Fixing function for bitsize structures.
Function:
(defun bitsize-fix$inline (x) (declare (xargs :guard (bitsizep x))) (let ((__function__ 'bitsize-fix)) (declare (ignorable __function__)) (mbe :logic (case (bitsize-kind x) (:|8| (cons :|8| (list))) (:|16| (cons :|16| (list))) (:|32| (cons :|32| (list))) (:|64| (cons :|64| (list))) (:|128| (cons :|128| (list)))) :exec x)))
Theorem:
(defthm bitsizep-of-bitsize-fix (b* ((new-x (bitsize-fix$inline x))) (bitsizep new-x)) :rule-classes :rewrite)
Theorem:
(defthm bitsize-fix-when-bitsizep (implies (bitsizep x) (equal (bitsize-fix x) x)))
Function:
(defun bitsize-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (bitsizep acl2::x) (bitsizep acl2::y)))) (equal (bitsize-fix acl2::x) (bitsize-fix acl2::y)))
Theorem:
(defthm bitsize-equiv-is-an-equivalence (and (booleanp (bitsize-equiv x y)) (bitsize-equiv x x) (implies (bitsize-equiv x y) (bitsize-equiv y x)) (implies (and (bitsize-equiv x y) (bitsize-equiv y z)) (bitsize-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm bitsize-equiv-implies-equal-bitsize-fix-1 (implies (bitsize-equiv acl2::x x-equiv) (equal (bitsize-fix acl2::x) (bitsize-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm bitsize-fix-under-bitsize-equiv (bitsize-equiv (bitsize-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-bitsize-fix-1-forward-to-bitsize-equiv (implies (equal (bitsize-fix acl2::x) acl2::y) (bitsize-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-bitsize-fix-2-forward-to-bitsize-equiv (implies (equal acl2::x (bitsize-fix acl2::y)) (bitsize-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm bitsize-equiv-of-bitsize-fix-1-forward (implies (bitsize-equiv (bitsize-fix acl2::x) acl2::y) (bitsize-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm bitsize-equiv-of-bitsize-fix-2-forward (implies (bitsize-equiv acl2::x (bitsize-fix acl2::y)) (bitsize-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm bitsize-kind$inline-of-bitsize-fix-x (equal (bitsize-kind$inline (bitsize-fix x)) (bitsize-kind$inline x)))
Theorem:
(defthm bitsize-kind$inline-bitsize-equiv-congruence-on-x (implies (bitsize-equiv x x-equiv) (equal (bitsize-kind$inline x) (bitsize-kind$inline x-equiv))) :rule-classes :congruence)
Theorem:
(defthm consp-of-bitsize-fix (consp (bitsize-fix x)) :rule-classes :type-prescription)