; Practice Midterm

; Implicitly below I am operating in the following environment:

; (include-book "m1-lemmas")

(in-package "M1")

; Problem 1

; Define the function execute-DUP to extend M1 with the following
; new instruction

;  Format: (DUP)
;  Stack:  ..., v => ..., v, v
;  Description:
;  Duplicates the topmost item on the stack.

; Note: To actually extend M1 we would have to change do-inst
; also, but that is not necessary here.

; Problem 2

; Given

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

; Prove the following theorem:

(defthm problem-2
  (implies (natp n)
           (equal (sum n) (/ (+ (* n n) n) 2))))

; Note:  I want to see a conventional proof, not an ACL2s
; abbreviated proof.  You may use conventional arithmetic
; notation, e.g., prove

; natp(n) -> sum(n) = (n^2 + n)/2


; Problem 3

; Suppose n is a natural number (a nonnegative integer).  Write
; an M1 program that sums the naturals from n down to 0 and halts
; with the result on top of the stack.  Let *prog* be the ACL2
; constant that contains your list of instructions.  Note: you
; know that the result of running *prog* will be (n^2 + n)/2, but
; the program should compute it by iterated additions.

; Problem 4

; Write the schedule function, sched, to drive your program to
; its HALT statement for an arbitrary given natural number n.

; Problem 5

; Write the additional definitions and theorems required to give
; the abbreviated proof of

; (implies (natp n)
;          (equal (top
;                  (stack
;                   (run (sched n)
;                        (make-state 0 (list n a) stack *prog*))))
;                 (/ (+ (* n n) n) 2)))