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• Bytelist-bytelist-map

Bytelist-bytelist-mfix

(bytelist-bytelist-mfix x) is a usual ACL2::fty omap fixing function.

Signature

(bytelist-bytelist-mfix x) → *

Arguments

x — Guard (bytelist-bytelist-mapp x).

Definitions and Theorems

Function: bytelist-bytelist-mfix

(defun bytelist-bytelist-mfix (x)
  (declare (xargs :guard (bytelist-bytelist-mapp x)))
  (mbe :logic (if (bytelist-bytelist-mapp x) x nil)
       :exec x))

Theorem: bytelist-bytelist-mapp-of-bytelist-bytelist-mfix

(defthm bytelist-bytelist-mapp-of-bytelist-bytelist-mfix
  (bytelist-bytelist-mapp (bytelist-bytelist-mfix x)))

Theorem: bytelist-bytelist-mfix-when-bytelist-bytelist-mapp

(defthm bytelist-bytelist-mfix-when-bytelist-bytelist-mapp
  (implies (bytelist-bytelist-mapp x)
           (equal (bytelist-bytelist-mfix x) x)))

Theorem: emptyp-bytelist-bytelist-mfix

(defthm emptyp-bytelist-bytelist-mfix
  (implies (or (omap::emptyp x)
               (not (bytelist-bytelist-mapp x)))
           (omap::emptyp (bytelist-bytelist-mfix x))))

Theorem: emptyp-of-bytelist-bytelist-mfix-to-not-bytelist-bytelist-map-or-emptyp

(defthm
 emptyp-of-bytelist-bytelist-mfix-to-not-bytelist-bytelist-map-or-emptyp
 (equal (omap::emptyp (bytelist-bytelist-mfix x))
        (or (not (bytelist-bytelist-mapp x))
            (omap::emptyp x))))

Function: bytelist-bytelist-mequiv$inline

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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