#|

(include-book "m1-lemmas")

|#

(in-package "M1")

; Here is a function to compute the square a natural number
; without using multiplication:

(defun square (n)
  (if (zp n)
      0
    (+ n n -1 (square (- n 1)))))

; Problem 7-1: Fill in the blank below with an M1 program that
; computes the square of a natural number by iterative addition
; and halts with (square n) on the stack, provided n is local
; 0. You may not use the JVM IMUL instruction. You may assume n
; is a natural number.

(defconst *square-program*
  ...)

; Problem 7-2: Prove *square-program* correct.

; Here is a function to compute the cube of a natural number:

(defun cube (n)
  (if (zp n)
      0
    (+ (* 3 n (- n 1)) 1 (cube (- n 1)))))

; Problem 7-3: Fill in the blank below with an M1 program that
; computes the cube of a natural number by iterative addition and
; multiplication and halts with (cube n) on the stack, provided
; than n is local 0. You may assume n is a natural number.

(defconst *cube-program*
  ...)

; Problem 7-4: Prove *cube-program* correct.

; Here is the definition of the Fibonacci function:
; fib(n) = 1 if n = 0 or n = 1
;          fib(n-1) + fib(n-2) if n is a natural number > 1.

; Problem 7-5: Fill in the blank below with an M1 program that
; halts with the nth Fibonacci number on the stack, provided n is
; local 0. You may assume n is a natural number.

(defconst *fib-program*
  ...)

; Problem 7-6: Prove *fib-program* correct.  Note: this is a hard
; problem! Pay close attention to your scheduler function and
; carefully state the theorem about the loop. Proving the
; equivalence of the recursive and iterative Fibonacci functions
; also requires some thought.  You may need to use the updated
; version of the M1 compiler which supports if expressions and
; testing with =. This version is contained in
; problem-set-4-answers.lisp.