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  • Math
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• Priopt

Priopt-equiv

Basic equivalence relation for priopt structures.

Definitions and Theorems

Function: priopt-equiv$inline

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

Theorem: priopt-equiv-is-an-equivalence

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

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

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

Theorem: priopt-fix-under-priopt-equiv

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

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

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

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

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

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

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

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

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