* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: ++(x,nil()) -> x ++(++(x,y),z) -> ++(x,++(y,z)) ++(nil(),y) -> y flatten(++(x,y)) -> ++(flatten(x),flatten(y)) flatten(++(unit(x),y)) -> ++(flatten(x),flatten(y)) flatten(flatten(x)) -> flatten(x) flatten(nil()) -> nil() flatten(unit(x)) -> flatten(x) rev(++(x,y)) -> ++(rev(y),rev(x)) rev(nil()) -> nil() rev(rev(x)) -> x rev(unit(x)) -> unit(x) - Signature: {++/2,flatten/1,rev/1} / {nil/0,unit/1} - Obligation: innermost runtime complexity wrt. defined symbols {++,flatten,rev} and constructors {nil,unit} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: ++(x,nil()) -> x ++(++(x,y),z) -> ++(x,++(y,z)) ++(nil(),y) -> y flatten(++(x,y)) -> ++(flatten(x),flatten(y)) flatten(++(unit(x),y)) -> ++(flatten(x),flatten(y)) flatten(flatten(x)) -> flatten(x) flatten(nil()) -> nil() flatten(unit(x)) -> flatten(x) rev(++(x,y)) -> ++(rev(y),rev(x)) rev(nil()) -> nil() rev(rev(x)) -> x rev(unit(x)) -> unit(x) - Signature: {++/2,flatten/1,rev/1} / {nil/0,unit/1} - Obligation: innermost runtime complexity wrt. defined symbols {++,flatten,rev} and constructors {nil,unit} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: flatten(x){x -> unit(x)} = flatten(unit(x)) ->^+ flatten(x) = C[flatten(x) = flatten(x){}] ** Step 1.b:1: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: ++(x,nil()) -> x ++(++(x,y),z) -> ++(x,++(y,z)) ++(nil(),y) -> y flatten(++(x,y)) -> ++(flatten(x),flatten(y)) flatten(++(unit(x),y)) -> ++(flatten(x),flatten(y)) flatten(flatten(x)) -> flatten(x) flatten(nil()) -> nil() flatten(unit(x)) -> flatten(x) rev(++(x,y)) -> ++(rev(y),rev(x)) rev(nil()) -> nil() rev(rev(x)) -> x rev(unit(x)) -> unit(x) - Signature: {++/2,flatten/1,rev/1} / {nil/0,unit/1} - Obligation: innermost runtime complexity wrt. defined symbols {++,flatten,rev} and constructors {nil,unit} + 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 flatten_0(2) -> 1 flatten_1(2) -> 1 nil_0() -> 1 nil_0() -> 2 nil_1() -> 1 rev_0(2) -> 1 unit_0(2) -> 1 unit_0(2) -> 2 unit_1(2) -> 1 2 -> 1 ** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: ++(x,nil()) -> x ++(++(x,y),z) -> ++(x,++(y,z)) ++(nil(),y) -> y flatten(++(x,y)) -> ++(flatten(x),flatten(y)) flatten(++(unit(x),y)) -> ++(flatten(x),flatten(y)) flatten(flatten(x)) -> flatten(x) flatten(nil()) -> nil() flatten(unit(x)) -> flatten(x) rev(++(x,y)) -> ++(rev(y),rev(x)) rev(nil()) -> nil() rev(rev(x)) -> x rev(unit(x)) -> unit(x) - Signature: {++/2,flatten/1,rev/1} / {nil/0,unit/1} - Obligation: innermost runtime complexity wrt. defined symbols {++,flatten,rev} and constructors {nil,unit} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))