```* Step 1: ToInnermost WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
g(x,s(y)) -> g(f(x,y),0())
g(0(),f(x,x)) -> x
g(f(x,y),0()) -> f(g(x,0()),g(y,0()))
g(s(x),y) -> g(f(x,y),0())
- Signature:
{g/2} / {0/0,f/2,s/1}
- Obligation:
runtime complexity wrt. defined symbols {g} and constructors {0,f,s}
+ Applied Processor:
ToInnermost
+ Details:
switch to innermost, as the system is overlay and right linear and does not contain weak rules
* Step 2: Bounds WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
g(x,s(y)) -> g(f(x,y),0())
g(0(),f(x,x)) -> x
g(f(x,y),0()) -> f(g(x,0()),g(y,0()))
g(s(x),y) -> g(f(x,y),0())
- Signature:
{g/2} / {0/0,f/2,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {g} and constructors {0,f,s}
+ Applied Processor:
Bounds {initialAutomaton = minimal, enrichment = match}
+ Details:
The problem is match-bounded by 2.
The enriched problem is compatible with follwoing automaton.
0_0() -> 1
0_0() -> 2
0_1() -> 4
0_2() -> 9
0_2() -> 10
f_0(2,2) -> 1
f_0(2,2) -> 2
f_1(2,2) -> 3
f_1(2,4) -> 1
f_1(2,4) -> 2
f_1(2,9) -> 11
f_1(2,10) -> 12
f_1(5,6) -> 1
f_1(6,6) -> 5
f_1(6,6) -> 6
f_1(6,6) -> 7
f_1(6,6) -> 8
f_2(7,8) -> 1
f_2(8,13) -> 5
f_2(8,13) -> 6
f_2(8,13) -> 7
f_2(8,13) -> 8
g_0(2,2) -> 1
g_1(2,4) -> 5
g_1(2,4) -> 6
g_1(3,4) -> 1
g_1(4,4) -> 6
g_1(11,4) -> 7
g_1(12,4) -> 8
g_2(2,9) -> 7
g_2(2,10) -> 8
g_2(4,10) -> 13
g_2(9,10) -> 13
g_2(10,10) -> 13
s_0(2) -> 1
s_0(2) -> 2
2 -> 1
* Step 3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
g(x,s(y)) -> g(f(x,y),0())
g(0(),f(x,x)) -> x
g(f(x,y),0()) -> f(g(x,0()),g(y,0()))
g(s(x),y) -> g(f(x,y),0())
- Signature:
{g/2} / {0/0,f/2,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {g} and constructors {0,f,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))
```