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• Ctrex-ruletable

Ctrex-ruletable-equiv

Basic equivalence relation for ctrex-ruletable structures.

Definitions and Theorems

Function: ctrex-ruletable-equiv$inline

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

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

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

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

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

Theorem: ctrex-ruletable-fix-under-ctrex-ruletable-equiv

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

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

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

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

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

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

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

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

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