Fixing function for rsh-of-concat-table structures.
(rsh-of-concat-table-fix x) → new-x
Function:
(defun rsh-of-concat-table-fix$inline (x) (declare (xargs :guard (rsh-of-concat-table-p x))) (let ((__function__ 'rsh-of-concat-table-fix)) (declare (ignorable __function__)) (mbe :logic (b* ((alist (rsh-of-concat-alist-fix (cdr (std::da-nth 0 x)))) (alist-width (nfix (cdr (std::da-nth 1 x)))) (tail (svex-fix (cdr (std::da-nth 2 x))))) (list (cons 'alist alist) (cons 'alist-width alist-width) (cons 'tail tail))) :exec x)))
Theorem:
(defthm rsh-of-concat-table-p-of-rsh-of-concat-table-fix (b* ((new-x (rsh-of-concat-table-fix$inline x))) (rsh-of-concat-table-p new-x)) :rule-classes :rewrite)
Theorem:
(defthm rsh-of-concat-table-fix-when-rsh-of-concat-table-p (implies (rsh-of-concat-table-p x) (equal (rsh-of-concat-table-fix x) x)))
Function:
(defun rsh-of-concat-table-equiv$inline (x y) (declare (xargs :guard (and (rsh-of-concat-table-p x) (rsh-of-concat-table-p y)))) (equal (rsh-of-concat-table-fix x) (rsh-of-concat-table-fix y)))
Theorem:
(defthm rsh-of-concat-table-equiv-is-an-equivalence (and (booleanp (rsh-of-concat-table-equiv x y)) (rsh-of-concat-table-equiv x x) (implies (rsh-of-concat-table-equiv x y) (rsh-of-concat-table-equiv y x)) (implies (and (rsh-of-concat-table-equiv x y) (rsh-of-concat-table-equiv y z)) (rsh-of-concat-table-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm rsh-of-concat-table-equiv-implies-equal-rsh-of-concat-table-fix-1 (implies (rsh-of-concat-table-equiv x x-equiv) (equal (rsh-of-concat-table-fix x) (rsh-of-concat-table-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm rsh-of-concat-table-fix-under-rsh-of-concat-table-equiv (rsh-of-concat-table-equiv (rsh-of-concat-table-fix x) x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-rsh-of-concat-table-fix-1-forward-to-rsh-of-concat-table-equiv (implies (equal (rsh-of-concat-table-fix x) y) (rsh-of-concat-table-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-rsh-of-concat-table-fix-2-forward-to-rsh-of-concat-table-equiv (implies (equal x (rsh-of-concat-table-fix y)) (rsh-of-concat-table-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm rsh-of-concat-table-equiv-of-rsh-of-concat-table-fix-1-forward (implies (rsh-of-concat-table-equiv (rsh-of-concat-table-fix x) y) (rsh-of-concat-table-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm rsh-of-concat-table-equiv-of-rsh-of-concat-table-fix-2-forward (implies (rsh-of-concat-table-equiv x (rsh-of-concat-table-fix y)) (rsh-of-concat-table-equiv x y)) :rule-classes :forward-chaining)