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• Ext-declon

Ext-declon-equiv

Basic equivalence relation for ext-declon structures.

Definitions and Theorems

Function: ext-declon-equiv$inline

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

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

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

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

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

Theorem: ext-declon-fix-under-ext-declon-equiv

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

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

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

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

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

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

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

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

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