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Lax132576.RationalOfPrimes

Compositions of prime rational functions are rational

concepts/Lax132576/RationalOfPrimes.lean · lax-132576

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    In the paper

    Theorem

    Every composition of prime rational functions is rational (Theorem B.2.6 of Transducers, the implication ⇐): each prime is rational — a Mealy machine and a homomorphism are read directly as automata with output, the separator function needs one empty-input transition, and a right-to-left Mealy machine is a bimachine with a trivial prefix automaton — and rational functions are closed under composition (Theorem B.1.4).

    Lean source view on GitHub

    1import Lax132576.RationalFunctions
    2import Lax132576.PrimeRationalFunctions
    3
    4/-!
    5---
    6title: Compositions of prime rational functions are rational
    7type: theorem
    8---
    9Every composition of prime rational functions is rational (Theorem B.2.6 of
    10*Transducers*, the implication ⇐): each prime is rational — a Mealy machine and
    11a homomorphism are read directly as automata with output, the separator
    12function needs one empty-input transition, and a right-to-left Mealy machine is
    13a bimachine with a trivial prefix automaton — and rational functions are closed
    14under composition (Theorem B.1.4).
    15
    16# Formalization notes
    17
    18The hypothesis is membership in `CompClosure PrimeRationalFam`; both alphabets
    19are assumed finite, as the bimachine of a right-to-left machine needs.
    20-/
    21
    22namespace Lax132576.RationalOfPrimes
    23
    24open Lax765601.CompositionClosure Lax132576.RationalFunctions Lax132576.PrimeRationalFunctions
    25
    26/-- A composition of prime rational functions is rational. -/
    27axiom isRationalFun_of_compClosure_primeRational {A B : Type} [Finite A] [Finite B]
    28 {f : List A → List B} (hf : CompClosure PrimeRationalFam A B f) : IsRationalFun f
    29
    30end Lax132576.RationalOfPrimes
    31
    Show Proof

    Formalization notes

    The hypothesis is membership in CompClosurePrimeRationalFamCompClosure PrimeRationalFam; both alphabets are assumed finite, as the bimachine of a right-to-left machine needs.

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