```------------------------------------------------------------------------
-- 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
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 5 _∷_

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 5 _∷′_

_∷′_ : ∀ {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 = [] , []
```