* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ 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: WeightGap WORST_CASE(?,O(n^2))
+ 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:
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__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(a__add) = [1] x1 + [1] x2 + [0]
p(a__from) = [1] x1 + [0]
p(a__fst) = [1] x1 + [1] x2 + [0]
p(a__len) = [1] x1 + [0]
p(add) = [1] x2 + [0]
p(cons) = [1] x1 + [0]
p(from) = [1] x1 + [0]
p(fst) = [1] x1 + [0]
p(len) = [1] x1 + [0]
p(mark) = [0]
p(nil) = [0]
p(s) = [0]
Following rules are strictly oriented:
a__add(0(),X) = [1] X + [1]
> [0]
= mark(X)
a__fst(0(),Z) = [1] Z + [1]
> [0]
= nil()
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X2 + [0]
= add(X1,X2)
a__add(s(X),Y) = [1] Y + [0]
>= [0]
= s(add(X,Y))
a__from(X) = [1] X + [0]
>= [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 + [0]
= fst(X1,X2)
a__fst(s(X),cons(Y,Z)) = [1] Y + [0]
>= [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]
>= [1]
= 0()
mark(0()) = [0]
>= [1]
= 0()
mark(add(X1,X2)) = [0]
>= [0]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [0]
>= [0]
= cons(mark(X1),X2)
mark(from(X)) = [0]
>= [0]
= a__from(mark(X))
mark(fst(X1,X2)) = [0]
>= [0]
= a__fst(mark(X1),mark(X2))
mark(len(X)) = [0]
>= [0]
= a__len(mark(X))
mark(nil()) = [0]
>= [0]
= nil()
mark(s(X)) = [0]
>= [0]
= s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__add(X1,X2) -> add(X1,X2)
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(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__fst(0(),Z) -> 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:
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__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [4]
p(a__add) = [1] x1 + [1] x2 + [1]
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 + [0]
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(X1,X2) = [1] X1 + [1] X2 + [1]
> [1] X1 + [1] X2 + [0]
= add(X1,X2)
a__add(s(X),Y) = [1] Y + [1]
> [0]
= s(add(X,Y))
Following rules are (at-least) weakly oriented:
a__add(0(),X) = [1] X + [5]
>= [1] X + [0]
= mark(X)
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 + [4]
>= [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]
>= [4]
= 0()
mark(0()) = [4]
>= [4]
= 0()
mark(add(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [1]
= 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)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__from(X) -> cons(mark(X),from(s(X)))
a__from(X) -> from(X)
a__fst(X1,X2) -> fst(X1,X2)
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(X1,X2) -> add(X1,X2)
a__add(0(),X) -> mark(X)
a__add(s(X),Y) -> s(add(X,Y))
a__fst(0(),Z) -> 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:
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__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__add) = [1] x1 + [1] x2 + [0]
p(a__from) = [1] x1 + [4]
p(a__fst) = [1] x1 + [1] x2 + [0]
p(a__len) = [1] x1 + [0]
p(add) = [1] x1 + [1] x2 + [0]
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__from(X) = [1] X + [4]
> [1] X + [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1] X + [4]
> [1] X + [0]
= from(X)
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= add(X1,X2)
a__add(0(),X) = [1] X + [0]
>= [1] X + [0]
= mark(X)
a__add(s(X),Y) = [1] Y + [0]
>= [0]
= s(add(X,Y))
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 + [0]
>= [1] X1 + [1] X2 + [0]
= 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 + [4]
= 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)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__fst(X1,X2) -> fst(X1,X2)
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(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(0(),Z) -> 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:
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__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__add) = [1] x1 + [1] x2 + [1]
p(a__from) = [1] x1 + [3]
p(a__fst) = [1] x1 + [1] x2 + [1]
p(a__len) = [1] x1 + [0]
p(add) = [1] x1 + [1] x2 + [1]
p(cons) = [1] x1 + [2]
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__fst(X1,X2) = [1] X1 + [1] X2 + [1]
> [1] X1 + [1] X2 + [0]
= fst(X1,X2)
a__fst(s(X),cons(Y,Z)) = [1] Y + [3]
> [1] Y + [2]
= cons(mark(Y),fst(X,Z))
a__len(cons(X,Z)) = [1] X + [2]
> [0]
= s(len(Z))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= add(X1,X2)
a__add(0(),X) = [1] X + [1]
>= [1] X + [0]
= mark(X)
a__add(s(X),Y) = [1] Y + [1]
>= [0]
= s(add(X,Y))
a__from(X) = [1] X + [3]
>= [1] X + [2]
= cons(mark(X),from(s(X)))
a__from(X) = [1] X + [3]
>= [1] X + [0]
= from(X)
a__fst(0(),Z) = [1] Z + [1]
>= [0]
= nil()
a__len(X) = [1] X + [0]
>= [1] X + [0]
= len(X)
a__len(nil()) = [0]
>= [0]
= 0()
mark(0()) = [0]
>= [0]
= 0()
mark(add(X1,X2)) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1] X1 + [2]
>= [1] X1 + [2]
= cons(mark(X1),X2)
mark(from(X)) = [1] X + [0]
>= [1] X + [3]
= a__from(mark(X))
mark(fst(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [1]
= 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)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:5: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__len(X) -> len(X)
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(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))
- 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:
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__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [4]
p(a__add) = [1] x1 + [1] x2 + [7]
p(a__from) = [1] x1 + [2]
p(a__fst) = [1] x1 + [1] x2 + [2]
p(a__len) = [1] x1 + [0]
p(add) = [1] x1 + [1] x2 + [7]
p(cons) = [1] x1 + [0]
p(from) = [0]
p(fst) = [0]
p(len) = [0]
p(mark) = [2]
p(nil) = [6]
p(s) = [0]
Following rules are strictly oriented:
a__len(nil()) = [6]
> [4]
= 0()
mark(s(X)) = [2]
> [0]
= s(X)
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1] X1 + [1] X2 + [7]
>= [1] X1 + [1] X2 + [7]
= add(X1,X2)
a__add(0(),X) = [1] X + [11]
>= [2]
= mark(X)
a__add(s(X),Y) = [1] Y + [7]
>= [0]
= s(add(X,Y))
a__from(X) = [1] X + [2]
>= [2]
= cons(mark(X),from(s(X)))
a__from(X) = [1] X + [2]
>= [0]
= from(X)
a__fst(X1,X2) = [1] X1 + [1] X2 + [2]
>= [0]
= fst(X1,X2)
a__fst(0(),Z) = [1] Z + [6]
>= [6]
= nil()
a__fst(s(X),cons(Y,Z)) = [1] Y + [2]
>= [2]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1] X + [0]
>= [0]
= len(X)
a__len(cons(X,Z)) = [1] X + [0]
>= [0]
= s(len(Z))
mark(0()) = [2]
>= [4]
= 0()
mark(add(X1,X2)) = [2]
>= [11]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [2]
>= [2]
= cons(mark(X1),X2)
mark(from(X)) = [2]
>= [4]
= a__from(mark(X))
mark(fst(X1,X2)) = [2]
>= [6]
= a__fst(mark(X1),mark(X2))
mark(len(X)) = [2]
>= [2]
= a__len(mark(X))
mark(nil()) = [2]
>= [6]
= nil()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:6: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
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()
- 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(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:
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__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(a__add) = [1] x1 + [1] x2 + [2]
p(a__from) = [1] x1 + [4]
p(a__fst) = [1] x1 + [1] x2 + [4]
p(a__len) = [1] x1 + [0]
p(add) = [0]
p(cons) = [1] x1 + [0]
p(from) = [0]
p(fst) = [0]
p(len) = [0]
p(mark) = [4]
p(nil) = [2]
p(s) = [0]
Following rules are strictly oriented:
mark(0()) = [4]
> [2]
= 0()
mark(nil()) = [4]
> [2]
= nil()
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1] X1 + [1] X2 + [2]
>= [0]
= add(X1,X2)
a__add(0(),X) = [1] X + [4]
>= [4]
= mark(X)
a__add(s(X),Y) = [1] Y + [2]
>= [0]
= s(add(X,Y))
a__from(X) = [1] X + [4]
>= [4]
= cons(mark(X),from(s(X)))
a__from(X) = [1] X + [4]
>= [0]
= from(X)
a__fst(X1,X2) = [1] X1 + [1] X2 + [4]
>= [0]
= fst(X1,X2)
a__fst(0(),Z) = [1] Z + [6]
>= [2]
= nil()
a__fst(s(X),cons(Y,Z)) = [1] Y + [4]
>= [4]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1] X + [0]
>= [0]
= len(X)
a__len(cons(X,Z)) = [1] X + [0]
>= [0]
= s(len(Z))
a__len(nil()) = [2]
>= [2]
= 0()
mark(add(X1,X2)) = [4]
>= [10]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [4]
>= [4]
= cons(mark(X1),X2)
mark(from(X)) = [4]
>= [8]
= a__from(mark(X))
mark(fst(X1,X2)) = [4]
>= [12]
= a__fst(mark(X1),mark(X2))
mark(len(X)) = [4]
>= [4]
= a__len(mark(X))
mark(s(X)) = [4]
>= [0]
= s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:7: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__len(X) -> len(X)
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))
- 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(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:
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__add) = {1,2},
uargs(a__from) = {1},
uargs(a__fst) = {1,2},
uargs(a__len) = {1},
uargs(cons) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(a__add) = [1] x1 + [1] x2 + [0]
p(a__from) = [1] x1 + [5]
p(a__fst) = [1] x1 + [1] x2 + [1]
p(a__len) = [1] x1 + [1]
p(add) = [0]
p(cons) = [1] x1 + [4]
p(from) = [0]
p(fst) = [1] x2 + [1]
p(len) = [0]
p(mark) = [1]
p(nil) = [0]
p(s) = [0]
Following rules are strictly oriented:
a__len(X) = [1] X + [1]
> [0]
= len(X)
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1] X1 + [1] X2 + [0]
>= [0]
= add(X1,X2)
a__add(0(),X) = [1] X + [1]
>= [1]
= mark(X)
a__add(s(X),Y) = [1] Y + [0]
>= [0]
= s(add(X,Y))
a__from(X) = [1] X + [5]
>= [5]
= cons(mark(X),from(s(X)))
a__from(X) = [1] X + [5]
>= [0]
= from(X)
a__fst(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X2 + [1]
= fst(X1,X2)
a__fst(0(),Z) = [1] Z + [2]
>= [0]
= nil()
a__fst(s(X),cons(Y,Z)) = [1] Y + [5]
>= [5]
= cons(mark(Y),fst(X,Z))
a__len(cons(X,Z)) = [1] X + [5]
>= [0]
= s(len(Z))
a__len(nil()) = [1]
>= [1]
= 0()
mark(0()) = [1]
>= [1]
= 0()
mark(add(X1,X2)) = [1]
>= [2]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1]
>= [5]
= cons(mark(X1),X2)
mark(from(X)) = [1]
>= [6]
= a__from(mark(X))
mark(fst(X1,X2)) = [1]
>= [3]
= a__fst(mark(X1),mark(X2))
mark(len(X)) = [1]
>= [2]
= a__len(mark(X))
mark(nil()) = [1]
>= [0]
= nil()
mark(s(X)) = [1]
>= [0]
= s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:8: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
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))
- 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(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:
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__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) = [4]
[1]
p(a__add) = [1 0] x_1 + [1 4] x_2 + [4]
[0 1] [0 1] [0]
p(a__from) = [1 4] x_1 + [0]
[0 1] [0]
p(a__fst) = [1 0] x_1 + [1 4] x_2 + [4]
[0 1] [0 1] [2]
p(a__len) = [1 0] x_1 + [0]
[0 1] [0]
p(add) = [1 0] x_1 + [1 4] x_2 + [4]
[0 1] [0 1] [0]
p(cons) = [1 0] x_1 + [0]
[0 1] [0]
p(from) = [1 4] x_1 + [0]
[0 1] [0]
p(fst) = [1 0] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [2]
p(len) = [1 0] x_1 + [0]
[0 1] [0]
p(mark) = [1 4] x_1 + [0]
[0 1] [0]
p(nil) = [4]
[1]
p(s) = [0]
[0]
Following rules are strictly oriented:
mark(fst(X1,X2)) = [1 4] X1 + [1 8] X2 + [8]
[0 1] [0 1] [2]
> [1 4] X1 + [1 8] X2 + [4]
[0 1] [0 1] [2]
= a__fst(mark(X1),mark(X2))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0] X1 + [1 4] X2 + [4]
[0 1] [0 1] [0]
>= [1 0] X1 + [1 4] X2 + [4]
[0 1] [0 1] [0]
= add(X1,X2)
a__add(0(),X) = [1 4] X + [8]
[0 1] [1]
>= [1 4] X + [0]
[0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 4] Y + [4]
[0 1] [0]
>= [0]
[0]
= s(add(X,Y))
a__from(X) = [1 4] X + [0]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 4] X + [0]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= from(X)
a__fst(X1,X2) = [1 0] X1 + [1 4] X2 + [4]
[0 1] [0 1] [2]
>= [1 0] X1 + [1 4] X2 + [0]
[0 1] [0 1] [2]
= fst(X1,X2)
a__fst(0(),Z) = [1 4] Z + [8]
[0 1] [3]
>= [4]
[1]
= nil()
a__fst(s(X),cons(Y,Z)) = [1 4] Y + [4]
[0 1] [2]
>= [1 4] Y + [0]
[0 1] [0]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= len(X)
a__len(cons(X,Z)) = [1 0] X + [0]
[0 1] [0]
>= [0]
[0]
= s(len(Z))
a__len(nil()) = [4]
[1]
>= [4]
[1]
= 0()
mark(0()) = [8]
[1]
>= [4]
[1]
= 0()
mark(add(X1,X2)) = [1 4] X1 + [1 8] X2 + [4]
[0 1] [0 1] [0]
>= [1 4] X1 + [1 8] X2 + [4]
[0 1] [0 1] [0]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 4] X1 + [0]
[0 1] [0]
>= [1 4] X1 + [0]
[0 1] [0]
= cons(mark(X1),X2)
mark(from(X)) = [1 8] X + [0]
[0 1] [0]
>= [1 8] X + [0]
[0 1] [0]
= a__from(mark(X))
mark(len(X)) = [1 4] X + [0]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= a__len(mark(X))
mark(nil()) = [8]
[1]
>= [4]
[1]
= nil()
mark(s(X)) = [0]
[0]
>= [0]
[0]
= s(X)
** Step 1.b:9: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
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(len(X)) -> a__len(mark(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(X) -> len(X)
a__len(cons(X,Z)) -> s(len(Z))
a__len(nil()) -> 0()
mark(0()) -> 0()
mark(fst(X1,X2)) -> a__fst(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:
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__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]
p(a__add) = [1 0] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [1]
p(a__from) = [1 6] x_1 + [2]
[0 1] [0]
p(a__fst) = [1 4] x_1 + [1 5] x_2 + [0]
[0 1] [0 1] [0]
p(a__len) = [1 0] x_1 + [6]
[0 1] [0]
p(add) = [1 0] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [1]
p(cons) = [1 1] x_1 + [2]
[0 1] [0]
p(from) = [1 6] x_1 + [2]
[0 1] [0]
p(fst) = [1 4] x_1 + [1 5] x_2 + [0]
[0 1] [0 1] [0]
p(len) = [1 0] x_1 + [6]
[0 1] [0]
p(mark) = [1 4] x_1 + [0]
[0 1] [0]
p(nil) = [0]
[0]
p(s) = [2]
[0]
Following rules are strictly oriented:
mark(add(X1,X2)) = [1 4] X1 + [1 8] X2 + [4]
[0 1] [0 1] [1]
> [1 4] X1 + [1 8] X2 + [0]
[0 1] [0 1] [1]
= a__add(mark(X1),mark(X2))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0] X1 + [1 4] X2 + [0]
[0 1] [0 1] [1]
>= [1 0] X1 + [1 4] X2 + [0]
[0 1] [0 1] [1]
= add(X1,X2)
a__add(0(),X) = [1 4] X + [1]
[0 1] [1]
>= [1 4] X + [0]
[0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 4] Y + [2]
[0 1] [1]
>= [2]
[0]
= s(add(X,Y))
a__from(X) = [1 6] X + [2]
[0 1] [0]
>= [1 5] X + [2]
[0 1] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 6] X + [2]
[0 1] [0]
>= [1 6] X + [2]
[0 1] [0]
= from(X)
a__fst(X1,X2) = [1 4] X1 + [1 5] X2 + [0]
[0 1] [0 1] [0]
>= [1 4] X1 + [1 5] X2 + [0]
[0 1] [0 1] [0]
= fst(X1,X2)
a__fst(0(),Z) = [1 5] Z + [1]
[0 1] [0]
>= [0]
[0]
= nil()
a__fst(s(X),cons(Y,Z)) = [1 6] Y + [4]
[0 1] [0]
>= [1 5] Y + [2]
[0 1] [0]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1 0] X + [6]
[0 1] [0]
>= [1 0] X + [6]
[0 1] [0]
= len(X)
a__len(cons(X,Z)) = [1 1] X + [8]
[0 1] [0]
>= [2]
[0]
= s(len(Z))
a__len(nil()) = [6]
[0]
>= [1]
[0]
= 0()
mark(0()) = [1]
[0]
>= [1]
[0]
= 0()
mark(cons(X1,X2)) = [1 5] X1 + [2]
[0 1] [0]
>= [1 5] X1 + [2]
[0 1] [0]
= cons(mark(X1),X2)
mark(from(X)) = [1 10] X + [2]
[0 1] [0]
>= [1 10] X + [2]
[0 1] [0]
= a__from(mark(X))
mark(fst(X1,X2)) = [1 8] X1 + [1 9] X2 + [0]
[0 1] [0 1] [0]
>= [1 8] X1 + [1 9] X2 + [0]
[0 1] [0 1] [0]
= a__fst(mark(X1),mark(X2))
mark(len(X)) = [1 4] X + [6]
[0 1] [0]
>= [1 4] X + [6]
[0 1] [0]
= a__len(mark(X))
mark(nil()) = [0]
[0]
>= [0]
[0]
= nil()
mark(s(X)) = [2]
[0]
>= [2]
[0]
= s(X)
** Step 1.b:10: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(X))
mark(len(X)) -> a__len(mark(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(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(fst(X1,X2)) -> a__fst(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:
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__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]
[2]
p(a__add) = [1 4] x_1 + [1 4] x_2 + [5]
[0 1] [0 1] [4]
p(a__from) = [1 2] x_1 + [0]
[0 1] [0]
p(a__fst) = [1 1] x_1 + [1 2] x_2 + [1]
[0 1] [0 1] [0]
p(a__len) = [1 1] x_1 + [0]
[0 1] [1]
p(add) = [1 4] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [4]
p(cons) = [1 0] x_1 + [0]
[0 1] [0]
p(from) = [1 2] x_1 + [0]
[0 1] [0]
p(fst) = [1 1] x_1 + [1 2] x_2 + [1]
[0 1] [0 1] [0]
p(len) = [1 1] x_1 + [0]
[0 1] [1]
p(mark) = [1 2] x_1 + [0]
[0 1] [0]
p(nil) = [0]
[1]
p(s) = [0]
[0]
Following rules are strictly oriented:
mark(len(X)) = [1 3] X + [2]
[0 1] [1]
> [1 3] X + [0]
[0 1] [1]
= a__len(mark(X))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 4] X1 + [1 4] X2 + [5]
[0 1] [0 1] [4]
>= [1 4] X1 + [1 4] X2 + [0]
[0 1] [0 1] [4]
= add(X1,X2)
a__add(0(),X) = [1 4] X + [13]
[0 1] [6]
>= [1 2] X + [0]
[0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 4] Y + [5]
[0 1] [4]
>= [0]
[0]
= s(add(X,Y))
a__from(X) = [1 2] X + [0]
[0 1] [0]
>= [1 2] X + [0]
[0 1] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 2] X + [0]
[0 1] [0]
>= [1 2] X + [0]
[0 1] [0]
= from(X)
a__fst(X1,X2) = [1 1] X1 + [1 2] X2 + [1]
[0 1] [0 1] [0]
>= [1 1] X1 + [1 2] X2 + [1]
[0 1] [0 1] [0]
= fst(X1,X2)
a__fst(0(),Z) = [1 2] Z + [3]
[0 1] [2]
>= [0]
[1]
= nil()
a__fst(s(X),cons(Y,Z)) = [1 2] Y + [1]
[0 1] [0]
>= [1 2] Y + [0]
[0 1] [0]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1 1] X + [0]
[0 1] [1]
>= [1 1] X + [0]
[0 1] [1]
= len(X)
a__len(cons(X,Z)) = [1 1] X + [0]
[0 1] [1]
>= [0]
[0]
= s(len(Z))
a__len(nil()) = [1]
[2]
>= [0]
[2]
= 0()
mark(0()) = [4]
[2]
>= [0]
[2]
= 0()
mark(add(X1,X2)) = [1 6] X1 + [1 6] X2 + [8]
[0 1] [0 1] [4]
>= [1 6] X1 + [1 6] X2 + [5]
[0 1] [0 1] [4]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 2] X1 + [0]
[0 1] [0]
>= [1 2] X1 + [0]
[0 1] [0]
= cons(mark(X1),X2)
mark(from(X)) = [1 4] X + [0]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= a__from(mark(X))
mark(fst(X1,X2)) = [1 3] X1 + [1 4] X2 + [1]
[0 1] [0 1] [0]
>= [1 3] X1 + [1 4] X2 + [1]
[0 1] [0 1] [0]
= a__fst(mark(X1),mark(X2))
mark(nil()) = [2]
[1]
>= [0]
[1]
= nil()
mark(s(X)) = [0]
[0]
>= [0]
[0]
= s(X)
** Step 1.b:11: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(from(X)) -> a__from(mark(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(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(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:
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__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]
p(a__add) = [1 0] x_1 + [1 4] x_2 + [1]
[0 1] [0 1] [0]
p(a__from) = [1 4] x_1 + [1]
[0 1] [1]
p(a__fst) = [1 4] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [0]
p(a__len) = [1 4] x_1 + [3]
[0 1] [1]
p(add) = [1 0] x_1 + [1 4] x_2 + [1]
[0 1] [0 1] [0]
p(cons) = [1 0] x_1 + [0]
[0 1] [0]
p(from) = [1 4] x_1 + [0]
[0 1] [1]
p(fst) = [1 4] x_1 + [1 4] x_2 + [0]
[0 1] [0 1] [0]
p(len) = [1 4] x_1 + [0]
[0 1] [1]
p(mark) = [1 4] x_1 + [0]
[0 1] [0]
p(nil) = [1]
[0]
p(s) = [3]
[0]
Following rules are strictly oriented:
mark(from(X)) = [1 8] X + [4]
[0 1] [1]
> [1 8] X + [1]
[0 1] [1]
= a__from(mark(X))
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0] X1 + [1 4] X2 + [1]
[0 1] [0 1] [0]
>= [1 0] X1 + [1 4] X2 + [1]
[0 1] [0 1] [0]
= add(X1,X2)
a__add(0(),X) = [1 4] X + [2]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 4] Y + [4]
[0 1] [0]
>= [3]
[0]
= s(add(X,Y))
a__from(X) = [1 4] X + [1]
[0 1] [1]
>= [1 4] X + [0]
[0 1] [0]
= cons(mark(X),from(s(X)))
a__from(X) = [1 4] X + [1]
[0 1] [1]
>= [1 4] X + [0]
[0 1] [1]
= from(X)
a__fst(X1,X2) = [1 4] X1 + [1 4] X2 + [0]
[0 1] [0 1] [0]
>= [1 4] X1 + [1 4] X2 + [0]
[0 1] [0 1] [0]
= fst(X1,X2)
a__fst(0(),Z) = [1 4] Z + [1]
[0 1] [0]
>= [1]
[0]
= nil()
a__fst(s(X),cons(Y,Z)) = [1 4] Y + [3]
[0 1] [0]
>= [1 4] Y + [0]
[0 1] [0]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1 4] X + [3]
[0 1] [1]
>= [1 4] X + [0]
[0 1] [1]
= len(X)
a__len(cons(X,Z)) = [1 4] X + [3]
[0 1] [1]
>= [3]
[0]
= s(len(Z))
a__len(nil()) = [4]
[1]
>= [1]
[0]
= 0()
mark(0()) = [1]
[0]
>= [1]
[0]
= 0()
mark(add(X1,X2)) = [1 4] X1 + [1 8] X2 + [1]
[0 1] [0 1] [0]
>= [1 4] X1 + [1 8] X2 + [1]
[0 1] [0 1] [0]
= a__add(mark(X1),mark(X2))
mark(cons(X1,X2)) = [1 4] X1 + [0]
[0 1] [0]
>= [1 4] X1 + [0]
[0 1] [0]
= cons(mark(X1),X2)
mark(fst(X1,X2)) = [1 8] X1 + [1 8] X2 + [0]
[0 1] [0 1] [0]
>= [1 8] X1 + [1 8] X2 + [0]
[0 1] [0 1] [0]
= a__fst(mark(X1),mark(X2))
mark(len(X)) = [1 8] X + [4]
[0 1] [1]
>= [1 8] X + [3]
[0 1] [1]
= a__len(mark(X))
mark(nil()) = [1]
[0]
>= [1]
[0]
= nil()
mark(s(X)) = [3]
[0]
>= [3]
[0]
= s(X)
** Step 1.b:12: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
mark(cons(X1,X2)) -> cons(mark(X1),X2)
- 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(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:
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__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) = [6]
[1]
p(a__add) = [1 0] x_1 + [1 5] x_2 + [0]
[0 1] [0 1] [0]
p(a__from) = [1 4] x_1 + [1]
[0 1] [2]
p(a__fst) = [1 2] x_1 + [1 6] x_2 + [0]
[0 1] [0 1] [3]
p(a__len) = [1 0] x_1 + [5]
[0 1] [1]
p(add) = [1 0] x_1 + [1 5] x_2 + [0]
[0 1] [0 1] [0]
p(cons) = [1 0] x_1 + [0]
[0 1] [2]
p(from) = [1 4] x_1 + [1]
[0 1] [2]
p(fst) = [1 2] x_1 + [1 6] x_2 + [0]
[0 1] [0 1] [3]
p(len) = [1 0] x_1 + [5]
[0 1] [1]
p(mark) = [1 4] x_1 + [0]
[0 1] [0]
p(nil) = [4]
[0]
p(s) = [0]
[1]
Following rules are strictly oriented:
mark(cons(X1,X2)) = [1 4] X1 + [8]
[0 1] [2]
> [1 4] X1 + [0]
[0 1] [2]
= cons(mark(X1),X2)
Following rules are (at-least) weakly oriented:
a__add(X1,X2) = [1 0] X1 + [1 5] X2 + [0]
[0 1] [0 1] [0]
>= [1 0] X1 + [1 5] X2 + [0]
[0 1] [0 1] [0]
= add(X1,X2)
a__add(0(),X) = [1 5] X + [6]
[0 1] [1]
>= [1 4] X + [0]
[0 1] [0]
= mark(X)
a__add(s(X),Y) = [1 5] Y + [0]
[0 1] [1]
>= [0]
[1]
= s(add(X,Y))
a__from(X) = [1 4] X + [1]
[0 1] [2]
>= [1 4] X + [0]
[0 1] [2]
= cons(mark(X),from(s(X)))
a__from(X) = [1 4] X + [1]
[0 1] [2]
>= [1 4] X + [1]
[0 1] [2]
= from(X)
a__fst(X1,X2) = [1 2] X1 + [1 6] X2 + [0]
[0 1] [0 1] [3]
>= [1 2] X1 + [1 6] X2 + [0]
[0 1] [0 1] [3]
= fst(X1,X2)
a__fst(0(),Z) = [1 6] Z + [8]
[0 1] [4]
>= [4]
[0]
= nil()
a__fst(s(X),cons(Y,Z)) = [1 6] Y + [14]
[0 1] [6]
>= [1 4] Y + [0]
[0 1] [2]
= cons(mark(Y),fst(X,Z))
a__len(X) = [1 0] X + [5]
[0 1] [1]
>= [1 0] X + [5]
[0 1] [1]
= len(X)
a__len(cons(X,Z)) = [1 0] X + [5]
[0 1] [3]
>= [0]
[1]
= s(len(Z))
a__len(nil()) = [9]
[1]
>= [6]
[1]
= 0()
mark(0()) = [10]
[1]
>= [6]
[1]
= 0()
mark(add(X1,X2)) = [1 4] X1 + [1 9] X2 + [0]
[0 1] [0 1] [0]
>= [1 4] X1 + [1 9] X2 + [0]
[0 1] [0 1] [0]
= a__add(mark(X1),mark(X2))
mark(from(X)) = [1 8] X + [9]
[0 1] [2]
>= [1 8] X + [1]
[0 1] [2]
= a__from(mark(X))
mark(fst(X1,X2)) = [1 6] X1 + [1 10] X2 + [12]
[0 1] [0 1] [3]
>= [1 6] X1 + [1 10] X2 + [0]
[0 1] [0 1] [3]
= a__fst(mark(X1),mark(X2))
mark(len(X)) = [1 4] X + [9]
[0 1] [1]
>= [1 4] X + [5]
[0 1] [1]
= a__len(mark(X))
mark(nil()) = [4]
[0]
>= [4]
[0]
= nil()
mark(s(X)) = [4]
[1]
>= [0]
[1]
= s(X)
** Step 1.b:13: 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^2))