(aig-varlist-fix x) is a usual ACL2::fty list fixing function.
(aig-varlist-fix x) → fty::newx
In the logic, we apply aig-var-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 aig-varlist-fix$inline (x) (declare (xargs :guard (aig-varlist-p x))) (let ((__function__ 'aig-varlist-fix)) (declare (ignorable __function__)) (mbe :logic (if (atom x) x (cons (aig-var-fix (car x)) (aig-varlist-fix (cdr x)))) :exec x)))
Theorem:
(defthm aig-varlist-p-of-aig-varlist-fix (b* ((fty::newx (aig-varlist-fix$inline x))) (aig-varlist-p fty::newx)) :rule-classes :rewrite)
Theorem:
(defthm aig-varlist-fix-when-aig-varlist-p (implies (aig-varlist-p x) (equal (aig-varlist-fix x) x)))
Function:
(defun aig-varlist-equiv$inline (x y) (declare (xargs :guard (and (aig-varlist-p x) (aig-varlist-p y)))) (equal (aig-varlist-fix x) (aig-varlist-fix y)))
Theorem:
(defthm aig-varlist-equiv-is-an-equivalence (and (booleanp (aig-varlist-equiv x y)) (aig-varlist-equiv x x) (implies (aig-varlist-equiv x y) (aig-varlist-equiv y x)) (implies (and (aig-varlist-equiv x y) (aig-varlist-equiv y z)) (aig-varlist-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm aig-varlist-equiv-implies-equal-aig-varlist-fix-1 (implies (aig-varlist-equiv x x-equiv) (equal (aig-varlist-fix x) (aig-varlist-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm aig-varlist-fix-under-aig-varlist-equiv (aig-varlist-equiv (aig-varlist-fix x) x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-aig-varlist-fix-1-forward-to-aig-varlist-equiv (implies (equal (aig-varlist-fix x) y) (aig-varlist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-aig-varlist-fix-2-forward-to-aig-varlist-equiv (implies (equal x (aig-varlist-fix y)) (aig-varlist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm aig-varlist-equiv-of-aig-varlist-fix-1-forward (implies (aig-varlist-equiv (aig-varlist-fix x) y) (aig-varlist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm aig-varlist-equiv-of-aig-varlist-fix-2-forward (implies (aig-varlist-equiv x (aig-varlist-fix y)) (aig-varlist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm car-of-aig-varlist-fix-x-under-aig-var-equiv (aig-var-equiv (car (aig-varlist-fix x)) (car x)))
Theorem:
(defthm car-aig-varlist-equiv-congruence-on-x-under-aig-var-equiv (implies (aig-varlist-equiv x x-equiv) (aig-var-equiv (car x) (car x-equiv))) :rule-classes :congruence)
Theorem:
(defthm cdr-of-aig-varlist-fix-x-under-aig-varlist-equiv (aig-varlist-equiv (cdr (aig-varlist-fix x)) (cdr x)))
Theorem:
(defthm cdr-aig-varlist-equiv-congruence-on-x-under-aig-varlist-equiv (implies (aig-varlist-equiv x x-equiv) (aig-varlist-equiv (cdr x) (cdr x-equiv))) :rule-classes :congruence)
Theorem:
(defthm cons-of-aig-var-fix-x-under-aig-varlist-equiv (aig-varlist-equiv (cons (aig-var-fix x) y) (cons x y)))
Theorem:
(defthm cons-aig-var-equiv-congruence-on-x-under-aig-varlist-equiv (implies (aig-var-equiv x x-equiv) (aig-varlist-equiv (cons x y) (cons x-equiv y))) :rule-classes :congruence)
Theorem:
(defthm cons-of-aig-varlist-fix-y-under-aig-varlist-equiv (aig-varlist-equiv (cons x (aig-varlist-fix y)) (cons x y)))
Theorem:
(defthm cons-aig-varlist-equiv-congruence-on-y-under-aig-varlist-equiv (implies (aig-varlist-equiv y y-equiv) (aig-varlist-equiv (cons x y) (cons x y-equiv))) :rule-classes :congruence)
Theorem:
(defthm consp-of-aig-varlist-fix (equal (consp (aig-varlist-fix x)) (consp x)))
Theorem:
(defthm aig-varlist-fix-of-cons (equal (aig-varlist-fix (cons a x)) (cons (aig-var-fix a) (aig-varlist-fix x))))
Theorem:
(defthm len-of-aig-varlist-fix (equal (len (aig-varlist-fix x)) (len x)))
Theorem:
(defthm aig-varlist-fix-of-append (equal (aig-varlist-fix (append std::a std::b)) (append (aig-varlist-fix std::a) (aig-varlist-fix std::b))))
Theorem:
(defthm aig-varlist-fix-of-repeat (equal (aig-varlist-fix (acl2::repeat n x)) (acl2::repeat n (aig-var-fix x))))
Theorem:
(defthm nth-of-aig-varlist-fix (equal (nth n (aig-varlist-fix x)) (if (< (nfix n) (len x)) (aig-var-fix (nth n x)) nil)))
Theorem:
(defthm aig-varlist-equiv-implies-aig-varlist-equiv-append-1 (implies (aig-varlist-equiv x fty::x-equiv) (aig-varlist-equiv (append x y) (append fty::x-equiv y))) :rule-classes (:congruence))
Theorem:
(defthm aig-varlist-equiv-implies-aig-varlist-equiv-append-2 (implies (aig-varlist-equiv y fty::y-equiv) (aig-varlist-equiv (append x y) (append x fty::y-equiv))) :rule-classes (:congruence))
Theorem:
(defthm aig-varlist-equiv-implies-aig-varlist-equiv-nthcdr-2 (implies (aig-varlist-equiv l l-equiv) (aig-varlist-equiv (nthcdr n l) (nthcdr n l-equiv))) :rule-classes (:congruence))
Theorem:
(defthm aig-varlist-equiv-implies-aig-varlist-equiv-take-2 (implies (aig-varlist-equiv l l-equiv) (aig-varlist-equiv (take n l) (take n l-equiv))) :rule-classes (:congruence))