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• Cstate

Cstate-equiv

Basic equivalence relation for cstate structures.

Definitions and Theorems

Function: cstate-equiv$inline

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

Theorem: cstate-equiv-is-an-equivalence

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

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

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

Theorem: cstate-fix-under-cstate-equiv

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

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

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

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

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

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

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

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

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