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Reduction to nonempty summands

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

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    Natural Language Statement

    Lemma

    Lemma 6.5. In the situation of Lemma 6.4, one can further assume without loss of generality that all the terms fif_i and gig_i are nonempty.

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

    Each proof establishes this claim relative to its assumptions.

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    1import Lax149335.GuardedTerms
    2
    3/-!
    4---
    5title: Reduction to nonempty summands
    6type: lemma
    7---
    8**Lemma 6.5.** In the situation of Lemma 6.4, one can further assume without loss of
    9generality that all the terms fif_i and gig_i are nonempty.
    10
    11# Formalization notes
    12
    13As for Lemma 6.4, the statement is a reduction: if the conjugacy rule holds for co-pairings
    14of kk-guarded nonempty terms whose input type has list nesting depth at most DD, then it
    15holds for all co-pairings of kk-guarded terms whose input type has list nesting depth at
    16most DD. The assumption that BB is a finite alphabet is not needed.
    17-/
    18
    19namespace Lax149335.NonemptyReduction
    20
    21open Lax149335.Types Lax149335.Terms Lax149335.DerivedTerms Lax149335.DerivedTerms.Term
    22 Lax149335.EquationalTheory Lax149335.SimpleRationalWithinSyntax Lax149335.GuardedTerms
    23
    24/-- **Lemma 6.5.** The conjugacy rule for co-pairings of `k`-guarded terms follows from the
    25conjugacy rule for co-pairings of `k`-guarded nonempty terms. -/
    26axiom wlog_nonempty {B : Ty} (w : Value B⋆) (hw : 0 < valLength w) (D : ℕ)
    27 (H : ∀ (A₀ : Ty) (As : List Ty) (f₀ g₀ : Term A₀ B⋆) (fs gs : Cases B⋆ As),
    28 Ty.listDepth (sumTyOf A₀ As) ≤ D →
    29 KGuarded (valLength w) B f₀ → KGuarded (valLength w) B g₀ →
    30 fs.All (fun _ f => KGuarded (valLength w) B f) →
    31 gs.All (fun _ g => KGuarded (valLength w) B g) →
    32 NonemptyOutput f₀ → NonemptyOutput g₀ →
    33 fs.All (fun _ f => NonemptyOutput f) → gs.All (fun _ g => NonemptyOutput g) →
    34 dotR (copair f₀ fs) w ≡ₜ dotL w (copair g₀ gs) →
    35 dotR (homT (copair f₀ fs)) w ≡ₜ dotL w (homT (copair g₀ gs)))
    36 (A₀ : Ty) (As : List Ty) (f₀ g₀ : Term A₀ B⋆) (fs gs : Cases B⋆ As)
    37 (hD : Ty.listDepth (sumTyOf A₀ As) ≤ D)
    38 (hf₀ : KGuarded (valLength w) B f₀) (hg₀ : KGuarded (valLength w) B g₀)
    39 (hfs : fs.All (fun _ f => KGuarded (valLength w) B f))
    40 (hgs : gs.All (fun _ g => KGuarded (valLength w) B g))
    41 (hfg : dotR (copair f₀ fs) w ≡ₜ dotL w (copair g₀ gs)) :
    42 dotR (homT (copair f₀ fs)) w ≡ₜ dotL w (homT (copair g₀ gs))
    43
    44end Lax149335.NonemptyReduction
    45
    Show Proof
    Formalization notes

    As for Lemma 6.4, the statement is a reduction: if the conjugacy rule holds for co-pairings of kk-guarded nonempty terms whose input type has list nesting depth at most DD, then it holds for all co-pairings of kk-guarded terms whose input type has list nesting depth at most DD. The assumption that BB is a finite alphabet is not needed.

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