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• Obj-declor

Obj-declor-equiv

Basic equivalence relation for obj-declor structures.

Definitions and Theorems

Function: obj-declor-equiv$inline

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

Theorem: obj-declor-equiv-is-an-equivalence

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

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

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

Theorem: obj-declor-fix-under-obj-declor-equiv

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

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

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

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

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

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

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

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

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