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• Cuts4-config

Cuts4-config-equiv

Basic equivalence relation for cuts4-config structures.

Definitions and Theorems

Function: cuts4-config-equiv$inline

(defun cuts4-config-equiv$inline (x acl2::y)
  (declare (xargs :guard (and (cuts4-config-p x)
                              (cuts4-config-p acl2::y))))
  (equal (cuts4-config-fix x)
         (cuts4-config-fix acl2::y)))

Theorem: cuts4-config-equiv-is-an-equivalence

(defthm cuts4-config-equiv-is-an-equivalence
  (and (booleanp (cuts4-config-equiv x y))
       (cuts4-config-equiv x x)
       (implies (cuts4-config-equiv x y)
                (cuts4-config-equiv y x))
       (implies (and (cuts4-config-equiv x y)
                     (cuts4-config-equiv y z))
                (cuts4-config-equiv x z)))
  :rule-classes (:equivalence))

Theorem: cuts4-config-equiv-implies-equal-cuts4-config-fix-1

(defthm cuts4-config-equiv-implies-equal-cuts4-config-fix-1
  (implies (cuts4-config-equiv x x-equiv)
           (equal (cuts4-config-fix x)
                  (cuts4-config-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: cuts4-config-fix-under-cuts4-config-equiv

(defthm cuts4-config-fix-under-cuts4-config-equiv
  (cuts4-config-equiv (cuts4-config-fix x)
                      x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-cuts4-config-fix-1-forward-to-cuts4-config-equiv

(defthm equal-of-cuts4-config-fix-1-forward-to-cuts4-config-equiv
  (implies (equal (cuts4-config-fix x) acl2::y)
           (cuts4-config-equiv x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-cuts4-config-fix-2-forward-to-cuts4-config-equiv

(defthm equal-of-cuts4-config-fix-2-forward-to-cuts4-config-equiv
  (implies (equal x (cuts4-config-fix acl2::y))
           (cuts4-config-equiv x acl2::y))
  :rule-classes :forward-chaining)

Theorem: cuts4-config-equiv-of-cuts4-config-fix-1-forward

(defthm cuts4-config-equiv-of-cuts4-config-fix-1-forward
  (implies (cuts4-config-equiv (cuts4-config-fix x)
                               acl2::y)
           (cuts4-config-equiv x acl2::y))
  :rule-classes :forward-chaining)

Theorem: cuts4-config-equiv-of-cuts4-config-fix-2-forward

(defthm cuts4-config-equiv-of-cuts4-config-fix-2-forward
  (implies (cuts4-config-equiv x (cuts4-config-fix acl2::y))
           (cuts4-config-equiv x acl2::y))
  :rule-classes :forward-chaining)