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Dirdeclor

Dirdeclor-equiv

Basic equivalence relation for dirdeclor structures.

Definitions and Theorems

Function: dirdeclor-equiv$inline

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

Theorem: dirdeclor-equiv-is-an-equivalence

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

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

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

Theorem: dirdeclor-fix-under-dirdeclor-equiv

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

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

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

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

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

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

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

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

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