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• Balance-config

Balance-config-equiv

Basic equivalence relation for balance-config structures.

Definitions and Theorems

Function: balance-config-equiv$inline

(defun balance-config-equiv$inline (x acl2::y)
  (declare (xargs :guard (and (balance-config-p x)
                              (balance-config-p acl2::y))))
  (equal (balance-config-fix x)
         (balance-config-fix acl2::y)))

Theorem: balance-config-equiv-is-an-equivalence

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

Theorem: balance-config-equiv-implies-equal-balance-config-fix-1

(defthm balance-config-equiv-implies-equal-balance-config-fix-1
  (implies (balance-config-equiv x x-equiv)
           (equal (balance-config-fix x)
                  (balance-config-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: balance-config-fix-under-balance-config-equiv

(defthm balance-config-fix-under-balance-config-equiv
  (balance-config-equiv (balance-config-fix x)
                        x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-balance-config-fix-1-forward-to-balance-config-equiv

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

Theorem: equal-of-balance-config-fix-2-forward-to-balance-config-equiv

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

Theorem: balance-config-equiv-of-balance-config-fix-1-forward

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

Theorem: balance-config-equiv-of-balance-config-fix-2-forward

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