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For-transducers are computed by pebble transducers

Lax194892.PebbleOfFor · concepts/Lax194892/PebbleOfFor.lean · lax-194892

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    Natural Language Statement

    Theorem

    Every function computed by a for-transducer is computed by a pebble transducer (Theorem D.2.4 of Transducers, the implication from for-transducer to pebble). In prenex form the loop variables are pushed as pebbles, in the order of the loops; the loop-free body is evaluated by a one-way pass with the pebbles in place, since it can only compare positions, read letters and read Boolean variables, which the state carries.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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    In the paper

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    1import Lax194892.PebbleTransducers
    2import Lax194892.ForTransducers
    3
    4/-!
    5---
    6title: For-transducers are computed by pebble transducers
    7type: theorem
    8---
    9Every function computed by a for-transducer is computed by a pebble
    10transducer (Theorem D.2.4 of *Transducers*, the implication from for-transducer
    11to pebble). In prenex form the loop variables are pushed as pebbles, in the
    12order of the loops; the loop-free body is evaluated by a one-way pass with the
    13pebbles in place, since it can only compare positions, read letters and read
    14Boolean variables, which the state carries.
    15
    16# Formalization notes
    17
    18Both alphabets are assumed finite, as in the book.
    19-/
    20
    21namespace Lax194892.PebbleOfFor
    22
    23open Lax194892.PebbleTransducers Lax194892.ForTransducers
    24
    25/-- A function computed by a for-transducer is computed by a pebble transducer. -/
    26axiom isPebbleTransducer_of_isForTransducer {A B : Type} [Finite A] [Finite B]
    27 {f : List A → List B} (hf : IsForTransducer f) : IsPebbleTransducer f
    28
    29end Lax194892.PebbleOfFor
    30
    Show Proof
    Formalization notes

    Both alphabets are assumed finite, as in the book.

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