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• Funarg

Funarg-equiv

Basic equivalence relation for funarg structures.

Definitions and Theorems

Function: funarg-equiv$inline

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

Theorem: funarg-equiv-is-an-equivalence

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

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

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

Theorem: funarg-fix-under-funarg-equiv

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

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

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

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

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

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

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

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

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