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;; Copyright (C) 2017, Regents of the University of Texas
;; Written by Cuong Chau
;; License: A 3-clause BSD license. See the LICENSE file distributed with
;; ACL2.
;; Cuong Chau <ckcuong@cs.utexas.edu>
;; September 2018
(in-package "ADE")
(include-book "std/lists/take" :dir :system)
(include-book "std/util/bstar" :dir :system)
;; ======================================================================
;; BINARY-AND*
(defun binary-and* (x y)
(declare (xargs :guard t))
(if x y nil))
(defthm booleanp-binary-and*
(implies (booleanp y)
(booleanp (binary-and* x y)))
:rule-classes :type-prescription)
(in-theory (disable binary-and*))
(defund and*-macro (x)
(declare (xargs :guard t))
(cond ((atom x)
t)
((atom (cdr x))
(car x))
(t
`(binary-and* ,(car x)
,(and*-macro (cdr x))))))
(defmacro and* (&rest args)
(and*-macro args))
;; BINARY-OR*
(defun binary-or* (x y)
(declare (xargs :guard t))
(if x x y))
(defthm booleanp-binary-or*
(implies (and (booleanp x)
(booleanp y))
(booleanp (binary-or* x y)))
:rule-classes :type-prescription)
(in-theory (disable binary-or*))
(defund or*-macro (x)
(declare (xargs :guard t))
(cond ((atom x)
nil)
((atom (cdr x))
(car x))
(t
`(binary-or* ,(car x)
,(or*-macro (cdr x))))))
(defmacro or* (&rest args)
(or*-macro args))
;; NOT*
(defun not* (x)
(declare (xargs :guard t))
(not x))
(defthm booleanp-not*
(booleanp (not* x))
:rule-classes :type-prescription)
(in-theory (disable not*))
;; ======================================================================
;; Shuffle two lists
(defun insert (x i l)
(declare (xargs :guard (natp i)))
(cond
((atom l)
(list x))
((zp i)
(cons x l))
(t (cons (car l)
(insert x (1- i) (cdr l))))))
(defthm true-listp-insert
(implies (true-listp l)
(true-listp (insert x i l)))
:rule-classes :type-prescription)
(in-theory (disable insert))
(defun insert-up-to-i (x i l)
(declare (xargs :guard (natp i)))
(if (zp i)
(list (insert x i l))
(cons (insert x i l)
(insert-up-to-i x (1- i) l))))
(defthm true-list-listp-insert-up-to-i
(implies (true-listp l)
(true-list-listp (insert-up-to-i x i l)))
:rule-classes :type-prescription)
(in-theory (disable insert-up-to-i))
(defund cons-or-append (x y)
(declare (xargs :guard t))
(if (atom x)
(if (atom y)
(cons x (list y))
(cons x y))
(if (atom y)
(append (list-fix x) (list y))
(append (list-fix x) y))))
(defun cons-or-append-at-i (x i l)
(declare (xargs :guard (integerp i)))
(cond
((atom l)
(list x))
((= i 0)
(cons (cons-or-append x (car l))
(cdr l)))
((not (natp i)) nil)
(t (cons (car l)
(cons-or-append-at-i x (1- i) (cdr l))))))
(defthm true-listp-cons-or-append-at-i
(implies (true-listp l)
(true-listp (cons-or-append-at-i x i l)))
:rule-classes :type-prescription)
(in-theory (disable cons-or-append-at-i))
(defun cons-or-append-up-to-i (x i l)
(declare (xargs :measure (acl2-count (1+ i))
:guard (integerp i)))
(if (not (natp i))
nil
(cons (cons-or-append-at-i x i l)
(cons-or-append-up-to-i x (1- i) l))))
(defthm true-list-listp-cons-or-append-up-to-i
(implies (true-listp l)
(true-list-listp (cons-or-append-up-to-i x i l)))
:rule-classes :type-prescription)
(in-theory (disable cons-or-append-up-to-i))
(defund index-of-nested (k x)
(declare (xargs :guard t))
(cond ((atom x) nil)
((equal k (car x)) 0)
((atom (car x))
(b* ((res (index-of-nested k (cdr x))))
(and res (1+ res))))
(t (b* ((res1 (index-of-nested k (car x))))
(if res1
0
(b* ((res (index-of-nested k (cdr x))))
(and res (1+ res))))))))
(defun insert-up-to-rec (x y l)
(declare (xargs :guard (true-list-listp l)))
(if (atom l)
nil
(b* ((i (index-of-nested y (car l)))
(i (if (natp i) i (len (car l)))))
(append (append (insert-up-to-i x i (car l))
(cons-or-append-up-to-i x (1- i) (car l)))
(insert-up-to-rec x y (cdr l))))))
(defthm true-list-listp-of-append
(implies (and (true-list-listp x)
(true-list-listp y))
(true-list-listp (append x y)))
:rule-classes :type-prescription)
(defthm true-list-listp-insert-up-to-rec
(implies (true-list-listp l)
(true-list-listp (insert-up-to-rec x y l)))
:rule-classes :type-prescription)
(in-theory (disable insert-up-to-rec))
(defun shuffle (l1 l2)
(declare (xargs :guard (and (true-listp l1)
(true-listp l2))))
(if (atom l1)
(list l2)
(insert-up-to-rec (car l1) (cadr l1)
(shuffle (cdr l1) l2))))
(defthm true-list-listp-shuffle
(implies (and (true-listp l1)
(true-listp l2))
(true-list-listp (shuffle l1 l2)))
:rule-classes :type-prescription)
(in-theory (disable shuffle))
(defun shuffle-rec1 (x y)
(declare (xargs :guard (and (true-list-listp x)
(true-listp y))))
(if (atom x)
nil
(append (shuffle (car x) y)
(shuffle-rec1 (cdr x) y))))
(defun shuffle-rec2 (x y)
(declare (xargs :guard (and (true-listp x)
(true-list-listp y))))
(if (atom y)
nil
(append (shuffle x (car y))
(shuffle-rec2 x (cdr y)))))
;; Compute a powerset
(defund combine (x y)
(declare (xargs :guard t))
(list y (cons x y)))
(defund combine-rec (x y)
(declare (xargs :guard (true-listp y)))
(if (atom y)
(list (list x))
(append (combine x (car y))
(combine-rec x (cdr y)))))
(defund no-empty-powerset (x)
(declare (xargs :guard t))
(if (atom x)
nil
(combine-rec (car x)
(no-empty-powerset (cdr x)))))
(defund powerset (x)
(declare (xargs :guard t))
(cons nil (no-empty-powerset x)))
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