YES Problem: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(add(s(X),Y)) -> add#(X,Y) active#(add(s(X),Y)) -> s#(add(X,Y)) active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(from(X)) -> s#(X) active#(from(X)) -> from#(s(X)) active#(from(X)) -> cons#(X,from(s(X))) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) active#(if(X1,X2,X3)) -> active#(X1) active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) active#(add(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> add#(active(X1),X2) active#(first(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> first#(active(X1),X2) active#(first(X1,X2)) -> active#(X2) active#(first(X1,X2)) -> first#(X1,active(X2)) and#(mark(X1),X2) -> and#(X1,X2) if#(mark(X1),X2,X3) -> if#(X1,X2,X3) add#(mark(X1),X2) -> add#(X1,X2) first#(mark(X1),X2) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(if(X1,X2,X3)) -> proper#(X3) proper#(if(X1,X2,X3)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X1) proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(add(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(first(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(from(X)) -> proper#(X) proper#(from(X)) -> from#(proper(X)) and#(ok(X1),ok(X2)) -> and#(X1,X2) if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) add#(ok(X1),ok(X2)) -> add#(X1,X2) s#(ok(X)) -> s#(X) first#(ok(X1),ok(X2)) -> first#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) from#(ok(X)) -> from#(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(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(add(s(X),Y)) -> add#(X,Y) active#(add(s(X),Y)) -> s#(add(X,Y)) active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(from(X)) -> s#(X) active#(from(X)) -> from#(s(X)) active#(from(X)) -> cons#(X,from(s(X))) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) active#(if(X1,X2,X3)) -> active#(X1) active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) active#(add(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> add#(active(X1),X2) active#(first(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> first#(active(X1),X2) active#(first(X1,X2)) -> active#(X2) active#(first(X1,X2)) -> first#(X1,active(X2)) and#(mark(X1),X2) -> and#(X1,X2) if#(mark(X1),X2,X3) -> if#(X1,X2,X3) add#(mark(X1),X2) -> add#(X1,X2) first#(mark(X1),X2) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(if(X1,X2,X3)) -> proper#(X3) proper#(if(X1,X2,X3)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X1) proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(add(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(first(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(from(X)) -> proper#(X) proper#(from(X)) -> from#(proper(X)) and#(ok(X1),ok(X2)) -> and#(X1,X2) if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) add#(ok(X1),ok(X2)) -> add#(X1,X2) s#(ok(X)) -> s#(X) first#(ok(X1),ok(X2)) -> first#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) from#(ok(X)) -> from#(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(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: 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) 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)) -> first#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(add(X1,X2)) -> add#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) top#(ok(X)) -> active#(X) -> active#(if(X1,X2,X3)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(and(X1,X2)) -> and#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(and(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(from(X)) -> cons#(X,from(s(X))) top#(ok(X)) -> active#(X) -> active#(from(X)) -> from#(s(X)) top#(ok(X)) -> active#(X) -> active#(from(X)) -> s#(X) 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#(add(s(X),Y)) -> s#(add(X,Y)) top#(ok(X)) -> active#(X) -> active#(add(s(X),Y)) -> add#(X,Y) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(from(X)) -> from#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(from(X)) -> proper#(X) 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#(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#(s(X)) -> s#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) top#(mark(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X3) top#(mark(X)) -> proper#(X) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X2) proper#(from(X)) -> proper#(X) -> proper#(from(X)) -> from#(proper(X)) proper#(from(X)) -> proper#(X) -> proper#(from(X)) -> proper#(X) proper#(from(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(from(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(from(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(from(X)) -> proper#(X) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(from(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X1) proper#(from(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X2) proper#(from(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(from(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(from(X)) -> proper#(X) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(from(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X1) proper#(from(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X2) proper#(from(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(from(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X1) proper#(from(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X2) proper#(from(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X3) proper#(from(X)) -> proper#(X) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(from(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X1) proper#(from(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X2) proper#(from(X)) -> from#(proper(X)) -> from#(ok(X)) -> from#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(from(X)) -> from#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(from(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X3) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(from(X)) -> from#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(from(X)) -> proper#(X) 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#(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) 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#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X3) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) proper#(first(X1,X2)) -> proper#(X2) -> proper#(from(X)) -> from#(proper(X)) proper#(first(X1,X2)) -> proper#(X2) -> proper#(from(X)) -> proper#(X) 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#(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#(s(X)) -> s#(proper(X)) proper#(first(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(first(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(first(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(first(X1,X2)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X1) proper#(first(X1,X2)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X3) proper#(first(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(first(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X1) -> proper#(from(X)) -> from#(proper(X)) proper#(first(X1,X2)) -> proper#(X1) -> proper#(from(X)) -> proper#(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#(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#(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#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(first(X1,X2)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X1) proper#(first(X1,X2)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X3) proper#(first(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(first(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(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#(X1,mark(X2)) -> first#(X1,X2) proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) -> first#(mark(X1),X2) -> first#(X1,X2) proper#(s(X)) -> proper#(X) -> proper#(from(X)) -> from#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(from(X)) -> proper#(X) 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#(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#(s(X)) -> s#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(s(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X3) proper#(s(X)) -> proper#(X) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X2) proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X) proper#(add(X1,X2)) -> proper#(X2) -> proper#(from(X)) -> from#(proper(X)) proper#(add(X1,X2)) -> proper#(X2) -> proper#(from(X)) -> proper#(X) 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#(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) 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#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(add(X1,X2)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X3) proper#(add(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) -> proper#(from(X)) -> from#(proper(X)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(from(X)) -> proper#(X) proper#(add(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(add(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X3) proper#(add(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) -> add#(mark(X1),X2) -> add#(X1,X2) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(from(X)) -> from#(proper(X)) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(from(X)) -> proper#(X) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(first(X1,X2)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(first(X1,X2)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(s(X)) -> s#(proper(X)) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(s(X)) -> proper#(X) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(add(X1,X2)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(add(X1,X2)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(if(X1,X2,X3)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(if(X1,X2,X3)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(if(X1,X2,X3)) -> proper#(X3) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(and(X1,X2)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X3) -> proper#(and(X1,X2)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(from(X)) -> from#(proper(X)) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(from(X)) -> proper#(X) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(first(X1,X2)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(first(X1,X2)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X3) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(from(X)) -> from#(proper(X)) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(from(X)) -> proper#(X) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(first(X1,X2)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(first(X1,X2)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X3) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X2) proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) -> if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) -> if#(mark(X1),X2,X3) -> if#(X1,X2,X3) proper#(and(X1,X2)) -> proper#(X2) -> proper#(from(X)) -> from#(proper(X)) proper#(and(X1,X2)) -> proper#(X2) -> proper#(from(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(and(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(and(X1,X2)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X2) -> proper#(if(X1,X2,X3)) -> proper#(X3) proper#(and(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) -> proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) -> proper#(from(X)) -> from#(proper(X)) proper#(and(X1,X2)) -> proper#(X1) -> proper#(from(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(and(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) proper#(and(X1,X2)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) -> proper#(if(X1,X2,X3)) -> proper#(X3) proper#(and(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X1) -> proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) -> and#(ok(X1),ok(X2)) -> and#(X1,X2) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) -> and#(mark(X1),X2) -> and#(X1,X2) if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) -> if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) -> if#(mark(X1),X2,X3) -> if#(X1,X2,X3) if#(mark(X1),X2,X3) -> if#(X1,X2,X3) -> if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) if#(mark(X1),X2,X3) -> if#(X1,X2,X3) -> if#(mark(X1),X2,X3) -> if#(X1,X2,X3) and#(ok(X1),ok(X2)) -> and#(X1,X2) -> and#(ok(X1),ok(X2)) -> and#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) -> and#(mark(X1),X2) -> and#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) -> and#(ok(X1),ok(X2)) -> and#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) -> and#(mark(X1),X2) -> and#(X1,X2) from#(ok(X)) -> from#(X) -> from#(ok(X)) -> from#(X) 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#(ok(X1),ok(X2)) -> first#(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#(mark(X1),X2) -> first#(X1,X2) -> first#(ok(X1),ok(X2)) -> first#(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#(X1,mark(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) s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X) add#(ok(X1),ok(X2)) -> add#(X1,X2) -> add#(ok(X1),ok(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#(ok(X1),ok(X2)) -> add#(X1,X2) add#(mark(X1),X2) -> add#(X1,X2) -> add#(mark(X1),X2) -> add#(X1,X2) active#(from(X)) -> from#(s(X)) -> from#(ok(X)) -> from#(X) active#(from(X)) -> cons#(X,from(s(X))) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(from(X)) -> s#(X) -> s#(ok(X)) -> s#(X) active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) 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#(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(X1,X2)) -> first#(active(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#(X1,active(X2)) -> first#(ok(X1),ok(X2)) -> first#(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)) -> active#(X2) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(first(X1,X2)) -> active#(X2) -> active#(first(X1,X2)) -> active#(X2) active#(first(X1,X2)) -> active#(X2) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(first(X1,X2)) -> active#(X2) -> active#(first(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(first(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X2) -> active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) active#(first(X1,X2)) -> active#(X2) -> active#(if(X1,X2,X3)) -> active#(X1) active#(first(X1,X2)) -> active#(X2) -> active#(and(X1,X2)) -> and#(active(X1),X2) active#(first(X1,X2)) -> active#(X2) -> active#(and(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X2) -> active#(from(X)) -> cons#(X,from(s(X))) active#(first(X1,X2)) -> active#(X2) -> active#(from(X)) -> from#(s(X)) active#(first(X1,X2)) -> active#(X2) -> active#(from(X)) -> s#(X) 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)) -> 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#(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)) -> first#(active(X1),X2) active#(first(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(first(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X1) -> active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) active#(first(X1,X2)) -> active#(X1) -> active#(if(X1,X2,X3)) -> active#(X1) active#(first(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> and#(active(X1),X2) active#(first(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X1) -> active#(from(X)) -> cons#(X,from(s(X))) active#(first(X1,X2)) -> active#(X1) -> active#(from(X)) -> from#(s(X)) active#(first(X1,X2)) -> active#(X1) -> active#(from(X)) -> s#(X) 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(s(X),cons(Y,Z))) -> first#(X,Z) active#(first(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(first(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> add#(X,Y) active#(add(s(X),Y)) -> s#(add(X,Y)) -> s#(ok(X)) -> s#(X) active#(add(s(X),Y)) -> add#(X,Y) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) active#(add(s(X),Y)) -> add#(X,Y) -> add#(mark(X1),X2) -> add#(X1,X2) 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)) -> active#(X1) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(add(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> active#(X2) active#(add(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(add(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(add(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X1) -> active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) active#(add(X1,X2)) -> active#(X1) -> active#(if(X1,X2,X3)) -> active#(X1) active#(add(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> and#(active(X1),X2) active#(add(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X1) -> active#(from(X)) -> cons#(X,from(s(X))) active#(add(X1,X2)) -> active#(X1) -> active#(from(X)) -> from#(s(X)) active#(add(X1,X2)) -> active#(X1) -> active#(from(X)) -> s#(X) 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#(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#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) -> if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) -> if#(mark(X1),X2,X3) -> if#(X1,X2,X3) active#(if(X1,X2,X3)) -> active#(X1) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(if(X1,X2,X3)) -> active#(X1) -> active#(first(X1,X2)) -> active#(X2) active#(if(X1,X2,X3)) -> active#(X1) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(if(X1,X2,X3)) -> active#(X1) -> active#(first(X1,X2)) -> active#(X1) active#(if(X1,X2,X3)) -> active#(X1) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(if(X1,X2,X3)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X1) active#(if(X1,X2,X3)) -> active#(X1) -> active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) active#(if(X1,X2,X3)) -> active#(X1) -> active#(if(X1,X2,X3)) -> active#(X1) active#(if(X1,X2,X3)) -> active#(X1) -> active#(and(X1,X2)) -> and#(active(X1),X2) active#(if(X1,X2,X3)) -> active#(X1) -> active#(and(X1,X2)) -> active#(X1) active#(if(X1,X2,X3)) -> active#(X1) -> active#(from(X)) -> cons#(X,from(s(X))) active#(if(X1,X2,X3)) -> active#(X1) -> active#(from(X)) -> from#(s(X)) active#(if(X1,X2,X3)) -> active#(X1) -> active#(from(X)) -> s#(X) active#(if(X1,X2,X3)) -> active#(X1) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(if(X1,X2,X3)) -> active#(X1) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(if(X1,X2,X3)) -> active#(X1) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(if(X1,X2,X3)) -> active#(X1) -> active#(add(s(X),Y)) -> add#(X,Y) active#(and(X1,X2)) -> and#(active(X1),X2) -> and#(ok(X1),ok(X2)) -> and#(X1,X2) active#(and(X1,X2)) -> and#(active(X1),X2) -> and#(mark(X1),X2) -> and#(X1,X2) active#(and(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(and(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> active#(X2) active#(and(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(and(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(and(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> active#(X1) -> active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) active#(and(X1,X2)) -> active#(X1) -> active#(if(X1,X2,X3)) -> active#(X1) active#(and(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> and#(active(X1),X2) active#(and(X1,X2)) -> active#(X1) -> active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> active#(X1) -> active#(from(X)) -> cons#(X,from(s(X))) active#(and(X1,X2)) -> active#(X1) -> active#(from(X)) -> from#(s(X)) active#(and(X1,X2)) -> active#(X1) -> active#(from(X)) -> s#(X) active#(and(X1,X2)) -> active#(X1) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(and(X1,X2)) -> active#(X1) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(and(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(and(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> add#(X,Y) SCC Processor: #sccs: 10 #rules: 32 #arcs: 448/2809 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [top#](x0) = x0, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [from](x0) = 1, [cons](x0, x1) = 0, [nil] = 0, [first](x0, x1) = x0 + x1 + 1, [s](x0) = 0, [add](x0, x1) = x0 + x1 + 1, [0] = 0, [if](x0, x1, x2) = x0 + x1 + x2, [false] = 1, [mark](x0) = x0 + 1, [active](x0) = x0, [and](x0, x1) = x0 + x1 + 1, [true] = 1 orientation: top#(ok(X)) = X >= X = top#(active(X)) top#(mark(X)) = X + 1 >= X = top#(proper(X)) active(and(true(),X)) = X + 2 >= X + 1 = mark(X) active(and(false(),Y)) = Y + 2 >= 2 = mark(false()) active(if(true(),X,Y)) = X + Y + 1 >= X + 1 = mark(X) active(if(false(),X,Y)) = X + Y + 1 >= Y + 1 = mark(Y) active(add(0(),X)) = X + 1 >= X + 1 = mark(X) active(add(s(X),Y)) = Y + 1 >= 1 = mark(s(add(X,Y))) active(first(0(),X)) = X + 1 >= 1 = mark(nil()) active(first(s(X),cons(Y,Z))) = 1 >= 1 = mark(cons(Y,first(X,Z))) active(from(X)) = 1 >= 1 = mark(cons(X,from(s(X)))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(if(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = if(active(X1),X2,X3) active(add(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = add(active(X1),X2) active(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(active(X1),X2) active(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(X1,active(X2)) and(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(and(X1,X2)) if(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(if(X1,X2,X3)) add(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(add(X1,X2)) first(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(first(X1,X2)) proper(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(proper(X1),proper(X2)) proper(true()) = 1 >= 1 = ok(true()) proper(false()) = 1 >= 1 = ok(false()) proper(if(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = add(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(from(X)) = 1 >= 1 = from(proper(X)) and(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(add(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) first(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(first(X1,X2)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) from(ok(X)) = 1 >= 1 = ok(from(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: top#(ok(X)) -> top#(active(X)) TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [top#](x0) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [from](x0) = x0, [cons](x0, x1) = x1 + 1, [nil] = 0, [first](x0, x1) = x0 + 1, [s](x0) = x0 + 1, [add](x0, x1) = x1 + 1, [0] = 0, [if](x0, x1, x2) = x2 + 1, [false] = 1, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = x1, [true] = 1 orientation: top#(ok(X)) = X + 1 >= X = top#(active(X)) active(and(true(),X)) = X >= 0 = mark(X) active(and(false(),Y)) = Y >= 0 = mark(false()) active(if(true(),X,Y)) = Y + 1 >= 0 = mark(X) active(if(false(),X,Y)) = Y + 1 >= 0 = mark(Y) active(add(0(),X)) = X + 1 >= 0 = mark(X) active(add(s(X),Y)) = Y + 1 >= 0 = mark(s(add(X,Y))) active(first(0(),X)) = 1 >= 0 = mark(nil()) active(first(s(X),cons(Y,Z))) = X + 2 >= 0 = mark(cons(Y,first(X,Z))) active(from(X)) = X >= 0 = mark(cons(X,from(s(X)))) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(if(X1,X2,X3)) = X3 + 1 >= X3 + 1 = if(active(X1),X2,X3) active(add(X1,X2)) = X2 + 1 >= X2 + 1 = add(active(X1),X2) active(first(X1,X2)) = X1 + 1 >= X1 + 1 = first(active(X1),X2) active(first(X1,X2)) = X1 + 1 >= X1 + 1 = first(X1,active(X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) if(mark(X1),X2,X3) = X3 + 1 >= 0 = mark(if(X1,X2,X3)) add(mark(X1),X2) = X2 + 1 >= 0 = mark(add(X1,X2)) first(mark(X1),X2) = 1 >= 0 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 + 1 >= 0 = mark(first(X1,X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(true()) = 2 >= 2 = ok(true()) proper(false()) = 2 >= 2 = ok(false()) proper(if(X1,X2,X3)) = X3 + 2 >= X3 + 2 = if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) = X2 + 2 >= X2 + 2 = add(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = X + 2 >= X + 2 = s(proper(X)) proper(first(X1,X2)) = X1 + 2 >= X1 + 2 = first(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(cons(X1,X2)) = X2 + 2 >= X2 + 2 = cons(proper(X1),proper(X2)) proper(from(X)) = X + 1 >= X + 1 = from(proper(X)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 2 = ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(add(X1,X2)) s(ok(X)) = X + 2 >= X + 2 = ok(s(X)) first(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(first(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(cons(X1,X2)) from(ok(X)) = X + 1 >= X + 1 = ok(from(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: active#(and(X1,X2)) -> active#(X1) active#(if(X1,X2,X3)) -> active#(X1) active#(add(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X2) TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [from](x0) = 1, [cons](x0, x1) = 0, [nil] = 0, [first](x0, x1) = x0 + x1 + 1, [s](x0) = 0, [add](x0, x1) = x0 + x1 + 1, [0] = 0, [if](x0, x1, x2) = x0 + x1 + x2 + 1, [false] = 1, [mark](x0) = x0 + 1, [active](x0) = x0, [and](x0, x1) = x0 + x1 + 1, [true] = 0 orientation: active#(and(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(if(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = active#(X1) active#(add(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(first(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(first(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = active#(X2) active(and(true(),X)) = X + 1 >= X + 1 = mark(X) active(and(false(),Y)) = Y + 2 >= 2 = mark(false()) active(if(true(),X,Y)) = X + Y + 1 >= X + 1 = mark(X) active(if(false(),X,Y)) = X + Y + 2 >= Y + 1 = mark(Y) active(add(0(),X)) = X + 1 >= X + 1 = mark(X) active(add(s(X),Y)) = Y + 1 >= 1 = mark(s(add(X,Y))) active(first(0(),X)) = X + 1 >= 1 = mark(nil()) active(first(s(X),cons(Y,Z))) = 1 >= 1 = mark(cons(Y,first(X,Z))) active(from(X)) = 1 >= 1 = mark(cons(X,from(s(X)))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(active(X1),X2,X3) active(add(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = add(active(X1),X2) active(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(active(X1),X2) active(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(X1,active(X2)) and(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(and(X1,X2)) if(mark(X1),X2,X3) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = mark(if(X1,X2,X3)) add(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(add(X1,X2)) first(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(first(X1,X2)) proper(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(proper(X1),proper(X2)) proper(true()) = 0 >= 0 = ok(true()) proper(false()) = 1 >= 0 = ok(false()) proper(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = add(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(from(X)) = 1 >= 1 = from(proper(X)) and(ok(X1),ok(X2)) = 1 >= 0 = ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) = 1 >= 0 = ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) = 1 >= 0 = ok(add(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) first(ok(X1),ok(X2)) = 1 >= 0 = ok(first(X1,X2)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) from(ok(X)) = 1 >= 0 = ok(from(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(if(X1,X2,X3)) -> proper#(X3) proper#(if(X1,X2,X3)) -> proper#(X2) proper#(if(X1,X2,X3)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) proper#(first(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(from(X)) -> proper#(X) TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [from](x0) = x0 + 1, [cons](x0, x1) = x0 + x1 + 1, [nil] = 0, [first](x0, x1) = x0 + x1 + 1, [s](x0) = x0 + 1, [add](x0, x1) = x0 + x1 + 1, [0] = 0, [if](x0, x1, x2) = x0 + x1 + x2 + 1, [false] = 0, [mark](x0) = 1, [active](x0) = x0, [and](x0, x1) = x0 + x1 + 1, [true] = 0 orientation: proper#(and(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(and(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(if(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X3 + 1 = proper#(X3) proper#(if(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X2 + 1 = proper#(X2) proper#(if(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = proper#(X1) proper#(add(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(add(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(s(X)) = X + 2 >= X + 1 = proper#(X) proper#(first(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(first(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(from(X)) = X + 2 >= X + 1 = proper#(X) active(and(true(),X)) = X + 1 >= 1 = mark(X) active(and(false(),Y)) = Y + 1 >= 1 = mark(false()) active(if(true(),X,Y)) = X + Y + 1 >= 1 = mark(X) active(if(false(),X,Y)) = X + Y + 1 >= 1 = mark(Y) active(add(0(),X)) = X + 1 >= 1 = mark(X) active(add(s(X),Y)) = X + Y + 2 >= 1 = mark(s(add(X,Y))) active(first(0(),X)) = X + 1 >= 1 = mark(nil()) active(first(s(X),cons(Y,Z))) = X + Y + Z + 3 >= 1 = mark(cons(Y,first(X,Z))) active(from(X)) = X + 1 >= 1 = mark(cons(X,from(s(X)))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(active(X1),X2,X3) active(add(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = add(active(X1),X2) active(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(active(X1),X2) active(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(X1,active(X2)) and(mark(X1),X2) = X2 + 2 >= 1 = mark(and(X1,X2)) if(mark(X1),X2,X3) = X2 + X3 + 2 >= 1 = mark(if(X1,X2,X3)) add(mark(X1),X2) = X2 + 2 >= 1 = mark(add(X1,X2)) first(mark(X1),X2) = X2 + 2 >= 1 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 + 2 >= 1 = mark(first(X1,X2)) proper(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(proper(X1),proper(X2)) proper(true()) = 0 >= 0 = ok(true()) proper(false()) = 0 >= 0 = ok(false()) proper(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = add(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(from(X)) = X + 1 >= X + 1 = from(proper(X)) and(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(add(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) first(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(first(X1,X2)) cons(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(cons(X1,X2)) from(ok(X)) = X + 1 >= X + 1 = ok(from(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: from#(ok(X)) -> from#(X) TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [from#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [from](x0) = x0, [cons](x0, x1) = x1 + 1, [nil] = 1, [first](x0, x1) = x0, [s](x0) = x0, [add](x0, x1) = x1 + 1, [0] = 0, [if](x0, x1, x2) = x2, [false] = 0, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = x0, [true] = 1 orientation: from#(ok(X)) = X + 2 >= X + 1 = from#(X) active(and(true(),X)) = 1 >= 0 = mark(X) active(and(false(),Y)) = 0 >= 0 = mark(false()) active(if(true(),X,Y)) = Y >= 0 = mark(X) active(if(false(),X,Y)) = Y >= 0 = mark(Y) active(add(0(),X)) = X + 1 >= 0 = mark(X) active(add(s(X),Y)) = Y + 1 >= 0 = mark(s(add(X,Y))) active(first(0(),X)) = 0 >= 0 = mark(nil()) active(first(s(X),cons(Y,Z))) = X >= 0 = mark(cons(Y,first(X,Z))) active(from(X)) = X >= 0 = mark(cons(X,from(s(X)))) active(and(X1,X2)) = X1 >= X1 = and(active(X1),X2) active(if(X1,X2,X3)) = X3 >= X3 = if(active(X1),X2,X3) active(add(X1,X2)) = X2 + 1 >= X2 + 1 = add(active(X1),X2) active(first(X1,X2)) = X1 >= X1 = first(active(X1),X2) active(first(X1,X2)) = X1 >= X1 = first(X1,active(X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) if(mark(X1),X2,X3) = X3 >= 0 = mark(if(X1,X2,X3)) add(mark(X1),X2) = X2 + 1 >= 0 = mark(add(X1,X2)) first(mark(X1),X2) = 0 >= 0 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 >= 0 = mark(first(X1,X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(true()) = 2 >= 2 = ok(true()) proper(false()) = 1 >= 1 = ok(false()) proper(if(X1,X2,X3)) = X3 + 1 >= X3 + 1 = if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) = X2 + 2 >= X2 + 2 = add(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(first(X1,X2)) = X1 + 1 >= X1 + 1 = first(proper(X1),proper(X2)) proper(nil()) = 2 >= 2 = ok(nil()) proper(cons(X1,X2)) = X2 + 2 >= X2 + 2 = cons(proper(X1),proper(X2)) proper(from(X)) = X + 1 >= X + 1 = from(proper(X)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(add(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) first(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(first(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(cons(X1,X2)) from(ok(X)) = X + 1 >= X + 1 = ok(from(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: cons#(ok(X1),ok(X2)) -> cons#(X1,X2) TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [from](x0) = x0, [cons](x0, x1) = x1, [nil] = 0, [first](x0, x1) = x1, [s](x0) = x0, [add](x0, x1) = x1, [0] = 0, [if](x0, x1, x2) = x1, [false] = 0, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = x1, [true] = 0 orientation: cons#(ok(X1),ok(X2)) = X2 + 1 >= X2 = cons#(X1,X2) active(and(true(),X)) = X >= 0 = mark(X) active(and(false(),Y)) = Y >= 0 = mark(false()) active(if(true(),X,Y)) = X >= 0 = mark(X) active(if(false(),X,Y)) = X >= 0 = mark(Y) active(add(0(),X)) = X >= 0 = mark(X) active(add(s(X),Y)) = Y >= 0 = mark(s(add(X,Y))) active(first(0(),X)) = X >= 0 = mark(nil()) active(first(s(X),cons(Y,Z))) = Z >= 0 = mark(cons(Y,first(X,Z))) active(from(X)) = X >= 0 = mark(cons(X,from(s(X)))) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(if(X1,X2,X3)) = X2 >= X2 = if(active(X1),X2,X3) active(add(X1,X2)) = X2 >= X2 = add(active(X1),X2) active(first(X1,X2)) = X2 >= X2 = first(active(X1),X2) active(first(X1,X2)) = X2 >= X2 = first(X1,active(X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) if(mark(X1),X2,X3) = X2 >= 0 = mark(if(X1,X2,X3)) add(mark(X1),X2) = X2 >= 0 = mark(add(X1,X2)) first(mark(X1),X2) = X2 >= 0 = mark(first(X1,X2)) first(X1,mark(X2)) = 0 >= 0 = mark(first(X1,X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(true()) = 1 >= 1 = ok(true()) proper(false()) = 1 >= 1 = ok(false()) proper(if(X1,X2,X3)) = 1 >= 1 = if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) = 1 >= 1 = add(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(first(X1,X2)) = 1 >= 1 = first(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(from(X)) = 1 >= 1 = from(proper(X)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 + 1 = ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(add(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) first(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(first(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) from(ok(X)) = X + 1 >= X + 1 = ok(from(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: first#(mark(X1),X2) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) first#(ok(X1),ok(X2)) -> first#(X1,X2) TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [first#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [from](x0) = 1, [cons](x0, x1) = 0, [nil] = 0, [first](x0, x1) = x0 + x1, [s](x0) = 0, [add](x0, x1) = x0 + x1, [0] = 0, [if](x0, x1, x2) = x0 + x1 + x2, [false] = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [true] = 1 orientation: first#(mark(X1),X2) = X1 + 1 >= X1 = first#(X1,X2) first#(X1,mark(X2)) = X1 >= X1 = first#(X1,X2) first#(ok(X1),ok(X2)) = X1 >= X1 = first#(X1,X2) active(and(true(),X)) = X + 2 >= X + 1 = mark(X) active(and(false(),Y)) = Y + 1 >= 1 = mark(false()) active(if(true(),X,Y)) = X + Y + 2 >= X + 1 = mark(X) active(if(false(),X,Y)) = X + Y + 1 >= Y + 1 = mark(Y) active(add(0(),X)) = X + 1 >= X + 1 = mark(X) active(add(s(X),Y)) = Y + 1 >= 1 = mark(s(add(X,Y))) active(first(0(),X)) = X + 1 >= 1 = mark(nil()) active(first(s(X),cons(Y,Z))) = 1 >= 1 = mark(cons(Y,first(X,Z))) active(from(X)) = 2 >= 1 = mark(cons(X,from(s(X)))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(active(X1),X2,X3) active(add(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = add(active(X1),X2) active(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(active(X1),X2) active(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(X1,active(X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) if(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(if(X1,X2,X3)) add(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(add(X1,X2)) first(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(first(X1,X2)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(true()) = 1 >= 1 = ok(true()) proper(false()) = 0 >= 0 = ok(false()) proper(if(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) = X1 + X2 >= X1 + X2 = add(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(first(X1,X2)) = X1 + X2 >= X1 + X2 = first(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(from(X)) = 1 >= 1 = from(proper(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(add(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) first(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(first(X1,X2)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) from(ok(X)) = 1 >= 1 = ok(from(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: first#(X1,mark(X2)) -> first#(X1,X2) first#(ok(X1),ok(X2)) -> first#(X1,X2) TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [first#](x0, x1) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [from](x0) = x0, [cons](x0, x1) = x0, [nil] = 0, [first](x0, x1) = x0, [s](x0) = x0, [add](x0, x1) = x0, [0] = 1, [if](x0, x1, x2) = x0 + 1, [false] = 0, [mark](x0) = 0, [active](x0) = x0 + 1, [and](x0, x1) = x1, [true] = 0 orientation: first#(X1,mark(X2)) = X1 + 1 >= X1 + 1 = first#(X1,X2) first#(ok(X1),ok(X2)) = X1 + 2 >= X1 + 1 = first#(X1,X2) active(and(true(),X)) = X + 1 >= 0 = mark(X) active(and(false(),Y)) = Y + 1 >= 0 = mark(false()) active(if(true(),X,Y)) = 2 >= 0 = mark(X) active(if(false(),X,Y)) = 2 >= 0 = mark(Y) active(add(0(),X)) = 2 >= 0 = mark(X) active(add(s(X),Y)) = X + 1 >= 0 = mark(s(add(X,Y))) active(first(0(),X)) = 2 >= 0 = mark(nil()) active(first(s(X),cons(Y,Z))) = X + 1 >= 0 = mark(cons(Y,first(X,Z))) active(from(X)) = X + 1 >= 0 = mark(cons(X,from(s(X)))) active(and(X1,X2)) = X2 + 1 >= X2 = and(active(X1),X2) active(if(X1,X2,X3)) = X1 + 2 >= X1 + 2 = if(active(X1),X2,X3) active(add(X1,X2)) = X1 + 1 >= X1 + 1 = add(active(X1),X2) active(first(X1,X2)) = X1 + 1 >= X1 + 1 = first(active(X1),X2) active(first(X1,X2)) = X1 + 1 >= X1 = first(X1,active(X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) if(mark(X1),X2,X3) = 1 >= 0 = mark(if(X1,X2,X3)) add(mark(X1),X2) = 0 >= 0 = mark(add(X1,X2)) first(mark(X1),X2) = 0 >= 0 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 >= 0 = mark(first(X1,X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(true()) = 1 >= 1 = ok(true()) proper(false()) = 1 >= 1 = ok(false()) proper(if(X1,X2,X3)) = X1 + 2 >= X1 + 2 = if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) = X1 + 1 >= X1 + 1 = add(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(first(X1,X2)) = X1 + 1 >= X1 + 1 = first(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(from(X)) = X + 1 >= X + 1 = from(proper(X)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) = X1 + 2 >= X1 + 2 = ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(add(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) first(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(first(X1,X2)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) from(ok(X)) = X + 1 >= X + 1 = ok(from(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: first#(X1,mark(X2)) -> first#(X1,X2) TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [first#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [from](x0) = 0, [cons](x0, x1) = 0, [nil] = 0, [first](x0, x1) = x0 + x1, [s](x0) = 1, [add](x0, x1) = x0 + x1 + 1, [0] = 1, [if](x0, x1, x2) = x0 + x1 + x2, [false] = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [true] = 0 orientation: first#(X1,mark(X2)) = X2 + 2 >= X2 + 1 = first#(X1,X2) active(and(true(),X)) = X + 1 >= X + 1 = mark(X) active(and(false(),Y)) = Y + 1 >= 1 = mark(false()) active(if(true(),X,Y)) = X + Y + 1 >= X + 1 = mark(X) active(if(false(),X,Y)) = X + Y + 1 >= Y + 1 = mark(Y) active(add(0(),X)) = X + 3 >= X + 1 = mark(X) active(add(s(X),Y)) = Y + 3 >= 2 = mark(s(add(X,Y))) active(first(0(),X)) = X + 2 >= 1 = mark(nil()) active(first(s(X),cons(Y,Z))) = 2 >= 1 = mark(cons(Y,first(X,Z))) active(from(X)) = 1 >= 1 = mark(cons(X,from(s(X)))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(active(X1),X2,X3) active(add(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = add(active(X1),X2) active(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(active(X1),X2) active(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(X1,active(X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) if(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(if(X1,X2,X3)) add(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(add(X1,X2)) first(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(first(X1,X2)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(true()) = 0 >= 0 = ok(true()) proper(false()) = 0 >= 0 = ok(false()) proper(if(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = add(proper(X1),proper(X2)) proper(0()) = 1 >= 0 = ok(0()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(first(X1,X2)) = X1 + X2 >= X1 + X2 = first(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(from(X)) = 0 >= 0 = from(proper(X)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) = 1 >= 0 = ok(add(X1,X2)) s(ok(X)) = 1 >= 0 = ok(s(X)) first(ok(X1),ok(X2)) = 0 >= 0 = ok(first(X1,X2)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) from(ok(X)) = 0 >= 0 = ok(from(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: s#(ok(X)) -> s#(X) TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [from](x0) = x0, [cons](x0, x1) = x1 + 1, [nil] = 1, [first](x0, x1) = x0, [s](x0) = x0, [add](x0, x1) = x1 + 1, [0] = 0, [if](x0, x1, x2) = x2, [false] = 0, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = x0, [true] = 1 orientation: s#(ok(X)) = X + 2 >= X + 1 = s#(X) active(and(true(),X)) = 1 >= 0 = mark(X) active(and(false(),Y)) = 0 >= 0 = mark(false()) active(if(true(),X,Y)) = Y >= 0 = mark(X) active(if(false(),X,Y)) = Y >= 0 = mark(Y) active(add(0(),X)) = X + 1 >= 0 = mark(X) active(add(s(X),Y)) = Y + 1 >= 0 = mark(s(add(X,Y))) active(first(0(),X)) = 0 >= 0 = mark(nil()) active(first(s(X),cons(Y,Z))) = X >= 0 = mark(cons(Y,first(X,Z))) active(from(X)) = X >= 0 = mark(cons(X,from(s(X)))) active(and(X1,X2)) = X1 >= X1 = and(active(X1),X2) active(if(X1,X2,X3)) = X3 >= X3 = if(active(X1),X2,X3) active(add(X1,X2)) = X2 + 1 >= X2 + 1 = add(active(X1),X2) active(first(X1,X2)) = X1 >= X1 = first(active(X1),X2) active(first(X1,X2)) = X1 >= X1 = first(X1,active(X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) if(mark(X1),X2,X3) = X3 >= 0 = mark(if(X1,X2,X3)) add(mark(X1),X2) = X2 + 1 >= 0 = mark(add(X1,X2)) first(mark(X1),X2) = 0 >= 0 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 >= 0 = mark(first(X1,X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(true()) = 2 >= 2 = ok(true()) proper(false()) = 1 >= 1 = ok(false()) proper(if(X1,X2,X3)) = X3 + 1 >= X3 + 1 = if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) = X2 + 2 >= X2 + 2 = add(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(first(X1,X2)) = X1 + 1 >= X1 + 1 = first(proper(X1),proper(X2)) proper(nil()) = 2 >= 2 = ok(nil()) proper(cons(X1,X2)) = X2 + 2 >= X2 + 2 = cons(proper(X1),proper(X2)) proper(from(X)) = X + 1 >= X + 1 = from(proper(X)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(add(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) first(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(first(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(cons(X1,X2)) from(ok(X)) = X + 1 >= X + 1 = ok(from(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: add#(mark(X1),X2) -> add#(X1,X2) add#(ok(X1),ok(X2)) -> add#(X1,X2) TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [add#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [from](x0) = x0 + 1, [cons](x0, x1) = x1, [nil] = 1, [first](x0, x1) = x0, [s](x0) = x0, [add](x0, x1) = x0, [0] = 0, [if](x0, x1, x2) = x2, [false] = 0, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = x0, [true] = 0 orientation: add#(mark(X1),X2) = X2 + 1 >= X2 + 1 = add#(X1,X2) add#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = add#(X1,X2) active(and(true(),X)) = 0 >= 0 = mark(X) active(and(false(),Y)) = 0 >= 0 = mark(false()) active(if(true(),X,Y)) = Y >= 0 = mark(X) active(if(false(),X,Y)) = Y >= 0 = mark(Y) active(add(0(),X)) = 0 >= 0 = mark(X) active(add(s(X),Y)) = X >= 0 = mark(s(add(X,Y))) active(first(0(),X)) = 0 >= 0 = mark(nil()) active(first(s(X),cons(Y,Z))) = X >= 0 = mark(cons(Y,first(X,Z))) active(from(X)) = X + 1 >= 0 = mark(cons(X,from(s(X)))) active(and(X1,X2)) = X1 >= X1 = and(active(X1),X2) active(if(X1,X2,X3)) = X3 >= X3 = if(active(X1),X2,X3) active(add(X1,X2)) = X1 >= X1 = add(active(X1),X2) active(first(X1,X2)) = X1 >= X1 = first(active(X1),X2) active(first(X1,X2)) = X1 >= X1 = first(X1,active(X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) if(mark(X1),X2,X3) = X3 >= 0 = mark(if(X1,X2,X3)) add(mark(X1),X2) = 0 >= 0 = mark(add(X1,X2)) first(mark(X1),X2) = 0 >= 0 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 >= 0 = mark(first(X1,X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(true()) = 1 >= 1 = ok(true()) proper(false()) = 1 >= 1 = ok(false()) proper(if(X1,X2,X3)) = X3 + 1 >= X3 + 1 = if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) = X1 + 1 >= X1 + 1 = add(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(first(X1,X2)) = X1 + 1 >= X1 + 1 = first(proper(X1),proper(X2)) proper(nil()) = 2 >= 2 = ok(nil()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(from(X)) = X + 2 >= X + 2 = from(proper(X)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(add(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) first(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(first(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) from(ok(X)) = X + 2 >= X + 2 = ok(from(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: add#(mark(X1),X2) -> add#(X1,X2) TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [add#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [from](x0) = 0, [cons](x0, x1) = 0, [nil] = 0, [first](x0, x1) = x0 + x1, [s](x0) = 1, [add](x0, x1) = x0 + x1 + 1, [0] = 1, [if](x0, x1, x2) = x0 + x1 + x2, [false] = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [true] = 0 orientation: add#(mark(X1),X2) = X1 + 2 >= X1 + 1 = add#(X1,X2) active(and(true(),X)) = X + 1 >= X + 1 = mark(X) active(and(false(),Y)) = Y + 1 >= 1 = mark(false()) active(if(true(),X,Y)) = X + Y + 1 >= X + 1 = mark(X) active(if(false(),X,Y)) = X + Y + 1 >= Y + 1 = mark(Y) active(add(0(),X)) = X + 3 >= X + 1 = mark(X) active(add(s(X),Y)) = Y + 3 >= 2 = mark(s(add(X,Y))) active(first(0(),X)) = X + 2 >= 1 = mark(nil()) active(first(s(X),cons(Y,Z))) = 2 >= 1 = mark(cons(Y,first(X,Z))) active(from(X)) = 1 >= 1 = mark(cons(X,from(s(X)))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(active(X1),X2,X3) active(add(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = add(active(X1),X2) active(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(active(X1),X2) active(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(X1,active(X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) if(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(if(X1,X2,X3)) add(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(add(X1,X2)) first(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(first(X1,X2)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(true()) = 0 >= 0 = ok(true()) proper(false()) = 0 >= 0 = ok(false()) proper(if(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = add(proper(X1),proper(X2)) proper(0()) = 1 >= 0 = ok(0()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(first(X1,X2)) = X1 + X2 >= X1 + X2 = first(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(from(X)) = 0 >= 0 = from(proper(X)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) = 1 >= 0 = ok(add(X1,X2)) s(ok(X)) = 1 >= 0 = ok(s(X)) first(ok(X1),ok(X2)) = 0 >= 0 = ok(first(X1,X2)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) from(ok(X)) = 0 >= 0 = ok(from(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: if#(mark(X1),X2,X3) -> if#(X1,X2,X3) if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [if#](x0, x1, x2) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [from](x0) = 0, [cons](x0, x1) = 0, [nil] = 0, [first](x0, x1) = x0 + x1 + 1, [s](x0) = 0, [add](x0, x1) = x0 + x1, [0] = 1, [if](x0, x1, x2) = x0 + x1 + x2 + 1, [false] = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [true] = 1 orientation: if#(mark(X1),X2,X3) = X1 + 2 >= X1 + 1 = if#(X1,X2,X3) if#(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = if#(X1,X2,X3) active(and(true(),X)) = X + 2 >= X + 1 = mark(X) active(and(false(),Y)) = Y + 1 >= 1 = mark(false()) active(if(true(),X,Y)) = X + Y + 3 >= X + 1 = mark(X) active(if(false(),X,Y)) = X + Y + 2 >= Y + 1 = mark(Y) active(add(0(),X)) = X + 2 >= X + 1 = mark(X) active(add(s(X),Y)) = Y + 1 >= 1 = mark(s(add(X,Y))) active(first(0(),X)) = X + 3 >= 1 = mark(nil()) active(first(s(X),cons(Y,Z))) = 2 >= 1 = mark(cons(Y,first(X,Z))) active(from(X)) = 1 >= 1 = mark(cons(X,from(s(X)))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(if(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = if(active(X1),X2,X3) active(add(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = add(active(X1),X2) active(first(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = first(active(X1),X2) active(first(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = first(X1,active(X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) if(mark(X1),X2,X3) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = mark(if(X1,X2,X3)) add(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(add(X1,X2)) first(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(first(X1,X2)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(true()) = 1 >= 1 = ok(true()) proper(false()) = 0 >= 0 = ok(false()) proper(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) = X1 + X2 >= X1 + X2 = add(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(from(X)) = 0 >= 0 = from(proper(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(add(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) first(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(first(X1,X2)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) from(ok(X)) = 0 >= 0 = ok(from(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [if#](x0, x1, x2) = x2, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [from](x0) = x0, [cons](x0, x1) = x0, [nil] = 1, [first](x0, x1) = x1, [s](x0) = x0, [add](x0, x1) = x1, [0] = 0, [if](x0, x1, x2) = x0 + 1, [false] = 1, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = x1, [true] = 0 orientation: if#(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 = if#(X1,X2,X3) active(and(true(),X)) = X >= 0 = mark(X) active(and(false(),Y)) = Y >= 0 = mark(false()) active(if(true(),X,Y)) = 1 >= 0 = mark(X) active(if(false(),X,Y)) = 2 >= 0 = mark(Y) active(add(0(),X)) = X >= 0 = mark(X) active(add(s(X),Y)) = Y >= 0 = mark(s(add(X,Y))) active(first(0(),X)) = X >= 0 = mark(nil()) active(first(s(X),cons(Y,Z))) = Y >= 0 = mark(cons(Y,first(X,Z))) active(from(X)) = X >= 0 = mark(cons(X,from(s(X)))) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(if(X1,X2,X3)) = X1 + 1 >= X1 + 1 = if(active(X1),X2,X3) active(add(X1,X2)) = X2 >= X2 = add(active(X1),X2) active(first(X1,X2)) = X2 >= X2 = first(active(X1),X2) active(first(X1,X2)) = X2 >= X2 = first(X1,active(X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) if(mark(X1),X2,X3) = 1 >= 0 = mark(if(X1,X2,X3)) add(mark(X1),X2) = X2 >= 0 = mark(add(X1,X2)) first(mark(X1),X2) = X2 >= 0 = mark(first(X1,X2)) first(X1,mark(X2)) = 0 >= 0 = mark(first(X1,X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(true()) = 1 >= 1 = ok(true()) proper(false()) = 2 >= 2 = ok(false()) proper(if(X1,X2,X3)) = X1 + 2 >= X1 + 2 = if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) = X2 + 1 >= X2 + 1 = add(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(first(X1,X2)) = X2 + 1 >= X2 + 1 = first(proper(X1),proper(X2)) proper(nil()) = 2 >= 2 = ok(nil()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(from(X)) = X + 1 >= X + 1 = from(proper(X)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) = X1 + 2 >= X1 + 2 = ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(add(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) first(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(first(X1,X2)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) from(ok(X)) = X + 1 >= X + 1 = ok(from(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: and#(mark(X1),X2) -> and#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [from](x0) = x0 + 1, [cons](x0, x1) = x1, [nil] = 1, [first](x0, x1) = x0, [s](x0) = x0, [add](x0, x1) = x0, [0] = 0, [if](x0, x1, x2) = x2, [false] = 0, [mark](x0) = 0, [active](x0) = x0, [and](x0, x1) = x0, [true] = 0 orientation: and#(mark(X1),X2) = X2 + 1 >= X2 + 1 = and#(X1,X2) and#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = and#(X1,X2) active(and(true(),X)) = 0 >= 0 = mark(X) active(and(false(),Y)) = 0 >= 0 = mark(false()) active(if(true(),X,Y)) = Y >= 0 = mark(X) active(if(false(),X,Y)) = Y >= 0 = mark(Y) active(add(0(),X)) = 0 >= 0 = mark(X) active(add(s(X),Y)) = X >= 0 = mark(s(add(X,Y))) active(first(0(),X)) = 0 >= 0 = mark(nil()) active(first(s(X),cons(Y,Z))) = X >= 0 = mark(cons(Y,first(X,Z))) active(from(X)) = X + 1 >= 0 = mark(cons(X,from(s(X)))) active(and(X1,X2)) = X1 >= X1 = and(active(X1),X2) active(if(X1,X2,X3)) = X3 >= X3 = if(active(X1),X2,X3) active(add(X1,X2)) = X1 >= X1 = add(active(X1),X2) active(first(X1,X2)) = X1 >= X1 = first(active(X1),X2) active(first(X1,X2)) = X1 >= X1 = first(X1,active(X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) if(mark(X1),X2,X3) = X3 >= 0 = mark(if(X1,X2,X3)) add(mark(X1),X2) = 0 >= 0 = mark(add(X1,X2)) first(mark(X1),X2) = 0 >= 0 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 >= 0 = mark(first(X1,X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(true()) = 1 >= 1 = ok(true()) proper(false()) = 1 >= 1 = ok(false()) proper(if(X1,X2,X3)) = X3 + 1 >= X3 + 1 = if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) = X1 + 1 >= X1 + 1 = add(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(first(X1,X2)) = X1 + 1 >= X1 + 1 = first(proper(X1),proper(X2)) proper(nil()) = 2 >= 2 = ok(nil()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(from(X)) = X + 2 >= X + 2 = from(proper(X)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(add(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) first(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(first(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) from(ok(X)) = X + 2 >= X + 2 = ok(from(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: and#(mark(X1),X2) -> and#(X1,X2) TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [from](x0) = 0, [cons](x0, x1) = 0, [nil] = 0, [first](x0, x1) = x0 + x1, [s](x0) = 1, [add](x0, x1) = x0 + x1 + 1, [0] = 1, [if](x0, x1, x2) = x0 + x1 + x2, [false] = 0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [true] = 0 orientation: and#(mark(X1),X2) = X1 + 2 >= X1 + 1 = and#(X1,X2) active(and(true(),X)) = X + 1 >= X + 1 = mark(X) active(and(false(),Y)) = Y + 1 >= 1 = mark(false()) active(if(true(),X,Y)) = X + Y + 1 >= X + 1 = mark(X) active(if(false(),X,Y)) = X + Y + 1 >= Y + 1 = mark(Y) active(add(0(),X)) = X + 3 >= X + 1 = mark(X) active(add(s(X),Y)) = Y + 3 >= 2 = mark(s(add(X,Y))) active(first(0(),X)) = X + 2 >= 1 = mark(nil()) active(first(s(X),cons(Y,Z))) = 2 >= 1 = mark(cons(Y,first(X,Z))) active(from(X)) = 1 >= 1 = mark(cons(X,from(s(X)))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(active(X1),X2,X3) active(add(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = add(active(X1),X2) active(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(active(X1),X2) active(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = first(X1,active(X2)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) if(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(if(X1,X2,X3)) add(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(add(X1,X2)) first(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(first(X1,X2)) first(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(first(X1,X2)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(true()) = 0 >= 0 = ok(true()) proper(false()) = 0 >= 0 = ok(false()) proper(if(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = add(proper(X1),proper(X2)) proper(0()) = 1 >= 0 = ok(0()) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(first(X1,X2)) = X1 + X2 >= X1 + X2 = first(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(from(X)) = 0 >= 0 = from(proper(X)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) = 1 >= 0 = ok(add(X1,X2)) s(ok(X)) = 1 >= 0 = ok(s(X)) first(ok(X1),ok(X2)) = 0 >= 0 = ok(first(X1,X2)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) from(ok(X)) = 0 >= 0 = ok(from(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) active(and(X1,X2)) -> and(active(X1),X2) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(add(X1,X2)) -> add(active(X1),X2) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) add(mark(X1),X2) -> mark(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(true()) -> ok(true()) proper(false()) -> ok(false()) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) s(ok(X)) -> ok(s(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) from(ok(X)) -> ok(from(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed