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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 "idemo" 1)
J Moore
Here is the Java for an iterative factorial method.
class IDemo {
public static int ifact(int n){
int temp = 1;
while (0<n) {
temp = n*temp;
n = n-1;
}
return temp;
}
}
The corresponding bytecode for ifact is shown below.
Method int ifact(int)
0 iconst_1
1 istore_1
2 goto 13
5 iload_0
6 iload_1
7 imul
8 istore_1
9 iload_0
10 iconst_1
11 isub
12 istore_0
13 iload_0
14 ifgt 5
17 iload_1
18 ireturn
|#
(in-package "M5")
; Note that we have not shown a main program for our IDemo
; class. We make one up below. The initial state is only used to
; play around. We don't care about the main program when we prove
; ifact correct.
(defconst *IDemo-thread-table*
(list
(cons 0
(make-thread
(push
(make-frame
0
nil
nil
'((ICONST\_4)
(ICONST\_4)
(IADD)
(INVOKESTATIC "IDemo" "ifact" 1)
(HALT))
'UNLOCKED
"IDemo")
nil)
'SCHEDULED
nil))))
(defconst *IDemo-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>" . "IDemo"))
("java.lang.Object"
("monitor" . 0)
("mcount" . 0)
("wait-set" . 0))))))
(defconst *IDemo-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))
("IDemo"
("java.lang.Object")
NIL
NIL
NIL
(("<init>" NIL NIL
(ALOAD\_0)
(INVOKESPECIAL "java.lang.Object" "<init>" 0)
(RETURN))
("ifact" (INT) NIL
(ICONST\_1)
(ISTORE\_1)
(GOTO 11)
(ILOAD\_0)
(ILOAD\_1)
(IMUL)
(ISTORE\_1)
(ILOAD\_0)
(ICONST\_1)
(ISUB)
(ISTORE\_0)
(ILOAD\_0)
(IFGT -9)
(ILOAD\_1)
(IRETURN)))
(REF 5))))
(defconst *IDemo-state*
(make-state *IDemo-thread-table*
*IDemo-heap*
*IDemo-class-table*))
; The Mathematical Function
(defun factorial (n)
(if (zp n)
1
(* n (factorial (- n 1)))))
(defthm integerp-factorial
(integerp (factorial n))
:rule-classes :type-prescription)
; A Schedule that Runs fact to Completion
; This subroutine computes a schedule suitable for starting at the
; (ILOAD_0) at pc 13, which we consider the "top" of the loop. The
; schedule drives the machine to (but not through) the (IRETURN).
(defun ifact-loop-sched (th n)
(if (zp n)
(repeat th 3)
(append (repeat th 10)
(ifact-loop-sched th (- n 1)))))
(defun ifact-sched (th n)
(append (repeat th 4)
(ifact-loop-sched th n)
(repeat th 1)))
; Playing Around with Main
; This schedule executes main to termination.
(defun imain-sched (th)
(append (repeat th 3)
(ifact-sched th 8)))
(defthm isample-execution
(equal (top
(stack
(top-frame 0
(run (imain-sched 0)
*IDemo-state*))))
(factorial 8))
:rule-classes nil)
; Proving Fact Correct
(defun ifactorial (n temp)
(if (zp n)
temp
(ifactorial (- n 1) (int-fix (* n temp)))))
(defconst *ifact-def*
(lookup-method "ifact" "IDemo" *IDemo-class-table*))
(defun poised-at-ifact-loop (th s n)
(and (equal (status th s) 'SCHEDULED)
(equal (pc (top-frame th s)) 13)
(equal (program (top-frame th s)) (method-program *ifact-def*))
(equal n (nth 0 (locals (top-frame th s))))
(intp n)
(intp (nth 1 (locals (top-frame th s))))))
(defun ifact-loop-induction-hint (th s n)
(if (zp n)
s
(ifact-loop-induction-hint
th
(make-state
(bind
th
(make-thread
(push
(make-frame 13
(update-nth 0 (- n 1)
(update-nth 1 (int-fix
(* n
(nth 1 (locals
(top-frame th s)))))
(locals (top-frame th s))))
(stack (top-frame th s))
(method-program *ifact-def*)
(sync-flg (top-frame th s))
(cur-class (top-frame th s)))
(pop (call-stack th s)))
'scheduled
(rref th s))
(thread-table s))
(heap s)
(class-table s))
(- n 1))))
(defthm ifact-loop-is-correct
(implies (poised-at-ifact-loop th s n)
(equal
(run (ifact-loop-sched th n) s)
(modify th s
:pc 18
:locals
(if (zp n)
(locals (top-frame th s))
(update-nth 0 0
(update-nth 1
(int-fix
(ifactorial
n
(nth 1 (locals (top-frame th s)))))
(locals (top-frame th s)))))
:stack
(push (int-fix
(ifactorial
n
(nth 1 (locals (top-frame th s)))))
(stack (top-frame th s))))))
:hints (("Goal"
:induct (ifact-loop-induction-hint th s n))))
(defun poised-to-invoke-ifact (th s n)
(and (equal (status th s) 'SCHEDULED)
(equal (next-inst th s) '(invokestatic "IDemo" "ifact" 1))
(equal n (top (stack (top-frame th s))))
(intp n)
(equal (lookup-method "ifact" "IDemo" (class-table s))
*ifact-def*)))
(defthm ifact-is-correct
(implies (poised-to-invoke-ifact th s n)
(equal
(run (ifact-sched th n) s)
(modify th s
:pc (+ 3 (pc (top-frame th s)))
:stack
(push (int-fix (ifactorial n 1))
(pop (stack (top-frame th s))))))))
(in-theory (disable ifact-sched))
(defthm ifact-is-fact
(implies (and (integerp n)
(integerp temp))
(equal (int-fix (ifactorial n temp))
(int-fix (* temp (factorial n))))))
(defthm ifact-main-result
(implies (poised-to-invoke-ifact th s n)
(equal
(run (ifact-sched th n) s)
(modify th s
:pc (+ 3 (pc (top-frame th s)))
:stack
(push (int-fix (factorial n))
(pop (stack (top-frame th s))))))))
|
