YES Time: 0.425293 TRS: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, top mark X -> top proper X, top ok X -> top active X} DP: DP: { 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)) -> first#(X, Z), active# first(s X, cons(Y, Z)) -> cons#(Y, first(X, Z)), active# cons(X1, X2) -> active# X1, active# cons(X1, X2) -> cons#(active X1, X2), active# s X -> active# X, active# s X -> s# active X, active# from X -> active# X, active# from X -> cons#(X, from s X), active# from X -> s# X, active# from X -> from# active X, active# from X -> from# s X, first#(X1, mark X2) -> first#(X1, X2), first#(mark X1, X2) -> first#(X1, X2), first#(ok X1, ok X2) -> first#(X1, X2), cons#(mark X1, X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2), s# mark X -> s# X, s# ok X -> s# X, from# mark X -> from# X, from# ok X -> from# X, proper# first(X1, X2) -> first#(proper X1, proper X2), proper# first(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X2, proper# cons(X1, X2) -> cons#(proper X1, proper X2), proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# s X -> s# proper X, proper# s X -> proper# X, proper# from X -> from# proper X, proper# from X -> proper# X, top# mark X -> proper# X, top# mark X -> top# proper X, top# ok X -> active# X, top# ok X -> top# active X} TRS: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, top mark X -> top proper X, top ok X -> top active X} EDG: { (active# s X -> s# active X, s# ok X -> s# X) (active# s X -> s# active X, s# mark X -> s# X) (active# from X -> from# s X, from# ok X -> from# X) (active# from X -> from# s X, from# mark X -> from# X) (proper# from X -> from# proper X, from# ok X -> from# X) (proper# from X -> from# proper X, from# mark X -> from# 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# cons(X1, X2) -> cons#(active X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)) (active# cons(X1, X2) -> cons#(active X1, X2), cons#(mark X1, X2) -> cons#(X1, X2)) (first#(mark X1, X2) -> first#(X1, X2), first#(ok X1, ok 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#(X1, mark X2) -> first#(X1, X2)) (cons#(mark X1, X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)) (cons#(mark X1, X2) -> cons#(X1, X2), cons#(mark X1, X2) -> cons#(X1, X2)) (active# first(X1, X2) -> first#(X1, active X2), first#(ok X1, ok 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#(X1, mark X2) -> first#(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)) (active# s X -> active# X, active# from X -> from# s X) (active# s X -> active# X, active# from X -> from# active X) (active# s X -> active# X, active# from X -> s# X) (active# s X -> active# X, active# from X -> cons#(X, from s X)) (active# s X -> active# X, active# from X -> active# X) (active# s X -> active# X, active# s X -> s# active X) (active# s X -> active# X, active# s X -> active# X) (active# s X -> active# X, active# cons(X1, X2) -> cons#(active X1, X2)) (active# s X -> active# X, active# cons(X1, X2) -> active# X1) (active# s X -> active# X, active# first(s X, cons(Y, Z)) -> cons#(Y, first(X, Z))) (active# s X -> active# X, active# first(s X, cons(Y, Z)) -> 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# from X -> s# X, s# ok X -> s# X) (active# from X -> s# X, s# mark X -> s# X) (s# ok X -> s# X, s# ok X -> s# X) (s# ok X -> s# X, s# mark X -> s# X) (from# ok X -> from# X, from# ok X -> from# X) (from# ok X -> from# X, from# mark X -> from# X) (proper# cons(X1, X2) -> proper# X2, proper# from X -> proper# X) (proper# cons(X1, X2) -> proper# X2, proper# from X -> from# proper X) (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# 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# 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# from X -> proper# X, proper# from X -> proper# X) (proper# from X -> proper# X, proper# from X -> from# proper X) (proper# from X -> proper# X, proper# s X -> proper# X) (proper# from X -> proper# X, proper# s X -> s# proper X) (proper# from X -> proper# X, proper# cons(X1, X2) -> proper# X2) (proper# from X -> proper# X, proper# cons(X1, X2) -> proper# X1) (proper# from X -> proper# X, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# from X -> proper# X, proper# first(X1, X2) -> proper# X2) (proper# from X -> proper# X, proper# first(X1, X2) -> proper# X1) (proper# from X -> proper# X, proper# first(X1, X2) -> first#(proper X1, proper X2)) (top# ok X -> active# X, active# from X -> from# s X) (top# ok X -> active# X, active# from X -> from# active X) (top# ok X -> active# X, active# from X -> s# X) (top# ok X -> active# X, active# from X -> cons#(X, from s X)) (top# ok X -> active# X, active# from 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# cons(X1, X2) -> cons#(active X1, X2)) (top# ok X -> active# X, active# cons(X1, X2) -> active# X1) (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)) (top# ok X -> active# X, active# first(X1, X2) -> first#(active X1, X2)) (top# ok X -> active# X, active# first(X1, X2) -> first#(X1, active X2)) (top# ok X -> active# X, active# first(X1, X2) -> active# X2) (top# ok X -> active# X, active# first(X1, X2) -> active# X1) (active# from X -> cons#(X, from s X), cons#(ok X1, ok X2) -> cons#(X1, X2)) (active# from X -> cons#(X, from s X), cons#(mark X1, X2) -> cons#(X1, X2)) (active# cons(X1, X2) -> active# X1, active# from X -> from# s X) (active# cons(X1, X2) -> active# X1, active# from X -> from# active X) (active# cons(X1, X2) -> active# X1, active# from X -> s# X) (active# cons(X1, X2) -> active# X1, active# from X -> cons#(X, from s X)) (active# cons(X1, X2) -> active# X1, active# from X -> active# X) (active# cons(X1, X2) -> active# X1, active# s X -> s# active X) (active# cons(X1, X2) -> active# X1, active# s X -> active# X) (active# cons(X1, X2) -> active# X1, active# cons(X1, X2) -> cons#(active X1, X2)) (active# cons(X1, X2) -> active# X1, active# cons(X1, X2) -> active# X1) (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(s X, cons(Y, Z)) -> 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) (proper# cons(X1, X2) -> proper# X1, proper# from X -> proper# X) (proper# cons(X1, X2) -> proper# X1, proper# from X -> from# proper X) (proper# cons(X1, X2) -> proper# X1, proper# s X -> proper# X) (proper# cons(X1, X2) -> proper# X1, proper# s X -> s# proper X) (proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2) (proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X1) (proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# cons(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X2) (proper# cons(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X1) (proper# cons(X1, X2) -> proper# X1, proper# first(X1, X2) -> first#(proper X1, proper X2)) (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# 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# s X -> s# proper X) (proper# first(X1, X2) -> proper# X1, proper# s X -> proper# X) (proper# first(X1, X2) -> proper# X1, proper# from X -> from# proper X) (proper# first(X1, X2) -> proper# X1, proper# from X -> proper# X) (active# first(X1, X2) -> active# X1, active# first(X1, X2) -> active# X1) (active# first(X1, X2) -> active# X1, active# first(X1, X2) -> active# 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) -> first#(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# cons(X1, X2) -> active# X1) (active# first(X1, X2) -> active# X1, active# cons(X1, X2) -> cons#(active X1, X2)) (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# from X -> active# X) (active# first(X1, X2) -> active# X1, active# from X -> cons#(X, from s X)) (active# first(X1, X2) -> active# X1, active# from X -> s# X) (active# first(X1, X2) -> active# X1, active# from X -> from# active X) (active# first(X1, X2) -> active# X1, active# from X -> from# s X) (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)) (top# mark X -> proper# X, proper# first(X1, X2) -> first#(proper X1, proper X2)) (top# mark X -> proper# X, proper# first(X1, X2) -> proper# X1) (top# mark X -> proper# X, proper# first(X1, X2) -> proper# X2) (top# mark X -> proper# X, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (top# mark X -> proper# X, proper# cons(X1, X2) -> proper# X1) (top# mark X -> proper# X, proper# cons(X1, X2) -> proper# X2) (top# mark X -> proper# X, proper# s X -> s# proper X) (top# mark X -> proper# X, proper# s X -> proper# X) (top# mark X -> proper# X, proper# from X -> from# proper X) (top# mark X -> proper# X, proper# from X -> proper# X) (proper# s X -> proper# X, proper# first(X1, X2) -> first#(proper X1, proper X2)) (proper# s X -> proper# X, proper# first(X1, X2) -> proper# X1) (proper# s X -> proper# X, proper# first(X1, X2) -> proper# X2) (proper# s X -> proper# X, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# s X -> proper# X, proper# cons(X1, X2) -> proper# X1) (proper# s X -> proper# X, proper# cons(X1, X2) -> proper# X2) (proper# s X -> proper# X, proper# s X -> s# proper X) (proper# s X -> proper# X, proper# s X -> proper# X) (proper# s X -> proper# X, proper# from X -> from# proper X) (proper# s X -> proper# X, proper# from X -> proper# X) (proper# first(X1, X2) -> proper# X2, proper# first(X1, X2) -> first#(proper X1, proper X2)) (proper# first(X1, X2) -> proper# X2, proper# first(X1, X2) -> proper# X1) (proper# first(X1, X2) -> proper# X2, proper# first(X1, X2) -> proper# X2) (proper# first(X1, X2) -> proper# X2, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# first(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X1) (proper# first(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X2) (proper# first(X1, X2) -> proper# X2, proper# s X -> s# proper X) (proper# first(X1, X2) -> proper# X2, proper# s X -> proper# X) (proper# first(X1, X2) -> proper# X2, proper# from X -> from# proper X) (proper# first(X1, X2) -> proper# X2, proper# from X -> proper# X) (from# mark X -> from# X, from# mark X -> from# X) (from# mark X -> from# X, from# ok X -> from# X) (s# mark X -> s# X, s# mark X -> s# X) (s# mark X -> s# X, s# ok X -> s# X) (active# from X -> active# X, active# first(X1, X2) -> active# X1) (active# from X -> active# X, active# first(X1, X2) -> active# X2) (active# from X -> active# X, active# first(X1, X2) -> first#(X1, active X2)) (active# from X -> active# X, active# first(X1, X2) -> first#(active X1, X2)) (active# from X -> active# X, active# first(s X, cons(Y, Z)) -> first#(X, Z)) (active# from X -> active# X, active# first(s X, cons(Y, Z)) -> cons#(Y, first(X, Z))) (active# from X -> active# X, active# cons(X1, X2) -> active# X1) (active# from X -> active# X, active# cons(X1, X2) -> cons#(active X1, X2)) (active# from X -> active# X, active# s X -> active# X) (active# from X -> active# X, active# s X -> s# active X) (active# from X -> active# X, active# from X -> active# X) (active# from X -> active# X, active# from X -> cons#(X, from s X)) (active# from X -> active# X, active# from X -> s# X) (active# from X -> active# X, active# from X -> from# active X) (active# from X -> active# X, active# from X -> from# s 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)) -> first#(X, Z)) (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# cons(X1, X2) -> active# X1) (active# first(X1, X2) -> active# X2, active# cons(X1, X2) -> cons#(active X1, X2)) (active# first(X1, X2) -> active# X2, active# s X -> active# X) (active# first(X1, X2) -> active# X2, active# s X -> s# active X) (active# first(X1, X2) -> active# X2, active# from X -> active# X) (active# first(X1, X2) -> active# X2, active# from X -> cons#(X, from s X)) (active# first(X1, X2) -> active# X2, active# from X -> s# X) (active# first(X1, X2) -> active# X2, active# from X -> from# active X) (active# first(X1, X2) -> active# X2, active# from X -> from# s X) (proper# first(X1, X2) -> first#(proper X1, proper X2), first#(X1, mark 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#(ok X1, ok X2) -> first#(X1, X2)) (cons#(ok X1, ok X2) -> cons#(X1, X2), cons#(mark X1, X2) -> cons#(X1, X2)) (cons#(ok X1, ok X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)) (first#(ok X1, ok X2) -> first#(X1, X2), first#(X1, mark 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#(ok X1, ok X2) -> first#(X1, X2)) (first#(X1, mark X2) -> first#(X1, X2), first#(X1, mark 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#(ok X1, ok X2) -> first#(X1, X2)) (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)) (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) (active# from X -> from# active X, from# mark X -> from# X) (active# from X -> from# active X, from# ok X -> from# X) (active# first(s X, cons(Y, Z)) -> first#(X, Z), first#(X1, mark 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#(ok X1, ok X2) -> first#(X1, X2)) } STATUS: arrows: 0.846953 SCCS (7): Scc: {top# mark X -> top# proper X, top# ok X -> top# active X} Scc: {active# first(X1, X2) -> active# X1, active# first(X1, X2) -> active# X2, active# cons(X1, X2) -> active# X1, active# s X -> active# X, active# from X -> active# X} Scc: {proper# first(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# s X -> proper# X, proper# from X -> proper# X} Scc: { cons#(mark X1, X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)} Scc: {from# mark X -> from# X, from# ok X -> from# X} Scc: {s# mark X -> s# X, s# ok X -> s# X} Scc: { first#(X1, mark X2) -> first#(X1, X2), first#(mark X1, X2) -> first#(X1, X2), first#(ok X1, ok X2) -> first#(X1, X2)} SCC (2): Strict: {top# mark X -> top# proper X, top# ok X -> top# active X} Weak: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = x0 + x1, [cons](x0, x1) = x0, [mark](x0) = x0 + 1, [active](x0) = x0, [s](x0) = x0 + 1, [from](x0) = x0 + 1, [proper](x0) = x0, [ok](x0) = x0, [top](x0) = 0, [nil] = 0, [0] = 1, [top#](x0) = x0 Strict: top# ok X -> top# active X 0 + 1X >= 0 + 1X top# mark X -> top# proper X 1 + 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 from X -> from proper X 1 + 1X >= 1 + 1X proper s X -> s proper X 1 + 1X >= 1 + 1X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 proper 0() -> ok 0() 1 >= 1 proper first(X1, X2) -> first(proper X1, proper X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 proper nil() -> ok nil() 0 >= 0 from ok X -> ok from X 1 + 1X >= 1 + 1X from mark X -> mark from X 2 + 1X >= 2 + 1X s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 active from X -> from active X 1 + 1X >= 1 + 1X active from X -> mark cons(X, from s X) 1 + 1X >= 1 + 1X active s X -> s active X 1 + 1X >= 1 + 1X active cons(X1, X2) -> cons(active X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 1 + 1X + 1Y + 0Z >= 1 + 0X + 1Y + 0Z active first(0(), X) -> mark nil() 1 + 1X >= 1 active first(X1, X2) -> first(active X1, X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 active first(X1, X2) -> first(X1, active X2) 0 + 1X1 + 1X2 >= 0 + 1X1 + 1X2 SCCS (1): Scc: {top# ok X -> top# active X} SCC (1): Strict: {top# ok X -> top# active X} Weak: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = x0 + 1, [cons](x0, x1) = x0 + x1 + 1, [mark](x0) = 0, [active](x0) = x0, [s](x0) = x0 + 1, [from](x0) = x0, [proper](x0) = x0 + 1, [ok](x0) = x0 + 1, [top](x0) = 0, [nil] = 1, [0] = 0, [top#](x0) = x0 Strict: top# ok X -> top# active X 1 + 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 from X -> from proper X 1 + 1X >= 1 + 1X proper s X -> s proper X 2 + 1X >= 2 + 1X proper cons(X1, X2) -> cons(proper X1, proper X2) 2 + 1X1 + 1X2 >= 3 + 1X1 + 1X2 proper 0() -> ok 0() 1 >= 1 proper first(X1, X2) -> first(proper X1, proper X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 proper nil() -> ok nil() 2 >= 2 from ok X -> ok from X 1 + 1X >= 1 + 1X from mark X -> mark from X 0 + 0X >= 0 + 0X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 1 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 first(mark X1, X2) -> mark first(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 1 + 1X1 + 0X2 >= 0 + 0X1 + 0X2 active from X -> from active X 0 + 1X >= 0 + 1X active from X -> mark cons(X, from s X) 0 + 1X >= 0 + 0X active s X -> s active X 1 + 1X >= 1 + 1X active cons(X1, X2) -> cons(active X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 2 + 1X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(0(), X) -> mark nil() 1 + 0X >= 0 active first(X1, X2) -> first(active X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active first(X1, X2) -> first(X1, active X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 Qed SCC (5): Strict: {active# first(X1, X2) -> active# X1, active# first(X1, X2) -> active# X2, active# cons(X1, X2) -> active# X1, active# s X -> active# X, active# from X -> active# X} Weak: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = x0 + x1, [cons](x0, x1) = x0, [mark](x0) = x0 + 1, [active](x0) = 0, [s](x0) = x0, [from](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [nil] = 1, [0] = 0, [active#](x0) = x0 Strict: active# from X -> active# X 1 + 1X >= 0 + 1X active# s X -> active# X 0 + 1X >= 0 + 1X active# cons(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# first(X1, X2) -> active# X2 0 + 1X1 + 1X2 >= 0 + 1X2 active# first(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 from X -> from proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 from ok X -> ok from X 2 + 1X >= 2 + 1X from mark X -> mark from X 2 + 1X >= 2 + 1X s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 1 + 1X >= 1 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 first(ok X1, ok X2) -> ok first(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 active from X -> from active X 0 + 0X >= 1 + 0X active from X -> mark cons(X, from s X) 0 + 0X >= 1 + 1X active s X -> s active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 1 + 0X + 1Y + 0Z active first(0(), X) -> mark nil() 0 + 0X >= 2 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 SCCS (1): Scc: {active# first(X1, X2) -> active# X1, active# first(X1, X2) -> active# X2, active# cons(X1, X2) -> active# X1, active# s X -> active# X} SCC (4): Strict: {active# first(X1, X2) -> active# X1, active# first(X1, X2) -> active# X2, active# cons(X1, X2) -> active# X1, active# s X -> active# X} Weak: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = x0 + x1, [cons](x0, x1) = x0, [mark](x0) = 0, [active](x0) = 0, [s](x0) = x0 + 1, [from](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [nil] = 0, [0] = 0, [active#](x0) = x0 Strict: active# s X -> active# X 1 + 1X >= 0 + 1X active# cons(X1, X2) -> active# X1 0 + 1X1 + 0X2 >= 0 + 1X1 active# first(X1, X2) -> active# X2 0 + 1X1 + 1X2 >= 0 + 1X2 active# first(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 from X -> from proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 1 from ok X -> ok from X 2 + 1X >= 2 + 1X from mark X -> mark from X 1 + 0X >= 0 + 0X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 1 + 0X >= 0 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, 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 active from X -> from active X 0 + 0X >= 1 + 0X active from X -> mark cons(X, from s X) 0 + 0X >= 0 + 0X active s X -> s active X 0 + 0X >= 1 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(0(), X) -> mark nil() 0 + 0X >= 0 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 SCCS (1): Scc: {active# first(X1, X2) -> active# X1, active# first(X1, X2) -> active# X2, active# cons(X1, X2) -> active# X1} SCC (3): Strict: {active# first(X1, X2) -> active# X1, active# first(X1, X2) -> active# X2, active# cons(X1, X2) -> active# X1} Weak: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = x0 + x1, [cons](x0, x1) = x0 + 1, [mark](x0) = x0 + 1, [active](x0) = 0, [s](x0) = 1, [from](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [nil] = 1, [0] = 0, [active#](x0) = x0 Strict: active# cons(X1, X2) -> active# X1 1 + 1X1 + 0X2 >= 0 + 1X1 active# first(X1, X2) -> active# X2 0 + 1X1 + 1X2 >= 0 + 1X2 active# first(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 from X -> from proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 from ok X -> ok from X 2 + 1X >= 2 + 1X from mark X -> mark from X 2 + 1X >= 2 + 1X s ok X -> ok s X 1 + 0X >= 2 + 0X s mark X -> mark s X 1 + 0X >= 2 + 0X 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 first(ok X1, ok X2) -> ok first(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 active from X -> from active X 0 + 0X >= 1 + 0X active from X -> mark cons(X, from s X) 0 + 0X >= 2 + 1X active s X -> s active X 0 + 0X >= 1 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 2 + 0X + 1Y + 0Z active first(0(), X) -> mark nil() 0 + 0X >= 2 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 SCCS (1): Scc: {active# first(X1, X2) -> active# X1, active# first(X1, X2) -> active# X2} SCC (2): Strict: {active# first(X1, X2) -> active# X1, active# first(X1, X2) -> active# X2} Weak: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = x0 + x1 + 1, [cons](x0, x1) = 0, [mark](x0) = x0 + 1, [active](x0) = 0, [s](x0) = 1, [from](x0) = x0, [proper](x0) = x0 + 1, [ok](x0) = 0, [top](x0) = x0, [nil] = 1, [0] = 0, [active#](x0) = x0 + 1 Strict: active# first(X1, X2) -> active# X2 2 + 1X1 + 1X2 >= 1 + 1X2 active# first(X1, X2) -> active# X1 2 + 1X1 + 1X2 >= 1 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 1 + 1X >= 1 + 1X proper from X -> from proper X 1 + 1X >= 1 + 1X proper s X -> s proper X 2 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper 0() -> ok 0() 1 >= 0 proper first(X1, X2) -> first(proper X1, proper X2) 2 + 1X1 + 1X2 >= 3 + 1X1 + 1X2 proper nil() -> ok nil() 2 >= 0 from ok X -> ok from X 0 + 0X >= 0 + 0X from mark X -> mark from X 1 + 1X >= 1 + 1X s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 1 + 0X >= 2 + 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 + 0X2 >= 1 + 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) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active from X -> from active X 0 + 0X >= 0 + 0X active from X -> mark cons(X, from s X) 0 + 0X >= 1 + 0X active s X -> s active X 0 + 0X >= 1 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 1 + 0X + 0Y + 0Z active first(0(), X) -> mark nil() 0 + 0X >= 2 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 Qed SCC (6): Strict: {proper# first(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# s X -> proper# X, proper# from X -> proper# X} Weak: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = x0 + x1, [cons](x0, x1) = x0 + x1, [mark](x0) = x0 + 1, [active](x0) = 0, [s](x0) = x0, [from](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [nil] = 1, [0] = 0, [proper#](x0) = x0 Strict: proper# from X -> proper# X 1 + 1X >= 0 + 1X proper# s 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 proper# first(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# first(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 from X -> from proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 from ok X -> ok from X 2 + 1X >= 2 + 1X from mark X -> mark from X 2 + 1X >= 2 + 1X s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 1 + 1X >= 1 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 first(ok X1, ok X2) -> ok first(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 active from X -> from active X 0 + 0X >= 1 + 0X active from X -> mark cons(X, from s X) 0 + 0X >= 2 + 2X active s X -> s active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 1 + 1X + 1Y + 1Z active first(0(), X) -> mark nil() 0 + 0X >= 2 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 SCCS (1): Scc: {proper# first(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# s X -> proper# X} SCC (5): Strict: {proper# first(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# s X -> proper# X} Weak: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = x0 + x1, [cons](x0, x1) = x0 + x1, [mark](x0) = 0, [active](x0) = 0, [s](x0) = x0 + 1, [from](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [nil] = 0, [0] = 0, [proper#](x0) = x0 Strict: proper# s 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 proper# first(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# first(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 from X -> from proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 1 from ok X -> ok from X 2 + 1X >= 2 + 1X from mark X -> mark from X 1 + 0X >= 0 + 0X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s 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 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 active from X -> from active X 0 + 0X >= 1 + 0X active from X -> mark cons(X, from s X) 0 + 0X >= 0 + 0X active s X -> s active X 0 + 0X >= 1 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 0 + 0X + 0Y + 0Z active first(0(), X) -> mark nil() 0 + 0X >= 0 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 SCCS (1): Scc: {proper# first(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2} SCC (4): Strict: {proper# first(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2} Weak: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = x0 + x1, [cons](x0, x1) = x0 + x1 + 1, [mark](x0) = x0 + 1, [active](x0) = 0, [s](x0) = 1, [from](x0) = 0, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [nil] = 1, [0] = 0, [proper#](x0) = x0 Strict: proper# cons(X1, X2) -> proper# X2 1 + 1X1 + 1X2 >= 0 + 1X2 proper# cons(X1, X2) -> proper# X1 1 + 1X1 + 1X2 >= 0 + 1X1 proper# first(X1, X2) -> proper# X2 0 + 1X1 + 1X2 >= 0 + 1X2 proper# first(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 from X -> from proper X 0 + 0X >= 0 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 1 proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 from ok X -> ok from X 0 + 0X >= 1 + 0X from mark X -> mark from X 0 + 0X >= 1 + 0X s ok X -> ok s X 1 + 0X >= 2 + 0X s mark X -> mark s X 1 + 0X >= 2 + 0X cons(ok X1, ok X2) -> ok cons(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(ok X1, ok X2) -> ok first(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 first(mark X1, X2) -> mark first(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 active from X -> from active X 0 + 0X >= 0 + 0X active from X -> mark cons(X, from s X) 0 + 0X >= 2 + 1X active s X -> s active X 0 + 0X >= 1 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 1X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 2 + 1X + 1Y + 1Z active first(0(), X) -> mark nil() 0 + 0X >= 2 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 SCCS (1): Scc: {proper# first(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X2} SCC (2): Strict: {proper# first(X1, X2) -> proper# X1, proper# first(X1, X2) -> proper# X2} Weak: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = x0 + x1 + 1, [cons](x0, x1) = 0, [mark](x0) = x0 + 1, [active](x0) = 0, [s](x0) = 1, [from](x0) = x0, [proper](x0) = x0 + 1, [ok](x0) = 0, [top](x0) = x0, [nil] = 1, [0] = 0, [proper#](x0) = x0 + 1 Strict: proper# first(X1, X2) -> proper# X2 2 + 1X1 + 1X2 >= 1 + 1X2 proper# first(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 1 + 1X >= 1 + 1X proper from X -> from proper X 1 + 1X >= 1 + 1X proper s X -> s proper X 2 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper 0() -> ok 0() 1 >= 0 proper first(X1, X2) -> first(proper X1, proper X2) 2 + 1X1 + 1X2 >= 3 + 1X1 + 1X2 proper nil() -> ok nil() 2 >= 0 from ok X -> ok from X 0 + 0X >= 0 + 0X from mark X -> mark from X 1 + 1X >= 1 + 1X s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 1 + 0X >= 2 + 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 + 0X2 >= 1 + 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) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active from X -> from active X 0 + 0X >= 0 + 0X active from X -> mark cons(X, from s X) 0 + 0X >= 1 + 0X active s X -> s active X 0 + 0X >= 1 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 1 + 0X + 0Y + 0Z active first(0(), X) -> mark nil() 0 + 0X >= 2 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 Qed SCC (2): Strict: { cons#(mark X1, X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)} Weak: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = x0 + 1, [cons](x0, x1) = x0 + 1, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [s](x0) = x0 + 1, [from](x0) = 0, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [nil] = 1, [0] = 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 from X -> from proper X 0 + 0X >= 0 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 2 proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 from ok X -> ok from X 0 + 0X >= 1 + 0X from mark X -> mark from X 0 + 0X >= 1 + 0X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 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 active from X -> from active X 1 + 0X >= 0 + 0X active from X -> mark cons(X, from s X) 1 + 0X >= 2 + 0X active s X -> s active X 2 + 1X >= 2 + 1X active cons(X1, X2) -> cons(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 3 + 0X + 0Y + 1Z >= 3 + 0X + 0Y + 1Z active first(0(), X) -> mark nil() 2 + 1X >= 2 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 SCCS (1): Scc: {cons#(mark X1, X2) -> cons#(X1, X2)} SCC (1): Strict: {cons#(mark X1, X2) -> cons#(X1, X2)} Weak: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = x0 + x1 + 1, [cons](x0, x1) = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [s](x0) = 1, [from](x0) = x0, [proper](x0) = 1, [ok](x0) = 0, [top](x0) = x0, [nil] = 0, [0] = 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 0 + 0X >= 1 + 1X top mark X -> top proper X 1 + 1X >= 1 + 0X proper from X -> from proper X 1 + 0X >= 1 + 0X proper s X -> s proper X 1 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper 0() -> ok 0() 1 >= 0 proper first(X1, X2) -> first(proper X1, proper X2) 1 + 0X1 + 0X2 >= 3 + 0X1 + 0X2 proper nil() -> ok nil() 1 >= 0 from ok X -> ok from X 0 + 0X >= 0 + 0X from mark X -> mark from X 1 + 1X >= 1 + 1X s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 1 + 0X >= 2 + 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 + 0X2 >= 1 + 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) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active from X -> from active X 1 + 1X >= 1 + 1X active from X -> mark cons(X, from s X) 1 + 1X >= 1 + 0X active s X -> s active X 2 + 0X >= 1 + 0X active cons(X1, X2) -> cons(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 3 + 0X + 0Y + 0Z >= 1 + 0X + 0Y + 0Z active first(0(), X) -> mark nil() 3 + 1X >= 1 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 Qed SCC (2): Strict: {from# mark X -> from# X, from# ok X -> from# X} Weak: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = x0 + 1, [cons](x0, x1) = x0 + 1, [mark](x0) = x0, [active](x0) = x0 + 1, [s](x0) = x0 + 1, [from](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [nil] = 0, [0] = 1, [from#](x0) = x0 Strict: from# ok X -> from# X 1 + 1X >= 0 + 1X from# mark X -> from# 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 from X -> from proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 2 proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 1 from ok X -> ok from X 1 + 0X >= 2 + 0X from mark X -> mark from X 1 + 0X >= 1 + 0X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 1 + 1X >= 1 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 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 >= 1 + 0X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active from X -> from active X 2 + 0X >= 1 + 0X active from X -> mark cons(X, from s X) 2 + 0X >= 2 + 0X active s X -> s active X 2 + 1X >= 2 + 1X active cons(X1, X2) -> cons(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 3 + 0X + 0Y + 1Z >= 2 + 0X + 0Y + 1Z active first(0(), X) -> mark nil() 2 + 1X >= 0 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 SCCS (1): Scc: {from# mark X -> from# X} SCC (1): Strict: {from# mark X -> from# X} Weak: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = x0 + x1 + 1, [cons](x0, x1) = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [s](x0) = 1, [from](x0) = x0, [proper](x0) = 1, [ok](x0) = 0, [top](x0) = x0, [nil] = 0, [0] = 1, [from#](x0) = x0 Strict: from# mark X -> from# X 1 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 1 + 1X top mark X -> top proper X 1 + 1X >= 1 + 0X proper from X -> from proper X 1 + 0X >= 1 + 0X proper s X -> s proper X 1 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper 0() -> ok 0() 1 >= 0 proper first(X1, X2) -> first(proper X1, proper X2) 1 + 0X1 + 0X2 >= 3 + 0X1 + 0X2 proper nil() -> ok nil() 1 >= 0 from ok X -> ok from X 0 + 0X >= 0 + 0X from mark X -> mark from X 1 + 1X >= 1 + 1X s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 1 + 0X >= 2 + 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 + 0X2 >= 1 + 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) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active from X -> from active X 1 + 1X >= 1 + 1X active from X -> mark cons(X, from s X) 1 + 1X >= 1 + 0X active s X -> s active X 2 + 0X >= 1 + 0X active cons(X1, X2) -> cons(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 3 + 0X + 0Y + 0Z >= 1 + 0X + 0Y + 0Z active first(0(), X) -> mark nil() 3 + 1X >= 1 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 Qed SCC (2): Strict: {s# mark X -> s# X, s# ok X -> s# X} Weak: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = x0 + 1, [cons](x0, x1) = x0 + 1, [mark](x0) = x0, [active](x0) = x0 + 1, [s](x0) = x0 + 1, [from](x0) = 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [nil] = 0, [0] = 1, [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 from X -> from proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 2 proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 1 from ok X -> ok from X 1 + 0X >= 2 + 0X from mark X -> mark from X 1 + 0X >= 1 + 0X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 1 + 1X >= 1 + 1X cons(ok X1, ok X2) -> ok cons(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 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 >= 1 + 0X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active from X -> from active X 2 + 0X >= 1 + 0X active from X -> mark cons(X, from s X) 2 + 0X >= 2 + 0X active s X -> s active X 2 + 1X >= 2 + 1X active cons(X1, X2) -> cons(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 3 + 0X + 0Y + 1Z >= 2 + 0X + 0Y + 1Z active first(0(), X) -> mark nil() 2 + 1X >= 0 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 SCCS (1): Scc: {s# mark X -> s# X} SCC (1): Strict: {s# mark X -> s# X} Weak: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = x0 + x1 + 1, [cons](x0, x1) = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [s](x0) = 1, [from](x0) = x0, [proper](x0) = 1, [ok](x0) = 0, [top](x0) = x0, [nil] = 0, [0] = 1, [s#](x0) = x0 Strict: s# mark X -> s# X 1 + 1X >= 0 + 1X Weak: top ok X -> top active X 0 + 0X >= 1 + 1X top mark X -> top proper X 1 + 1X >= 1 + 0X proper from X -> from proper X 1 + 0X >= 1 + 0X proper s X -> s proper X 1 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper 0() -> ok 0() 1 >= 0 proper first(X1, X2) -> first(proper X1, proper X2) 1 + 0X1 + 0X2 >= 3 + 0X1 + 0X2 proper nil() -> ok nil() 1 >= 0 from ok X -> ok from X 0 + 0X >= 0 + 0X from mark X -> mark from X 1 + 1X >= 1 + 1X s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 1 + 0X >= 2 + 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 + 0X2 >= 1 + 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) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active from X -> from active X 1 + 1X >= 1 + 1X active from X -> mark cons(X, from s X) 1 + 1X >= 1 + 0X active s X -> s active X 2 + 0X >= 1 + 0X active cons(X1, X2) -> cons(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 3 + 0X + 0Y + 0Z >= 1 + 0X + 0Y + 0Z active first(0(), X) -> mark nil() 3 + 1X >= 1 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 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: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = 0, [cons](x0, x1) = 0, [mark](x0) = x0 + 1, [active](x0) = 0, [s](x0) = x0, [from](x0) = 0, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [nil] = 1, [0] = 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 from X -> from proper X 0 + 0X >= 0 + 0X proper s X -> s proper X 0 + 0X >= 0 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper 0() -> ok 0() 0 >= 2 proper first(X1, X2) -> first(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 from ok X -> ok from X 0 + 0X >= 1 + 0X from mark X -> mark from X 0 + 0X >= 1 + 0X s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 1 + 1X >= 1 + 1X 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 >= 1 + 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 >= 1 + 0X1 + 0X2 first(X1, mark X2) -> mark first(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active from X -> from active X 0 + 0X >= 0 + 0X active from X -> mark cons(X, from s X) 0 + 0X >= 1 + 0X active s X -> s active X 0 + 0X >= 0 + 0X active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 0 + 0X + 0Y + 0Z >= 1 + 0X + 0Y + 0Z active first(0(), X) -> mark nil() 0 + 0X >= 2 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 SCCS (1): Scc: {first#(X1, mark X2) -> first#(X1, X2)} SCC (1): Strict: {first#(X1, mark X2) -> first#(X1, X2)} Weak: { active first(X1, X2) -> first(X1, active X2), active first(X1, X2) -> first(active X1, X2), active first(0(), X) -> mark nil(), active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)), active cons(X1, X2) -> cons(active X1, X2), active s X -> s active X, active from X -> mark cons(X, from s X), active from X -> from active 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), cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), s mark X -> mark s X, s ok X -> ok s X, from mark X -> mark from X, from ok X -> ok from X, proper nil() -> ok nil(), proper first(X1, X2) -> first(proper X1, proper X2), proper 0() -> ok 0(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper s X -> s proper X, proper from X -> from proper X, 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: [first](x0, x1) = x0 + x1 + 1, [cons](x0, x1) = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [s](x0) = 1, [from](x0) = x0, [proper](x0) = 1, [ok](x0) = 0, [top](x0) = x0, [nil] = 0, [0] = 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 0 + 0X >= 1 + 1X top mark X -> top proper X 1 + 1X >= 1 + 0X proper from X -> from proper X 1 + 0X >= 1 + 0X proper s X -> s proper X 1 + 0X >= 1 + 0X proper cons(X1, X2) -> cons(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper 0() -> ok 0() 1 >= 0 proper first(X1, X2) -> first(proper X1, proper X2) 1 + 0X1 + 0X2 >= 3 + 0X1 + 0X2 proper nil() -> ok nil() 1 >= 0 from ok X -> ok from X 0 + 0X >= 0 + 0X from mark X -> mark from X 1 + 1X >= 1 + 1X s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 1 + 0X >= 2 + 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 + 0X2 >= 1 + 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) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 first(X1, mark X2) -> mark first(X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active from X -> from active X 1 + 1X >= 1 + 1X active from X -> mark cons(X, from s X) 1 + 1X >= 1 + 0X active s X -> s active X 2 + 0X >= 1 + 0X active cons(X1, X2) -> cons(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active first(s X, cons(Y, Z)) -> mark cons(Y, first(X, Z)) 3 + 0X + 0Y + 0Z >= 1 + 0X + 0Y + 0Z active first(0(), X) -> mark nil() 3 + 1X >= 1 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 Qed