------------------------------------------------------------------------
-- The Agda standard library
--
-- Lists where every consecutative pair of elements is related.
------------------------------------------------------------------------
 
{-# OPTIONS --cubical-compatible --safe #-}
 
module Data.List.Relation.Unary.Linked {a} {A : Set a} where
 
open import Data.List.Base as List using (List; []; _∷_)
open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
open import Data.Product.Base as Prod using (_,_; _×_; uncurry; <_,_>)
open import Data.Maybe.Base using (just)
open import Data.Maybe.Relation.Binary.Connected
  using (Connected; just; just-nothing)
open import Function.Base using (id; _∘_)
open import Level using (Level; _⊔_)
open import Relation.Binary.Definitions as B
open import Relation.Binary.Core using (Rel; _⇒_)
open import Relation.Binary.Construct.Intersection renaming (_∩_ to _∩ᵇ_)
open import Relation.Binary.PropositionalEquality.Core using (refl; cong₂)
open import Relation.Unary as U renaming (_∩_ to _∩ᵘ_) hiding (_⇒_)
open import Relation.Nullary.Decidable using (yes; no; map′; _×-dec_)
 
private
  variable
    p q r ℓ : Level
 
------------------------------------------------------------------------
-- Definition
 
-- Linked R xs means that every consecutative pair of elements in xs is
-- a member of relation R.
 
infixr  _∷_
 
data Linked (R : Rel A ℓ) : List A → Set (a ⊔ ℓ) where
  []  : Linked R []
  [-] : ∀ {x} → Linked R (x ∷ [])
  _∷_ : ∀ {x y xs} → R x y → Linked R (y ∷ xs) → Linked R (x ∷ y ∷ xs)
 
------------------------------------------------------------------------
-- Operations
 
module _ {R : Rel A p} where
 
  head : ∀ {x y xs} → Linked R (x ∷ y ∷ xs) → R x y
  head (Rxy ∷ Rxs) = Rxy
 
  tail : ∀ {x xs} → Linked R (x ∷ xs) → Linked R xs
  tail [-]       = []
  tail (_ ∷ Rxs) = Rxs
 
  head′ : ∀ {x xs} → Linked R (x ∷ xs) → Connected R (just x) (List.head xs)
  head′ [-]       = just-nothing
  head′ (Rxy ∷ _) = just Rxy
 
  infixr  _∷′_
 
  _∷′_ : ∀ {x xs} →
         Connected R (just x) (List.head xs) →
         Linked R xs →
         Linked R (x ∷ xs)
  _∷′_ {xs = []}     _  _            = [-]
  _∷′_ {xs = y ∷ xs} (just Rxy) Ryxs = Rxy ∷ Ryxs
 
module _ {R : Rel A p} {S : Rel A q} where
 
  map : R ⇒ S → Linked R ⊆ Linked S
  map R⇒S []           = []
  map R⇒S [-]          = [-]
  map R⇒S (x~xs ∷ pxs) = R⇒S x~xs ∷ map R⇒S pxs
 
module _ {P : Rel A p} {Q : Rel A q} {R : Rel A r} where
 
  zipWith : P ∩ᵇ Q ⇒ R → Linked P ∩ᵘ Linked Q ⊆ Linked R
  zipWith f ([] , [])             = []
  zipWith f ([-] , [-])           = [-]
  zipWith f (px ∷ pxs , qx ∷ qxs) = f (px , qx) ∷ zipWith f (pxs , qxs)
 
  unzipWith : R ⇒ P ∩ᵇ Q → Linked R ⊆ Linked P ∩ᵘ Linked Q
  unzipWith f []         = [] , []
  unzipWith f [-]        = [-] , [-]
  unzipWith f (rx ∷ rxs) = Prod.zip _∷_ _∷_ (f rx) (unzipWith f rxs)
 
module _ {P : Rel A p} {Q : Rel A q} where
 
  zip : Linked P ∩ᵘ Linked Q ⊆ Linked (P ∩ᵇ Q)
  zip = zipWith id
 
  unzip : Linked (P ∩ᵇ Q) ⊆ Linked P ∩ᵘ Linked Q
  unzip = unzipWith id
 
------------------------------------------------------------------------
-- Properties of predicates preserved by Linked
 
module _ {R : Rel A ℓ} where
 
  linked? : B.Decidable R → U.Decidable (Linked R)
  linked? R? []           = yes []
  linked? R? (x ∷ [])     = yes [-]
  linked? R? (x ∷ y ∷ xs) =
    map′ (uncurry _∷_) < head , tail > (R? x y ×-dec linked? R? (y ∷ xs))
 
  irrelevant : B.Irrelevant R → U.Irrelevant (Linked R)
  irrelevant irr []           []           = refl
  irrelevant irr [-]          [-]          = refl
  irrelevant irr (px₁ ∷ pxs₁) (px₂ ∷ pxs₂) =
    cong₂ _∷_ (irr px₁ px₂) (irrelevant irr pxs₁ pxs₂)
 
  satisfiable : U.Satisfiable (Linked R)
  satisfiable = [] , []