Fixing function for typed-variable structures.
(typed-variable-fix x) → new-x
Function:
(defun typed-variable-fix$inline (x) (declare (xargs :guard (typed-variablep x))) (let ((__function__ 'typed-variable-fix)) (declare (ignorable __function__)) (mbe :logic (b* ((name (identifier-fix (cdr (std::da-nth 0 x)))) (type (type-fix (cdr (std::da-nth 1 x))))) (list (cons 'name name) (cons 'type type))) :exec x)))
Theorem:
(defthm typed-variablep-of-typed-variable-fix (b* ((new-x (typed-variable-fix$inline x))) (typed-variablep new-x)) :rule-classes :rewrite)
Theorem:
(defthm typed-variable-fix-when-typed-variablep (implies (typed-variablep x) (equal (typed-variable-fix x) x)))
Function:
(defun typed-variable-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (typed-variablep acl2::x) (typed-variablep acl2::y)))) (equal (typed-variable-fix acl2::x) (typed-variable-fix acl2::y)))
Theorem:
(defthm typed-variable-equiv-is-an-equivalence (and (booleanp (typed-variable-equiv x y)) (typed-variable-equiv x x) (implies (typed-variable-equiv x y) (typed-variable-equiv y x)) (implies (and (typed-variable-equiv x y) (typed-variable-equiv y z)) (typed-variable-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm typed-variable-equiv-implies-equal-typed-variable-fix-1 (implies (typed-variable-equiv acl2::x x-equiv) (equal (typed-variable-fix acl2::x) (typed-variable-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm typed-variable-fix-under-typed-variable-equiv (typed-variable-equiv (typed-variable-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-typed-variable-fix-1-forward-to-typed-variable-equiv (implies (equal (typed-variable-fix acl2::x) acl2::y) (typed-variable-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-typed-variable-fix-2-forward-to-typed-variable-equiv (implies (equal acl2::x (typed-variable-fix acl2::y)) (typed-variable-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm typed-variable-equiv-of-typed-variable-fix-1-forward (implies (typed-variable-equiv (typed-variable-fix acl2::x) acl2::y) (typed-variable-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm typed-variable-equiv-of-typed-variable-fix-2-forward (implies (typed-variable-equiv acl2::x (typed-variable-fix acl2::y)) (typed-variable-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)