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• Pristate

Pristate-equiv

Basic equivalence relation for pristate structures.

Definitions and Theorems

Function: pristate-equiv$inline

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

Theorem: pristate-equiv-is-an-equivalence

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

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

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

Theorem: pristate-fix-under-pristate-equiv

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

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

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

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

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

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

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

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

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