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