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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)