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;; Copyright (C) 2018, 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>
;; May 2019
;; Utility functions for indices
(in-package "ADE")
(include-book "utils")
(include-book "std/strings/decimal" :dir :system)
(defthm acl2::equal-of-appends-when-true-listps
(implies (and (true-listp x1)
(true-listp x2))
(equal (equal (append x1 y)
(append x2 y))
(equal x1 x2))))
(in-theory (disable acl2::equal-of-append-and-append-same-arg2))
(include-book "std/strings/istrprefixp" :dir :system)
;; ======================================================================
(defun str-append-symbol-underscore (s)
(declare (xargs :guard (symbolp s)))
(string-append (symbol-name s) "_"))
(defun append-symbol-string (s str)
(declare (xargs :guard (and (symbolp s)
(stringp str))))
(string-append (str-append-symbol-underscore s)
str))
;; SI
(defun si (s idx)
(declare (xargs :guard (and (symbolp s)
(natp idx))))
(intern$ (append-symbol-string s (nat-to-dec-string (nfix idx)))
"ADE"))
(local
(defthm symbol-inequality
(implies (and (not (equal (symbol-name s1)
(symbol-name s2)))
(symbolp s1)
(symbolp s2))
(not (equal s1 s2)))))
(defthm not-iprefixp-append
(implies (not (iprefixp x y))
(not (iprefixp (append x a) y)))
:hints (("Goal" :in-theory (enable iprefixp))))
(defthm not-istrprefixp-str-append-symbol-underscore
(implies (not (istrprefixp (str-append-symbol-underscore s2)
(symbol-name s1)))
(not (istrprefixp (str-append-symbol-underscore (si s2 n))
(symbol-name s1))))
:hints (("Goal"
:use (:instance not-iprefixp-append
(x (append (explode (symbol-name s2))
'(#\_)))
(y (explode (symbol-name s1)))
(a (append (explode (nat-to-dec-string (nfix n)))
'(#\_))))
:in-theory (disable not-iprefixp-append))))
(defthm not-equal-with-si-of-diff-symbol
(implies (not (istrprefixp (str-append-symbol-underscore s2)
(symbol-name s1)))
(and (not (equal (symbol-name s1)
(symbol-name (si s2 n))))
(not (equal s1 (si s2 n)))))
:hints (("Goal" :in-theory (disable str-append-symbol-underscore)))
:rule-classes ((:rewrite
:corollary
(implies (not (istrprefixp (str-append-symbol-underscore s2)
(symbol-name s1)))
(not (equal (symbol-name s1)
(symbol-name (si s2 n))))))
(:rewrite
:corollary
(implies (not (istrprefixp (str-append-symbol-underscore s2)
(symbol-name s1)))
(not (equal s1 (si s2 n)))))))
(encapsulate
()
(local
(defthm not-member-underscore-nat-to-dec-string
(not (member #\_ (explode (nat-to-dec-string n))))
:hints (("Goal" :in-theory (enable basic-nat-to-dec-chars
nat-to-dec-chars
nat-to-dec-string)))))
(local
(defthmd si-of-diff-symbols-aux-1
(implies (and (< (length str1) (length str2))
(equal (char str1 (1- (length str1)))
(char str2 (1- (length str2))))
(char str1 (1- (length str1)))
(not (member (char str1 (1- (length str1)))
(explode str3))))
(not (equal (char (string-append str1 str3)
(1- (length str2)))
(char (string-append str2 str4)
(1- (length str2))))))
:hints (("Goal" :in-theory (enable not-member=>not-equal-nth)))))
(local
(defthmd si-of-diff-symbols-aux-2
(implies (not (equal (char str1 n) (char str2 n)))
(not (equal str1 str2)))))
(local
(defthm si-of-diff-symbols-aux-3
(implies (not (equal (length (symbol-name s1))
(length (symbol-name s2))))
(not (equal (symbol-name (si s1 n1))
(symbol-name (si s2 n2)))))
:hints (("Goal"
:use ((:instance
si-of-diff-symbols-aux-1
(str1 (str-append-symbol-underscore s1))
(str2 (str-append-symbol-underscore s2))
(str3 (nat-to-dec-string (nfix n1)))
(str4 (nat-to-dec-string (nfix n2))))
(:instance
si-of-diff-symbols-aux-1
(str1 (str-append-symbol-underscore s2))
(str2 (str-append-symbol-underscore s1))
(str3 (nat-to-dec-string (nfix n2)))
(str4 (nat-to-dec-string (nfix n1))))
(:instance
si-of-diff-symbols-aux-2
(n (length (symbol-name s2)))
(str1 (append-symbol-string s1 (nat-to-dec-string (nfix n1))))
(str2 (append-symbol-string s2 (nat-to-dec-string (nfix n2)))))
(:instance
si-of-diff-symbols-aux-2
(n (length (symbol-name s1)))
(str1 (append-symbol-string s2 (nat-to-dec-string (nfix n2))))
(str2 (append-symbol-string s1 (nat-to-dec-string (nfix n1))))))))))
(local
(defun diff-elementp (x y)
(declare (xargs :guard t))
(cond ((atom x) (consp y))
((atom y) (consp x))
(t (or (not (equal (car x) (car y)))
(diff-elementp (cdr x) (cdr y)))))))
(local
(defthmd not-equal-equiv-diff-elementp
(implies (and (true-listp x)
(true-listp y))
(equal (not (equal x y))
(diff-elementp x y)))))
(local
(defthmd diff-elementp-append
(implies (and (diff-elementp x1 y1)
(equal (len x1) (len y1)))
(diff-elementp (append x1 x2) (append y1 y2)))))
(local
(defthm si-of-diff-symbols-aux-4
(implies (and (equal (length (symbol-name s1))
(length (symbol-name s2)))
(not (equal (symbol-name s1) (symbol-name s2))))
(not (equal (symbol-name (si s1 n1))
(symbol-name (si s2 n2)))))
:hints (("Goal"
:use ((:instance
not-equal-equiv-diff-elementp
(x (explode (symbol-name s1)))
(y (explode (symbol-name s2))))
(:instance
not-equal-equiv-diff-elementp
(x (append (explode (symbol-name s1))
(cons #\_ (explode (nat-to-dec-string (nfix n1))))))
(y (append (explode (symbol-name s2))
(cons #\_ (explode (nat-to-dec-string (nfix n2)))))))
(:instance
diff-elementp-append
(x1 (explode (symbol-name s1)))
(x2 (cons #\_ (explode (nat-to-dec-string (nfix n1)))))
(y1 (explode (symbol-name s2)))
(y2 (cons #\_ (explode (nat-to-dec-string (nfix n2)))))))))))
(in-theory (disable istrprefixp
str-append-symbol-underscore
append-symbol-string
si))
(defthm si-of-diff-symbols
(implies (not (equal (symbol-name s1) (symbol-name s2)))
(and (not (equal (symbol-name (si s1 n1))
(symbol-name (si s2 n2))))
(not (equal (si s1 n1) (si s2 n2)))))
:hints (("Goal" :cases ((equal (length (symbol-name s1))
(length (symbol-name s2))))))
:rule-classes ((:rewrite
:corollary
(implies (not (equal (symbol-name s1) (symbol-name s2)))
(not (equal (symbol-name (si s1 n1))
(symbol-name (si s2 n2))))))
(:rewrite
:corollary
(implies (not (equal (symbol-name s1) (symbol-name s2)))
(not (equal (si s1 n1) (si s2 n2)))))))
)
(defthm append-symbol-diff-strings
(implies (and (not (equal str1 str2))
(stringp str1)
(stringp str2))
(not (equal (append-symbol-string s str1)
(append-symbol-string s str2))))
:hints (("Goal" :in-theory (enable append-symbol-string))))
(defthm si-of-diff-idxes
(implies (and (natp m)
(natp n)
(not (equal m n)))
(and (not (equal (symbol-name (si s m))
(symbol-name (si s n))))
(not (equal (si s m)
(si s n)))))
:hints (("Goal" :in-theory (enable si)))
:rule-classes ((:rewrite
:corollary
(implies (and (natp m)
(natp n)
(not (equal m n)))
(not (equal (symbol-name (si s m))
(symbol-name (si s n))))))
(:rewrite
:corollary
(implies (and (natp m)
(natp n)
(not (equal m n)))
(not (equal (si s m)
(si s n)))))))
;; ======================================================================
;; SIS
(defun sis (s start-idx count)
(declare (xargs :guard (and (symbolp s)
(natp start-idx)
(natp count))))
(if (zp count)
nil
(cons (si s start-idx)
(sis s (1+ start-idx) (1- count)))))
(defthm consp-sis
(implies (posp n)
(consp (sis s m n)))
:hints (("Goal" :expand (sis s m n)))
:rule-classes :type-prescription)
(defthm not-member-with-sis-of-diff-symbol
(implies (not (istrprefixp (str-append-symbol-underscore s2)
(symbol-name s1)))
(not (member s1 (sis s2 m n)))))
(defthm si-sis-diff-symbols
(implies (not (equal (symbol-name s1) (symbol-name s2)))
(not (member (si s1 k) (sis s2 m n)))))
(defthm si-sis-diff-idxes
(implies (and (natp k)
(natp m)
(< k m))
(not (member (si s k) (sis s m n)))))
(defthm len-sis
(equal (len (sis s m n))
(nfix n)))
(defthm no-duplicatesp-sis
(implies (natp m)
(no-duplicatesp (sis s m n))))
(defthm sis-of-singleton
(equal (sis s m 1)
(list (si s m)))
:hints (("Goal" :expand (sis s m 1))))
(defthm disjoint-sis-diff-syms
(implies (not (equal (symbol-name s1) (symbol-name s2)))
(disjoint (sis s1 m1 n1) (sis s2 m2 n2))))
(defthm disjoint-sis-same-sym-1
(implies (and (natp i)
(natp m)
(<= (+ i j) m))
(disjoint (sis s i j) (sis s m n))))
(defthm disjoint-sis-same-sym-2
(implies (and (natp i)
(natp m)
(<= (+ m n) i))
(disjoint (sis s i j) (sis s m n)))
:hints (("Goal" :in-theory (enable disjoint-commutative))))
(defthm si-member-sis
(implies (and (natp i)
(natp m)
(natp n)
(<= m i)
(< i (+ m n)))
(member (si s i) (sis s m n)))
:hints (("Goal"
:expand ((sis s 0 n)
(sis s i 1)
(sis s i n)))))
(defthm sis-subset-sis
(implies (and (natp i)
(natp m)
(natp n)
(<= m i)
(<= (+ i j) (+ m n)))
(subsetp (sis s i j)
(sis s m n))))
(defthmd si-is-nth-of-sis
(implies (and (natp i)
(natp m)
(natp n)
(<= m i)
(< i (+ m n)))
(equal (si s i)
(nth (- i m) (sis s m n))))
:hints (("Goal"
:expand ((sis s 0 n)
(sis s i 1)
(sis s i n)))))
(defthmd nth-of-sis-is-si
(implies (and (natp i)
(natp m)
(natp n)
(<= m i)
(< i (+ m n)))
(equal (nth (- i m) (sis s m n))
(si s i)))
:hints (("Goal" :use si-is-nth-of-sis)))
(defthm position1-si-sis
(implies (and (natp i)
(natp m)
(natp n)
(<= m i)
(< i (+ m n)))
(equal (position1 (si s i) (sis s m n))
(- i m)))
:hints (("Goal" :in-theory (enable position1))))
(defthmd sis-of-subset
(implies (and (natp i)
(natp m)
(natp n)
(<= m i)
(<= (+ i j) (+ m n)))
(equal (sis s i j)
(take j (nthcdr (- i m) (sis s m n)))))
:hints (("Goal"
:in-theory (enable nth-of-sis-is-si
car-nthcdr
cdr-nthcdr))))
(in-theory (disable sis))
|
