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; (include-book "m1")
; (certify-book "g-invariant" 1)
(in-package "M1")
(include-book "misc/defpun" :dir :system)
(defmacro defpun (g args &rest tail)
`(acl2::defpun ,g ,args ,@tail))
(defmacro defspec (name prog inputs pre-pc post-pc annotation-alist)
(let ((Inv
(intern-in-package-of-symbol
(concatenate 'string (symbol-name name) "-INV")
'run))
(Inv-def
(intern-in-package-of-symbol
(concatenate 'string (symbol-name name) "-INV-DEF")
'run))
(Inv-opener
(intern-in-package-of-symbol
(concatenate 'string (symbol-name name) "-INV-OPENER")
'run))
(Inv-step
(intern-in-package-of-symbol
(concatenate 'string (symbol-name name) "-INV-STEP")
'run))
(Inv-run
(intern-in-package-of-symbol
(concatenate 'string (symbol-name name) "-INV-RUN")
'run))
(Correctness
(intern-in-package-of-symbol
(concatenate 'string "PARTIAL-CORRECTNESS-OF-PROGRAM-"
(symbol-name name))
'run)))
`(progn
(defpun ,Inv (,@inputs s)
(if (member (pc s)
',(strip-cars annotation-alist))
(and (equal (program s)
,prog)
(case (pc s)
,@annotation-alist))
(,Inv ,@inputs (step s))))
(defthm ,Inv-opener
(implies (and (equal pc (pc s))
(syntaxp (quotep pc))
(not
(member pc
',(strip-cars annotation-alist))))
(equal (,Inv ,@inputs s)
(,Inv ,@inputs (step s)))))
(defthm ,Inv-step
(implies (,Inv ,@inputs s)
(,Inv ,@inputs (step s))))
(defthm ,Inv-run
(implies (,Inv ,@inputs s)
(,Inv ,@inputs (run sched s)))
:rule-classes nil
:hints (("Goal" :in-theory (e/d (run)(,Inv-def)))))
(defthm ,Correctness
(let* ((sk (run sched s0)))
(implies
(and (let ((s s0)) ,(cadr (assoc pre-pc annotation-alist)))
(equal (pc s0) ,pre-pc)
(equal (locals s0) (list* ,@inputs any))
(equal (program s0) ,prog)
(equal (pc sk) ,post-pc))
(let ((s sk)) ,(cadr (assoc post-pc annotation-alist)))))
:hints (("Goal" :use
(:instance ,Inv-run
,@(pairlis$ inputs (acl2::pairlis-x2 inputs nil))
(s s0)
(sched sched))))
:rule-classes nil))))
(defun f (n)
(if (zp n)
1
(* n (f (- n 1)))))
(defconst *g*
'((PUSH 1)
(STORE 1)
(LOAD 0)
(IFLE 10)
(LOAD 0)
(LOAD 1)
(MUL)
(STORE 1)
(LOAD 0)
(PUSH 1)
(SUB)
(STORE 0)
(GOTO -10)
(LOAD 1)
(RETURN)))
(defun n (s) (nth 0 (locals s)))
(defun a (s) (nth 1 (locals s)))
(defspec g *g* (n0 a0) 0 14
((0 ; Pre-condition
(and (equal n0 (n s))
(natp n0)))
(2 ; Loop Invariant
(and (natp n0)
(natp (n s))
(natp (nth 1 (locals s)))
(<= (n s) n0)
(equal (f n0) (* (f (n s)) (a s)))))
(14 ; Post-condition
(equal (top (stack s)) (f n0)))))
(defthm corollary
(let ((sk (run sched s0)))
(implies (and (equal n0 (n s0))
(natp n0)
(equal (pc s0) 0)
(equal (locals s0) (list* n0 a0 any))
(equal (program s0) *g*)
(equal (pc sk) 14))
(equal (top (stack sk))
(f n0))))
:hints (("Goal" :use PARTIAL-CORRECTNESS-OF-PROGRAM-G)))
; Here is a proof of the half method.
(defconst *h*
; public static int half(int n){
; int a = 0;
; while (n!=0){a=a+1;n=n-2;};
; return a;
; }
'((push 0) ; 0
(store 1) ; 1
(goto 9) ; 2
(load 1) ; 3
(push 1) ; 4
(add) ; 5
(store 1) ; 6
(load 0) ; 7
(push 2) ; 8
(sub) ; 9
(store 0) ; 10
(load 0) ; 11
(ifne -9) ; 12
(load 1) ; 13
(return)) ; 14
)
(defun run-h (n)
(run (repeat 0 10000)
(make-state 0
(list n 0)
nil
*h*)))
(defspec h *h* (n0) 0 14
(; Pre-Condition:
(0 (and (equal (n s) n0)
(natp n0)))
; Loop Invariant:
(11 (and (natp n0)
(integerp (n s))
(if (and (natp (a s))
(evenp (n s)))
(equal (+ (a s) (/ (n s) 2))
(/ n0 2))
(not (evenp (n s))))
(iff (evenp n0) (evenp (n s)))))
; Post-condition:
(14 (and (evenp n0)
(equal (top (stack s)) (/ n0 2))))))
(defthm h-corollary
(implies (and (natp (n s0))
(equal (pc s0) 0)
(equal (program s0) *h*)
(equal sk (run sched s0))
(equal (pc sk) 14))
(and (evenp (n s0))
(equal (top (stack sk)) (/ (n s0) 2))))
:hints
(("Goal"
:use (:instance PARTIAL-CORRECTNESS-OF-PROGRAM-H
(n0 (n s0))
(any (cdr (locals s0)))
))))
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