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|
(in-package "HANOI")
(include-book "state")
(include-book "move")
(include-book "hanoi-model")
(include-book "hanoi-solution")
(include-book "perm")
;; (defun disks-perm (st l)
;; (perm (app (get-peg 'A st)
;; (app (get-peg 'B st)
;; (get-peg 'C st)))
;; l))
(defun disks-perm (st l)
(perm (app (get-peg 'A st)
(app (get-peg 'B st)
(get-peg 'C st)))
l))
;----------------------------------------------------------------------
(defund wff-move (m)
(and (movep m)
(not (equal (src m)
(dest m)))))
(defund legal-movep1 (m s)
(let ((from-stk (get-peg (src m) s)))
(and (wff-move m)
(has-morep from-stk))))
;----------------------------------------------------------------------
(defthmd do-move-preserve-disk-perm
(implies (and (legal-movep1 m st)
(disks-perm st l))
(disks-perm (do-move m st) l))
:hints (("Goal" :in-theory (e/d (push top pop
has-morep
wff-move
movep
pegp
perm-app-cons-norm-1
perm-app-cons-norm-0
legal-movep1
perm-app-reduce-1
perm-app-reduce-2)
(perm)))))
;----------------------------------------------------------------------
(defun stack-inorder (stk)
(declare (xargs :hints (("Goal" :in-theory (enable has-morep pop push)))))
(if (has-morep stk)
(if (has-morep (pop stk))
(and (< (top stk)
(top (pop stk)))
(stack-inorder (pop stk)))
t)
t))
(defund disks-inorder (st)
(and (stack-inorder (g 'A st))
(stack-inorder (g 'B st))
(stack-inorder (g 'C st))))
(defund legal-movep2 (m s)
(let ((from-stk (get-peg (src m) s))
(to-stk (get-peg (dest m) s)))
(and (wff-move m)
(has-morep from-stk)
(or (not (has-morep to-stk))
(< (top from-stk)
(top to-stk))))))
;----------------------------------------------------------------------
(defthmd do-move-preserve-disks-inorder
(implies (and (legal-movep2 m st)
(disks-inorder st))
(disks-inorder (do-move m st)))
:hints (("Goal" :in-theory (e/d (push top pop
has-morep
disks-inorder
wff-move
movep
pegp
legal-movep2)
()))))
;----------------------------------------------------------------------
(defun safe-state (st n)
(and (disks-perm st (tower n))
(disks-inorder st)))
(defund legal-movep (m s)
(and (legal-movep1 m s)
(legal-movep2 m s)))
(defthmd do-move-preserve-safe-state
(implies (and (legal-movep m st)
(safe-state st n))
(safe-state (do-move m st) n))
:hints (("Goal" :in-theory (e/d (legal-movep
safe-state
do-move-preserve-disk-perm
do-move-preserve-disks-inorder)
(do-move disks-perm disks-inorder)))))
;----------------------------------------------------------------------
(defun safe-moves (moves s)
(if (endp moves)
t
(and (legal-movep (car moves) s)
(safe-moves (cdr moves)
(do-move (car moves) s)))))
(defthm safe-moves-app
(implies (safe-moves a s)
(equal (safe-moves (app a b) s)
(safe-moves b (play a s)))))
;----------------------------------------------------------------------
(defthm safe-moves-preserve-safe-state
(implies (and (safe-state st n)
(safe-moves moves st))
(safe-state (play moves st) n))
:hints (("Goal" :in-theory (e/d (do-move-preserve-safe-state)
(safe-state do-move)))))
(local
(defthmd h-lemma-2
(implies (and (natp n)
(pegp from)
(pegp to)
(pegp temp)
(not (equal from to))
(not (equal from temp))
(not (equal to temp))
(big-tops from to temp (s from anylist s) n))
(equal (play (h from to temp n)
(s from (app (tower n) anylist) s))
(s from anylist
(s to (app (tower n)
(g to s)) s))))
:hints (("Goal" :use ((:instance h-lemma
(s (s from anylist s))))))))
(local
(defthmd h-is-safe-lemma
(implies (and (natp n)
(pegp from)
(pegp to)
(pegp temp)
(not (equal from to))
(not (equal from temp))
(not (equal to temp))
(big-tops from to temp s n))
(safe-moves (h from to temp n)
(s from (app (tower n)
(g from s)) s)))
:hints (("Goal" :do-not '(generalize fertilize)
:in-theory (enable wff-move has-morep
legal-movep h-lemma-2
h-lemma new-move movep
legal-movep1 legal-movep2)
:induct (induction-hint from to temp n s)))))
;----------------------------------------------------------------------
(defthm safe-play
(safe-moves (Hanoi n) (init-state n))
:hints (("Goal" :use ((:instance h-is-safe-lemma
(from 'A)
(to 'B)
(temp 'C)
(s nil))))))
;----------------------------------------------------------------------
(defun collect-states (moves st)
(if (endp moves)
nil
(let ((nx-state (do-move (car moves) st)))
(cons nx-state (collect-states (cdr moves) nx-state)))))
(defun all-safe-states (stl n)
(if (endp stl) t
(and (safe-state (car stl) n)
(all-safe-states (cdr stl) n))))
(defthm safe-moves-implies-safe-state
(implies (and (safe-moves moves st)
(safe-state st n))
(all-safe-states (collect-states moves st) n))
:hints (("Goal" :do-not '(generalize)
:in-theory (e/d (do-move-preserve-safe-state)
(safe-state legal-movep do-move)))))
;----------------------------------------------------------------------
(defthm stack-inorder-app
(implies (and (stack-inorder stk)
(case-split (consp stk))
(case-split (< (car (last stk)) n)))
(stack-inorder (app stk (list n))))
:hints (("Goal" :in-theory (e/d (stack-inorder
pop push top has-morep)
()))))
(defthm stack-inorder-list-n
(stack-inorder (list n))
:hints (("Goal" :in-theory (enable has-morep pop top
stack-inorder))))
;; (defthm stack-inorder-app-2
;; (implies (and (stack-inorder stk)
;; (not (consp stk))
;; (stack-inorder (app stk (list n))))
;; :hints (("Goal" :in-theory (e/d (stack-inorder
;; pop push top has-morep)
;; ()))))
(defthm last-app
(equal (car (last (app l (list n))))
n))
(defthm last-tower-n
(implies (and (natp n)
(case-split (> n 0)))
(equal (car (last (tower n))) n))
:hints (("Goal" :in-theory (enable tower))))
(defthm consp-tower-n
(implies (natp n)
(iff (consp (tower n))
(> n 0))))
(defthm stack-inorder-tower-n
(implies (natp n)
(stack-inorder (tower n)))
:hints (("Goal" :do-not '(generalize)
:in-theory (e/d (stack-inorder
pop push top has-morep)
()))))
(defthm safe-state-init-state
(safe-state (init-state n) n)
:hints (("Goal" :in-theory (enable disks-inorder stack-inorder))))
(defthm safe-play-1
(all-safe-states (collect-states (Hanoi n) (init-state n)) n)
:hints (("Goal" :in-theory (e/d () (Hanoi init-state)))))
;----------------------------------------------------------------------
|
