Fixing function for primitive-type structures.
(primitive-type-fix x) → new-x
Function:
(defun primitive-type-fix$inline (x) (declare (xargs :guard (primitive-typep x))) (let ((__function__ 'primitive-type-fix)) (declare (ignorable __function__)) (mbe :logic (case (primitive-type-kind x) (:boolean (cons :boolean (list))) (:char (cons :char (list))) (:byte (cons :byte (list))) (:short (cons :short (list))) (:int (cons :int (list))) (:long (cons :long (list))) (:float (cons :float (list))) (:double (cons :double (list)))) :exec x)))
Theorem:
(defthm primitive-typep-of-primitive-type-fix (b* ((new-x (primitive-type-fix$inline x))) (primitive-typep new-x)) :rule-classes :rewrite)
Theorem:
(defthm primitive-type-fix-when-primitive-typep (implies (primitive-typep x) (equal (primitive-type-fix x) x)))
Function:
(defun primitive-type-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (primitive-typep acl2::x) (primitive-typep acl2::y)))) (equal (primitive-type-fix acl2::x) (primitive-type-fix acl2::y)))
Theorem:
(defthm primitive-type-equiv-is-an-equivalence (and (booleanp (primitive-type-equiv x y)) (primitive-type-equiv x x) (implies (primitive-type-equiv x y) (primitive-type-equiv y x)) (implies (and (primitive-type-equiv x y) (primitive-type-equiv y z)) (primitive-type-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm primitive-type-equiv-implies-equal-primitive-type-fix-1 (implies (primitive-type-equiv acl2::x x-equiv) (equal (primitive-type-fix acl2::x) (primitive-type-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm primitive-type-fix-under-primitive-type-equiv (primitive-type-equiv (primitive-type-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-primitive-type-fix-1-forward-to-primitive-type-equiv (implies (equal (primitive-type-fix acl2::x) acl2::y) (primitive-type-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-primitive-type-fix-2-forward-to-primitive-type-equiv (implies (equal acl2::x (primitive-type-fix acl2::y)) (primitive-type-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm primitive-type-equiv-of-primitive-type-fix-1-forward (implies (primitive-type-equiv (primitive-type-fix acl2::x) acl2::y) (primitive-type-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm primitive-type-equiv-of-primitive-type-fix-2-forward (implies (primitive-type-equiv acl2::x (primitive-type-fix acl2::y)) (primitive-type-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm primitive-type-kind$inline-of-primitive-type-fix-x (equal (primitive-type-kind$inline (primitive-type-fix x)) (primitive-type-kind$inline x)))
Theorem:
(defthm primitive-type-kind$inline-primitive-type-equiv-congruence-on-x (implies (primitive-type-equiv x x-equiv) (equal (primitive-type-kind$inline x) (primitive-type-kind$inline x-equiv))) :rule-classes :congruence)
Theorem:
(defthm consp-of-primitive-type-fix (consp (primitive-type-fix x)) :rule-classes :type-prescription)