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• Linkage

Linkage-equiv

Basic equivalence relation for linkage structures.

Definitions and Theorems

Function: linkage-equiv$inline

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

Theorem: linkage-equiv-is-an-equivalence

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

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

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

Theorem: linkage-fix-under-linkage-equiv

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

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

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

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

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

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

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

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

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