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• Truth-idx

Truth-idx-fix

Fixing function for truth-idx bit structures.

Signature

(truth-idx-fix x) → fty::fixed

Arguments

x — Guard (truth-idx-p x).

Returns

fty::fixed — Type (truth-idx-p fty::fixed).

Definitions and Theorems

Function: truth-idx-fix

(defun truth-idx-fix (x)
  (declare (xargs :guard (truth-idx-p x)))
  (let ((__function__ 'truth-idx-fix))
    (declare (ignorable __function__))
    (mbe :logic (loghead 16 x) :exec x)))

Theorem: truth-idx-p-of-truth-idx-fix

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

Theorem: truth-idx-fix-when-truth-idx-p

(defthm truth-idx-fix-when-truth-idx-p
  (implies (truth-idx-p x)
           (equal (truth-idx-fix x) x)))

Function: truth-idx-equiv$inline

(defun truth-idx-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (truth-idx-p acl2::x)
                              (truth-idx-p acl2::y))))
  (equal (truth-idx-fix acl2::x)
         (truth-idx-fix acl2::y)))

Theorem: truth-idx-equiv-is-an-equivalence

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

Theorem: truth-idx-equiv-implies-equal-truth-idx-fix-1

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

Theorem: truth-idx-fix-under-truth-idx-equiv

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

Theorem: truth-idx-fix-of-truth-idx-fix-x

(defthm truth-idx-fix-of-truth-idx-fix-x
  (equal (truth-idx-fix (truth-idx-fix x))
         (truth-idx-fix x)))

Theorem: truth-idx-fix-truth-idx-equiv-congruence-on-x

(defthm truth-idx-fix-truth-idx-equiv-congruence-on-x
  (implies (truth-idx-equiv x x-equiv)
           (equal (truth-idx-fix x)
                  (truth-idx-fix x-equiv)))
  :rule-classes :congruence)