* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: concat(cons(U,V),Y) -> cons(U,concat(V,Y)) concat(leaf(),Y) -> Y lessleaves(X,leaf()) -> false() lessleaves(cons(U,V),cons(W,Z)) -> lessleaves(concat(U,V),concat(W,Z)) lessleaves(leaf(),cons(W,Z)) -> true() - Signature: {concat/2,lessleaves/2} / {cons/2,false/0,leaf/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {concat,lessleaves} and constructors {cons,false,leaf ,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: concat(cons(U,V),Y) -> cons(U,concat(V,Y)) concat(leaf(),Y) -> Y lessleaves(X,leaf()) -> false() lessleaves(cons(U,V),cons(W,Z)) -> lessleaves(concat(U,V),concat(W,Z)) lessleaves(leaf(),cons(W,Z)) -> true() - Signature: {concat/2,lessleaves/2} / {cons/2,false/0,leaf/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {concat,lessleaves} and constructors {cons,false,leaf ,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: concat(y,z){y -> cons(x,y)} = concat(cons(x,y),z) ->^+ cons(x,concat(y,z)) = C[concat(y,z) = concat(y,z){}] ** Step 1.b:1: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: concat(cons(U,V),Y) -> cons(U,concat(V,Y)) concat(leaf(),Y) -> Y lessleaves(X,leaf()) -> false() lessleaves(cons(U,V),cons(W,Z)) -> lessleaves(concat(U,V),concat(W,Z)) lessleaves(leaf(),cons(W,Z)) -> true() - Signature: {concat/2,lessleaves/2} / {cons/2,false/0,leaf/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {concat,lessleaves} and constructors {cons,false,leaf ,true} + Applied Processor: Bounds {initialAutomaton = minimal, enrichment = match} + Details: The problem is match-bounded by 2. The enriched problem is compatible with follwoing automaton. concat_0(2,2) -> 1 concat_1(2,2) -> 3 concat_1(2,2) -> 4 concat_1(2,2) -> 5 concat_1(2,3) -> 3 concat_1(2,3) -> 4 concat_1(2,3) -> 5 concat_2(2,3) -> 4 concat_2(2,3) -> 5 cons_0(2,2) -> 1 cons_0(2,2) -> 2 cons_0(2,2) -> 3 cons_0(2,2) -> 4 cons_0(2,2) -> 5 cons_1(2,3) -> 1 cons_1(2,3) -> 3 cons_1(2,3) -> 4 cons_1(2,3) -> 5 false_0() -> 1 false_0() -> 2 false_0() -> 3 false_0() -> 4 false_0() -> 5 false_1() -> 1 leaf_0() -> 1 leaf_0() -> 2 leaf_0() -> 3 leaf_0() -> 4 leaf_0() -> 5 lessleaves_0(2,2) -> 1 lessleaves_1(3,3) -> 1 lessleaves_2(4,5) -> 1 true_0() -> 1 true_0() -> 2 true_0() -> 3 true_0() -> 4 true_0() -> 5 true_1() -> 1 2 -> 1 2 -> 3 2 -> 4 2 -> 5 3 -> 4 3 -> 5 ** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: concat(cons(U,V),Y) -> cons(U,concat(V,Y)) concat(leaf(),Y) -> Y lessleaves(X,leaf()) -> false() lessleaves(cons(U,V),cons(W,Z)) -> lessleaves(concat(U,V),concat(W,Z)) lessleaves(leaf(),cons(W,Z)) -> true() - Signature: {concat/2,lessleaves/2} / {cons/2,false/0,leaf/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {concat,lessleaves} and constructors {cons,false,leaf ,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))