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|
; Copyright (C) 2013, Regents of the University of Texas
; Written by: J Strother Moore and Qiang Zhang
; Department of Computer Sciences
; Univesity of Texas at Austin
; October, 2004
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
; (certify-book "script")
; A Proof of the Correctness of Dijkstra's Shortest Path Algorithm in
; ACL2
; See the paper ``Dijkstra's Shortest Path Algorithm Verified with ACL2''
; by the authors for a description of this script.
; Historical Notes: The first version of this proof was completed by
; Moore in September, 2003. See /u/moore/shortest-path/dsp6.lisp.
; That file contained 92 defthms, 35 :hints, about 75 clause
; identifiers like "Goal" and "Subgoal" (some occur in comments), and
; 40 :use hints. In order to learn ACL2, Zhang was given the task by
; Moore, first, to discover (independently) a proof (given the
; invariant), and, then, to clean it up.
; Zhang finished his first proof in December, 2003, using ACL2 Version
; 2.7. In October, 2004, he cleaned up the proof, eliminating some of
; his hints. This file is his second proof script. In February,
; 2005, the authors wrote the paper above and in doing so renamed some
; of Zhang's functions and theorems.
; The current file contains 39 defuns, 126 defthms, 51 hints, 23 of
; which are :use hints mentioning 31 lemma instances.
(in-package "ACL2")
;All definitions
(defun del (x y)
(if (endp y) nil
(if (equal x (car y))
(cdr y)
(cons (car y) (del x (cdr y))))))
(defun memp (e x)
(if (endp x) nil
(or (equal e (car x))
(memp e (cdr x)))))
(defun setp (s)
(cond ((endp s) t)
((memp (car s) (cdr s)) nil)
(t (setp (cdr s)))))
(defun my-union (s1 s2)
(cond ((endp s1) s2)
((memp (car s1) s2)
(my-union (cdr s1) s2))
(t (cons (car s1)
(my-union (cdr s1) s2)))))
(defun my-subsetp (s1 s2)
(cond ((endp s1) t)
(t (and (memp (car s1) s2)
(my-subsetp (cdr s1) s2)))))
(defun infinitep (x) (null x))
(defun lt (x y)
(cond ((infinitep x) nil)
((infinitep y) t)
(t (< x y))))
(defun plus (x y)
(cond ((or (infinitep x)
(infinitep y))
nil)
(t (+ x y))))
(defun edge-listp (lst)
(cond ((endp lst) (equal lst nil))
((and (consp (car lst))
(rationalp (cdar lst))
(<= 0 (cdar lst))
(not (assoc-equal (caar lst)
(cdr lst))))
(edge-listp (cdr lst)))
(t nil)))
(defun graphp (g)
(cond ((endp g) (equal g nil))
((and (consp (car g))
(edge-listp (cdar g)))
(graphp (cdr g)))
(t nil)))
(defun cons-set (e s)
(if (memp e s) s
(cons e s)))
(defun all-nodes (g)
(cond ((endp g) nil)
(t (cons-set (caar g)
(my-union (strip-cars (cdar g))
(all-nodes (cdr g)))))))
(defun nodep (n g) (memp n (all-nodes g)))
(defun neighbors (n g)
(strip-cars (cdr (assoc-equal n g))))
(defun pathp-aux (path g)
(cond ((endp path) nil)
((endp (cdr path))
(nodep (car path) g))
(t (and (memp (cadr path)
(neighbors (car path) g))
(pathp-aux (cdr path) g)))))
(defun pathp (path g)
(and (true-listp path)
(pathp-aux path g)))
(defun edge-len (a b g)
(cdr (assoc-equal b (cdr (assoc-equal a g)))))
(defun path-len (path g)
(cond ((endp path) nil)
((endp (cdr path))
(if (nodep (car path) g) 0 nil))
(t (plus (edge-len (car path) (cadr path) g)
(path-len (cdr path) g)))))
(defun path (n pt) (cdr (assoc-equal n pt)))
(defun d (n pt g)
(path-len (path n pt) g))
#|
(defun choose-next (ts pt g)
(cond ((endp ts) '(non-node))
((endp (cdr ts)) (car ts))
((lt (d (car ts) pt g)
(d (choose-next (cdr ts) pt g) pt g))
(car ts))
(t (choose-next (cdr ts) pt g))))
|#
(defun choose-next (ts pt g)
(cond ((endp ts) '(non-node))
((endp (cdr ts)) (car ts))
(t (let ((u (choose-next (cdr ts) pt g)))
(if (lt (d (car ts) pt g) (d u pt g))
(car ts)
u)))))
(defun reassign (u v-lst pt g)
(cond ((endp v-lst) pt)
(t (let* ((v (car v-lst))
(w (edge-len u v g)))
(cond ((lt (plus (d u pt g) w)
(d v pt g))
(cons (cons v (append (path u pt) (list v)))
(reassign u (cdr v-lst) pt g)))
(t (reassign u (cdr v-lst) pt g)))))))
(defun dsp (ts pt g)
(cond ((endp ts) pt)
(t (let ((u (choose-next ts pt g)))
(dsp (del u ts)
(reassign u (neighbors u g) pt g)
g)))))
(defun dijkstra-shortest-path (a b g)
(let ((p (dsp (all-nodes g)
(list (cons a (list a))) g)))
(path b p)))
;===================================================================
(defun pathp-from-to (p a b g)
(and (pathp p g)
(equal (car p) a)
(equal (car (last p)) b)))
(defun pt-propertyp (a pt g)
(if (endp pt) t
(and (or (null (cdar pt))
(pathp-from-to (cdar pt) a (caar pt) g))
(pt-propertyp a (cdr pt) g))))
(defun lte (n1 n2)
(not (lt n2 n1)))
(defun shorterp (p1 p2 g)
(lte (path-len p1 g)
(path-len p2 g)))
(defun confinedp (p fs)
(if (endp p) t
(if (endp (cdr p)) t
(and (memp (car p) fs)
(confinedp (cdr p) fs)))))
#|
(defun-sk shortest-confined-pathp (a b p fs g)
(forall path (implies (and (pathp-from-to path a b g)
(confinedp path fs))
(shorterp p path g))))
|#
(ENCAPSULATE
NIL (LOGIC)
(SET-MATCH-FREE-DEFAULT :ALL)
(SET-INHIBIT-WARNINGS "Theory"
"Use" "Free" "Non-rec" "Infected")
(ENCAPSULATE
((SHORTEST-CONFINED-PATHP-WITNESS (A B P FS G) PATH)
(SHORTEST-CONFINED-PATHP (A B P FS G) T))
(LOCAL (IN-THEORY '(IMPLIES)))
(LOCAL (DEFCHOOSE SHORTEST-CONFINED-PATHP-WITNESS (PATH)
(A B P FS G)
(NOT (IMPLIES (AND (PATHP-FROM-TO PATH A B G)
(CONFINEDP PATH FS))
(SHORTERP P PATH G)))))
(LOCAL (DEFUN SHORTEST-CONFINED-PATHP (A B P FS G)
(LET ((PATH (SHORTEST-CONFINED-PATHP-WITNESS A B P FS G)))
(IMPLIES (AND (PATHP-FROM-TO PATH A B G)
(CONFINEDP PATH FS))
(SHORTERP P PATH G)))))
;(IN-THEORY (DISABLE (SHORTEST-CONFINED-PATHP)))
(DEFTHM SHORTEST-CONFINED-PATHP-DEF
(EQUAL (SHORTEST-CONFINED-PATHP A B P FS G)
(LET ((PATH (SHORTEST-CONFINED-PATHP-WITNESS A B P FS G)))
(IMPLIES (AND (PATHP-FROM-TO PATH A B G)
(CONFINEDP PATH FS))
(SHORTERP P PATH G)))))
(DEFTHM SHORTEST-CONFINED-PATHP-NECC
(IMPLIES (NOT (IMPLIES (AND (PATHP-FROM-TO PATH A B G)
(CONFINEDP PATH FS))
(SHORTERP P PATH G)))
(NOT (SHORTEST-CONFINED-PATHP A B P FS G)))
:HINTS (("Goal" :USE (SHORTEST-CONFINED-PATHP-WITNESS SHORTEST-CONFINED-PATHP)
:IN-THEORY (THEORY 'MINIMAL-THEORY))))))
(include-book "data-structures/utilities" :dir :system)
(include-book "tools/bstar" :dir :system)
(include-book "ordinals/e0-ordinal" :dir :system)
(encapsulate ()
;Auxiliary functions for fixers and for instantiating defun-sk
(defun count-non-members (x y)
(cond ((endp x) 0)
((member (car x) y) (count-non-members (cdr x) y))
(t (+ 1 (count-non-members (cdr x) y)))))
(defun measure (c stack g)
(cons (+ 1 (count-non-members (strip-cars g) stack))
(len c)))
(defun neighbors2 (node g)
(cond ((endp g) nil)
((equal node (car (car g))) (cdr (car g)))
(t (neighbors2 node (cdr g)))))
(set-well-founded-relation e0-ord-<)
(local (in-theory (enable strip-cars)))
(defun find-all-next-steps (c stack b g)
(declare (xargs :measure (measure c stack g)))
(cond
((endp c) nil)
((member (car c) stack)
(find-all-next-steps (cdr c) stack b g))
((equal (car c) b)
(cons (reverse (cons b stack))
(find-all-next-steps (cdr c) stack b g)))
(t (append (find-all-next-steps (neighbors2 (car c) g)
(cons (car c) stack)
b g)
(find-all-next-steps (cdr c) stack b g)))))
(defun change-rep (vs g)
;remove weights and also add empty entries for non-neighbour vertices
(if (endp vs)
g
(b* ((entry (assoc-equal (car vs) g))
((when (null entry)) (change-rep (cdr vs) (put-assoc-equal (car vs) nil g)))
(edges-with-weights (cdr entry))
(edges (strip-cars edges-with-weights)))
(change-rep (cdr vs) (put-assoc-equal (car vs) edges g)))))
(defun find-all-simple-paths (a b g)
(if (and (equal a b) (nodep a g)) ; added for type-safety and less-restrictive fixer rules
(list (list a))
(b* ((g1 (change-rep (all-nodes g) g)))
(find-all-next-steps (neighbors2 a g1) ;(neighbours a g1) bug
(list a) b g1))))
)
(u::defloop shortest-confined-pathp/exec1 (paths a b p fs g)
(for ((path in paths)) (always (implies (and (pathp-from-to path a b g)
(confinedp path fs))
(shorterp p path g)))))
(defun shortest-confined-pathp/exec (a b p fs g)
(shortest-confined-pathp/exec1 (find-all-simple-paths a b g) a b p fs g))
(defttag t)
;(defattach (shortest-confined-pathp shortest-confined-pathp/exec)) guards not verified
(defattach (shortest-confined-pathp shortest-confined-pathp/exec) :skip-checks t)
(defttag nil)
#|
(defun-sk shortest-pathp (a b p g)
(forall path (implies (pathp-from-to path a b g)
(shorterp p path g))))
|#
(ENCAPSULATE NIL (LOGIC)
(SET-MATCH-FREE-DEFAULT :ALL)
(SET-INHIBIT-WARNINGS "Theory"
"Use" "Free" "Non-rec" "Infected")
(ENCAPSULATE ((SHORTEST-PATHP-WITNESS (A B P G) PATH)
(SHORTEST-PATHP (A B P G) T))
(LOCAL (IN-THEORY '(IMPLIES)))
(LOCAL (DEFCHOOSE SHORTEST-PATHP-WITNESS (PATH)
(A B P G)
(NOT (IMPLIES (PATHP-FROM-TO PATH A B G)
(SHORTERP P PATH G)))))
(LOCAL (DEFUN SHORTEST-PATHP (A B P G)
(LET ((PATH (SHORTEST-PATHP-WITNESS A B P G)))
(IMPLIES (PATHP-FROM-TO PATH A B G)
(SHORTERP P PATH G)))))
;(IN-THEORY (DISABLE (SHORTEST-PATHP)))
(DEFTHM SHORTEST-PATHP-DEF
(EQUAL (SHORTEST-PATHP A B P G)
(LET ((PATH (SHORTEST-PATHP-WITNESS A B P G)))
(IMPLIES (PATHP-FROM-TO PATH A B G)
(SHORTERP P PATH G)))))
(DEFTHM SHORTEST-PATHP-NECC
(IMPLIES (NOT (IMPLIES (PATHP-FROM-TO PATH A B G)
(SHORTERP P PATH G)))
(NOT (SHORTEST-PATHP A B P G)))
:HINTS (("Goal" :USE (SHORTEST-PATHP-WITNESS SHORTEST-PATHP)
:IN-THEORY (THEORY 'MINIMAL-THEORY))))))
(u::defloop shortest-pathp/exec1 (paths a b p g)
(for ((path in paths)) (always (implies (pathp-from-to path a b g)
(shorterp p path g)))))
(defun shortest-pathp/exec (a b p g)
(shortest-pathp/exec1 (find-all-simple-paths a b g) a b p g))
(defttag t)
(defattach (shortest-pathp shortest-pathp/exec) :skip-checks t)
(defttag nil)
(defun ts-propertyp (a ts fs pt g)
(if (endp ts) t
(and (shortest-confined-pathp a (car ts) (path (car ts) pt) fs g)
(confinedp (path (car ts) pt) fs)
(ts-propertyp a (cdr ts) fs pt g))))
(defun fs-propertyp (a fs fs0 pt g)
(if (endp fs) t
(and (shortest-pathp a (car fs) (path (car fs) pt) g)
(confinedp (path (car fs) pt) fs0)
(fs-propertyp a (cdr fs) fs0 pt g))))
(defun comp-set (ts s)
(if (endp s) nil
(if (memp (car s) ts)
(comp-set ts (cdr s))
(cons (car s) (comp-set ts (cdr s))))))
(defun invp (ts pt g a)
(let ((fs (comp-set ts (all-nodes g))))
(and (ts-propertyp a ts fs pt g)
(fs-propertyp a fs fs pt g)
(pt-propertyp a pt g))))
(defun find-partial-path (p s)
(if (endp p) nil
(if (memp (car p) s)
(cons (car p) (find-partial-path (cdr p) s))
(list (car p)))))
(defun find-last-next-path (p)
(if (or (endp p) (endp (cdr p))) nil
(cons (car p) (find-last-next-path (cdr p)))))
(defun last-node (p)
(car (last (find-last-next-path p))))
(defun find-partial-path-to-u (p u)
(cond ((not (memp u p)) nil)
((equal (car p) u) (list u))
(t (cons (car p)
(find-partial-path-to-u (cdr p) u)))))
|
