Closed Modality #970
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Closed Modality #970
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ecavallo
reviewed
Jan 14, 2023
Cubical/HITs/Join/Properties.agda
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| join-leftUnit : {A : Type ℓ} → isContr (join (Unit* {ℓ'}) A) | ||
| fst join-leftUnit = inl tt* | ||
| snd join-leftUnit (inl tt*) = refl | ||
| snd join-leftUnit (inr a) = push tt* a | ||
| snd join-leftUnit (push tt* a i) j = push tt* a (i ∧ j) |
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The name leftUnit is a bit ambiguous---it could also refer to the left unit law join ⊥ A ≃ A. How about instead calling this joinAnnihilL, following the algebra naming conventions (Cubical/Algebra/NAMING.md)?
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| Modality.◯-elim closedModality {B = B} B-modal f (push x a i) = | ||
| hcomp | ||
| (λ where | ||
| j (i = i0) → contr (transport (λ k → B (push x a (~ k))) (f a)) (~ j) -- Contractibilty | ||
| j (i = i1) → f a) | ||
| (transport-filler (sym (cong B (push x a))) (f a) (~ i)) | ||
| where | ||
| contr : (y : B (inl x)) → fst (B-modal (inl x) x) ≡ y | ||
| contr = snd (B-modal (inl x) x) |
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A more compact (if less "efficient") proof for this case is
Modality.◯-elim closedModality B-modal f (push x a i) =
isProp→PathP (λ i → isContr→isProp (B-modal (push x a i) x)) (B-modal (inl x) x .fst) (f a) i
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Thanks so much for the suggestions! I've implemented them now.
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Implement the closed modality, also left unit law for join of types.