* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: comp_f_g#1(comp_f_g(x4,x5),comp_f_g(x2,x3),x1) -> comp_f_g#1(x4,x5,comp_f_g#1(x2,x3,x1)) comp_f_g#1(comp_f_g(x7,x9),cons_x(x2),x4) -> comp_f_g#1(x7,x9,Cons(x2,x4)) comp_f_g#1(cons_x(x2),comp_f_g(x5,x7),x3) -> Cons(x2,comp_f_g#1(x5,x7,x3)) comp_f_g#1(cons_x(x5),cons_x(x2),x4) -> Cons(x5,Cons(x2,x4)) main(Leaf(x4)) -> Cons(x4,Nil()) main(Node(x9,x5)) -> comp_f_g#1(walk#1(x9),walk#1(x5),Nil()) walk#1(Leaf(x2)) -> cons_x(x2) walk#1(Node(x5,x3)) -> comp_f_g(walk#1(x5),walk#1(x3)) - Signature: {comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Leaf/1,Nil/0,Node/2,comp_f_g/2,cons_x/1} - Obligation: innermost runtime complexity wrt. defined symbols {comp_f_g#1,main,walk#1} and constructors {Cons,Leaf,Nil ,Node,comp_f_g,cons_x} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: comp_f_g#1(comp_f_g(x4,x5),comp_f_g(x2,x3),x1) -> comp_f_g#1(x4,x5,comp_f_g#1(x2,x3,x1)) comp_f_g#1(comp_f_g(x7,x9),cons_x(x2),x4) -> comp_f_g#1(x7,x9,Cons(x2,x4)) comp_f_g#1(cons_x(x2),comp_f_g(x5,x7),x3) -> Cons(x2,comp_f_g#1(x5,x7,x3)) comp_f_g#1(cons_x(x5),cons_x(x2),x4) -> Cons(x5,Cons(x2,x4)) main(Leaf(x4)) -> Cons(x4,Nil()) main(Node(x9,x5)) -> comp_f_g#1(walk#1(x9),walk#1(x5),Nil()) walk#1(Leaf(x2)) -> cons_x(x2) walk#1(Node(x5,x3)) -> comp_f_g(walk#1(x5),walk#1(x3)) - Signature: {comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Leaf/1,Nil/0,Node/2,comp_f_g/2,cons_x/1} - Obligation: innermost runtime complexity wrt. defined symbols {comp_f_g#1,main,walk#1} and constructors {Cons,Leaf,Nil ,Node,comp_f_g,cons_x} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: walk#1(x){x -> Node(x,y)} = walk#1(Node(x,y)) ->^+ comp_f_g(walk#1(x),walk#1(y)) = C[walk#1(x) = walk#1(x){}] ** Step 1.b:1: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: comp_f_g#1(comp_f_g(x4,x5),comp_f_g(x2,x3),x1) -> comp_f_g#1(x4,x5,comp_f_g#1(x2,x3,x1)) comp_f_g#1(comp_f_g(x7,x9),cons_x(x2),x4) -> comp_f_g#1(x7,x9,Cons(x2,x4)) comp_f_g#1(cons_x(x2),comp_f_g(x5,x7),x3) -> Cons(x2,comp_f_g#1(x5,x7,x3)) comp_f_g#1(cons_x(x5),cons_x(x2),x4) -> Cons(x5,Cons(x2,x4)) main(Leaf(x4)) -> Cons(x4,Nil()) main(Node(x9,x5)) -> comp_f_g#1(walk#1(x9),walk#1(x5),Nil()) walk#1(Leaf(x2)) -> cons_x(x2) walk#1(Node(x5,x3)) -> comp_f_g(walk#1(x5),walk#1(x3)) - Signature: {comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Leaf/1,Nil/0,Node/2,comp_f_g/2,cons_x/1} - Obligation: innermost runtime complexity wrt. defined symbols {comp_f_g#1,main,walk#1} and constructors {Cons,Leaf,Nil ,Node,comp_f_g,cons_x} + Applied Processor: Bounds {initialAutomaton = minimal, enrichment = match} + Details: The problem is match-bounded by 2. The enriched problem is compatible with follwoing automaton. Cons_0(2,2) -> 2 Cons_1(2,2) -> 2 Cons_1(2,2) -> 3 Cons_1(2,2) -> 4 Cons_1(2,3) -> 1 Cons_1(2,3) -> 3 Cons_1(2,3) -> 4 Cons_1(2,3) -> 5 Cons_1(2,4) -> 2 Cons_1(2,5) -> 3 Cons_1(2,5) -> 5 Cons_2(2,2) -> 4 Cons_2(2,3) -> 5 Cons_2(2,4) -> 2 Cons_2(2,4) -> 3 Cons_2(2,4) -> 4 Cons_2(2,5) -> 1 Cons_2(2,5) -> 3 Cons_2(2,5) -> 4 Cons_2(2,5) -> 5 Leaf_0(2) -> 2 Nil_0() -> 2 Nil_1() -> 3 Node_0(2,2) -> 2 comp_f_g_0(2,2) -> 2 comp_f_g_1(2,2) -> 1 comp_f_g_1(2,2) -> 2 comp_f_g#1_0(2,2,2) -> 1 comp_f_g#1_1(2,2,2) -> 2 comp_f_g#1_1(2,2,2) -> 3 comp_f_g#1_1(2,2,2) -> 4 comp_f_g#1_1(2,2,3) -> 1 comp_f_g#1_1(2,2,3) -> 3 comp_f_g#1_1(2,2,3) -> 4 comp_f_g#1_1(2,2,3) -> 5 comp_f_g#1_1(2,2,4) -> 2 comp_f_g#1_1(2,2,5) -> 3 comp_f_g#1_2(2,2,2) -> 4 comp_f_g#1_2(2,2,3) -> 5 comp_f_g#1_2(2,2,4) -> 2 comp_f_g#1_2(2,2,4) -> 3 comp_f_g#1_2(2,2,4) -> 4 comp_f_g#1_2(2,2,5) -> 1 comp_f_g#1_2(2,2,5) -> 3 comp_f_g#1_2(2,2,5) -> 4 comp_f_g#1_2(2,2,5) -> 5 cons_x_0(2) -> 2 cons_x_1(2) -> 1 cons_x_1(2) -> 2 main_0(2) -> 1 walk#1_0(2) -> 1 walk#1_1(2) -> 2 ** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: comp_f_g#1(comp_f_g(x4,x5),comp_f_g(x2,x3),x1) -> comp_f_g#1(x4,x5,comp_f_g#1(x2,x3,x1)) comp_f_g#1(comp_f_g(x7,x9),cons_x(x2),x4) -> comp_f_g#1(x7,x9,Cons(x2,x4)) comp_f_g#1(cons_x(x2),comp_f_g(x5,x7),x3) -> Cons(x2,comp_f_g#1(x5,x7,x3)) comp_f_g#1(cons_x(x5),cons_x(x2),x4) -> Cons(x5,Cons(x2,x4)) main(Leaf(x4)) -> Cons(x4,Nil()) main(Node(x9,x5)) -> comp_f_g#1(walk#1(x9),walk#1(x5),Nil()) walk#1(Leaf(x2)) -> cons_x(x2) walk#1(Node(x5,x3)) -> comp_f_g(walk#1(x5),walk#1(x3)) - Signature: {comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Leaf/1,Nil/0,Node/2,comp_f_g/2,cons_x/1} - Obligation: innermost runtime complexity wrt. defined symbols {comp_f_g#1,main,walk#1} and constructors {Cons,Leaf,Nil ,Node,comp_f_g,cons_x} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))