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• Rulelist

Rulelist-equiv

Basic equivalence relation for rulelist structures.

Definitions and Theorems

Function: rulelist-equiv$inline

(defun rulelist-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (rulelistp acl2::x)
                              (rulelistp acl2::y))))
  (equal (rulelist-fix acl2::x)
         (rulelist-fix acl2::y)))

Theorem: rulelist-equiv-is-an-equivalence

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

Theorem: rulelist-equiv-implies-equal-rulelist-fix-1

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

Theorem: rulelist-fix-under-rulelist-equiv

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

Theorem: equal-of-rulelist-fix-1-forward-to-rulelist-equiv

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

Theorem: equal-of-rulelist-fix-2-forward-to-rulelist-equiv

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

Theorem: rulelist-equiv-of-rulelist-fix-1-forward

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

Theorem: rulelist-equiv-of-rulelist-fix-2-forward

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