* 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(from(X)) -> from(active(X)) active(from(X)) -> mark(cons(X,from(s(X)))) active(fst(X1,X2)) -> fst(X1,active(X2)) active(fst(X1,X2)) -> fst(active(X1),X2) active(fst(0(),Z)) -> mark(nil()) active(fst(s(X),cons(Y,Z))) -> mark(cons(Y,fst(X,Z))) active(len(X)) -> len(active(X)) active(len(cons(X,Z))) -> mark(s(len(Z))) active(len(nil())) -> mark(0()) 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)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) fst(X1,mark(X2)) -> mark(fst(X1,X2)) fst(mark(X1),X2) -> mark(fst(X1,X2)) fst(ok(X1),ok(X2)) -> ok(fst(X1,X2)) len(mark(X)) -> mark(len(X)) len(ok(X)) -> ok(len(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(fst(X1,X2)) -> fst(proper(X1),proper(X2)) proper(len(X)) -> len(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,add/2,cons/2,from/1,fst/2,len/1,proper/1,s/1,top/1} / {0/0,mark/1,nil/0,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,add,cons,from,fst,len,proper,s ,top} and constructors {0,mark,nil,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(from(X)) -> from(active(X)) active(from(X)) -> mark(cons(X,from(s(X)))) active(fst(X1,X2)) -> fst(X1,active(X2)) active(fst(X1,X2)) -> fst(active(X1),X2) active(fst(0(),Z)) -> mark(nil()) active(fst(s(X),cons(Y,Z))) -> mark(cons(Y,fst(X,Z))) active(len(X)) -> len(active(X)) active(len(cons(X,Z))) -> mark(s(len(Z))) active(len(nil())) -> mark(0()) 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)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) fst(X1,mark(X2)) -> mark(fst(X1,X2)) fst(mark(X1),X2) -> mark(fst(X1,X2)) fst(ok(X1),ok(X2)) -> ok(fst(X1,X2)) len(mark(X)) -> mark(len(X)) len(ok(X)) -> ok(len(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(fst(X1,X2)) -> fst(proper(X1),proper(X2)) proper(len(X)) -> len(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,add/2,cons/2,from/1,fst/2,len/1,proper/1,s/1,top/1} / {0/0,mark/1,nil/0,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,add,cons,from,fst,len,proper,s ,top} and constructors {0,mark,nil,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),?)