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Declor

Declor-equiv

Basic equivalence relation for declor structures.

Definitions and Theorems

Function: declor-equiv$inline

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

Theorem: declor-equiv-is-an-equivalence

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

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

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

Theorem: declor-fix-under-declor-equiv

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

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

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

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

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

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

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

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

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