* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: +(0(),y) -> y +(s(x),y) -> +(x,s(y)) +(s(x),y) -> s(+(x,y)) - Signature: {+/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+} 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: +(0(),y) -> y +(s(x),y) -> +(x,s(y)) +(s(x),y) -> s(+(x,y)) - Signature: {+/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+} and constructors {0,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: +(x,y){x -> s(x)} = +(s(x),y) ->^+ +(x,s(y)) = C[+(x,s(y)) = +(x,y){y -> s(y)}] ** Step 1.b:1: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: +(0(),y) -> y +(s(x),y) -> +(x,s(y)) +(s(x),y) -> s(+(x,y)) - Signature: {+/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+} 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(2,2) -> 4 +_1(2,3) -> 1 +_1(2,3) -> 4 0_0() -> 1 0_0() -> 2 0_0() -> 4 s_0(2) -> 1 s_0(2) -> 2 s_0(2) -> 4 s_1(1) -> 1 s_1(2) -> 1 s_1(2) -> 3 s_1(2) -> 4 s_1(3) -> 1 s_1(3) -> 3 s_1(3) -> 4 s_1(4) -> 1 s_1(4) -> 4 2 -> 1 2 -> 4 3 -> 1 3 -> 4 ** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(0(),y) -> y +(s(x),y) -> +(x,s(y)) +(s(x),y) -> s(+(x,y)) - Signature: {+/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+} 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))