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Regular functions are continuous

Lax916827.RegularContinuity · concepts/Lax916827/RegularContinuity.lean · lax-916827

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    Natural Language Statement

    Theorem

    Regular functions are continuous (Theorem C.1.1 of Transducers, the continuity half). Continuous functions are closed under composition and rational functions are continuous (Theorem B.1.5), so it remains to see that map reverse and map duplicate are continuous, which is Lemma C.1.2 (reversal and duplication are continuous) lifted through Lemma C.1.3 (the map lifting of a continuous function is continuous).

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    In the paper

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    1import Lax765601.Continuity
    2import Lax916827.RegularFunctions
    3
    4/-!
    5---
    6title: Regular functions are continuous
    7type: theorem
    8---
    9Regular functions are continuous (Theorem C.1.1 of *Transducers*, the
    10continuity half). Continuous functions are closed under composition and
    11rational functions are continuous (Theorem B.1.5), so it remains to see that
    12map reverse and map duplicate are continuous, which is Lemma C.1.2 (reversal
    13and duplication are continuous) lifted through Lemma C.1.3 (the map lifting of
    14a continuous function is continuous).
    15
    16# Formalization notes
    17
    18The finiteness hypotheses on the alphabets are those of the book; the prime
    19regular functions are in fact continuous over any alphabets.
    20-/
    21
    22namespace Lax916827.RegularContinuity
    23
    24open Lax765601.Continuity Lax916827.RegularFunctions
    25
    26/-- A regular function is continuous. -/
    27axiom continuous_of_isRegularFun {A B : Type} [Finite A] [Finite B] {f : List A → List B}
    28 (hf : IsRegularFun f) : Continuous f
    29
    30end Lax916827.RegularContinuity
    31
    Show Proof
    Formalization notes

    The finiteness hypotheses on the alphabets are those of the book; the prime regular functions are in fact continuous over any alphabets.

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