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Desiniter

Desiniter-equiv

Basic equivalence relation for desiniter structures.

Definitions and Theorems

Function: desiniter-equiv$inline

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

Theorem: desiniter-equiv-is-an-equivalence

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

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

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

Theorem: desiniter-fix-under-desiniter-equiv

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

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

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

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

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

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

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

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

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