```* Step 1: DependencyPairs WORST_CASE(?,O(n^5))
+ Considered Problem:
- Strict TRS:
f_0(x) -> a()
f_1(x) -> g_1(x,x)
f_2(x) -> g_2(x,x)
f_3(x) -> g_3(x,x)
f_4(x) -> g_4(x,x)
f_5(x) -> g_5(x,x)
g_1(s(x),y) -> b(f_0(y),g_1(x,y))
g_2(s(x),y) -> b(f_1(y),g_2(x,y))
g_3(s(x),y) -> b(f_2(y),g_3(x,y))
g_4(s(x),y) -> b(f_3(y),g_4(x,y))
g_5(s(x),y) -> b(f_4(y),g_5(x,y))
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2} / {a/0,b/2,s/1}
- Obligation:
runtime complexity wrt. defined symbols {f_0,f_1,f_2,f_3,f_4,f_5,g_1,g_2,g_3,g_4,g_5} and constructors {a,b
,s}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak dependency pairs:

Strict DPs
f_0#(x) -> c_1()
f_1#(x) -> c_2(g_1#(x,x))
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> c_4(g_3#(x,x))
f_4#(x) -> c_5(g_4#(x,x))
f_5#(x) -> c_6(g_5#(x,x))
g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^5))
+ Considered Problem:
- Strict DPs:
f_0#(x) -> c_1()
f_1#(x) -> c_2(g_1#(x,x))
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> c_4(g_3#(x,x))
f_4#(x) -> c_5(g_4#(x,x))
f_5#(x) -> c_6(g_5#(x,x))
g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
- Strict TRS:
f_0(x) -> a()
f_1(x) -> g_1(x,x)
f_2(x) -> g_2(x,x)
f_3(x) -> g_3(x,x)
f_4(x) -> g_4(x,x)
f_5(x) -> g_5(x,x)
g_1(s(x),y) -> b(f_0(y),g_1(x,y))
g_2(s(x),y) -> b(f_1(y),g_2(x,y))
g_3(s(x),y) -> b(f_2(y),g_3(x,y))
g_4(s(x),y) -> b(f_3(y),g_4(x,y))
g_5(s(x),y) -> b(f_4(y),g_5(x,y))
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/2,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f_0#(x) -> c_1()
f_1#(x) -> c_2(g_1#(x,x))
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> c_4(g_3#(x,x))
f_4#(x) -> c_5(g_4#(x,x))
f_5#(x) -> c_6(g_5#(x,x))
g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^5))
+ Considered Problem:
- Strict DPs:
f_0#(x) -> c_1()
f_1#(x) -> c_2(g_1#(x,x))
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> c_4(g_3#(x,x))
f_4#(x) -> c_5(g_4#(x,x))
f_5#(x) -> c_6(g_5#(x,x))
g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/2,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1}
by application of
Pre({1}) = {7}.
Here rules are labelled as follows:
1: f_0#(x) -> c_1()
2: f_1#(x) -> c_2(g_1#(x,x))
3: f_2#(x) -> c_3(g_2#(x,x))
4: f_3#(x) -> c_4(g_3#(x,x))
5: f_4#(x) -> c_5(g_4#(x,x))
6: f_5#(x) -> c_6(g_5#(x,x))
7: g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
8: g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
9: g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
10: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
11: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^5))
+ Considered Problem:
- Strict DPs:
f_1#(x) -> c_2(g_1#(x,x))
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> c_4(g_3#(x,x))
f_4#(x) -> c_5(g_4#(x,x))
f_5#(x) -> c_6(g_5#(x,x))
g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
- Weak DPs:
f_0#(x) -> c_1()
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/2,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f_1#(x) -> c_2(g_1#(x,x))
-->_1 g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)):6

2:S:f_2#(x) -> c_3(g_2#(x,x))
-->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7

3:S:f_3#(x) -> c_4(g_3#(x,x))
-->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8

4:S:f_4#(x) -> c_5(g_4#(x,x))
-->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9

5:S:f_5#(x) -> c_6(g_5#(x,x))
-->_1 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10

6:S:g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
-->_1 f_0#(x) -> c_1():11
-->_2 g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)):6

7:S:g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
-->_2 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7
-->_1 f_1#(x) -> c_2(g_1#(x,x)):1

8:S:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
-->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8
-->_1 f_2#(x) -> c_3(g_2#(x,x)):2

9:S:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
-->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9
-->_1 f_3#(x) -> c_4(g_3#(x,x)):3

10:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
-->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10
-->_1 f_4#(x) -> c_5(g_4#(x,x)):4

11:W:f_0#(x) -> c_1()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
11: f_0#(x) -> c_1()
* Step 5: SimplifyRHS WORST_CASE(?,O(n^5))
+ Considered Problem:
- Strict DPs:
f_1#(x) -> c_2(g_1#(x,x))
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> c_4(g_3#(x,x))
f_4#(x) -> c_5(g_4#(x,x))
f_5#(x) -> c_6(g_5#(x,x))
g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/2,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:f_1#(x) -> c_2(g_1#(x,x))
-->_1 g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)):6

2:S:f_2#(x) -> c_3(g_2#(x,x))
-->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7

3:S:f_3#(x) -> c_4(g_3#(x,x))
-->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8

4:S:f_4#(x) -> c_5(g_4#(x,x))
-->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9

5:S:f_5#(x) -> c_6(g_5#(x,x))
-->_1 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10

6:S:g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y))
-->_2 g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)):6

7:S:g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
-->_2 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7
-->_1 f_1#(x) -> c_2(g_1#(x,x)):1

8:S:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
-->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8
-->_1 f_2#(x) -> c_3(g_2#(x,x)):2

9:S:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
-->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9
-->_1 f_3#(x) -> c_4(g_3#(x,x)):3

10:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
-->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10
-->_1 f_4#(x) -> c_5(g_4#(x,x)):4

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
g_1#(s(x),y) -> c_7(g_1#(x,y))
+ Considered Problem:
- Strict DPs:
f_1#(x) -> c_2(g_1#(x,x))
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> c_4(g_3#(x,x))
f_4#(x) -> c_5(g_4#(x,x))
f_5#(x) -> c_6(g_5#(x,x))
g_1#(s(x),y) -> c_7(g_1#(x,y))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
+ Details:
Consider the dependency graph

1:S:f_1#(x) -> c_2(g_1#(x,x))
-->_1 g_1#(s(x),y) -> c_7(g_1#(x,y)):6

2:S:f_2#(x) -> c_3(g_2#(x,x))
-->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7

3:S:f_3#(x) -> c_4(g_3#(x,x))
-->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8

4:S:f_4#(x) -> c_5(g_4#(x,x))
-->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9

5:S:f_5#(x) -> c_6(g_5#(x,x))
-->_1 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10

6:S:g_1#(s(x),y) -> c_7(g_1#(x,y))
-->_1 g_1#(s(x),y) -> c_7(g_1#(x,y)):6

7:S:g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
-->_2 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7
-->_1 f_1#(x) -> c_2(g_1#(x,x)):1

8:S:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
-->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8
-->_1 f_2#(x) -> c_3(g_2#(x,x)):2

9:S:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
-->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9
-->_1 f_3#(x) -> c_4(g_3#(x,x)):3

10:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
-->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10
-->_1 f_4#(x) -> c_5(g_4#(x,x)):4

Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).

[(5,f_5#(x) -> c_6(g_5#(x,x)))]
* Step 7: DecomposeDG WORST_CASE(?,O(n^5))
+ Considered Problem:
- Strict DPs:
f_1#(x) -> c_2(g_1#(x,x))
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> c_4(g_3#(x,x))
f_4#(x) -> c_5(g_4#(x,x))
g_1#(s(x),y) -> c_7(g_1#(x,y))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
and a lower component
f_1#(x) -> c_2(g_1#(x,x))
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> c_4(g_3#(x,x))
f_4#(x) -> c_5(g_4#(x,x))
g_1#(s(x),y) -> c_7(g_1#(x,y))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
Further, following extension rules are added to the lower component.
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
** Step 7.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))

The strictly oriented rules are moved into the weak component.
*** Step 7.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_11) = {2}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a) = [1]
p(b) = [1] x1 + [1] x2 + [0]
p(f_0) = [0]
p(f_1) = [0]
p(f_2) = [0]
p(f_3) = [0]
p(f_4) = [0]
p(f_5) = [0]
p(g_1) = [0]
p(g_2) = [0]
p(g_3) = [0]
p(g_4) = [0]
p(g_5) = [1] x1 + [2]
p(s) = [1] x1 + [8]
p(f_0#) = [1] x1 + [1]
p(f_1#) = [2] x1 + [1]
p(f_2#) = [4] x1 + [0]
p(f_3#) = [2] x1 + [1]
p(f_4#) = [8]
p(f_5#) = [4] x1 + [0]
p(g_1#) = [1] x1 + [1] x2 + [0]
p(g_2#) = [2] x1 + [1] x2 + [2]
p(g_3#) = [8] x1 + [1] x2 + [1]
p(g_4#) = [1] x2 + [2]
p(g_5#) = [1] x1 + [6] x2 + [4]
p(c_1) = [1]
p(c_2) = [1] x1 + [1]
p(c_3) = [1]
p(c_4) = [1] x1 + [1]
p(c_5) = [0]
p(c_6) = [1] x1 + [2]
p(c_7) = [1] x1 + [1]
p(c_8) = [2] x1 + [1] x2 + [1]
p(c_9) = [1] x1 + [1] x2 + [1]
p(c_10) = [1] x1 + [0]
p(c_11) = [1] x2 + [4]

Following rules are strictly oriented:
g_5#(s(x),y) = [1] x + [6] y + [12]
> [1] x + [6] y + [8]
= c_11(f_4#(y),g_5#(x,y))

Following rules are (at-least) weakly oriented:

*** Step 7.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

*** Step 7.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
-->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y))
*** Step 7.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

** Step 7.b:1: DecomposeDG WORST_CASE(?,O(n^4))
+ Considered Problem:
- Strict DPs:
f_1#(x) -> c_2(g_1#(x,x))
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> c_4(g_3#(x,x))
f_4#(x) -> c_5(g_4#(x,x))
g_1#(s(x),y) -> c_7(g_1#(x,y))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
- Weak DPs:
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
f_4#(x) -> c_5(g_4#(x,x))
g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
and a lower component
f_1#(x) -> c_2(g_1#(x,x))
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> c_4(g_3#(x,x))
g_1#(s(x),y) -> c_7(g_1#(x,y))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
Further, following extension rules are added to the lower component.
f_4#(x) -> g_4#(x,x)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
*** Step 7.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_4#(x) -> c_5(g_4#(x,x))
g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
- Weak DPs:
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: f_4#(x) -> c_5(g_4#(x,x))
2: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))

The strictly oriented rules are moved into the weak component.
**** Step 7.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_4#(x) -> c_5(g_4#(x,x))
g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
- Weak DPs:
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_5) = {1},
uargs(c_10) = {2}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a) = [0]
p(b) = [1] x2 + [1]
p(f_0) = [2] x1 + [0]
p(f_1) = [8]
p(f_2) = [1] x1 + [2]
p(f_3) = [1] x1 + [2]
p(f_4) = [2] x1 + [0]
p(f_5) = [2] x1 + [8]
p(g_1) = [1] x1 + [8] x2 + [8]
p(g_2) = [1]
p(g_3) = [8] x1 + [2] x2 + [2]
p(g_4) = [1] x1 + [1]
p(g_5) = [1] x1 + [1] x2 + [4]
p(s) = [1] x1 + [1]
p(f_0#) = [1] x1 + [1]
p(f_1#) = [1] x1 + [1]
p(f_2#) = [0]
p(f_3#) = [1] x1 + [8]
p(f_4#) = [5] x1 + [12]
p(f_5#) = [2] x1 + [1]
p(g_1#) = [1] x1 + [1]
p(g_2#) = [2] x1 + [1] x2 + [1]
p(g_3#) = [1] x1 + [1] x2 + [0]
p(g_4#) = [1] x1 + [4] x2 + [0]
p(g_5#) = [8] x1 + [8] x2 + [8]
p(c_1) = [1]
p(c_2) = [8] x1 + [2]
p(c_3) = [1] x1 + [1]
p(c_4) = [2] x1 + [1]
p(c_5) = [1] x1 + [2]
p(c_6) = [1] x1 + [0]
p(c_7) = [2] x1 + [0]
p(c_8) = [1] x2 + [0]
p(c_9) = [8] x1 + [2] x2 + [1]
p(c_10) = [1] x2 + [0]
p(c_11) = [1] x1 + [1] x2 + [2]

Following rules are strictly oriented:
f_4#(x) = [5] x + [12]
> [5] x + [2]
= c_5(g_4#(x,x))

g_4#(s(x),y) = [1] x + [4] y + [1]
> [1] x + [4] y + [0]
= c_10(f_3#(y),g_4#(x,y))

Following rules are (at-least) weakly oriented:
g_5#(s(x),y) =  [8] x + [8] y + [16]
>= [5] y + [12]
=  f_4#(y)

g_5#(s(x),y) =  [8] x + [8] y + [16]
>= [8] x + [8] y + [8]
=  g_5#(x,y)

**** Step 7.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f_4#(x) -> c_5(g_4#(x,x))
g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

**** Step 7.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f_4#(x) -> c_5(g_4#(x,x))
g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:f_4#(x) -> c_5(g_4#(x,x))
-->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):2

2:W:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
-->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):2

3:W:g_5#(s(x),y) -> f_4#(y)
-->_1 f_4#(x) -> c_5(g_4#(x,x)):1

4:W:g_5#(s(x),y) -> g_5#(x,y)
-->_1 g_5#(s(x),y) -> g_5#(x,y):4
-->_1 g_5#(s(x),y) -> f_4#(y):3

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: g_5#(s(x),y) -> g_5#(x,y)
3: g_5#(s(x),y) -> f_4#(y)
1: f_4#(x) -> c_5(g_4#(x,x))
2: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y))
**** Step 7.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

*** Step 7.b:1.b:1: DecomposeDG WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
f_1#(x) -> c_2(g_1#(x,x))
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> c_4(g_3#(x,x))
g_1#(s(x),y) -> c_7(g_1#(x,y))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
- Weak DPs:
f_4#(x) -> g_4#(x,x)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
f_3#(x) -> c_4(g_3#(x,x))
f_4#(x) -> g_4#(x,x)
g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
and a lower component
f_1#(x) -> c_2(g_1#(x,x))
f_2#(x) -> c_3(g_2#(x,x))
g_1#(s(x),y) -> c_7(g_1#(x,y))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
Further, following extension rules are added to the lower component.
f_3#(x) -> g_3#(x,x)
f_4#(x) -> g_4#(x,x)
g_3#(s(x),y) -> f_2#(y)
g_3#(s(x),y) -> g_3#(x,y)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
**** Step 7.b:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_3#(x) -> c_4(g_3#(x,x))
g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
- Weak DPs:
f_4#(x) -> g_4#(x,x)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
2: g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))

The strictly oriented rules are moved into the weak component.
***** Step 7.b:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_3#(x) -> c_4(g_3#(x,x))
g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
- Weak DPs:
f_4#(x) -> g_4#(x,x)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_4) = {1},
uargs(c_9) = {2}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a) = [2]
p(b) = [1] x1 + [1] x2 + [8]
p(f_0) = [1]
p(f_1) = [0]
p(f_2) = [0]
p(f_3) = [0]
p(f_4) = [0]
p(f_5) = [0]
p(g_1) = [0]
p(g_2) = [0]
p(g_3) = [0]
p(g_4) = [0]
p(g_5) = [0]
p(s) = [1] x1 + [8]
p(f_0#) = [0]
p(f_1#) = [0]
p(f_2#) = [1]
p(f_3#) = [4] x1 + [0]
p(f_4#) = [6] x1 + [8]
p(f_5#) = [0]
p(g_1#) = [0]
p(g_2#) = [1] x1 + [2]
p(g_3#) = [2] x1 + [0]
p(g_4#) = [2] x1 + [4] x2 + [8]
p(g_5#) = [1] x1 + [9] x2 + [15]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [2] x1 + [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [8] x1 + [1]
p(c_8) = [1] x2 + [2]
p(c_9) = [4] x1 + [1] x2 + [11]
p(c_10) = [1] x1 + [0]
p(c_11) = [2] x2 + [0]

Following rules are strictly oriented:
g_3#(s(x),y) = [2] x + [16]
> [2] x + [15]
= c_9(f_2#(y),g_3#(x,y))

Following rules are (at-least) weakly oriented:
f_3#(x) =  [4] x + [0]
>= [4] x + [0]
=  c_4(g_3#(x,x))

f_4#(x) =  [6] x + [8]
>= [6] x + [8]
=  g_4#(x,x)

g_4#(s(x),y) =  [2] x + [4] y + [24]
>= [4] y + [0]
=  f_3#(y)

g_4#(s(x),y) =  [2] x + [4] y + [24]
>= [2] x + [4] y + [8]
=  g_4#(x,y)

g_5#(s(x),y) =  [1] x + [9] y + [23]
>= [6] y + [8]
=  f_4#(y)

g_5#(s(x),y) =  [1] x + [9] y + [23]
>= [1] x + [9] y + [15]
=  g_5#(x,y)

***** Step 7.b:1.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f_3#(x) -> c_4(g_3#(x,x))
- Weak DPs:
f_4#(x) -> g_4#(x,x)
g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

***** Step 7.b:1.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_3#(x) -> c_4(g_3#(x,x))
- Weak DPs:
f_4#(x) -> g_4#(x,x)
g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f_3#(x) -> c_4(g_3#(x,x))
-->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):3

2:W:f_4#(x) -> g_4#(x,x)
-->_1 g_4#(s(x),y) -> g_4#(x,y):5
-->_1 g_4#(s(x),y) -> f_3#(y):4

3:W:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
-->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):3

4:W:g_4#(s(x),y) -> f_3#(y)
-->_1 f_3#(x) -> c_4(g_3#(x,x)):1

5:W:g_4#(s(x),y) -> g_4#(x,y)
-->_1 g_4#(s(x),y) -> g_4#(x,y):5
-->_1 g_4#(s(x),y) -> f_3#(y):4

6:W:g_5#(s(x),y) -> f_4#(y)
-->_1 f_4#(x) -> g_4#(x,x):2

7:W:g_5#(s(x),y) -> g_5#(x,y)
-->_1 g_5#(s(x),y) -> g_5#(x,y):7
-->_1 g_5#(s(x),y) -> f_4#(y):6

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y))
***** Step 7.b:1.b:1.a:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_3#(x) -> c_4(g_3#(x,x))
- Weak DPs:
f_4#(x) -> g_4#(x,x)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:f_3#(x) -> c_4(g_3#(x,x))

2:W:f_4#(x) -> g_4#(x,x)
-->_1 g_4#(s(x),y) -> g_4#(x,y):5
-->_1 g_4#(s(x),y) -> f_3#(y):4

4:W:g_4#(s(x),y) -> f_3#(y)
-->_1 f_3#(x) -> c_4(g_3#(x,x)):1

5:W:g_4#(s(x),y) -> g_4#(x,y)
-->_1 g_4#(s(x),y) -> g_4#(x,y):5
-->_1 g_4#(s(x),y) -> f_3#(y):4

6:W:g_5#(s(x),y) -> f_4#(y)
-->_1 f_4#(x) -> g_4#(x,x):2

7:W:g_5#(s(x),y) -> g_5#(x,y)
-->_1 g_5#(s(x),y) -> g_5#(x,y):7
-->_1 g_5#(s(x),y) -> f_4#(y):6

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
f_3#(x) -> c_4()
***** Step 7.b:1.b:1.a:1.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_3#(x) -> c_4()
- Weak DPs:
f_4#(x) -> g_4#(x,x)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: f_3#(x) -> c_4()

The strictly oriented rules are moved into the weak component.
****** Step 7.b:1.b:1.a:1.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_3#(x) -> c_4()
- Weak DPs:
f_4#(x) -> g_4#(x,x)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
none

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a) = [0]
p(b) = [1] x1 + [1] x2 + [0]
p(f_0) = [0]
p(f_1) = [0]
p(f_2) = [0]
p(f_3) = [0]
p(f_4) = [0]
p(f_5) = [1]
p(g_1) = [0]
p(g_2) = [0]
p(g_3) = [0]
p(g_4) = [0]
p(g_5) = [0]
p(s) = [1] x1 + [0]
p(f_0#) = [0]
p(f_1#) = [0]
p(f_2#) = [0]
p(f_3#) = [2]
p(f_4#) = [2]
p(f_5#) = [0]
p(g_1#) = [0]
p(g_2#) = [0]
p(g_3#) = [0]
p(g_4#) = [2]
p(g_5#) = [2]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [0]
p(c_10) = [0]
p(c_11) = [0]

Following rules are strictly oriented:
f_3#(x) = [2]
> [0]
= c_4()

Following rules are (at-least) weakly oriented:
f_4#(x) =  [2]
>= [2]
=  g_4#(x,x)

g_4#(s(x),y) =  [2]
>= [2]
=  f_3#(y)

g_4#(s(x),y) =  [2]
>= [2]
=  g_4#(x,y)

g_5#(s(x),y) =  [2]
>= [2]
=  f_4#(y)

g_5#(s(x),y) =  [2]
>= [2]
=  g_5#(x,y)

****** Step 7.b:1.b:1.a:1.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f_3#(x) -> c_4()
f_4#(x) -> g_4#(x,x)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

****** Step 7.b:1.b:1.a:1.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f_3#(x) -> c_4()
f_4#(x) -> g_4#(x,x)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:f_3#(x) -> c_4()

2:W:f_4#(x) -> g_4#(x,x)
-->_1 g_4#(s(x),y) -> g_4#(x,y):4
-->_1 g_4#(s(x),y) -> f_3#(y):3

3:W:g_4#(s(x),y) -> f_3#(y)
-->_1 f_3#(x) -> c_4():1

4:W:g_4#(s(x),y) -> g_4#(x,y)
-->_1 g_4#(s(x),y) -> g_4#(x,y):4
-->_1 g_4#(s(x),y) -> f_3#(y):3

5:W:g_5#(s(x),y) -> f_4#(y)
-->_1 f_4#(x) -> g_4#(x,x):2

6:W:g_5#(s(x),y) -> g_5#(x,y)
-->_1 g_5#(s(x),y) -> g_5#(x,y):6
-->_1 g_5#(s(x),y) -> f_4#(y):5

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: g_5#(s(x),y) -> g_5#(x,y)
5: g_5#(s(x),y) -> f_4#(y)
2: f_4#(x) -> g_4#(x,x)
4: g_4#(s(x),y) -> g_4#(x,y)
3: g_4#(s(x),y) -> f_3#(y)
1: f_3#(x) -> c_4()
****** Step 7.b:1.b:1.a:1.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

**** Step 7.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
f_1#(x) -> c_2(g_1#(x,x))
f_2#(x) -> c_3(g_2#(x,x))
g_1#(s(x),y) -> c_7(g_1#(x,y))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
- Weak DPs:
f_3#(x) -> g_3#(x,x)
f_4#(x) -> g_4#(x,x)
g_3#(s(x),y) -> f_2#(y)
g_3#(s(x),y) -> g_3#(x,y)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
2: f_2#(x) -> c_3(g_2#(x,x))
3: g_1#(s(x),y) -> c_7(g_1#(x,y))

The strictly oriented rules are moved into the weak component.
***** Step 7.b:1.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
f_1#(x) -> c_2(g_1#(x,x))
f_2#(x) -> c_3(g_2#(x,x))
g_1#(s(x),y) -> c_7(g_1#(x,y))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
- Weak DPs:
f_3#(x) -> g_3#(x,x)
f_4#(x) -> g_4#(x,x)
g_3#(s(x),y) -> f_2#(y)
g_3#(s(x),y) -> g_3#(x,y)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_7) = {1},
uargs(c_8) = {1,2}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a) = 0
p(b) = 1
p(f_0) = 0
p(f_1) = 1 + x1
p(f_2) = 0
p(f_3) = 2*x1^2
p(f_4) = 4 + x1^2
p(f_5) = 4 + x1^2
p(g_1) = x1
p(g_2) = 1 + x1
p(g_3) = 1 + x1 + 2*x1^2 + 2*x2 + 4*x2^2
p(g_4) = 2*x1^2
p(g_5) = 4*x1 + 2*x2
p(s) = 1 + x1
p(f_0#) = x1 + x1^2
p(f_1#) = 1 + x1
p(f_2#) = 3 + 4*x1 + 3*x1^2
p(f_3#) = 5 + 5*x1 + 4*x1^2
p(f_4#) = 5 + 6*x1 + 6*x1^2
p(f_5#) = 1 + 2*x1 + x1^2
p(g_1#) = x1
p(g_2#) = 2*x1 + x1*x2
p(g_3#) = 4 + 4*x2 + 3*x2^2
p(g_4#) = 5 + 2*x1*x2 + 6*x2 + 4*x2^2
p(g_5#) = 7 + 3*x1 + 2*x1^2 + 6*x2 + 7*x2^2
p(c_1) = 0
p(c_2) = 1 + x1
p(c_3) = x1
p(c_4) = 1 + x1
p(c_5) = 0
p(c_6) = 0
p(c_7) = x1
p(c_8) = 1 + x1 + x2
p(c_9) = 1 + x1 + x2
p(c_10) = x1
p(c_11) = 1 + x1

Following rules are strictly oriented:
f_2#(x) = 3 + 4*x + 3*x^2
> 2*x + x^2
= c_3(g_2#(x,x))

g_1#(s(x),y) = 1 + x
> x
= c_7(g_1#(x,y))

Following rules are (at-least) weakly oriented:
f_1#(x) =  1 + x
>= 1 + x
=  c_2(g_1#(x,x))

f_3#(x) =  5 + 5*x + 4*x^2
>= 4 + 4*x + 3*x^2
=  g_3#(x,x)

f_4#(x) =  5 + 6*x + 6*x^2
>= 5 + 6*x + 6*x^2
=  g_4#(x,x)

g_2#(s(x),y) =  2 + 2*x + x*y + y
>= 2 + 2*x + x*y + y
=  c_8(f_1#(y),g_2#(x,y))

g_3#(s(x),y) =  4 + 4*y + 3*y^2
>= 3 + 4*y + 3*y^2
=  f_2#(y)

g_3#(s(x),y) =  4 + 4*y + 3*y^2
>= 4 + 4*y + 3*y^2
=  g_3#(x,y)

g_4#(s(x),y) =  5 + 2*x*y + 8*y + 4*y^2
>= 5 + 5*y + 4*y^2
=  f_3#(y)

g_4#(s(x),y) =  5 + 2*x*y + 8*y + 4*y^2
>= 5 + 2*x*y + 6*y + 4*y^2
=  g_4#(x,y)

g_5#(s(x),y) =  12 + 7*x + 2*x^2 + 6*y + 7*y^2
>= 5 + 6*y + 6*y^2
=  f_4#(y)

g_5#(s(x),y) =  12 + 7*x + 2*x^2 + 6*y + 7*y^2
>= 7 + 3*x + 2*x^2 + 6*y + 7*y^2
=  g_5#(x,y)

***** Step 7.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f_1#(x) -> c_2(g_1#(x,x))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
- Weak DPs:
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> g_3#(x,x)
f_4#(x) -> g_4#(x,x)
g_1#(s(x),y) -> c_7(g_1#(x,y))
g_3#(s(x),y) -> f_2#(y)
g_3#(s(x),y) -> g_3#(x,y)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

***** Step 7.b:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_1#(x) -> c_2(g_1#(x,x))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
- Weak DPs:
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> g_3#(x,x)
f_4#(x) -> g_4#(x,x)
g_1#(s(x),y) -> c_7(g_1#(x,y))
g_3#(s(x),y) -> f_2#(y)
g_3#(s(x),y) -> g_3#(x,y)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f_1#(x) -> c_2(g_1#(x,x))
-->_1 g_1#(s(x),y) -> c_7(g_1#(x,y)):6

2:S:g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
-->_2 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):2
-->_1 f_1#(x) -> c_2(g_1#(x,x)):1

3:W:f_2#(x) -> c_3(g_2#(x,x))
-->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):2

4:W:f_3#(x) -> g_3#(x,x)
-->_1 g_3#(s(x),y) -> g_3#(x,y):8
-->_1 g_3#(s(x),y) -> f_2#(y):7

5:W:f_4#(x) -> g_4#(x,x)
-->_1 g_4#(s(x),y) -> g_4#(x,y):10
-->_1 g_4#(s(x),y) -> f_3#(y):9

6:W:g_1#(s(x),y) -> c_7(g_1#(x,y))
-->_1 g_1#(s(x),y) -> c_7(g_1#(x,y)):6

7:W:g_3#(s(x),y) -> f_2#(y)
-->_1 f_2#(x) -> c_3(g_2#(x,x)):3

8:W:g_3#(s(x),y) -> g_3#(x,y)
-->_1 g_3#(s(x),y) -> g_3#(x,y):8
-->_1 g_3#(s(x),y) -> f_2#(y):7

9:W:g_4#(s(x),y) -> f_3#(y)
-->_1 f_3#(x) -> g_3#(x,x):4

10:W:g_4#(s(x),y) -> g_4#(x,y)
-->_1 g_4#(s(x),y) -> g_4#(x,y):10
-->_1 g_4#(s(x),y) -> f_3#(y):9

11:W:g_5#(s(x),y) -> f_4#(y)
-->_1 f_4#(x) -> g_4#(x,x):5

12:W:g_5#(s(x),y) -> g_5#(x,y)
-->_1 g_5#(s(x),y) -> g_5#(x,y):12
-->_1 g_5#(s(x),y) -> f_4#(y):11

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: g_1#(s(x),y) -> c_7(g_1#(x,y))
***** Step 7.b:1.b:1.b:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_1#(x) -> c_2(g_1#(x,x))
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
- Weak DPs:
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> g_3#(x,x)
f_4#(x) -> g_4#(x,x)
g_3#(s(x),y) -> f_2#(y)
g_3#(s(x),y) -> g_3#(x,y)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:f_1#(x) -> c_2(g_1#(x,x))

2:S:g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
-->_2 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):2
-->_1 f_1#(x) -> c_2(g_1#(x,x)):1

3:W:f_2#(x) -> c_3(g_2#(x,x))
-->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):2

4:W:f_3#(x) -> g_3#(x,x)
-->_1 g_3#(s(x),y) -> g_3#(x,y):8
-->_1 g_3#(s(x),y) -> f_2#(y):7

5:W:f_4#(x) -> g_4#(x,x)
-->_1 g_4#(s(x),y) -> g_4#(x,y):10
-->_1 g_4#(s(x),y) -> f_3#(y):9

7:W:g_3#(s(x),y) -> f_2#(y)
-->_1 f_2#(x) -> c_3(g_2#(x,x)):3

8:W:g_3#(s(x),y) -> g_3#(x,y)
-->_1 g_3#(s(x),y) -> g_3#(x,y):8
-->_1 g_3#(s(x),y) -> f_2#(y):7

9:W:g_4#(s(x),y) -> f_3#(y)
-->_1 f_3#(x) -> g_3#(x,x):4

10:W:g_4#(s(x),y) -> g_4#(x,y)
-->_1 g_4#(s(x),y) -> g_4#(x,y):10
-->_1 g_4#(s(x),y) -> f_3#(y):9

11:W:g_5#(s(x),y) -> f_4#(y)
-->_1 f_4#(x) -> g_4#(x,x):5

12:W:g_5#(s(x),y) -> g_5#(x,y)
-->_1 g_5#(s(x),y) -> g_5#(x,y):12
-->_1 g_5#(s(x),y) -> f_4#(y):11

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
f_1#(x) -> c_2()
***** Step 7.b:1.b:1.b:1.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_1#(x) -> c_2()
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
- Weak DPs:
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> g_3#(x,x)
f_4#(x) -> g_4#(x,x)
g_3#(s(x),y) -> f_2#(y)
g_3#(s(x),y) -> g_3#(x,y)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
2: g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))

Consider the set of all dependency pairs
1: f_1#(x) -> c_2()
2: g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
3: f_2#(x) -> c_3(g_2#(x,x))
4: f_3#(x) -> g_3#(x,x)
5: f_4#(x) -> g_4#(x,x)
6: g_3#(s(x),y) -> f_2#(y)
7: g_3#(s(x),y) -> g_3#(x,y)
8: g_4#(s(x),y) -> f_3#(y)
9: g_4#(s(x),y) -> g_4#(x,y)
10: g_5#(s(x),y) -> f_4#(y)
11: g_5#(s(x),y) -> g_5#(x,y)
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{2}
These cover all (indirect) predecessors of dependency pairs
{1,2}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
****** Step 7.b:1.b:1.b:1.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_1#(x) -> c_2()
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
- Weak DPs:
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> g_3#(x,x)
f_4#(x) -> g_4#(x,x)
g_3#(s(x),y) -> f_2#(y)
g_3#(s(x),y) -> g_3#(x,y)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_3) = {1},
uargs(c_8) = {1,2}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a) = [0]
p(b) = [1] x1 + [1]
p(f_0) = [1]
p(f_1) = [1]
p(f_2) = [0]
p(f_3) = [4]
p(f_4) = [8]
p(f_5) = [1]
p(g_1) = [2] x1 + [1] x2 + [0]
p(g_2) = [1] x1 + [2]
p(g_3) = [8] x1 + [8]
p(g_4) = [2] x1 + [2] x2 + [1]
p(g_5) = [2] x2 + [0]
p(s) = [1] x1 + [2]
p(f_0#) = [1] x1 + [1]
p(f_1#) = [0]
p(f_2#) = [2] x1 + [0]
p(f_3#) = [2] x1 + [0]
p(f_4#) = [2] x1 + [0]
p(f_5#) = [1] x1 + [2]
p(g_1#) = [2] x2 + [1]
p(g_2#) = [1] x1 + [1] x2 + [0]
p(g_3#) = [2] x2 + [0]
p(g_4#) = [2] x2 + [0]
p(g_5#) = [2] x2 + [0]
p(c_1) = [2]
p(c_2) = [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [2] x1 + [0]
p(c_5) = [2] x1 + [1]
p(c_6) = [1] x1 + [8]
p(c_7) = [1]
p(c_8) = [4] x1 + [1] x2 + [0]
p(c_9) = [1]
p(c_10) = [0]
p(c_11) = [1] x1 + [2] x2 + [1]

Following rules are strictly oriented:
g_2#(s(x),y) = [1] x + [1] y + [2]
> [1] x + [1] y + [0]
= c_8(f_1#(y),g_2#(x,y))

Following rules are (at-least) weakly oriented:
f_1#(x) =  [0]
>= [0]
=  c_2()

f_2#(x) =  [2] x + [0]
>= [2] x + [0]
=  c_3(g_2#(x,x))

f_3#(x) =  [2] x + [0]
>= [2] x + [0]
=  g_3#(x,x)

f_4#(x) =  [2] x + [0]
>= [2] x + [0]
=  g_4#(x,x)

g_3#(s(x),y) =  [2] y + [0]
>= [2] y + [0]
=  f_2#(y)

g_3#(s(x),y) =  [2] y + [0]
>= [2] y + [0]
=  g_3#(x,y)

g_4#(s(x),y) =  [2] y + [0]
>= [2] y + [0]
=  f_3#(y)

g_4#(s(x),y) =  [2] y + [0]
>= [2] y + [0]
=  g_4#(x,y)

g_5#(s(x),y) =  [2] y + [0]
>= [2] y + [0]
=  f_4#(y)

g_5#(s(x),y) =  [2] y + [0]
>= [2] y + [0]
=  g_5#(x,y)

****** Step 7.b:1.b:1.b:1.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f_1#(x) -> c_2()
- Weak DPs:
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> g_3#(x,x)
f_4#(x) -> g_4#(x,x)
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
g_3#(s(x),y) -> f_2#(y)
g_3#(s(x),y) -> g_3#(x,y)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

****** Step 7.b:1.b:1.b:1.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f_1#(x) -> c_2()
f_2#(x) -> c_3(g_2#(x,x))
f_3#(x) -> g_3#(x,x)
f_4#(x) -> g_4#(x,x)
g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
g_3#(s(x),y) -> f_2#(y)
g_3#(s(x),y) -> g_3#(x,y)
g_4#(s(x),y) -> f_3#(y)
g_4#(s(x),y) -> g_4#(x,y)
g_5#(s(x),y) -> f_4#(y)
g_5#(s(x),y) -> g_5#(x,y)
- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:f_1#(x) -> c_2()

2:W:f_2#(x) -> c_3(g_2#(x,x))
-->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):5

3:W:f_3#(x) -> g_3#(x,x)
-->_1 g_3#(s(x),y) -> g_3#(x,y):7
-->_1 g_3#(s(x),y) -> f_2#(y):6

4:W:f_4#(x) -> g_4#(x,x)
-->_1 g_4#(s(x),y) -> g_4#(x,y):9
-->_1 g_4#(s(x),y) -> f_3#(y):8

5:W:g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
-->_2 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):5
-->_1 f_1#(x) -> c_2():1

6:W:g_3#(s(x),y) -> f_2#(y)
-->_1 f_2#(x) -> c_3(g_2#(x,x)):2

7:W:g_3#(s(x),y) -> g_3#(x,y)
-->_1 g_3#(s(x),y) -> g_3#(x,y):7
-->_1 g_3#(s(x),y) -> f_2#(y):6

8:W:g_4#(s(x),y) -> f_3#(y)
-->_1 f_3#(x) -> g_3#(x,x):3

9:W:g_4#(s(x),y) -> g_4#(x,y)
-->_1 g_4#(s(x),y) -> g_4#(x,y):9
-->_1 g_4#(s(x),y) -> f_3#(y):8

10:W:g_5#(s(x),y) -> f_4#(y)
-->_1 f_4#(x) -> g_4#(x,x):4

11:W:g_5#(s(x),y) -> g_5#(x,y)
-->_1 g_5#(s(x),y) -> g_5#(x,y):11
-->_1 g_5#(s(x),y) -> f_4#(y):10

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
11: g_5#(s(x),y) -> g_5#(x,y)
10: g_5#(s(x),y) -> f_4#(y)
4: f_4#(x) -> g_4#(x,x)
9: g_4#(s(x),y) -> g_4#(x,y)
8: g_4#(s(x),y) -> f_3#(y)
3: f_3#(x) -> g_3#(x,x)
7: g_3#(s(x),y) -> g_3#(x,y)
6: g_3#(s(x),y) -> f_2#(y)
2: f_2#(x) -> c_3(g_2#(x,x))
5: g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y))
1: f_1#(x) -> c_2()
****** Step 7.b:1.b:1.b:1.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1
,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2
,c_10/2,c_11/2}
- Obligation:
runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#
,g_5#} and constructors {a,b,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^5))
```