;; Unify
;; Takes two FOL expressions and returns a list of substitutions.
;; FOL expressions can have:
;; not, and, or, var, forall, exists, and functions
;; (Predicates, Equality, and Functions are all treated identically)
;;
;; Returns the assignment (empty assignment means sentences are identical)
;; Returns the list ("Failed") on failiure.
(defun unify (s1 s2 &optional (try2 nil))
  (reverse
    ((lambda (assignment) ;; Cleanup: Check for failure.
       (if (member "Failed" assignment :test #'equal)
	 (list "Failed")
	 (do* ((a assignment (subst-fol a a)) ;; Make sure all assignments are in their final form.
	       (b (subst-fol a a) (subst-fol a b)))
	   ((equal a b) b))))
     (if (equal s1 s2)
       nil
       (if (listp s1)
	 (if (eq (car s1) 'var) ;; If s1 is a variable just set it to s2.
	   (list (list (cadr s1) s2))
	   (if (listp s2)
	     (if (= (length s1) (length s2))
	       ((lambda (a) (if (member "Failed" a :test #'equal)
			      (unify-compound s2 s1) ;; Recalculate on failiure
			      a))
		(unify-compound s1 s2))
	       (if (not try2)
		 (unify s2 s1 T) ;; Maybe s2 is a variable?
		 (list "Failed")))))
	 (if (listp s2) (unify s2 s1) (list "Failed")))))))

;; Helper function:
;; unify-compound unifies compound sentences/terms s1 and s2
;; (both must be lists)
(defun unify-compound (s1 s2)
  (if (and s1 s2)
    ((lambda (assignment)
       (append (unify-compound (subst-fol assignment (cdr s1))
			      (subst-fol assignment (cdr s2)))
	      assignment))
     (unify (car s1) (car s2)))))

;; Helper function: Applies a stubstituion based on an assignment to a sentence
(defun subst-fol (assignment sentence)
  (if (and assignment sentence (listp sentence))
    (if (eq (car sentence) 'var)
      ((lambda (x) (if x (cadar x) sentence))
       (member (cadr sentence) assignment :test (lambda (a b) (eq a (car b)))))
      (cons (subst-fol assignment (car sentence))
	    (subst-fol assignment (cdr sentence))))
    sentence ;; Nothing we can do here.
    ))