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|
; This book proves the correctness of a recursive static method
; for factorial on M5.
#|
; Certification Instructions.
(include-book "utilities")
(certify-book "jvm-fact-setup" 1)
J Moore
Here is the Java for a factorial method.
class Demo {
static int ans;
public static int fact(int n){
if (n>0)
{return n*fact(n-1);}
else return 1;
}
public static void main(String[] args){
int k = 4;
ans = fact(k+1);
return;
}
}
If you put this into Demo.java and run
% javac Demo.java
% javap -o Demo
you get the following:
Compiled from Demo.java
synchronized class Demo extends java.lang.Object
/* ACC_SUPER bit set */
{
static int ans;
public static int fact(int);
public static void main(java.lang.String[]);
Demo();
}
Method int fact(int)
0 iload_0
1 ifle 13
4 iload_0
5 iload_0
6 iconst_1
7 isub
8 invokestatic #5 <Method int fact(int)>
11 imul
12 ireturn
13 iconst_1
14 ireturn
Method void main(java.lang.String[])
0 iconst_4
1 istore_1
2 iload_1
3 iconst_1
4 iadd
5 invokestatic #5 <Method int fact(int)>
8 putstatic #4 <Field int ans>
11 return
Method Demo()
0 aload_0
1 invokespecial #3 <Method java.lang.Object()>
4 return
Below is the output of jvm2acl2 for M5.
|#
(in-package "M5")
(defconst *Demo-class-table-in-tagged-form*
(make-class-def
(list
(make-class-decl
"Demo"
'("java.lang.Object")
'()
'("ans")
'()
(list
'("<init>" () nil
(aload_0)
(invokespecial "java.lang.Object" "<init>" 0)
(return))
'("fact" (int) nil
(iload_0)
(ifle LABEL::TAG_0)
(iload_0)
(iload_0)
(iconst_1)
(isub)
(invokestatic "Demo" "fact" 1)
(imul)
(ireturn)
(LABEL::TAG_0 iconst_1)
(ireturn))
'("main" (java.lang.String[]) nil
(iconst_4)
(istore_1)
(iload_1)
(iconst_1)
(iadd)
(invokestatic "Demo" "fact" 1)
(putstatic "Demo" "ans" nil)
(return)))
'(REF -1)))))
(defconst *Demo-main*
'((iconst_4)
(istore_1)
(iload_1)
(iconst_1)
(iadd)
(invokestatic "Demo" "fact" 1)
(putstatic "Demo" "ans" nil)
(return)))
(defun Demo-ms ()
(make-state
(make-tt (push (make-frame 0
nil
nil
*Demo-main*
'UNLOCKED
"Demo")
nil))
nil
*Demo-class-table-in-tagged-form*))
(defun Demo ()
(m5_load (Demo-ms)))
; Here is the state created by (Demo):
#|
(((0 ((0 NIL NIL
((ICONST_4)
(ISTORE_1)
(ILOAD_1)
(ICONST_1)
(IADD)
(INVOKESTATIC "Demo" "fact" 1)
(PUTSTATIC "Demo" "ans" NIL)
(RETURN))
UNLOCKED "Demo"))
SCHEDULED NIL))
((0 ("java.lang.Class" ("<name>" . "java.lang.Object"))
("java.lang.Object" ("monitor" . 0)
("mcount" . 0)
("wait-set" . 0)))
(1 ("java.lang.Class" ("<name>" . "ARRAY"))
("java.lang.Object" ("monitor" . 0)
("mcount" . 0)
("wait-set" . 0)))
(2 ("java.lang.Class" ("<name>" . "java.lang.Thread"))
("java.lang.Object" ("monitor" . 0)
("mcount" . 0)
("wait-set" . 0)))
(3 ("java.lang.Class" ("<name>" . "java.lang.String"))
("java.lang.Object" ("monitor" . 0)
("mcount" . 0)
("wait-set" . 0)))
(4 ("java.lang.Class" ("<name>" . "java.lang.Class"))
("java.lang.Object" ("monitor" . 0)
("mcount" . 0)
("wait-set" . 0)))
(5 ("java.lang.Class" ("<name>" . "Demo")
("ans" . 0))
("java.lang.Object" ("monitor" . 0)
("mcount" . 0)
("wait-set" . 0))))
(("java.lang.Object" NIL ("monitor" "mcount" "wait-set")
NIL NIL (("<init>" NIL NIL (RETURN)))
(REF 0))
("ARRAY" ("java.lang.Object")
(("<array>" . *ARRAY*))
NIL NIL NIL (REF 1))
("java.lang.Thread"
("java.lang.Object")
NIL NIL NIL
(("run" NIL NIL (RETURN))
("start" NIL NIL NIL)
("stop" NIL NIL NIL)
("<init>" NIL NIL (ALOAD_0)
(INVOKESPECIAL "java.lang.Object" "<init>" 0)
(RETURN)))
(REF 2))
("java.lang.String"
("java.lang.Object")
("strcontents")
NIL NIL
(("<init>" NIL NIL (ALOAD_0)
(INVOKESPECIAL "java.lang.Object" "<init>" 0)
(RETURN)))
(REF 3))
("java.lang.Class" ("java.lang.Object")
NIL NIL NIL
(("<init>" NIL NIL (ALOAD_0)
(INVOKESPECIAL "java.lang.Object" "<init>" 0)
(RETURN)))
(REF 4))
("Demo" ("java.lang.Object")
NIL ("ans")
NIL
(("<init>" NIL NIL (ALOAD_0)
(INVOKESPECIAL "java.lang.Object" "<init>" 0)
(RETURN))
("fact" (INT)
NIL (ILOAD_0)
(IFLE 12)
(ILOAD_0)
(ILOAD_0)
(ICONST_1)
(ISUB)
(INVOKESTATIC "Demo" "fact" 1)
(IMUL)
(IRETURN)
(ICONST_1)
(IRETURN))
("main" (|JAVA.LANG.STRING[]|)
NIL (ICONST_4)
(ISTORE_1)
(ILOAD_1)
(ICONST_1)
(IADD)
(INVOKESTATIC "Demo" "fact" 1)
(PUTSTATIC "Demo" "ans" NIL)
(RETURN)))
(REF 5))))
|#
; But in the paper we discuss it component by component and
; define constants for each. Note that we can write ICONST_4 or
; ICONST\_4 in Common Lisp. We use the latter so that we can
; pick these forms up and dump them into LaTeX.
(defconst *Demo-thread-table*
(list
(cons 0
(make-thread
(push
(make-frame
0
nil
nil
'((ICONST\_4)
(ISTORE\_1)
(ILOAD\_1)
(ICONST\_1)
(IADD)
(INVOKESTATIC "Demo" "fact" 1)
(PUTSTATIC "Demo" "ans" NIL)
(RETURN))
'UNLOCKED
"Demo")
nil)
'SCHEDULED
nil))))
(defconst *Demo-heap*
'((0 . (("java.lang.Class"
("<name>" . "java.lang.Object"))
("java.lang.Object"
("monitor" . 0)
("mcount" . 0)
("wait-set" . 0))))
(1 . (("java.lang.Class"
("<name>" . "ARRAY"))
("java.lang.Object"
("monitor" . 0)
("mcount" . 0)
("wait-set" . 0))))
(2 . (("java.lang.Class"
("<name>" . "java.lang.Thread"))
("java.lang.Object"
("monitor" . 0)
("mcount" . 0)
("wait-set" . 0))))
(3 . (("java.lang.Class"
("<name>" . "java.lang.String"))
("java.lang.Object"
("monitor" . 0)
("mcount" . 0)
("wait-set" . 0))))
(4 . (("java.lang.Class"
("<name>" . "java.lang.Class"))
("java.lang.Object"
("monitor" . 0)
("mcount" . 0)
("wait-set" . 0))))
(5 . (("java.lang.Class"
("<name>" . "Demo")
("ans" . 0))
("java.lang.Object"
("monitor" . 0)
("mcount" . 0)
("wait-set" . 0))))))
(defconst *Demo-class-table*
'(("java.lang.Object"
NIL
("monitor" "mcount" "wait-set")
NIL
NIL
(("<init>" NIL NIL (RETURN)))
(REF 0))
("ARRAY"
("java.lang.Object")
(("<array>" . *ARRAY*))
NIL
NIL
NIL
(REF 1))
("java.lang.Thread"
("java.lang.Object")
NIL
NIL
NIL
(("run" NIL NIL (RETURN))
("start" NIL NIL NIL)
("stop" NIL NIL NIL)
("<init>" NIL NIL (ALOAD\_0)
(INVOKESPECIAL "java.lang.Object" "<init>" 0)
(RETURN)))
(REF 2))
("java.lang.String"
("java.lang.Object")
("strcontents")
NIL
NIL
(("<init>" NIL NIL
(ALOAD\_0)
(INVOKESPECIAL "java.lang.Object" "<init>" 0)
(RETURN)))
(REF 3))
("java.lang.Class"
("java.lang.Object")
NIL
NIL
NIL
(("<init>" NIL NIL
(ALOAD\_0)
(INVOKESPECIAL "java.lang.Object" "<init>" 0)
(RETURN)))
(REF 4))
("Demo"
("java.lang.Object")
NIL
("ans")
NIL
(("<init>" NIL NIL
(ALOAD\_0)
(INVOKESPECIAL "java.lang.Object" "<init>" 0)
(RETURN))
("fact" (INT) NIL
(ILOAD\_0)
(IFLE 12)
(ILOAD\_0)
(ILOAD\_0)
(ICONST\_1)
(ISUB)
(INVOKESTATIC "Demo" "fact" 1)
(IMUL)
(IRETURN)
(ICONST\_1)
(IRETURN))
("main" (|JAVA.LANG.STRING[]|) NIL
(ICONST\_4)
(ISTORE\_1)
(ILOAD\_1)
(ICONST\_1)
(IADD)
(INVOKESTATIC "Demo" "fact" 1)
(PUTSTATIC "Demo" "ans" NIL)
(RETURN)))
(REF 5))))
(defconst *Demo-state*
(make-state *demo-thread-table*
*demo-heap*
*demo-class-table*))
(defthm demo-state-is-demo
(equal (Demo)
*Demo-state*)
:rule-classes nil)
; The Mathematical Function
(defun ! (n)
(if (zp n)
1
(* n (! (- n 1)))))
(defthm integerp-factorial
(integerp (! n))
:rule-classes :type-prescription)
; A Schedule that Runs fact to Completion
(defun fact-sched (th n)
(if (zp n)
(repeat th 5)
(append (repeat th 7)
(fact-sched th (- n 1))
(repeat th 2))))
(defthm len-repeat
(equal (len (repeat th n)) (nfix n)))
(defthm len-append
(equal (len (append a b))
(+ (len a) (len b))))
(defthm len-fact-sched
(equal (len (fact-sched th n))
(+ 5 (* 9 (nfix n)))))
; Playing Around with Main
; This schedule executes main to termination.
(defun main-sched (th)
(append (repeat th 5)
(fact-sched th 5)
(repeat th 2)))
(defthm sample-execution
(equal (static-field-value "Demo" "ans"
(run (main-sched 0) *Demo-state*))
120)
:rule-classes nil)
#|
; Below is a fact-test function. I define it in raw Lisp rather
; than ACL2 so that I can time the execution of the JVM model
; without timing the construction of the schedule. To define
; this function, exit the loop with :q and do these two forms.
(in-package "M5")
(compile
(defun fact-test (n)
(format t "Computing schedule for ~a.~%" n)
(let ((sched (append (repeat 0 1)
(fact-sched 0 n)
(repeat 0 2))))
(format t "Schedule length: ~a.~%" (len sched))
(time
(static-field-value
"Demo" "ans"
(run sched
(make-state
(list
(cons 0
(make-thread
(push
(make-frame
0
(list n)
nil
'((ILOAD\_0)
(INVOKESTATIC "Demo" "fact" 1)
(PUTSTATIC "Demo" "ans" NIL)
(RETURN))
'UNLOCKED
"Demo")
nil)
'SCHEDULED
nil)))
*demo-heap*
*demo-class-table*)))))))
; Allocate additional bignum space
(si::allocate 'lisp::bignum 400 t)
T
; Then do things like (fact-test 17) or (fact-test 1000). On a 797
; MHz Pentium III, the latter requires a schedule of length 9008 and
; takes 0.100 seconds to execute, provided no (BIGNUM) gcs occur.
; This gives a simulation speed of 90K JVM bytecodes per second.
|#
; Proving Fact Correct
(defconst *fact-def*
'("fact" (INT) NIL
(ILOAD_0) ;;; 0
(IFLE 12) ;;; 1
(ILOAD_0) ;;; 4
(ILOAD_0) ;;; 5
(ICONST_1) ;;; 6
(ISUB) ;;; 7
(INVOKESTATIC "Demo" "fact" 1) ;;; 8
(IMUL) ;;; 11
(IRETURN) ;;; 12
(ICONST_1) ;;; 13
(IRETURN))) ;;; 14
(defun poised-to-invoke-fact (th s n)
(and (equal (status th s) 'SCHEDULED)
(equal (next-inst th s) '(invokestatic "Demo" "fact" 1))
(equal n (top (stack (top-frame th s))))
(intp n)
(equal (lookup-method "fact" "Demo" (class-table s))
*fact-def*)))
(defun induction-hint (th s n)
(if (zp n)
s
(induction-hint
th
(make-state ;;; (run (repeat th 7) s)
(bind
th
(make-thread
(push
(make-frame
8
(list (top (stack (top-frame th s))))
(push (- (top (stack (top-frame th s))) 1)
(push (top (stack (top-frame th s)))
nil))
(method-program *fact-def*)
'UNLOCKED
"Demo")
(push (make-frame (+ 3 (pc (top (call-stack th s))))
(locals (top (call-stack th s)))
(pop (stack (top (call-stack th s))))
(program (top (call-stack th s)))
(sync-flg (top (call-stack th s)))
(cur-class (top (call-stack th s))))
(pop (call-stack th s))))
'scheduled
(rref th s))
(thread-table s))
(heap s)
(class-table s))
(- n 1))))
; The make-state in the induction-hint above is equivalent to
; (run (repeat th 7) s), under the hypotheses that s is poised to
; invoke fact and that n is non-0. We prove that below, just to
; demonstrate this claim. The import of this claim is that we
; could use this to help generate the induction hint, i.e., the
; make-state is not "magic." Indeed, the theorem below is
; generated by the following forms:
#||
(include-book "misc/expander" :dir :system)
(acl2::defthm? induction-hint-explanation
(implies (and (poised-to-invoke-fact th s n)
(not (zp n)))
(equal (run (repeat th 7) s)
acl2::?)))
||#
(defthm induction-hint-explanation
(implies (and (poised-to-invoke-fact th s n)
(not (zp n)))
(equal (run (repeat th 7) s)
(MAKE-STATE
(BIND
TH
(LIST
(PUSH
(MAKE-FRAME 8 (LIST N)
(PUSH (+ -1 N) (PUSH N NIL))
'((ILOAD_0)
(IFLE 12)
(ILOAD_0)
(ILOAD_0)
(ICONST_1)
(ISUB)
(INVOKESTATIC "Demo" "fact" 1)
(IMUL)
(IRETURN)
(ICONST_1)
(IRETURN))
'UNLOCKED
"Demo")
(PUSH
(MAKE-FRAME
(+ 3
(PC (TOP (CADR (ASSOC-EQUAL TH (THREAD-TABLE S))))))
(LOCALS (TOP (CADR (ASSOC-EQUAL TH (THREAD-TABLE S)))))
(POP (STACK (TOP (CADR (ASSOC-EQUAL TH (THREAD-TABLE S))))))
(PROGRAM (TOP (CADR (ASSOC-EQUAL TH (THREAD-TABLE S)))))
(SYNC-FLG (TOP (CADR (ASSOC-EQUAL TH (THREAD-TABLE S)))))
(CUR-CLASS (TOP (CADR (ASSOC-EQUAL TH (THREAD-TABLE S))))))
(POP (CADR (ASSOC-EQUAL TH (THREAD-TABLE S))))))
'SCHEDULED
(CADDDR (ASSOC-EQUAL TH (THREAD-TABLE S))))
(THREAD-TABLE S))
(HEAP S)
(CLASS-TABLE S))))
:rule-classes nil)
|
