* Step 1: ToInnermost WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(+(x,y)) -> *(f(x),f(y)) f(+(x,s(0()))) -> +(s(s(0())),f(x)) f(0()) -> s(0()) f(s(0())) -> *(s(s(0())),f(0())) f(s(0())) -> s(s(0())) - Signature: {f/1} / {*/2,+/2,0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {f} and constructors {*,+,0,s} + Applied Processor: ToInnermost + Details: switch to innermost, as the system is overlay and right linear and does not contain weak rules * Step 2: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(+(x,y)) -> *(f(x),f(y)) f(+(x,s(0()))) -> +(s(s(0())),f(x)) f(0()) -> s(0()) f(s(0())) -> *(s(s(0())),f(0())) f(s(0())) -> s(s(0())) - Signature: {f/1} / {*/2,+/2,0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f} and constructors {*,+,0,s} + Applied Processor: Bounds {initialAutomaton = minimal, enrichment = match} + Details: The problem is match-bounded by 2. The enriched problem is compatible with follwoing automaton. *_0(2,2) -> 2 *_1(3,4) -> 1 *_1(4,4) -> 3 *_1(4,4) -> 4 *_1(5,8) -> 1 *_1(5,8) -> 3 *_1(5,8) -> 4 +_0(2,2) -> 2 +_1(5,4) -> 1 +_1(5,4) -> 3 +_1(5,4) -> 4 0_0() -> 2 0_1() -> 7 0_2() -> 9 f_0(2) -> 1 f_1(2) -> 3 f_1(2) -> 4 f_1(7) -> 8 s_0(2) -> 2 s_1(6) -> 1 s_1(6) -> 3 s_1(6) -> 4 s_1(6) -> 5 s_1(7) -> 1 s_1(7) -> 3 s_1(7) -> 4 s_1(7) -> 6 s_2(9) -> 8 * Step 3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: f(+(x,y)) -> *(f(x),f(y)) f(+(x,s(0()))) -> +(s(s(0())),f(x)) f(0()) -> s(0()) f(s(0())) -> *(s(s(0())),f(0())) f(s(0())) -> s(s(0())) - Signature: {f/1} / {*/2,+/2,0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f} and constructors {*,+,0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))