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• Stat

Stat-equiv

Basic equivalence relation for stat structures.

Definitions and Theorems

Function: stat-equiv$inline

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

Theorem: stat-equiv-is-an-equivalence

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

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

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

Theorem: stat-fix-under-stat-equiv

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

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

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

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

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

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

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

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

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