```* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
average(x,s(s(s(y)))) -> s(average(s(x),y))
average(0(),0()) -> 0()
average(0(),s(0())) -> 0()
average(0(),s(s(0()))) -> s(0())
average(s(x),y) -> average(x,s(y))
- Signature:
{average/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {average} and constructors {0,s}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:

Strict DPs
average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
average#(0(),0()) -> c_2()
average#(0(),s(0())) -> c_3()
average#(0(),s(s(0()))) -> c_4()
average#(s(x),y) -> c_5(average#(x,s(y)))
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
average#(0(),0()) -> c_2()
average#(0(),s(0())) -> c_3()
average#(0(),s(s(0()))) -> c_4()
average#(s(x),y) -> c_5(average#(x,s(y)))
- Strict TRS:
average(x,s(s(s(y)))) -> s(average(s(x),y))
average(0(),0()) -> 0()
average(0(),s(0())) -> 0()
average(0(),s(s(0()))) -> s(0())
average(s(x),y) -> average(x,s(y))
- Signature:
{average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
average#(0(),0()) -> c_2()
average#(0(),s(0())) -> c_3()
average#(0(),s(s(0()))) -> c_4()
average#(s(x),y) -> c_5(average#(x,s(y)))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
average#(0(),0()) -> c_2()
average#(0(),s(0())) -> c_3()
average#(0(),s(s(0()))) -> c_4()
average#(s(x),y) -> c_5(average#(x,s(y)))
- Signature:
{average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2,3,4}
by application of
Pre({2,3,4}) = {5}.
Here rules are labelled as follows:
1: average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
2: average#(0(),0()) -> c_2()
3: average#(0(),s(0())) -> c_3()
4: average#(0(),s(s(0()))) -> c_4()
5: average#(s(x),y) -> c_5(average#(x,s(y)))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
average#(s(x),y) -> c_5(average#(x,s(y)))
- Weak DPs:
average#(0(),0()) -> c_2()
average#(0(),s(0())) -> c_3()
average#(0(),s(s(0()))) -> c_4()
- Signature:
{average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
-->_1 average#(s(x),y) -> c_5(average#(x,s(y))):2
-->_1 average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)):1

2:S:average#(s(x),y) -> c_5(average#(x,s(y)))
-->_1 average#(0(),s(s(0()))) -> c_4():5
-->_1 average#(0(),s(0())) -> c_3():4
-->_1 average#(s(x),y) -> c_5(average#(x,s(y))):2
-->_1 average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)):1

3:W:average#(0(),0()) -> c_2()

4:W:average#(0(),s(0())) -> c_3()

5:W:average#(0(),s(s(0()))) -> c_4()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: average#(0(),0()) -> c_2()
4: average#(0(),s(0())) -> c_3()
5: average#(0(),s(s(0()))) -> c_4()
* Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
average#(s(x),y) -> c_5(average#(x,s(y)))
- Signature:
{average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {average#} and constructors {0,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: average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
2: average#(s(x),y) -> c_5(average#(x,s(y)))

The strictly oriented rules are moved into the weak component.
** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
average#(s(x),y) -> c_5(average#(x,s(y)))
- Signature:
{average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {average#} and constructors {0,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_1) = {1},
uargs(c_5) = {1}

Following symbols are considered usable:
{average#}
TcT has computed the following interpretation:
p(0) = 
p(average) =  x2 + 
p(s) =  x1 + 
p(average#) =  x1 +  x2 + 
p(c_1) =  x1 + 
p(c_2) = 
p(c_3) = 
p(c_4) = 
p(c_5) =  x1 + 

Following rules are strictly oriented:
average#(x,s(s(s(y)))) =  x +  y + 
>  x +  y + 
= c_1(average#(s(x),y))

average#(s(x),y) =  x +  y + 
>  x +  y + 
= c_5(average#(x,s(y)))

Following rules are (at-least) weakly oriented:

** Step 5.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
average#(s(x),y) -> c_5(average#(x,s(y)))
- Signature:
{average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
average#(s(x),y) -> c_5(average#(x,s(y)))
- Signature:
{average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
-->_1 average#(s(x),y) -> c_5(average#(x,s(y))):2
-->_1 average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)):1

2:W:average#(s(x),y) -> c_5(average#(x,s(y)))
-->_1 average#(s(x),y) -> c_5(average#(x,s(y))):2
-->_1 average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
2: average#(s(x),y) -> c_5(average#(x,s(y)))
** Step 5.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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