MAYBE Time: 2.026364 TRS: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} DP: DP: { cons#(mark X1, X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2), recip# mark X -> recip# X, recip# ok X -> recip# X, sqr# mark X -> sqr# X, sqr# ok X -> sqr# X, terms# mark X -> terms# X, terms# ok X -> terms# X, s# mark X -> s# X, s# ok X -> s# X, active# cons(X1, X2) -> cons#(active X1, X2), active# cons(X1, X2) -> active# X1, active# recip X -> recip# active X, active# recip X -> active# X, active# sqr X -> sqr# active X, active# sqr X -> active# X, active# sqr s X -> sqr# X, active# sqr s X -> s# add(sqr X, dbl X), active# sqr s X -> add#(sqr X, dbl X), active# sqr s X -> dbl# X, active# terms N -> cons#(recip sqr N, terms s N), active# terms N -> recip# sqr N, active# terms N -> sqr# N, active# terms N -> terms# s N, active# terms N -> s# N, active# terms X -> terms# active X, active# terms X -> active# X, active# s X -> s# active X, active# s X -> active# X, active# add(X1, X2) -> active# X1, active# add(X1, X2) -> active# X2, active# add(X1, X2) -> add#(X1, active X2), active# add(X1, X2) -> add#(active X1, X2), active# add(s X, Y) -> s# add(X, Y), active# add(s X, Y) -> add#(X, Y), active# dbl X -> active# X, active# dbl X -> dbl# active X, active# dbl s X -> s# s dbl X, active# dbl s X -> s# dbl X, active# dbl s X -> dbl# X, active# first(X1, X2) -> active# X1, active# first(X1, X2) -> active# X2, active# first(X1, X2) -> first#(X1, active X2), active# first(X1, X2) -> first#(active X1, X2), active# first(s X, cons(Y, Z)) -> cons#(Y, first(X, Z)), active# first(s X, cons(Y, Z)) -> first#(X, Z), add#(X1, mark X2) -> add#(X1, X2), add#(mark X1, X2) -> add#(X1, X2), add#(ok X1, ok X2) -> add#(X1, X2), dbl# mark X -> dbl# X, dbl# ok X -> dbl# X, first#(X1, mark X2) -> first#(X1, X2), first#(mark X1, X2) -> first#(X1, X2), first#(ok X1, ok X2) -> first#(X1, X2), proper# cons(X1, X2) -> cons#(proper X1, proper X2), proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# recip X -> recip# proper X, proper# recip X -> proper# X, proper# sqr X -> sqr# proper X, proper# sqr X -> proper# X, proper# terms X -> terms# proper X, proper# terms X -> proper# X, proper# s X -> s# proper X, proper# s X -> proper# X, proper# add(X1, X2) -> add#(proper X1, proper X2), proper# add(X1, X2) -> proper# X1, proper# add(X1, X2) -> proper# X2, proper# dbl X -> dbl# proper X, proper# dbl X -> proper# X, proper# first(X1, X2) -> first#(proper X1, proper X2), proper# first(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X2, top# mark X -> proper# X, top# mark X -> top# proper X, top# ok X -> active# X, top# ok X -> top# active X} TRS: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} UR: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), a(x, y) -> x, a(x, y) -> y} EDG: { (active# add(X1, X2) -> add#(X1, active X2), add#(ok X1, ok X2) -> add#(X1, X2)) (active# add(X1, X2) -> add#(X1, active X2), add#(mark X1, X2) -> add#(X1, X2)) (active# add(X1, X2) -> add#(X1, active X2), add#(X1, mark X2) -> add#(X1, X2)) (proper# cons(X1, X2) -> cons#(proper X1, proper X2), cons#(ok X1, ok X2) -> cons#(X1, X2)) (proper# cons(X1, X2) -> cons#(proper X1, proper X2), cons#(mark X1, X2) -> cons#(X1, X2)) (proper# first(X1, X2) -> first#(proper X1, proper X2), first#(ok X1, ok X2) -> first#(X1, X2)) (proper# first(X1, X2) -> first#(proper X1, proper X2), first#(mark X1, X2) -> first#(X1, X2)) (proper# first(X1, X2) -> first#(proper X1, proper X2), first#(X1, mark X2) -> first#(X1, X2)) (recip# ok X -> recip# X, recip# ok X -> recip# X) (recip# ok X -> recip# X, recip# mark X -> recip# X) (sqr# ok X -> sqr# X, sqr# ok X -> sqr# X) (sqr# ok X -> sqr# X, sqr# mark X -> sqr# X) (terms# ok X -> terms# X, terms# ok X -> terms# X) (terms# ok X -> terms# X, terms# mark X -> terms# X) (s# ok X -> s# X, s# ok X -> s# X) (s# ok X -> s# X, s# mark X -> s# X) (active# sqr X -> active# X, active# first(s X, cons(Y, Z)) -> first#(X, Z)) (active# sqr X -> active# X, active# first(s X, cons(Y, Z)) -> cons#(Y, first(X, Z))) (active# sqr X -> active# X, active# first(X1, X2) -> first#(active X1, X2)) (active# sqr X -> active# X, active# first(X1, X2) -> first#(X1, active X2)) (active# sqr X -> active# X, active# first(X1, X2) -> active# X2) (active# sqr X -> active# X, active# first(X1, X2) -> active# X1) (active# sqr X -> active# X, active# dbl s X -> dbl# X) (active# sqr X -> active# X, active# dbl s X -> s# dbl X) (active# sqr X -> active# X, active# dbl s X -> s# s dbl X) (active# sqr X -> active# X, active# dbl X -> dbl# active X) (active# sqr X -> active# X, active# dbl X -> active# X) (active# sqr X -> active# X, active# add(s X, Y) -> add#(X, Y)) (active# sqr X -> active# X, active# add(s X, Y) -> s# add(X, Y)) (active# sqr X -> active# X, active# add(X1, X2) -> add#(active X1, X2)) (active# sqr X -> active# X, active# add(X1, X2) -> add#(X1, active X2)) (active# sqr X -> active# X, active# add(X1, X2) -> active# X2) (active# sqr X -> active# X, active# add(X1, X2) -> active# X1) (active# sqr X -> active# X, active# s X -> active# X) (active# sqr X -> active# X, active# s X -> s# active X) (active# sqr X -> active# X, active# terms X -> active# X) (active# sqr X -> active# X, active# terms X -> terms# active X) (active# sqr X -> active# X, active# terms N -> s# N) (active# sqr X -> active# X, active# terms N -> terms# s N) (active# sqr X -> active# X, active# terms N -> sqr# N) (active# sqr X -> active# X, active# terms N -> recip# sqr N) (active# sqr X -> active# X, active# terms N -> cons#(recip sqr N, terms s N)) (active# sqr X -> active# X, active# sqr s X -> dbl# X) (active# sqr X -> active# X, active# sqr s X -> add#(sqr X, dbl X)) (active# sqr X -> active# X, active# sqr s X -> s# add(sqr X, dbl X)) (active# sqr X -> active# X, active# sqr s X -> sqr# X) (active# sqr X -> active# X, active# sqr X -> active# X) (active# sqr X -> active# X, active# sqr X -> sqr# active X) (active# sqr X -> active# X, active# recip X -> active# X) (active# sqr X -> active# X, active# recip X -> recip# active X) (active# sqr X -> active# X, active# cons(X1, X2) -> active# X1) (active# sqr X -> active# X, active# cons(X1, X2) -> cons#(active X1, X2)) (active# sqr s X -> dbl# X, dbl# ok X -> dbl# X) (active# sqr s X -> dbl# X, dbl# mark X -> dbl# X) (active# s X -> active# X, active# first(s X, cons(Y, Z)) -> first#(X, Z)) (active# s X -> active# X, active# first(s X, cons(Y, Z)) -> cons#(Y, first(X, Z))) (active# s X -> active# X, active# first(X1, X2) -> first#(active X1, X2)) (active# s X -> active# X, active# first(X1, X2) -> first#(X1, active X2)) (active# s X -> active# X, active# first(X1, X2) -> active# X2) (active# s X -> active# X, active# first(X1, X2) -> active# X1) (active# s X -> active# X, active# dbl s X -> dbl# X) (active# s X -> active# X, active# dbl s X -> s# dbl X) (active# s X -> active# X, active# dbl s X -> s# s dbl X) (active# s X -> active# X, active# dbl X -> dbl# active X) (active# s X -> active# X, active# dbl X -> active# X) (active# s X -> active# X, active# add(s X, Y) -> add#(X, Y)) (active# s X -> active# X, active# add(s X, Y) -> s# add(X, Y)) (active# s X -> active# X, active# add(X1, X2) -> add#(active X1, X2)) (active# s X -> active# X, active# add(X1, X2) -> add#(X1, active X2)) (active# s X -> active# X, active# add(X1, X2) -> active# X2) (active# s X -> active# X, active# add(X1, X2) -> active# X1) (active# s X -> active# X, active# s X -> active# X) (active# s X -> active# X, active# s X -> s# active X) (active# s X -> active# X, active# terms X -> active# X) (active# s X -> active# X, active# terms X -> terms# active X) (active# s X -> active# X, active# terms N -> s# N) (active# s X -> active# X, active# terms N -> terms# s N) (active# s X -> active# X, active# terms N -> sqr# N) (active# s X -> active# X, active# terms N -> recip# sqr N) (active# s X -> active# X, active# terms N -> cons#(recip sqr N, terms s N)) (active# s X -> active# X, active# sqr s X -> dbl# X) (active# s X -> active# X, active# sqr s X -> add#(sqr X, dbl X)) (active# s X -> active# X, active# sqr s X -> s# add(sqr X, dbl X)) (active# s X -> active# X, active# sqr s X -> sqr# X) (active# s X -> active# X, active# sqr X -> active# X) (active# s X -> active# X, active# sqr X -> sqr# active X) (active# s X -> active# X, active# recip X -> active# X) (active# s X -> active# X, active# recip X -> recip# active X) (active# s X -> active# X, active# cons(X1, X2) -> active# X1) (active# s X -> active# X, active# cons(X1, X2) -> cons#(active X1, X2)) (active# dbl s X -> dbl# X, dbl# ok X -> dbl# X) (active# dbl s X -> dbl# X, dbl# mark X -> dbl# X) (dbl# ok X -> dbl# X, dbl# ok X -> dbl# X) (dbl# ok X -> dbl# X, dbl# mark X -> dbl# X) (proper# sqr X -> proper# X, proper# first(X1, X2) -> proper# X2) (proper# sqr X -> proper# X, proper# first(X1, X2) -> proper# X1) (proper# sqr X -> proper# X, proper# first(X1, X2) -> first#(proper X1, proper X2)) (proper# sqr X -> proper# X, proper# dbl X -> proper# X) (proper# sqr X -> proper# X, proper# dbl X -> dbl# proper X) (proper# sqr X -> proper# X, proper# add(X1, X2) -> proper# X2) (proper# sqr X -> proper# X, proper# add(X1, X2) -> proper# X1) (proper# sqr X -> proper# X, proper# add(X1, X2) -> add#(proper X1, proper X2)) (proper# sqr X -> proper# X, proper# s X -> proper# X) (proper# sqr X -> proper# X, proper# s X -> s# proper X) (proper# sqr X -> proper# X, proper# terms X -> proper# X) (proper# sqr X -> proper# X, proper# terms X -> terms# proper X) (proper# sqr X -> proper# X, proper# sqr X -> proper# X) (proper# sqr X -> proper# X, proper# sqr X -> sqr# proper X) (proper# sqr X -> proper# X, proper# recip X -> proper# X) (proper# sqr X -> proper# X, proper# recip X -> recip# proper X) (proper# sqr X -> proper# X, proper# cons(X1, X2) -> proper# X2) (proper# sqr X -> proper# X, proper# cons(X1, X2) -> proper# X1) (proper# sqr X -> proper# X, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# s X -> proper# X, proper# first(X1, X2) -> proper# X2) (proper# s X -> proper# X, proper# first(X1, X2) -> proper# X1) (proper# s X -> proper# X, proper# first(X1, X2) -> first#(proper X1, proper X2)) (proper# s X -> proper# X, proper# dbl X -> proper# X) (proper# s X -> proper# X, proper# dbl X -> dbl# proper X) (proper# s X -> proper# X, proper# add(X1, X2) -> proper# X2) (proper# s X -> proper# X, proper# add(X1, X2) -> proper# X1) (proper# s X -> proper# X, proper# add(X1, X2) -> add#(proper X1, proper X2)) (proper# s X -> proper# X, proper# s X -> proper# X) (proper# s X -> proper# X, proper# s X -> s# proper X) (proper# s X -> proper# X, proper# terms X -> proper# X) (proper# s X -> proper# X, proper# terms X -> terms# proper X) (proper# s X -> proper# X, proper# sqr X -> proper# X) (proper# s X -> proper# X, proper# sqr X -> sqr# proper X) (proper# s X -> proper# X, proper# recip X -> proper# X) (proper# s X -> proper# X, proper# recip X -> recip# proper X) (proper# s X -> proper# X, proper# cons(X1, X2) -> proper# X2) (proper# s X -> proper# X, proper# cons(X1, X2) -> proper# X1) (proper# s X -> proper# X, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (top# mark X -> proper# X, proper# first(X1, X2) -> proper# X2) (top# mark X -> proper# X, proper# first(X1, X2) -> proper# X1) (top# mark X -> proper# X, proper# first(X1, X2) -> first#(proper X1, proper X2)) (top# mark X -> proper# X, proper# dbl X -> proper# X) (top# mark X -> proper# X, proper# dbl X -> dbl# proper X) (top# mark X -> proper# X, proper# add(X1, X2) -> proper# X2) (top# mark X -> proper# X, proper# add(X1, X2) -> proper# X1) (top# mark X -> proper# X, proper# add(X1, X2) -> add#(proper X1, proper X2)) (top# mark X -> proper# X, proper# s X -> proper# X) (top# mark X -> proper# X, proper# s X -> s# proper X) (top# mark X -> proper# X, proper# terms X -> proper# X) (top# mark X -> proper# X, proper# terms X -> terms# proper X) (top# mark X -> proper# X, proper# sqr X -> proper# X) (top# mark X -> proper# X, proper# sqr X -> sqr# proper X) (top# mark X -> proper# X, proper# recip X -> proper# X) (top# mark X -> proper# X, proper# recip X -> recip# proper X) (top# mark X -> proper# X, proper# cons(X1, X2) -> proper# X2) (top# mark X -> proper# X, proper# cons(X1, X2) -> proper# X1) (top# mark X -> proper# X, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (active# sqr s X -> s# add(sqr X, dbl X), s# ok X -> s# X) (active# sqr s X -> s# add(sqr X, dbl X), s# mark X -> s# X) (active# cons(X1, X2) -> active# X1, active# first(s X, cons(Y, Z)) -> first#(X, Z)) (active# cons(X1, X2) -> active# X1, active# first(s X, cons(Y, Z)) -> cons#(Y, first(X, Z))) (active# cons(X1, X2) -> active# X1, active# first(X1, X2) -> first#(active X1, X2)) (active# cons(X1, X2) -> active# X1, active# first(X1, X2) -> first#(X1, active X2)) (active# cons(X1, X2) -> active# X1, active# first(X1, X2) -> active# X2) (active# cons(X1, X2) -> active# X1, active# first(X1, X2) -> active# X1) (active# cons(X1, X2) -> active# X1, active# dbl s X -> dbl# X) (active# cons(X1, X2) -> active# X1, active# dbl s X -> s# dbl X) (active# cons(X1, X2) -> active# X1, active# dbl s X -> s# s dbl X) (active# cons(X1, X2) -> active# X1, active# dbl X -> dbl# active X) (active# cons(X1, X2) -> active# X1, active# dbl X -> active# X) (active# cons(X1, X2) -> active# X1, active# add(s X, Y) -> add#(X, Y)) (active# cons(X1, X2) -> active# X1, active# add(s X, Y) -> s# add(X, Y)) (active# cons(X1, X2) -> active# X1, active# add(X1, X2) -> add#(active X1, X2)) (active# cons(X1, X2) -> active# X1, active# add(X1, X2) -> add#(X1, active X2)) (active# cons(X1, X2) -> active# X1, active# add(X1, X2) -> active# X2) (active# cons(X1, X2) -> active# X1, active# add(X1, X2) -> active# X1) (active# cons(X1, X2) -> active# X1, active# s X -> active# X) (active# cons(X1, X2) -> active# X1, active# s X -> s# active X) (active# cons(X1, X2) -> active# X1, active# terms X -> active# X) (active# cons(X1, X2) -> active# X1, active# terms X -> terms# active X) (active# cons(X1, X2) -> active# X1, active# terms N -> s# N) (active# cons(X1, X2) -> active# X1, active# terms N -> terms# s N) (active# cons(X1, X2) -> active# X1, active# terms N -> sqr# N) (active# cons(X1, X2) -> active# X1, active# terms N -> recip# sqr N) (active# cons(X1, X2) -> active# X1, active# terms N -> cons#(recip sqr N, terms s N)) (active# cons(X1, X2) -> active# X1, active# sqr s X -> dbl# X) (active# cons(X1, X2) -> active# X1, active# sqr s X -> add#(sqr X, dbl X)) (active# cons(X1, X2) -> active# X1, active# sqr s X -> s# add(sqr X, dbl X)) (active# cons(X1, X2) -> active# X1, active# sqr s X -> sqr# X) (active# cons(X1, X2) -> active# X1, active# sqr X -> active# X) (active# cons(X1, X2) -> active# X1, active# sqr X -> sqr# active X) (active# cons(X1, X2) -> active# X1, active# recip X -> active# X) (active# cons(X1, X2) -> active# X1, active# recip X -> recip# active X) (active# cons(X1, X2) -> active# X1, active# cons(X1, X2) -> active# X1) (active# cons(X1, X2) -> active# X1, active# cons(X1, X2) -> cons#(active X1, X2)) (active# first(X1, X2) -> active# X1, active# first(s X, cons(Y, Z)) -> first#(X, Z)) (active# first(X1, X2) -> active# X1, active# first(s X, cons(Y, Z)) -> cons#(Y, first(X, Z))) (active# first(X1, X2) -> active# X1, active# first(X1, X2) -> first#(active X1, X2)) (active# first(X1, X2) -> active# X1, active# first(X1, X2) -> first#(X1, active X2)) (active# first(X1, X2) -> active# X1, active# first(X1, X2) -> active# X2) (active# first(X1, X2) -> active# X1, active# first(X1, X2) -> active# X1) (active# first(X1, X2) -> active# X1, active# dbl s X -> dbl# X) (active# first(X1, X2) -> active# X1, active# dbl s X -> s# dbl X) (active# first(X1, X2) -> active# X1, active# dbl s X -> s# s dbl X) (active# first(X1, X2) -> active# X1, active# dbl X -> dbl# active X) (active# first(X1, X2) -> active# X1, active# dbl X -> active# X) (active# first(X1, X2) -> active# X1, active# add(s X, Y) -> add#(X, Y)) (active# first(X1, X2) -> active# X1, active# add(s X, Y) -> s# add(X, Y)) (active# first(X1, X2) -> active# X1, active# add(X1, X2) -> add#(active X1, X2)) (active# first(X1, X2) -> active# X1, active# add(X1, X2) -> add#(X1, active X2)) (active# first(X1, X2) -> active# X1, active# add(X1, X2) -> active# X2) (active# first(X1, X2) -> active# X1, active# add(X1, X2) -> active# X1) (active# first(X1, X2) -> active# X1, active# s X -> active# X) (active# first(X1, X2) -> active# X1, active# s X -> s# active X) (active# first(X1, X2) -> active# X1, active# terms X -> active# X) (active# first(X1, X2) -> active# X1, active# terms X -> terms# active X) (active# first(X1, X2) -> active# X1, active# terms N -> s# N) (active# first(X1, X2) -> active# X1, active# terms N -> terms# s N) (active# first(X1, X2) -> active# X1, active# terms N -> sqr# N) (active# first(X1, X2) -> active# X1, active# terms N -> recip# sqr N) (active# first(X1, X2) -> active# X1, active# terms N -> cons#(recip sqr N, terms s N)) (active# first(X1, X2) -> active# X1, active# sqr s X -> dbl# X) (active# first(X1, X2) -> active# X1, active# sqr s X -> add#(sqr X, dbl X)) (active# first(X1, X2) -> active# X1, active# sqr s X -> s# add(sqr X, dbl X)) (active# first(X1, X2) -> active# X1, active# sqr s X -> sqr# X) (active# first(X1, X2) -> active# X1, active# sqr X -> active# X) (active# first(X1, X2) -> active# X1, active# sqr X -> sqr# active X) (active# first(X1, X2) -> active# X1, active# recip X -> active# X) (active# first(X1, X2) -> active# X1, active# recip X -> recip# active X) (active# first(X1, X2) -> active# X1, active# cons(X1, X2) -> active# X1) (active# first(X1, X2) -> active# X1, active# cons(X1, X2) -> cons#(active X1, X2)) (proper# add(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X2) (proper# add(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X1) (proper# add(X1, X2) -> proper# X1, proper# first(X1, X2) -> first#(proper X1, proper X2)) (proper# add(X1, X2) -> proper# X1, proper# dbl X -> proper# X) (proper# add(X1, X2) -> proper# X1, proper# dbl X -> dbl# proper X) (proper# add(X1, X2) -> proper# X1, proper# add(X1, X2) -> proper# X2) (proper# add(X1, X2) -> proper# X1, proper# add(X1, X2) -> proper# X1) (proper# add(X1, X2) -> proper# X1, proper# add(X1, X2) -> add#(proper X1, proper X2)) (proper# add(X1, X2) -> proper# X1, proper# s X -> proper# X) (proper# add(X1, X2) -> proper# X1, proper# s X -> s# proper X) (proper# add(X1, X2) -> proper# X1, proper# terms X -> proper# X) (proper# add(X1, X2) -> proper# X1, proper# terms X -> terms# proper X) (proper# add(X1, X2) -> proper# X1, proper# sqr X -> proper# X) (proper# add(X1, X2) -> proper# X1, proper# sqr X -> sqr# proper X) (proper# add(X1, X2) -> proper# X1, proper# recip X -> proper# X) (proper# add(X1, X2) -> proper# X1, proper# recip X -> recip# proper X) (proper# add(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2) (proper# add(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X1) (proper# add(X1, X2) -> proper# X1, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (active# terms N -> sqr# N, sqr# ok X -> sqr# X) (active# terms N -> sqr# N, sqr# mark X -> sqr# X) (active# add(s X, Y) -> s# add(X, Y), s# ok X -> s# X) (active# add(s X, Y) -> s# add(X, Y), s# mark X -> s# X) (active# first(s X, cons(Y, Z)) -> first#(X, Z), first#(ok X1, ok X2) -> first#(X1, X2)) (active# first(s X, cons(Y, Z)) -> first#(X, Z), first#(mark X1, X2) -> first#(X1, X2)) (active# first(s X, cons(Y, Z)) -> first#(X, Z), first#(X1, mark X2) -> first#(X1, X2)) (active# recip X -> recip# active X, recip# ok X -> recip# X) (active# recip X -> recip# active X, recip# mark X -> recip# X) (active# terms X -> terms# active X, terms# ok X -> terms# X) (active# terms X -> terms# active X, terms# mark X -> terms# X) (active# dbl X -> dbl# active X, dbl# ok X -> dbl# X) (active# dbl X -> dbl# active X, dbl# mark X -> dbl# X) (proper# recip X -> recip# proper X, recip# ok X -> recip# X) (proper# recip X -> recip# proper X, recip# mark X -> recip# X) (proper# terms X -> terms# proper X, terms# ok X -> terms# X) (proper# terms X -> terms# proper X, terms# mark X -> terms# X) (proper# dbl X -> dbl# proper X, dbl# ok X -> dbl# X) (proper# dbl X -> dbl# proper X, dbl# mark X -> dbl# X) (top# ok X -> top# active X, top# ok X -> top# active X) (top# ok X -> top# active X, top# ok X -> active# X) (top# ok X -> top# active X, top# mark X -> top# proper X) (top# ok X -> top# active X, top# mark X -> proper# X) (active# terms N -> terms# s N, terms# ok X -> terms# X) (active# terms N -> terms# s N, terms# mark X -> terms# X) (active# add(X1, X2) -> add#(active X1, X2), add#(ok X1, ok X2) -> add#(X1, X2)) (active# add(X1, X2) -> add#(active X1, X2), add#(mark X1, X2) -> add#(X1, X2)) (active# add(X1, X2) -> add#(active X1, X2), add#(X1, mark X2) -> add#(X1, X2)) (active# add(X1, X2) -> active# X2, active# first(s X, cons(Y, Z)) -> first#(X, Z)) (active# add(X1, X2) -> active# X2, active# first(s X, cons(Y, Z)) -> cons#(Y, first(X, Z))) (active# add(X1, X2) -> active# X2, active# first(X1, X2) -> first#(active X1, X2)) (active# add(X1, X2) -> active# X2, active# first(X1, X2) -> first#(X1, active X2)) (active# add(X1, X2) -> active# X2, active# first(X1, X2) -> active# X2) (active# add(X1, X2) -> active# X2, active# first(X1, X2) -> active# X1) (active# add(X1, X2) -> active# X2, active# dbl s X -> dbl# X) (active# add(X1, X2) -> active# X2, active# dbl s X -> s# dbl X) (active# add(X1, X2) -> active# X2, active# dbl s X -> s# s dbl X) (active# add(X1, X2) -> active# X2, active# dbl X -> dbl# active X) (active# add(X1, X2) -> active# X2, active# dbl X -> active# X) (active# add(X1, X2) -> active# X2, active# add(s X, Y) -> add#(X, Y)) (active# add(X1, X2) -> active# X2, active# add(s X, Y) -> s# add(X, Y)) (active# add(X1, X2) -> active# X2, active# add(X1, X2) -> add#(active X1, X2)) (active# add(X1, X2) -> active# X2, active# add(X1, X2) -> add#(X1, active X2)) (active# add(X1, X2) -> active# X2, active# add(X1, X2) -> active# X2) (active# add(X1, X2) -> active# X2, active# add(X1, X2) -> active# X1) (active# add(X1, X2) -> active# X2, active# s X -> active# X) (active# add(X1, X2) -> active# X2, active# s X -> s# active X) (active# add(X1, X2) -> active# X2, active# terms X -> active# X) (active# add(X1, X2) -> active# X2, active# terms X -> terms# active X) (active# add(X1, X2) -> active# X2, active# terms N -> s# N) (active# add(X1, X2) -> active# X2, active# terms N -> terms# s N) (active# add(X1, X2) -> active# X2, active# terms N -> sqr# N) (active# add(X1, X2) -> active# X2, active# terms N -> recip# sqr N) (active# add(X1, X2) -> active# X2, active# terms N -> cons#(recip sqr N, terms s N)) (active# add(X1, X2) -> active# X2, active# sqr s X -> dbl# X) (active# add(X1, X2) -> active# X2, active# sqr s X -> add#(sqr X, dbl X)) (active# add(X1, X2) -> active# X2, active# sqr s X -> s# add(sqr X, dbl X)) (active# add(X1, X2) -> active# X2, active# sqr s X -> sqr# X) (active# add(X1, X2) -> active# X2, active# sqr X -> active# X) (active# add(X1, X2) -> active# X2, active# sqr X -> sqr# active X) (active# add(X1, X2) -> active# X2, active# recip X -> active# X) (active# add(X1, X2) -> active# X2, active# recip X -> recip# active X) (active# add(X1, X2) -> active# X2, active# cons(X1, X2) -> active# X1) (active# add(X1, X2) -> active# X2, active# cons(X1, X2) -> cons#(active X1, X2)) (proper# cons(X1, X2) -> proper# X2, proper# first(X1, X2) -> proper# X2) (proper# cons(X1, X2) -> proper# X2, proper# first(X1, X2) -> proper# X1) (proper# cons(X1, X2) -> proper# X2, proper# first(X1, X2) -> first#(proper X1, proper X2)) (proper# cons(X1, X2) -> proper# X2, proper# dbl X -> proper# X) (proper# cons(X1, X2) -> proper# X2, proper# dbl X -> dbl# proper X) (proper# cons(X1, X2) -> proper# X2, proper# add(X1, X2) -> proper# X2) (proper# cons(X1, X2) -> proper# X2, proper# add(X1, X2) -> proper# X1) (proper# cons(X1, X2) -> proper# X2, proper# add(X1, X2) -> add#(proper X1, proper X2)) (proper# cons(X1, X2) -> proper# X2, proper# s X -> proper# X) (proper# cons(X1, X2) -> proper# X2, proper# s X -> s# proper X) (proper# cons(X1, X2) -> proper# X2, proper# terms X -> proper# X) (proper# cons(X1, X2) -> proper# X2, proper# terms X -> terms# proper X) (proper# cons(X1, X2) -> proper# X2, proper# sqr X -> proper# X) (proper# cons(X1, X2) -> proper# X2, proper# sqr X -> sqr# proper X) (proper# cons(X1, X2) -> proper# X2, proper# recip X -> proper# X) (proper# cons(X1, X2) -> proper# X2, proper# recip X -> recip# proper X) (proper# cons(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X2) (proper# cons(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X1) (proper# cons(X1, X2) -> proper# X2, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# first(X1, X2) -> proper# X2, proper# first(X1, X2) -> proper# X2) (proper# first(X1, X2) -> proper# X2, proper# first(X1, X2) -> proper# X1) (proper# first(X1, X2) -> proper# X2, proper# first(X1, X2) -> first#(proper X1, proper X2)) (proper# first(X1, X2) -> proper# X2, proper# dbl X -> proper# X) (proper# first(X1, X2) -> proper# X2, proper# dbl X -> dbl# proper X) (proper# first(X1, X2) -> proper# X2, proper# add(X1, X2) -> proper# X2) (proper# first(X1, X2) -> proper# X2, proper# add(X1, X2) -> proper# X1) (proper# first(X1, X2) -> proper# X2, proper# add(X1, X2) -> add#(proper X1, proper X2)) (proper# first(X1, X2) -> proper# X2, proper# s X -> proper# X) (proper# first(X1, X2) -> proper# X2, proper# s X -> s# proper X) (proper# first(X1, X2) -> proper# X2, proper# terms X -> proper# X) (proper# first(X1, X2) -> proper# X2, proper# terms X -> terms# proper X) (proper# first(X1, X2) -> proper# X2, proper# sqr X -> proper# X) (proper# first(X1, X2) -> proper# X2, proper# sqr X -> sqr# proper X) (proper# first(X1, X2) -> proper# X2, proper# recip X -> proper# X) (proper# first(X1, X2) -> proper# X2, proper# recip X -> recip# proper X) (proper# first(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X2) (proper# first(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X1) (proper# first(X1, X2) -> proper# X2, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (cons#(ok X1, ok X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)) (cons#(ok X1, ok X2) -> cons#(X1, X2), cons#(mark X1, X2) -> cons#(X1, X2)) (add#(mark X1, X2) -> add#(X1, X2), add#(ok X1, ok X2) -> add#(X1, X2)) (add#(mark X1, X2) -> add#(X1, X2), add#(mark X1, X2) -> add#(X1, X2)) (add#(mark X1, X2) -> add#(X1, X2), add#(X1, mark X2) -> add#(X1, X2)) (first#(X1, mark X2) -> first#(X1, X2), first#(ok X1, ok X2) -> first#(X1, X2)) (first#(X1, mark X2) -> first#(X1, X2), first#(mark X1, X2) -> first#(X1, X2)) (first#(X1, mark X2) -> first#(X1, X2), first#(X1, mark X2) -> first#(X1, X2)) (first#(ok X1, ok X2) -> first#(X1, X2), first#(ok X1, ok X2) -> first#(X1, X2)) (first#(ok X1, ok X2) -> first#(X1, X2), first#(mark X1, X2) -> first#(X1, X2)) (first#(ok X1, ok X2) -> first#(X1, X2), first#(X1, mark X2) -> first#(X1, X2)) (active# add(s X, Y) -> add#(X, Y), add#(X1, mark X2) -> add#(X1, X2)) (active# add(s X, Y) -> add#(X, Y), add#(mark X1, X2) -> add#(X1, X2)) (active# add(s X, Y) -> add#(X, Y), add#(ok X1, ok X2) -> add#(X1, X2)) (first#(mark X1, X2) -> first#(X1, X2), first#(X1, mark X2) -> first#(X1, X2)) (first#(mark X1, X2) -> first#(X1, X2), first#(mark X1, X2) -> first#(X1, X2)) (first#(mark X1, X2) -> first#(X1, X2), first#(ok X1, ok X2) -> first#(X1, X2)) (add#(ok X1, ok X2) -> add#(X1, X2), add#(X1, mark X2) -> add#(X1, X2)) (add#(ok X1, ok X2) -> add#(X1, X2), add#(mark X1, X2) -> add#(X1, X2)) (add#(ok X1, ok X2) -> add#(X1, X2), add#(ok X1, ok X2) -> add#(X1, X2)) (add#(X1, mark X2) -> add#(X1, X2), add#(X1, mark X2) -> add#(X1, X2)) (add#(X1, mark X2) -> add#(X1, X2), add#(mark X1, X2) -> add#(X1, X2)) (add#(X1, mark X2) -> add#(X1, X2), add#(ok X1, ok X2) -> add#(X1, X2)) (cons#(mark X1, X2) -> cons#(X1, X2), cons#(mark X1, X2) -> cons#(X1, X2)) (cons#(mark X1, X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)) (proper# add(X1, X2) -> proper# X2, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# add(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X1) (proper# add(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X2) (proper# add(X1, X2) -> proper# X2, proper# recip X -> recip# proper X) (proper# add(X1, X2) -> proper# X2, proper# recip X -> proper# X) (proper# add(X1, X2) -> proper# X2, proper# sqr X -> sqr# proper X) (proper# add(X1, X2) -> proper# X2, proper# sqr X -> proper# X) (proper# add(X1, X2) -> proper# X2, proper# terms X -> terms# proper X) (proper# add(X1, X2) -> proper# X2, proper# terms X -> proper# X) (proper# add(X1, X2) -> proper# X2, proper# s X -> s# proper X) (proper# add(X1, X2) -> proper# X2, proper# s X -> proper# X) (proper# add(X1, X2) -> proper# X2, proper# add(X1, X2) -> add#(proper X1, proper X2)) (proper# add(X1, X2) -> proper# X2, proper# add(X1, X2) -> proper# X1) (proper# add(X1, X2) -> proper# X2, proper# add(X1, X2) -> proper# X2) (proper# add(X1, X2) -> proper# X2, proper# dbl X -> dbl# proper X) (proper# add(X1, X2) -> proper# X2, proper# dbl X -> proper# X) (proper# add(X1, X2) -> proper# X2, proper# first(X1, X2) -> first#(proper X1, proper X2)) (proper# add(X1, X2) -> proper# X2, proper# first(X1, X2) -> proper# X1) (proper# add(X1, X2) -> proper# X2, proper# first(X1, X2) -> proper# X2) (active# first(X1, X2) -> active# X2, active# cons(X1, X2) -> cons#(active X1, X2)) (active# first(X1, X2) -> active# X2, active# cons(X1, X2) -> active# X1) (active# first(X1, X2) -> active# X2, active# recip X -> recip# active X) (active# first(X1, X2) -> active# X2, active# recip X -> active# X) (active# first(X1, X2) -> active# X2, active# sqr X -> sqr# active X) (active# first(X1, X2) -> active# X2, active# sqr X -> active# X) (active# first(X1, X2) -> active# X2, active# sqr s X -> sqr# X) (active# first(X1, X2) -> active# X2, active# sqr s X -> s# add(sqr X, dbl X)) (active# first(X1, X2) -> active# X2, active# sqr s X -> add#(sqr X, dbl X)) (active# first(X1, X2) -> active# X2, active# sqr s X -> dbl# X) (active# first(X1, X2) -> active# X2, active# terms N -> cons#(recip sqr N, terms s N)) (active# first(X1, X2) -> active# X2, active# terms N -> recip# sqr N) (active# first(X1, X2) -> active# X2, active# terms N -> sqr# N) (active# first(X1, X2) -> active# X2, active# terms N -> terms# s N) (active# first(X1, X2) -> active# X2, active# terms N -> s# N) (active# first(X1, X2) -> active# X2, active# terms X -> terms# active X) (active# first(X1, X2) -> active# X2, active# terms X -> active# X) (active# first(X1, X2) -> active# X2, active# s X -> s# active X) (active# first(X1, X2) -> active# X2, active# s X -> active# X) (active# first(X1, X2) -> active# X2, active# add(X1, X2) -> active# X1) (active# first(X1, X2) -> active# X2, active# add(X1, X2) -> active# X2) (active# first(X1, X2) -> active# X2, active# add(X1, X2) -> add#(X1, active X2)) (active# first(X1, X2) -> active# X2, active# add(X1, X2) -> add#(active X1, X2)) (active# first(X1, X2) -> active# X2, active# add(s X, Y) -> s# add(X, Y)) (active# first(X1, X2) -> active# X2, active# add(s X, Y) -> add#(X, Y)) (active# first(X1, X2) -> active# X2, active# dbl X -> active# X) (active# first(X1, X2) -> active# X2, active# dbl X -> dbl# active X) (active# first(X1, X2) -> active# X2, active# dbl s X -> s# s dbl X) (active# first(X1, X2) -> active# X2, active# dbl s X -> s# dbl X) (active# first(X1, X2) -> active# X2, active# dbl s X -> dbl# X) (active# first(X1, X2) -> active# X2, active# first(X1, X2) -> active# X1) (active# first(X1, X2) -> active# X2, active# first(X1, X2) -> active# X2) (active# first(X1, X2) -> active# X2, active# first(X1, X2) -> first#(X1, active X2)) (active# first(X1, X2) -> active# X2, active# first(X1, X2) -> first#(active X1, X2)) (active# first(X1, X2) -> active# X2, active# first(s X, cons(Y, Z)) -> cons#(Y, first(X, Z))) (active# first(X1, X2) -> active# X2, active# first(s X, cons(Y, Z)) -> first#(X, Z)) (active# first(X1, X2) -> first#(active X1, X2), first#(X1, mark X2) -> first#(X1, X2)) (active# first(X1, X2) -> first#(active X1, X2), first#(mark X1, X2) -> first#(X1, X2)) (active# first(X1, X2) -> first#(active X1, X2), first#(ok X1, ok X2) -> first#(X1, X2)) (active# cons(X1, X2) -> cons#(active X1, X2), cons#(mark X1, X2) -> cons#(X1, X2)) (active# cons(X1, X2) -> cons#(active X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)) (active# terms N -> recip# sqr N, recip# mark X -> recip# X) (active# terms N -> recip# sqr N, recip# ok X -> recip# X) (top# mark X -> top# proper X, top# mark X -> proper# X) (top# mark X -> top# proper X, top# mark X -> top# proper X) (top# mark X -> top# proper X, top# ok X -> active# X) (top# mark X -> top# proper X, top# ok X -> top# active X) (proper# s X -> s# proper X, s# mark X -> s# X) (proper# s X -> s# proper X, s# ok X -> s# X) (proper# sqr X -> sqr# proper X, sqr# mark X -> sqr# X) (proper# sqr X -> sqr# proper X, sqr# ok X -> sqr# X) (active# dbl s X -> s# dbl X, s# mark X -> s# X) (active# dbl s X -> s# dbl X, s# ok X -> s# X) (active# s X -> s# active X, s# mark X -> s# X) (active# s X -> s# active X, s# ok X -> s# X) (active# sqr X -> sqr# active X, sqr# mark X -> sqr# X) (active# sqr X -> sqr# active X, sqr# ok X -> sqr# X) (active# terms N -> cons#(recip sqr N, terms s N), cons#(mark X1, X2) -> cons#(X1, X2)) (active# terms N -> cons#(recip sqr N, terms s N), cons#(ok X1, ok X2) -> cons#(X1, X2)) (active# first(s X, cons(Y, Z)) -> cons#(Y, first(X, Z)), cons#(mark X1, X2) -> cons#(X1, X2)) (active# first(s X, cons(Y, Z)) -> cons#(Y, first(X, Z)), cons#(ok X1, ok X2) -> cons#(X1, X2)) (active# terms N -> s# N, s# mark X -> s# X) (active# terms N -> s# N, s# ok X -> s# X) (proper# first(X1, X2) -> proper# X1, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# first(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X1) (proper# first(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2) (proper# first(X1, X2) -> proper# X1, proper# recip X -> recip# proper X) (proper# first(X1, X2) -> proper# X1, proper# recip X -> proper# X) (proper# first(X1, X2) -> proper# X1, proper# sqr X -> sqr# proper X) (proper# first(X1, X2) -> proper# X1, proper# sqr X -> proper# X) (proper# first(X1, X2) -> proper# X1, proper# terms X -> terms# proper X) (proper# first(X1, X2) -> proper# X1, proper# terms X -> proper# X) (proper# first(X1, X2) -> proper# X1, proper# s X -> s# proper X) (proper# first(X1, X2) -> proper# X1, proper# s X -> proper# X) (proper# first(X1, X2) -> proper# X1, proper# add(X1, X2) -> add#(proper X1, proper X2)) (proper# first(X1, X2) -> proper# X1, proper# add(X1, X2) -> proper# X1) (proper# first(X1, X2) -> proper# X1, proper# add(X1, X2) -> proper# X2) (proper# first(X1, X2) -> proper# X1, proper# dbl X -> dbl# proper X) (proper# first(X1, X2) -> proper# X1, proper# dbl X -> proper# X) (proper# first(X1, X2) -> proper# X1, proper# first(X1, X2) -> first#(proper X1, proper X2)) (proper# first(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X1) (proper# first(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X2) (proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X1) (proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2) (proper# cons(X1, X2) -> proper# X1, proper# recip X -> recip# proper X) (proper# cons(X1, X2) -> proper# X1, proper# recip X -> proper# X) (proper# cons(X1, X2) -> proper# X1, proper# sqr X -> sqr# proper X) (proper# cons(X1, X2) -> proper# X1, proper# sqr X -> proper# X) (proper# cons(X1, X2) -> proper# X1, proper# terms X -> terms# proper X) (proper# cons(X1, X2) -> proper# X1, proper# terms X -> proper# X) (proper# cons(X1, X2) -> proper# X1, proper# s X -> s# proper X) (proper# cons(X1, X2) -> proper# X1, proper# s X -> proper# X) (proper# cons(X1, X2) -> proper# X1, proper# add(X1, X2) -> add#(proper X1, proper X2)) (proper# cons(X1, X2) -> proper# X1, proper# add(X1, X2) -> proper# X1) (proper# cons(X1, X2) -> proper# X1, proper# add(X1, X2) -> proper# X2) (proper# cons(X1, X2) -> proper# X1, proper# dbl X -> dbl# proper X) (proper# cons(X1, X2) -> proper# X1, proper# dbl X -> proper# X) (proper# cons(X1, X2) -> proper# X1, proper# first(X1, X2) -> first#(proper X1, proper X2)) (proper# cons(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X1) (proper# cons(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X2) (active# add(X1, X2) -> active# X1, active# cons(X1, X2) -> cons#(active X1, X2)) (active# add(X1, X2) -> active# X1, active# cons(X1, X2) -> active# X1) (active# add(X1, X2) -> active# X1, active# recip X -> recip# active X) (active# add(X1, X2) -> active# X1, active# recip X -> active# X) (active# add(X1, X2) -> active# X1, active# sqr X -> sqr# active X) (active# add(X1, X2) -> active# X1, active# sqr X -> active# X) (active# add(X1, X2) -> active# X1, active# sqr s X -> sqr# X) (active# add(X1, X2) -> active# X1, active# sqr s X -> s# add(sqr X, dbl X)) (active# add(X1, X2) -> active# X1, active# sqr s X -> add#(sqr X, dbl X)) (active# add(X1, X2) -> active# X1, active# sqr s X -> dbl# X) (active# add(X1, X2) -> active# X1, active# terms N -> cons#(recip sqr N, terms s N)) (active# add(X1, X2) -> active# X1, active# terms N -> recip# sqr N) (active# add(X1, X2) -> active# X1, active# terms N -> sqr# N) (active# add(X1, X2) -> active# X1, active# terms N -> terms# s N) (active# add(X1, X2) -> active# X1, active# terms N -> s# N) (active# add(X1, X2) -> active# X1, active# terms X -> terms# active X) (active# add(X1, X2) -> active# X1, active# terms X -> active# X) (active# add(X1, X2) -> active# X1, active# s X -> s# active X) (active# add(X1, X2) -> active# X1, active# s X -> active# X) (active# add(X1, X2) -> active# X1, active# add(X1, X2) -> active# X1) (active# add(X1, X2) -> active# X1, active# add(X1, X2) -> active# X2) (active# add(X1, X2) -> active# X1, active# add(X1, X2) -> add#(X1, active X2)) (active# add(X1, X2) -> active# X1, active# add(X1, X2) -> add#(active X1, X2)) (active# add(X1, X2) -> active# X1, active# add(s X, Y) -> s# add(X, Y)) (active# add(X1, X2) -> active# X1, active# add(s X, Y) -> add#(X, Y)) (active# add(X1, X2) -> active# X1, active# dbl X -> active# X) (active# add(X1, X2) -> active# X1, active# dbl X -> dbl# active X) (active# add(X1, X2) -> active# X1, active# dbl s X -> s# s dbl X) (active# add(X1, X2) -> active# X1, active# dbl s X -> s# dbl X) (active# add(X1, X2) -> active# X1, active# dbl s X -> dbl# X) (active# add(X1, X2) -> active# X1, active# first(X1, X2) -> active# X1) (active# add(X1, X2) -> active# X1, active# first(X1, X2) -> active# X2) (active# add(X1, X2) -> active# X1, active# first(X1, X2) -> first#(X1, active X2)) (active# add(X1, X2) -> active# X1, active# first(X1, X2) -> first#(active X1, X2)) (active# add(X1, X2) -> active# X1, active# first(s X, cons(Y, Z)) -> cons#(Y, first(X, Z))) (active# add(X1, X2) -> active# X1, active# first(s X, cons(Y, Z)) -> first#(X, Z)) (active# dbl s X -> s# s dbl X, s# mark X -> s# X) (active# dbl s X -> s# s dbl X, s# ok X -> s# X) (top# ok X -> active# X, active# cons(X1, X2) -> cons#(active X1, X2)) (top# ok X -> active# X, active# cons(X1, X2) -> active# X1) (top# ok X -> active# X, active# recip X -> recip# active X) (top# ok X -> active# X, active# recip X -> active# X) (top# ok X -> active# X, active# sqr X -> sqr# active X) (top# ok X -> active# X, active# sqr X -> active# X) (top# ok X -> active# X, active# sqr s X -> sqr# X) (top# ok X -> active# X, active# sqr s X -> s# add(sqr X, dbl X)) (top# ok X -> active# X, active# sqr s X -> add#(sqr X, dbl X)) (top# ok X -> active# X, active# sqr s X -> dbl# X) (top# ok X -> active# X, active# terms N -> cons#(recip sqr N, terms s N)) (top# ok X -> active# X, active# terms N -> recip# sqr N) (top# ok X -> active# X, active# terms N -> sqr# N) (top# ok X -> active# X, active# terms N -> terms# s N) (top# ok X -> active# X, active# terms N -> s# N) (top# ok X -> active# X, active# terms X -> terms# active X) (top# ok X -> active# X, active# terms X -> active# X) (top# ok X -> active# X, active# s X -> s# active X) (top# ok X -> active# X, active# s X -> active# X) (top# ok X -> active# X, active# add(X1, X2) -> active# X1) (top# ok X -> active# X, active# add(X1, X2) -> active# X2) (top# ok X -> active# X, active# add(X1, X2) -> add#(X1, active X2)) (top# ok X -> active# X, active# add(X1, X2) -> add#(active X1, X2)) (top# ok X -> active# X, active# add(s X, Y) -> s# add(X, Y)) (top# ok X -> active# X, active# add(s X, Y) -> add#(X, Y)) (top# ok X -> active# X, active# dbl X -> active# X) (top# ok X -> active# X, active# dbl X -> dbl# active X) (top# ok X -> active# X, active# dbl s X -> s# s dbl X) (top# ok X -> active# X, active# dbl s X -> s# dbl X) (top# ok X -> active# X, active# dbl s X -> dbl# X) (top# ok X -> active# X, active# first(X1, X2) -> active# X1) (top# ok X -> active# X, active# first(X1, X2) -> active# X2) (top# ok X -> active# X, active# first(X1, X2) -> first#(X1, active X2)) (top# ok X -> active# X, active# first(X1, X2) -> first#(active X1, X2)) (top# ok X -> active# X, active# first(s X, cons(Y, Z)) -> cons#(Y, first(X, Z))) (top# ok X -> active# X, active# first(s X, cons(Y, Z)) -> first#(X, Z)) (proper# dbl X -> proper# X, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# dbl X -> proper# X, proper# cons(X1, X2) -> proper# X1) (proper# dbl X -> proper# X, proper# cons(X1, X2) -> proper# X2) (proper# dbl X -> proper# X, proper# recip X -> recip# proper X) (proper# dbl X -> proper# X, proper# recip X -> proper# X) (proper# dbl X -> proper# X, proper# sqr X -> sqr# proper X) (proper# dbl X -> proper# X, proper# sqr X -> proper# X) (proper# dbl X -> proper# X, proper# terms X -> terms# proper X) (proper# dbl X -> proper# X, proper# terms X -> proper# X) (proper# dbl X -> proper# X, proper# s X -> s# proper X) (proper# dbl X -> proper# X, proper# s X -> proper# X) (proper# dbl X -> proper# X, proper# add(X1, X2) -> add#(proper X1, proper X2)) (proper# dbl X -> proper# X, proper# add(X1, X2) -> proper# X1) (proper# dbl X -> proper# X, proper# add(X1, X2) -> proper# X2) (proper# dbl X -> proper# X, proper# dbl X -> dbl# proper X) (proper# dbl X -> proper# X, proper# dbl X -> proper# X) (proper# dbl X -> proper# X, proper# first(X1, X2) -> first#(proper X1, proper X2)) (proper# dbl X -> proper# X, proper# first(X1, X2) -> proper# X1) (proper# dbl X -> proper# X, proper# first(X1, X2) -> proper# X2) (proper# terms X -> proper# X, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# terms X -> proper# X, proper# cons(X1, X2) -> proper# X1) (proper# terms X -> proper# X, proper# cons(X1, X2) -> proper# X2) (proper# terms X -> proper# X, proper# recip X -> recip# proper X) (proper# terms X -> proper# X, proper# recip X -> proper# X) (proper# terms X -> proper# X, proper# sqr X -> sqr# proper X) (proper# terms X -> proper# X, proper# sqr X -> proper# X) (proper# terms X -> proper# X, proper# terms X -> terms# proper X) (proper# terms X -> proper# X, proper# terms X -> proper# X) (proper# terms X -> proper# X, proper# s X -> s# proper X) (proper# terms X -> proper# X, proper# s X -> proper# X) (proper# terms X -> proper# X, proper# add(X1, X2) -> add#(proper X1, proper X2)) (proper# terms X -> proper# X, proper# add(X1, X2) -> proper# X1) (proper# terms X -> proper# X, proper# add(X1, X2) -> proper# X2) (proper# terms X -> proper# X, proper# dbl X -> dbl# proper X) (proper# terms X -> proper# X, proper# dbl X -> proper# X) (proper# terms X -> proper# X, proper# first(X1, X2) -> first#(proper X1, proper X2)) (proper# terms X -> proper# X, proper# first(X1, X2) -> proper# X1) (proper# terms X -> proper# X, proper# first(X1, X2) -> proper# X2) (proper# recip X -> proper# X, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# recip X -> proper# X, proper# cons(X1, X2) -> proper# X1) (proper# recip X -> proper# X, proper# cons(X1, X2) -> proper# X2) (proper# recip X -> proper# X, proper# recip X -> recip# proper X) (proper# recip X -> proper# X, proper# recip X -> proper# X) (proper# recip X -> proper# X, proper# sqr X -> sqr# proper X) (proper# recip X -> proper# X, proper# sqr X -> proper# X) (proper# recip X -> proper# X, proper# terms X -> terms# proper X) (proper# recip X -> proper# X, proper# terms X -> proper# X) (proper# recip X -> proper# X, proper# s X -> s# proper X) (proper# recip X -> proper# X, proper# s X -> proper# X) (proper# recip X -> proper# X, proper# add(X1, X2) -> add#(proper X1, proper X2)) (proper# recip X -> proper# X, proper# add(X1, X2) -> proper# X1) (proper# recip X -> proper# X, proper# add(X1, X2) -> proper# X2) (proper# recip X -> proper# X, proper# dbl X -> dbl# proper X) (proper# recip X -> proper# X, proper# dbl X -> proper# X) (proper# recip X -> proper# X, proper# first(X1, X2) -> first#(proper X1, proper X2)) (proper# recip X -> proper# X, proper# first(X1, X2) -> proper# X1) (proper# recip X -> proper# X, proper# first(X1, X2) -> proper# X2) (dbl# mark X -> dbl# X, dbl# mark X -> dbl# X) (dbl# mark X -> dbl# X, dbl# ok X -> dbl# X) (active# dbl X -> active# X, active# cons(X1, X2) -> cons#(active X1, X2)) (active# dbl X -> active# X, active# cons(X1, X2) -> active# X1) (active# dbl X -> active# X, active# recip X -> recip# active X) (active# dbl X -> active# X, active# recip X -> active# X) (active# dbl X -> active# X, active# sqr X -> sqr# active X) (active# dbl X -> active# X, active# sqr X -> active# X) (active# dbl X -> active# X, active# sqr s X -> sqr# X) (active# dbl X -> active# X, active# sqr s X -> s# add(sqr X, dbl X)) (active# dbl X -> active# X, active# sqr s X -> add#(sqr X, dbl X)) (active# dbl X -> active# X, active# sqr s X -> dbl# X) (active# dbl X -> active# X, active# terms N -> cons#(recip sqr N, terms s N)) (active# dbl X -> active# X, active# terms N -> recip# sqr N) (active# dbl X -> active# X, active# terms N -> sqr# N) (active# dbl X -> active# X, active# terms N -> terms# s N) (active# dbl X -> active# X, active# terms N -> s# N) (active# dbl X -> active# X, active# terms X -> terms# active X) (active# dbl X -> active# X, active# terms X -> active# X) (active# dbl X -> active# X, active# s X -> s# active X) (active# dbl X -> active# X, active# s X -> active# X) (active# dbl X -> active# X, active# add(X1, X2) -> active# X1) (active# dbl X -> active# X, active# add(X1, X2) -> active# X2) (active# dbl X -> active# X, active# add(X1, X2) -> add#(X1, active X2)) (active# dbl X -> active# X, active# add(X1, X2) -> add#(active X1, X2)) (active# dbl X -> active# X, active# add(s X, Y) -> s# add(X, Y)) (active# dbl X -> active# X, active# add(s X, Y) -> add#(X, Y)) (active# dbl X -> active# X, active# dbl X -> active# X) (active# dbl X -> active# X, active# dbl X -> dbl# active X) (active# dbl X -> active# X, active# dbl s X -> s# s dbl X) (active# dbl X -> active# X, active# dbl s X -> s# dbl X) (active# dbl X -> active# X, active# dbl s X -> dbl# X) (active# dbl X -> active# X, active# first(X1, X2) -> active# X1) (active# dbl X -> active# X, active# first(X1, X2) -> active# X2) (active# dbl X -> active# X, active# first(X1, X2) -> first#(X1, active X2)) (active# dbl X -> active# X, active# first(X1, X2) -> first#(active X1, X2)) (active# dbl X -> active# X, active# first(s X, cons(Y, Z)) -> cons#(Y, first(X, Z))) (active# dbl X -> active# X, active# first(s X, cons(Y, Z)) -> first#(X, Z)) (active# terms X -> active# X, active# cons(X1, X2) -> cons#(active X1, X2)) (active# terms X -> active# X, active# cons(X1, X2) -> active# X1) (active# terms X -> active# X, active# recip X -> recip# active X) (active# terms X -> active# X, active# recip X -> active# X) (active# terms X -> active# X, active# sqr X -> sqr# active X) (active# terms X -> active# X, active# sqr X -> active# X) (active# terms X -> active# X, active# sqr s X -> sqr# X) (active# terms X -> active# X, active# sqr s X -> s# add(sqr X, dbl X)) (active# terms X -> active# X, active# sqr s X -> add#(sqr X, dbl X)) (active# terms X -> active# X, active# sqr s X -> dbl# X) (active# terms X -> active# X, active# terms N -> cons#(recip sqr N, terms s N)) (active# terms X -> active# X, active# terms N -> recip# sqr N) (active# terms X -> active# X, active# terms N -> sqr# N) (active# terms X -> active# X, active# terms N -> terms# s N) (active# terms X -> active# X, active# terms N -> s# N) (active# terms X -> active# X, active# terms X -> terms# active X) (active# terms X -> active# X, active# terms X -> active# X) (active# terms X -> active# X, active# s X -> s# active X) (active# terms X -> active# X, active# s X -> active# X) (active# terms X -> active# X, active# add(X1, X2) -> active# X1) (active# terms X -> active# X, active# add(X1, X2) -> active# X2) (active# terms X -> active# X, active# add(X1, X2) -> add#(X1, active X2)) (active# terms X -> active# X, active# add(X1, X2) -> add#(active X1, X2)) (active# terms X -> active# X, active# add(s X, Y) -> s# add(X, Y)) (active# terms X -> active# X, active# add(s X, Y) -> add#(X, Y)) (active# terms X -> active# X, active# dbl X -> active# X) (active# terms X -> active# X, active# dbl X -> dbl# active X) (active# terms X -> active# X, active# dbl s X -> s# s dbl X) (active# terms X -> active# X, active# dbl s X -> s# dbl X) (active# terms X -> active# X, active# dbl s X -> dbl# X) (active# terms X -> active# X, active# first(X1, X2) -> active# X1) (active# terms X -> active# X, active# first(X1, X2) -> active# X2) (active# terms X -> active# X, active# first(X1, X2) -> first#(X1, active X2)) (active# terms X -> active# X, active# first(X1, X2) -> first#(active X1, X2)) (active# terms X -> active# X, active# first(s X, cons(Y, Z)) -> cons#(Y, first(X, Z))) (active# terms X -> active# X, active# first(s X, cons(Y, Z)) -> first#(X, Z)) (active# sqr s X -> sqr# X, sqr# mark X -> sqr# X) (active# sqr s X -> sqr# X, sqr# ok X -> sqr# X) (active# recip X -> active# X, active# cons(X1, X2) -> cons#(active X1, X2)) (active# recip X -> active# X, active# cons(X1, X2) -> active# X1) (active# recip X -> active# X, active# recip X -> recip# active X) (active# recip X -> active# X, active# recip X -> active# X) (active# recip X -> active# X, active# sqr X -> sqr# active X) (active# recip X -> active# X, active# sqr X -> active# X) (active# recip X -> active# X, active# sqr s X -> sqr# X) (active# recip X -> active# X, active# sqr s X -> s# add(sqr X, dbl X)) (active# recip X -> active# X, active# sqr s X -> add#(sqr X, dbl X)) (active# recip X -> active# X, active# sqr s X -> dbl# X) (active# recip X -> active# X, active# terms N -> cons#(recip sqr N, terms s N)) (active# recip X -> active# X, active# terms N -> recip# sqr N) (active# recip X -> active# X, active# terms N -> sqr# N) (active# recip X -> active# X, active# terms N -> terms# s N) (active# recip X -> active# X, active# terms N -> s# N) (active# recip X -> active# X, active# terms X -> terms# active X) (active# recip X -> active# X, active# terms X -> active# X) (active# recip X -> active# X, active# s X -> s# active X) (active# recip X -> active# X, active# s X -> active# X) (active# recip X -> active# X, active# add(X1, X2) -> active# X1) (active# recip X -> active# X, active# add(X1, X2) -> active# X2) (active# recip X -> active# X, active# add(X1, X2) -> add#(X1, active X2)) (active# recip X -> active# X, active# add(X1, X2) -> add#(active X1, X2)) (active# recip X -> active# X, active# add(s X, Y) -> s# add(X, Y)) (active# recip X -> active# X, active# add(s X, Y) -> add#(X, Y)) (active# recip X -> active# X, active# dbl X -> active# X) (active# recip X -> active# X, active# dbl X -> dbl# active X) (active# recip X -> active# X, active# dbl s X -> s# s dbl X) (active# recip X -> active# X, active# dbl s X -> s# dbl X) (active# recip X -> active# X, active# dbl s X -> dbl# X) (active# recip X -> active# X, active# first(X1, X2) -> active# X1) (active# recip X -> active# X, active# first(X1, X2) -> active# X2) (active# recip X -> active# X, active# first(X1, X2) -> first#(X1, active X2)) (active# recip X -> active# X, active# first(X1, X2) -> first#(active X1, X2)) (active# recip X -> active# X, active# first(s X, cons(Y, Z)) -> cons#(Y, first(X, Z))) (active# recip X -> active# X, active# first(s X, cons(Y, Z)) -> first#(X, Z)) (s# mark X -> s# X, s# mark X -> s# X) (s# mark X -> s# X, s# ok X -> s# X) (terms# mark X -> terms# X, terms# mark X -> terms# X) (terms# mark X -> terms# X, terms# ok X -> terms# X) (sqr# mark X -> sqr# X, sqr# mark X -> sqr# X) (sqr# mark X -> sqr# X, sqr# ok X -> sqr# X) (recip# mark X -> recip# X, recip# mark X -> recip# X) (recip# mark X -> recip# X, recip# ok X -> recip# X) (proper# add(X1, X2) -> add#(proper X1, proper X2), add#(X1, mark X2) -> add#(X1, X2)) (proper# add(X1, X2) -> add#(proper X1, proper X2), add#(mark X1, X2) -> add#(X1, X2)) (proper# add(X1, X2) -> add#(proper X1, proper X2), add#(ok X1, ok X2) -> add#(X1, X2)) (active# first(X1, X2) -> first#(X1, active X2), first#(X1, mark X2) -> first#(X1, X2)) (active# first(X1, X2) -> first#(X1, active X2), first#(mark X1, X2) -> first#(X1, X2)) (active# first(X1, X2) -> first#(X1, active X2), first#(ok X1, ok X2) -> first#(X1, X2)) (active# sqr s X -> add#(sqr X, dbl X), add#(X1, mark X2) -> add#(X1, X2)) (active# sqr s X -> add#(sqr X, dbl X), add#(mark X1, X2) -> add#(X1, X2)) (active# sqr s X -> add#(sqr X, dbl X), add#(ok X1, ok X2) -> add#(X1, X2)) } STATUS: arrows: 0.873334 SCCS (11): Scc: {top# mark X -> top# proper X, top# ok X -> top# active X} Scc: { proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# recip X -> proper# X, proper# sqr X -> proper# X, proper# terms X -> proper# X, proper# s X -> proper# X, proper# add(X1, X2) -> proper# X1, proper# add(X1, X2) -> proper# X2, proper# dbl X -> proper# X, proper# first(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X2} Scc: { active# cons(X1, X2) -> active# X1, active# recip X -> active# X, active# sqr X -> active# X, active# terms X -> active# X, active# s X -> active# X, active# add(X1, X2) -> active# X1, active# add(X1, X2) -> active# X2, active# dbl X -> active# X, active# first(X1, X2) -> active# X1, active# first(X1, X2) -> active# X2} Scc: {dbl# mark X -> dbl# X, dbl# ok X -> dbl# X} Scc: {s# mark X -> s# X, s# ok X -> s# X} Scc: {terms# mark X -> terms# X, terms# ok X -> terms# X} Scc: {sqr# mark X -> sqr# X, sqr# ok X -> sqr# X} Scc: {recip# mark X -> recip# X, recip# ok X -> recip# X} Scc: { cons#(mark X1, X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)} Scc: { first#(X1, mark X2) -> first#(X1, X2), first#(mark X1, X2) -> first#(X1, X2), first#(ok X1, ok X2) -> first#(X1, X2)} Scc: { add#(X1, mark X2) -> add#(X1, X2), add#(mark X1, X2) -> add#(X1, X2), add#(ok X1, ok X2) -> add#(X1, X2)} SCC (2): Strict: {top# mark X -> top# proper X, top# ok X -> top# active X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} Fail SCC (11): Strict: { proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# recip X -> proper# X, proper# sqr X -> proper# X, proper# terms X -> proper# X, proper# s X -> proper# X, proper# add(X1, X2) -> proper# X1, proper# add(X1, X2) -> proper# X2, proper# dbl X -> proper# X, proper# first(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X2} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + x1, [add](x0, x1) = x0 + x1, [first](x0, x1) = x0 + x1 + 1, [mark](x0) = 0, [recip](x0) = x0, [sqr](x0) = x0, [terms](x0) = x0, [s](x0) = x0, [active](x0) = 0, [dbl](x0) = x0, [proper](x0) = 0, [ok](x0) = 0, [top](x0) = 0, [0] = 0, [nil] = 0, [proper#](x0) = x0 Strict: proper# first(X1, X2) -> proper# X2 1 + 1X1 + 1X2 >= 0 + 1X2 proper# first(X1, X2) -> proper# X1 1 + 1X1 + 1X2 >= 0 + 1X1 proper# dbl X -> proper# X 0 + 1X >= 0 + 1X proper# add(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# add(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# s X -> proper# X 0 + 1X >= 0 + 1X proper# terms X -> proper# X 0 + 1X >= 0 + 1X proper# sqr X -> proper# X 0 + 1X >= 0 + 1X proper# recip X -> proper# X 0 + 1X >= 0 + 1X proper# cons(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# cons(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 0 proper dbl X -> dbl proper X 0 + 0X >= 0 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 0 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 0 + 0X proper sqr X -> sqr proper X 0 + 0X >= 0 + 0X proper recip X -> recip proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 1 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 0 + 0X >= 0 + 0X dbl mark X -> mark dbl X 0 + 0X >= 0 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 add(mark X1, X2) -> mark add(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 add(X1, mark X2) -> mark add(X1, X2) 0 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 1X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 0 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 0X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 active s X -> s active X 0 + 0X >= 0 + 0X active terms X -> terms active X 0 + 0X >= 0 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 0 + 0X active recip X -> recip active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 s ok X -> ok s X 0 + 0X >= 0 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X terms ok X -> ok terms X 0 + 0X >= 0 + 0X terms mark X -> mark terms X 0 + 0X >= 0 + 0X sqr ok X -> ok sqr X 0 + 0X >= 0 + 0X sqr mark X -> mark sqr X 0 + 0X >= 0 + 0X recip ok X -> ok recip X 0 + 0X >= 0 + 0X recip mark X -> mark recip X 0 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# recip X -> proper# X, proper# sqr X -> proper# X, proper# terms X -> proper# X, proper# s X -> proper# X, proper# add(X1, X2) -> proper# X1, proper# add(X1, X2) -> proper# X2, proper# dbl X -> proper# X} SCC (9): Strict: {proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# recip X -> proper# X, proper# sqr X -> proper# X, proper# terms X -> proper# X, proper# s X -> proper# X, proper# add(X1, X2) -> proper# X1, proper# add(X1, X2) -> proper# X2, proper# dbl X -> proper# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + x1, [add](x0, x1) = x0 + x1, [first](x0, x1) = x0 + x1, [mark](x0) = 0, [recip](x0) = x0, [sqr](x0) = x0, [terms](x0) = x0, [s](x0) = x0, [active](x0) = 0, [dbl](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [nil] = 1, [proper#](x0) = x0 Strict: proper# dbl X -> proper# X 1 + 1X >= 0 + 1X proper# add(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# add(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 proper# s X -> proper# X 0 + 1X >= 0 + 1X proper# terms X -> proper# X 0 + 1X >= 0 + 1X proper# sqr X -> proper# X 0 + 1X >= 0 + 1X proper# recip X -> proper# X 0 + 1X >= 0 + 1X proper# cons(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# cons(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 0 + 0X proper sqr X -> sqr proper X 0 + 0X >= 0 + 0X proper recip X -> recip proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 2 + 1X >= 2 + 1X dbl mark X -> mark dbl X 1 + 0X >= 0 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 add(X1, mark X2) -> mark add(X1, X2) 0 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 1 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 0X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 active s X -> s active X 0 + 0X >= 0 + 0X active terms X -> terms active X 0 + 0X >= 0 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 0 + 0X active recip X -> recip active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 0 + 0X >= 0 + 0X terms ok X -> ok terms X 1 + 1X >= 1 + 1X terms mark X -> mark terms X 0 + 0X >= 0 + 0X sqr ok X -> ok sqr X 1 + 1X >= 1 + 1X sqr mark X -> mark sqr X 0 + 0X >= 0 + 0X recip ok X -> ok recip X 1 + 1X >= 1 + 1X recip mark X -> mark recip X 0 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# recip X -> proper# X, proper# sqr X -> proper# X, proper# terms X -> proper# X, proper# s X -> proper# X, proper# add(X1, X2) -> proper# X1, proper# add(X1, X2) -> proper# X2} SCC (8): Strict: {proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# recip X -> proper# X, proper# sqr X -> proper# X, proper# terms X -> proper# X, proper# s X -> proper# X, proper# add(X1, X2) -> proper# X1, proper# add(X1, X2) -> proper# X2} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + x1, [add](x0, x1) = x0 + x1 + 1, [first](x0, x1) = 0, [mark](x0) = 0, [recip](x0) = x0, [sqr](x0) = x0, [terms](x0) = x0, [s](x0) = x0, [active](x0) = 0, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [nil] = 1, [proper#](x0) = x0 Strict: proper# add(X1, X2) -> proper# X2 1 + 1X1 + 1X2 >= 0 + 1X2 proper# add(X1, X2) -> proper# X1 1 + 1X1 + 1X2 >= 0 + 1X1 proper# s X -> proper# X 0 + 1X >= 0 + 1X proper# terms X -> proper# X 0 + 1X >= 0 + 1X proper# sqr X -> proper# X 0 + 1X >= 0 + 1X proper# recip X -> proper# X 0 + 1X >= 0 + 1X proper# cons(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# cons(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 0 + 0X proper sqr X -> sqr proper X 0 + 0X >= 0 + 0X proper recip X -> recip proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 0 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 add(X1, mark X2) -> mark add(X1, X2) 1 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 1 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 0X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 1X1 + 0X2 active s X -> s active X 0 + 0X >= 0 + 0X active terms X -> terms active X 0 + 0X >= 0 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 0 + 0X active recip X -> recip active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 0 + 0X >= 0 + 0X terms ok X -> ok terms X 1 + 1X >= 1 + 1X terms mark X -> mark terms X 0 + 0X >= 0 + 0X sqr ok X -> ok sqr X 1 + 1X >= 1 + 1X sqr mark X -> mark sqr X 0 + 0X >= 0 + 0X recip ok X -> ok recip X 1 + 1X >= 1 + 1X recip mark X -> mark recip X 0 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# recip X -> proper# X, proper# sqr X -> proper# X, proper# terms X -> proper# X, proper# s X -> proper# X} SCC (6): Strict: {proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# recip X -> proper# X, proper# sqr X -> proper# X, proper# terms X -> proper# X, proper# s X -> proper# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + x1, [add](x0, x1) = 1, [first](x0, x1) = 0, [mark](x0) = 0, [recip](x0) = x0, [sqr](x0) = x0, [terms](x0) = x0, [s](x0) = x0 + 1, [active](x0) = 0, [dbl](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [nil] = 1, [proper#](x0) = x0 Strict: proper# s X -> proper# X 1 + 1X >= 0 + 1X proper# terms X -> proper# X 0 + 1X >= 0 + 1X proper# sqr X -> proper# X 0 + 1X >= 0 + 1X proper# recip X -> proper# X 0 + 1X >= 0 + 1X proper# cons(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# cons(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 1 + 0X proper terms X -> terms proper X 0 + 0X >= 0 + 0X proper sqr X -> sqr proper X 0 + 0X >= 0 + 0X proper recip X -> recip proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 2 + 1X >= 2 + 1X dbl mark X -> mark dbl X 1 + 0X >= 0 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 add(X1, mark X2) -> mark add(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 1 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 0X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active s X -> s active X 0 + 0X >= 1 + 0X active terms X -> terms active X 0 + 0X >= 0 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 0 + 0X active recip X -> recip active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 1 + 0X >= 0 + 0X terms ok X -> ok terms X 1 + 1X >= 1 + 1X terms mark X -> mark terms X 0 + 0X >= 0 + 0X sqr ok X -> ok sqr X 1 + 1X >= 1 + 1X sqr mark X -> mark sqr X 0 + 0X >= 0 + 0X recip ok X -> ok recip X 1 + 1X >= 1 + 1X recip mark X -> mark recip X 0 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# recip X -> proper# X, proper# sqr X -> proper# X, proper# terms X -> proper# X} SCC (5): Strict: {proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# recip X -> proper# X, proper# sqr X -> proper# X, proper# terms X -> proper# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + x1, [add](x0, x1) = x0 + x1 + 1, [first](x0, x1) = 0, [mark](x0) = 0, [recip](x0) = x0, [sqr](x0) = x0, [terms](x0) = x0 + 1, [s](x0) = 0, [active](x0) = 0, [dbl](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [nil] = 1, [proper#](x0) = x0 Strict: proper# terms X -> proper# X 1 + 1X >= 0 + 1X proper# sqr X -> proper# X 0 + 1X >= 0 + 1X proper# recip X -> proper# X 0 + 1X >= 0 + 1X proper# cons(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# cons(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 1 + 0X proper sqr X -> sqr proper X 0 + 0X >= 0 + 0X proper recip X -> recip proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 2 + 1X >= 2 + 1X dbl mark X -> mark dbl X 1 + 0X >= 0 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 add(X1, mark X2) -> mark add(X1, X2) 1 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 1 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 0X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 1X1 + 0X2 active s X -> s active X 0 + 0X >= 0 + 0X active terms X -> terms active X 0 + 0X >= 1 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 0 + 0X active recip X -> recip active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X terms ok X -> ok terms X 2 + 1X >= 2 + 1X terms mark X -> mark terms X 1 + 0X >= 0 + 0X sqr ok X -> ok sqr X 1 + 1X >= 1 + 1X sqr mark X -> mark sqr X 0 + 0X >= 0 + 0X recip ok X -> ok recip X 1 + 1X >= 1 + 1X recip mark X -> mark recip X 0 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# recip X -> proper# X, proper# sqr X -> proper# X} SCC (4): Strict: {proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# recip X -> proper# X, proper# sqr X -> proper# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + x1, [add](x0, x1) = x0 + 1, [first](x0, x1) = 0, [mark](x0) = 0, [recip](x0) = x0, [sqr](x0) = x0 + 1, [terms](x0) = 0, [s](x0) = 0, [active](x0) = 0, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 1, [nil] = 1, [proper#](x0) = x0 Strict: proper# sqr X -> proper# X 1 + 1X >= 0 + 1X proper# recip X -> proper# X 0 + 1X >= 0 + 1X proper# cons(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# cons(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 0 + 0X proper sqr X -> sqr proper X 0 + 0X >= 1 + 0X proper recip X -> recip proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 0 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 add(X1, mark X2) -> mark add(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 1 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 0X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active s X -> s active X 0 + 0X >= 0 + 0X active terms X -> terms active X 0 + 0X >= 0 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 1 + 0X active recip X -> recip active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X terms ok X -> ok terms X 0 + 0X >= 1 + 0X terms mark X -> mark terms X 0 + 0X >= 0 + 0X sqr ok X -> ok sqr X 2 + 1X >= 2 + 1X sqr mark X -> mark sqr X 1 + 0X >= 0 + 0X recip ok X -> ok recip X 1 + 1X >= 1 + 1X recip mark X -> mark recip X 0 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# recip X -> proper# X} SCC (3): Strict: {proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# recip X -> proper# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + x1, [add](x0, x1) = x0 + x1 + 1, [first](x0, x1) = 0, [mark](x0) = 0, [recip](x0) = x0 + 1, [sqr](x0) = 0, [terms](x0) = 0, [s](x0) = 0, [active](x0) = 0, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 1, [nil] = 1, [proper#](x0) = x0 Strict: proper# recip X -> proper# X 1 + 1X >= 0 + 1X proper# cons(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# cons(X1, X2) -> proper# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 0 + 0X proper sqr X -> sqr proper X 0 + 0X >= 0 + 0X proper recip X -> recip proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 0 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 add(X1, mark X2) -> mark add(X1, X2) 1 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 1 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 0X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 1X1 + 0X2 active s X -> s active X 0 + 0X >= 0 + 0X active terms X -> terms active X 0 + 0X >= 0 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 0 + 0X active recip X -> recip active X 0 + 0X >= 1 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X terms ok X -> ok terms X 0 + 0X >= 1 + 0X terms mark X -> mark terms X 0 + 0X >= 0 + 0X sqr ok X -> ok sqr X 0 + 0X >= 1 + 0X sqr mark X -> mark sqr X 0 + 0X >= 0 + 0X recip ok X -> ok recip X 2 + 1X >= 2 + 1X recip mark X -> mark recip X 1 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2} SCC (2): Strict: {proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + x1 + 1, [add](x0, x1) = x0 + 1, [first](x0, x1) = x0 + 1, [mark](x0) = x0 + 1, [recip](x0) = x0 + 1, [sqr](x0) = x0 + 1, [terms](x0) = x0 + 1, [s](x0) = x0 + 1, [active](x0) = 0, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0, [top](x0) = 0, [0] = 1, [nil] = 1, [proper#](x0) = x0 + 1 Strict: proper# cons(X1, X2) -> proper# X2 2 + 1X1 + 1X2 >= 1 + 1X2 proper# cons(X1, X2) -> proper# X1 2 + 1X1 + 1X2 >= 1 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 1 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 1 + 0X proper terms X -> terms proper X 0 + 0X >= 1 + 0X proper sqr X -> sqr proper X 0 + 0X >= 1 + 0X proper recip X -> recip proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 dbl ok X -> ok dbl X 1 + 0X >= 1 + 0X dbl mark X -> mark dbl X 1 + 0X >= 2 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active first(0(), X) -> mark nil() 0 + 0X >= 2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 3 + 0X + 1Y + 1Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active dbl 0() -> mark 0() 0 >= 2 active dbl s X -> mark s s dbl X 0 + 0X >= 4 + 0X active dbl X -> dbl active X 0 + 0X >= 1 + 0X active add(0(), X) -> mark X 0 + 0X >= 1 + 1X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 3 + 0X + 1Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active s X -> s active X 0 + 0X >= 1 + 0X active terms X -> terms active X 0 + 0X >= 1 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 6 + 2N active sqr 0() -> mark 0() 0 >= 2 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 4 + 0X active sqr X -> sqr active X 0 + 0X >= 1 + 0X active recip X -> recip active X 0 + 0X >= 1 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X terms ok X -> ok terms X 1 + 1X >= 1 + 1X terms mark X -> mark terms X 2 + 1X >= 2 + 1X sqr ok X -> ok sqr X 1 + 1X >= 1 + 1X sqr mark X -> mark sqr X 2 + 1X >= 2 + 1X recip ok X -> ok recip X 1 + 1X >= 1 + 1X recip mark X -> mark recip X 2 + 1X >= 2 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 Qed SCC (10): Strict: { active# cons(X1, X2) -> active# X1, active# recip X -> active# X, active# sqr X -> active# X, active# terms X -> active# X, active# s X -> active# X, active# add(X1, X2) -> active# X1, active# add(X1, X2) -> active# X2, active# dbl X -> active# X, active# first(X1, X2) -> active# X1, active# first(X1, X2) -> active# X2} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + x1, [add](x0, x1) = x0 + x1, [first](x0, x1) = x0 + x1 + 1, [mark](x0) = 0, [recip](x0) = x0, [sqr](x0) = x0, [terms](x0) = x0, [s](x0) = x0, [active](x0) = 0, [dbl](x0) = x0, [proper](x0) = 0, [ok](x0) = 0, [top](x0) = 0, [0] = 0, [nil] = 0, [active#](x0) = x0 Strict: active# first(X1, X2) -> active# X2 1 + 1X1 + 1X2 >= 0 + 1X2 active# first(X1, X2) -> active# X1 1 + 1X1 + 1X2 >= 0 + 1X1 active# dbl X -> active# X 0 + 1X >= 0 + 1X active# add(X1, X2) -> active# X2 0 + 1X1 + 1X2 >= 0 + 1X2 active# add(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 active# s X -> active# X 0 + 1X >= 0 + 1X active# terms X -> active# X 0 + 1X >= 0 + 1X active# sqr X -> active# X 0 + 1X >= 0 + 1X active# recip X -> active# X 0 + 1X >= 0 + 1X active# cons(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 0 proper dbl X -> dbl proper X 0 + 0X >= 0 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 0 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 0 + 0X proper sqr X -> sqr proper X 0 + 0X >= 0 + 0X proper recip X -> recip proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 1 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 0 + 0X >= 0 + 0X dbl mark X -> mark dbl X 0 + 0X >= 0 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 add(mark X1, X2) -> mark add(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 add(X1, mark X2) -> mark add(X1, X2) 0 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 1X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 0 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 0X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 active s X -> s active X 0 + 0X >= 0 + 0X active terms X -> terms active X 0 + 0X >= 0 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 0 + 0X active recip X -> recip active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 s ok X -> ok s X 0 + 0X >= 0 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X terms ok X -> ok terms X 0 + 0X >= 0 + 0X terms mark X -> mark terms X 0 + 0X >= 0 + 0X sqr ok X -> ok sqr X 0 + 0X >= 0 + 0X sqr mark X -> mark sqr X 0 + 0X >= 0 + 0X recip ok X -> ok recip X 0 + 0X >= 0 + 0X recip mark X -> mark recip X 0 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {active# cons(X1, X2) -> active# X1, active# recip X -> active# X, active# sqr X -> active# X, active# terms X -> active# X, active# s X -> active# X, active# add(X1, X2) -> active# X1, active# add(X1, X2) -> active# X2, active# dbl X -> active# X} SCC (8): Strict: {active# cons(X1, X2) -> active# X1, active# recip X -> active# X, active# sqr X -> active# X, active# terms X -> active# X, active# s X -> active# X, active# add(X1, X2) -> active# X1, active# add(X1, X2) -> active# X2, active# dbl X -> active# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + x1, [add](x0, x1) = x0 + x1, [first](x0, x1) = x0 + x1, [mark](x0) = 0, [recip](x0) = x0, [sqr](x0) = x0, [terms](x0) = x0, [s](x0) = x0, [active](x0) = 0, [dbl](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [nil] = 1, [active#](x0) = x0 Strict: active# dbl X -> active# X 1 + 1X >= 0 + 1X active# add(X1, X2) -> active# X2 0 + 1X1 + 1X2 >= 0 + 1X2 active# add(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 active# s X -> active# X 0 + 1X >= 0 + 1X active# terms X -> active# X 0 + 1X >= 0 + 1X active# sqr X -> active# X 0 + 1X >= 0 + 1X active# recip X -> active# X 0 + 1X >= 0 + 1X active# cons(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 0 + 0X proper sqr X -> sqr proper X 0 + 0X >= 0 + 0X proper recip X -> recip proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 2 + 1X >= 2 + 1X dbl mark X -> mark dbl X 1 + 0X >= 0 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 add(X1, mark X2) -> mark add(X1, X2) 0 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 1 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 0X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 active s X -> s active X 0 + 0X >= 0 + 0X active terms X -> terms active X 0 + 0X >= 0 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 0 + 0X active recip X -> recip active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 0 + 0X >= 0 + 0X terms ok X -> ok terms X 1 + 1X >= 1 + 1X terms mark X -> mark terms X 0 + 0X >= 0 + 0X sqr ok X -> ok sqr X 1 + 1X >= 1 + 1X sqr mark X -> mark sqr X 0 + 0X >= 0 + 0X recip ok X -> ok recip X 1 + 1X >= 1 + 1X recip mark X -> mark recip X 0 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {active# cons(X1, X2) -> active# X1, active# recip X -> active# X, active# sqr X -> active# X, active# terms X -> active# X, active# s X -> active# X, active# add(X1, X2) -> active# X1, active# add(X1, X2) -> active# X2} SCC (7): Strict: {active# cons(X1, X2) -> active# X1, active# recip X -> active# X, active# sqr X -> active# X, active# terms X -> active# X, active# s X -> active# X, active# add(X1, X2) -> active# X1, active# add(X1, X2) -> active# X2} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + x1, [add](x0, x1) = x0 + x1 + 1, [first](x0, x1) = 0, [mark](x0) = 0, [recip](x0) = x0, [sqr](x0) = x0, [terms](x0) = x0, [s](x0) = x0, [active](x0) = 0, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [nil] = 1, [active#](x0) = x0 Strict: active# add(X1, X2) -> active# X2 1 + 1X1 + 1X2 >= 0 + 1X2 active# add(X1, X2) -> active# X1 1 + 1X1 + 1X2 >= 0 + 1X1 active# s X -> active# X 0 + 1X >= 0 + 1X active# terms X -> active# X 0 + 1X >= 0 + 1X active# sqr X -> active# X 0 + 1X >= 0 + 1X active# recip X -> active# X 0 + 1X >= 0 + 1X active# cons(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 0 + 0X proper sqr X -> sqr proper X 0 + 0X >= 0 + 0X proper recip X -> recip proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 0 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 add(X1, mark X2) -> mark add(X1, X2) 1 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 1 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 0X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 1X1 + 0X2 active s X -> s active X 0 + 0X >= 0 + 0X active terms X -> terms active X 0 + 0X >= 0 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 0 + 0X active recip X -> recip active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 0 + 0X >= 0 + 0X terms ok X -> ok terms X 1 + 1X >= 1 + 1X terms mark X -> mark terms X 0 + 0X >= 0 + 0X sqr ok X -> ok sqr X 1 + 1X >= 1 + 1X sqr mark X -> mark sqr X 0 + 0X >= 0 + 0X recip ok X -> ok recip X 1 + 1X >= 1 + 1X recip mark X -> mark recip X 0 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {active# cons(X1, X2) -> active# X1, active# recip X -> active# X, active# sqr X -> active# X, active# terms X -> active# X, active# s X -> active# X} SCC (5): Strict: {active# cons(X1, X2) -> active# X1, active# recip X -> active# X, active# sqr X -> active# X, active# terms X -> active# X, active# s X -> active# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + x1, [add](x0, x1) = 1, [first](x0, x1) = 0, [mark](x0) = 0, [recip](x0) = x0, [sqr](x0) = x0, [terms](x0) = x0, [s](x0) = x0 + 1, [active](x0) = 0, [dbl](x0) = 0, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [nil] = 1, [active#](x0) = x0 Strict: active# s X -> active# X 1 + 1X >= 0 + 1X active# terms X -> active# X 0 + 1X >= 0 + 1X active# sqr X -> active# X 0 + 1X >= 0 + 1X active# recip X -> active# X 0 + 1X >= 0 + 1X active# cons(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 0 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 1 + 0X proper terms X -> terms proper X 0 + 0X >= 0 + 0X proper sqr X -> sqr proper X 0 + 0X >= 0 + 0X proper recip X -> recip proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 0 + 0X >= 1 + 0X dbl mark X -> mark dbl X 0 + 0X >= 0 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 add(X1, mark X2) -> mark add(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 0 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 0X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active s X -> s active X 0 + 0X >= 1 + 0X active terms X -> terms active X 0 + 0X >= 0 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 0 + 0X active recip X -> recip active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 1 + 0X >= 0 + 0X terms ok X -> ok terms X 1 + 1X >= 1 + 1X terms mark X -> mark terms X 0 + 0X >= 0 + 0X sqr ok X -> ok sqr X 1 + 1X >= 1 + 1X sqr mark X -> mark sqr X 0 + 0X >= 0 + 0X recip ok X -> ok recip X 1 + 1X >= 1 + 1X recip mark X -> mark recip X 0 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {active# cons(X1, X2) -> active# X1, active# recip X -> active# X, active# sqr X -> active# X, active# terms X -> active# X} SCC (4): Strict: {active# cons(X1, X2) -> active# X1, active# recip X -> active# X, active# sqr X -> active# X, active# terms X -> active# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + x1, [add](x0, x1) = x0 + x1 + 1, [first](x0, x1) = 0, [mark](x0) = 0, [recip](x0) = x0, [sqr](x0) = x0, [terms](x0) = x0 + 1, [s](x0) = 0, [active](x0) = 0, [dbl](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [nil] = 1, [active#](x0) = x0 Strict: active# terms X -> active# X 1 + 1X >= 0 + 1X active# sqr X -> active# X 0 + 1X >= 0 + 1X active# recip X -> active# X 0 + 1X >= 0 + 1X active# cons(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 1 + 0X proper sqr X -> sqr proper X 0 + 0X >= 0 + 0X proper recip X -> recip proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 2 + 1X >= 2 + 1X dbl mark X -> mark dbl X 1 + 0X >= 0 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 add(X1, mark X2) -> mark add(X1, X2) 1 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 1 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 0X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 1X1 + 0X2 active s X -> s active X 0 + 0X >= 0 + 0X active terms X -> terms active X 0 + 0X >= 1 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 0 + 0X active recip X -> recip active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X terms ok X -> ok terms X 2 + 1X >= 2 + 1X terms mark X -> mark terms X 1 + 0X >= 0 + 0X sqr ok X -> ok sqr X 1 + 1X >= 1 + 1X sqr mark X -> mark sqr X 0 + 0X >= 0 + 0X recip ok X -> ok recip X 1 + 1X >= 1 + 1X recip mark X -> mark recip X 0 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {active# cons(X1, X2) -> active# X1, active# recip X -> active# X, active# sqr X -> active# X} SCC (3): Strict: {active# cons(X1, X2) -> active# X1, active# recip X -> active# X, active# sqr X -> active# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + x1, [add](x0, x1) = x0 + 1, [first](x0, x1) = 0, [mark](x0) = 0, [recip](x0) = x0, [sqr](x0) = x0 + 1, [terms](x0) = 0, [s](x0) = 0, [active](x0) = 0, [dbl](x0) = 1, [proper](x0) = x0 + 1, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 1, [nil] = 1, [active#](x0) = x0 Strict: active# sqr X -> active# X 1 + 1X >= 0 + 1X active# recip X -> active# X 0 + 1X >= 0 + 1X active# cons(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 2 >= 2 proper dbl X -> dbl proper X 2 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 proper 0() -> ok 0() 2 >= 2 proper s X -> s proper X 1 + 0X >= 0 + 0X proper terms X -> terms proper X 1 + 0X >= 0 + 0X proper sqr X -> sqr proper X 2 + 1X >= 2 + 1X proper recip X -> recip proper X 1 + 1X >= 1 + 1X proper cons(X1, X2) -> cons(proper X1, proper X2) 1 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(ok X1, ok X2) -> ok first(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 0 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 add(X1, mark X2) -> mark add(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 1 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 0X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active s X -> s active X 0 + 0X >= 0 + 0X active terms X -> terms active X 0 + 0X >= 0 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 1 + 0X active recip X -> recip active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X terms ok X -> ok terms X 0 + 0X >= 1 + 0X terms mark X -> mark terms X 0 + 0X >= 0 + 0X sqr ok X -> ok sqr X 2 + 1X >= 2 + 1X sqr mark X -> mark sqr X 1 + 0X >= 0 + 0X recip ok X -> ok recip X 1 + 1X >= 1 + 1X recip mark X -> mark recip X 0 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {active# cons(X1, X2) -> active# X1, active# recip X -> active# X} SCC (2): Strict: {active# cons(X1, X2) -> active# X1, active# recip X -> active# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + x1, [add](x0, x1) = x0 + x1 + 1, [first](x0, x1) = 0, [mark](x0) = 0, [recip](x0) = x0 + 1, [sqr](x0) = 0, [terms](x0) = 0, [s](x0) = 0, [active](x0) = 0, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 1, [nil] = 1, [active#](x0) = x0 Strict: active# recip X -> active# X 1 + 1X >= 0 + 1X active# cons(X1, X2) -> active# X1 0 + 1X1 + 1X2 >= 0 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 2 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 0 + 0X proper sqr X -> sqr proper X 0 + 0X >= 0 + 0X proper recip X -> recip proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 0 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 add(X1, mark X2) -> mark add(X1, X2) 1 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 1 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 0X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 1X1 + 0X2 active s X -> s active X 0 + 0X >= 0 + 0X active terms X -> terms active X 0 + 0X >= 0 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 0 + 0X active recip X -> recip active X 0 + 0X >= 1 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X terms ok X -> ok terms X 0 + 0X >= 1 + 0X terms mark X -> mark terms X 0 + 0X >= 0 + 0X sqr ok X -> ok sqr X 0 + 0X >= 1 + 0X sqr mark X -> mark sqr X 0 + 0X >= 0 + 0X recip ok X -> ok recip X 2 + 1X >= 2 + 1X recip mark X -> mark recip X 1 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {active# cons(X1, X2) -> active# X1} SCC (1): Strict: {active# cons(X1, X2) -> active# X1} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + 1, [add](x0, x1) = x0 + 1, [first](x0, x1) = x0 + x1 + 1, [mark](x0) = x0 + 1, [recip](x0) = x0 + 1, [sqr](x0) = x0 + 1, [terms](x0) = x0 + 1, [s](x0) = x0 + 1, [active](x0) = x0 + 1, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = x0 + 1, [0] = 0, [nil] = 1, [active#](x0) = x0 Strict: active# cons(X1, X2) -> active# X1 1 + 1X1 + 0X2 >= 0 + 1X1 Weak: top ok X -> top active X 2 + 1X >= 2 + 1X top mark X -> top proper X 2 + 1X >= 1 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 1 + 0X proper terms X -> terms proper X 0 + 0X >= 1 + 0X proper sqr X -> sqr proper X 0 + 0X >= 1 + 0X proper recip X -> recip proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 2 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active first(0(), X) -> mark nil() 2 + 1X >= 2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 4 + 1X + 1Y + 0Z >= 2 + 0X + 1Y + 0Z active first(X1, X2) -> first(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active dbl 0() -> mark 0() 2 >= 1 active dbl s X -> mark s s dbl X 2 + 0X >= 4 + 0X active dbl X -> dbl active X 2 + 0X >= 1 + 0X active add(0(), X) -> mark X 2 + 1X >= 1 + 1X active add(s X, Y) -> mark s add(X, Y) 2 + 0X + 1Y >= 3 + 0X + 1Y active add(X1, X2) -> add(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active s X -> s active X 2 + 1X >= 2 + 1X active terms X -> terms active X 2 + 1X >= 2 + 1X active terms N -> mark cons(recip sqr N, terms s N) 2 + 1N >= 4 + 1N active sqr 0() -> mark 0() 2 >= 1 active sqr s X -> mark s add(sqr X, dbl X) 3 + 1X >= 4 + 0X active sqr X -> sqr active X 2 + 1X >= 2 + 1X active recip X -> recip active X 2 + 1X >= 2 + 1X active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X terms ok X -> ok terms X 2 + 1X >= 2 + 1X terms mark X -> mark terms X 2 + 1X >= 2 + 1X sqr ok X -> ok sqr X 2 + 1X >= 2 + 1X sqr mark X -> mark sqr X 2 + 1X >= 2 + 1X recip ok X -> ok recip X 2 + 1X >= 2 + 1X recip mark X -> mark recip X 2 + 1X >= 2 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 Qed SCC (2): Strict: {dbl# mark X -> dbl# X, dbl# ok X -> dbl# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = 0, [add](x0, x1) = x0 + 1, [first](x0, x1) = 0, [mark](x0) = x0, [recip](x0) = 0, [sqr](x0) = 0, [terms](x0) = 0, [s](x0) = 0, [active](x0) = 0, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [nil] = 0, [dbl#](x0) = x0 Strict: dbl# ok X -> dbl# X 1 + 1X >= 0 + 1X dbl# mark X -> dbl# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 1 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 0 + 0X proper sqr X -> sqr proper X 0 + 0X >= 0 + 0X proper recip X -> recip proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 1 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 1 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 1X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active s X -> s active X 0 + 0X >= 0 + 0X active terms X -> terms active X 0 + 0X >= 0 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 0 + 0X active recip X -> recip active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X terms ok X -> ok terms X 0 + 0X >= 1 + 0X terms mark X -> mark terms X 0 + 0X >= 0 + 0X sqr ok X -> ok sqr X 0 + 0X >= 1 + 0X sqr mark X -> mark sqr X 0 + 0X >= 0 + 0X recip ok X -> ok recip X 0 + 0X >= 1 + 0X recip mark X -> mark recip X 0 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {dbl# mark X -> dbl# X} SCC (1): Strict: {dbl# mark X -> dbl# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + 1, [add](x0, x1) = x0 + 1, [first](x0, x1) = x0 + x1 + 1, [mark](x0) = x0 + 1, [recip](x0) = x0 + 1, [sqr](x0) = x0 + 1, [terms](x0) = x0 + 1, [s](x0) = x0 + 1, [active](x0) = x0 + 1, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = x0 + 1, [0] = 0, [nil] = 1, [dbl#](x0) = x0 Strict: dbl# mark X -> dbl# X 1 + 1X >= 0 + 1X Weak: top ok X -> top active X 2 + 1X >= 2 + 1X top mark X -> top proper X 2 + 1X >= 1 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 1 + 0X proper terms X -> terms proper X 0 + 0X >= 1 + 0X proper sqr X -> sqr proper X 0 + 0X >= 1 + 0X proper recip X -> recip proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 2 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active first(0(), X) -> mark nil() 2 + 1X >= 2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 4 + 1X + 1Y + 0Z >= 2 + 0X + 1Y + 0Z active first(X1, X2) -> first(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active dbl 0() -> mark 0() 2 >= 1 active dbl s X -> mark s s dbl X 2 + 0X >= 4 + 0X active dbl X -> dbl active X 2 + 0X >= 1 + 0X active add(0(), X) -> mark X 2 + 1X >= 1 + 1X active add(s X, Y) -> mark s add(X, Y) 2 + 0X + 1Y >= 3 + 0X + 1Y active add(X1, X2) -> add(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active s X -> s active X 2 + 1X >= 2 + 1X active terms X -> terms active X 2 + 1X >= 2 + 1X active terms N -> mark cons(recip sqr N, terms s N) 2 + 1N >= 4 + 1N active sqr 0() -> mark 0() 2 >= 1 active sqr s X -> mark s add(sqr X, dbl X) 3 + 1X >= 4 + 0X active sqr X -> sqr active X 2 + 1X >= 2 + 1X active recip X -> recip active X 2 + 1X >= 2 + 1X active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X terms ok X -> ok terms X 2 + 1X >= 2 + 1X terms mark X -> mark terms X 2 + 1X >= 2 + 1X sqr ok X -> ok sqr X 2 + 1X >= 2 + 1X sqr mark X -> mark sqr X 2 + 1X >= 2 + 1X recip ok X -> ok recip X 2 + 1X >= 2 + 1X recip mark X -> mark recip X 2 + 1X >= 2 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 Qed SCC (2): Strict: {s# mark X -> s# X, s# ok X -> s# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = 0, [add](x0, x1) = x0 + 1, [first](x0, x1) = 0, [mark](x0) = x0, [recip](x0) = 0, [sqr](x0) = 0, [terms](x0) = 0, [s](x0) = 0, [active](x0) = 0, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [nil] = 0, [s#](x0) = x0 Strict: s# ok X -> s# X 1 + 1X >= 0 + 1X s# mark X -> s# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 1 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 0 + 0X proper sqr X -> sqr proper X 0 + 0X >= 0 + 0X proper recip X -> recip proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 1 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 1 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 1X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active s X -> s active X 0 + 0X >= 0 + 0X active terms X -> terms active X 0 + 0X >= 0 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 0 + 0X active recip X -> recip active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X terms ok X -> ok terms X 0 + 0X >= 1 + 0X terms mark X -> mark terms X 0 + 0X >= 0 + 0X sqr ok X -> ok sqr X 0 + 0X >= 1 + 0X sqr mark X -> mark sqr X 0 + 0X >= 0 + 0X recip ok X -> ok recip X 0 + 0X >= 1 + 0X recip mark X -> mark recip X 0 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {s# mark X -> s# X} SCC (1): Strict: {s# mark X -> s# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + 1, [add](x0, x1) = x0 + 1, [first](x0, x1) = x0 + x1 + 1, [mark](x0) = x0 + 1, [recip](x0) = x0 + 1, [sqr](x0) = x0 + 1, [terms](x0) = x0 + 1, [s](x0) = x0 + 1, [active](x0) = x0 + 1, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = x0 + 1, [0] = 0, [nil] = 1, [s#](x0) = x0 Strict: s# mark X -> s# X 1 + 1X >= 0 + 1X Weak: top ok X -> top active X 2 + 1X >= 2 + 1X top mark X -> top proper X 2 + 1X >= 1 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 1 + 0X proper terms X -> terms proper X 0 + 0X >= 1 + 0X proper sqr X -> sqr proper X 0 + 0X >= 1 + 0X proper recip X -> recip proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 2 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active first(0(), X) -> mark nil() 2 + 1X >= 2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 4 + 1X + 1Y + 0Z >= 2 + 0X + 1Y + 0Z active first(X1, X2) -> first(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active dbl 0() -> mark 0() 2 >= 1 active dbl s X -> mark s s dbl X 2 + 0X >= 4 + 0X active dbl X -> dbl active X 2 + 0X >= 1 + 0X active add(0(), X) -> mark X 2 + 1X >= 1 + 1X active add(s X, Y) -> mark s add(X, Y) 2 + 0X + 1Y >= 3 + 0X + 1Y active add(X1, X2) -> add(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active s X -> s active X 2 + 1X >= 2 + 1X active terms X -> terms active X 2 + 1X >= 2 + 1X active terms N -> mark cons(recip sqr N, terms s N) 2 + 1N >= 4 + 1N active sqr 0() -> mark 0() 2 >= 1 active sqr s X -> mark s add(sqr X, dbl X) 3 + 1X >= 4 + 0X active sqr X -> sqr active X 2 + 1X >= 2 + 1X active recip X -> recip active X 2 + 1X >= 2 + 1X active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X terms ok X -> ok terms X 2 + 1X >= 2 + 1X terms mark X -> mark terms X 2 + 1X >= 2 + 1X sqr ok X -> ok sqr X 2 + 1X >= 2 + 1X sqr mark X -> mark sqr X 2 + 1X >= 2 + 1X recip ok X -> ok recip X 2 + 1X >= 2 + 1X recip mark X -> mark recip X 2 + 1X >= 2 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 Qed SCC (2): Strict: {terms# mark X -> terms# X, terms# ok X -> terms# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = 0, [add](x0, x1) = x0 + 1, [first](x0, x1) = 0, [mark](x0) = x0, [recip](x0) = 0, [sqr](x0) = 0, [terms](x0) = 0, [s](x0) = 0, [active](x0) = 0, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [nil] = 0, [terms#](x0) = x0 Strict: terms# ok X -> terms# X 1 + 1X >= 0 + 1X terms# mark X -> terms# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 1 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 0 + 0X proper sqr X -> sqr proper X 0 + 0X >= 0 + 0X proper recip X -> recip proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 1 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 1 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 1X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active s X -> s active X 0 + 0X >= 0 + 0X active terms X -> terms active X 0 + 0X >= 0 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 0 + 0X active recip X -> recip active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X terms ok X -> ok terms X 0 + 0X >= 1 + 0X terms mark X -> mark terms X 0 + 0X >= 0 + 0X sqr ok X -> ok sqr X 0 + 0X >= 1 + 0X sqr mark X -> mark sqr X 0 + 0X >= 0 + 0X recip ok X -> ok recip X 0 + 0X >= 1 + 0X recip mark X -> mark recip X 0 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {terms# mark X -> terms# X} SCC (1): Strict: {terms# mark X -> terms# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + 1, [add](x0, x1) = x0 + 1, [first](x0, x1) = x0 + x1 + 1, [mark](x0) = x0 + 1, [recip](x0) = x0 + 1, [sqr](x0) = x0 + 1, [terms](x0) = x0 + 1, [s](x0) = x0 + 1, [active](x0) = x0 + 1, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = x0 + 1, [0] = 0, [nil] = 1, [terms#](x0) = x0 Strict: terms# mark X -> terms# X 1 + 1X >= 0 + 1X Weak: top ok X -> top active X 2 + 1X >= 2 + 1X top mark X -> top proper X 2 + 1X >= 1 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 1 + 0X proper terms X -> terms proper X 0 + 0X >= 1 + 0X proper sqr X -> sqr proper X 0 + 0X >= 1 + 0X proper recip X -> recip proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 2 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active first(0(), X) -> mark nil() 2 + 1X >= 2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 4 + 1X + 1Y + 0Z >= 2 + 0X + 1Y + 0Z active first(X1, X2) -> first(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active dbl 0() -> mark 0() 2 >= 1 active dbl s X -> mark s s dbl X 2 + 0X >= 4 + 0X active dbl X -> dbl active X 2 + 0X >= 1 + 0X active add(0(), X) -> mark X 2 + 1X >= 1 + 1X active add(s X, Y) -> mark s add(X, Y) 2 + 0X + 1Y >= 3 + 0X + 1Y active add(X1, X2) -> add(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active s X -> s active X 2 + 1X >= 2 + 1X active terms X -> terms active X 2 + 1X >= 2 + 1X active terms N -> mark cons(recip sqr N, terms s N) 2 + 1N >= 4 + 1N active sqr 0() -> mark 0() 2 >= 1 active sqr s X -> mark s add(sqr X, dbl X) 3 + 1X >= 4 + 0X active sqr X -> sqr active X 2 + 1X >= 2 + 1X active recip X -> recip active X 2 + 1X >= 2 + 1X active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X terms ok X -> ok terms X 2 + 1X >= 2 + 1X terms mark X -> mark terms X 2 + 1X >= 2 + 1X sqr ok X -> ok sqr X 2 + 1X >= 2 + 1X sqr mark X -> mark sqr X 2 + 1X >= 2 + 1X recip ok X -> ok recip X 2 + 1X >= 2 + 1X recip mark X -> mark recip X 2 + 1X >= 2 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 Qed SCC (2): Strict: {sqr# mark X -> sqr# X, sqr# ok X -> sqr# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = 0, [add](x0, x1) = x0 + 1, [first](x0, x1) = 0, [mark](x0) = x0, [recip](x0) = 0, [sqr](x0) = 0, [terms](x0) = 0, [s](x0) = 0, [active](x0) = 0, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [nil] = 0, [sqr#](x0) = x0 Strict: sqr# ok X -> sqr# X 1 + 1X >= 0 + 1X sqr# mark X -> sqr# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 1 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 0 + 0X proper sqr X -> sqr proper X 0 + 0X >= 0 + 0X proper recip X -> recip proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 1 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 1 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 1X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active s X -> s active X 0 + 0X >= 0 + 0X active terms X -> terms active X 0 + 0X >= 0 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 0 + 0X active recip X -> recip active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X terms ok X -> ok terms X 0 + 0X >= 1 + 0X terms mark X -> mark terms X 0 + 0X >= 0 + 0X sqr ok X -> ok sqr X 0 + 0X >= 1 + 0X sqr mark X -> mark sqr X 0 + 0X >= 0 + 0X recip ok X -> ok recip X 0 + 0X >= 1 + 0X recip mark X -> mark recip X 0 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {sqr# mark X -> sqr# X} SCC (1): Strict: {sqr# mark X -> sqr# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + 1, [add](x0, x1) = x0 + 1, [first](x0, x1) = x0 + x1 + 1, [mark](x0) = x0 + 1, [recip](x0) = x0 + 1, [sqr](x0) = x0 + 1, [terms](x0) = x0 + 1, [s](x0) = x0 + 1, [active](x0) = x0 + 1, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = x0 + 1, [0] = 0, [nil] = 1, [sqr#](x0) = x0 Strict: sqr# mark X -> sqr# X 1 + 1X >= 0 + 1X Weak: top ok X -> top active X 2 + 1X >= 2 + 1X top mark X -> top proper X 2 + 1X >= 1 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 1 + 0X proper terms X -> terms proper X 0 + 0X >= 1 + 0X proper sqr X -> sqr proper X 0 + 0X >= 1 + 0X proper recip X -> recip proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 2 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active first(0(), X) -> mark nil() 2 + 1X >= 2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 4 + 1X + 1Y + 0Z >= 2 + 0X + 1Y + 0Z active first(X1, X2) -> first(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active dbl 0() -> mark 0() 2 >= 1 active dbl s X -> mark s s dbl X 2 + 0X >= 4 + 0X active dbl X -> dbl active X 2 + 0X >= 1 + 0X active add(0(), X) -> mark X 2 + 1X >= 1 + 1X active add(s X, Y) -> mark s add(X, Y) 2 + 0X + 1Y >= 3 + 0X + 1Y active add(X1, X2) -> add(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active s X -> s active X 2 + 1X >= 2 + 1X active terms X -> terms active X 2 + 1X >= 2 + 1X active terms N -> mark cons(recip sqr N, terms s N) 2 + 1N >= 4 + 1N active sqr 0() -> mark 0() 2 >= 1 active sqr s X -> mark s add(sqr X, dbl X) 3 + 1X >= 4 + 0X active sqr X -> sqr active X 2 + 1X >= 2 + 1X active recip X -> recip active X 2 + 1X >= 2 + 1X active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X terms ok X -> ok terms X 2 + 1X >= 2 + 1X terms mark X -> mark terms X 2 + 1X >= 2 + 1X sqr ok X -> ok sqr X 2 + 1X >= 2 + 1X sqr mark X -> mark sqr X 2 + 1X >= 2 + 1X recip ok X -> ok recip X 2 + 1X >= 2 + 1X recip mark X -> mark recip X 2 + 1X >= 2 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 Qed SCC (2): Strict: {recip# mark X -> recip# X, recip# ok X -> recip# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = 0, [add](x0, x1) = x0 + 1, [first](x0, x1) = 0, [mark](x0) = x0, [recip](x0) = 0, [sqr](x0) = 0, [terms](x0) = 0, [s](x0) = 0, [active](x0) = 0, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [nil] = 0, [recip#](x0) = x0 Strict: recip# ok X -> recip# X 1 + 1X >= 0 + 1X recip# mark X -> recip# X 0 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 1 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 0 + 0X proper sqr X -> sqr proper X 0 + 0X >= 0 + 0X proper recip X -> recip proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 1 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active first(0(), X) -> mark nil() 0 + 0X >= 0 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(X1, X2) -> first(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(X1, X2) -> first(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active dbl 0() -> mark 0() 0 >= 0 active dbl s X -> mark s s dbl X 0 + 0X >= 0 + 0X active dbl X -> dbl active X 0 + 0X >= 1 + 0X active add(0(), X) -> mark X 0 + 0X >= 0 + 1X active add(s X, Y) -> mark s add(X, Y) 0 + 0X + 0Y >= 0 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active s X -> s active X 0 + 0X >= 0 + 0X active terms X -> terms active X 0 + 0X >= 0 + 0X active terms N -> mark cons(recip sqr N, terms s N) 0 + 0N >= 0 + 0N active sqr 0() -> mark 0() 0 >= 0 active sqr s X -> mark s add(sqr X, dbl X) 0 + 0X >= 0 + 0X active sqr X -> sqr active X 0 + 0X >= 0 + 0X active recip X -> recip active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 0 + 0X terms ok X -> ok terms X 0 + 0X >= 1 + 0X terms mark X -> mark terms X 0 + 0X >= 0 + 0X sqr ok X -> ok sqr X 0 + 0X >= 1 + 0X sqr mark X -> mark sqr X 0 + 0X >= 0 + 0X recip ok X -> ok recip X 0 + 0X >= 1 + 0X recip mark X -> mark recip X 0 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 SCCS (1): Scc: {recip# mark X -> recip# X} SCC (1): Strict: {recip# mark X -> recip# X} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + 1, [add](x0, x1) = x0 + 1, [first](x0, x1) = x0 + x1 + 1, [mark](x0) = x0 + 1, [recip](x0) = x0 + 1, [sqr](x0) = x0 + 1, [terms](x0) = x0 + 1, [s](x0) = x0 + 1, [active](x0) = x0 + 1, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = x0 + 1, [0] = 0, [nil] = 1, [recip#](x0) = x0 Strict: recip# mark X -> recip# X 1 + 1X >= 0 + 1X Weak: top ok X -> top active X 2 + 1X >= 2 + 1X top mark X -> top proper X 2 + 1X >= 1 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 1 + 0X proper terms X -> terms proper X 0 + 0X >= 1 + 0X proper sqr X -> sqr proper X 0 + 0X >= 1 + 0X proper recip X -> recip proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 2 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active first(0(), X) -> mark nil() 2 + 1X >= 2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 4 + 1X + 1Y + 0Z >= 2 + 0X + 1Y + 0Z active first(X1, X2) -> first(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active dbl 0() -> mark 0() 2 >= 1 active dbl s X -> mark s s dbl X 2 + 0X >= 4 + 0X active dbl X -> dbl active X 2 + 0X >= 1 + 0X active add(0(), X) -> mark X 2 + 1X >= 1 + 1X active add(s X, Y) -> mark s add(X, Y) 2 + 0X + 1Y >= 3 + 0X + 1Y active add(X1, X2) -> add(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active s X -> s active X 2 + 1X >= 2 + 1X active terms X -> terms active X 2 + 1X >= 2 + 1X active terms N -> mark cons(recip sqr N, terms s N) 2 + 1N >= 4 + 1N active sqr 0() -> mark 0() 2 >= 1 active sqr s X -> mark s add(sqr X, dbl X) 3 + 1X >= 4 + 0X active sqr X -> sqr active X 2 + 1X >= 2 + 1X active recip X -> recip active X 2 + 1X >= 2 + 1X active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X terms ok X -> ok terms X 2 + 1X >= 2 + 1X terms mark X -> mark terms X 2 + 1X >= 2 + 1X sqr ok X -> ok sqr X 2 + 1X >= 2 + 1X sqr mark X -> mark sqr X 2 + 1X >= 2 + 1X recip ok X -> ok recip X 2 + 1X >= 2 + 1X recip mark X -> mark recip X 2 + 1X >= 2 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 Qed SCC (2): Strict: { cons#(mark X1, X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + 1, [add](x0, x1) = x0 + 1, [first](x0, x1) = x0 + x1 + 1, [mark](x0) = x0 + 1, [recip](x0) = x0 + 1, [sqr](x0) = x0 + 1, [terms](x0) = x0 + 1, [s](x0) = x0 + 1, [active](x0) = x0 + 1, [dbl](x0) = 1, [proper](x0) = x0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [nil] = 1, [cons#](x0, x1) = x0 Strict: cons#(ok X1, ok X2) -> cons#(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 cons#(mark X1, X2) -> cons#(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 proper nil() -> ok nil() 1 >= 2 proper dbl X -> dbl proper X 1 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 1 + 1X >= 1 + 1X proper terms X -> terms proper X 1 + 1X >= 1 + 1X proper sqr X -> sqr proper X 1 + 1X >= 1 + 1X proper recip X -> recip proper X 1 + 1X >= 1 + 1X proper cons(X1, X2) -> cons(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 2 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active first(0(), X) -> mark nil() 2 + 1X >= 2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 4 + 1X + 1Y + 0Z >= 2 + 0X + 1Y + 0Z active first(X1, X2) -> first(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active dbl 0() -> mark 0() 2 >= 1 active dbl s X -> mark s s dbl X 2 + 0X >= 4 + 0X active dbl X -> dbl active X 2 + 0X >= 1 + 0X active add(0(), X) -> mark X 2 + 1X >= 1 + 1X active add(s X, Y) -> mark s add(X, Y) 2 + 0X + 1Y >= 3 + 0X + 1Y active add(X1, X2) -> add(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active s X -> s active X 2 + 1X >= 2 + 1X active terms X -> terms active X 2 + 1X >= 2 + 1X active terms N -> mark cons(recip sqr N, terms s N) 2 + 1N >= 4 + 1N active sqr 0() -> mark 0() 2 >= 1 active sqr s X -> mark s add(sqr X, dbl X) 3 + 1X >= 4 + 0X active sqr X -> sqr active X 2 + 1X >= 2 + 1X active recip X -> recip active X 2 + 1X >= 2 + 1X active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X terms ok X -> ok terms X 2 + 1X >= 2 + 1X terms mark X -> mark terms X 2 + 1X >= 2 + 1X sqr ok X -> ok sqr X 2 + 1X >= 2 + 1X sqr mark X -> mark sqr X 2 + 1X >= 2 + 1X recip ok X -> ok recip X 2 + 1X >= 2 + 1X recip mark X -> mark recip X 2 + 1X >= 2 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 SCCS (1): Scc: {cons#(mark X1, X2) -> cons#(X1, X2)} SCC (1): Strict: {cons#(mark X1, X2) -> cons#(X1, X2)} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + 1, [add](x0, x1) = x0 + 1, [first](x0, x1) = x0 + x1 + 1, [mark](x0) = x0 + 1, [recip](x0) = x0 + 1, [sqr](x0) = x0 + 1, [terms](x0) = x0 + 1, [s](x0) = x0 + 1, [active](x0) = x0 + 1, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = x0 + 1, [0] = 0, [nil] = 1, [cons#](x0, x1) = x0 Strict: cons#(mark X1, X2) -> cons#(X1, X2) 1 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 Weak: top ok X -> top active X 2 + 1X >= 2 + 1X top mark X -> top proper X 2 + 1X >= 1 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 1 + 0X proper terms X -> terms proper X 0 + 0X >= 1 + 0X proper sqr X -> sqr proper X 0 + 0X >= 1 + 0X proper recip X -> recip proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 2 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active first(0(), X) -> mark nil() 2 + 1X >= 2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 4 + 1X + 1Y + 0Z >= 2 + 0X + 1Y + 0Z active first(X1, X2) -> first(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active dbl 0() -> mark 0() 2 >= 1 active dbl s X -> mark s s dbl X 2 + 0X >= 4 + 0X active dbl X -> dbl active X 2 + 0X >= 1 + 0X active add(0(), X) -> mark X 2 + 1X >= 1 + 1X active add(s X, Y) -> mark s add(X, Y) 2 + 0X + 1Y >= 3 + 0X + 1Y active add(X1, X2) -> add(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active s X -> s active X 2 + 1X >= 2 + 1X active terms X -> terms active X 2 + 1X >= 2 + 1X active terms N -> mark cons(recip sqr N, terms s N) 2 + 1N >= 4 + 1N active sqr 0() -> mark 0() 2 >= 1 active sqr s X -> mark s add(sqr X, dbl X) 3 + 1X >= 4 + 0X active sqr X -> sqr active X 2 + 1X >= 2 + 1X active recip X -> recip active X 2 + 1X >= 2 + 1X active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X terms ok X -> ok terms X 2 + 1X >= 2 + 1X terms mark X -> mark terms X 2 + 1X >= 2 + 1X sqr ok X -> ok sqr X 2 + 1X >= 2 + 1X sqr mark X -> mark sqr X 2 + 1X >= 2 + 1X recip ok X -> ok recip X 2 + 1X >= 2 + 1X recip mark X -> mark recip X 2 + 1X >= 2 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 Qed SCC (3): Strict: { first#(X1, mark X2) -> first#(X1, X2), first#(mark X1, X2) -> first#(X1, X2), first#(ok X1, ok X2) -> first#(X1, X2)} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + 1, [add](x0, x1) = x0 + 1, [first](x0, x1) = x0 + 1, [mark](x0) = x0 + 1, [recip](x0) = x0 + 1, [sqr](x0) = x0 + 1, [terms](x0) = x0 + 1, [s](x0) = 0, [active](x0) = x0 + 1, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [nil] = 1, [first#](x0, x1) = x0 + 1 Strict: first#(ok X1, ok X2) -> first#(X1, X2) 2 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 first#(mark X1, X2) -> first#(X1, X2) 2 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 first#(X1, mark X2) -> first#(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 1 + 0X proper sqr X -> sqr proper X 0 + 0X >= 1 + 0X proper recip X -> recip proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 2 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active first(0(), X) -> mark nil() 2 + 1X >= 2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 3 + 0X + 1Y + 0Z >= 2 + 0X + 1Y + 0Z active first(X1, X2) -> first(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active dbl 0() -> mark 0() 2 >= 1 active dbl s X -> mark s s dbl X 2 + 0X >= 1 + 0X active dbl X -> dbl active X 2 + 0X >= 1 + 0X active add(0(), X) -> mark X 2 + 1X >= 1 + 1X active add(s X, Y) -> mark s add(X, Y) 2 + 0X + 1Y >= 1 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active s X -> s active X 1 + 0X >= 0 + 0X active terms X -> terms active X 2 + 1X >= 2 + 1X active terms N -> mark cons(recip sqr N, terms s N) 2 + 1N >= 4 + 1N active sqr 0() -> mark 0() 2 >= 1 active sqr s X -> mark s add(sqr X, dbl X) 2 + 0X >= 1 + 0X active sqr X -> sqr active X 2 + 1X >= 2 + 1X active recip X -> recip active X 2 + 1X >= 2 + 1X active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X terms ok X -> ok terms X 2 + 1X >= 2 + 1X terms mark X -> mark terms X 2 + 1X >= 2 + 1X sqr ok X -> ok sqr X 2 + 1X >= 2 + 1X sqr mark X -> mark sqr X 2 + 1X >= 2 + 1X recip ok X -> ok recip X 2 + 1X >= 2 + 1X recip mark X -> mark recip X 2 + 1X >= 2 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 SCCS (1): Scc: {first#(X1, mark X2) -> first#(X1, X2)} SCC (1): Strict: {first#(X1, mark X2) -> first#(X1, X2)} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + 1, [add](x0, x1) = x0 + 1, [first](x0, x1) = x0 + x1 + 1, [mark](x0) = x0 + 1, [recip](x0) = x0 + 1, [sqr](x0) = x0 + 1, [terms](x0) = x0 + 1, [s](x0) = x0 + 1, [active](x0) = x0 + 1, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = x0 + 1, [0] = 0, [nil] = 1, [first#](x0, x1) = x0 Strict: first#(X1, mark X2) -> first#(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 Weak: top ok X -> top active X 2 + 1X >= 2 + 1X top mark X -> top proper X 2 + 1X >= 1 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 1 + 0X proper terms X -> terms proper X 0 + 0X >= 1 + 0X proper sqr X -> sqr proper X 0 + 0X >= 1 + 0X proper recip X -> recip proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 2 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active first(0(), X) -> mark nil() 2 + 1X >= 2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 4 + 1X + 1Y + 0Z >= 2 + 0X + 1Y + 0Z active first(X1, X2) -> first(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active dbl 0() -> mark 0() 2 >= 1 active dbl s X -> mark s s dbl X 2 + 0X >= 4 + 0X active dbl X -> dbl active X 2 + 0X >= 1 + 0X active add(0(), X) -> mark X 2 + 1X >= 1 + 1X active add(s X, Y) -> mark s add(X, Y) 2 + 0X + 1Y >= 3 + 0X + 1Y active add(X1, X2) -> add(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active s X -> s active X 2 + 1X >= 2 + 1X active terms X -> terms active X 2 + 1X >= 2 + 1X active terms N -> mark cons(recip sqr N, terms s N) 2 + 1N >= 4 + 1N active sqr 0() -> mark 0() 2 >= 1 active sqr s X -> mark s add(sqr X, dbl X) 3 + 1X >= 4 + 0X active sqr X -> sqr active X 2 + 1X >= 2 + 1X active recip X -> recip active X 2 + 1X >= 2 + 1X active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X terms ok X -> ok terms X 2 + 1X >= 2 + 1X terms mark X -> mark terms X 2 + 1X >= 2 + 1X sqr ok X -> ok sqr X 2 + 1X >= 2 + 1X sqr mark X -> mark sqr X 2 + 1X >= 2 + 1X recip ok X -> ok recip X 2 + 1X >= 2 + 1X recip mark X -> mark recip X 2 + 1X >= 2 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 Qed SCC (3): Strict: { add#(X1, mark X2) -> add#(X1, X2), add#(mark X1, X2) -> add#(X1, X2), add#(ok X1, ok X2) -> add#(X1, X2)} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + 1, [add](x0, x1) = x0 + 1, [first](x0, x1) = x0 + 1, [mark](x0) = x0 + 1, [recip](x0) = x0 + 1, [sqr](x0) = x0 + 1, [terms](x0) = x0 + 1, [s](x0) = 0, [active](x0) = x0 + 1, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [0] = 0, [nil] = 1, [add#](x0, x1) = x0 + 1 Strict: add#(ok X1, ok X2) -> add#(X1, X2) 2 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 add#(mark X1, X2) -> add#(X1, X2) 2 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 add#(X1, mark X2) -> add#(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 0 + 0X proper terms X -> terms proper X 0 + 0X >= 1 + 0X proper sqr X -> sqr proper X 0 + 0X >= 1 + 0X proper recip X -> recip proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 2 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active first(0(), X) -> mark nil() 2 + 1X >= 2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 3 + 0X + 1Y + 0Z >= 2 + 0X + 1Y + 0Z active first(X1, X2) -> first(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active dbl 0() -> mark 0() 2 >= 1 active dbl s X -> mark s s dbl X 2 + 0X >= 1 + 0X active dbl X -> dbl active X 2 + 0X >= 1 + 0X active add(0(), X) -> mark X 2 + 1X >= 1 + 1X active add(s X, Y) -> mark s add(X, Y) 2 + 0X + 1Y >= 1 + 0X + 0Y active add(X1, X2) -> add(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active s X -> s active X 1 + 0X >= 0 + 0X active terms X -> terms active X 2 + 1X >= 2 + 1X active terms N -> mark cons(recip sqr N, terms s N) 2 + 1N >= 4 + 1N active sqr 0() -> mark 0() 2 >= 1 active sqr s X -> mark s add(sqr X, dbl X) 2 + 0X >= 1 + 0X active sqr X -> sqr active X 2 + 1X >= 2 + 1X active recip X -> recip active X 2 + 1X >= 2 + 1X active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X terms ok X -> ok terms X 2 + 1X >= 2 + 1X terms mark X -> mark terms X 2 + 1X >= 2 + 1X sqr ok X -> ok sqr X 2 + 1X >= 2 + 1X sqr mark X -> mark sqr X 2 + 1X >= 2 + 1X recip ok X -> ok recip X 2 + 1X >= 2 + 1X recip mark X -> mark recip X 2 + 1X >= 2 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 SCCS (1): Scc: {add#(X1, mark X2) -> add#(X1, X2)} SCC (1): Strict: {add#(X1, mark X2) -> add#(X1, X2)} Weak: { cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), recip mark X -> mark recip X, recip ok X -> ok recip X, sqr mark X -> mark sqr X, sqr ok X -> ok sqr X, terms mark X -> mark terms X, terms ok X -> ok terms X, s mark X -> mark s X, s ok X -> ok s X, active cons(X1, X2) -> cons(active X1, X2), active recip X -> recip active X, active sqr X -> sqr active X, active sqr s X -> mark s add(sqr X, dbl X), active sqr 0() -> mark 0(), active terms N -> mark cons(recip sqr N, terms s N), active terms X -> terms active X, active s X -> s active X, active add(X1, X2) -> add(X1, active X2), active add(X1, X2) -> add(active X1, X2), active add(s X, Y) -> mark s add(X, Y), active add(0(), X) -> mark X, active dbl X -> dbl active X, active dbl s X -> mark s s dbl X, active dbl 0() -> mark 0(), active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active first(0(), X) -> mark nil(), add(X1, mark X2) -> mark add(X1, X2), add(mark X1, X2) -> mark add(X1, X2), add(ok X1, ok X2) -> ok add(X1, X2), dbl mark X -> mark dbl X, dbl ok X -> ok dbl X, first(X1, mark X2) -> mark first(X1, X2), first(mark X1, X2) -> mark first(X1, X2), first(ok X1, ok X2) -> ok first(X1, X2), proper cons(X1, X2) -> cons(proper X1, proper X2), proper recip X -> recip proper X, proper sqr X -> sqr proper X, proper terms X -> terms proper X, proper s X -> s proper X, proper 0() -> ok 0(), proper add(X1, X2) -> add(proper X1, proper X2), proper dbl X -> dbl proper X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [cons](x0, x1) = x0 + 1, [add](x0, x1) = x0 + 1, [first](x0, x1) = x0 + x1 + 1, [mark](x0) = x0 + 1, [recip](x0) = x0 + 1, [sqr](x0) = x0 + 1, [terms](x0) = x0 + 1, [s](x0) = x0 + 1, [active](x0) = x0 + 1, [dbl](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = x0 + 1, [0] = 0, [nil] = 1, [add#](x0, x1) = x0 Strict: add#(X1, mark X2) -> add#(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 Weak: top ok X -> top active X 2 + 1X >= 2 + 1X top mark X -> top proper X 2 + 1X >= 1 + 0X proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper dbl X -> dbl proper X 0 + 0X >= 1 + 0X proper add(X1, X2) -> add(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper s X -> s proper X 0 + 0X >= 1 + 0X proper terms X -> terms proper X 0 + 0X >= 1 + 0X proper sqr X -> sqr proper X 0 + 0X >= 1 + 0X proper recip X -> recip proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 dbl ok X -> ok dbl X 1 + 0X >= 2 + 0X dbl mark X -> mark dbl X 1 + 0X >= 2 + 0X add(ok X1, ok X2) -> ok add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(mark X1, X2) -> mark add(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 add(X1, mark X2) -> mark add(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active first(0(), X) -> mark nil() 2 + 1X >= 2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 4 + 1X + 1Y + 0Z >= 2 + 0X + 1Y + 0Z active first(X1, X2) -> first(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active dbl 0() -> mark 0() 2 >= 1 active dbl s X -> mark s s dbl X 2 + 0X >= 4 + 0X active dbl X -> dbl active X 2 + 0X >= 1 + 0X active add(0(), X) -> mark X 2 + 1X >= 1 + 1X active add(s X, Y) -> mark s add(X, Y) 2 + 0X + 1Y >= 3 + 0X + 1Y active add(X1, X2) -> add(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active add(X1, X2) -> add(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active s X -> s active X 2 + 1X >= 2 + 1X active terms X -> terms active X 2 + 1X >= 2 + 1X active terms N -> mark cons(recip sqr N, terms s N) 2 + 1N >= 4 + 1N active sqr 0() -> mark 0() 2 >= 1 active sqr s X -> mark s add(sqr X, dbl X) 3 + 1X >= 4 + 0X active sqr X -> sqr active X 2 + 1X >= 2 + 1X active recip X -> recip active X 2 + 1X >= 2 + 1X active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X terms ok X -> ok terms X 2 + 1X >= 2 + 1X terms mark X -> mark terms X 2 + 1X >= 2 + 1X sqr ok X -> ok sqr X 2 + 1X >= 2 + 1X sqr mark X -> mark sqr X 2 + 1X >= 2 + 1X recip ok X -> ok recip X 2 + 1X >= 2 + 1X recip mark X -> mark recip X 2 + 1X >= 2 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 Qed