;;;;;;;;;
;; Nat ;;
;;;;;;;;;

;; Define Nat
(: Nat Type)
(: Z Nat)
(: S (-> Nat Nat))

;; Define <=
(: <= (-> $a $a Bool))
(= (<= $x $y) (or (< $x $y) (== $x $y)))

;; Define cast functions between Nat and Number
(: fromNumber (-> Number Nat))
(= (fromNumber $n) (if (<= $n 0) Z (S (fromNumber (- $n 1)))))
(: fromNat (-> Nat Number))
(= (fromNat Z) 0)
(= (fromNat (S $k)) (+ 1 (fromNat $k)))

;;;;;;;;;;;;;;;;;
;; Bare Entail ;;
;;;;;;;;;;;;;;;;;

;; Variant of entail where ⊢ is not wrapped around the knowledge base.

! "=== Bare Entail ==="

;; Backward chainer
(: bc_bare (-> Atom (-> Atom) (-> Atom) Nat Atom))
;; Base case
(= (bc_bare $conclusion $kb $rb $depth)
   (let $conclusion ($kb) $conclusion))
;; Recursive step
(= (bc_bare $conclusion $kb $rb (S $k))
   (let* (((⊢ $premise1 $premise2 $conclusion) ($rb))
          ($premise1 (bc_bare $premise1 $kb $rb $k))
          ($premise2 (bc_bare $premise2 $kb $rb $k)))
     $conclusion))

;; Knowledge base
(: kb_bare (-> Atom))
(= (kb_bare) (superpose ((→ A B)
                         (→ B C)
                         A)))

;; Rule base
(: rb_bare (-> Atom))
(= (rb_bare) (⊢
              ;; Premises
              (→ $p $q)
              $p
              ;; Conclusion
              $q))

;; Test backward chainer
!(assertEqual
  (bc_bare A kb_bare rb_bare Z)
  A)
!(assertEqual
  (bc_bare B kb_bare rb_bare (fromNumber 1))
  B)
!(assertEqual
  (bc_bare C kb_bare rb_bare (fromNumber 2))
  C)
!(assertEqualToResult
  (bc_bare D kb_bare rb_bare (fromNumber 4))
  ())

;;;;;;;;;;;;;;
;; Equality ;;
;;;;;;;;;;;;;;

! "=== Equality ==="

;; Knowledge base
(: kb_eq (-> Atom))
(= (kb_eq) (superpose ((→ A B)
                       (→ B C)
                       A)))

;; Backward chainer.  The rule based is directly embedded in the
;; backward chainer function.
(: bc_eq (-> Atom (-> Atom) Nat Atom))
;; Base case
(= (bc_eq $conclusion $kb $depth)
   (let $conclusion ($kb) $conclusion))
;; Recursive step (modus ponens)
(= (bc_eq $q $kb (S $k))
   (let* (((→ $p $q) (bc_eq (→ $p $q) $kb $k)) ; premise1
          ($p (bc_eq $p $kb $k)))              ; premise2
     $q))                                      ; conclusion
;; Recursive step (deduction)
(= (bc_eq (→ $p $r) $kb (S $k))
   (let* (((→ $p $q) (bc_eq (→ $p $q) $kb $k))  ; premise1
          ((→ $q $r) (bc_eq (→ $q $r) $kb $k))) ; premise2
     (→ $p $r)))                                ; conclusion

;; Test backward chainer
!(assertEqual
  (bc_eq A kb_eq Z)
  A)
!(assertEqual
  (bc_eq B kb_eq (fromNumber 1))
  B)
!(assertEqualToResult
  (bc_eq C kb_eq (fromNumber 2))
  (C   ; 2 modus ponens
   C)) ; deduction + modus ponens
!(assertEqual
  (bc_eq (→ A B) kb_eq (fromNumber 0))
  (→ A B))
!(assertEqual
  (bc_eq (→ B C) kb_eq (fromNumber 0))
  (→ B C))
!(assertEqual
  (bc_eq (→ A C) kb_eq (fromNumber 2))
  (→ A C))

;;;;;;;;;;;;;;;;;;;;;;;
;; Bare Entail Match ;;
;;;;;;;;;;;;;;;;;;;;;;;

;; Like bare entail but let is replaced by match

! "=== Bare Entail Match ==="

;; Knowledge base
!(bind! &kb_bem (new-space))
!(add-atom &kb_bem (→ A B))
!(add-atom &kb_bem (→ B C))
!(add-atom &kb_bem A)

;; Rule base
!(bind! &rb_bem (new-space))
!(add-atom &rb_bem (⊢
                    ;; Premises
                    (→ $p $q)
                    $p
                    ;; Conclusion
                    $q))
!(add-atom &rb_bem (⊢
                    ;; Premises
                    (→ $p $q)
                    (→ $q $r)
                    ;; Conclusion
                    (→ $p $r)))

;; Backward chainer
(: bc_bem (-> Atom Nat Atom))
;; Base case
(= (bc_bem $conclusion $depth)
   (match &kb_bem $conclusion $conclusion))
;; Recursive step
(= (bc_bem $conclusion (S $k))
   (match &rb_bem (⊢ $premise1 $premise2 $conclusion)
          (let* (($premise1 (bc_bem $premise1 $k))
                 ($premise2 (bc_bem $premise2 $k)))
            $conclusion)))

;; Test backward chainer
!(assertEqual
  (bc_bem A Z)
  A)
!(assertEqual
  (bc_bem B (fromNumber 1))
  B)
!(assertEqualToResult
  (bc_bem C (fromNumber 2))
  (C   ; 2 modus ponens
   C)) ; deduction + modus ponens
!(assertEqual
  (bc_bem (→ A B) (fromNumber 0))
  (→ A B))
!(assertEqual
  (bc_bem (→ B C) (fromNumber 0))
  (→ B C))
!(assertEqual
  (bc_bem (→ A C) (fromNumber 2))
  (→ A C))

;;;;;;;;;;;;;;;;;;;;
;; Equality Match ;;
;;;;;;;;;;;;;;;;;;;;

;; Like equality but let is replaced by match

! "=== Equality Match ==="

;; Knowledge base
!(bind! &kb_em (new-space))
!(add-atom &kb_em (→ A B))
!(add-atom &kb_em (→ B C))
!(add-atom &kb_em A)

;; Backward chainer.  The rule based is directly embedded in the
;; backward chainer function.
(: bc_em (-> Atom Nat Atom))
;; Base case
(= (bc_em $conclusion $depth)
   (match &kb_em $conclusion $conclusion))
;; Recursive step (modus ponens)
(= (bc_em $q (S $k))
   (let* (((→ $p $q) (bc_em (→ $p $q) $k)) ; premise1
          ($p (bc_em $p $k)))              ; premise2
     $q))                                  ; conclusion
;; Recursive step (deduction)
(= (bc_em (→ $p $r) (S $k))
   (let* (((→ $p $q) (bc_em (→ $p $q) $k))  ; premise1
          ((→ $q $r) (bc_em (→ $q $r) $k))) ; premise2
     (→ $p $r)))                            ; conclusion

;; Test backward chainer
!(assertEqual
  (bc_em A Z)
  A)
!(assertEqual
  (bc_em B (fromNumber 1))
  B)
!(assertEqualToResult
  (bc_em C (fromNumber 2))
  (C   ; 2 modus ponens
   C)) ; deduction + modus ponens
!(assertEqual
  (bc_em (→ A B) (fromNumber 0))
  (→ A B))
!(assertEqual
  (bc_em (→ B C) (fromNumber 0))
  (→ B C))
!(assertEqual
  (bc_em (→ A C) (fromNumber 2))
  (→ A C))

;;;;;;;;;;;;;;;;;;;;;;;;
;; DTL Equality Match ;;
;;;;;;;;;;;;;;;;;;;;;;;;

;; Like DTL (Synthesizer) but the rules are hard wired in the backward
;; chainer and match is used instead of let (as far as matching
;; premises is concerned).

! "=== DTL Equality Match ==="

;; Knowledge base
!(bind! &kb_dem (new-space))
!(add-atom &kb_dem (: ab (→ A B)))
!(add-atom &kb_dem (: bc (→ B C)))
!(add-atom &kb_dem (: a A))

;; Backward chainer.  The rule based is directly embedded in the
;; backward chainer function.
(: bc_dem (-> Atom Nat Atom))
;; Base case
(= (bc_dem (: $prf $conclusion) $depth)
   (match &kb_dem (: $prf $conclusion) (: $prf $conclusion)))
;; Recursive step (modus ponens)
(= (bc_dem (: $prf_q $q) (S $k))
   (let* (((: $prf_pq (→ $p $q)) (bc_dem (: $prf_pq (→ $p $q)) $k)) ; premise1
          ((: $prf_p $p) (bc_dem (: $prf_p $p) $k))                 ; premise2
          ((: (ModusPonens $prf_pq $prf_p) $q) (: $prf_q $q)))      ; conclusion
     (: $prf_q $q)))
;; Recursive step (deduction)
(= (bc_dem (: $prf_pr (→ $p $r)) (S $k))
   (let* (((: $prf_pq (→ $p $q)) (bc_dem (: $prf_pq (→ $p $q)) $k))          ; premise1
          ((: $prf_qr (→ $q $r)) (bc_dem (: $prf_qr (→ $q $r)) $k))          ; premise2
          ((: (Deduction $prf_pq $prf_qr) (→ $p $r)) (: $prf_pr (→ $p $r)))) ; conclusion
     (: $prf_pr (→ $p $r))))

;; Test backward chainer
!(assertEqual
  (bc_dem (: $prf A) Z)
  (: a A))
!(assertEqual
  (bc_dem (: $prf (→ A B)) Z)
  (: ab (→ A B)))
!(assertEqual
  (bc_dem (: $prf B) (fromNumber 1))
  (: (ModusPonens ab a) B))
!(assertEqualToResult
  (bc_dem (: $prf C) (fromNumber 2))
  ((: (ModusPonens bc (ModusPonens ab a)) C)
   (: (ModusPonens (Deduction ab bc) a) C)))
!(assertEqual
  (bc_dem (: $prf (→ A B)) (fromNumber 0))
  (: ab (→ A B)))
!(assertEqual
  (bc_dem (: $prf (→ B C)) (fromNumber 0))
  (: bc (→ B C)))
!(assertEqual
  (bc_dem (: $prf (→ A C)) (fromNumber 2))
  (: (Deduction ab bc) (→ A C)))