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Lax765601.ReversibleComposition

Reversible Mealy machines are closed under composition

concepts/Lax765601/ReversibleComposition.lean · lax-765601

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    In the paper

    Theorem

    The composition of two functions computed by reversible Mealy machines is computed by a reversible Mealy machine (Lemma A.2.6 of Transducers). Hence the class (Reversible)∗(\mathrm{Reversible})^* of compositions of reversible machines is just the class of reversible machines: the product machine of the composition (Theorem A.1.3) is reversible, since the state transformation of a letter in it is a permutation in each coordinate.

    Lean source view on GitHub

    1import Lax765601.PrimeMealyMachines
    2
    3/-!
    4---
    5title: Reversible Mealy machines are closed under composition
    6type: theorem
    7---
    8The composition of two functions computed by reversible Mealy machines is
    9computed by a reversible Mealy machine (Lemma A.2.6 of *Transducers*). Hence the
    10class (Reversible)∗(\mathrm{Reversible})^* of compositions of reversible machines is just
    11the class of reversible machines: the product machine of the composition
    12(Theorem A.1.3) is reversible, since the state transformation of a letter in it
    13is a permutation in each coordinate.
    14
    15# Formalization notes
    16
    17As for closure under composition of Mealy machines, the intermediate alphabet is
    18assumed finite.
    19-/
    20
    21namespace Lax765601.ReversibleComposition
    22
    23open Lax765601.PrimeMealyMachines
    24
    25/-- The composition of two reversible Mealy machines is a reversible Mealy
    26machine. -/
    27axiom isReversibleMealy_comp {A B C : Type} [Finite B]
    28 {f : List A → List B} {g : List B → List C}
    29 (hf : IsReversibleMealy f) (hg : IsReversibleMealy g) : IsReversibleMealy (g ∘ f)
    30
    31end Lax765601.ReversibleComposition
    32
    Show Proof

    Formalization notes

    As for closure under composition of Mealy machines, the intermediate alphabet is assumed finite.

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