```* Step 1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__fst(X1,X2)) -> fst(X1,X2)
activate(n__len(X)) -> len(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
fst(X1,X2) -> n__fst(X1,X2)
fst(0(),Z) -> nil()
fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
len(X) -> n__len(X)
len(cons(X,Z)) -> s(n__len(activate(Z)))
len(nil()) -> 0()
- Signature:
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons
+ 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(cons) = {2},
uargs(n__fst) = {1,2},
uargs(n__len) = {1},
uargs(s) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(activate) = [10] x1 + [0]
p(add) = [10] x1 + [10] x2 + [4]
p(cons) = [1] x2 + [0]
p(from) = [0]
p(fst) = [10] x1 + [10] x2 + [0]
p(len) = [10] x1 + [0]
p(n__add) = [1] x1 + [1] x2 + [0]
p(n__from) = [0]
p(n__fst) = [1] x1 + [1] x2 + [1]
p(n__len) = [1] x1 + [0]
p(nil) = [0]
p(s) = [1] x1 + [1]

Following rules are strictly oriented:
activate(n__fst(X1,X2)) = [10] X1 + [10] X2 + [10]
> [10] X1 + [10] X2 + [0]
= fst(X1,X2)

add(X1,X2) = [10] X1 + [10] X2 + [4]
> [1] X1 + [1] X2 + [0]

add(0(),X) = [10] X + [14]
> [1] X + [0]
= X

add(s(X),Y) = [10] X + [10] Y + [14]
> [10] X + [1] Y + [1]

fst(0(),Z) = [10] Z + [10]
> [0]
= nil()

fst(s(X),cons(Y,Z)) = [10] X + [10] Z + [10]
> [10] X + [10] Z + [1]
= cons(Y,n__fst(activate(X),activate(Z)))

Following rules are (at-least) weakly oriented:
activate(X) =  [10] X + [0]
>= [1] X + [0]
=  X

activate(n__add(X1,X2)) =  [10] X1 + [10] X2 + [0]
>= [10] X1 + [10] X2 + [4]

activate(n__from(X)) =  [0]
>= [0]
=  from(X)

activate(n__len(X)) =  [10] X + [0]
>= [10] X + [0]
=  len(X)

from(X) =  [0]
>= [0]
=  cons(X,n__from(s(X)))

from(X) =  [0]
>= [0]
=  n__from(X)

fst(X1,X2) =  [10] X1 + [10] X2 + [0]
>= [1] X1 + [1] X2 + [1]
=  n__fst(X1,X2)

len(X) =  [10] X + [0]
>= [1] X + [0]
=  n__len(X)

len(cons(X,Z)) =  [10] Z + [0]
>= [10] Z + [1]
=  s(n__len(activate(Z)))

len(nil()) =  [0]
>= [1]
=  0()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__len(X)) -> len(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
fst(X1,X2) -> n__fst(X1,X2)
len(X) -> n__len(X)
len(cons(X,Z)) -> s(n__len(activate(Z)))
len(nil()) -> 0()
- Weak TRS:
activate(n__fst(X1,X2)) -> fst(X1,X2)
fst(0(),Z) -> nil()
fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
- Signature:
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons
+ 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(cons) = {2},
uargs(n__fst) = {1,2},
uargs(n__len) = {1},
uargs(s) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [4]
p(activate) = [1] x1 + [0]
p(add) = [1] x1 + [1] x2 + [4]
p(cons) = [1] x2 + [0]
p(from) = [0]
p(fst) = [1] x1 + [1] x2 + [7]
p(len) = [1] x1 + [3]
p(n__add) = [1] x1 + [1] x2 + [2]
p(n__from) = [0]
p(n__fst) = [1] x1 + [1] x2 + [7]
p(n__len) = [1] x1 + [5]
p(nil) = [0]
p(s) = [1] x1 + [0]

Following rules are strictly oriented:
activate(n__len(X)) = [1] X + [5]
> [1] X + [3]
= len(X)

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

activate(n__add(X1,X2)) =  [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [4]

activate(n__from(X)) =  [0]
>= [0]
=  from(X)

activate(n__fst(X1,X2)) =  [1] X1 + [1] X2 + [7]
>= [1] X1 + [1] X2 + [7]
=  fst(X1,X2)

add(X1,X2) =  [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [2]

add(0(),X) =  [1] X + [8]
>= [1] X + [0]
=  X

add(s(X),Y) =  [1] X + [1] Y + [4]
>= [1] X + [1] Y + [2]

from(X) =  [0]
>= [0]
=  cons(X,n__from(s(X)))

from(X) =  [0]
>= [0]
=  n__from(X)

fst(X1,X2) =  [1] X1 + [1] X2 + [7]
>= [1] X1 + [1] X2 + [7]
=  n__fst(X1,X2)

fst(0(),Z) =  [1] Z + [11]
>= [0]
=  nil()

fst(s(X),cons(Y,Z)) =  [1] X + [1] Z + [7]
>= [1] X + [1] Z + [7]
=  cons(Y,n__fst(activate(X),activate(Z)))

len(X) =  [1] X + [3]
>= [1] X + [5]
=  n__len(X)

len(cons(X,Z)) =  [1] Z + [3]
>= [1] Z + [5]
=  s(n__len(activate(Z)))

len(nil()) =  [3]
>= [4]
=  0()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
fst(X1,X2) -> n__fst(X1,X2)
len(X) -> n__len(X)
len(cons(X,Z)) -> s(n__len(activate(Z)))
len(nil()) -> 0()
- Weak TRS:
activate(n__fst(X1,X2)) -> fst(X1,X2)
activate(n__len(X)) -> len(X)
fst(0(),Z) -> nil()
fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
- Signature:
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons
+ 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(cons) = {2},
uargs(n__fst) = {1,2},
uargs(n__len) = {1},
uargs(s) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(activate) = [2] x1 + [2]
p(add) = [2] x1 + [1] x2 + [1]
p(cons) = [1] x1 + [1] x2 + [0]
p(from) = [2] x1 + [0]
p(fst) = [2] x1 + [2] x2 + [2]
p(len) = [2] x1 + [0]
p(n__add) = [1] x1 + [1] x2 + [0]
p(n__from) = [1] x1 + [0]
p(n__fst) = [1] x1 + [1] x2 + [0]
p(n__len) = [1] x1 + [0]
p(nil) = [0]
p(s) = [1] x1 + [1]

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

activate(n__add(X1,X2)) = [2] X1 + [2] X2 + [2]
> [2] X1 + [1] X2 + [1]

activate(n__from(X)) = [2] X + [2]
> [2] X + [0]
= from(X)

fst(X1,X2) = [2] X1 + [2] X2 + [2]
> [1] X1 + [1] X2 + [0]
= n__fst(X1,X2)

Following rules are (at-least) weakly oriented:
activate(n__fst(X1,X2)) =  [2] X1 + [2] X2 + [2]
>= [2] X1 + [2] X2 + [2]
=  fst(X1,X2)

activate(n__len(X)) =  [2] X + [2]
>= [2] X + [0]
=  len(X)

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

add(0(),X) =  [1] X + [5]
>= [1] X + [0]
=  X

add(s(X),Y) =  [2] X + [1] Y + [3]
>= [2] X + [1] Y + [3]

from(X) =  [2] X + [0]
>= [2] X + [1]
=  cons(X,n__from(s(X)))

from(X) =  [2] X + [0]
>= [1] X + [0]
=  n__from(X)

fst(0(),Z) =  [2] Z + [6]
>= [0]
=  nil()

fst(s(X),cons(Y,Z)) =  [2] X + [2] Y + [2] Z + [4]
>= [2] X + [1] Y + [2] Z + [4]
=  cons(Y,n__fst(activate(X),activate(Z)))

len(X) =  [2] X + [0]
>= [1] X + [0]
=  n__len(X)

len(cons(X,Z)) =  [2] X + [2] Z + [0]
>= [2] Z + [3]
=  s(n__len(activate(Z)))

len(nil()) =  [0]
>= [2]
=  0()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
len(X) -> n__len(X)
len(cons(X,Z)) -> s(n__len(activate(Z)))
len(nil()) -> 0()
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__fst(X1,X2)) -> fst(X1,X2)
activate(n__len(X)) -> len(X)
fst(X1,X2) -> n__fst(X1,X2)
fst(0(),Z) -> nil()
fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
- Signature:
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons
+ 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(cons) = {2},
uargs(n__fst) = {1,2},
uargs(n__len) = {1},
uargs(s) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(activate) = [2] x1 + [0]
p(add) = [2] x1 + [1] x2 + [3]
p(cons) = [1] x2 + [1]
p(from) = [7]
p(fst) = [2] x1 + [2] x2 + [0]
p(len) = [2] x1 + [0]
p(n__add) = [1] x1 + [1] x2 + [2]
p(n__from) = [4]
p(n__fst) = [1] x1 + [1] x2 + [0]
p(n__len) = [1] x1 + [3]
p(nil) = [4]
p(s) = [1] x1 + [2]

Following rules are strictly oriented:
from(X) = [7]
> [5]
= cons(X,n__from(s(X)))

from(X) = [7]
> [4]
= n__from(X)

len(nil()) = [8]
> [2]
= 0()

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

activate(n__add(X1,X2)) =  [2] X1 + [2] X2 + [4]
>= [2] X1 + [1] X2 + [3]

activate(n__from(X)) =  [8]
>= [7]
=  from(X)

activate(n__fst(X1,X2)) =  [2] X1 + [2] X2 + [0]
>= [2] X1 + [2] X2 + [0]
=  fst(X1,X2)

activate(n__len(X)) =  [2] X + [6]
>= [2] X + [0]
=  len(X)

add(X1,X2) =  [2] X1 + [1] X2 + [3]
>= [1] X1 + [1] X2 + [2]

add(0(),X) =  [1] X + [7]
>= [1] X + [0]
=  X

add(s(X),Y) =  [2] X + [1] Y + [7]
>= [2] X + [1] Y + [4]

fst(X1,X2) =  [2] X1 + [2] X2 + [0]
>= [1] X1 + [1] X2 + [0]
=  n__fst(X1,X2)

fst(0(),Z) =  [2] Z + [4]
>= [4]
=  nil()

fst(s(X),cons(Y,Z)) =  [2] X + [2] Z + [6]
>= [2] X + [2] Z + [1]
=  cons(Y,n__fst(activate(X),activate(Z)))

len(X) =  [2] X + [0]
>= [1] X + [3]
=  n__len(X)

len(cons(X,Z)) =  [2] Z + [2]
>= [2] Z + [5]
=  s(n__len(activate(Z)))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
len(X) -> n__len(X)
len(cons(X,Z)) -> s(n__len(activate(Z)))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__fst(X1,X2)) -> fst(X1,X2)
activate(n__len(X)) -> len(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
fst(X1,X2) -> n__fst(X1,X2)
fst(0(),Z) -> nil()
fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
len(nil()) -> 0()
- Signature:
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons
+ 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(cons) = {2},
uargs(n__fst) = {1,2},
uargs(n__len) = {1},
uargs(s) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [6]
p(activate) = [1] x1 + [1]
p(add) = [1] x1 + [1] x2 + [1]
p(cons) = [1] x2 + [0]
p(from) = [1]
p(fst) = [1] x1 + [1] x2 + [0]
p(len) = [1] x1 + [7]
p(n__add) = [1] x1 + [1] x2 + [0]
p(n__from) = [1]
p(n__fst) = [1] x1 + [1] x2 + [0]
p(n__len) = [1] x1 + [6]
p(nil) = [0]
p(s) = [1] x1 + [2]

Following rules are strictly oriented:
len(X) = [1] X + [7]
> [1] X + [6]
= n__len(X)

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

activate(n__add(X1,X2)) =  [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]

activate(n__from(X)) =  [2]
>= [1]
=  from(X)

activate(n__fst(X1,X2)) =  [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [0]
=  fst(X1,X2)

activate(n__len(X)) =  [1] X + [7]
>= [1] X + [7]
=  len(X)

add(X1,X2) =  [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [0]

add(0(),X) =  [1] X + [7]
>= [1] X + [0]
=  X

add(s(X),Y) =  [1] X + [1] Y + [3]
>= [1] X + [1] Y + [3]

from(X) =  [1]
>= [1]
=  cons(X,n__from(s(X)))

from(X) =  [1]
>= [1]
=  n__from(X)

fst(X1,X2) =  [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
=  n__fst(X1,X2)

fst(0(),Z) =  [1] Z + [6]
>= [0]
=  nil()

fst(s(X),cons(Y,Z)) =  [1] X + [1] Z + [2]
>= [1] X + [1] Z + [2]
=  cons(Y,n__fst(activate(X),activate(Z)))

len(cons(X,Z)) =  [1] Z + [7]
>= [1] Z + [9]
=  s(n__len(activate(Z)))

len(nil()) =  [7]
>= [6]
=  0()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
len(cons(X,Z)) -> s(n__len(activate(Z)))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__fst(X1,X2)) -> fst(X1,X2)
activate(n__len(X)) -> len(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
fst(X1,X2) -> n__fst(X1,X2)
fst(0(),Z) -> nil()
fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
len(X) -> n__len(X)
len(nil()) -> 0()
- Signature:
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons
+ 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(cons) = {2},
uargs(n__fst) = {1,2},
uargs(n__len) = {1},
uargs(s) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [4] x1 + [0]
p(add) = [4] x1 + [4] x2 + [0]
p(cons) = [1] x2 + [0]
p(from) = [0]
p(fst) = [4] x1 + [4] x2 + [5]
p(len) = [4] x1 + [4]
p(n__add) = [1] x1 + [1] x2 + [0]
p(n__from) = [0]
p(n__fst) = [1] x1 + [1] x2 + [2]
p(n__len) = [1] x1 + [1]
p(nil) = [2]
p(s) = [1] x1 + [2]

Following rules are strictly oriented:
len(cons(X,Z)) = [4] Z + [4]
> [4] Z + [3]
= s(n__len(activate(Z)))

Following rules are (at-least) weakly oriented:
activate(X) =  [4] X + [0]
>= [1] X + [0]
=  X

activate(n__add(X1,X2)) =  [4] X1 + [4] X2 + [0]
>= [4] X1 + [4] X2 + [0]

activate(n__from(X)) =  [0]
>= [0]
=  from(X)

activate(n__fst(X1,X2)) =  [4] X1 + [4] X2 + [8]
>= [4] X1 + [4] X2 + [5]
=  fst(X1,X2)

activate(n__len(X)) =  [4] X + [4]
>= [4] X + [4]
=  len(X)

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

add(0(),X) =  [4] X + [0]
>= [1] X + [0]
=  X

add(s(X),Y) =  [4] X + [4] Y + [8]
>= [4] X + [1] Y + [2]

from(X) =  [0]
>= [0]
=  cons(X,n__from(s(X)))

from(X) =  [0]
>= [0]
=  n__from(X)

fst(X1,X2) =  [4] X1 + [4] X2 + [5]
>= [1] X1 + [1] X2 + [2]
=  n__fst(X1,X2)

fst(0(),Z) =  [4] Z + [5]
>= [2]
=  nil()

fst(s(X),cons(Y,Z)) =  [4] X + [4] Z + [13]
>= [4] X + [4] Z + [2]
=  cons(Y,n__fst(activate(X),activate(Z)))

len(X) =  [4] X + [4]
>= [1] X + [1]
=  n__len(X)

len(nil()) =  [12]
>= [0]
=  0()

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
activate(n__fst(X1,X2)) -> fst(X1,X2)
activate(n__len(X)) -> len(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
fst(X1,X2) -> n__fst(X1,X2)
fst(0(),Z) -> nil()
fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
len(X) -> n__len(X)
len(cons(X,Z)) -> s(n__len(activate(Z)))
len(nil()) -> 0()
- Signature: