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(in-package "ACL2")
(set-state-ok t)
;; This clause processor lets one replace one of the hyps (i.e. the negation of
;; one of the literals of the clause) with something that hyp provably
;; implies. See the comment above (defun replace-impl ... ) below.
(defevaluator use-impl-ev use-impl-ev-list
((not x) (if a b c) (implies a b)))
(defthm conjoin-clauses-of-one
(equal (conjoin-clauses (list x))
(disjoin x)))
(defthm use-impl-ev-conjoin-clauses-of-two-or-more
(iff (use-impl-ev (conjoin-clauses (cons x (cons y z))) a)
(and (use-impl-ev (disjoin x) a)
(use-impl-ev (conjoin-clauses (cons y z)) a))))
(in-theory (disable conjoin-clauses))
(defthm use-impl-ev-of-disjoin1
(iff (use-impl-ev (disjoin (cons x y)) a)
(or (use-impl-ev x a)
(use-impl-ev (disjoin y) a))))
(defthm use-impl-ev-of-conjoin1
(iff (use-impl-ev (conjoin (cons x y)) a)
(and (use-impl-ev x a)
(use-impl-ev (conjoin y) a))))
(defthm use-impl-ev-of-disjoin2
(implies (not (consp x))
(equal (use-impl-ev (disjoin x) a) nil)))
(defthm use-impl-ev-of-conjoin2
(implies (not (consp x))
(equal (use-impl-ev (conjoin x) a) t)))
(defthm pseudo-term-listp-cons
(iff (pseudo-term-listp (cons x y))
(and (pseudo-termp x)
(pseudo-term-listp y))))
(in-theory (disable conjoin disjoin pseudo-term-listp))
(defun replace-impl-fn (hyp sense repl clause)
(declare (xargs :guard t))
(if (atom clause)
(mv t clause)
(let ((lit (car clause)))
(if sense
(if (equal lit hyp)
(mv nil (cons (list 'not repl) (cdr clause)))
(mv-let (samep rest)
(replace-impl-fn hyp sense repl (cdr clause))
(if samep
(mv t clause)
(mv nil (cons (car clause) rest)))))
(if (and (consp lit)
(eq (car lit) 'not)
(consp (cdr lit))
(equal (cadr lit) hyp))
(mv nil (cons (list 'not repl) (cdr clause)))
(mv-let (samep rest)
(replace-impl-fn hyp sense repl (cdr clause))
(if samep
(mv t clause)
(mv nil (cons (car clause) rest)))))))))
(defthm replace-impl-fn-same
(implies (mv-nth 0 (replace-impl-fn hyp sense repl clause))
(equal (mv-nth 1 (replace-impl-fn hyp sense repl clause))
clause)))
(defthm replace-impl-fn-correct
(implies (and (use-impl-ev (disjoin (mv-nth 1 (replace-impl-fn
hyp sense repl clause))) a)
(or (if sense
(use-impl-ev hyp a)
(not (use-impl-ev hyp a)))
(use-impl-ev repl a)))
(use-impl-ev (disjoin clause) a))
:rule-classes :forward-chaining)
(defun replace-impl-cp (clause term)
(declare (xargs :guard t))
(if (eql (len term) 2)
(let ((hyp (car term))
(repl (cadr term)))
(mv-let (samep cl)
(if (and (consp hyp) (eq (car hyp) 'not) (consp (cdr hyp)))
(replace-impl-fn (cadr hyp) t repl clause)
(replace-impl-fn hyp nil repl clause))
(if samep
(list clause)
(list cl (list `(not ,hyp) repl)))))
(list clause)))
(defthm replace-impl-cp-correct
(implies (and (pseudo-term-listp cl)
(alistp a)
(use-impl-ev (conjoin-clauses
(replace-impl-cp cl term))
a))
(use-impl-ev (disjoin cl) a))
:rule-classes :clause-processor)
(mutual-recursion
(defun term-sublis (al x)
(declare (xargs :guard (and (alistp al)
(pseudo-termp x))
:mode :program))
(cond ((atom x)
(let ((entry (assoc x al)))
(if entry (cdr entry) x)))
((eq (car x) 'quote) x)
(t
;; Don't dive into lambdas, but do term-sublis the args.
(cons (car x) (term-list-sublis al (cdr x))))))
(defun term-list-sublis (al x)
(declare (xargs :guard (and (alistp al)
(pseudo-term-listp x))))
(if (atom x)
nil
(cons (term-sublis al (car x))
(term-list-sublis al (cdr x))))))
(defun find-instance-fn (x sense cl)
(declare (xargs :mode :program))
(if (atom cl)
(mv nil nil nil)
(if sense
(mv-let (okp al)
(one-way-unify x (car cl))
(if okp
(mv okp (list 'not (car cl)) al)
(find-instance-fn x sense (cdr cl))))
(if (and (consp (car cl))
(eq (car (car cl)) 'not))
(mv-let (okp al)
(one-way-unify x (cadr (car cl)))
(if okp
(mv okp (cadr (car cl)) al)
(find-instance-fn x sense (cdr cl))))
(find-instance-fn x sense (cdr cl))))))
(defun find-instance (hyp-pat concl-pat cl)
(declare (xargs :mode :program))
(mv-let (ok hyp alst)
(if (and (consp hyp-pat)
(eq (car hyp-pat) 'not))
(find-instance-fn (cadr hyp-pat) t cl)
(find-instance-fn hyp-pat nil cl))
(mv ok hyp (and ok (term-sublis alst concl-pat)))))
;; This is a computed hint which searches cl for a negated literal HYP that
;; unifies with (the translated version of) hyp-pat. It then replaces this
;; literal with the result CONCL of substituting the variables of concl-pat for
;; the ones that matched in hyp-pat. It is then also necessary to prove that
;; HYP implies CONCL.
(defun replace-impl (hyp-pat concl-pat cl state)
(declare (xargs :mode :program))
(er-let* ((hyp-pat (translate hyp-pat t t t 'replace-impl (w state) state))
(concl-pat (translate concl-pat t t t 'replace-impl (w state) state)))
(mv-let (ok hyp concl)
(find-instance hyp-pat concl-pat cl)
(value
(and ok
`(:clause-processor (replace-impl-cp clause
',(list hyp concl))))))))
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