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(in-package "ACL2")
(include-book "remove-nth")
(include-book "perm")
(include-book "std/util/bstar" :dir :system)
(local (include-book "system/random" :dir :system))
(local (include-book "tools/mv-nth" :dir :system))
(local (include-book "std/lists/len" :dir :system))
(defun shuffle (x state)
(declare (xargs :stobjs state
:measure (len x)
:guard (true-listp x)))
(if (endp x)
(mv nil state)
(b* (((mv i state) (random$ (len x) state))
((mv result state) (shuffle (remove-nth i x) state)))
(mv (cons (nth i x) result) state))))
(local
(defthm nth-member
(implies (and (consp x)
(equal (nth i x) e)
(< i (len x)))
(member-equal e x))
:rule-classes (:rewrite :forward-chaining)))
(local
(defthm remove-nth-remove1
(implies (and (consp x)
(< i (len x)))
(perm (remove-nth i x)
(remove1 (nth i x) x)))
:hints (("Goal" :induct (remove-nth i x))
("Subgoal *1/3.8'"
:cases ((consp (cdr x)))))))
(local
(defthm member-nth
(implies (and (consp x)
(< i (len x)))
(member (nth i x) x))))
(defthm perm-shuffle
(perm (mv-nth 0 (shuffle x state)) x))
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