WORST_CASE(Omega(n^1),O(n^3))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^3))
+ Considered Problem:
- Strict TRS:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__fst(X1,X2) -> fst(X1,X2)
a__fst(0(),Z) -> nil()
a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
a__len(X) -> len(X)
a__len(cons(X,Z)) -> s(len(Z))
a__len(nil()) -> 0()
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
- Signature:
{a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0
,add,cons,from,fst,len,nil,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__fst(X1,X2) -> fst(X1,X2)
a__fst(0(),Z) -> nil()
a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
a__len(X) -> len(X)
a__len(cons(X,Z)) -> s(len(Z))
a__len(nil()) -> 0()
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
- Signature:
{a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0
,add,cons,from,fst,len,nil,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
mark(x){x -> add(x,y)} =
mark(add(x,y)) ->^+ a__add(mark(x),mark(y))
= C[mark(x) = mark(x){}]
** Step 1.b:1: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__fst(X1,X2) -> fst(X1,X2)
a__fst(0(),Z) -> nil()
a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
a__len(X) -> len(X)
a__len(cons(X,Z)) -> s(len(Z))
a__len(nil()) -> 0()
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
- Signature:
{a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0
,add,cons,from,fst,len,nil,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
{a__add,a__from,a__fst,a__len,mark}
TcT has computed the following interpretation:
p(0) = [0]
p(a__add) = [1] x1 + [1] x2 + [2]
p(a__from) = [1] x1 + [0]
p(a__fst) = [1] x1 + [1] x2 + [0]
p(a__len) = [1] x1 + [0]
p(add) = [1] x1 + [1] x2 + [2]
p(cons) = [1] x1 + [0]
p(from) = [1] x1 + [0]
p(fst) = [1] x1 + [1] x2 + [0]
p(len) = [1] x1 + [0]
p(mark) = [1] x1 + [0]
p(nil) = [0]
p(s) = [0]
Following rules are strictly oriented:
a__add(0(),X) = [1] X + [2]
> [1] X + [0]
= mark(X)
a__add(s(X),Y) = [1] Y + [2]
> [0]
= s(add(X,Y))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= add(X1,X2)
a__from(X) = [1] X + [0]
>= [1] X + [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1] X + [0]
>= [1] X + [0]
= from(X)
a__fst(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= fst(X1,X2)
a__fst(0(),Z) = [1] Z + [0]
>= [0]
= nil()
a__fst(s(X),cons(Y,Z)) = [1] Y + [0]
>= [1] Y + [0]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1] X + [0]
>= [1] X + [0]
= len(X)
a__len(cons(X,Z)) = [1] X + [0]
>= [0]
= s(len(Z))
a__len(nil()) = [0]
>= [0]
= 0()
mark(0()) = [0]
>= [0]
= 0()
mark(add(X1,X2)) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1] X1 + [0]
>= [1] X1 + [0]
= cons(mark(X1),X2)
mark(from(X)) = [1] X + [0]
>= [1] X + [0]
= a__from(mark(X))
mark(fst(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= a__fst(mark(X1),mark(X2))
mark(len(X)) = [1] X + [0]
>= [1] X + [0]
= a__len(mark(X))
mark(nil()) = [0]
>= [0]
= nil()
mark(s(X)) = [0]
>= [0]
= s(X)
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__add(X1,X2) -> add(X1,X2)
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__fst(X1,X2) -> fst(X1,X2)
a__fst(0(),Z) -> nil()
a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
a__len(X) -> len(X)
a__len(cons(X,Z)) -> s(len(Z))
a__len(nil()) -> 0()
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
- Weak TRS:
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
- Signature:
{a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0
,add,cons,from,fst,len,nil,s}
+ Applied Processor:
NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(linear):
The following argument positions are considered usable:
uargs(a__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
{a__add,a__from,a__fst,a__len,mark}
TcT has computed the following interpretation:
p(0) = 3
p(a__add) = x1 + x2
p(a__from) = 7 + x1
p(a__fst) = 4 + x1 + x2
p(a__len) = 4 + x1
p(add) = x1 + x2
p(cons) = 3 + x1
p(from) = 7 + x1
p(fst) = 4 + x1 + x2
p(len) = 4 + x1
p(mark) = x1
p(nil) = 2
p(s) = 4
Following rules are strictly oriented:
a__from(X) = 7 + X
> 3 + X
= cons(mark(X),from(s(X)))
a__fst(0(),Z) = 7 + Z
> 2
= nil()
a__fst(s(X),cons(Y,Z)) = 11 + Y
> 3 + Y
= cons(mark(Y),fst(X,Z))
a__len(cons(X,Z)) = 7 + X
> 4
= s(len(Z))
a__len(nil()) = 6
> 3
= 0()
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = X1 + X2
>= X1 + X2
= add(X1,X2)
a__add(0(),X) = 3 + X
>= X
= mark(X)
a__add(s(X),Y) = 4 + Y
>= 4
= s(add(X,Y))
a__from(X) = 7 + X
>= 7 + X
= from(X)
a__fst(X1,X2) = 4 + X1 + X2
>= 4 + X1 + X2
= fst(X1,X2)
a__len(X) = 4 + X
>= 4 + X
= len(X)
mark(0()) = 3
>= 3
= 0()
mark(add(X1,X2)) = X1 + X2
>= X1 + X2
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = 3 + X1
>= 3 + X1
= cons(mark(X1),X2)
mark(from(X)) = 7 + X
>= 7 + X
= a__from(mark(X))
mark(fst(X1,X2)) = 4 + X1 + X2
>= 4 + X1 + X2
= a__fst(mark(X1),mark(X2))
mark(len(X)) = 4 + X
>= 4 + X
= a__len(mark(X))
mark(nil()) = 2
>= 2
= nil()
mark(s(X)) = 4
>= 4
= s(X)
** Step 1.b:3: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__add(X1,X2) -> add(X1,X2)
a__from(X) -> from(X)
a__fst(X1,X2) -> fst(X1,X2)
a__len(X) -> len(X)
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
- Weak TRS:
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__from(X) -> cons(mark(X),from(s(X)))
a__fst(0(),Z) -> nil()
a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
a__len(cons(X,Z)) -> s(len(Z))
a__len(nil()) -> 0()
- Signature:
{a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0
,add,cons,from,fst,len,nil,s}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
{a__add,a__from,a__fst,a__len,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__add) = [1 1 0] [1 1 0] [0]
[0 0 1] x1 + [0 0 1] x2 + [1]
[0 0 1] [0 0 1] [1]
p(a__from) = [1 1 0] [0]
[0 0 1] x1 + [0]
[0 0 1] [0]
p(a__fst) = [1 1 0] [1 1 0] [0]
[0 0 1] x1 + [0 0 1] x2 + [0]
[0 0 1] [0 0 1] [1]
p(a__len) = [1 1 0] [0]
[0 0 1] x1 + [1]
[0 0 1] [1]
p(add) = [1 1 0] [1 1 0] [0]
[0 0 1] x1 + [0 0 1] x2 + [1]
[0 0 1] [0 0 1] [1]
p(cons) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(from) = [1 1 0] [0]
[0 0 1] x1 + [0]
[0 0 1] [0]
p(fst) = [1 1 0] [1 1 0] [0]
[0 0 1] x1 + [0 0 1] x2 + [0]
[0 0 1] [0 0 1] [1]
p(len) = [1 1 0] [0]
[0 0 1] x1 + [0]
[0 0 1] [1]
p(mark) = [1 1 0] [0]
[0 0 1] x1 + [0]
[0 0 1] [0]
p(nil) = [0]
[0]
[0]
p(s) = [0]
[0]
[0]
Following rules are strictly oriented:
mark(add(X1,X2)) = [1 1 1] [1 1 1] [1]
[0 0 1] X1 + [0 0 1] X2 + [1]
[0 0 1] [0 0 1] [1]
> [1 1 1] [1 1 1] [0]
[0 0 1] X1 + [0 0 1] X2 + [1]
[0 0 1] [0 0 1] [1]
= a__add(mark(X1),mark(X2))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 1 0] [1 1 0] [0]
[0 0 1] X1 + [0 0 1] X2 + [1]
[0 0 1] [0 0 1] [1]
>= [1 1 0] [1 1 0] [0]
[0 0 1] X1 + [0 0 1] X2 + [1]
[0 0 1] [0 0 1] [1]
= add(X1,X2)
a__add(0(),X) = [1 1 0] [0]
[0 0 1] X + [1]
[0 0 1] [1]
>= [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 1 0] [0]
[0 0 1] Y + [1]
[0 0 1] [1]
>= [0]
[0]
[0]
= s(add(X,Y))
a__from(X) = [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [0]
>= [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [0]
>= [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [0]
= from(X)
a__fst(X1,X2) = [1 1 0] [1 1 0] [0]
[0 0 1] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [1]
>= [1 1 0] [1 1 0] [0]
[0 0 1] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [1]
= fst(X1,X2)
a__fst(0(),Z) = [1 1 0] [0]
[0 0 1] Z + [0]
[0 0 1] [1]
>= [0]
[0]
[0]
= nil()
a__fst(s(X),cons(Y,Z)) = [1 1 0] [0]
[0 0 1] Y + [0]
[0 0 1] [1]
>= [1 1 0] [0]
[0 0 1] Y + [0]
[0 0 1] [0]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1 1 0] [0]
[0 0 1] X + [1]
[0 0 1] [1]
>= [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [1]
= len(X)
a__len(cons(X,Z)) = [1 1 0] [0]
[0 0 1] X + [1]
[0 0 1] [1]
>= [0]
[0]
[0]
= s(len(Z))
a__len(nil()) = [0]
[1]
[1]
>= [0]
[0]
[0]
= 0()
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(cons(X1,X2)) = [1 1 0] [0]
[0 0 1] X1 + [0]
[0 0 1] [0]
>= [1 1 0] [0]
[0 0 1] X1 + [0]
[0 0 1] [0]
= cons(mark(X1),X2)
mark(from(X)) = [1 1 1] [0]
[0 0 1] X + [0]
[0 0 1] [0]
>= [1 1 1] [0]
[0 0 1] X + [0]
[0 0 1] [0]
= a__from(mark(X))
mark(fst(X1,X2)) = [1 1 1] [1 1 1] [0]
[0 0 1] X1 + [0 0 1] X2 + [1]
[0 0 1] [0 0 1] [1]
>= [1 1 1] [1 1 1] [0]
[0 0 1] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [1]
= a__fst(mark(X1),mark(X2))
mark(len(X)) = [1 1 1] [0]
[0 0 1] X + [1]
[0 0 1] [1]
>= [1 1 1] [0]
[0 0 1] X + [1]
[0 0 1] [1]
= a__len(mark(X))
mark(nil()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= nil()
mark(s(X)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= s(X)
** Step 1.b:4: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__add(X1,X2) -> add(X1,X2)
a__from(X) -> from(X)
a__fst(X1,X2) -> fst(X1,X2)
a__len(X) -> len(X)
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
- Weak TRS:
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__from(X) -> cons(mark(X),from(s(X)))
a__fst(0(),Z) -> nil()
a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
a__len(cons(X,Z)) -> s(len(Z))
a__len(nil()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
- Signature:
{a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0
,add,cons,from,fst,len,nil,s}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
{a__add,a__from,a__fst,a__len,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[1]
p(a__add) = [1 0 0] [1 0 1] [0]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 1] [0 0 1] [0]
p(a__from) = [1 0 1] [0]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(a__fst) = [1 0 0] [1 0 1] [0]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 1] [0 0 1] [0]
p(a__len) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(add) = [1 0 0] [1 0 1] [0]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 1] [0 0 1] [0]
p(cons) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(from) = [1 0 1] [0]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(fst) = [1 0 0] [1 0 1] [0]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 1] [0 0 1] [0]
p(len) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(mark) = [1 0 1] [0]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(nil) = [0]
[0]
[1]
p(s) = [0]
[0]
[0]
Following rules are strictly oriented:
mark(0()) = [1]
[0]
[1]
> [0]
[0]
[1]
= 0()
mark(nil()) = [1]
[0]
[1]
> [0]
[0]
[1]
= nil()
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0 0] [1 0 1] [0]
[0 0 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 0] [1 0 1] [0]
[0 0 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
= add(X1,X2)
a__add(0(),X) = [1 0 1] [0]
[0 0 0] X + [0]
[0 0 1] [1]
>= [1 0 1] [0]
[0 0 0] X + [0]
[0 0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 0 1] [0]
[0 0 0] Y + [0]
[0 0 1] [0]
>= [0]
[0]
[0]
= s(add(X,Y))
a__from(X) = [1 0 1] [0]
[0 0 0] X + [0]
[0 0 1] [0]
>= [1 0 1] [0]
[0 0 0] X + [0]
[0 0 1] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 0 1] [0]
[0 0 0] X + [0]
[0 0 1] [0]
>= [1 0 1] [0]
[0 0 0] X + [0]
[0 0 1] [0]
= from(X)
a__fst(X1,X2) = [1 0 0] [1 0 1] [0]
[0 0 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 0] [1 0 1] [0]
[0 0 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
= fst(X1,X2)
a__fst(0(),Z) = [1 0 1] [0]
[0 0 0] Z + [0]
[0 0 1] [1]
>= [0]
[0]
[1]
= nil()
a__fst(s(X),cons(Y,Z)) = [1 0 1] [0]
[0 0 0] Y + [0]
[0 0 1] [0]
>= [1 0 1] [0]
[0 0 0] Y + [0]
[0 0 1] [0]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1 0 0] [0]
[0 1 0] X + [0]
[0 0 1] [0]
>= [1 0 0] [0]
[0 0 0] X + [0]
[0 0 1] [0]
= len(X)
a__len(cons(X,Z)) = [1 0 0] [0]
[0 0 0] X + [0]
[0 0 1] [0]
>= [0]
[0]
[0]
= s(len(Z))
a__len(nil()) = [0]
[0]
[1]
>= [0]
[0]
[1]
= 0()
mark(add(X1,X2)) = [1 0 1] [1 0 2] [0]
[0 0 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 1] [1 0 2] [0]
[0 0 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 0 1] [0]
[0 0 0] X1 + [0]
[0 0 1] [0]
>= [1 0 1] [0]
[0 0 0] X1 + [0]
[0 0 1] [0]
= cons(mark(X1),X2)
mark(from(X)) = [1 0 2] [0]
[0 0 0] X + [0]
[0 0 1] [0]
>= [1 0 2] [0]
[0 0 0] X + [0]
[0 0 1] [0]
= a__from(mark(X))
mark(fst(X1,X2)) = [1 0 1] [1 0 2] [0]
[0 0 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 1] [1 0 2] [0]
[0 0 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
= a__fst(mark(X1),mark(X2))
mark(len(X)) = [1 0 1] [0]
[0 0 0] X + [0]
[0 0 1] [0]
>= [1 0 1] [0]
[0 0 0] X + [0]
[0 0 1] [0]
= a__len(mark(X))
mark(s(X)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= s(X)
** Step 1.b:5: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__add(X1,X2) -> add(X1,X2)
a__from(X) -> from(X)
a__fst(X1,X2) -> fst(X1,X2)
a__len(X) -> len(X)
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(s(X)) -> s(X)
- Weak TRS:
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__from(X) -> cons(mark(X),from(s(X)))
a__fst(0(),Z) -> nil()
a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
a__len(cons(X,Z)) -> s(len(Z))
a__len(nil()) -> 0()
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(nil()) -> nil()
- Signature:
{a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0
,add,cons,from,fst,len,nil,s}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
{a__add,a__from,a__fst,a__len,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__add) = [1 0 1] [1 1 0] [0]
[0 1 0] x1 + [0 0 1] x2 + [0]
[0 0 1] [0 0 1] [0]
p(a__from) = [1 1 0] [1]
[0 0 1] x1 + [1]
[0 0 1] [1]
p(a__fst) = [1 1 0] [1 1 0] [0]
[0 0 1] x1 + [0 0 1] x2 + [0]
[0 0 1] [0 0 1] [0]
p(a__len) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(add) = [1 0 1] [1 1 0] [0]
[0 1 0] x1 + [0 0 1] x2 + [0]
[0 0 1] [0 0 1] [0]
p(cons) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(from) = [1 1 0] [0]
[0 0 1] x1 + [1]
[0 0 1] [1]
p(fst) = [1 1 0] [1 1 0] [0]
[0 0 1] x1 + [0 0 1] x2 + [0]
[0 0 1] [0 0 1] [0]
p(len) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(mark) = [1 1 0] [0]
[0 0 1] x1 + [0]
[0 0 1] [0]
p(nil) = [0]
[0]
[0]
p(s) = [0]
[0]
[0]
Following rules are strictly oriented:
a__from(X) = [1 1 0] [1]
[0 0 1] X + [1]
[0 0 1] [1]
> [1 1 0] [0]
[0 0 1] X + [1]
[0 0 1] [1]
= from(X)
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0 1] [1 1 0] [0]
[0 1 0] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 1] [1 1 0] [0]
[0 1 0] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [0]
= add(X1,X2)
a__add(0(),X) = [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [0]
>= [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 1 0] [0]
[0 0 1] Y + [0]
[0 0 1] [0]
>= [0]
[0]
[0]
= s(add(X,Y))
a__from(X) = [1 1 0] [1]
[0 0 1] X + [1]
[0 0 1] [1]
>= [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [0]
= cons(mark(X),from(s(X)))
a__fst(X1,X2) = [1 1 0] [1 1 0] [0]
[0 0 1] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 1 0] [1 1 0] [0]
[0 0 1] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [0]
= fst(X1,X2)
a__fst(0(),Z) = [1 1 0] [0]
[0 0 1] Z + [0]
[0 0 1] [0]
>= [0]
[0]
[0]
= nil()
a__fst(s(X),cons(Y,Z)) = [1 1 0] [0]
[0 0 1] Y + [0]
[0 0 1] [0]
>= [1 1 0] [0]
[0 0 1] Y + [0]
[0 0 1] [0]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1 0 0] [0]
[0 1 0] X + [0]
[0 0 1] [0]
>= [1 0 0] [0]
[0 1 0] X + [0]
[0 0 1] [0]
= len(X)
a__len(cons(X,Z)) = [1 0 0] [0]
[0 1 0] X + [0]
[0 0 1] [0]
>= [0]
[0]
[0]
= s(len(Z))
a__len(nil()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(add(X1,X2)) = [1 1 1] [1 1 1] [0]
[0 0 1] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 1 1] [1 1 1] [0]
[0 0 1] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [0]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 1 0] [0]
[0 0 1] X1 + [0]
[0 0 1] [0]
>= [1 1 0] [0]
[0 0 1] X1 + [0]
[0 0 1] [0]
= cons(mark(X1),X2)
mark(from(X)) = [1 1 1] [1]
[0 0 1] X + [1]
[0 0 1] [1]
>= [1 1 1] [1]
[0 0 1] X + [1]
[0 0 1] [1]
= a__from(mark(X))
mark(fst(X1,X2)) = [1 1 1] [1 1 1] [0]
[0 0 1] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 1 1] [1 1 1] [0]
[0 0 1] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [0]
= a__fst(mark(X1),mark(X2))
mark(len(X)) = [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [0]
>= [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [0]
= a__len(mark(X))
mark(nil()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= nil()
mark(s(X)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= s(X)
** Step 1.b:6: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__add(X1,X2) -> add(X1,X2)
a__fst(X1,X2) -> fst(X1,X2)
a__len(X) -> len(X)
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(s(X)) -> s(X)
- Weak TRS:
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__fst(0(),Z) -> nil()
a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
a__len(cons(X,Z)) -> s(len(Z))
a__len(nil()) -> 0()
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(nil()) -> nil()
- Signature:
{a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0
,add,cons,from,fst,len,nil,s}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
{a__add,a__from,a__fst,a__len,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__add) = [1 0 0] [1 1 0] [0]
[0 1 0] x1 + [0 0 1] x2 + [0]
[0 0 1] [0 0 1] [0]
p(a__from) = [1 1 0] [0]
[0 0 1] x1 + [0]
[0 0 1] [1]
p(a__fst) = [1 1 0] [1 1 0] [0]
[0 0 1] x1 + [0 0 1] x2 + [0]
[0 0 1] [0 0 1] [0]
p(a__len) = [1 1 0] [0]
[0 0 1] x1 + [0]
[0 0 1] [0]
p(add) = [1 0 0] [1 1 0] [0]
[0 1 0] x1 + [0 0 1] x2 + [0]
[0 0 1] [0 0 1] [0]
p(cons) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [1]
p(from) = [1 1 0] [0]
[0 0 1] x1 + [0]
[0 0 1] [1]
p(fst) = [1 1 0] [1 1 0] [0]
[0 0 1] x1 + [0 0 1] x2 + [0]
[0 0 1] [0 0 1] [0]
p(len) = [1 1 0] [0]
[0 0 1] x1 + [0]
[0 0 1] [0]
p(mark) = [1 1 0] [0]
[0 0 1] x1 + [0]
[0 0 1] [0]
p(nil) = [0]
[0]
[0]
p(s) = [0]
[1]
[1]
Following rules are strictly oriented:
mark(s(X)) = [1]
[1]
[1]
> [0]
[1]
[1]
= s(X)
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0 0] [1 1 0] [0]
[0 1 0] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 0] [1 1 0] [0]
[0 1 0] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [0]
= add(X1,X2)
a__add(0(),X) = [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [0]
>= [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 1 0] [0]
[0 0 1] Y + [1]
[0 0 1] [1]
>= [0]
[1]
[1]
= s(add(X,Y))
a__from(X) = [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [1]
>= [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [1]
= cons(mark(X),from(s(X)))
a__from(X) = [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [1]
>= [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [1]
= from(X)
a__fst(X1,X2) = [1 1 0] [1 1 0] [0]
[0 0 1] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 1 0] [1 1 0] [0]
[0 0 1] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [0]
= fst(X1,X2)
a__fst(0(),Z) = [1 1 0] [0]
[0 0 1] Z + [0]
[0 0 1] [0]
>= [0]
[0]
[0]
= nil()
a__fst(s(X),cons(Y,Z)) = [1 1 0] [1]
[0 0 1] Y + [2]
[0 0 1] [2]
>= [1 1 0] [0]
[0 0 1] Y + [0]
[0 0 1] [1]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [0]
>= [1 1 0] [0]
[0 0 1] X + [0]
[0 0 1] [0]
= len(X)
a__len(cons(X,Z)) = [1 1 0] [0]
[0 0 1] X + [1]
[0 0 1] [1]
>= [0]
[1]
[1]
= s(len(Z))
a__len(nil()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(add(X1,X2)) = [1 1 0] [1 1 1] [0]
[0 0 1] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 1 0] [1 1 1] [0]
[0 0 1] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [0]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 1 0] [0]
[0 0 1] X1 + [1]
[0 0 1] [1]
>= [1 1 0] [0]
[0 0 1] X1 + [0]
[0 0 1] [1]
= cons(mark(X1),X2)
mark(from(X)) = [1 1 1] [0]
[0 0 1] X + [1]
[0 0 1] [1]
>= [1 1 1] [0]
[0 0 1] X + [0]
[0 0 1] [1]
= a__from(mark(X))
mark(fst(X1,X2)) = [1 1 1] [1 1 1] [0]
[0 0 1] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 1 1] [1 1 1] [0]
[0 0 1] X1 + [0 0 1] X2 + [0]
[0 0 1] [0 0 1] [0]
= a__fst(mark(X1),mark(X2))
mark(len(X)) = [1 1 1] [0]
[0 0 1] X + [0]
[0 0 1] [0]
>= [1 1 1] [0]
[0 0 1] X + [0]
[0 0 1] [0]
= a__len(mark(X))
mark(nil()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= nil()
** Step 1.b:7: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__add(X1,X2) -> add(X1,X2)
a__fst(X1,X2) -> fst(X1,X2)
a__len(X) -> len(X)
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
mark(len(X)) -> a__len(mark(X))
- Weak TRS:
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__fst(0(),Z) -> nil()
a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
a__len(cons(X,Z)) -> s(len(Z))
a__len(nil()) -> 0()
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
- Signature:
{a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0
,add,cons,from,fst,len,nil,s}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
{a__add,a__from,a__fst,a__len,mark}
TcT has computed the following interpretation:
p(0) = [1]
[0]
[1]
p(a__add) = [1 0 0] [1 0 1] [0]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 1] [0 0 1] [0]
p(a__from) = [1 0 1] [1]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(a__fst) = [1 0 0] [1 1 1] [0]
[0 1 0] x1 + [0 0 0] x2 + [0]
[0 0 1] [0 0 1] [0]
p(a__len) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [1]
p(add) = [1 0 0] [1 0 1] [0]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 1] [0 0 1] [0]
p(cons) = [1 0 0] [1]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(from) = [1 0 1] [1]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(fst) = [1 0 0] [1 0 1] [0]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 1] [0 0 1] [0]
p(len) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 1] [1]
p(mark) = [1 0 1] [0]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(nil) = [1]
[0]
[1]
p(s) = [1]
[0]
[0]
Following rules are strictly oriented:
mark(len(X)) = [1 0 1] [1]
[0 0 0] X + [0]
[0 0 1] [1]
> [1 0 1] [0]
[0 0 0] X + [0]
[0 0 1] [1]
= a__len(mark(X))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0 0] [1 0 1] [0]
[0 0 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 0] [1 0 1] [0]
[0 0 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
= add(X1,X2)
a__add(0(),X) = [1 0 1] [1]
[0 0 0] X + [0]
[0 0 1] [1]
>= [1 0 1] [0]
[0 0 0] X + [0]
[0 0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 0 1] [1]
[0 0 0] Y + [0]
[0 0 1] [0]
>= [1]
[0]
[0]
= s(add(X,Y))
a__from(X) = [1 0 1] [1]
[0 0 0] X + [0]
[0 0 1] [0]
>= [1 0 1] [1]
[0 0 0] X + [0]
[0 0 1] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 0 1] [1]
[0 0 0] X + [0]
[0 0 1] [0]
>= [1 0 1] [1]
[0 0 0] X + [0]
[0 0 1] [0]
= from(X)
a__fst(X1,X2) = [1 0 0] [1 1 1] [0]
[0 1 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 0] [1 0 1] [0]
[0 0 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
= fst(X1,X2)
a__fst(0(),Z) = [1 1 1] [1]
[0 0 0] Z + [0]
[0 0 1] [1]
>= [1]
[0]
[1]
= nil()
a__fst(s(X),cons(Y,Z)) = [1 0 1] [2]
[0 0 0] Y + [0]
[0 0 1] [0]
>= [1 0 1] [1]
[0 0 0] Y + [0]
[0 0 1] [0]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1 0 0] [0]
[0 1 0] X + [0]
[0 0 1] [1]
>= [1 0 0] [0]
[0 0 0] X + [0]
[0 0 1] [1]
= len(X)
a__len(cons(X,Z)) = [1 0 0] [1]
[0 0 0] X + [0]
[0 0 1] [1]
>= [1]
[0]
[0]
= s(len(Z))
a__len(nil()) = [1]
[0]
[2]
>= [1]
[0]
[1]
= 0()
mark(0()) = [2]
[0]
[1]
>= [1]
[0]
[1]
= 0()
mark(add(X1,X2)) = [1 0 1] [1 0 2] [0]
[0 0 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 1] [1 0 2] [0]
[0 0 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 0 1] [1]
[0 0 0] X1 + [0]
[0 0 1] [0]
>= [1 0 1] [1]
[0 0 0] X1 + [0]
[0 0 1] [0]
= cons(mark(X1),X2)
mark(from(X)) = [1 0 2] [1]
[0 0 0] X + [0]
[0 0 1] [0]
>= [1 0 2] [1]
[0 0 0] X + [0]
[0 0 1] [0]
= a__from(mark(X))
mark(fst(X1,X2)) = [1 0 1] [1 0 2] [0]
[0 0 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 1] [1 0 2] [0]
[0 0 0] X1 + [0 0 0] X2 + [0]
[0 0 1] [0 0 1] [0]
= a__fst(mark(X1),mark(X2))
mark(nil()) = [2]
[0]
[1]
>= [1]
[0]
[1]
= nil()
mark(s(X)) = [1]
[0]
[0]
>= [1]
[0]
[0]
= s(X)
** Step 1.b:8: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__add(X1,X2) -> add(X1,X2)
a__fst(X1,X2) -> fst(X1,X2)
a__len(X) -> len(X)
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
- Weak TRS:
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__fst(0(),Z) -> nil()
a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
a__len(cons(X,Z)) -> s(len(Z))
a__len(nil()) -> 0()
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
- Signature:
{a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0
,add,cons,from,fst,len,nil,s}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
{a__add,a__from,a__fst,a__len,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__add) = [1 0 0] [1 1 0] [0]
[0 1 0] x1 + [0 1 1] x2 + [0]
[0 0 1] [0 0 1] [0]
p(a__from) = [1 1 0] [1]
[0 1 1] x1 + [1]
[0 0 1] [1]
p(a__fst) = [1 0 0] [1 1 0] [0]
[0 1 1] x1 + [0 1 1] x2 + [0]
[0 0 1] [0 0 1] [0]
p(a__len) = [1 1 0] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(add) = [1 0 0] [1 1 0] [0]
[0 1 0] x1 + [0 1 1] x2 + [0]
[0 0 1] [0 0 1] [0]
p(cons) = [1 0 0] [1]
[0 1 0] x1 + [1]
[0 0 1] [0]
p(from) = [1 1 0] [1]
[0 1 1] x1 + [0]
[0 0 1] [1]
p(fst) = [1 0 0] [1 1 0] [0]
[0 1 1] x1 + [0 1 1] x2 + [0]
[0 0 1] [0 0 1] [0]
p(len) = [1 1 0] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(mark) = [1 1 0] [0]
[0 1 1] x1 + [0]
[0 0 1] [0]
p(nil) = [0]
[0]
[0]
p(s) = [0]
[0]
[0]
Following rules are strictly oriented:
mark(cons(X1,X2)) = [1 1 0] [2]
[0 1 1] X1 + [1]
[0 0 1] [0]
> [1 1 0] [1]
[0 1 1] X1 + [1]
[0 0 1] [0]
= cons(mark(X1),X2)
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0 0] [1 1 0] [0]
[0 1 0] X1 + [0 1 1] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 0] [1 1 0] [0]
[0 1 0] X1 + [0 1 1] X2 + [0]
[0 0 1] [0 0 1] [0]
= add(X1,X2)
a__add(0(),X) = [1 1 0] [0]
[0 1 1] X + [0]
[0 0 1] [0]
>= [1 1 0] [0]
[0 1 1] X + [0]
[0 0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 1 0] [0]
[0 1 1] Y + [0]
[0 0 1] [0]
>= [0]
[0]
[0]
= s(add(X,Y))
a__from(X) = [1 1 0] [1]
[0 1 1] X + [1]
[0 0 1] [1]
>= [1 1 0] [1]
[0 1 1] X + [1]
[0 0 1] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 1 0] [1]
[0 1 1] X + [1]
[0 0 1] [1]
>= [1 1 0] [1]
[0 1 1] X + [0]
[0 0 1] [1]
= from(X)
a__fst(X1,X2) = [1 0 0] [1 1 0] [0]
[0 1 1] X1 + [0 1 1] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 0 0] [1 1 0] [0]
[0 1 1] X1 + [0 1 1] X2 + [0]
[0 0 1] [0 0 1] [0]
= fst(X1,X2)
a__fst(0(),Z) = [1 1 0] [0]
[0 1 1] Z + [0]
[0 0 1] [0]
>= [0]
[0]
[0]
= nil()
a__fst(s(X),cons(Y,Z)) = [1 1 0] [2]
[0 1 1] Y + [1]
[0 0 1] [0]
>= [1 1 0] [1]
[0 1 1] Y + [1]
[0 0 1] [0]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1 1 0] [0]
[0 1 1] X + [0]
[0 0 1] [0]
>= [1 1 0] [0]
[0 1 1] X + [0]
[0 0 1] [0]
= len(X)
a__len(cons(X,Z)) = [1 1 0] [2]
[0 1 1] X + [1]
[0 0 1] [0]
>= [0]
[0]
[0]
= s(len(Z))
a__len(nil()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(add(X1,X2)) = [1 1 0] [1 2 1] [0]
[0 1 1] X1 + [0 1 2] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 1 0] [1 2 1] [0]
[0 1 1] X1 + [0 1 2] X2 + [0]
[0 0 1] [0 0 1] [0]
= a__add(mark(X1),mark(X2))
mark(from(X)) = [1 2 1] [1]
[0 1 2] X + [1]
[0 0 1] [1]
>= [1 2 1] [1]
[0 1 2] X + [1]
[0 0 1] [1]
= a__from(mark(X))
mark(fst(X1,X2)) = [1 1 1] [1 2 1] [0]
[0 1 2] X1 + [0 1 2] X2 + [0]
[0 0 1] [0 0 1] [0]
>= [1 1 0] [1 2 1] [0]
[0 1 2] X1 + [0 1 2] X2 + [0]
[0 0 1] [0 0 1] [0]
= a__fst(mark(X1),mark(X2))
mark(len(X)) = [1 2 1] [0]
[0 1 2] X + [0]
[0 0 1] [0]
>= [1 2 1] [0]
[0 1 2] X + [0]
[0 0 1] [0]
= a__len(mark(X))
mark(nil()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= nil()
mark(s(X)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= s(X)
** Step 1.b:9: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__add(X1,X2) -> add(X1,X2)
a__fst(X1,X2) -> fst(X1,X2)
a__len(X) -> len(X)
mark(from(X)) -> a__from(mark(X))
mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
- Weak TRS:
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__fst(0(),Z) -> nil()
a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
a__len(cons(X,Z)) -> s(len(Z))
a__len(nil()) -> 0()
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(len(X)) -> a__len(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
- Signature:
{a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0
,add,cons,from,fst,len,nil,s}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
{a__add,a__from,a__fst,a__len,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__add) = [1 0 0] [1 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0]
[0 0 1] [0 0 0] [0]
p(a__from) = [1 1 1] [0]
[0 1 1] x1 + [0]
[0 0 0] [0]
p(a__fst) = [1 1 0] [1 1 0] [1]
[0 1 1] x1 + [0 1 1] x2 + [1]
[0 0 0] [0 0 1] [0]
p(a__len) = [1 1 0] [0]
[0 1 0] x1 + [0]
[0 0 1] [0]
p(add) = [1 0 0] [1 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 0] [0]
p(cons) = [1 0 0] [0 1 0] [0]
[0 1 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
p(from) = [1 1 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(fst) = [1 1 0] [1 1 0] [1]
[0 1 0] x1 + [0 1 0] x2 + [1]
[0 0 0] [0 0 0] [0]
p(len) = [1 1 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(mark) = [1 1 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(nil) = [0]
[0]
[0]
p(s) = [0 1 0] [0]
[0 0 1] x1 + [0]
[0 0 0] [0]
Following rules are strictly oriented:
mark(fst(X1,X2)) = [1 2 0] [1 2 0] [2]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 0] [0]
> [1 2 0] [1 2 0] [1]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 0] [0]
= a__fst(mark(X1),mark(X2))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0 0] [1 1 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [0]
[0 0 1] [0 0 0] [0]
>= [1 0 0] [1 1 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [0]
[0 0 0] [0 0 0] [0]
= add(X1,X2)
a__add(0(),X) = [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
>= [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= mark(X)
a__add(s(X),Y) = [0 1 0] [1 1 0] [0]
[0 0 1] X + [0 1 0] Y + [0]
[0 0 0] [0 0 0] [0]
>= [0 1 0] [0 1 0] [0]
[0 0 0] X + [0 0 0] Y + [0]
[0 0 0] [0 0 0] [0]
= s(add(X,Y))
a__from(X) = [1 1 1] [0]
[0 1 1] X + [0]
[0 0 0] [0]
>= [1 1 1] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 1 1] [0]
[0 1 1] X + [0]
[0 0 0] [0]
>= [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= from(X)
a__fst(X1,X2) = [1 1 0] [1 1 0] [1]
[0 1 1] X1 + [0 1 1] X2 + [1]
[0 0 0] [0 0 1] [0]
>= [1 1 0] [1 1 0] [1]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 0] [0]
= fst(X1,X2)
a__fst(0(),Z) = [1 1 0] [1]
[0 1 1] Z + [1]
[0 0 1] [0]
>= [0]
[0]
[0]
= nil()
a__fst(s(X),cons(Y,Z)) = [0 1 1] [1 1 0] [0 1 0] [1]
[0 0 1] X + [0 1 0] Y + [0 0 0] Z + [1]
[0 0 0] [0 0 0] [0 0 0] [0]
>= [0 1 0] [1 1 0] [0 1 0] [1]
[0 0 0] X + [0 1 0] Y + [0 0 0] Z + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1 1 0] [0]
[0 1 0] X + [0]
[0 0 1] [0]
>= [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= len(X)
a__len(cons(X,Z)) = [1 1 0] [0 1 0] [0]
[0 1 0] X + [0 0 0] Z + [0]
[0 0 0] [0 0 0] [0]
>= [0 1 0] [0]
[0 0 0] Z + [0]
[0 0 0] [0]
= s(len(Z))
a__len(nil()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(add(X1,X2)) = [1 1 0] [1 2 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [0]
[0 0 0] [0 0 0] [0]
>= [1 1 0] [1 2 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [0]
[0 0 0] [0 0 0] [0]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 1 0] [0 1 0] [0]
[0 1 0] X1 + [0 0 0] X2 + [0]
[0 0 0] [0 0 0] [0]
>= [1 1 0] [0 1 0] [0]
[0 1 0] X1 + [0 0 0] X2 + [0]
[0 0 0] [0 0 0] [0]
= cons(mark(X1),X2)
mark(from(X)) = [1 2 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
>= [1 2 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= a__from(mark(X))
mark(len(X)) = [1 2 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
>= [1 2 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= a__len(mark(X))
mark(nil()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= nil()
mark(s(X)) = [0 1 1] [0]
[0 0 1] X + [0]
[0 0 0] [0]
>= [0 1 0] [0]
[0 0 1] X + [0]
[0 0 0] [0]
= s(X)
** Step 1.b:10: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__add(X1,X2) -> add(X1,X2)
a__fst(X1,X2) -> fst(X1,X2)
a__len(X) -> len(X)
mark(from(X)) -> a__from(mark(X))
- Weak TRS:
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__fst(0(),Z) -> nil()
a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
a__len(cons(X,Z)) -> s(len(Z))
a__len(nil()) -> 0()
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
- Signature:
{a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0
,add,cons,from,fst,len,nil,s}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
{a__add,a__from,a__fst,a__len,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__add) = [1 0 1] [1 1 0] [0]
[0 1 1] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 0] [0]
p(a__from) = [1 1 0] [0]
[0 1 0] x1 + [1]
[0 0 1] [0]
p(a__fst) = [1 1 1] [1 1 0] [0]
[0 1 1] x1 + [0 1 0] x2 + [0]
[0 0 1] [0 0 0] [0]
p(a__len) = [1 0 0] [0]
[0 1 1] x1 + [0]
[0 0 0] [0]
p(add) = [1 0 1] [1 1 0] [0]
[0 1 1] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 0] [0]
p(cons) = [1 0 0] [0 0 1] [0]
[0 1 0] x1 + [0 0 1] x2 + [0]
[0 0 0] [0 0 0] [0]
p(from) = [1 1 0] [0]
[0 1 0] x1 + [1]
[0 0 0] [0]
p(fst) = [1 1 1] [1 1 0] [0]
[0 1 1] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 0] [0]
p(len) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(mark) = [1 1 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(nil) = [0]
[0]
[0]
p(s) = [0]
[0]
[0]
Following rules are strictly oriented:
mark(from(X)) = [1 2 0] [1]
[0 1 0] X + [1]
[0 0 0] [0]
> [1 2 0] [0]
[0 1 0] X + [1]
[0 0 0] [0]
= a__from(mark(X))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0 1] [1 1 0] [0]
[0 1 1] X1 + [0 1 0] X2 + [0]
[0 0 0] [0 0 0] [0]
>= [1 0 1] [1 1 0] [0]
[0 1 1] X1 + [0 1 0] X2 + [0]
[0 0 0] [0 0 0] [0]
= add(X1,X2)
a__add(0(),X) = [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
>= [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= mark(X)
a__add(s(X),Y) = [1 1 0] [0]
[0 1 0] Y + [0]
[0 0 0] [0]
>= [0]
[0]
[0]
= s(add(X,Y))
a__from(X) = [1 1 0] [0]
[0 1 0] X + [1]
[0 0 1] [0]
>= [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 1 0] [0]
[0 1 0] X + [1]
[0 0 1] [0]
>= [1 1 0] [0]
[0 1 0] X + [1]
[0 0 0] [0]
= from(X)
a__fst(X1,X2) = [1 1 1] [1 1 0] [0]
[0 1 1] X1 + [0 1 0] X2 + [0]
[0 0 1] [0 0 0] [0]
>= [1 1 1] [1 1 0] [0]
[0 1 1] X1 + [0 1 0] X2 + [0]
[0 0 0] [0 0 0] [0]
= fst(X1,X2)
a__fst(0(),Z) = [1 1 0] [0]
[0 1 0] Z + [0]
[0 0 0] [0]
>= [0]
[0]
[0]
= nil()
a__fst(s(X),cons(Y,Z)) = [1 1 0] [0 0 2] [0]
[0 1 0] Y + [0 0 1] Z + [0]
[0 0 0] [0 0 0] [0]
>= [1 1 0] [0]
[0 1 0] Y + [0]
[0 0 0] [0]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1 0 0] [0]
[0 1 1] X + [0]
[0 0 0] [0]
>= [1 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= len(X)
a__len(cons(X,Z)) = [1 0 0] [0 0 1] [0]
[0 1 0] X + [0 0 1] Z + [0]
[0 0 0] [0 0 0] [0]
>= [0]
[0]
[0]
= s(len(Z))
a__len(nil()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(add(X1,X2)) = [1 1 2] [1 2 0] [0]
[0 1 1] X1 + [0 1 0] X2 + [0]
[0 0 0] [0 0 0] [0]
>= [1 1 0] [1 2 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [0]
[0 0 0] [0 0 0] [0]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 1 0] [0 0 2] [0]
[0 1 0] X1 + [0 0 1] X2 + [0]
[0 0 0] [0 0 0] [0]
>= [1 1 0] [0 0 1] [0]
[0 1 0] X1 + [0 0 1] X2 + [0]
[0 0 0] [0 0 0] [0]
= cons(mark(X1),X2)
mark(fst(X1,X2)) = [1 2 2] [1 2 0] [0]
[0 1 1] X1 + [0 1 0] X2 + [0]
[0 0 0] [0 0 0] [0]
>= [1 2 0] [1 2 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [0]
[0 0 0] [0 0 0] [0]
= a__fst(mark(X1),mark(X2))
mark(len(X)) = [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
>= [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= a__len(mark(X))
mark(nil()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= nil()
mark(s(X)) = [0]
[0]
[0]
>= [0]
[0]
[0]
= s(X)
** Step 1.b:11: NaturalMI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
a__add(X1,X2) -> add(X1,X2)
a__fst(X1,X2) -> fst(X1,X2)
a__len(X) -> len(X)
- Weak TRS:
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__fst(0(),Z) -> nil()
a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
a__len(cons(X,Z)) -> s(len(Z))
a__len(nil()) -> 0()
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
- Signature:
{a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0
,add,cons,from,fst,len,nil,s}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
{a__add,a__from,a__fst,a__len,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(a__add) = [1 0 1] [1 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [1]
[0 0 0] [0 0 0] [0]
p(a__from) = [1 1 0] [0]
[0 1 0] x1 + [1]
[0 0 1] [0]
p(a__fst) = [1 0 0] [1 1 0] [1]
[0 1 1] x1 + [0 1 1] x2 + [1]
[0 0 0] [0 0 0] [0]
p(a__len) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(add) = [1 0 0] [1 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [1]
[0 0 0] [0 0 0] [0]
p(cons) = [1 0 1] [0 0 1] [0]
[0 1 0] x1 + [0 0 0] x2 + [1]
[0 0 1] [0 0 0] [0]
p(from) = [1 1 0] [0]
[0 1 0] x1 + [1]
[0 0 1] [0]
p(fst) = [1 0 0] [1 1 0] [0]
[0 1 1] x1 + [0 1 1] x2 + [1]
[0 0 0] [0 0 0] [0]
p(len) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(mark) = [1 1 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(nil) = [0]
[1]
[0]
p(s) = [0 0 0] [0]
[0 0 1] x1 + [0]
[0 0 0] [0]
Following rules are strictly oriented:
a__fst(X1,X2) = [1 0 0] [1 1 0] [1]
[0 1 1] X1 + [0 1 1] X2 + [1]
[0 0 0] [0 0 0] [0]
> [1 0 0] [1 1 0] [0]
[0 1 1] X1 + [0 1 1] X2 + [1]
[0 0 0] [0 0 0] [0]
= fst(X1,X2)
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0 1] [1 1 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 0] [0]
>= [1 0 0] [1 1 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 0] [0]
= add(X1,X2)
a__add(0(),X) = [1 1 0] [0]
[0 1 0] X + [1]
[0 0 0] [0]
>= [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= mark(X)
a__add(s(X),Y) = [0 0 0] [1 1 0] [0]
[0 0 1] X + [0 1 0] Y + [1]
[0 0 0] [0 0 0] [0]
>= [0]
[0]
[0]
= s(add(X,Y))
a__from(X) = [1 1 0] [0]
[0 1 0] X + [1]
[0 0 1] [0]
>= [1 1 0] [0]
[0 1 0] X + [1]
[0 0 0] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 1 0] [0]
[0 1 0] X + [1]
[0 0 1] [0]
>= [1 1 0] [0]
[0 1 0] X + [1]
[0 0 1] [0]
= from(X)
a__fst(0(),Z) = [1 1 0] [1]
[0 1 1] Z + [1]
[0 0 0] [0]
>= [0]
[1]
[0]
= nil()
a__fst(s(X),cons(Y,Z)) = [0 0 0] [1 1 1] [0 0 1] [2]
[0 0 1] X + [0 1 1] Y + [0 0 0] Z + [2]
[0 0 0] [0 0 0] [0 0 0] [0]
>= [1 1 0] [0]
[0 1 0] Y + [1]
[0 0 0] [0]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
>= [1 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= len(X)
a__len(cons(X,Z)) = [1 0 1] [0 0 1] [0]
[0 1 0] X + [0 0 0] Z + [1]
[0 0 0] [0 0 0] [0]
>= [0]
[0]
[0]
= s(len(Z))
a__len(nil()) = [0]
[1]
[0]
>= [0]
[0]
[0]
= 0()
mark(0()) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
mark(add(X1,X2)) = [1 1 0] [1 2 0] [1]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 0] [0]
>= [1 1 0] [1 2 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 0] [0]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 1 1] [0 0 1] [1]
[0 1 0] X1 + [0 0 0] X2 + [1]
[0 0 0] [0 0 0] [0]
>= [1 1 0] [0 0 1] [0]
[0 1 0] X1 + [0 0 0] X2 + [1]
[0 0 0] [0 0 0] [0]
= cons(mark(X1),X2)
mark(from(X)) = [1 2 0] [1]
[0 1 0] X + [1]
[0 0 0] [0]
>= [1 2 0] [0]
[0 1 0] X + [1]
[0 0 0] [0]
= a__from(mark(X))
mark(fst(X1,X2)) = [1 1 1] [1 2 1] [1]
[0 1 1] X1 + [0 1 1] X2 + [1]
[0 0 0] [0 0 0] [0]
>= [1 1 0] [1 2 0] [1]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 0] [0]
= a__fst(mark(X1),mark(X2))
mark(len(X)) = [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
>= [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= a__len(mark(X))
mark(nil()) = [1]
[1]
[0]
>= [0]
[1]
[0]
= nil()
mark(s(X)) = [0 0 1] [0]
[0 0 1] X + [0]
[0 0 0] [0]
>= [0 0 0] [0]
[0 0 1] X + [0]
[0 0 0] [0]
= s(X)
** Step 1.b:12: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__add(X1,X2) -> add(X1,X2)
a__len(X) -> len(X)
- Weak TRS:
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__fst(X1,X2) -> fst(X1,X2)
a__fst(0(),Z) -> nil()
a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
a__len(cons(X,Z)) -> s(len(Z))
a__len(nil()) -> 0()
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
- Signature:
{a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0
,add,cons,from,fst,len,nil,s}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
{a__add,a__from,a__fst,a__len,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[1]
p(a__add) = [1 0 0] [1 1 0] [1]
[0 1 0] x1 + [0 1 0] x2 + [1]
[0 0 1] [0 0 0] [0]
p(a__from) = [1 1 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(a__fst) = [1 0 0] [1 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0]
[0 0 1] [0 0 0] [0]
p(a__len) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [1]
p(add) = [1 0 0] [1 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [1]
[0 0 0] [0 0 0] [0]
p(cons) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(from) = [1 1 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(fst) = [1 0 0] [1 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0]
[0 0 1] [0 0 0] [0]
p(len) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(mark) = [1 1 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [1]
p(nil) = [0]
[0]
[1]
p(s) = [0]
[0]
[0]
Following rules are strictly oriented:
a__add(X1,X2) = [1 0 0] [1 1 0] [1]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 1] [0 0 0] [0]
> [1 0 0] [1 1 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 0] [0]
= add(X1,X2)
Following rules are (at-least) weakly oriented:
a__add(0(),X) = [1 1 0] [1]
[0 1 0] X + [1]
[0 0 0] [1]
>= [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [1]
= mark(X)
a__add(s(X),Y) = [1 1 0] [1]
[0 1 0] Y + [1]
[0 0 0] [0]
>= [0]
[0]
[0]
= s(add(X,Y))
a__from(X) = [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
>= [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
>= [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= from(X)
a__fst(X1,X2) = [1 0 0] [1 1 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [0]
[0 0 1] [0 0 0] [0]
>= [1 0 0] [1 1 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [0]
[0 0 1] [0 0 0] [0]
= fst(X1,X2)
a__fst(0(),Z) = [1 1 0] [0]
[0 1 0] Z + [0]
[0 0 0] [1]
>= [0]
[0]
[1]
= nil()
a__fst(s(X),cons(Y,Z)) = [1 1 0] [0]
[0 1 0] Y + [0]
[0 0 0] [0]
>= [1 1 0] [0]
[0 1 0] Y + [0]
[0 0 0] [0]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [1]
>= [1 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= len(X)
a__len(cons(X,Z)) = [1 0 0] [0]
[0 1 0] X + [0]
[0 0 0] [1]
>= [0]
[0]
[0]
= s(len(Z))
a__len(nil()) = [0]
[0]
[1]
>= [0]
[0]
[1]
= 0()
mark(0()) = [0]
[0]
[1]
>= [0]
[0]
[1]
= 0()
mark(add(X1,X2)) = [1 1 0] [1 2 0] [1]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 0] [1]
>= [1 1 0] [1 2 0] [1]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 0] [1]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 1 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [1]
>= [1 1 0] [0]
[0 1 0] X1 + [0]
[0 0 0] [0]
= cons(mark(X1),X2)
mark(from(X)) = [1 2 0] [0]
[0 1 0] X + [0]
[0 0 0] [1]
>= [1 2 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= a__from(mark(X))
mark(fst(X1,X2)) = [1 1 0] [1 2 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [0]
[0 0 0] [0 0 0] [1]
>= [1 1 0] [1 2 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [0]
[0 0 0] [0 0 0] [1]
= a__fst(mark(X1),mark(X2))
mark(len(X)) = [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [1]
>= [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [1]
= a__len(mark(X))
mark(nil()) = [0]
[0]
[1]
>= [0]
[0]
[1]
= nil()
mark(s(X)) = [0]
[0]
[1]
>= [0]
[0]
[0]
= s(X)
** Step 1.b:13: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__len(X) -> len(X)
- Weak TRS:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__fst(X1,X2) -> fst(X1,X2)
a__fst(0(),Z) -> nil()
a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
a__len(cons(X,Z)) -> s(len(Z))
a__len(nil()) -> 0()
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
- Signature:
{a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0
,add,cons,from,fst,len,nil,s}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
{a__add,a__from,a__fst,a__len,mark}
TcT has computed the following interpretation:
p(0) = [1]
[0]
[0]
p(a__add) = [1 0 1] [1 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [1]
[0 0 1] [0 0 0] [0]
p(a__from) = [1 1 0] [1]
[0 1 0] x1 + [1]
[0 0 0] [0]
p(a__fst) = [1 0 0] [1 1 0] [0]
[0 1 0] x1 + [0 1 1] x2 + [0]
[0 0 0] [0 0 1] [0]
p(a__len) = [1 1 0] [1]
[0 1 0] x1 + [1]
[0 0 0] [0]
p(add) = [1 0 1] [1 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [1]
[0 0 0] [0 0 0] [0]
p(cons) = [1 0 1] [0]
[0 1 0] x1 + [1]
[0 0 0] [0]
p(from) = [1 1 0] [1]
[0 1 0] x1 + [1]
[0 0 0] [0]
p(fst) = [1 0 0] [1 1 0] [0]
[0 1 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 0] [0]
p(len) = [1 1 0] [0]
[0 1 0] x1 + [1]
[0 0 0] [0]
p(mark) = [1 1 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(nil) = [1]
[0]
[0]
p(s) = [1]
[1]
[0]
Following rules are strictly oriented:
a__len(X) = [1 1 0] [1]
[0 1 0] X + [1]
[0 0 0] [0]
> [1 1 0] [0]
[0 1 0] X + [1]
[0 0 0] [0]
= len(X)
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0 1] [1 1 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 1] [0 0 0] [0]
>= [1 0 1] [1 1 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 0] [0]
= add(X1,X2)
a__add(0(),X) = [1 1 0] [1]
[0 1 0] X + [1]
[0 0 0] [0]
>= [1 1 0] [0]
[0 1 0] X + [0]
[0 0 0] [0]
= mark(X)
a__add(s(X),Y) = [1 1 0] [1]
[0 1 0] Y + [2]
[0 0 0] [0]
>= [1]
[1]
[0]
= s(add(X,Y))
a__from(X) = [1 1 0] [1]
[0 1 0] X + [1]
[0 0 0] [0]
>= [1 1 0] [0]
[0 1 0] X + [1]
[0 0 0] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 1 0] [1]
[0 1 0] X + [1]
[0 0 0] [0]
>= [1 1 0] [1]
[0 1 0] X + [1]
[0 0 0] [0]
= from(X)
a__fst(X1,X2) = [1 0 0] [1 1 0] [0]
[0 1 0] X1 + [0 1 1] X2 + [0]
[0 0 0] [0 0 1] [0]
>= [1 0 0] [1 1 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [0]
[0 0 0] [0 0 0] [0]
= fst(X1,X2)
a__fst(0(),Z) = [1 1 0] [1]
[0 1 1] Z + [0]
[0 0 1] [0]
>= [1]
[0]
[0]
= nil()
a__fst(s(X),cons(Y,Z)) = [1 1 1] [2]
[0 1 0] Y + [2]
[0 0 0] [0]
>= [1 1 0] [0]
[0 1 0] Y + [1]
[0 0 0] [0]
= cons(mark(Y),fst(X,Z))
a__len(cons(X,Z)) = [1 1 1] [2]
[0 1 0] X + [2]
[0 0 0] [0]
>= [1]
[1]
[0]
= s(len(Z))
a__len(nil()) = [2]
[1]
[0]
>= [1]
[0]
[0]
= 0()
mark(0()) = [1]
[0]
[0]
>= [1]
[0]
[0]
= 0()
mark(add(X1,X2)) = [1 1 1] [1 2 0] [1]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 0] [0]
>= [1 1 0] [1 2 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [1]
[0 0 0] [0 0 0] [0]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 1 1] [1]
[0 1 0] X1 + [1]
[0 0 0] [0]
>= [1 1 0] [0]
[0 1 0] X1 + [1]
[0 0 0] [0]
= cons(mark(X1),X2)
mark(from(X)) = [1 2 0] [2]
[0 1 0] X + [1]
[0 0 0] [0]
>= [1 2 0] [1]
[0 1 0] X + [1]
[0 0 0] [0]
= a__from(mark(X))
mark(fst(X1,X2)) = [1 1 0] [1 2 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [0]
[0 0 0] [0 0 0] [0]
>= [1 1 0] [1 2 0] [0]
[0 1 0] X1 + [0 1 0] X2 + [0]
[0 0 0] [0 0 0] [0]
= a__fst(mark(X1),mark(X2))
mark(len(X)) = [1 2 0] [1]
[0 1 0] X + [1]
[0 0 0] [0]
>= [1 2 0] [1]
[0 1 0] X + [1]
[0 0 0] [0]
= a__len(mark(X))
mark(nil()) = [1]
[0]
[0]
>= [1]
[0]
[0]
= nil()
mark(s(X)) = [2]
[1]
[0]
>= [1]
[1]
[0]
= s(X)
** Step 1.b:14: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a__add(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__fst(X1,X2) -> fst(X1,X2)
a__fst(0(),Z) -> nil()
a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
a__len(X) -> len(X)
a__len(cons(X,Z)) -> s(len(Z))
a__len(nil()) -> 0()
mark(0()) -> 0()
mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(nil()) -> nil()
mark(s(X)) -> s(X)
- Signature:
{a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0
,add,cons,from,fst,len,nil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^3))