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Concatenation with values

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

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    Natural Language Statement

    Lemma

    Lemma 5.2 (Concatenation with values). For terms f:A→B∗f : A \to B^* and g:A′→B∗g : A' \to B^* and list values w,v:1→B∗w, v : 1 \to B^*,

    (f⋅w)⋅v≡f⋅(w⋅v),w⋅(v⋅f)≡(w⋅v)⋅f,f⋅[ ]≡f≡[ ]⋅f,(w⋅f)⋅v≡w⋅(f⋅v),(f⋅w)⊙g≡f⊙(w⋅g).\begin{aligned} (f \cdot w) \cdot v &\equiv f \cdot (w \cdot v), & w \cdot (v \cdot f) &\equiv (w \cdot v) \cdot f, & f \cdot [\,] &\equiv f \equiv [\,] \cdot f, \\ (w \cdot f) \cdot v &\equiv w \cdot (f \cdot v), & (f \cdot w) \odot g &\equiv f \odot (w \cdot g). \end{aligned}
    Concept map
    5 concepts
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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.EquationalTheory
    2
    3/-!
    4---
    5title: Concatenation with values
    6type: lemma
    7---
    8**Lemma 5.2 (Concatenation with values).** For terms f:A→B∗f : A \to B^* and
    9g:A′→B∗g : A' \to B^* and list values w,v:1→B∗w, v : 1 \to B^*,
    10
    11(f⋅w)⋅v≡f⋅(w⋅v),w⋅(v⋅f)≡(w⋅v)⋅f,f⋅[ ]≡f≡[ ]⋅f,(w⋅f)⋅v≡w⋅(f⋅v),(f⋅w)⊙g≡f⊙(w⋅g).\begin{aligned} (f \cdot w) \cdot v &\equiv f \cdot (w \cdot v), & w \cdot (v \cdot f) &\equiv (w \cdot v) \cdot f, & f \cdot [\,] &\equiv f \equiv [\,] \cdot f, \\ (w \cdot f) \cdot v &\equiv w \cdot (f \cdot v), & (f \cdot w) \odot g &\equiv f \odot (w \cdot g). \end{aligned}
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    18
    19-/
    20
    21namespace Lax149335.ValueConcatenation
    22
    23open Lax149335.Types Lax149335.Terms Lax149335.DerivedTerms Lax149335.DerivedTerms.Term
    24 Lax149335.EquationalTheory
    25
    26/-- **Lemma 5.2.** The associativity and unit laws for concatenation with values. -/
    27axiom value_assoc {A A' B : Ty} (f : Term A B⋆) (g : Term A' B⋆) (w v : Value B⋆) :
    28 (dotR (dotR f w) v ≡ₜ dotR f (dotL w v)) ∧
    29 (dotL w (dotL v f) ≡ₜ dotL (dotL w v) f) ∧
    30 (dotR f (nilT B) ≡ₜ f) ∧
    31 (dotL (nilT B) f ≡ₜ f) ∧
    32 (dotR (dotL w f) v ≡ₜ dotL w (dotR f v)) ∧
    33 (fdot (dotR f w) g ≡ₜ fdot f (dotL w g))
    34
    35end Lax149335.ValueConcatenation
    36
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