```* Step 1: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
if(false(),x,y) -> y
if(true(),x,y) -> x
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(if) = {1,3},
uargs(minus) = {2},
uargs(p) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(false) = 
p(if) =  x1 +  x2 +  x3 + 
p(le) = 
p(minus) =  x1 +  x2 + 
p(p) =  x1 + 
p(s) =  x1 + 
p(true) = 

Following rules are strictly oriented:
p(0()) = 
> 
= 0()

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

Following rules are (at-least) weakly oriented:
if(false(),x,y) =   x +  y + 
>=  y + 
=  y

if(true(),x,y) =   x +  y + 
>=  x + 
=  x

le(0(),y) =  
>= 
=  true()

le(s(x),0()) =  
>= 
=  false()

le(s(x),s(y)) =  
>= 
=  le(x,y)

minus(x,0()) =   x + 
>=  x + 
=  x

minus(x,s(y)) =   x +  y + 
>=  x +  y + 
=  if(le(x,s(y)),0(),p(minus(x,p(s(y)))))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
if(false(),x,y) -> y
if(true(),x,y) -> x
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
- Weak TRS:
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(if) = {1,3},
uargs(minus) = {2},
uargs(p) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(false) = 
p(if) =  x1 +  x2 +  x3 + 
p(le) = 
p(minus) =  x1 +  x2 + 
p(p) =  x1 + 
p(s) =  x1 + 
p(true) = 

Following rules are strictly oriented:
le(0(),y) = 
> 
= true()

le(s(x),0()) = 
> 
= false()

Following rules are (at-least) weakly oriented:
if(false(),x,y) =   x +  y + 
>=  y + 
=  y

if(true(),x,y) =   x +  y + 
>=  x + 
=  x

le(s(x),s(y)) =  
>= 
=  le(x,y)

minus(x,0()) =   x + 
>=  x + 
=  x

minus(x,s(y)) =   x +  y + 
>=  x +  y + 
=  if(le(x,s(y)),0(),p(minus(x,p(s(y)))))

p(0()) =  
>= 
=  0()

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

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
if(false(),x,y) -> y
if(true(),x,y) -> x
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(if) = {1,3},
uargs(minus) = {2},
uargs(p) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(false) = 
p(if) =  x1 +  x2 +  x3 + 
p(le) = 
p(minus) =  x1 +  x2 + 
p(p) =  x1 + 
p(s) =  x1 + 
p(true) = 

Following rules are strictly oriented:
minus(x,0()) =  x + 
>  x + 
= x

Following rules are (at-least) weakly oriented:
if(false(),x,y) =   x +  y + 
>=  y + 
=  y

if(true(),x,y) =   x +  y + 
>=  x + 
=  x

le(0(),y) =  
>= 
=  true()

le(s(x),0()) =  
>= 
=  false()

le(s(x),s(y)) =  
>= 
=  le(x,y)

minus(x,s(y)) =   x +  y + 
>=  x +  y + 
=  if(le(x,s(y)),0(),p(minus(x,p(s(y)))))

p(0()) =  
>= 
=  0()

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

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
if(false(),x,y) -> y
if(true(),x,y) -> x
le(s(x),s(y)) -> le(x,y)
minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
minus(x,0()) -> x
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(if) = {1,3},
uargs(minus) = {2},
uargs(p) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(false) = 
p(if) =  x1 +  x2 +  x3 + 
p(le) = 
p(minus) =  x1 +  x2 + 
p(p) =  x1 + 
p(s) =  x1 + 
p(true) = 

Following rules are strictly oriented:
if(false(),x,y) =  x +  y + 
>  y + 
= y

if(true(),x,y) =  x +  y + 
>  x + 
= x

Following rules are (at-least) weakly oriented:
le(0(),y) =  
>= 
=  true()

le(s(x),0()) =  
>= 
=  false()

le(s(x),s(y)) =  
>= 
=  le(x,y)

minus(x,0()) =   x + 
>=  x + 
=  x

minus(x,s(y)) =   x +  y + 
>=  x +  y + 
=  if(le(x,s(y)),0(),p(minus(x,p(s(y)))))

p(0()) =  
>= 
=  0()

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

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
le(s(x),s(y)) -> le(x,y)
minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
- Weak TRS:
if(false(),x,y) -> y
if(true(),x,y) -> x
le(0(),y) -> true()
le(s(x),0()) -> false()
minus(x,0()) -> x
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 2))), miDimension = 3, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 2))):

The following argument positions are considered usable:
uargs(if) = {1,3},
uargs(minus) = {2},
uargs(p) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 


p(false) = 


p(if) = [1 0 0]       [2 0 2]       [1 0 0]       
[0 0 0] x_1 + [0 2 1] x_2 + [0 1 0] x_3 + 
[1 0 0]       [1 1 2]       [0 0 1]       
p(le) = [0 0 0]       [0 0 0]       
[2 1 0] x_1 + [0 0 2] x_2 + 
[1 0 0]       [0 0 2]       
p(minus) = [3 2 0]       [2 0 1]       
[3 2 0] x_1 + [2 1 3] x_2 + 
[3 2 1]       [3 0 3]       
p(p) = [1 0 0]       
[1 0 0] x_1 + 
[0 1 0]       
p(s) = [1 1 0]       
[0 0 1] x_1 + 
[0 0 1]       
p(true) = 



Following rules are strictly oriented:
minus(x,s(y)) = [3 2 0]     [2 2 1]     
[3 2 0] x + [2 2 4] y + 
[3 2 1]     [3 3 3]     
> [3 2 0]     [2 2 1]     
[3 2 0] x + [2 2 1] y + 
[3 2 0]     [3 3 3]     
= if(le(x,s(y)),0(),p(minus(x,p(s(y)))))

Following rules are (at-least) weakly oriented:
if(false(),x,y) =  [2 0 2]     [1 0 0]     
[0 2 1] x + [0 1 0] y + 
[1 1 2]     [0 0 1]     
>= [1 0 0]     
[0 1 0] y + 
[0 0 1]     
=  y

if(true(),x,y) =  [2 0 2]     [1 0 0]     
[0 2 1] x + [0 1 0] y + 
[1 1 2]     [0 0 1]     
>= [1 0 0]     
[0 1 0] x + 
[0 0 1]     
=  x

le(0(),y) =  [0 0 0]     
[0 0 2] y + 
[0 0 2]     
>= 


=  true()

le(s(x),0()) =  [0 0 0]     
[2 2 1] x + 
[1 1 0]     
>= 


=  false()

le(s(x),s(y)) =  [0 0 0]     [0 0 0]     
[2 2 1] x + [0 0 2] y + 
[1 1 0]     [0 0 2]     
>= [0 0 0]     [0 0 0]     
[2 1 0] x + [0 0 2] y + 
[1 0 0]     [0 0 2]     
=  le(x,y)

minus(x,0()) =  [3 2 0]     
[3 2 0] x + 
[3 2 1]     
>= [1 0 0]     
[0 1 0] x + 
[0 0 1]     
=  x

p(0()) =  


>= 


=  0()

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

* Step 6: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
le(s(x),s(y)) -> le(x,y)
- Weak TRS:
if(false(),x,y) -> y
if(true(),x,y) -> x
le(0(),y) -> true()
le(s(x),0()) -> false()
minus(x,0()) -> x
minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 3))), miDimension = 4, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 3))):

The following argument positions are considered usable:
uargs(if) = {1,3},
uargs(minus) = {2},
uargs(p) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 



p(false) = 



p(if) = [1 0 0 0]       [1 0 0 0]       [1 0 0 0]       
[0 0 0 0] x_1 + [0 1 0 0] x_2 + [0 1 0 0] x_3 + 
[0 0 0 0]       [0 0 1 0]       [0 0 1 0]       
[0 0 0 0]       [0 0 0 1]       [0 0 0 1]       
p(le) = [0 0 0 0]       [0 0 0 1]       
[1 0 0 1] x_1 + [0 0 0 0] x_2 + 
[0 0 0 0]       [0 0 1 1]       
[0 0 1 0]       [0 0 0 1]       
p(minus) = [1 1 0 0]       [1 0 1 0]       
[1 1 1 0] x_1 + [1 0 1 0] x_2 + 
[1 1 1 0]       [1 0 1 0]       
[1 1 1 1]       [1 1 0 0]       
p(p) = [1 0 0 0]       
[1 0 0 0] x_1 + 
[0 1 0 0]       
[0 0 1 0]       
p(s) = [1 1 0 1]       
[0 0 1 0] x_1 + 
[0 0 1 1]       
[0 0 0 1]       
p(true) = 




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

Following rules are (at-least) weakly oriented:
if(false(),x,y) =  [1 0 0 0]     [1 0 0 0]     
[0 1 0 0] x + [0 1 0 0] y + 
[0 0 1 0]     [0 0 1 0]     
[0 0 0 1]     [0 0 0 1]     
>= [1 0 0 0]     
[0 1 0 0] y + 
[0 0 1 0]     
[0 0 0 1]     
=  y

if(true(),x,y) =  [1 0 0 0]     [1 0 0 0]     
[0 1 0 0] x + [0 1 0 0] y + 
[0 0 1 0]     [0 0 1 0]     
[0 0 0 1]     [0 0 0 1]     
>= [1 0 0 0]     
[0 1 0 0] x + 
[0 0 1 0]     
[0 0 0 1]     
=  x

le(0(),y) =  [0 0 0 1]     
[0 0 0 0] y + 
[0 0 1 1]     
[0 0 0 1]     
>= 



=  true()

le(s(x),0()) =  [0 0 0 0]     
[1 1 0 2] x + 
[0 0 0 0]     
[0 0 1 1]     
>= 



=  false()

minus(x,0()) =  [1 1 0 0]     
[1 1 1 0] x + 
[1 1 1 0]     
[1 1 1 1]     
>= [1 0 0 0]     
[0 1 0 0] x + 
[0 0 1 0]     
[0 0 0 1]     
=  x

minus(x,s(y)) =  [1 1 0 0]     [1 1 1 2]     
[1 1 1 0] x + [1 1 1 2] y + 
[1 1 1 0]     [1 1 1 2]     
[1 1 1 1]     [1 1 1 1]     
>= [1 1 0 0]     [1 1 1 2]     
[1 1 0 0] x + [1 1 1 1] y + 
[1 1 1 0]     [1 1 1 1]     
[1 1 1 0]     [1 1 1 1]     
=  if(le(x,s(y)),0(),p(minus(x,p(s(y)))))

p(0()) =  



>= 



=  0()

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

* Step 7: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
if(false(),x,y) -> y
if(true(),x,y) -> x
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
p(0()) -> 0()
p(s(x)) -> x
- Signature:
{if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^3))
```