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• Vardecl

Vardecl-equiv

Basic equivalence relation for vardecl structures.

Definitions and Theorems

Function: vardecl-equiv$inline

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

Theorem: vardecl-equiv-is-an-equivalence

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

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

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

Theorem: vardecl-fix-under-vardecl-equiv

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

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

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

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

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

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

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

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

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