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Completeness for simple rational terms

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

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    Natural Language Statement

    Lemma

    Lemma 6.1 (Completeness for simple rational terms). For all simple rational terms f,g:A→B∗f, g : A \to B^*, where BB is a finite alphabet, and every value w:1→B∗w : 1 \to B^*,

    [ ⁣[f] ⁣]=[ ⁣[g] ⁣]impliesf≡g,f⋅w≡w⋅gimplies(hom f)⋅w≡w⋅(hom g).\begin{aligned} [\![ f ]\!] = [\![ g ]\!] &\quad\text{implies}\quad f \equiv g, \\ f \cdot w \equiv w \cdot g &\quad\text{implies}\quad (\mathrm{hom}\, f) \cdot w \equiv w \cdot (\mathrm{hom}\, g). \end{aligned}
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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    Lean source view on GitHub

    1import Lax149335.SimpleRationalWithinSyntax
    2
    3/-!
    4---
    5title: Completeness for simple rational terms
    6type: lemma
    7---
    8**Lemma 6.1 (Completeness for simple rational terms).** For all simple rational terms
    9f,g:A→B∗f, g : A \to B^*, where BB is a finite alphabet, and every value w:1→B∗w : 1 \to B^*,
    10
    11[ ⁣[f] ⁣]=[ ⁣[g] ⁣]impliesf≡g,f⋅w≡w⋅gimplies(hom f)⋅w≡w⋅(hom g).\begin{aligned} [\![ f ]\!] = [\![ g ]\!] &\quad\text{implies}\quad f \equiv g, \\ f \cdot w \equiv w \cdot g &\quad\text{implies}\quad (\mathrm{hom}\, f) \cdot w \equiv w \cdot (\mathrm{hom}\, g). \end{aligned}
    12
    13
    14
    15
    16
    17-/
    18
    19namespace Lax149335.SimpleCompleteness
    20
    21open Lax149335.Types Lax149335.Terms Lax149335.DerivedTerms Lax149335.DerivedTerms.Term
    22 Lax149335.EquationalTheory Lax149335.SimpleRationalWithinSyntax
    23
    24/-- **Lemma 6.1.** The completeness rule and the conjugacy rule for simple rational terms. -/
    25axiom completeness_simple {A B : Ty} (hB : Ty.IsFinAlph B) (f g : Term A B⋆)
    26 (hf : IsSimple B f) (hg : IsSimple B g) (w : Value B⋆) :
    27 (f.den = g.den → f ≡ₜ g) ∧
    28 (dotR f w ≡ₜ dotL w g → dotR (homT f) w ≡ₜ dotL w (homT g))
    29
    30end Lax149335.SimpleCompleteness
    31
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