(sd-keylist-fix x) is a usual fty list fixing function.
(sd-keylist-fix x) → fty::newx
In the logic, we apply sd-key-fix to each member of the x. In the execution, none of that is actually necessary and this is just an inlined identity function.
Function:
(defun sd-keylist-fix$inline (x) (declare (xargs :guard (sd-keylist-p x))) (let ((__function__ 'sd-keylist-fix)) (declare (ignorable __function__)) (mbe :logic (if (atom x) x (cons (sd-key-fix (car x)) (sd-keylist-fix (cdr x)))) :exec x)))
Theorem:
(defthm sd-keylist-p-of-sd-keylist-fix (b* ((fty::newx (sd-keylist-fix$inline x))) (sd-keylist-p fty::newx)) :rule-classes :rewrite)
Theorem:
(defthm sd-keylist-fix-when-sd-keylist-p (implies (sd-keylist-p x) (equal (sd-keylist-fix x) x)))
Function:
(defun sd-keylist-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (sd-keylist-p acl2::x) (sd-keylist-p acl2::y)))) (equal (sd-keylist-fix acl2::x) (sd-keylist-fix acl2::y)))
Theorem:
(defthm sd-keylist-equiv-is-an-equivalence (and (booleanp (sd-keylist-equiv x y)) (sd-keylist-equiv x x) (implies (sd-keylist-equiv x y) (sd-keylist-equiv y x)) (implies (and (sd-keylist-equiv x y) (sd-keylist-equiv y z)) (sd-keylist-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm sd-keylist-equiv-implies-equal-sd-keylist-fix-1 (implies (sd-keylist-equiv acl2::x x-equiv) (equal (sd-keylist-fix acl2::x) (sd-keylist-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm sd-keylist-fix-under-sd-keylist-equiv (sd-keylist-equiv (sd-keylist-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-sd-keylist-fix-1-forward-to-sd-keylist-equiv (implies (equal (sd-keylist-fix acl2::x) acl2::y) (sd-keylist-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-sd-keylist-fix-2-forward-to-sd-keylist-equiv (implies (equal acl2::x (sd-keylist-fix acl2::y)) (sd-keylist-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm sd-keylist-equiv-of-sd-keylist-fix-1-forward (implies (sd-keylist-equiv (sd-keylist-fix acl2::x) acl2::y) (sd-keylist-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm sd-keylist-equiv-of-sd-keylist-fix-2-forward (implies (sd-keylist-equiv acl2::x (sd-keylist-fix acl2::y)) (sd-keylist-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm car-of-sd-keylist-fix-x-under-sd-key-equiv (sd-key-equiv (car (sd-keylist-fix acl2::x)) (car acl2::x)))
Theorem:
(defthm car-sd-keylist-equiv-congruence-on-x-under-sd-key-equiv (implies (sd-keylist-equiv acl2::x x-equiv) (sd-key-equiv (car acl2::x) (car x-equiv))) :rule-classes :congruence)
Theorem:
(defthm cdr-of-sd-keylist-fix-x-under-sd-keylist-equiv (sd-keylist-equiv (cdr (sd-keylist-fix acl2::x)) (cdr acl2::x)))
Theorem:
(defthm cdr-sd-keylist-equiv-congruence-on-x-under-sd-keylist-equiv (implies (sd-keylist-equiv acl2::x x-equiv) (sd-keylist-equiv (cdr acl2::x) (cdr x-equiv))) :rule-classes :congruence)
Theorem:
(defthm cons-of-sd-key-fix-x-under-sd-keylist-equiv (sd-keylist-equiv (cons (sd-key-fix acl2::x) acl2::y) (cons acl2::x acl2::y)))
Theorem:
(defthm cons-sd-key-equiv-congruence-on-x-under-sd-keylist-equiv (implies (sd-key-equiv acl2::x x-equiv) (sd-keylist-equiv (cons acl2::x acl2::y) (cons x-equiv acl2::y))) :rule-classes :congruence)
Theorem:
(defthm cons-of-sd-keylist-fix-y-under-sd-keylist-equiv (sd-keylist-equiv (cons acl2::x (sd-keylist-fix acl2::y)) (cons acl2::x acl2::y)))
Theorem:
(defthm cons-sd-keylist-equiv-congruence-on-y-under-sd-keylist-equiv (implies (sd-keylist-equiv acl2::y y-equiv) (sd-keylist-equiv (cons acl2::x acl2::y) (cons acl2::x y-equiv))) :rule-classes :congruence)
Theorem:
(defthm consp-of-sd-keylist-fix (equal (consp (sd-keylist-fix acl2::x)) (consp acl2::x)))
Theorem:
(defthm sd-keylist-fix-of-cons (equal (sd-keylist-fix (cons a x)) (cons (sd-key-fix a) (sd-keylist-fix x))))
Theorem:
(defthm len-of-sd-keylist-fix (equal (len (sd-keylist-fix acl2::x)) (len acl2::x)))
Theorem:
(defthm sd-keylist-fix-of-append (equal (sd-keylist-fix (append std::a std::b)) (append (sd-keylist-fix std::a) (sd-keylist-fix std::b))))
Theorem:
(defthm sd-keylist-fix-of-repeat (equal (sd-keylist-fix (repeat acl2::n acl2::x)) (repeat acl2::n (sd-key-fix acl2::x))))
Theorem:
(defthm nth-of-sd-keylist-fix (equal (nth acl2::n (sd-keylist-fix acl2::x)) (if (< (nfix acl2::n) (len acl2::x)) (sd-key-fix (nth acl2::n acl2::x)) nil)))
Theorem:
(defthm sd-keylist-equiv-implies-sd-keylist-equiv-append-1 (implies (sd-keylist-equiv acl2::x fty::x-equiv) (sd-keylist-equiv (append acl2::x acl2::y) (append fty::x-equiv acl2::y))) :rule-classes (:congruence))
Theorem:
(defthm sd-keylist-equiv-implies-sd-keylist-equiv-append-2 (implies (sd-keylist-equiv acl2::y fty::y-equiv) (sd-keylist-equiv (append acl2::x acl2::y) (append acl2::x fty::y-equiv))) :rule-classes (:congruence))
Theorem:
(defthm sd-keylist-equiv-implies-sd-keylist-equiv-nthcdr-2 (implies (sd-keylist-equiv acl2::l l-equiv) (sd-keylist-equiv (nthcdr acl2::n acl2::l) (nthcdr acl2::n l-equiv))) :rule-classes (:congruence))
Theorem:
(defthm sd-keylist-equiv-implies-sd-keylist-equiv-take-2 (implies (sd-keylist-equiv acl2::l l-equiv) (sd-keylist-equiv (take acl2::n acl2::l) (take acl2::n l-equiv))) :rule-classes (:congruence))