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Prefixes

Prefixes-fix

Fixing function for prefixes bit structures.

Signature
(prefixes-fix x) → fty::fixed
Arguments: x — Guard (prefixes-p x).
Returns: fty::fixed — Type (prefixes-p fty::fixed).

Definitions and Theorems

Function: prefixes-fix$inline

(defun prefixes-fix$inline (x)
  (declare (xargs :guard (prefixes-p x)))
  (mbe :logic (loghead 52 x) :exec x))

Theorem: prefixes-p-of-prefixes-fix

(defthm prefixes-p-of-prefixes-fix
  (b* ((fty::fixed (prefixes-fix$inline x)))
    (prefixes-p fty::fixed))
  :rule-classes :rewrite)

Theorem: prefixes-fix-when-prefixes-p

(defthm prefixes-fix-when-prefixes-p
  (implies (prefixes-p x)
           (equal (prefixes-fix x) x)))

Function: prefixes-equiv$inline

(defun prefixes-equiv$inline (x y)
  (declare (xargs :guard (and (prefixes-p x) (prefixes-p y))))
  (equal (prefixes-fix x)
         (prefixes-fix y)))

Theorem: prefixes-equiv-is-an-equivalence

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

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

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

Theorem: prefixes-fix-under-prefixes-equiv

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