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• Finalize-command

Finalize-command-equiv

Basic equivalence relation for finalize-command structures.

Definitions and Theorems

Function: finalize-command-equiv$inline

(defun finalize-command-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (finalize-commandp acl2::x)
                              (finalize-commandp acl2::y))))
  (equal (finalize-command-fix acl2::x)
         (finalize-command-fix acl2::y)))

Theorem: finalize-command-equiv-is-an-equivalence

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

Theorem: finalize-command-equiv-implies-equal-finalize-command-fix-1

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

Theorem: finalize-command-fix-under-finalize-command-equiv

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

Theorem: equal-of-finalize-command-fix-1-forward-to-finalize-command-equiv

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

Theorem: equal-of-finalize-command-fix-2-forward-to-finalize-command-equiv

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

Theorem: finalize-command-equiv-of-finalize-command-fix-1-forward

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

Theorem: finalize-command-equiv-of-finalize-command-fix-2-forward

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