* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: 2ndsneg(X1,mark(X2)) -> mark(2ndsneg(X1,X2)) 2ndsneg(mark(X1),X2) -> mark(2ndsneg(X1,X2)) 2ndsneg(ok(X1),ok(X2)) -> ok(2ndsneg(X1,X2)) 2ndspos(X1,mark(X2)) -> mark(2ndspos(X1,X2)) 2ndspos(mark(X1),X2) -> mark(2ndspos(X1,X2)) 2ndspos(ok(X1),ok(X2)) -> ok(2ndspos(X1,X2)) active(2ndsneg(X1,X2)) -> 2ndsneg(X1,active(X2)) active(2ndsneg(X1,X2)) -> 2ndsneg(active(X1),X2) active(2ndsneg(0(),Z)) -> mark(rnil()) active(2ndsneg(s(N),cons(X,Z))) -> mark(2ndsneg(s(N),cons2(X,Z))) active(2ndsneg(s(N),cons2(X,cons(Y,Z)))) -> mark(rcons(negrecip(Y),2ndspos(N,Z))) active(2ndspos(X1,X2)) -> 2ndspos(X1,active(X2)) active(2ndspos(X1,X2)) -> 2ndspos(active(X1),X2) active(2ndspos(0(),Z)) -> mark(rnil()) active(2ndspos(s(N),cons(X,Z))) -> mark(2ndspos(s(N),cons2(X,Z))) active(2ndspos(s(N),cons2(X,cons(Y,Z)))) -> mark(rcons(posrecip(Y),2ndsneg(N,Z))) active(cons(X1,X2)) -> cons(active(X1),X2) active(cons2(X1,X2)) -> cons2(X1,active(X2)) active(from(X)) -> from(active(X)) active(from(X)) -> mark(cons(X,from(s(X)))) active(negrecip(X)) -> negrecip(active(X)) active(pi(X)) -> mark(2ndspos(X,from(0()))) active(pi(X)) -> pi(active(X)) active(plus(X1,X2)) -> plus(X1,active(X2)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(0(),Y)) -> mark(Y) active(plus(s(X),Y)) -> mark(s(plus(X,Y))) active(posrecip(X)) -> posrecip(active(X)) active(rcons(X1,X2)) -> rcons(X1,active(X2)) active(rcons(X1,X2)) -> rcons(active(X1),X2) active(s(X)) -> s(active(X)) active(square(X)) -> mark(times(X,X)) active(square(X)) -> square(active(X)) active(times(X1,X2)) -> times(X1,active(X2)) active(times(X1,X2)) -> times(active(X1),X2) active(times(0(),Y)) -> mark(0()) active(times(s(X),Y)) -> mark(plus(Y,times(X,Y))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) cons2(X1,mark(X2)) -> mark(cons2(X1,X2)) cons2(ok(X1),ok(X2)) -> ok(cons2(X1,X2)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) negrecip(mark(X)) -> mark(negrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) pi(mark(X)) -> mark(pi(X)) pi(ok(X)) -> ok(pi(X)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) posrecip(mark(X)) -> mark(posrecip(X)) posrecip(ok(X)) -> ok(posrecip(X)) proper(0()) -> ok(0()) proper(2ndsneg(X1,X2)) -> 2ndsneg(proper(X1),proper(X2)) proper(2ndspos(X1,X2)) -> 2ndspos(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(cons2(X1,X2)) -> cons2(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil()) -> ok(nil()) proper(pi(X)) -> pi(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(posrecip(X)) -> posrecip(proper(X)) proper(rcons(X1,X2)) -> rcons(proper(X1),proper(X2)) proper(rnil()) -> ok(rnil()) proper(s(X)) -> s(proper(X)) proper(square(X)) -> square(proper(X)) proper(times(X1,X2)) -> times(proper(X1),proper(X2)) rcons(X1,mark(X2)) -> mark(rcons(X1,X2)) rcons(mark(X1),X2) -> mark(rcons(X1,X2)) rcons(ok(X1),ok(X2)) -> ok(rcons(X1,X2)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) square(mark(X)) -> mark(square(X)) square(ok(X)) -> ok(square(X)) times(X1,mark(X2)) -> mark(times(X1,X2)) times(mark(X1),X2) -> mark(times(X1,X2)) times(ok(X1),ok(X2)) -> ok(times(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {2ndsneg/2,2ndspos/2,active/1,cons/2,cons2/2,from/1,negrecip/1,pi/1,plus/2,posrecip/1,proper/1,rcons/2,s/1 ,square/1,times/2,top/1} / {0/0,mark/1,nil/0,ok/1,rnil/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,active,cons,cons2,from,negrecip,pi,plus ,posrecip,proper,rcons,s,square,times,top} and constructors {0,mark,nil,ok,rnil} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: 2ndsneg(X1,mark(X2)) -> mark(2ndsneg(X1,X2)) 2ndsneg(mark(X1),X2) -> mark(2ndsneg(X1,X2)) 2ndsneg(ok(X1),ok(X2)) -> ok(2ndsneg(X1,X2)) 2ndspos(X1,mark(X2)) -> mark(2ndspos(X1,X2)) 2ndspos(mark(X1),X2) -> mark(2ndspos(X1,X2)) 2ndspos(ok(X1),ok(X2)) -> ok(2ndspos(X1,X2)) active(2ndsneg(X1,X2)) -> 2ndsneg(X1,active(X2)) active(2ndsneg(X1,X2)) -> 2ndsneg(active(X1),X2) active(2ndsneg(0(),Z)) -> mark(rnil()) active(2ndsneg(s(N),cons(X,Z))) -> mark(2ndsneg(s(N),cons2(X,Z))) active(2ndsneg(s(N),cons2(X,cons(Y,Z)))) -> mark(rcons(negrecip(Y),2ndspos(N,Z))) active(2ndspos(X1,X2)) -> 2ndspos(X1,active(X2)) active(2ndspos(X1,X2)) -> 2ndspos(active(X1),X2) active(2ndspos(0(),Z)) -> mark(rnil()) active(2ndspos(s(N),cons(X,Z))) -> mark(2ndspos(s(N),cons2(X,Z))) active(2ndspos(s(N),cons2(X,cons(Y,Z)))) -> mark(rcons(posrecip(Y),2ndsneg(N,Z))) active(cons(X1,X2)) -> cons(active(X1),X2) active(cons2(X1,X2)) -> cons2(X1,active(X2)) active(from(X)) -> from(active(X)) active(from(X)) -> mark(cons(X,from(s(X)))) active(negrecip(X)) -> negrecip(active(X)) active(pi(X)) -> mark(2ndspos(X,from(0()))) active(pi(X)) -> pi(active(X)) active(plus(X1,X2)) -> plus(X1,active(X2)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(0(),Y)) -> mark(Y) active(plus(s(X),Y)) -> mark(s(plus(X,Y))) active(posrecip(X)) -> posrecip(active(X)) active(rcons(X1,X2)) -> rcons(X1,active(X2)) active(rcons(X1,X2)) -> rcons(active(X1),X2) active(s(X)) -> s(active(X)) active(square(X)) -> mark(times(X,X)) active(square(X)) -> square(active(X)) active(times(X1,X2)) -> times(X1,active(X2)) active(times(X1,X2)) -> times(active(X1),X2) active(times(0(),Y)) -> mark(0()) active(times(s(X),Y)) -> mark(plus(Y,times(X,Y))) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) cons2(X1,mark(X2)) -> mark(cons2(X1,X2)) cons2(ok(X1),ok(X2)) -> ok(cons2(X1,X2)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) negrecip(mark(X)) -> mark(negrecip(X)) negrecip(ok(X)) -> ok(negrecip(X)) pi(mark(X)) -> mark(pi(X)) pi(ok(X)) -> ok(pi(X)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) posrecip(mark(X)) -> mark(posrecip(X)) posrecip(ok(X)) -> ok(posrecip(X)) proper(0()) -> ok(0()) proper(2ndsneg(X1,X2)) -> 2ndsneg(proper(X1),proper(X2)) proper(2ndspos(X1,X2)) -> 2ndspos(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(cons2(X1,X2)) -> cons2(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(negrecip(X)) -> negrecip(proper(X)) proper(nil()) -> ok(nil()) proper(pi(X)) -> pi(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(posrecip(X)) -> posrecip(proper(X)) proper(rcons(X1,X2)) -> rcons(proper(X1),proper(X2)) proper(rnil()) -> ok(rnil()) proper(s(X)) -> s(proper(X)) proper(square(X)) -> square(proper(X)) proper(times(X1,X2)) -> times(proper(X1),proper(X2)) rcons(X1,mark(X2)) -> mark(rcons(X1,X2)) rcons(mark(X1),X2) -> mark(rcons(X1,X2)) rcons(ok(X1),ok(X2)) -> ok(rcons(X1,X2)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) square(mark(X)) -> mark(square(X)) square(ok(X)) -> ok(square(X)) times(X1,mark(X2)) -> mark(times(X1,X2)) times(mark(X1),X2) -> mark(times(X1,X2)) times(ok(X1),ok(X2)) -> ok(times(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {2ndsneg/2,2ndspos/2,active/1,cons/2,cons2/2,from/1,negrecip/1,pi/1,plus/2,posrecip/1,proper/1,rcons/2,s/1 ,square/1,times/2,top/1} / {0/0,mark/1,nil/0,ok/1,rnil/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,active,cons,cons2,from,negrecip,pi,plus ,posrecip,proper,rcons,s,square,times,top} and constructors {0,mark,nil,ok,rnil} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: 2ndsneg(x,y){y -> mark(y)} = 2ndsneg(x,mark(y)) ->^+ mark(2ndsneg(x,y)) = C[2ndsneg(x,y) = 2ndsneg(x,y){}] WORST_CASE(Omega(n^1),?)