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The rational functions are the compositions of prime rational functions

Lax132576.RationalPrimes · concepts/Lax132576/RationalPrimes.lean · lax-132576

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    Natural Language Statement

    Theorem

    A string-to-string function is rational if and only if it is a composition of prime rational functions: prime Mealy machines, their right-to-left variants, string homomorphisms and the separator function w↦w#w \mapsto w\# (Theorem B.2.6 of Transducers). This is the Krohn–Rhodes theorem one step up the transducer ladder. The two implications are the separate statements PrimesOfRationalPrimesOfRational and RationalOfPrimesRationalOfPrimes; this statement is their conjunction.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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    1import Lax132576.RationalFunctions
    2import Lax132576.PrimeRationalFunctions
    3
    4/-!
    5---
    6title: The rational functions are the compositions of prime rational functions
    7type: theorem
    8---
    9A string-to-string function is rational if and only if it is a composition of
    10prime rational functions: prime Mealy machines, their right-to-left variants,
    11string homomorphisms and the separator function w↦w#w \mapsto w\# (Theorem B.2.6
    12of *Transducers*). This is the Krohn–Rhodes theorem one step up the transducer
    13ladder. The two implications are the separate statements `PrimesOfRational` and
    14`RationalOfPrimes`; this statement is their conjunction.
    15
    16# Formalization notes
    17
    18Both alphabets are assumed finite.
    19-/
    20
    21namespace Lax132576.RationalPrimes
    22
    23open Lax765601.CompositionClosure Lax132576.RationalFunctions Lax132576.PrimeRationalFunctions
    24
    25/-- A function is rational if and only if it is a composition of prime rational
    26functions. -/
    27axiom isRationalFun_iff_compClosure_primeRational {A B : Type} [Finite A] [Finite B]
    28 (f : List A → List B) : IsRationalFun f ↔ CompClosure PrimeRationalFam A B f
    29
    30end Lax132576.RationalPrimes
    31
    Show Proof
    Formalization notes

    Both alphabets are assumed finite.

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