Fixing function for op/en-p structures.
Function:
(defun op/en-p-fix$inline (x) (declare (xargs :guard (op/en-p-p x))) (let ((__function__ 'op/en-p-fix)) (declare (ignorable __function__)) (mbe :logic (b* ((src1 (keyword-list-fix (cdr (std::da-nth 0 x)))) (src2 (keyword-list-fix (cdr (std::da-nth 1 x)))) (src3 (keyword-list-fix (cdr (std::da-nth 2 x)))) (src4 (keyword-list-fix (cdr (std::da-nth 3 x))))) (list (cons 'src1 src1) (cons 'src2 src2) (cons 'src3 src3) (cons 'src4 src4))) :exec x)))
Theorem:
(defthm op/en-p-p-of-op/en-p-fix (b* ((new-x (op/en-p-fix$inline x))) (op/en-p-p new-x)) :rule-classes :rewrite)
Theorem:
(defthm op/en-p-fix-when-op/en-p-p (implies (op/en-p-p x) (equal (op/en-p-fix x) x)))
Function:
(defun op/en-p-equiv$inline (x y) (declare (xargs :guard (and (op/en-p-p x) (op/en-p-p y)))) (equal (op/en-p-fix x) (op/en-p-fix y)))
Theorem:
(defthm op/en-p-equiv-is-an-equivalence (and (booleanp (op/en-p-equiv x y)) (op/en-p-equiv x x) (implies (op/en-p-equiv x y) (op/en-p-equiv y x)) (implies (and (op/en-p-equiv x y) (op/en-p-equiv y z)) (op/en-p-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm op/en-p-equiv-implies-equal-op/en-p-fix-1 (implies (op/en-p-equiv x x-equiv) (equal (op/en-p-fix x) (op/en-p-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm op/en-p-fix-under-op/en-p-equiv (op/en-p-equiv (op/en-p-fix x) x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-op/en-p-fix-1-forward-to-op/en-p-equiv (implies (equal (op/en-p-fix x) y) (op/en-p-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-op/en-p-fix-2-forward-to-op/en-p-equiv (implies (equal x (op/en-p-fix y)) (op/en-p-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm op/en-p-equiv-of-op/en-p-fix-1-forward (implies (op/en-p-equiv (op/en-p-fix x) y) (op/en-p-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm op/en-p-equiv-of-op/en-p-fix-2-forward (implies (op/en-p-equiv x (op/en-p-fix y)) (op/en-p-equiv x y)) :rule-classes :forward-chaining)