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|
#|
; Certification instructions:
(certify-book "perm")
JSM June, 2000
|#
(in-package "ACL2")
(defun del (x y)
(if (consp y)
(if (equal x (car y))
(cdr y)
(cons (car y) (del x (cdr y))))
y))
(defun mem (e x)
(if (consp x)
(or (equal e (car x))
(mem e (cdr x)))
nil))
(defun perm (x y)
(if (consp x)
(and (mem (car x) y)
(perm (cdr x) (del (car x) y)))
(not (consp y))))
; The following could be proved right now.
; (local
; (defthm perm-reflexive
; (perm x x)))
(local
(defthm perm-cons
(implies (mem a x)
(equal (perm x (cons a y))
(perm (del a x) y)))
:hints (("Goal" :induct (perm x y)))))
(local
(defthm perm-symmetric
(implies (perm x y) (perm y x))))
(local
(defthm mem-del
(implies (mem a (del b x)) (mem a x))))
(local
(defthm perm-mem
(implies (and (perm x y)
(mem a x))
(mem a y))))
(local
(defthm comm-del
(equal (del a (del b x)) (del b (del a x)))))
(local
(defthm perm-del
(implies (perm x y)
(perm (del a x) (del a y)))))
; We now prove this because we give a hint.
(local
(defthm perm-transitive
(implies (and (perm x y) (perm y z)) (perm x z))
:hints (("Goal" :induct (and (perm x y) (perm x z))))))
(defequiv perm)
(defcong perm perm (cons x y) 2)
(defcong perm perm (append x y) 1)
(defcong perm perm (append x y) 2)
|
