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• Rule-option

Rule-option-equiv

Basic equivalence relation for rule-option structures.

Definitions and Theorems

Function: rule-option-equiv$inline

(defun rule-option-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (rule-optionp acl2::x)
                              (rule-optionp acl2::y))))
  (equal (rule-option-fix acl2::x)
         (rule-option-fix acl2::y)))

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

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

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

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

Theorem: rule-option-fix-under-rule-option-equiv

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

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

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

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

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

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

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

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

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