* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: active(add(X1,X2)) -> add(X1,active(X2)) active(add(X1,X2)) -> add(active(X1),X2) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(cons(X1,X2)) -> cons(active(X1),X2) active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib(X)) -> fib(active(X)) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) add(X1,mark(X2)) -> mark(add(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) fib(mark(X)) -> mark(fib(X)) fib(ok(X)) -> ok(fib(X)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(fib(X)) -> fib(proper(X)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,add/2,cons/2,fib/1,fib1/2,proper/1,s/1,sel/2,top/1} / {0/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,add,cons,fib,fib1,proper,s,sel ,top} and constructors {0,mark,ok} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: active(add(X1,X2)) -> add(X1,active(X2)) active(add(X1,X2)) -> add(active(X1),X2) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(cons(X1,X2)) -> cons(active(X1),X2) active(fib(N)) -> mark(sel(N,fib1(s(0()),s(0())))) active(fib(X)) -> fib(active(X)) active(fib1(X,Y)) -> mark(cons(X,fib1(Y,add(X,Y)))) active(fib1(X1,X2)) -> fib1(X1,active(X2)) active(fib1(X1,X2)) -> fib1(active(X1),X2) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(0(),cons(X,XS))) -> mark(X) active(sel(s(N),cons(X,XS))) -> mark(sel(N,XS)) add(X1,mark(X2)) -> mark(add(X1,X2)) add(mark(X1),X2) -> mark(add(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) fib(mark(X)) -> mark(fib(X)) fib(ok(X)) -> ok(fib(X)) fib1(X1,mark(X2)) -> mark(fib1(X1,X2)) fib1(mark(X1),X2) -> mark(fib1(X1,X2)) fib1(ok(X1),ok(X2)) -> ok(fib1(X1,X2)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(fib(X)) -> fib(proper(X)) proper(fib1(X1,X2)) -> fib1(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,add/2,cons/2,fib/1,fib1/2,proper/1,s/1,sel/2,top/1} / {0/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,add,cons,fib,fib1,proper,s,sel ,top} and constructors {0,mark,ok} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: add(x,y){y -> mark(y)} = add(x,mark(y)) ->^+ mark(add(x,y)) = C[add(x,y) = add(x,y){}] WORST_CASE(Omega(n^1),?)