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(defconst *aces* '(A))
(defconst *faces* '(J Q K))
(defun acep (x)
(declare (xargs :guard t))
(and (member x *aces*) t))
(defun facep (x)
(declare (xargs :guard t))
(and (member x *faces*) t))
(defun rankp (x)
(declare (xargs :guard t))
(or (acep x)
(facep x)
(and (natp x)
(<= 2 x)
(<= x 10))))
(defun scorep (x)
(declare (xargs :guard t))
(case x
((BUST NATURAL-BLACKJACK) t)
(t (and (natp x)
(<= x 21)))))
(defun rank-list-p (x)
(declare (xargs :guard t))
(cond ((consp x)
(and (rankp (car x))
(rank-list-p (cdr x))))
(t (null x))))
(defun filter-natp (x)
(declare (xargs :guard (true-listp x)))
(cond ((endp x) nil)
((natp (car x))
(cons (car x) (filter-natp (cdr x))))
(t (filter-natp (cdr x)))))
(defun nat-list-p (x)
(declare (xargs :guard t))
(cond ((consp x)
(and (natp (car x))
(nat-list-p (cdr x))))
(t (null x))))
(defun sum (x)
(declare (xargs :guard (nat-list-p x)))
(cond ((endp x) 0)
(t (+ (car x) (sum (cdr x))))))
(defun filter-facep (x)
(declare (xargs :guard (true-listp x)))
(cond ((endp x) nil)
((facep (car x))
(cons (car x) (filter-facep (cdr x))))
(t (filter-facep (cdr x)))))
(defun face-list-p (x)
(declare (xargs :guard t))
(cond ((consp x)
(and (facep (car x))
(face-list-p (cdr x))))
(t (null x))))
(defun face-list-to-score (x)
(declare (xargs :guard (face-list-p x)))
(* 10 (length x)))
(defun score-without-aces (rl)
(declare (xargs :guard (rank-list-p rl)))
(+ (sum (filter-natp rl))
(face-list-to-score (filter-facep rl))))
(defthm natp/score-without-aces
(implies (rank-list-p x)
(natp (score-without-aces x))))
(defun ace-count (x)
(declare (xargs :guard (true-listp x)))
(cond ((endp x) 0)
((acep (car x)) (+ 1 (ace-count (cdr x))))
(t (ace-count (cdr x)))))
(defthm natp/ace-count
(implies (rank-list-p x)
(natp (ace-count x))))
(defun add-ace-scores (s ac)
(declare (xargs :guard (and (natp s)
(natp ac))))
(cond
((zerop ac) s)
((<= (+ s 11) 21) (+ s 11 (- ac 1)))
(t (+ s 1 (- ac 1)))))
(defthm natp/add-ace-scores
(implies (and (natp x) (natp y))
(natp (add-ace-scores x y))))
(defun score* (rl)
(declare (xargs :guard (rank-list-p rl)))
(add-ace-scores (score-without-aces rl)
(ace-count rl)))
(defthm natp/score*-1
(implies (rank-list-p rl)
(natp (+ (sum (filter-natp rl))
(* 10 (len (filter-facep rl)))))))
(defthm natp/score*
(implies (rank-list-p rl)
(natp (score* rl))))
(defun score (rl)
(declare (xargs :guard (rank-list-p rl)))
(let ((s (score* rl)))
(cond
((and (< 21 s)) 'BUST)
((and (= (len rl) 2) (= s 21))
'NATURAL-BLACKJACK)
(t s))))
(defthm scorep/score-1
(implies (rank-list-p x)
(natp (+ (sum (filter-natp x))
(* 10 (len (filter-facep x)))))))
(defthm scorep/score
(implies (rank-list-p x)
(scorep (score x)))
:hints (("Goal" :hands-off (add-ace-scores))))
(defun judge/score (player dealer)
(declare (xargs :guard (and (scorep player)
(scorep dealer))))
(cond
((and (equal player 'NATURAL-BLACKJACK)
(equal dealer 'NATURAL-BLACKJACK))
'PUSH)
((equal player 'NATURAL-BLACKJACK) 'WIN)
((equal dealer 'NATURAL-BLACKJACK) 'LOSE)
((equal player 'BUST) 'LOSE)
((equal dealer 'BUST) 'WIN)
((< player dealer) 'LOSE)
((> player dealer) 'WIN)
(t 'PUSH)))
(defun judge/ranks (player dealer)
(declare (xargs :guard (and (rank-list-p player)
(rank-list-p dealer))
:guard-hints (("Goal"
:hands-off (score)
:use (scorep/score)))))
(judge/score (score player)
(score dealer)))
(defthm bust-and-lose
(equal (judge/score 'BUST x) 'LOSE))
(defthm draw-a-face-and-blackjack
(implies (facep r)
(equal (score (list 'A r))
'NATURAL-BLACKJACK)))
(defthm bust-12
(let ((rs '(K 2))
(new 'J))
(and (= (score* rs) 12)
(equal (score (cons new rs)) 'BUST))))
(defthm safe-hit
(implies (and (rankp r1)
(rankp r2)
(rankp new)
(<= (score* (list r1 r2)) 11))
(not (equal (score (list new r1 r2))
'BUST))))
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