* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: comp_f_g#1(comp_f_g(x7,x9),walk_xs_3(x8),x12) -> comp_f_g#1(x7,x9,Cons(x8,x12)) comp_f_g#1(walk_xs(),walk_xs_3(x8),x12) -> Cons(x8,x12) main(Cons(x4,x5)) -> comp_f_g#1(walk#1(x5),walk_xs_3(x4),Nil()) main(Nil()) -> Nil() walk#1(Cons(x4,x3)) -> comp_f_g(walk#1(x3),walk_xs_3(x4)) walk#1(Nil()) -> walk_xs() - Signature: {comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Nil/0,comp_f_g/2,walk_xs/0,walk_xs_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {comp_f_g#1,main,walk#1} and constructors {Cons,Nil ,comp_f_g,walk_xs,walk_xs_3} + 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(x7,x9),walk_xs_3(x8),x12) -> comp_f_g#1(x7,x9,Cons(x8,x12)) comp_f_g#1(walk_xs(),walk_xs_3(x8),x12) -> Cons(x8,x12) main(Cons(x4,x5)) -> comp_f_g#1(walk#1(x5),walk_xs_3(x4),Nil()) main(Nil()) -> Nil() walk#1(Cons(x4,x3)) -> comp_f_g(walk#1(x3),walk_xs_3(x4)) walk#1(Nil()) -> walk_xs() - Signature: {comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Nil/0,comp_f_g/2,walk_xs/0,walk_xs_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {comp_f_g#1,main,walk#1} and constructors {Cons,Nil ,comp_f_g,walk_xs,walk_xs_3} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: walk#1(y){y -> Cons(x,y)} = walk#1(Cons(x,y)) ->^+ comp_f_g(walk#1(y),walk_xs_3(x)) = C[walk#1(y) = walk#1(y){}] ** Step 1.b:1: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: comp_f_g#1(comp_f_g(x7,x9),walk_xs_3(x8),x12) -> comp_f_g#1(x7,x9,Cons(x8,x12)) comp_f_g#1(walk_xs(),walk_xs_3(x8),x12) -> Cons(x8,x12) main(Cons(x4,x5)) -> comp_f_g#1(walk#1(x5),walk_xs_3(x4),Nil()) main(Nil()) -> Nil() walk#1(Cons(x4,x3)) -> comp_f_g(walk#1(x3),walk_xs_3(x4)) walk#1(Nil()) -> walk_xs() - Signature: {comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Nil/0,comp_f_g/2,walk_xs/0,walk_xs_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {comp_f_g#1,main,walk#1} and constructors {Cons,Nil ,comp_f_g,walk_xs,walk_xs_3} + 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(2,1) -> 1 Cons_1(2,2) -> 1 Cons_1(2,2) -> 3 Cons_1(2,3) -> 1 Cons_1(2,4) -> 1 Cons_2(2,1) -> 1 Cons_2(2,3) -> 1 Cons_2(2,3) -> 4 Cons_2(2,4) -> 1 Nil_0() -> 2 Nil_1() -> 1 Nil_1() -> 3 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(2,2,1) -> 1 comp_f_g#1_1(2,2,3) -> 1 comp_f_g#1_2(2,2,1) -> 1 comp_f_g#1_2(2,2,4) -> 1 main_0(2) -> 1 walk#1_0(2) -> 1 walk#1_1(2) -> 2 walk_xs_0() -> 2 walk_xs_1() -> 1 walk_xs_1() -> 2 walk_xs_3_0(2) -> 2 walk_xs_3_1(2) -> 2 ** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: comp_f_g#1(comp_f_g(x7,x9),walk_xs_3(x8),x12) -> comp_f_g#1(x7,x9,Cons(x8,x12)) comp_f_g#1(walk_xs(),walk_xs_3(x8),x12) -> Cons(x8,x12) main(Cons(x4,x5)) -> comp_f_g#1(walk#1(x5),walk_xs_3(x4),Nil()) main(Nil()) -> Nil() walk#1(Cons(x4,x3)) -> comp_f_g(walk#1(x3),walk_xs_3(x4)) walk#1(Nil()) -> walk_xs() - Signature: {comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Nil/0,comp_f_g/2,walk_xs/0,walk_xs_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {comp_f_g#1,main,walk#1} and constructors {Cons,Nil ,comp_f_g,walk_xs,walk_xs_3} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))