is computed by an aperiodic bimachine (Theorem C.4.16 of Transducers, the implication from relabelling to bimachine). For a first-order formula with one free variable, the prefixes and the suffixes on which it holds at the marked position are first-order definable languages, recognised by aperiodic automata (Theorem C.4.11), which become the prefix and suffix automata of the bimachine.

    Lean source view on GitHub

    1import Lax132576.Bimachines
    2import Lax314295.MSORelabellings
    3
    4/-!
    5---
    6title: First-order relabellings are computed by aperiodic bimachines
    7type: theorem
    8---
    9Every first-order relabelling is computed by an aperiodic bimachine (Theorem
    10C.4.16 of *Transducers*, the implication from relabelling to bimachine). For a
    11first-order formula with one free variable, the prefixes and the suffixes on
    12which it holds at the marked position are first-order definable languages,
    13recognised by aperiodic automata (Theorem C.4.11), which become the prefix and
    14suffix automata of the bimachine.
    15
    16# Formalization notes
    17
    18Both alphabets are assumed finite, as in the book.
    19-/
    20
    21namespace Lax314295.AperiodicBimachineOfFORelabelling
    22
    23open Lax132576.Bimachines Lax314295.MSORelabellings
    24
    25/-- A first-order relabelling is computed by an aperiodic bimachine. -/
    26axiom isAperiodicBimachine_of_isFORelabelling {A B : Type} [Finite A] [Finite B]
    27 {f : List A → List B} (hf : IsFORelabelling f) : IsAperiodicBimachine f
    28
    29end Lax314295.AperiodicBimachineOfFORelabelling
    30
    Show Proof

    Formalization notes

    Both alphabets are assumed finite, as in the book.

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