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Lax194892.PebbleContinuity

Pebble transducers are continuous

concepts/Lax194892/PebbleContinuity.lean · lax-194892

proven

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    In the paper

    Theorem

    Pebble transducers compute continuous functions (Theorem D.2.1 of Transducers). The languages recognised by pebble automata — pebble transducers with a yes/no answer — are regular, by induction on the number of pebbles; running an automaton for the target language on the output turns the transducer into a pebble automaton.

    Lean source view on GitHub

    1import Lax765601.Continuity
    2import Lax194892.PebbleTransducers
    3
    4/-!
    5---
    6title: Pebble transducers are continuous
    7type: theorem
    8---
    9Pebble transducers compute continuous functions (Theorem D.2.1 of
    10*Transducers*). The languages recognised by pebble automata — pebble
    11transducers with a yes/no answer — are regular, by induction on the number of
    12pebbles; running an automaton for the target language on the output turns the
    13transducer into a pebble automaton.
    14
    15# Formalization notes
    16
    17Both alphabets are assumed finite, as in the book.
    18-/
    19
    20namespace Lax194892.PebbleContinuity
    21
    22open Lax765601.Continuity Lax194892.PebbleTransducers
    23
    24/-- A function computed by a pebble transducer is continuous. -/
    25axiom continuous_of_isPebbleTransducer {A B : Type} [Finite A] [Finite B] {f : List A → List B}
    26 (hf : IsPebbleTransducer f) : Continuous f
    27
    28end Lax194892.PebbleContinuity
    29
    Show Proof

    Formalization notes

    Both alphabets are assumed finite, as in the book.

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