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• Stack

Stack-equiv

Basic equivalence relation for stack structures.

Definitions and Theorems

Function: stack-equiv$inline

(defun stack-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (stackp acl2::x)
                              (stackp acl2::y))))
  (equal (stack-fix acl2::x)
         (stack-fix acl2::y)))

Theorem: stack-equiv-is-an-equivalence

(defthm stack-equiv-is-an-equivalence
  (and (booleanp (stack-equiv x y))
       (stack-equiv x x)
       (implies (stack-equiv x y)
                (stack-equiv y x))
       (implies (and (stack-equiv x y)
                     (stack-equiv y z))
                (stack-equiv x z)))
  :rule-classes (:equivalence))

Theorem: stack-equiv-implies-equal-stack-fix-1

(defthm stack-equiv-implies-equal-stack-fix-1
  (implies (stack-equiv acl2::x x-equiv)
           (equal (stack-fix acl2::x)
                  (stack-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: stack-fix-under-stack-equiv

(defthm stack-fix-under-stack-equiv
  (stack-equiv (stack-fix acl2::x)
               acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-stack-fix-1-forward-to-stack-equiv

(defthm equal-of-stack-fix-1-forward-to-stack-equiv
  (implies (equal (stack-fix acl2::x) acl2::y)
           (stack-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-stack-fix-2-forward-to-stack-equiv

(defthm equal-of-stack-fix-2-forward-to-stack-equiv
  (implies (equal acl2::x (stack-fix acl2::y))
           (stack-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: stack-equiv-of-stack-fix-1-forward

(defthm stack-equiv-of-stack-fix-1-forward
  (implies (stack-equiv (stack-fix acl2::x)
                        acl2::y)
           (stack-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: stack-equiv-of-stack-fix-2-forward

(defthm stack-equiv-of-stack-fix-2-forward
  (implies (stack-equiv acl2::x (stack-fix acl2::y))
           (stack-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)