```* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
-(x,0()) -> x
-(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0())
-(0(),y) -> 0()
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{-/2,p/1} / {0/0,greater/2,if/3,s/1}
- Obligation:
runtime complexity wrt. defined symbols {-,p} and constructors {0,greater,if,s}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak dependency pairs:

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

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
-#(0(),y) -> c_3()
p#(0()) -> c_4()
p#(s(x)) -> c_5(x)
- Strict TRS:
-(x,0()) -> x
-(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0())
-(0(),y) -> 0()
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
- Obligation:
runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
p(s(x)) -> x
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
-#(0(),y) -> c_3()
p#(0()) -> c_4()
p#(s(x)) -> c_5(x)
* Step 3: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
-#(0(),y) -> c_3()
p#(0()) -> c_4()
p#(s(x)) -> c_5(x)
- Strict TRS:
p(s(x)) -> x
- Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
- Obligation:
runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(-#) = {2},
uargs(c_2) = {3}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(-) = 
p(0) = 
p(greater) =  x1 +  x2 + 
p(if) =  x1 +  x2 +  x3 + 
p(p) =  x1 + 
p(s) =  x1 + 
p(-#) =  x2 + 
p(p#) = 
p(c_1) = 
p(c_2) =  x3 + 
p(c_3) = 
p(c_4) = 
p(c_5) = 

Following rules are strictly oriented:
p(s(x)) =  x + 
>  x + 
= x

Following rules are (at-least) weakly oriented:
-#(x,0()) =  
>= 
=  c_1(x)

-#(x,s(y)) =   y + 
>=  y + 
=  c_2(x,y,-#(x,p(s(y))))

-#(0(),y) =   y + 
>= 
=  c_3()

p#(0()) =  
>= 
=  c_4()

p#(s(x)) =  
>= 
=  c_5(x)

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
-#(0(),y) -> c_3()
p#(0()) -> c_4()
p#(s(x)) -> c_5(x)
- Weak TRS:
p(s(x)) -> x
- Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
- Obligation:
runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{3,4}
by application of
Pre({3,4}) = {1,2,5}.
Here rules are labelled as follows:
1: -#(x,0()) -> c_1(x)
2: -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
3: -#(0(),y) -> c_3()
4: p#(0()) -> c_4()
5: p#(s(x)) -> c_5(x)
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
p#(s(x)) -> c_5(x)
- Weak DPs:
-#(0(),y) -> c_3()
p#(0()) -> c_4()
- Weak TRS:
p(s(x)) -> x
- Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
- Obligation:
runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:-#(x,0()) -> c_1(x)
-->_1 p#(s(x)) -> c_5(x):3
-->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_1 p#(0()) -> c_4():5
-->_1 -#(0(),y) -> c_3():4
-->_1 -#(x,0()) -> c_1(x):1

2:S:-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
-->_2 p#(s(x)) -> c_5(x):3
-->_1 p#(s(x)) -> c_5(x):3
-->_2 p#(0()) -> c_4():5
-->_1 p#(0()) -> c_4():5
-->_3 -#(0(),y) -> c_3():4
-->_2 -#(0(),y) -> c_3():4
-->_1 -#(0(),y) -> c_3():4
-->_3 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_2 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_3 -#(x,0()) -> c_1(x):1
-->_2 -#(x,0()) -> c_1(x):1
-->_1 -#(x,0()) -> c_1(x):1

3:S:p#(s(x)) -> c_5(x)
-->_1 p#(0()) -> c_4():5
-->_1 -#(0(),y) -> c_3():4
-->_1 p#(s(x)) -> c_5(x):3
-->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_1 -#(x,0()) -> c_1(x):1

4:W:-#(0(),y) -> c_3()

5:W:p#(0()) -> c_4()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: -#(0(),y) -> c_3()
5: p#(0()) -> c_4()
* Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
p#(s(x)) -> c_5(x)
- Weak TRS:
p(s(x)) -> x
- Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
- Obligation:
runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,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:
3: p#(s(x)) -> c_5(x)

The strictly oriented rules are moved into the weak component.
** Step 6.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
p#(s(x)) -> c_5(x)
- Weak TRS:
p(s(x)) -> x
- Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
- Obligation:
runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,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_2) = {3}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(-) = 
p(0) = 
p(greater) =  x1 +  x2 + 
p(if) =  x1 +  x2 +  x3 + 
p(p) =  x1 + 
p(s) =  x1 + 
p(-#) = 
p(p#) = 
p(c_1) = 
p(c_2) =  x3 + 
p(c_3) = 
p(c_4) = 
p(c_5) = 

Following rules are strictly oriented:
p#(s(x)) = 
> 
= c_5(x)

Following rules are (at-least) weakly oriented:
-#(x,0()) =  
>= 
=  c_1(x)

-#(x,s(y)) =  
>= 
=  c_2(x,y,-#(x,p(s(y))))

p(s(x)) =   x + 
>=  x + 
=  x

** Step 6.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
- Weak DPs:
p#(s(x)) -> c_5(x)
- Weak TRS:
p(s(x)) -> x
- Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
- Obligation:
runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 6.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
- Weak DPs:
p#(s(x)) -> c_5(x)
- Weak TRS:
p(s(x)) -> x
- Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
- Obligation:
runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,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: -#(x,0()) -> c_1(x)

The strictly oriented rules are moved into the weak component.
*** Step 6.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
- Weak DPs:
p#(s(x)) -> c_5(x)
- Weak TRS:
p(s(x)) -> x
- Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
- Obligation:
runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,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_2) = {3}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(-) =  x1 + 
p(0) = 
p(greater) =  x2 + 
p(if) =  x3 + 
p(p) =  x1 + 
p(s) =  x1 + 
p(-#) = 
p(p#) =  x1 + 
p(c_1) = 
p(c_2) =  x3 + 
p(c_3) = 
p(c_4) = 
p(c_5) =  x1 + 

Following rules are strictly oriented:
-#(x,0()) = 
> 
= c_1(x)

Following rules are (at-least) weakly oriented:
-#(x,s(y)) =  
>= 
=  c_2(x,y,-#(x,p(s(y))))

p#(s(x)) =   x + 
>=  x + 
=  c_5(x)

p(s(x)) =   x + 
>=  x + 
=  x

*** Step 6.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
- Weak DPs:
-#(x,0()) -> c_1(x)
p#(s(x)) -> c_5(x)
- Weak TRS:
p(s(x)) -> x
- Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
- Obligation:
runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

*** Step 6.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
- Weak DPs:
-#(x,0()) -> c_1(x)
p#(s(x)) -> c_5(x)
- Weak TRS:
p(s(x)) -> x
- Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
- Obligation:
runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))

The strictly oriented rules are moved into the weak component.
**** Step 6.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
- Weak DPs:
-#(x,0()) -> c_1(x)
p#(s(x)) -> c_5(x)
- Weak TRS:
p(s(x)) -> x
- Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
- Obligation:
runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 2, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_2) = {3}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(-) = [1 1 1]      [2 0 2]      
[2 2 0] x1 + [2 0 2] x2 + 
[2 2 1]      [1 0 2]      
p(0) = 


p(greater) = [0 0 0]      [0 0 2]      
[0 0 2] x1 + [0 0 0] x2 + 
[0 0 0]      [0 0 0]      
p(if) = [0 0 1]      [0 0 0]      [0 0 1]      
[0 0 2] x1 + [0 0 1] x2 + [0 0 1] x3 + 
[0 0 0]      [0 0 0]      [0 0 0]      
p(p) = [2 3 0]      
[2 0 0] x1 + 
[0 1 0]      
p(s) = [1 1 0]      
[0 0 1] x1 + 
[0 0 1]      
p(-#) = [0 0 1]      [0 0 2]      
[0 0 3] x1 + [0 1 0] x2 + 
[3 2 0]      [2 1 1]      
p(p#) = [0 0 2]      
[1 0 0] x1 + 
[1 3 1]      
p(c_1) = [0 0 0]      
[0 0 0] x1 + 
[1 0 0]      
p(c_2) = [0 0 0]      [0 0 0]      [1 0 0]      
[0 0 0] x1 + [0 0 1] x2 + [0 0 0] x3 + 
[3 0 0]      [2 0 0]      [0 0 0]      
p(c_3) = 


p(c_4) = 


p(c_5) = [0 0 0]      
[1 0 0] x1 + 
[0 0 0]      

Following rules are strictly oriented:
-#(x,s(y)) = [0 0 1]     [0 0 2]     
[0 0 3] x + [0 0 1] y + 
[3 2 0]     [2 2 2]     
> [0 0 1]     [0 0 2]     
[0 0 0] x + [0 0 1] y + 
[3 0 0]     [2 0 0]     
= c_2(x,y,-#(x,p(s(y))))

Following rules are (at-least) weakly oriented:
-#(x,0()) =  [0 0 1]     
[0 0 3] x + 
[3 2 0]     
>= [0 0 0]     
[0 0 0] x + 
[1 0 0]     
=  c_1(x)

p#(s(x)) =  [0 0 2]     
[1 1 0] x + 
[1 1 4]     
>= [0 0 0]     
[1 0 0] x + 
[0 0 0]     
=  c_5(x)

p(s(x)) =  [2 2 3]     
[2 2 0] x + 
[0 0 1]     
>= [1 0 0]     
[0 1 0] x + 
[0 0 1]     
=  x

**** Step 6.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
p#(s(x)) -> c_5(x)
- Weak TRS:
p(s(x)) -> x
- Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
- Obligation:
runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

**** Step 6.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
p#(s(x)) -> c_5(x)
- Weak TRS:
p(s(x)) -> x
- Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
- Obligation:
runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:-#(x,0()) -> c_1(x)
-->_1 p#(s(x)) -> c_5(x):3
-->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_1 -#(x,0()) -> c_1(x):1

2:W:-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
-->_2 p#(s(x)) -> c_5(x):3
-->_1 p#(s(x)) -> c_5(x):3
-->_3 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_2 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_3 -#(x,0()) -> c_1(x):1
-->_2 -#(x,0()) -> c_1(x):1
-->_1 -#(x,0()) -> c_1(x):1

3:W:p#(s(x)) -> c_5(x)
-->_1 p#(s(x)) -> c_5(x):3
-->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_1 -#(x,0()) -> c_1(x):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: -#(x,0()) -> c_1(x)
3: p#(s(x)) -> c_5(x)
2: -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
**** Step 6.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
p(s(x)) -> x
- Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
- Obligation:
runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))
```