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Ignoring empty summands

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

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    Natural Language Statement

    Claim

    Claim 6.6. Let f=⟨fE,fN⟩:E+N→B∗f = \langle f_E, f_N \rangle : E + N \to B^*, where fEf_E is the empty constant, i.e. fE≡(E→1);[ ]f_E \equiv (E \to 1) \mathbin{;} [\,]. Then hom f\mathrm{hom}\, f is provably equivalent to the composition

    (E+N)∗→ split E∗×(N×E∗)∗→ snd ; fst∗ N∗→ hom fN B∗.(E + N)^* \xrightarrow{\ \mathsf{split}\ } E^* \times (N \times E^*)^* \xrightarrow{\ \mathsf{snd} \,;\, \mathsf{fst}^*\ } N^* \xrightarrow{\ \mathrm{hom}\, f_N\ } B^* .
    Concept map
    5 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.EquationalTheory
    2
    3/-!
    4---
    5title: Ignoring empty summands
    6type: claim
    7---
    8**Claim 6.6.** Let f=⟨fE,fN⟩:E+N→B∗f = \langle f_E, f_N \rangle : E + N \to B^*, where fEf_E is the
    9empty constant, i.e. fE≡(E→1);[ ]f_E \equiv (E \to 1) \mathbin{;} [\,]. Then hom f\mathrm{hom}\, f is
    10provably equivalent to the composition
    11
    12(E+N)∗→ split E∗×(N×E∗)∗→ snd ; fst∗ N∗→ hom fN B∗.(E + N)^* \xrightarrow{\ \mathsf{split}\ } E^* \times (N \times E^*)^* \xrightarrow{\ \mathsf{snd} \,;\, \mathsf{fst}^*\ } N^* \xrightarrow{\ \mathrm{hom}\, f_N\ } B^* .
    13
    14
    15
    16-/
    17
    18namespace Lax149335.EmptySummands
    19
    20open Lax149335.Types Lax149335.Terms Lax149335.Terms.Term Lax149335.DerivedTerms
    21 Lax149335.DerivedTerms.Term Lax149335.EquationalTheory
    22
    23/-- **Claim 6.6.** If the case `f_E` of a co-pairing is the empty constant, then `hom f`
    24factors through `hom f_N`. -/
    25axiom hom_factors {B E N : Ty} (fE : Term E B⋆) (fN : Term N B⋆)
    26 (hfE : fE ≡ₜ toUnit E ⨟ nilT B) :
    27 homT (Term.either fE fN) ≡ₜ
    28 (split E N ⨟ snd E⋆ ((N ⊗ E⋆)⋆) ⨟ listF (fst N E⋆) ⨟ homT fN)
    29
    30end Lax149335.EmptySummands
    31
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