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• Ctrex-rule

Ctrex-rule-equiv

Basic equivalence relation for ctrex-rule structures.

Definitions and Theorems

Function: ctrex-rule-equiv$inline

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

Theorem: ctrex-rule-equiv-is-an-equivalence

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

Theorem: ctrex-rule-equiv-implies-equal-ctrex-rule-fix-1

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

Theorem: ctrex-rule-fix-under-ctrex-rule-equiv

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

Theorem: equal-of-ctrex-rule-fix-1-forward-to-ctrex-rule-equiv

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

Theorem: equal-of-ctrex-rule-fix-2-forward-to-ctrex-rule-equiv

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

Theorem: ctrex-rule-equiv-of-ctrex-rule-fix-1-forward

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

Theorem: ctrex-rule-equiv-of-ctrex-rule-fix-2-forward

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