* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: eq0(0(),0()) -> S(0()) eq0(0(),S(x)) -> 0() eq0(S(x),0()) -> 0() eq0(S(x'),S(x)) -> eq0(x',x) - Signature: {eq0/2} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {eq0} 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: eq0(0(),0()) -> S(0()) eq0(0(),S(x)) -> 0() eq0(S(x),0()) -> 0() eq0(S(x'),S(x)) -> eq0(x',x) - Signature: {eq0/2} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {eq0} and constructors {0,S} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: eq0(x,y){x -> S(x),y -> S(y)} = eq0(S(x),S(y)) ->^+ eq0(x,y) = C[eq0(x,y) = eq0(x,y){}] ** Step 1.b:1: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: eq0(0(),0()) -> S(0()) eq0(0(),S(x)) -> 0() eq0(S(x),0()) -> 0() eq0(S(x'),S(x)) -> eq0(x',x) - Signature: {eq0/2} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {eq0} and constructors {0,S} + Applied Processor: Bounds {initialAutomaton = perSymbol, enrichment = match} + Details: The problem is match-bounded by 1. The enriched problem is compatible with follwoing automaton. 0_0() -> 1 0_1() -> 3 0_1() -> 4 S_0(1) -> 2 S_0(2) -> 2 S_1(4) -> 3 eq0_0(1,1) -> 3 eq0_0(1,2) -> 3 eq0_0(2,1) -> 3 eq0_0(2,2) -> 3 eq0_1(1,1) -> 3 eq0_1(1,2) -> 3 eq0_1(2,1) -> 3 eq0_1(2,2) -> 3 ** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: eq0(0(),0()) -> S(0()) eq0(0(),S(x)) -> 0() eq0(S(x),0()) -> 0() eq0(S(x'),S(x)) -> eq0(x',x) - Signature: {eq0/2} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {eq0} 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))