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Lax132576.RationalComposition

Rational relations are closed under composition

concepts/Lax132576/RationalComposition.lean · lax-132576

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

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    In the paper

    Theorem

    If R⊆A∗×B∗R \subseteq A^* \times B^* and S⊆B∗×C∗S \subseteq B^* \times C^* are rational relations, then so is their relational composition

    R⋅S={(u,v)∈A∗×C∗∣u R w and w S v for some w∈B∗}R \cdot S = \{(u, v) \in A^* \times C^* \mid u\,R\,w \text{ and } w\,S\,v \text{ for some } w \in B^*\}

    (Theorem B.1.4 of Transducers). The proof is the product construction of Theorem A.1.3, after splitting transitions so that each produces at most one letter of input or output and adding empty transitions around every state, so that the two runs can be synchronised on the intermediate string.

    Lean source view on GitHub

    1import Lax132576.RationalRelations
    2
    3/-!
    4---
    5title: Rational relations are closed under composition
    6type: theorem
    7---
    8If R⊆A∗×B∗R \subseteq A^* \times B^* and S⊆B∗×C∗S \subseteq B^* \times C^* are rational
    9relations, then so is their relational composition
    10R⋅S={(u,v)∈A∗×C∗∣u R w and w S v for some w∈B∗}R \cdot S = \{(u, v) \in A^* \times C^* \mid u\,R\,w \text{ and } w\,S\,v \text{ for some } w \in B^*\}
    11(Theorem B.1.4 of *Transducers*). The proof is the product construction of
    12Theorem A.1.3, after splitting transitions so that each produces at most one
    13letter of input or output and adding empty transitions around every state, so
    14that the two runs can be synchronised on the intermediate string.
    15
    16# Formalization notes
    17
    18No finiteness of the alphabets is needed: the product construction only uses
    19the finiteness of the two state spaces.
    20-/
    21
    22namespace Lax132576.RationalComposition
    23
    24open Lax132576.RationalRelations
    25
    26/-- The composition of two rational relations is rational. -/
    27axiom isRationalRel_comp {A B C : Type}
    28 {R : List A → List B → Prop} {S : List B → List C → Prop}
    29 (hR : IsRationalRel R) (hS : IsRationalRel S) :
    30 IsRationalRel (fun w v => ∃ u, R w u ∧ S u v)
    31
    32end Lax132576.RationalComposition
    33
    Show Proof

    Formalization notes

    No finiteness of the alphabets is needed: the product construction only uses the finiteness of the two state spaces.

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