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Simplification of rational terms

Lax149335.Simplification · concepts/Lax149335/Simplification.lean · lax-149335

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    Natural Language Statement

    Theorem

    Theorem 2.9 (Simplification of rational terms). Consider rational terms f,g:N∗→M∗f, g : N^* \to M^* where NN and MM are finite alphabets. There is a type CC and a rational term π:C→N∗\pi : C \to N^* with surjective denotation, such that [ ⁣[π;f] ⁣][\![ \pi \mathbin{;} f ]\!] and [ ⁣[π;g] ⁣][\![ \pi \mathbin{;} g ]\!] are represented by simple rational terms.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    Lean source view on GitHub

    1import Lax149335.SimpleRationalTerms
    2import Lax149335.Terms
    3
    4/-!
    5---
    6title: Simplification of rational terms
    7type: theorem
    8---
    9**Theorem 2.9 (Simplification of rational terms).** Consider rational terms
    10f,g:N∗→M∗f, g : N^* \to M^* where NN and MM are finite alphabets. There is a type CC and a
    11rational term π:C→N∗\pi : C \to N^* with surjective denotation, such that
    12[ ⁣[π;f] ⁣][\![ \pi \mathbin{;} f ]\!] and [ ⁣[π;g] ⁣][\![ \pi \mathbin{;} g ]\!] are represented by simple
    13rational terms.
    14
    15# Formalization notes
    16
    17"Represented by a simple rational term" refers to the simple rational terms of
    18Definition 2.5, with output alphabet the set of letters of MM.
    19-/
    20
    21namespace Lax149335.Simplification
    22
    23open Lax149335.Types Lax149335.Terms Lax149335.SimpleRationalTerms
    24
    25/-- **Theorem 2.9.** After precomposing with a rational term with surjective denotation, two
    26rational terms `N* → M*` are represented by simple rational terms. -/
    27axiom simplification_of_rational_terms {N M : Ty} (hN : Ty.IsFinAlph N)
    28 (hM : Ty.IsFinAlph M) (f g : Term N⋆ M⋆) :
    29 ∃ (C : Ty) (π : Term C N⋆), Function.Surjective π.den ∧
    30 SimpleTerm.IsSimpleFun (Bs := ⟦M⟧) (π ⨟ f).den ∧
    31 SimpleTerm.IsSimpleFun (Bs := ⟦M⟧) (π ⨟ g).den
    32
    33end Lax149335.Simplification
    34
    Show Proof
    Formalization notes

    "Represented by a simple rational term" refers to the simple rational terms of Definition 2.5, with output alphabet the set of letters of MM.

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