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• Nibblelist-bytelist-map

Nibblelist-bytelist-mequiv

Basic equivalence relation for nibblelist-bytelist-map structures.

Definitions and Theorems

Function: nibblelist-bytelist-mequiv$inline

(defun nibblelist-bytelist-mequiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (nibblelist-bytelist-mapp acl2::x)
                              (nibblelist-bytelist-mapp acl2::y))))
  (equal (nibblelist-bytelist-mfix acl2::x)
         (nibblelist-bytelist-mfix acl2::y)))

Theorem: nibblelist-bytelist-mequiv-is-an-equivalence

(defthm nibblelist-bytelist-mequiv-is-an-equivalence
  (and (booleanp (nibblelist-bytelist-mequiv x y))
       (nibblelist-bytelist-mequiv x x)
       (implies (nibblelist-bytelist-mequiv x y)
                (nibblelist-bytelist-mequiv y x))
       (implies (and (nibblelist-bytelist-mequiv x y)
                     (nibblelist-bytelist-mequiv y z))
                (nibblelist-bytelist-mequiv x z)))
  :rule-classes (:equivalence))

Theorem: nibblelist-bytelist-mequiv-implies-equal-nibblelist-bytelist-mfix-1

(defthm
 nibblelist-bytelist-mequiv-implies-equal-nibblelist-bytelist-mfix-1
 (implies (nibblelist-bytelist-mequiv acl2::x x-equiv)
          (equal (nibblelist-bytelist-mfix acl2::x)
                 (nibblelist-bytelist-mfix x-equiv)))
 :rule-classes (:congruence))

Theorem: nibblelist-bytelist-mfix-under-nibblelist-bytelist-mequiv

(defthm nibblelist-bytelist-mfix-under-nibblelist-bytelist-mequiv
  (nibblelist-bytelist-mequiv (nibblelist-bytelist-mfix acl2::x)
                              acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-nibblelist-bytelist-mfix-1-forward-to-nibblelist-bytelist-mequiv

(defthm
 equal-of-nibblelist-bytelist-mfix-1-forward-to-nibblelist-bytelist-mequiv
 (implies (equal (nibblelist-bytelist-mfix acl2::x)
                 acl2::y)
          (nibblelist-bytelist-mequiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: equal-of-nibblelist-bytelist-mfix-2-forward-to-nibblelist-bytelist-mequiv

(defthm
 equal-of-nibblelist-bytelist-mfix-2-forward-to-nibblelist-bytelist-mequiv
 (implies (equal acl2::x
                 (nibblelist-bytelist-mfix acl2::y))
          (nibblelist-bytelist-mequiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: nibblelist-bytelist-mequiv-of-nibblelist-bytelist-mfix-1-forward

(defthm
   nibblelist-bytelist-mequiv-of-nibblelist-bytelist-mfix-1-forward
 (implies
      (nibblelist-bytelist-mequiv (nibblelist-bytelist-mfix acl2::x)
                                  acl2::y)
      (nibblelist-bytelist-mequiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: nibblelist-bytelist-mequiv-of-nibblelist-bytelist-mfix-2-forward

(defthm
   nibblelist-bytelist-mequiv-of-nibblelist-bytelist-mfix-2-forward
 (implies
     (nibblelist-bytelist-mequiv acl2::x
                                 (nibblelist-bytelist-mfix acl2::y))
     (nibblelist-bytelist-mequiv acl2::x acl2::y))
 :rule-classes :forward-chaining)