;; lib/minikanren/relations.sx — Phase 4 standard relations.
;;
;; Programs use native SX lists as data. Relations decompose lists via the
;; tagged cons-cell shape `(:cons h t)` because SX has no improper pairs;
;; the unifier treats `(:cons h t)` and the native list `(h . t)` as
;; equivalent, and `mk-walk*` flattens cons cells back to flat lists for
;; reification.

;; --- pair / list shape relations ---

(define nullo (fn (l) (== l (list))))

(define pairo (fn (p) (fresh (a d) (== p (mk-cons a d)))))

(define caro (fn (p a) (fresh (d) (== p (mk-cons a d)))))

(define cdro (fn (p d) (fresh (a) (== p (mk-cons a d)))))

(define conso (fn (a d p) (== p (mk-cons a d))))

(define firsto caro)
(define resto cdro)

(define
  listo
  (fn (l) (conde ((nullo l)) ((fresh (a d) (conso a d l) (listo d))))))

;; --- appendo: the canary ---
;;
;; (appendo l s ls) — `ls` is the concatenation of `l` and `s`.
;; Runs forwards (l, s known → ls), backwards (ls known → all (l, s) pairs),
;; and bidirectionally (mix of bound + unbound).

(define
  appendo
  (fn
    (l s ls)
    (conde
      ((nullo l) (== s ls))
      ((fresh (a d res) (conso a d l) (conso a res ls) (appendo d s res))))))

;; --- membero ---
;; (membero x l) — x appears (at least once) in l.

(define
  membero
  (fn
    (x l)
    (conde
      ((fresh (d) (conso x d l)))
      ((fresh (a d) (conso a d l) (membero x d))))))

(define
  reverseo
  (fn
    (l r)
    (conde
      ((nullo l) (nullo r))
      ((fresh (a d res-rev) (conso a d l) (reverseo d res-rev) (appendo res-rev (list a) r))))))

(define
  lengtho
  (fn
    (l n)
    (conde
      ((nullo l) (== n :z))
      ((fresh (a d n-1) (conso a d l) (== n (list :s n-1)) (lengtho d n-1))))))

(define
  inserto
  (fn
    (a l p)
    (conde
      ((conso a l p))
      ((fresh (h t pt) (conso h t l) (conso h pt p) (inserto a t pt))))))

(define
  permuteo
  (fn
    (l p)
    (conde
      ((nullo l) (nullo p))
      ((fresh (a d perm-d) (conso a d l) (permuteo d perm-d) (inserto a perm-d p))))))

(define
  flatteno
  (fn
    (tree flat)
    (conde
      ((nullo tree) (nullo flat))
      ((pairo tree)
        (fresh
          (h t hf tf)
          (conso h t tree)
          (flatteno h hf)
          (flatteno t tf)
          (appendo hf tf flat)))
      ((nafc (nullo tree)) (nafc (pairo tree)) (== flat (list tree))))))


(define
  rembero
  (fn
    (x l out)
    (conde
      ((nullo l) (nullo out))
      ((fresh (a d) (conso a d l) (== a x) (== out d)))
      ((fresh (a d res) (conso a d l) (nafc (== a x)) (conso a res out) (rembero x d res))))))

(define
  assoco
  (fn
    (key pairs val)
    (fresh
      (rest)
      (conde
        ((conso (list key val) rest pairs))
        ((fresh (other) (conso other rest pairs) (assoco key rest val)))))))

(define
  nth-o
  (fn
    (n l elem)
    (conde
      ((== n :z) (fresh (d) (conso elem d l)))
      ((fresh (n-1 a d) (== n (list :s n-1)) (conso a d l) (nth-o n-1 d elem))))))

(define
  samelengtho
  (fn
    (l1 l2)
    (conde
      ((nullo l1) (nullo l2))
      ((fresh (a d a-prime d-prime) (conso a d l1) (conso a-prime d-prime l2) (samelengtho d d-prime))))))