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• Alternation

Alternation-equiv

Basic equivalence relation for alternation structures.

Definitions and Theorems

Function: alternation-equiv$inline

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

Theorem: alternation-equiv-is-an-equivalence

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

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

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

Theorem: alternation-fix-under-alternation-equiv

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

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

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

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

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

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

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

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

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