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