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Tlb-key

Tlb-key-fix

Fixing function for tlb-key bit structures.

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

Definitions and Theorems

Function: tlb-key-fix

(defun tlb-key-fix (x)
  (declare (xargs :guard (tlb-key-p x)))
  (let ((__function__ 'tlb-key-fix))
    (declare (ignorable __function__))
    (mbe :logic (loghead 46 x) :exec x)))

Theorem: tlb-key-p-of-tlb-key-fix

(defthm tlb-key-p-of-tlb-key-fix
  (b* ((fty::fixed (tlb-key-fix x)))
    (tlb-key-p fty::fixed))
  :rule-classes :rewrite)

Theorem: tlb-key-fix-when-tlb-key-p

(defthm tlb-key-fix-when-tlb-key-p
  (implies (tlb-key-p x)
           (equal (tlb-key-fix x) x)))

Function: tlb-key-equiv$inline

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

Theorem: tlb-key-equiv-is-an-equivalence

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

Theorem: tlb-key-equiv-implies-equal-tlb-key-fix-1

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

Theorem: tlb-key-fix-under-tlb-key-equiv

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