------------------------------------------------------------------------
-- The Agda standard library
--
-- Pointers into star-lists
------------------------------------------------------------------------
{-# OPTIONS --with-K --safe #-}
module Data.Star.Pointer {ℓ} {I : Set ℓ} where
open import Data.Maybe.Base using (Maybe; nothing; just)
open import Data.Star.Decoration
open import Data.Unit.Base using (tt)
open import Function.Base using (const; case_of_)
open import Level using (Level; _⊔_; lift)
open import Relation.Binary.Core using (Rel)
open import Relation.Binary.Definitions using (NonEmpty; nonEmpty)
open import Relation.Binary.Construct.Closure.ReflexiveTransitive
using (Star; ε; _◅_; gmap; map; _◅◅_; _▻▻_; _⋆)
private
variable
r p q : Level
-- Pointers into star-lists. The edge pointed to is decorated with Q,
-- while other edges are decorated with P.
data Pointer {T : Rel I r}
(P : EdgePred p T) (Q : EdgePred q T)
: Rel (Maybe (NonEmpty (Star T))) (ℓ ⊔ r ⊔ p ⊔ q) where
step : ∀ {i j k} {x : T i j} {xs : Star T j k}
(p : P x) → Pointer P Q (just (nonEmpty (x ◅ xs)))
(just (nonEmpty xs))
done : ∀ {i j k} {x : T i j} {xs : Star T j k}
(q : Q x) → Pointer P Q (just (nonEmpty (x ◅ xs))) nothing
-- Any P Q xs means that some edge in xs satisfies Q, while all
-- preceding edges satisfy P. A star-list of type Any Always Always xs
-- is basically a prefix of xs; the existence of such a prefix
-- guarantees that xs is non-empty.
Any : {T : Rel I r} (P : EdgePred p T) (Q : EdgePred q T) →
EdgePred (ℓ ⊔ (r ⊔ (p ⊔ q))) (Star T)
Any P Q xs = Star (Pointer P Q) (just (nonEmpty xs)) nothing
module _ {T : Rel I r} {P : EdgePred p T} {Q : EdgePred q T} where
this : ∀ {i j k} {x : T i j} {xs : Star T j k} →
Q x → Any P Q (x ◅ xs)
this q = done q ◅ ε
that : ∀ {i j k} {x : T i j} {xs : Star T j k} →
P x → Any P Q xs → Any P Q (x ◅ xs)
that p = _◅_ (step p)
-- Safe lookup.
data Result (T : Rel I r)
(P : EdgePred p T) (Q : EdgePred q T) : Set (ℓ ⊔ r ⊔ p ⊔ q) where
result : ∀ {i j} {x : T i j} (p : P x) (q : Q x) → Result T P Q
-- The first argument points out which edge to extract. The edge is
-- returned, together with proofs that it satisfies Q and R.
module _ {T : Rel I r} {P : EdgePred p T} {Q : EdgePred q T} where
lookup : ∀ {r} {R : EdgePred r T} {i j} {xs : Star T i j} →
All R xs → Any P Q xs → Result T Q R
lookup (↦ r ◅ _) (done q ◅ ε) = result q r
lookup (↦ _ ◅ rs) (step p ◅ ps) = lookup rs ps
-- We can define something resembling init.
prefixIndex : ∀ {i j} {xs : Star T i j} → Any P Q xs → I
prefixIndex (done {i = i} q ◅ _) = i
prefixIndex (step p ◅ ps) = prefixIndex ps
prefix : ∀ {i j} {xs : Star T i j} →
(ps : Any P Q xs) → Star T i (prefixIndex ps)
prefix (done q ◅ _) = ε
prefix (step {x = x} p ◅ ps) = x ◅ prefix ps
-- Here we are taking the initial segment of ps (all elements but the
-- last, i.e. all edges satisfying P).
init : ∀ {i j} {xs : Star T i j} →
(ps : Any P Q xs) → All P (prefix ps)
init (done q ◅ _) = ε
init (step p ◅ ps) = ↦ p ◅ init ps
-- One can simplify the implementation by not carrying around the
-- indices in the type:
last : ∀ {i j} {xs : Star T i j} →
Any P Q xs → NonEmptyEdgePred T Q
last ps with result q _ ← lookup {r = p} (decorate (const (lift tt)) _) ps =
nonEmptyEdgePred q
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