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