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Svexl-node

Svexl-node-equiv

Basic equivalence relation for svexl-node structures.

Definitions and Theorems

Function: svexl-node-equiv$inline

(defun svexl-node-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (svexl-node-p acl2::x)
                              (svexl-node-p acl2::y))))
  (equal (svexl-node-fix acl2::x)
         (svexl-node-fix acl2::y)))

Theorem: svexl-node-equiv-is-an-equivalence

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

Theorem: svexl-node-equiv-implies-equal-svexl-node-fix-1

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

Theorem: svexl-node-fix-under-svexl-node-equiv

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

Theorem: equal-of-svexl-node-fix-1-forward-to-svexl-node-equiv

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

Theorem: equal-of-svexl-node-fix-2-forward-to-svexl-node-equiv

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

Theorem: svexl-node-equiv-of-svexl-node-fix-1-forward

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

Theorem: svexl-node-equiv-of-svexl-node-fix-2-forward

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