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; Correctness of Less Than
; Problem: Define an M1 program to compute whether natural number x is less than
; natural number y. Indicate true with 1 and false with 0.
; Design Plan: I will count x and y both down by 1 and stop when either reaches
; 0. If y reaches 0 first (or at the same time), x < y was false. If x
; reaches 0 before y, x < y was true.
; (0) Start ACL2
; (include-book "m1")
(in-package "M1")
; (1) Write your specification, i.e., define the expected inputs and the
; desired output, theta.
(defun ok-inputs (x y)
(and (natp x)
(natp y)))
(defun theta (x y)
(if (< x y) 1 0))
; (2) Write your algorithm. This will consist of a tail-recursive helper
; function and a wrapper, fn.
; Note: In this case, the helper and the top-level fn are the same. We
; don't need both, but we'll stick with the template.
(defun helper (x y)
(cond ((zp y) 0)
((zp x) 1)
(t (helper (- x 1) (- y 1)))))
(defun fn (x y) (helper x y))
; (3) Prove that the algorithm satisfies the spec, by proving first that the
; helper is appropriately related to theta and then that fn is theta on ok
; inputs.
; Important Note: When you verify your helper function, you must consider the
; most general case. For example, if helper is defined with formal parameters
; n, m, a, and rest and fn calls helper initializing a to 0 and rest to nil,
; your helper theorem must be about (helper n m a rest), not just about the
; special case (helper n m 0 nil).
(defthm helper-is-theta
(implies (and (natp x)
(natp y))
(equal (helper x y)
(theta x y))))
(defthm fn-is-theta
(implies (ok-inputs x y)
(equal (fn x y)
(theta x y))))
; Disable these two lemmas because they confuse the theorem prover when it is
; dealing with the code versus fn.
(in-theory (disable helper-is-theta fn-is-theta))
; (4) Write your M1 program with the intention of implementing your algorithm.
(defconst *pi*
'((iload 1) ; 0 lessp-loop: -- the code for lessp
(ifeq 12) ; 1 if y=0, goto false
(iload 0) ; 2
(ifeq 12) ; 3 if x=0, goto to true
(iload 0) ; 4
(iconst 1) ; 5
(isub) ; 6
(istore 0) ; 7 x = x-1
(iload 1) ; 8
(iconst 1) ; 9
(isub) ; 10
(istore 1) ; 11 y = y-1
(goto -12) ; 12 goto lessp-loop
(iconst 0) ; 13 lessp is false
(halt) ; 14
(iconst 1) ; 15 lessp is true
(halt) ; 16 return a
))
; (5) Define the ACL2 function that clocks your program, starting with the
; loop clock and then using it to clock the whole program. The clock
; should take the program from pc 0 to a HALT statement. (Sometimes your
; clocks will require multiple inputs or other locals, but our example only
; requires the first local.)
(defun loop-clk (x y)
(if (zp y)
4
(if (zp x)
5
(clk+ 13
(loop-clk (- x 1) (- y 1))))))
(defun clk (x y)
(loop-clk x y))
; (6) Prove that the code implements your algorithm, starting with the lemma
; that the loop implements the helper. Each time you prove a lemma relating
; code to algorithm, disable the corresponding clock function so the theorem
; prover doesn't look any deeper into subsequent code.
; Important Note: Your lemmas about your loops must consider the general case.
; For example, if a loop uses the locals n, m, and a, you must characterize
; its behavior for arbitrary, legal n, m, and a, not just a special case (e.g.,
; where a is 0).
(defthm loop-is-helper
(implies (and (natp x)
(natp y))
(equal (m1 (make-state 0
(list x y)
nil
*pi*)
(loop-clk x y))
(make-state (if (< x y) 16 14)
(if (< x y)
(list 0 (- y x))
(list (- x y) 0))
(push (helper x y) nil)
*pi*))))
(in-theory (disable loop-clk))
(defthm program-is-fn
(implies (ok-inputs x y)
(equal (m1 (make-state 0
(list x y)
nil
*pi*)
(clk x y))
(make-state (if (< x y) 16 14)
(if (< x y)
(list 0 (- y x))
(list (- x y) 0))
(push (fn x y) nil)
*pi*))))
(in-theory (disable clk))
; (7) Put the two steps together to get correctness.
(in-theory (enable fn-is-theta))
(defthm program-correct
(implies (ok-inputs x y)
(equal (m1 (make-state 0
(list x y)
nil
*pi*)
(clk x y))
(make-state (if (< x y) 16 14)
(if (< x y)
(list 0 (- y x))
(list (- x y) 0))
(push (theta x y)
nil)
*pi*))))
; This corollary just shows we did what we set out to do:
(defthm total-correctness
(implies (and (natp x)
(natp y)
(equal sf (m1 (make-state 0
(list x y)
nil
*pi*)
(clk x y))))
(and (equal (next-inst sf) '(HALT))
(equal (top (stack sf)) (if (< x y) 1 0))))
:rule-classes nil)
; Think of the above theorem as saying: for all natural numbers x and y, there
; exists a clock (for example, the one constructed by (clk x y)) such that
; running *pi* with (list x y) as input produces a state, sf, that is halted
; and which contains 1 or 0 on top of the stack depending on whether x < y.
; Note that the algorithm used by *pi* is not specified or derivable from this
; formula.
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