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• Unreachability-config

Unreachability-config-equiv

Basic equivalence relation for unreachability-config structures.

Definitions and Theorems

Function: unreachability-config-equiv$inline

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

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

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

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

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

Theorem: unreachability-config-fix-under-unreachability-config-equiv

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

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

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

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

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

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

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

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

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