* Step 1: ToInnermost WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(.(.(x,y),z)) -> f(.(x,.(y,z))) f(.(nil(),y)) -> .(nil(),f(y)) f(nil()) -> nil() g(.(x,.(y,z))) -> g(.(.(x,y),z)) g(.(x,nil())) -> .(g(x),nil()) g(nil()) -> nil() - Signature: {f/1,g/1} / {./2,nil/0} - Obligation: runtime complexity wrt. defined symbols {f,g} and constructors {.,nil} + 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),z)) -> f(.(x,.(y,z))) f(.(nil(),y)) -> .(nil(),f(y)) f(nil()) -> nil() g(.(x,.(y,z))) -> g(.(.(x,y),z)) g(.(x,nil())) -> .(g(x),nil()) g(nil()) -> nil() - Signature: {f/1,g/1} / {./2,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {f,g} and constructors {.,nil} + Applied Processor: Bounds {initialAutomaton = perSymbol, enrichment = match} + Details: The problem is match-bounded by 1. The enriched problem is compatible with follwoing automaton. ._0(1,1) -> 1 ._0(1,4) -> 1 ._0(4,1) -> 1 ._0(4,4) -> 1 ._1(1,1) -> 6 ._1(1,4) -> 6 ._1(1,5) -> 5 ._1(1,6) -> 5 ._1(4,1) -> 6 ._1(4,4) -> 6 ._1(4,5) -> 5 ._1(4,6) -> 5 ._1(6,1) -> 9 ._1(6,4) -> 9 ._1(7,8) -> 2 ._1(9,1) -> 9 ._1(9,4) -> 9 ._1(10,8) -> 3 ._1(10,8) -> 8 ._1(10,10) -> 10 f_0(1) -> 2 f_0(4) -> 2 f_1(1) -> 8 f_1(4) -> 8 f_1(5) -> 2 f_1(5) -> 8 f_1(6) -> 8 g_0(1) -> 3 g_0(4) -> 3 g_1(1) -> 10 g_1(4) -> 10 g_1(6) -> 10 g_1(9) -> 3 g_1(9) -> 10 nil_0() -> 4 nil_1() -> 2 nil_1() -> 3 nil_1() -> 7 nil_1() -> 8 nil_1() -> 10 * Step 3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: f(.(.(x,y),z)) -> f(.(x,.(y,z))) f(.(nil(),y)) -> .(nil(),f(y)) f(nil()) -> nil() g(.(x,.(y,z))) -> g(.(.(x,y),z)) g(.(x,nil())) -> .(g(x),nil()) g(nil()) -> nil() - Signature: {f/1,g/1} / {./2,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {f,g} and constructors {.,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))