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; Copyright (C) 2004, Regents of the University of Texas
; Written by Sol Swords
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
(in-package "ACL2")
;; (include-book "equal-based-set")
;; (local (include-book "defset-encapsulates"))
(include-book "std/util/deflist" :dir :system)
(defun string-index-measure (idx str)
(declare (xargs :guard t))
(if (and (stringp str)
(natp idx))
(nfix (+ 1 (length str) (- idx)))
0))
(defun string-index-end (idx str)
(declare (xargs :guard (and (stringp str)
(natp idx))))
(or (not (mbt (natp idx)))
(not (mbt (stringp str)))
(>= idx (length str))))
(defthm not-string-index-end-characterp
(implies (not (string-index-end idx str))
(characterp (char str idx))))
(in-theory (disable char))
(defun string-index-past-end (idx str)
(declare (xargs :guard (and (stringp str)
(natp idx))))
(or (not (mbt (natp idx)))
(not (mbt (stringp str)))
(> idx (length str))))
(defun length-equiv (a b)
(declare (xargs :guard (and (or (stringp a)
(true-listp a))
(or (stringp b)
(true-listp b)))))
(= (length a)
(length b)))
(defequiv length-equiv
:hints (("Goal" :in-theory (disable length))))
(defthm length-equiv-linear
(implies (length-equiv a b)
(= (length a)
(length b)))
:rule-classes :linear)
(defthm length-equiv-implies
(implies (length-equiv a b)
(equal (equal (length a)
(length b))
t)))
(defthm implies-length-equiv
(implies (= (length a)
(length b))
(equal (length-equiv a b) t)))
(defun indexp (n str)
(declare (xargs :guard (stringp str)))
(and (natp n)
(<= n (length str))))
(defthm indexp-characterp
(implies (and (stringp str)
(indexp n str)
(not (= n (length str))))
(characterp (char str n))))
(defcong length-equiv equal (indexp n str) 2
:hints (("Goal" :in-theory (disable length))))
(defun backrefp (x str)
(declare (xargs :guard (stringp str)))
(and (consp x)
(indexp (car x) str)
(indexp (cdr x) str)
(<= (car x) (cdr x))))
(defthm backref-ints
(implies (backrefp x str)
(and (integerp (car x))
(integerp (cdr x)))))
(defcong length-equiv equal (backrefp x str) 2
:hints (("Goal" :in-theory (disable length))))
(defun pseudo-backrefp (x)
(declare (xargs :guard t))
(and (consp x)
(natp (car x))
(natp (cdr x))))
(defthm backrefp-pseudo
(implies (backrefp x str)
(pseudo-backrefp x)))
(defun get-backref-string (br str)
(declare (xargs :guard (and (stringp str)
(backrefp br str))))
(subseq str (car br) (cdr br)))
(defthm backrefp-stringp
(implies (and (stringp str)
(backrefp br str))
(stringp (get-backref-string br str))))
(defun backref-listp (l str)
(declare (xargs :guard (stringp str)))
(if (atom l)
(equal l nil)
(and (or (not (car l))
(backrefp (car l) str))
(backref-listp (cdr l) str))))
(defthm backref-listp-nth
(implies (and (backref-listp l str)
(nth n l))
(and (consp (nth n l))
(integerp (car (nth n l)))
(integerp (cdr (nth n l))))))
(defthm backref-listp-nth-linear
(implies (and (backref-listp l str)
(nth n l))
(and (<= (car (nth n l))
(cdr (nth n l)))
(<= 0 (car (nth n l)))
(<= 0 (cdr (nth n l)))
(<= (cdr (nth n l)) (length str))))
:rule-classes (:rewrite :linear))
(defcong length-equiv equal (backref-listp x str) 2
:hints (("Goal" :in-theory (disable length length-equiv))))
(defun pseudo-backref-listp (l)
(declare (xargs :guard t))
(if (atom l)
(null l)
(and (or (not (car l))
(pseudo-backrefp (car l)))
(pseudo-backref-listp (cdr l)))))
(defthm pseudo-backref-listp-nth
(implies (and (pseudo-backref-listp l)
(nth n l))
(and (consp (nth n l))
(integerp (car (nth n l)))
(integerp (cdr (nth n l))))))
(defthm backref-listp-pseudo
(implies (backref-listp l str)
(pseudo-backref-listp l)))
(defthm backref-listp-true-listp
(implies (backref-listp l str)
(true-listp l)))
(defun input-list-eltp (x str)
(declare (xargs :guard (stringp str)))
(and (consp x)
(indexp (car x) str)
(backref-listp (cdr x) str)))
(defthm input-list-eltp-thm
(implies (input-list-eltp x str)
(and (integerp (car x))
(<= 0 (car x))
(<= (car x) (length str))
(backref-listp (cdr x) str))))
(defthm input-list-eltp-suffic
(implies (and (consp x)
(indexp (car x) str)
(backref-listp (cdr x) str))
(input-list-eltp x str)))
(defcong length-equiv equal (input-list-eltp x str) 2
:hints (("Goal" :in-theory (disable length length-equiv))))
(defun pseudo-input-list-eltp (x)
(declare (xargs :guard t))
(and (consp x)
(natp (car x))
(pseudo-backref-listp (cdr x))))
(defthm input-list-eltp-pseudo
(implies (input-list-eltp x str)
(pseudo-input-list-eltp x)))
(std::deflist input-listp (x str)
(input-list-eltp x str)
:guard (stringp str))
(defcong length-equiv equal (input-listp l str) 2
:hints (("Goal" :in-theory (e/d (input-listp)
(length length-equiv)))))
(std::deflist pseudo-input-listp (x)
(pseudo-input-list-eltp x))
(defthm input-listp-pseudo
(implies (input-listp x str)
(pseudo-input-listp x))
:hints(("Goal" :in-theory (enable input-listp pseudo-input-listp))))
(defthm backref-listp-update-nth
(implies (and (backrefp x str)
(backref-listp brl str))
(backref-listp (update-nth n x brl) str)))
(defthm pseudo-backref-listp-update-nth
(implies (and (pseudo-backrefp x)
(pseudo-backref-listp brl))
(pseudo-backref-listp (update-nth n x brl))))
;; Functions on input lists
(defun min-idx-il (il str)
(declare (xargs :guard (and (stringp str)
(input-listp il str))))
(if (atom il)
(length str)
(min (caar il)
(min-idx-il (cdr il) str))))
(defthm min-idx-il-type
(implies (pseudo-input-listp il)
(and (integerp (min-idx-il il str))
(<= 0 (min-idx-il il str))))
:rule-classes (:rewrite :type-prescription))
(defthm min-idx-il-lte-car
(implies (and (pseudo-input-listp ilist)
(consp ilist)
(stringp str))
(>= (caar ilist)
(min-idx-il ilist str))))
(defthm min-idx-il-max
(<= (min-idx-il l str) (length str))
:rule-classes (:rewrite :linear))
(defthm member-min-idx
(implies (member-equal x l)
(<= (min-idx-il l str) (car x)))
:rule-classes (:rewrite :linear))
;; (defthm insert-min-idx-il
;; (<= (min-idx-il (set-insert-equal e l) str)
;; (min-idx-il l str))
;; :rule-classes (:rewrite :linear)
;; :hints (("Goal" :in-theory (enable set-insert-equal))))
;; (defthm insert-min-idx
;; (<= (min-idx-il (set-insert-equal e l) str)
;; (car e))
;; :rule-classes (:rewrite :linear)
;; :hints (("Goal" :in-theory (enable set-insert-equal))))
;; (defthm insert-min-idx2
;; (equal (min-idx-il (set-insert-equal e l) str)
;; (min (car e) (min-idx-il l str)))
;; :hints (("Goal" :in-theory (enable set-insert-equal))))
(defthm integerp-numberp
(implies (integerp x)
(acl2-numberp x)))
(defthm min-idx-append
(implies (and (pseudo-input-listp a)
(pseudo-input-listp b)
(stringp str))
(equal (min-idx-il (append a b) str)
(min (min-idx-il a str) (min-idx-il b str)))))
(defun longest-il (il)
(declare (xargs :guard (and (consp il)
(pseudo-input-listp il))))
(if (atom (cdr il))
(car il)
(let ((l-rest (longest-il (cdr il))))
(if (>= (caar il) (car l-rest))
(car il)
l-rest))))
(defthm longest-il-type-weak
(implies (and (pseudo-input-listp il)
(consp il))
(and (consp (longest-il il))
(integerp (car (longest-il il)))
(<= 0 (car (longest-il il)))
(pseudo-backref-listp (cdr (longest-il il)))))
:rule-classes (:rewrite :type-prescription))
(defthm longest-il-type-strong
(implies (and (stringp str)
(input-listp il str)
(consp il))
(and (<= (car (longest-il il)) (length str))
(backref-listp (cdr (longest-il il)) str))))
(defthm longest-il-max-len
(implies (and (stringp str)
(input-listp il str)
(consp il))
(<= (car (longest-il il)) (length str)))
:rule-classes (:rewrite :linear))
(defthm longest-il-min-len
(implies (and (stringp str)
(input-listp il str)
(consp il))
(<= (min-idx-il il str) (car (longest-il il))))
:rule-classes (:rewrite :linear))
(defthm longest-il-greater-compare-to-min
(implies (and (stringp str)
(input-listp il str)
(consp il)
(<= idx (min-idx-il il str)))
(<= idx (car (longest-il il))))
:rule-classes (:rewrite :linear))
(defun remove-all-longer-equal-fn (x minidx)
(declare (xargs :guard (and (integerp minidx)
(pseudo-input-list-eltp x))))
(if (> (car x) minidx)
(list x)
nil))
(defun remove-all-longer-equal-il (s minidx)
(declare (xargs :guard (and (pseudo-input-listp s)
(integerp minidx))))
(if (atom s)
nil
(append (remove-all-longer-equal-fn (car s) minidx)
(remove-all-longer-equal-il (cdr s) minidx))))
;; (prove-set-congruence-of-f remove-all-longer-equal-fn
;; :f-on-set remove-all-longer-equal-il
;; :equiv equal
;; :hard-guard nil
;; :additional-param t)
(defthm remove-all-longer-equal-shorter
(implies (and (integerp minidx)
(stringp str)
(< minidx (length str)))
(> (min-idx-il
(remove-all-longer-equal-il il minidx) str)
minidx))
:rule-classes (:rewrite :linear))
(in-theory (disable length length-equiv))
;; Type theorems about input lists
(defthm remove-all-longer-equal-il-il
(implies (input-listp l str)
(input-listp (remove-all-longer-equal-il l maxlen) str)))
(defthm remove-all-longer-equal-il-pseudo
(implies (pseudo-input-listp l)
(pseudo-input-listp (remove-all-longer-equal-il l maxlen))))
|
