#|$ACL2s-Preamble$;


(include-book "primitives")
(begin-book t :ttags :all);$ACL2s-Preamble$|#


(in-package "M5")

; Arithmetic

(include-book "arithmetic-3/extra/top-ext" :dir :system)

; Basic List Processing

(defthm put-nth-opener
  (and (equal (put-nth 0 v lst) (cons v (cdr lst)))
       (implies (natp n)
                (equal (put-nth (+ 1 n) v lst)
                       (cons (car lst) (put-nth n v (cdr lst))))))
  :hints (("Goal" :induct (put-nth n v lst))))

; Stacks

(defthm stacks
  (and (equal (top (push x s)) x)
       (equal (pop (push x s)) s)
       (equal (top (cons x s)) x)
       (equal (pop (cons x s)) s)))

(in-theory (disable push top pop))

; Instructions

; Binding Environments

(defthm bind-bind-3
  (implies (and (not (equal var1 var2))
                (boundp var2 alist))
           (equal (bind var1 val1 (bind var2 val2 alist))
                  (bind var2 val2 (bind var1 val1 alist))))
  :rule-classes
  ((:rewrite :loop-stopper ((var1 var2)))))

(defthm bind-bind-2
  (implies (and (not (equal var1 var2))
                (boundp var1 alist))
           (equal (bind var1 val1 (bind var2 val2 alist))
                  (bind var2 val2 (bind var1 val1 alist))))
  :rule-classes
  ((:rewrite :loop-stopper ((var1 var2)))))

(defthm bind-bind-1
  (equal (bind var val1 (bind var val2 alist))
         (bind var val1 alist)))

(defthm binding-bind
  (equal (binding var1 (bind var2 val2 alist))
         (if (equal var1 var2)
             val2
           (binding var1 alist))))#|ACL2s-ToDo-Line|#