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(in-package "ACL2")
(include-book "textbook/chap11/perm-append" :dir :system)
;; --------------------------------------------------------------
;; bv-add definition and some theorems about it
(include-book "arithmetic-3/floor-mod/floor-mod" :dir :system)
(defund bv-add (x y)
(mod (+ (nfix x) (nfix y)) 32))
(defun bv-add-lex-< (x y)
(lexorder x y))
(defthm bv-commute
(implies (syntaxp (not (bv-add-lex-< x y)))
(equal (bv-add x y)
(bv-add y x)))
:hints (("Goal" :in-theory (enable bv-add))))
(defthm bv-commute-2
(equal (bv-add x (bv-add y z))
(bv-add y (bv-add x z)))
:hints (("Goal" :in-theory (enable bv-add)))
:rule-classes nil)
(defun make-bv-add (sym-list)
(cond
((atom sym-list)
nil)
((atom (cdr sym-list))
(car sym-list))
(t
`(bv-add ,(car sym-list)
,(make-bv-add (cdr sym-list))))))
(defevaluator evl evl-list
((if a b c) (bv-add x y)))
(defthm not-consp-cddr-perm
(implies
(and (perm x y)
(not (consp (cddr x))))
(not (consp (cddr y))))
:rule-classes nil)
(defthm perm-cdr
(implies
(and (perm x y)
(not (consp (cdr x))))
(not (consp (cdr y))))
:rule-classes nil)
(defthm del-implies-bv-add-equality-2-0
(IMPLIES (AND (CONSP X)
(CONSP (CDR X))
(IN A (CDR X))
(NOT (EQUAL A (CAR X)))
(NOT (CONSP (DEL A (CDR X)))))
(EQUAL (BV-ADD (EVL (CAR X) ENV)
(EVL (MAKE-BV-ADD (CDR X)) ENV))
(BV-ADD (EVL A ENV)
(EVL (CAR X) ENV))))
:hints (("Goal" :expand (in a (cdr x)))))
(defthm del-implies-bv-add-equality-2
(implies (and (in a x)
(consp x)
(consp (cdr x)))
(equal (evl (make-bv-add x) env)
(bv-add (evl a env) (evl (make-bv-add (del a x)) env))))
:hints (("Subgoal *1/4.2'" :use ((:instance bv-commute-2 (x (evl a env))
(y (evl (car x) env))
(z (evl (make-bv-add (del a
(cdr
x)))
env)))))
("Subgoal *1/4.1" :in-theory (disable make-bv-add evl-constraint-1)))
:rule-classes nil)
(defthm perm-implies-bv-add-equality
(implies
(perm sym-list perm-list)
(equal (evl (make-bv-add sym-list) env)
(evl (make-bv-add perm-list) env)))
:hints (("Goal" :induct (perm sym-list perm-list))
("Subgoal *1/2"
:use ((:instance del-implies-bv-add-equality-2
(a (car sym-list))
(x perm-list)
(env env))
(:instance perm-cdr
(x sym-list)
(y perm-list))))))
;; --------------------------------------------------------------
;; Definition and proof of Clause Processor
;; For now, lexorder will do.
(defun bv-add-lex-< (x y)
(lexorder x y))
(defun split-list (x)
(cond ((atom x) (mv nil nil))
((atom (cdr x)) (mv x nil))
(t (mv-let (a b)
(split-list (cddr x))
(mv (cons (car x) a) (cons (cadr x) b))))))
(defun merge2 (x y)
(declare (xargs :measure (+ (acl2-count x) (acl2-count y))))
(cond ((atom x) y)
((atom y) x)
((bv-add-lex-< (car x) (car y))
(cons (car x) (merge2 (cdr x) y)))
(t (cons (car y) (merge2 x (cdr y))))))
(defthm split-list-smaller
(implies (and (consp x) (consp (cdr x)))
(and (< (acl2-count (car (split-list x)))
(acl2-count x))
;; Originally we used (cadr ..) instead of (mv-nth 1 ..) below,
;; but the mv-nth didn't open up to cadr in the termination
;; proof of mergesort, so we are going with mv-nth below.
(< (acl2-count (mv-nth 1 (split-list x)))
(acl2-count x)))))
(defun mergesort (x)
(cond ((atom x) nil)
((atom (cdr x)) x)
(t (mv-let (a b)
(split-list x)
(merge2 (mergesort a) (mergesort b))))))
(defthm perm-append
(perm (append x y) (append y x)))
(defthm perm-append-del
(implies (and (consp y)
(in (car y) x))
(perm (append (del (car y) x) y)
(append x (cdr y)))))
(defthm merge2-is-append
(perm (merge2 x y)
(append x y))
:hints (("Goal" :induct (merge2 x y))))
(defthm perm-append-cons-2
(perm (append x (cons a y))
(cons a (append x y))))
(defthm perm-append-split-list
(perm (append (car (split-list lst)) (mv-nth 1 (split-list lst)))
lst))
(defthm perm-mergesort
(perm (mergesort lst) lst))
(defun simplify-bv-adds-in-flg-num (flg)
(cond
((equal 'expr-list flg)
0)
((equal 'expr flg)
1)
(t ;;(equal 'bv-add flg)
0)))
(defun simplify-bv-adds-in (flg x)
(declare (xargs :measure
(list* (cons 1 (1+ (acl2-count x)))
(simplify-bv-adds-in-flg-num flg))))
(cond
((equal 'expr-list flg)
(cond
((atom x)
nil)
(t
(cons (simplify-bv-adds-in 'expr (car x))
(simplify-bv-adds-in 'expr-list (cdr x))))))
((equal 'expr flg)
(cond
((atom x)
x)
((quotep x)
x)
((equal (car x) 'bv-add)
(let* ((add-lst (simplify-bv-adds-in 'bv-add x))
(add-lst (mergesort add-lst)))
(make-bv-add add-lst)))
(t
(cons (car x)
(simplify-bv-adds-in 'expr-list (cdr x))))))
(t ;;(equal 'bv-add flg)
;; x=(bv-add a0 a1)
(cond
((or (atom x) (not (equal (car x) 'bv-add)))
;; This case makes no sense but I added
;; it to make the termination proof easy.
nil)
(t
(let ((a0 (cadr x))
(a1 (caddr x)))
(cond
((and (consp a1)
(equal 'bv-add (car a1)))
(cons (simplify-bv-adds-in 'expr a0)
(simplify-bv-adds-in 'bv-add a1)))
(t
(list (simplify-bv-adds-in 'expr a0)
(simplify-bv-adds-in 'expr a1))))))))))
(defun bv-add-sort-cp (clause)
(list (simplify-bv-adds-in 'expr-list clause)))
(in-theory (disable mergesort perm))
(defthm bv-add-merge-sort-ok
(implies
(and (consp sym-lst)
(consp (cdr sym-lst)))
(equal (evl (make-bv-add (mergesort sym-lst)) env)
(evl (make-bv-add sym-lst) env)))
:hints (("Goal" :use ((:instance perm-implies-bv-add-equality
(sym-list sym-lst)
(perm-list (mergesort sym-lst))
(env env))))))
(defthm lemma0
(implies (and (consp x1)
(equal fn (car x1))
(not (equal fn 'quote))
(equal (evl-list args0 env) (evl-list (cdr x1) env)))
(equal (equal (evl (cons fn args0) env) (evl x1 env))
t))
:hints (("Goal" :in-theory (enable evl-constraint-0))))
(defthm correctness-of-bv-adds-in
(and (implies
(equal flg 'expr)
(equal (evl (simplify-bv-adds-in flg x) env)
(evl x env)))
(implies
(equal flg 'expr-list)
(equal (evl-list (simplify-bv-adds-in flg x) env)
(evl-list x env)))
(implies
(and (not (equal flg 'expr))
(not (equal flg 'expr-list))
(consp x)
(equal (car x) 'bv-add))
(equal (evl (make-bv-add (simplify-bv-adds-in flg x)) env)
(evl x env))))
:hints (;;("Goal" :in-theory (enable evl-constraint-0)))
("Subgoal *1/6" :use ((:instance evl-constraint-0
(x x) (a env)))))
:rule-classes nil)
(defthm correctness-of-bv-adds-in-expr
(equal (evl (simplify-bv-adds-in 'expr x) env)
(evl x env))
:hints (("Goal" :use ((:instance correctness-of-bv-adds-in
(x x) (env env) (flg 'expr)))))
:rule-classes nil)
(defthm correctness-of-bv-add-sort-cp
(implies (and (pseudo-term-listp clause)
(alistp env)
(evl (conjoin-clauses (bv-add-sort-cp clause)) env))
(evl (disjoin clause) env))
:hints (("Goal" :induct (disjoin clause))
("Subgoal *1/3" :use ((:instance correctness-of-bv-adds-in-expr
(x (car clause))
(env env))))
("Subgoal *1/2" :use ((:instance correctness-of-bv-adds-in-expr
(x (car clause))
(env env)))))
:rule-classes :clause-processor)
(in-theory (disable bv-commute))
|
