* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: div2(0()) -> 0() div2(S(0())) -> 0() div2(S(S(x))) -> +(S(0()),div2(x)) - Weak TRS: +(x,S(0())) -> S(x) +(S(0()),y) -> S(y) - Signature: {+/2,div2/1} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,div2} 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: div2(0()) -> 0() div2(S(0())) -> 0() div2(S(S(x))) -> +(S(0()),div2(x)) - Weak TRS: +(x,S(0())) -> S(x) +(S(0()),y) -> S(y) - Signature: {+/2,div2/1} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,div2} and constructors {0,S} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: div2(x){x -> S(S(x))} = div2(S(S(x))) ->^+ +(S(0()),div2(x)) = C[div2(x) = div2(x){}] ** Step 1.b:1: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: div2(0()) -> 0() div2(S(0())) -> 0() div2(S(S(x))) -> +(S(0()),div2(x)) - Weak TRS: +(x,S(0())) -> S(x) +(S(0()),y) -> S(y) - Signature: {+/2,div2/1} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,div2} 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(2,2) -> 1 +_1(3,4) -> 1 +_1(3,4) -> 4 0_0() -> 2 0_1() -> 1 0_1() -> 4 S_0(2) -> 1 S_0(2) -> 2 S_1(1) -> 3 S_1(3) -> 1 S_1(3) -> 4 S_1(4) -> 1 S_1(4) -> 4 div2_0(2) -> 1 div2_1(2) -> 4 ** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(x,S(0())) -> S(x) +(S(0()),y) -> S(y) div2(0()) -> 0() div2(S(0())) -> 0() div2(S(S(x))) -> +(S(0()),div2(x)) - Signature: {+/2,div2/1} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,div2} 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))