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• Struct-declon

Struct-declon-equiv

Basic equivalence relation for struct-declon structures.

Definitions and Theorems

Function: struct-declon-equiv$inline

(defun struct-declon-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (struct-declonp acl2::x)
                              (struct-declonp acl2::y))))
  (equal (struct-declon-fix acl2::x)
         (struct-declon-fix acl2::y)))

Theorem: struct-declon-equiv-is-an-equivalence

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

Theorem: struct-declon-equiv-implies-equal-struct-declon-fix-1

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

Theorem: struct-declon-fix-under-struct-declon-equiv

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

Theorem: equal-of-struct-declon-fix-1-forward-to-struct-declon-equiv

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

Theorem: equal-of-struct-declon-fix-2-forward-to-struct-declon-equiv

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

Theorem: struct-declon-equiv-of-struct-declon-fix-1-forward

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

Theorem: struct-declon-equiv-of-struct-declon-fix-2-forward

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