* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: dbl(0(),y) -> y dbl(S(0()),S(0())) -> S(S(S(S(0())))) save(0()) -> 0() save(S(x)) -> dbl(0(),save(x)) - Signature: {dbl/2,save/1} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {dbl,save} and constructors {0,S} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: dbl(0(),y) -> y dbl(S(0()),S(0())) -> S(S(S(S(0())))) save(0()) -> 0() save(S(x)) -> dbl(0(),save(x)) - Signature: {dbl/2,save/1} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {dbl,save} and constructors {0,S} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: save(x){x -> S(x)} = save(S(x)) ->^+ dbl(0(),save(x)) = C[save(x) = save(x){}] ** Step 1.b:1: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: dbl(0(),y) -> y dbl(S(0()),S(0())) -> S(S(S(S(0())))) save(0()) -> 0() save(S(x)) -> dbl(0(),save(x)) - Signature: {dbl/2,save/1} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {dbl,save} and constructors {0,S} + Applied Processor: Bounds {initialAutomaton = minimal, enrichment = match} + Details: The problem is match-bounded by 1. The enriched problem is compatible with follwoing automaton. 0_0() -> 1 0_0() -> 2 0_1() -> 1 0_1() -> 6 0_1() -> 7 S_0(2) -> 1 S_0(2) -> 2 S_1(3) -> 1 S_1(4) -> 3 S_1(5) -> 4 S_1(6) -> 5 dbl_0(2,2) -> 1 dbl_1(6,7) -> 1 dbl_1(7,7) -> 1 dbl_1(7,7) -> 7 save_0(2) -> 1 save_1(2) -> 1 save_1(2) -> 7 2 -> 1 7 -> 1 ** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: dbl(0(),y) -> y dbl(S(0()),S(0())) -> S(S(S(S(0())))) save(0()) -> 0() save(S(x)) -> dbl(0(),save(x)) - Signature: {dbl/2,save/1} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {dbl,save} and constructors {0,S} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))