* Step 1: Sum WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(X1,X2)
activate(n__from(X)) -> from(X)
activate(n__fst(X1,X2)) -> fst(X1,X2)
activate(n__len(X)) -> len(X)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
add(s(X),Y) -> s(n__add(activate(X),Y))
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:
{activate/1,add/2,from/1,fst/2,len/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons
,n__add,n__from,n__fst,n__len,nil,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(X1,X2)
activate(n__from(X)) -> from(X)
activate(n__fst(X1,X2)) -> fst(X1,X2)
activate(n__len(X)) -> len(X)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
add(s(X),Y) -> s(n__add(activate(X),Y))
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:
{activate/1,add/2,from/1,fst/2,len/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons
,n__add,n__from,n__fst,n__len,nil,s}
+ 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__add) = {1},
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]
= n__add(X1,X2)
add(0(),X) = [10] X + [14]
> [1] X + [0]
= X
add(s(X),Y) = [10] X + [10] Y + [14]
> [10] X + [1] Y + [1]
= s(n__add(activate(X),Y))
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]
= add(X1,X2)
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 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(X1,X2)
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)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
add(s(X),Y) -> s(n__add(activate(X),Y))
fst(0(),Z) -> nil()
fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
- Signature:
{activate/1,add/2,from/1,fst/2,len/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons
,n__add,n__from,n__fst,n__len,nil,s}
+ 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__add) = {1},
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]
= add(X1,X2)
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]
= n__add(X1,X2)
add(0(),X) = [1] X + [8]
>= [1] X + [0]
= X
add(s(X),Y) = [1] X + [1] Y + [4]
>= [1] X + [1] Y + [2]
= s(n__add(activate(X),Y))
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 4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(X1,X2)
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)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
add(s(X),Y) -> s(n__add(activate(X),Y))
fst(0(),Z) -> nil()
fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
- Signature:
{activate/1,add/2,from/1,fst/2,len/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons
,n__add,n__from,n__fst,n__len,nil,s}
+ 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__add) = {1},
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]
= add(X1,X2)
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]
= n__add(X1,X2)
add(0(),X) = [1] X + [5]
>= [1] X + [0]
= X
add(s(X),Y) = [2] X + [1] Y + [3]
>= [2] X + [1] Y + [3]
= s(n__add(activate(X),Y))
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 5: 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__add(X1,X2)) -> add(X1,X2)
activate(n__from(X)) -> from(X)
activate(n__fst(X1,X2)) -> fst(X1,X2)
activate(n__len(X)) -> len(X)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
add(s(X),Y) -> s(n__add(activate(X),Y))
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:
{activate/1,add/2,from/1,fst/2,len/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons
,n__add,n__from,n__fst,n__len,nil,s}
+ 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__add) = {1},
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]
= add(X1,X2)
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]
= n__add(X1,X2)
add(0(),X) = [1] X + [7]
>= [1] X + [0]
= X
add(s(X),Y) = [2] X + [1] Y + [7]
>= [2] X + [1] Y + [4]
= s(n__add(activate(X),Y))
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 6: 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__add(X1,X2)) -> add(X1,X2)
activate(n__from(X)) -> from(X)
activate(n__fst(X1,X2)) -> fst(X1,X2)
activate(n__len(X)) -> len(X)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
add(s(X),Y) -> s(n__add(activate(X),Y))
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:
{activate/1,add/2,from/1,fst/2,len/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons
,n__add,n__from,n__fst,n__len,nil,s}
+ 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__add) = {1},
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]
= add(X1,X2)
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]
= n__add(X1,X2)
add(0(),X) = [1] X + [7]
>= [1] X + [0]
= X
add(s(X),Y) = [1] X + [1] Y + [3]
>= [1] X + [1] Y + [3]
= s(n__add(activate(X),Y))
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 7: 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__add(X1,X2)) -> add(X1,X2)
activate(n__from(X)) -> from(X)
activate(n__fst(X1,X2)) -> fst(X1,X2)
activate(n__len(X)) -> len(X)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
add(s(X),Y) -> s(n__add(activate(X),Y))
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:
{activate/1,add/2,from/1,fst/2,len/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons
,n__add,n__from,n__fst,n__len,nil,s}
+ 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__add) = {1},
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]
= add(X1,X2)
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]
= n__add(X1,X2)
add(0(),X) = [4] X + [0]
>= [1] X + [0]
= X
add(s(X),Y) = [4] X + [4] Y + [8]
>= [4] X + [1] Y + [2]
= s(n__add(activate(X),Y))
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 8: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(X1,X2)
activate(n__from(X)) -> from(X)
activate(n__fst(X1,X2)) -> fst(X1,X2)
activate(n__len(X)) -> len(X)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> X
add(s(X),Y) -> s(n__add(activate(X),Y))
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:
{activate/1,add/2,from/1,fst/2,len/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons
,n__add,n__from,n__fst,n__len,nil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))