* 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))