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|
#|-*-Lisp-*-=================================================================|#
#| |#
#| Copyright © 2005-7 Rockwell Collins, Inc. All Rights Reserved. |#
#| |#
#|===========================================================================|#
(in-package "ACL2")
(include-book "defminterm")
;;(xxclude-book "basic" :dir :lists)
(defun natural-listp (list)
(declare (type t list))
(if (consp list)
(and (natp (car list))
(natural-listp (cdr list)))
(null list)))
(defthm true-listp-from-natural-listp
(implies (natural-listp x)
(true-listp x))
:rule-classes (:forward-chaining))
(defminterm ack (x y xargs)
(declare (xargs :verify-guards nil)
(type (integer 0 *) x y)
(type (satisfies natural-listp) xargs))
(if (and (not (consp xargs)) (zp x)) (+ y 1)
(if (zp x) (ack (car xargs) (+ y 1) (cdr xargs))
(if (zp y) (ack (1- x) 1 xargs)
(ack x (1- y) (cons (1- x) xargs))))))
(defun ack_induction (x y r s)
(declare (xargs :measure (ack_measure x y r)))
(if (ack_terminates x y r)
(if (and (not (consp r)) (zp x)) (+ y 1)
(if (zp x) (ack_induction (car r) (+ y 1) (cdr r) (cdr s))
(if (zp y) (ack_induction (1- x) 1 r s)
(ack_induction x (1- y) (cons (1- x) r) (cons (1- x) s)))))
(cons s (+ y 1))))
;;
;;
;;
(defun head-equal (s r)
(if (consp s)
(and (consp r)
(equal (car s) (car r))
(head-equal (cdr s) (cdr r)))
t))
;; jcd - Removing list-equal -- use list::equiv instead, it is provably
;; equal and already has lots of nice rules.
(defun list-equal (x y)
(if (consp x)
(and (consp y)
(equal (car x) (car y))
(list-equal (cdr x) (cdr y)))
(not (consp y))))
(DEFTHM OPEN-list-equal
(IMPLIES (AND (CONSP X) (CONSP Y))
(EQUAL (LIST-EQUAL X Y)
(AND (EQUAL (CAR X) (CAR Y))
(LIST-EQUAL (CDR X) (CDR Y)))))
:HINTS (("Goal" :IN-THEORY (ENABLE LIST-EQUAL))))
(defequiv list-equal)
(defcong list-equal equal (consp x) 1)
(defcong list-equal list-equal (cons a b) 2)
(defcong list-equal equal (car x) 1)
(defcong list-equal list-equal (cdr x) 1)
(in-theory (disable list-equal))
;;
(DEFTHM CDR-APPEND-CONSP
(IMPLIES (CONSP X)
(EQUAL (CDR (APPEND X Y))
(APPEND (CDR X) Y)))
:HINTS (("goal" :IN-THEORY (ENABLE APPEND))))
(DEFTHM CAR-APPEND-CONSP
(IMPLIES (CONSP X)
(EQUAL (CAR (APPEND X Y)) (CAR X)))
:HINTS (("goal" :IN-THEORY (ENABLE APPEND))))
(DEFTHM LEN-APPEND
(EQUAL (LEN (APPEND X Y))
(+ (LEN X) (LEN Y)))
:HINTS (("Goal" :IN-THEORY (ENABLE APPEND))))
(DEFTHM LEN-CONS
(EQUAL (LEN (CONS A X)) (+ 1 (LEN X)))
:HINTS (("Goal" :IN-THEORY (ENABLE LEN))))
(DEFTHM APPEND-CONSP-TYPE-TWO
(IMPLIES (CONSP Y) (CONSP (APPEND X Y)))
:RULE-CLASSES ((:TYPE-PRESCRIPTION)))
(DEFTHM APPEND-OF-CONS
(EQUAL (APPEND (CONS A X) Y)
(CONS A (APPEND X Y))))
(DEFTHM APPEND-OF-NON-CONSP-ONE
(IMPLIES (NOT (CONSP X))
(EQUAL (APPEND X Y) Y))
:HINTS (("Goal" :IN-THEORY (ENABLE APPEND))))
;;
(defthm list-equal-implication
(implies (list-equal r s)
(and (head-equal r s)
(<= (len r) (len s))))
:rule-classes (:forward-chaining))
(defthm not-consp-implication
(implies (not (consp r))
(and (head-equal r s)
(<= (len r) (len s)))))
(defthm head-equal-append
(head-equal x (append x y)))
(in-theory (disable append))
;; we get this from lists
;; (defthm len-append
;; (<= (len x) (len (append x y))))
(defthm ack_terminates_from_ack_terminates
(implies (and (ack_terminates x y s)
(head-equal r s)
(<= (len r) (len s)))
(ack_terminates x y r))
:hints (("goal" :do-not '(generalize eliminate-destructors)
:induct (ack_induction x y s r))))
(encapsulate
()
(local
(encapsulate
()
;; jcd - note, if you can prove that ack_terminates returns a
;; boolean, then you can instead immediately prove
(defthm ack_terminates_list_equal
(implies (and (ack_terminates x y s)
(list-equal r s))
(ack_terminates x y r)))
(defthm not_ack_terminates_list_equal
(implies (and (not (ack_terminates x y r))
(list-equal r s))
(not (ack_terminates x y s))))
))
(defcong list-equal iff (ack_terminates x y r) 3)
)
(defthm ack_terminates_nil
(implies (and (ack_terminates x y s)
(syntaxp (not (quotep s))))
(ack_terminates x y nil))
:rule-classes (:forward-chaining))
(defcong list-equal equal (ack x y r) 3
:hints (("goal" :induct (ack_induction x y r r-equiv))))
;; jcd - removing this, it seems redundant with the congruence rule
;; above.
;; (defthm ack_reduction_free
;; (implies (and (ack_terminates x y r)
;; (syntaxp (not (acl2::term-order r s)))
;; (list-equal s r))
;; (equal (ack x y s)
;; (ack x y r))))
(defthmd open_ack_terminates-1
(implies
(syntaxp (and (or (symbolp x) (quotep x))
(or (symbolp y) (quotep y))))
(and
(implies
(not (acktest_body x y xargs))
(iff
(ack_terminates x y xargs)
(let
((x (car (ackstep_body_1 (list x y xargs))))
(y (car (cdr (ackstep_body_1 (list x y xargs)))))
(xargs
(car (cdr (cdr (ackstep_body_1 (list x y xargs)))))))
(ack_terminates x y xargs))))
(implies (acktest_body x y xargs)
(ack_terminates x y xargs)))))
(defthm ack_termiantes_on_ack
(implies
(and
(consp r)
(ack_terminates x y (append s r)))
(ack_terminates (car r) (ack x y s) (cdr r)))
:hints (("goal" :in-theory (enable open_ack_terminates-1)
:induct (ack x y s))))
(defthm ack_measure_on_ack
(implies
(and
(consp r)
(ack_terminates x y (append s r)))
(equal (ack_measure x y (append s r))
(+ (ack_measure x y s)
(ack_measure (car r) (ack x y s) (cdr r)))))
:rule-classes nil
:hints (("goal" :in-theory (enable open_ack_terminates-1)
:induct (ack x y s))))
(defthm ack_measure_reduction
(implies
(ack_terminates x y (cons a r))
(equal (ack_measure x y (cons a r))
(+ (ack_measure x y nil)
(ack_measure a (ack x y nil) r))))
:hints (("goal" :in-theory (disable OPEN_ACK_MEASURE)
:use (:instance ack_measure_on_ack
(s nil)
(r (cons a r))))))
(defthm ack_on_ack
(implies
(and
(consp r)
(ack_terminates x y (append s r)))
(equal (ack x y (append s r))
(ack (car r) (ack x y s) (cdr r))))
:rule-classes nil
:hints (("goal" :in-theory (enable open_ack_terminates-1)
:induct (ack x y s))))
(defthm ack_reduction
(implies
(ack_terminates x y (cons a r))
(equal (ack x y (cons a r))
(ack a (ack x y nil) r)))
:hints (("goal" :use (:instance ack_on_ack
(s nil)
(r (cons a r))))))
(defun ak (x y)
(ack x y nil))
(defun ak_terminates (x y)
(ack_terminates x y nil))
(defun ak_measure (x y)
(ack_measure x y nil))
(defthm ak_spec
(equal (ak x y)
(if (ak_terminates x y)
(if (zp x) (+ y 1)
(if (zp y) (ak (1- x) 1)
(ak (1- x) (ak x (1- y)))))
(+ y 1)))
:rule-classes nil)
(defthm ak_measure_spec
(implies
(ak_terminates x y)
(equal (ak_measure x y)
(if (zp x) (o)
(if (zp y) (1+ (ak_measure (1- x) 1))
(+ 1 (ak_measure x (1- y))
(ak_measure (1- x) (ak x (1- y))))))))
:hints (("goal" :in-theory (disable OPEN_ACK_TERMINATES-1))))
(in-theory
(disable
ak
ak_measure
ak_terminates
))
|
