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Lax916827.RegularConcatenation

Regular functions are closed under concatenation

concepts/Lax916827/RegularConcatenation.lean · lax-916827

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    In the paper

    Theorem

    If f,g:A∗→B∗f, g : A^* \to B^* are regular, then so is their concatenation w↦f(w)⋅g(w)w \mapsto f(w) \cdot g(w) (Lemma C.2.10 of Transducers, second item): map duplicate produces two copies of the input, and the map liftings of ff and of gg are applied to the two copies, selected by a rational marking.

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    1import Lax916827.RegularFunctions
    2
    3/-!
    4---
    5title: Regular functions are closed under concatenation
    6type: theorem
    7---
    8If f,g:A∗→B∗f, g : A^* \to B^* are regular, then so is their concatenation
    9w↦f(w)⋅g(w)w \mapsto f(w) \cdot g(w) (Lemma C.2.10 of *Transducers*, second item): map
    10duplicate produces two copies of the input, and the map liftings of ff and of
    11gg are applied to the two copies, selected by a rational marking.
    12
    13# Formalization notes
    14
    15Both alphabets are assumed finite, as in the book.
    16-/
    17
    18namespace Lax916827.RegularConcatenation
    19
    20open Lax916827.RegularFunctions
    21
    22/-- The concatenation of two regular functions is regular. -/
    23axiom isRegularFun_concat {A B : Type} [Finite A] [Finite B] {f g : List A → List B}
    24 (hf : IsRegularFun f) (hg : IsRegularFun g) : IsRegularFun (fun w => f w ++ g w)
    25
    26end Lax916827.RegularConcatenation
    27
    Show Proof

    Formalization notes

    Both alphabets are assumed finite, as in the book.

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