```* Step 1: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__nats() -> nats()
a__tl(X) -> tl(X)
a__tl(cons(X,Y)) -> mark(Y)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(incr(X)) -> a__incr(mark(X))
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
mark(zeros()) -> a__zeros()
- Signature:
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
+ 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(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(a__adx) =  x1 + 
p(a__hd) =  x1 + 
p(a__incr) =  x1 + 
p(a__nats) = 
p(a__tl) =  x1 + 
p(a__zeros) = 
p(cons) = 
p(hd) =  x1 + 
p(incr) =  x1 + 
p(mark) = 
p(nats) = 
p(s) = 
p(tl) =  x1 + 
p(zeros) = 

Following rules are strictly oriented:
a__adx(X) =  X + 
> 

> 

Following rules are (at-least) weakly oriented:
a__hd(X) =   X + 
>=  X + 
=  hd(X)

a__hd(cons(X,Y)) =  
>= 
=  mark(X)

a__incr(X) =   X + 
>=  X + 
=  incr(X)

a__incr(cons(X,Y)) =  
>= 
=  cons(s(X),incr(Y))

a__nats() =  
>= 

a__nats() =  
>= 
=  nats()

a__tl(X) =   X + 
>=  X + 
=  tl(X)

a__tl(cons(X,Y)) =  
>= 
=  mark(Y)

a__zeros() =  
>= 
=  cons(0(),zeros())

a__zeros() =  
>= 
=  zeros()

mark(0()) =  
>= 
=  0()

>= 

mark(cons(X1,X2)) =  
>= 
=  cons(X1,X2)

mark(hd(X)) =  
>= 
=  a__hd(mark(X))

mark(incr(X)) =  
>= 
=  a__incr(mark(X))

mark(nats()) =  
>= 
=  a__nats()

mark(s(X)) =  
>= 
=  s(X)

mark(tl(X)) =  
>= 
=  a__tl(mark(X))

mark(zeros()) =  
>= 
=  a__zeros()

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:
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__nats() -> nats()
a__tl(X) -> tl(X)
a__tl(cons(X,Y)) -> mark(Y)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(incr(X)) -> a__incr(mark(X))
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
mark(zeros()) -> a__zeros()
- Weak TRS:
- Signature:
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
+ 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(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(a__adx) =  x1 + 
p(a__hd) =  x1 + 
p(a__incr) =  x1 + 
p(a__nats) = 
p(a__tl) =  x1 + 
p(a__zeros) = 
p(adx) =  x1 + 
p(cons) =  x1 +  x2 + 
p(hd) =  x1 + 
p(incr) =  x1 + 
p(mark) =  x1 + 
p(nats) = 
p(s) =  x1 + 
p(tl) =  x1 + 
p(zeros) = 

Following rules are strictly oriented:
a__hd(cons(X,Y)) =  X +  Y + 
>  X + 
= mark(X)

a__tl(cons(X,Y)) =  X +  Y + 
>  Y + 
= mark(Y)

mark(zeros()) = 
> 
= a__zeros()

Following rules are (at-least) weakly oriented:
a__adx(X) =   X + 
>=  X + 

a__adx(cons(X,Y)) =   X +  Y + 
>=  X +  Y + 

a__hd(X) =   X + 
>=  X + 
=  hd(X)

a__incr(X) =   X + 
>=  X + 
=  incr(X)

a__incr(cons(X,Y)) =   X +  Y + 
>=  X +  Y + 
=  cons(s(X),incr(Y))

a__nats() =  
>= 

a__nats() =  
>= 
=  nats()

a__tl(X) =   X + 
>=  X + 
=  tl(X)

a__zeros() =  
>= 
=  cons(0(),zeros())

a__zeros() =  
>= 
=  zeros()

mark(0()) =  
>= 
=  0()

mark(adx(X)) =   X + 
>=  X + 

mark(cons(X1,X2)) =   X1 +  X2 + 
>=  X1 +  X2 + 
=  cons(X1,X2)

mark(hd(X)) =   X + 
>=  X + 
=  a__hd(mark(X))

mark(incr(X)) =   X + 
>=  X + 
=  a__incr(mark(X))

mark(nats()) =  
>= 
=  a__nats()

mark(s(X)) =   X + 
>=  X + 
=  s(X)

mark(tl(X)) =   X + 
>=  X + 
=  a__tl(mark(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:
a__hd(X) -> hd(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__nats() -> nats()
a__tl(X) -> tl(X)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(incr(X)) -> a__incr(mark(X))
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
- Weak TRS:
a__hd(cons(X,Y)) -> mark(X)
a__tl(cons(X,Y)) -> mark(Y)
mark(zeros()) -> a__zeros()
- Signature:
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
+ 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(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(a__adx) =  x1 + 
p(a__hd) =  x1 + 
p(a__incr) =  x1 + 
p(a__nats) = 
p(a__tl) =  x1 + 
p(a__zeros) = 
p(adx) =  x1 + 
p(cons) = 
p(hd) = 
p(incr) = 
p(mark) = 
p(nats) = 
p(s) = 
p(tl) =  x1 + 
p(zeros) = 

Following rules are strictly oriented:
a__hd(X) =  X + 
> 
= hd(X)

Following rules are (at-least) weakly oriented:
a__adx(X) =   X + 
>=  X + 

>= 

a__hd(cons(X,Y)) =  
>= 
=  mark(X)

a__incr(X) =   X + 
>= 
=  incr(X)

a__incr(cons(X,Y)) =  
>= 
=  cons(s(X),incr(Y))

a__nats() =  
>= 

a__nats() =  
>= 
=  nats()

a__tl(X) =   X + 
>=  X + 
=  tl(X)

a__tl(cons(X,Y)) =  
>= 
=  mark(Y)

a__zeros() =  
>= 
=  cons(0(),zeros())

a__zeros() =  
>= 
=  zeros()

mark(0()) =  
>= 
=  0()

>= 

mark(cons(X1,X2)) =  
>= 
=  cons(X1,X2)

mark(hd(X)) =  
>= 
=  a__hd(mark(X))

mark(incr(X)) =  
>= 
=  a__incr(mark(X))

mark(nats()) =  
>= 
=  a__nats()

mark(s(X)) =  
>= 
=  s(X)

mark(tl(X)) =  
>= 
=  a__tl(mark(X))

mark(zeros()) =  
>= 
=  a__zeros()

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:
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__nats() -> nats()
a__tl(X) -> tl(X)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(incr(X)) -> a__incr(mark(X))
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
- Weak TRS:
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__tl(cons(X,Y)) -> mark(Y)
mark(zeros()) -> a__zeros()
- Signature:
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
+ 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(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(a__adx) =  x1 + 
p(a__hd) =  x1 + 
p(a__incr) =  x1 + 
p(a__nats) = 
p(a__tl) =  x1 + 
p(a__zeros) = 
p(cons) = 
p(hd) = 
p(incr) =  x1 + 
p(mark) = 
p(nats) = 
p(s) = 
p(tl) =  x1 + 
p(zeros) = 

Following rules are strictly oriented:
a__incr(X) =  X + 
>  X + 
= incr(X)

a__incr(cons(X,Y)) = 
> 
= cons(s(X),incr(Y))

mark(0()) = 
> 
= 0()

mark(nats()) = 
> 
= a__nats()

mark(s(X)) = 
> 
= s(X)

Following rules are (at-least) weakly oriented:
a__adx(X) =   X + 
>= 

>= 

a__hd(X) =   X + 
>= 
=  hd(X)

a__hd(cons(X,Y)) =  
>= 
=  mark(X)

a__nats() =  
>= 

a__nats() =  
>= 
=  nats()

a__tl(X) =   X + 
>=  X + 
=  tl(X)

a__tl(cons(X,Y)) =  
>= 
=  mark(Y)

a__zeros() =  
>= 
=  cons(0(),zeros())

a__zeros() =  
>= 
=  zeros()

>= 

mark(cons(X1,X2)) =  
>= 
=  cons(X1,X2)

mark(hd(X)) =  
>= 
=  a__hd(mark(X))

mark(incr(X)) =  
>= 
=  a__incr(mark(X))

mark(tl(X)) =  
>= 
=  a__tl(mark(X))

mark(zeros()) =  
>= 
=  a__zeros()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__nats() -> nats()
a__tl(X) -> tl(X)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(incr(X)) -> a__incr(mark(X))
mark(tl(X)) -> a__tl(mark(X))
- Weak TRS:
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__tl(cons(X,Y)) -> mark(Y)
mark(0()) -> 0()
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(zeros()) -> a__zeros()
- Signature:
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
+ 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(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(a__adx) =  x1 + 
p(a__hd) =  x1 + 
p(a__incr) =  x1 + 
p(a__nats) = 
p(a__tl) =  x1 + 
p(a__zeros) = 
p(adx) =  x1 + 
p(cons) =  x1 +  x2 + 
p(hd) =  x1 + 
p(incr) =  x1 + 
p(mark) =  x1 + 
p(nats) = 
p(s) =  x1 + 
p(tl) =  x1 + 
p(zeros) = 

Following rules are strictly oriented:
mark(tl(X)) =  X + 
>  X + 
= a__tl(mark(X))

Following rules are (at-least) weakly oriented:
a__adx(X) =   X + 
>=  X + 

a__adx(cons(X,Y)) =   X +  Y + 
>=  X +  Y + 

a__hd(X) =   X + 
>=  X + 
=  hd(X)

a__hd(cons(X,Y)) =   X +  Y + 
>=  X + 
=  mark(X)

a__incr(X) =   X + 
>=  X + 
=  incr(X)

a__incr(cons(X,Y)) =   X +  Y + 
>=  X +  Y + 
=  cons(s(X),incr(Y))

a__nats() =  
>= 

a__nats() =  
>= 
=  nats()

a__tl(X) =   X + 
>=  X + 
=  tl(X)

a__tl(cons(X,Y)) =   X +  Y + 
>=  Y + 
=  mark(Y)

a__zeros() =  
>= 
=  cons(0(),zeros())

a__zeros() =  
>= 
=  zeros()

mark(0()) =  
>= 
=  0()

mark(adx(X)) =   X + 
>=  X + 

mark(cons(X1,X2)) =   X1 +  X2 + 
>=  X1 +  X2 + 
=  cons(X1,X2)

mark(hd(X)) =   X + 
>=  X + 
=  a__hd(mark(X))

mark(incr(X)) =   X + 
>=  X + 
=  a__incr(mark(X))

mark(nats()) =  
>= 
=  a__nats()

mark(s(X)) =   X + 
>=  X + 
=  s(X)

mark(zeros()) =  
>= 
=  a__zeros()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: WeightGap WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__nats() -> nats()
a__tl(X) -> tl(X)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(incr(X)) -> a__incr(mark(X))
- Weak TRS:
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__tl(cons(X,Y)) -> mark(Y)
mark(0()) -> 0()
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
mark(zeros()) -> a__zeros()
- Signature:
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
+ 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(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(a__adx) =  x1 + 
p(a__hd) =  x1 + 
p(a__incr) =  x1 + 
p(a__nats) = 
p(a__tl) =  x1 + 
p(a__zeros) = 
p(adx) =  x1 + 
p(cons) =  x1 +  x2 + 
p(hd) =  x1 + 
p(incr) =  x1 + 
p(mark) =  x1 + 
p(nats) = 
p(s) =  x1 + 
p(tl) =  x1 + 
p(zeros) = 

Following rules are strictly oriented:
a__nats() = 
> 

a__nats() = 
> 
= nats()

a__zeros() = 
> 
= cons(0(),zeros())

a__zeros() = 
> 
= zeros()

mark(cons(X1,X2)) =  X1 +  X2 + 
>  X1 +  X2 + 
= cons(X1,X2)

Following rules are (at-least) weakly oriented:
a__adx(X) =   X + 
>=  X + 

a__adx(cons(X,Y)) =   X +  Y + 
>=  X +  Y + 

a__hd(X) =   X + 
>=  X + 
=  hd(X)

a__hd(cons(X,Y)) =   X +  Y + 
>=  X + 
=  mark(X)

a__incr(X) =   X + 
>=  X + 
=  incr(X)

a__incr(cons(X,Y)) =   X +  Y + 
>=  X +  Y + 
=  cons(s(X),incr(Y))

a__tl(X) =   X + 
>=  X + 
=  tl(X)

a__tl(cons(X,Y)) =   X +  Y + 
>=  Y + 
=  mark(Y)

mark(0()) =  
>= 
=  0()

mark(adx(X)) =   X + 
>=  X + 

mark(hd(X)) =   X + 
>=  X + 
=  a__hd(mark(X))

mark(incr(X)) =   X + 
>=  X + 
=  a__incr(mark(X))

mark(nats()) =  
>= 
=  a__nats()

mark(s(X)) =   X + 
>=  X + 
=  s(X)

mark(tl(X)) =   X + 
>=  X + 
=  a__tl(mark(X))

mark(zeros()) =  
>= 
=  a__zeros()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__tl(X) -> tl(X)
mark(hd(X)) -> a__hd(mark(X))
mark(incr(X)) -> a__incr(mark(X))
- Weak TRS:
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__nats() -> nats()
a__tl(cons(X,Y)) -> mark(Y)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
mark(zeros()) -> a__zeros()
- Signature:
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 1))), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 1))):

The following argument positions are considered usable:
uargs(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

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

p(a__adx) = [1 0] x_1 + 
[0 1]       
p(a__hd) = [1 0] x_1 + 
[0 1]       
p(a__incr) = [1 0] x_1 + 
[0 1]       
p(a__nats) = 

p(a__tl) = [1 0] x_1 + 
[0 1]       
p(a__zeros) = 

p(adx) = [0 0] x_1 + 
[0 1]       
p(cons) = [0 2] x_1 + [0 2] x_2 + 
[0 1]       [0 1]       
p(hd) = [0 0] x_1 + 
[0 1]       
p(incr) = [0 0] x_1 + 
[0 1]       
p(mark) = [0 2] x_1 + 
[0 1]       
p(nats) = 

p(s) = 

p(tl) = [0 0] x_1 + 
[0 1]       
p(zeros) = 


Following rules are strictly oriented:
a__tl(X) = [1 0] X + 
[0 1]     
> [0 0] X + 
[0 1]     
= tl(X)

mark(hd(X)) = [0 2] X + 
[0 1]     
> [0 2] X + 
[0 1]     
= a__hd(mark(X))

Following rules are (at-least) weakly oriented:
a__adx(X) =  [1 0] X + 
[0 1]     
>= [0 0] X + 
[0 1]     

a__adx(cons(X,Y)) =  [0 2] X + [0 2] Y + 
[0 1]     [0 1]     
>= [0 2] X + [0 2] Y + 
[0 1]     [0 1]     

a__hd(X) =  [1 0] X + 
[0 1]     
>= [0 0] X + 
[0 1]     
=  hd(X)

a__hd(cons(X,Y)) =  [0 2] X + [0 2] Y + 
[0 1]     [0 1]     
>= [0 2] X + 
[0 1]     
=  mark(X)

a__incr(X) =  [1 0] X + 
[0 1]     
>= [0 0] X + 
[0 1]     
=  incr(X)

a__incr(cons(X,Y)) =  [0 2] X + [0 2] Y + 
[0 1]     [0 1]     
>= [0 2] Y + 
[0 1]     
=  cons(s(X),incr(Y))

a__nats() =  

>= 


a__nats() =  

>= 

=  nats()

a__tl(cons(X,Y)) =  [0 2] X + [0 2] Y + 
[0 1]     [0 1]     
>= [0 2] Y + 
[0 1]     
=  mark(Y)

a__zeros() =  

>= 

=  cons(0(),zeros())

a__zeros() =  

>= 

=  zeros()

mark(0()) =  

>= 

=  0()

mark(adx(X)) =  [0 2] X + 
[0 1]     
>= [0 2] X + 
[0 1]     

mark(cons(X1,X2)) =  [0 2] X1 + [0 2] X2 + 
[0 1]      [0 1]      
>= [0 2] X1 + [0 2] X2 + 
[0 1]      [0 1]      
=  cons(X1,X2)

mark(incr(X)) =  [0 2] X + 
[0 1]     
>= [0 2] X + 
[0 1]     
=  a__incr(mark(X))

mark(nats()) =  

>= 

=  a__nats()

mark(s(X)) =  

>= 

=  s(X)

mark(tl(X)) =  [0 2] X + 
[0 1]     
>= [0 2] X + 
[0 1]     
=  a__tl(mark(X))

mark(zeros()) =  

>= 

=  a__zeros()

* Step 8: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
mark(incr(X)) -> a__incr(mark(X))
- Weak TRS:
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__nats() -> nats()
a__tl(X) -> tl(X)
a__tl(cons(X,Y)) -> mark(Y)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
mark(zeros()) -> a__zeros()
- Signature:
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):

The following argument positions are considered usable:
uargs(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

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

p(a__adx) = [1 0] x_1 + 
[0 1]       
p(a__hd) = [1 4] x_1 + 
[0 1]       
p(a__incr) = [1 0] x_1 + 
[0 1]       
p(a__nats) = 

p(a__tl) = [1 4] x_1 + 
[0 1]       
p(a__zeros) = 

p(adx) = [1 0] x_1 + 
[0 1]       
p(cons) = [1 0] x_1 + [1 0] x_2 + 
[0 1]       [0 1]       
p(hd) = [1 4] x_1 + 
[0 1]       
p(incr) = [1 0] x_1 + 
[0 1]       
p(mark) = [1 4] x_1 + 
[0 1]       
p(nats) = 

p(s) = [0 0] x_1 + 
[0 1]       
p(tl) = [1 4] x_1 + 
[0 1]       
p(zeros) = 


Following rules are strictly oriented:
mark(adx(X)) = [1 4] X + 
[0 1]     
> [1 4] X + 
[0 1]     

Following rules are (at-least) weakly oriented:
a__adx(X) =  [1 0] X + 
[0 1]     
>= [1 0] X + 
[0 1]     

a__adx(cons(X,Y)) =  [1 0] X + [1 0] Y + 
[0 1]     [0 1]     
>= [1 0] X + [1 0] Y + 
[0 1]     [0 1]     

a__hd(X) =  [1 4] X + 
[0 1]     
>= [1 4] X + 
[0 1]     
=  hd(X)

a__hd(cons(X,Y)) =  [1 4] X + [1 4] Y + 
[0 1]     [0 1]     
>= [1 4] X + 
[0 1]     
=  mark(X)

a__incr(X) =  [1 0] X + 
[0 1]     
>= [1 0] X + 
[0 1]     
=  incr(X)

a__incr(cons(X,Y)) =  [1 0] X + [1 0] Y + 
[0 1]     [0 1]     
>= [0 0] X + [1 0] Y + 
[0 1]     [0 1]     
=  cons(s(X),incr(Y))

a__nats() =  

>= 


a__nats() =  

>= 

=  nats()

a__tl(X) =  [1 4] X + 
[0 1]     
>= [1 4] X + 
[0 1]     
=  tl(X)

a__tl(cons(X,Y)) =  [1 4] X + [1 4] Y + 
[0 1]     [0 1]     
>= [1 4] Y + 
[0 1]     
=  mark(Y)

a__zeros() =  

>= 

=  cons(0(),zeros())

a__zeros() =  

>= 

=  zeros()

mark(0()) =  

>= 

=  0()

mark(cons(X1,X2)) =  [1 4] X1 + [1 4] X2 + 
[0 1]      [0 1]      
>= [1 0] X1 + [1 0] X2 + 
[0 1]      [0 1]      
=  cons(X1,X2)

mark(hd(X)) =  [1 8] X + 
[0 1]     
>= [1 8] X + 
[0 1]     
=  a__hd(mark(X))

mark(incr(X)) =  [1 4] X + 
[0 1]     
>= [1 4] X + 
[0 1]     
=  a__incr(mark(X))

mark(nats()) =  

>= 

=  a__nats()

mark(s(X)) =  [0 4] X + 
[0 1]     
>= [0 0] X + 
[0 1]     
=  s(X)

mark(tl(X)) =  [1 8] X + 
[0 1]     
>= [1 8] X + 
[0 1]     
=  a__tl(mark(X))

mark(zeros()) =  

>= 

=  a__zeros()

* Step 9: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
mark(incr(X)) -> a__incr(mark(X))
- Weak TRS:
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__nats() -> nats()
a__tl(X) -> tl(X)
a__tl(cons(X,Y)) -> mark(Y)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
mark(zeros()) -> a__zeros()
- Signature:
,tl/1,zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__adx,a__hd,a__incr,a__nats,a__tl,a__zeros
+ 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(a__hd) = {1},
uargs(a__incr) = {1},
uargs(a__tl) = {1}

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



p(a__adx) = [1 0 0 2]       
[0 1 0 2] x_1 + 
[0 0 1 1]       
[0 0 0 1]       
p(a__hd) = [1 0 1 0]       
[0 1 1 0] x_1 + 
[0 0 1 2]       
[0 0 0 1]       
p(a__incr) = [1 0 0 1]       
[0 1 0 1] x_1 + 
[0 0 1 0]       
[0 0 0 1]       
p(a__nats) = 



p(a__tl) = [1 0 1 0]       
[0 1 1 0] x_1 + 
[0 0 1 2]       
[0 0 0 1]       
p(a__zeros) = 



p(adx) = [0 0 0 0]       
[0 1 0 1] x_1 + 
[0 0 1 1]       
[0 0 0 1]       
p(cons) = [0 1 0 0]       [0 1 0 0]       
[0 1 0 0] x_1 + [0 1 0 0] x_2 + 
[0 0 1 0]       [0 0 1 0]       
[0 0 0 1]       [0 0 0 1]       
p(hd) = [0 0 1 0]       
[0 1 1 0] x_1 + 
[0 0 1 2]       
[0 0 0 1]       
p(incr) = [0 0 0 0]       
[0 1 0 1] x_1 + 
[0 0 1 0]       
[0 0 0 1]       
p(mark) = [0 1 1 0]       
[0 1 1 0] x_1 + 
[0 0 1 2]       
[0 0 0 1]       
p(nats) = 



p(s) = [0 0 0 0]       
[0 0 0 0] x_1 + 
[0 0 0 0]       
[0 0 0 1]       
p(tl) = [0 0 0 0]       
[0 1 1 0] x_1 + 
[0 0 1 2]       
[0 0 0 1]       
p(zeros) = 




Following rules are strictly oriented:
mark(incr(X)) = [0 1 1 1]     
[0 1 1 1] X + 
[0 0 1 2]     
[0 0 0 1]     
> [0 1 1 1]     
[0 1 1 1] X + 
[0 0 1 2]     
[0 0 0 1]     
= a__incr(mark(X))

Following rules are (at-least) weakly oriented:
a__adx(X) =  [1 0 0 2]     
[0 1 0 2] X + 
[0 0 1 1]     
[0 0 0 1]     
>= [0 0 0 0]     
[0 1 0 1] X + 
[0 0 1 1]     
[0 0 0 1]     

a__adx(cons(X,Y)) =  [0 1 0 2]     [0 1 0 2]     
[0 1 0 2] X + [0 1 0 2] Y + 
[0 0 1 1]     [0 0 1 1]     
[0 0 0 1]     [0 0 0 1]     
>= [0 1 0 1]     [0 1 0 2]     
[0 1 0 1] X + [0 1 0 2] Y + 
[0 0 1 0]     [0 0 1 1]     
[0 0 0 1]     [0 0 0 1]     

a__hd(X) =  [1 0 1 0]     
[0 1 1 0] X + 
[0 0 1 2]     
[0 0 0 1]     
>= [0 0 1 0]     
[0 1 1 0] X + 
[0 0 1 2]     
[0 0 0 1]     
=  hd(X)

a__hd(cons(X,Y)) =  [0 1 1 0]     [0 1 1 0]     
[0 1 1 0] X + [0 1 1 0] Y + 
[0 0 1 2]     [0 0 1 2]     
[0 0 0 1]     [0 0 0 1]     
>= [0 1 1 0]     
[0 1 1 0] X + 
[0 0 1 2]     
[0 0 0 1]     
=  mark(X)

a__incr(X) =  [1 0 0 1]     
[0 1 0 1] X + 
[0 0 1 0]     
[0 0 0 1]     
>= [0 0 0 0]     
[0 1 0 1] X + 
[0 0 1 0]     
[0 0 0 1]     
=  incr(X)

a__incr(cons(X,Y)) =  [0 1 0 1]     [0 1 0 1]     
[0 1 0 1] X + [0 1 0 1] Y + 
[0 0 1 0]     [0 0 1 0]     
[0 0 0 1]     [0 0 0 1]     
>= [0 0 0 0]     [0 1 0 1]     
[0 0 0 0] X + [0 1 0 1] Y + 
[0 0 0 0]     [0 0 1 0]     
[0 0 0 1]     [0 0 0 1]     
=  cons(s(X),incr(Y))

a__nats() =  



>= 




a__nats() =  



>= 



=  nats()

a__tl(X) =  [1 0 1 0]     
[0 1 1 0] X + 
[0 0 1 2]     
[0 0 0 1]     
>= [0 0 0 0]     
[0 1 1 0] X + 
[0 0 1 2]     
[0 0 0 1]     
=  tl(X)

a__tl(cons(X,Y)) =  [0 1 1 0]     [0 1 1 0]     
[0 1 1 0] X + [0 1 1 0] Y + 
[0 0 1 2]     [0 0 1 2]     
[0 0 0 1]     [0 0 0 1]     
>= [0 1 1 0]     
[0 1 1 0] Y + 
[0 0 1 2]     
[0 0 0 1]     
=  mark(Y)

a__zeros() =  



>= 



=  cons(0(),zeros())

a__zeros() =  



>= 



=  zeros()

mark(0()) =  



>= 



=  0()

mark(adx(X)) =  [0 1 1 2]     
[0 1 1 2] X + 
[0 0 1 3]     
[0 0 0 1]     
>= [0 1 1 2]     
[0 1 1 2] X + 
[0 0 1 3]     
[0 0 0 1]     

mark(cons(X1,X2)) =  [0 1 1 0]      [0 1 1 0]      
[0 1 1 0] X1 + [0 1 1 0] X2 + 
[0 0 1 2]      [0 0 1 2]      
[0 0 0 1]      [0 0 0 1]      
>= [0 1 0 0]      [0 1 0 0]      
[0 1 0 0] X1 + [0 1 0 0] X2 + 
[0 0 1 0]      [0 0 1 0]      
[0 0 0 1]      [0 0 0 1]      
=  cons(X1,X2)

mark(hd(X)) =  [0 1 2 2]     
[0 1 2 2] X + 
[0 0 1 4]     
[0 0 0 1]     
>= [0 1 2 2]     
[0 1 2 2] X + 
[0 0 1 4]     
[0 0 0 1]     
=  a__hd(mark(X))

mark(nats()) =  



>= 



=  a__nats()

mark(s(X)) =  [0 0 0 0]     
[0 0 0 0] X + 
[0 0 0 2]     
[0 0 0 1]     
>= [0 0 0 0]     
[0 0 0 0] X + 
[0 0 0 0]     
[0 0 0 1]     
=  s(X)

mark(tl(X)) =  [0 1 2 2]     
[0 1 2 2] X + 
[0 0 1 4]     
[0 0 0 1]     
>= [0 1 2 2]     
[0 1 2 2] X + 
[0 0 1 4]     
[0 0 0 1]     
=  a__tl(mark(X))

mark(zeros()) =  



>= 



=  a__zeros()

* Step 10: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a__hd(X) -> hd(X)
a__hd(cons(X,Y)) -> mark(X)
a__incr(X) -> incr(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__nats() -> nats()
a__tl(X) -> tl(X)
a__tl(cons(X,Y)) -> mark(Y)
a__zeros() -> cons(0(),zeros())
a__zeros() -> zeros()
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(incr(X)) -> a__incr(mark(X))
mark(nats()) -> a__nats()
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
mark(zeros()) -> a__zeros()
- Signature: