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The conjugacy rule for syntactic conjugates

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

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    Natural Language Statement

    Lemma

    Lemma 6.2. For a simple rational term h:A→B∗h : A \to B^* and a value w:1→B∗w : 1 \to B^*,

    hom(w⋅h)⋅w  ≡  w⋅hom(h⋅w).\mathrm{hom}(w \cdot h) \cdot w \;\equiv\; w \cdot \mathrm{hom}(h \cdot w).

    That is, the conjugacy rule holds when its assumption is strengthened from provable to syntactic conjugacy.

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

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax149335.SimpleRationalWithinSyntax
    2
    3/-!
    4---
    5title: The conjugacy rule for syntactic conjugates
    6type: lemma
    7---
    8**Lemma 6.2.** For a simple rational term h:A→B∗h : A \to B^* and a value w:1→B∗w : 1 \to B^*,
    9
    10hom(w⋅h)⋅w  ≡  w⋅hom(h⋅w).\mathrm{hom}(w \cdot h) \cdot w \;\equiv\; w \cdot \mathrm{hom}(h \cdot w).
    11
    12That is, the conjugacy rule holds when its assumption is strengthened from provable to
    13syntactic conjugacy.
    14
    15# Formalization notes
    16
    17The paper states the lemma for a simple rational term over a finite alphabet. Neither
    18assumption is used in the proof, so the statement is formalised for an arbitrary rational
    19term h:A→B∗h : A \to B^*, which makes it stronger.
    20-/
    21
    22namespace Lax149335.ConjugacyRule
    23
    24open Lax149335.Types Lax149335.Terms Lax149335.DerivedTerms Lax149335.DerivedTerms.Term
    25 Lax149335.EquationalTheory
    26
    27/-- **Lemma 6.2.** `hom(w · h) · w ≡ w · hom(h · w)`. -/
    28axiom conjugacy_of_syntactic {A B : Ty} (h : Term A B⋆) (w : Value B⋆) :
    29 dotR (homT (dotL w h)) w ≡ₜ dotL w (homT (dotR h w))
    30
    31end Lax149335.ConjugacyRule
    32
    Show Proof
    Formalization notes

    The paper states the lemma for a simple rational term over a finite alphabet. Neither assumption is used in the proof, so the statement is formalised for an arbitrary rational term h:A→B∗h : A \to B^*, which makes it stronger.

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