Lax Lean Archive

Draft — mutable and not usable as a dependency; its citation marks the draft state.

Lax132576.RationalPrimes

The rational functions are the compositions of prime rational functions

concepts/Lax132576/RationalPrimes.lean · lax-132576

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Concept map

    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

    Evidence

    Each proof establishes this claim relative to its assumptions.

    In the paper

    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.

    Lean source view on GitHub

    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.

    Builds on

    Used by

    none

    From Mathlib

    none

    Community review

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above; your ORCID profile must share a public name.

    0 comments

    Loading discussion…