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Svstack

Svstack-equiv

Basic equivalence relation for svstack structures.

Definitions and Theorems

Function: svstack-equiv$inline

(defun svstack-equiv$inline (x y)
  (declare (xargs :guard (and (svstack-p x) (svstack-p y))))
  (equal (svstack-fix x) (svstack-fix y)))

Theorem: svstack-equiv-is-an-equivalence

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

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

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

Theorem: svstack-fix-under-svstack-equiv

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

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

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

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

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

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

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

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

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