* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
a__filter(X1,X2,X3) -> filter(X1,X2,X3)
a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
a__nats(N) -> cons(mark(N),nats(s(N)))
a__nats(X) -> nats(X)
a__sieve(X) -> sieve(X)
a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() -> a__sieve(a__nats(s(s(0()))))
a__zprimes() -> zprimes()
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) -> a__nats(mark(X))
mark(s(X)) -> s(mark(X))
mark(sieve(X)) -> a__sieve(mark(X))
mark(zprimes()) -> a__zprimes()
- Signature:
{a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes
,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a__filter(X1,X2,X3) -> filter(X1,X2,X3)
a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
a__nats(N) -> cons(mark(N),nats(s(N)))
a__nats(X) -> nats(X)
a__sieve(X) -> sieve(X)
a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() -> a__sieve(a__nats(s(s(0()))))
a__zprimes() -> zprimes()
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) -> a__nats(mark(X))
mark(s(X)) -> s(mark(X))
mark(sieve(X)) -> a__sieve(mark(X))
mark(zprimes()) -> a__zprimes()
- Signature:
{a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes
,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
mark(x){x -> cons(x,y)} =
mark(cons(x,y)) ->^+ cons(mark(x),y)
= C[mark(x) = mark(x){}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__filter(X1,X2,X3) -> filter(X1,X2,X3)
a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
a__nats(N) -> cons(mark(N),nats(s(N)))
a__nats(X) -> nats(X)
a__sieve(X) -> sieve(X)
a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() -> a__sieve(a__nats(s(s(0()))))
a__zprimes() -> zprimes()
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) -> a__nats(mark(X))
mark(s(X)) -> s(mark(X))
mark(sieve(X)) -> a__sieve(mark(X))
mark(zprimes()) -> a__zprimes()
- Signature:
{a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes
,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes}
+ 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__filter) = {1,2,3},
uargs(a__nats) = {1},
uargs(a__sieve) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__filter) = [1] x1 + [1] x2 + [1] x3 + [0]
p(a__nats) = [1] x1 + [0]
p(a__sieve) = [1] x1 + [0]
p(a__zprimes) = [0]
p(cons) = [1] x1 + [0]
p(filter) = [1] x1 + [1] x2 + [1] x3 + [0]
p(mark) = [1] x1 + [1]
p(nats) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(sieve) = [1] x1 + [0]
p(zprimes) = [0]
Following rules are strictly oriented:
mark(0()) = [1]
> [0]
= 0()
mark(zprimes()) = [1]
> [0]
= a__zprimes()
Following rules are (at-least) weakly oriented:
a__filter(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0]
>= [1] X1 + [1] X2 + [1] X3 + [0]
= filter(X1,X2,X3)
a__filter(cons(X,Y),0(),M) = [1] M + [1] X + [0]
>= [0]
= cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) = [1] M + [1] N + [1] X + [0]
>= [1] X + [1]
= cons(mark(X),filter(Y,N,M))
a__nats(N) = [1] N + [0]
>= [1] N + [1]
= cons(mark(N),nats(s(N)))
a__nats(X) = [1] X + [0]
>= [1] X + [0]
= nats(X)
a__sieve(X) = [1] X + [0]
>= [1] X + [0]
= sieve(X)
a__sieve(cons(0(),Y)) = [0]
>= [0]
= cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) = [1] N + [0]
>= [1] N + [1]
= cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() = [0]
>= [0]
= a__sieve(a__nats(s(s(0()))))
a__zprimes() = [0]
>= [0]
= zprimes()
mark(cons(X1,X2)) = [1] X1 + [1]
>= [1] X1 + [1]
= cons(mark(X1),X2)
mark(filter(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [1]
>= [1] X1 + [1] X2 + [1] X3 + [3]
= a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) = [1] X + [1]
>= [1] X + [1]
= a__nats(mark(X))
mark(s(X)) = [1] X + [1]
>= [1] X + [1]
= s(mark(X))
mark(sieve(X)) = [1] X + [1]
>= [1] X + [1]
= a__sieve(mark(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__filter(X1,X2,X3) -> filter(X1,X2,X3)
a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
a__nats(N) -> cons(mark(N),nats(s(N)))
a__nats(X) -> nats(X)
a__sieve(X) -> sieve(X)
a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() -> a__sieve(a__nats(s(s(0()))))
a__zprimes() -> zprimes()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) -> a__nats(mark(X))
mark(s(X)) -> s(mark(X))
mark(sieve(X)) -> a__sieve(mark(X))
- Weak TRS:
mark(0()) -> 0()
mark(zprimes()) -> a__zprimes()
- Signature:
{a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes
,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes}
+ 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__filter) = {1,2,3},
uargs(a__nats) = {1},
uargs(a__sieve) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__filter) = [1] x1 + [1] x2 + [1] x3 + [0]
p(a__nats) = [1] x1 + [0]
p(a__sieve) = [1] x1 + [0]
p(a__zprimes) = [0]
p(cons) = [1] x1 + [0]
p(filter) = [1] x1 + [1] x2 + [1] x3 + [0]
p(mark) = [1] x1 + [0]
p(nats) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(sieve) = [1] x1 + [7]
p(zprimes) = [0]
Following rules are strictly oriented:
mark(sieve(X)) = [1] X + [7]
> [1] X + [0]
= a__sieve(mark(X))
Following rules are (at-least) weakly oriented:
a__filter(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0]
>= [1] X1 + [1] X2 + [1] X3 + [0]
= filter(X1,X2,X3)
a__filter(cons(X,Y),0(),M) = [1] M + [1] X + [0]
>= [0]
= cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) = [1] M + [1] N + [1] X + [0]
>= [1] X + [0]
= cons(mark(X),filter(Y,N,M))
a__nats(N) = [1] N + [0]
>= [1] N + [0]
= cons(mark(N),nats(s(N)))
a__nats(X) = [1] X + [0]
>= [1] X + [0]
= nats(X)
a__sieve(X) = [1] X + [0]
>= [1] X + [7]
= sieve(X)
a__sieve(cons(0(),Y)) = [0]
>= [0]
= cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) = [1] N + [0]
>= [1] N + [0]
= cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() = [0]
>= [0]
= a__sieve(a__nats(s(s(0()))))
a__zprimes() = [0]
>= [0]
= zprimes()
mark(0()) = [0]
>= [0]
= 0()
mark(cons(X1,X2)) = [1] X1 + [0]
>= [1] X1 + [0]
= cons(mark(X1),X2)
mark(filter(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0]
>= [1] X1 + [1] X2 + [1] X3 + [0]
= a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) = [1] X + [0]
>= [1] X + [0]
= a__nats(mark(X))
mark(s(X)) = [1] X + [0]
>= [1] X + [0]
= s(mark(X))
mark(zprimes()) = [0]
>= [0]
= a__zprimes()
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__filter(X1,X2,X3) -> filter(X1,X2,X3)
a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
a__nats(N) -> cons(mark(N),nats(s(N)))
a__nats(X) -> nats(X)
a__sieve(X) -> sieve(X)
a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() -> a__sieve(a__nats(s(s(0()))))
a__zprimes() -> zprimes()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) -> a__nats(mark(X))
mark(s(X)) -> s(mark(X))
- Weak TRS:
mark(0()) -> 0()
mark(sieve(X)) -> a__sieve(mark(X))
mark(zprimes()) -> a__zprimes()
- Signature:
{a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes
,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes}
+ 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__filter) = {1,2,3},
uargs(a__nats) = {1},
uargs(a__sieve) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [5]
p(a__filter) = [1] x1 + [1] x2 + [1] x3 + [3]
p(a__nats) = [1] x1 + [0]
p(a__sieve) = [1] x1 + [0]
p(a__zprimes) = [1]
p(cons) = [1] x1 + [0]
p(filter) = [1] x1 + [1] x2 + [1] x3 + [7]
p(mark) = [1] x1 + [1]
p(nats) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(sieve) = [1] x1 + [0]
p(zprimes) = [0]
Following rules are strictly oriented:
a__filter(cons(X,Y),0(),M) = [1] M + [1] X + [8]
> [5]
= cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) = [1] M + [1] N + [1] X + [3]
> [1] X + [1]
= cons(mark(X),filter(Y,N,M))
a__zprimes() = [1]
> [0]
= zprimes()
mark(filter(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [8]
> [1] X1 + [1] X2 + [1] X3 + [6]
= a__filter(mark(X1),mark(X2),mark(X3))
Following rules are (at-least) weakly oriented:
a__filter(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [3]
>= [1] X1 + [1] X2 + [1] X3 + [7]
= filter(X1,X2,X3)
a__nats(N) = [1] N + [0]
>= [1] N + [1]
= cons(mark(N),nats(s(N)))
a__nats(X) = [1] X + [0]
>= [1] X + [0]
= nats(X)
a__sieve(X) = [1] X + [0]
>= [1] X + [0]
= sieve(X)
a__sieve(cons(0(),Y)) = [5]
>= [5]
= cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) = [1] N + [0]
>= [1] N + [1]
= cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() = [1]
>= [5]
= a__sieve(a__nats(s(s(0()))))
mark(0()) = [6]
>= [5]
= 0()
mark(cons(X1,X2)) = [1] X1 + [1]
>= [1] X1 + [1]
= cons(mark(X1),X2)
mark(nats(X)) = [1] X + [1]
>= [1] X + [1]
= a__nats(mark(X))
mark(s(X)) = [1] X + [1]
>= [1] X + [1]
= s(mark(X))
mark(sieve(X)) = [1] X + [1]
>= [1] X + [1]
= a__sieve(mark(X))
mark(zprimes()) = [1]
>= [1]
= a__zprimes()
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__filter(X1,X2,X3) -> filter(X1,X2,X3)
a__nats(N) -> cons(mark(N),nats(s(N)))
a__nats(X) -> nats(X)
a__sieve(X) -> sieve(X)
a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() -> a__sieve(a__nats(s(s(0()))))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(nats(X)) -> a__nats(mark(X))
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
a__zprimes() -> zprimes()
mark(0()) -> 0()
mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
mark(sieve(X)) -> a__sieve(mark(X))
mark(zprimes()) -> a__zprimes()
- Signature:
{a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes
,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes}
+ 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__filter) = {1,2,3},
uargs(a__nats) = {1},
uargs(a__sieve) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(a__filter) = [1] x1 + [1] x2 + [1] x3 + [0]
p(a__nats) = [1] x1 + [0]
p(a__sieve) = [1] x1 + [0]
p(a__zprimes) = [3]
p(cons) = [1] x1 + [4]
p(filter) = [1] x1 + [1] x2 + [1] x3 + [0]
p(mark) = [1] x1 + [0]
p(nats) = [1] x1 + [6]
p(s) = [1] x1 + [3]
p(sieve) = [1] x1 + [0]
p(zprimes) = [3]
Following rules are strictly oriented:
mark(nats(X)) = [1] X + [6]
> [1] X + [0]
= a__nats(mark(X))
Following rules are (at-least) weakly oriented:
a__filter(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0]
>= [1] X1 + [1] X2 + [1] X3 + [0]
= filter(X1,X2,X3)
a__filter(cons(X,Y),0(),M) = [1] M + [1] X + [5]
>= [5]
= cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) = [1] M + [1] N + [1] X + [7]
>= [1] X + [4]
= cons(mark(X),filter(Y,N,M))
a__nats(N) = [1] N + [0]
>= [1] N + [4]
= cons(mark(N),nats(s(N)))
a__nats(X) = [1] X + [0]
>= [1] X + [6]
= nats(X)
a__sieve(X) = [1] X + [0]
>= [1] X + [0]
= sieve(X)
a__sieve(cons(0(),Y)) = [5]
>= [5]
= cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) = [1] N + [7]
>= [1] N + [7]
= cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() = [3]
>= [7]
= a__sieve(a__nats(s(s(0()))))
a__zprimes() = [3]
>= [3]
= zprimes()
mark(0()) = [1]
>= [1]
= 0()
mark(cons(X1,X2)) = [1] X1 + [4]
>= [1] X1 + [4]
= cons(mark(X1),X2)
mark(filter(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0]
>= [1] X1 + [1] X2 + [1] X3 + [0]
= a__filter(mark(X1),mark(X2),mark(X3))
mark(s(X)) = [1] X + [3]
>= [1] X + [3]
= s(mark(X))
mark(sieve(X)) = [1] X + [0]
>= [1] X + [0]
= a__sieve(mark(X))
mark(zprimes()) = [3]
>= [3]
= a__zprimes()
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__filter(X1,X2,X3) -> filter(X1,X2,X3)
a__nats(N) -> cons(mark(N),nats(s(N)))
a__nats(X) -> nats(X)
a__sieve(X) -> sieve(X)
a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() -> a__sieve(a__nats(s(s(0()))))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
a__zprimes() -> zprimes()
mark(0()) -> 0()
mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) -> a__nats(mark(X))
mark(sieve(X)) -> a__sieve(mark(X))
mark(zprimes()) -> a__zprimes()
- Signature:
{a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes
,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes}
+ 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__filter) = {1,2,3},
uargs(a__nats) = {1},
uargs(a__sieve) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [7]
p(a__filter) = [1] x1 + [1] x2 + [1] x3 + [0]
p(a__nats) = [1] x1 + [4]
p(a__sieve) = [1] x1 + [0]
p(a__zprimes) = [4]
p(cons) = [1] x1 + [1]
p(filter) = [1] x1 + [1] x2 + [1] x3 + [7]
p(mark) = [1] x1 + [1]
p(nats) = [1] x1 + [4]
p(s) = [1] x1 + [2]
p(sieve) = [1] x1 + [0]
p(zprimes) = [4]
Following rules are strictly oriented:
a__nats(N) = [1] N + [4]
> [1] N + [2]
= cons(mark(N),nats(s(N)))
Following rules are (at-least) weakly oriented:
a__filter(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0]
>= [1] X1 + [1] X2 + [1] X3 + [7]
= filter(X1,X2,X3)
a__filter(cons(X,Y),0(),M) = [1] M + [1] X + [8]
>= [8]
= cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) = [1] M + [1] N + [1] X + [3]
>= [1] X + [2]
= cons(mark(X),filter(Y,N,M))
a__nats(X) = [1] X + [4]
>= [1] X + [4]
= nats(X)
a__sieve(X) = [1] X + [0]
>= [1] X + [0]
= sieve(X)
a__sieve(cons(0(),Y)) = [8]
>= [8]
= cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) = [1] N + [3]
>= [1] N + [4]
= cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() = [4]
>= [15]
= a__sieve(a__nats(s(s(0()))))
a__zprimes() = [4]
>= [4]
= zprimes()
mark(0()) = [8]
>= [7]
= 0()
mark(cons(X1,X2)) = [1] X1 + [2]
>= [1] X1 + [2]
= cons(mark(X1),X2)
mark(filter(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [8]
>= [1] X1 + [1] X2 + [1] X3 + [3]
= a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) = [1] X + [5]
>= [1] X + [5]
= a__nats(mark(X))
mark(s(X)) = [1] X + [3]
>= [1] X + [3]
= s(mark(X))
mark(sieve(X)) = [1] X + [1]
>= [1] X + [1]
= a__sieve(mark(X))
mark(zprimes()) = [5]
>= [4]
= a__zprimes()
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__filter(X1,X2,X3) -> filter(X1,X2,X3)
a__nats(X) -> nats(X)
a__sieve(X) -> sieve(X)
a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() -> a__sieve(a__nats(s(s(0()))))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
a__nats(N) -> cons(mark(N),nats(s(N)))
a__zprimes() -> zprimes()
mark(0()) -> 0()
mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) -> a__nats(mark(X))
mark(sieve(X)) -> a__sieve(mark(X))
mark(zprimes()) -> a__zprimes()
- Signature:
{a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes
,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes}
+ 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__filter) = {1,2,3},
uargs(a__nats) = {1},
uargs(a__sieve) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [6]
p(a__filter) = [1] x1 + [1] x2 + [1] x3 + [0]
p(a__nats) = [1] x1 + [5]
p(a__sieve) = [1] x1 + [1]
p(a__zprimes) = [0]
p(cons) = [1] x1 + [0]
p(filter) = [1] x1 + [1] x2 + [1] x3 + [0]
p(mark) = [1] x1 + [0]
p(nats) = [1] x1 + [5]
p(s) = [1] x1 + [1]
p(sieve) = [1] x1 + [6]
p(zprimes) = [0]
Following rules are strictly oriented:
a__sieve(cons(0(),Y)) = [7]
> [6]
= cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) = [1] N + [2]
> [1] N + [1]
= cons(s(mark(N)),sieve(filter(Y,N,N)))
Following rules are (at-least) weakly oriented:
a__filter(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0]
>= [1] X1 + [1] X2 + [1] X3 + [0]
= filter(X1,X2,X3)
a__filter(cons(X,Y),0(),M) = [1] M + [1] X + [6]
>= [6]
= cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) = [1] M + [1] N + [1] X + [1]
>= [1] X + [0]
= cons(mark(X),filter(Y,N,M))
a__nats(N) = [1] N + [5]
>= [1] N + [0]
= cons(mark(N),nats(s(N)))
a__nats(X) = [1] X + [5]
>= [1] X + [5]
= nats(X)
a__sieve(X) = [1] X + [1]
>= [1] X + [6]
= sieve(X)
a__zprimes() = [0]
>= [14]
= a__sieve(a__nats(s(s(0()))))
a__zprimes() = [0]
>= [0]
= zprimes()
mark(0()) = [6]
>= [6]
= 0()
mark(cons(X1,X2)) = [1] X1 + [0]
>= [1] X1 + [0]
= cons(mark(X1),X2)
mark(filter(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0]
>= [1] X1 + [1] X2 + [1] X3 + [0]
= a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) = [1] X + [5]
>= [1] X + [5]
= a__nats(mark(X))
mark(s(X)) = [1] X + [1]
>= [1] X + [1]
= s(mark(X))
mark(sieve(X)) = [1] X + [6]
>= [1] X + [1]
= a__sieve(mark(X))
mark(zprimes()) = [0]
>= [0]
= a__zprimes()
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__filter(X1,X2,X3) -> filter(X1,X2,X3)
a__nats(X) -> nats(X)
a__sieve(X) -> sieve(X)
a__zprimes() -> a__sieve(a__nats(s(s(0()))))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
a__nats(N) -> cons(mark(N),nats(s(N)))
a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() -> zprimes()
mark(0()) -> 0()
mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) -> a__nats(mark(X))
mark(sieve(X)) -> a__sieve(mark(X))
mark(zprimes()) -> a__zprimes()
- Signature:
{a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes
,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes}
+ 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__filter) = {1,2,3},
uargs(a__nats) = {1},
uargs(a__sieve) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a__filter) = [1] x1 + [1] x2 + [1] x3 + [1]
p(a__nats) = [1] x1 + [0]
p(a__sieve) = [1] x1 + [0]
p(a__zprimes) = [1]
p(cons) = [1] x1 + [0]
p(filter) = [1] x1 + [1] x2 + [1] x3 + [2]
p(mark) = [1] x1 + [0]
p(nats) = [1] x1 + [4]
p(s) = [1] x1 + [0]
p(sieve) = [1] x1 + [1]
p(zprimes) = [1]
Following rules are strictly oriented:
a__zprimes() = [1]
> [0]
= a__sieve(a__nats(s(s(0()))))
Following rules are (at-least) weakly oriented:
a__filter(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [1]
>= [1] X1 + [1] X2 + [1] X3 + [2]
= filter(X1,X2,X3)
a__filter(cons(X,Y),0(),M) = [1] M + [1] X + [1]
>= [0]
= cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) = [1] M + [1] N + [1] X + [1]
>= [1] X + [0]
= cons(mark(X),filter(Y,N,M))
a__nats(N) = [1] N + [0]
>= [1] N + [0]
= cons(mark(N),nats(s(N)))
a__nats(X) = [1] X + [0]
>= [1] X + [4]
= nats(X)
a__sieve(X) = [1] X + [0]
>= [1] X + [1]
= sieve(X)
a__sieve(cons(0(),Y)) = [0]
>= [0]
= cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) = [1] N + [0]
>= [1] N + [0]
= cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() = [1]
>= [1]
= zprimes()
mark(0()) = [0]
>= [0]
= 0()
mark(cons(X1,X2)) = [1] X1 + [0]
>= [1] X1 + [0]
= cons(mark(X1),X2)
mark(filter(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [2]
>= [1] X1 + [1] X2 + [1] X3 + [1]
= a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) = [1] X + [4]
>= [1] X + [0]
= a__nats(mark(X))
mark(s(X)) = [1] X + [0]
>= [1] X + [0]
= s(mark(X))
mark(sieve(X)) = [1] X + [1]
>= [1] X + [0]
= a__sieve(mark(X))
mark(zprimes()) = [1]
>= [1]
= a__zprimes()
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:
a__filter(X1,X2,X3) -> filter(X1,X2,X3)
a__nats(X) -> nats(X)
a__sieve(X) -> sieve(X)
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
a__nats(N) -> cons(mark(N),nats(s(N)))
a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() -> a__sieve(a__nats(s(s(0()))))
a__zprimes() -> zprimes()
mark(0()) -> 0()
mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) -> a__nats(mark(X))
mark(sieve(X)) -> a__sieve(mark(X))
mark(zprimes()) -> a__zprimes()
- Signature:
{a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes
,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes}
+ 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__filter) = {1,2,3},
uargs(a__nats) = {1},
uargs(a__sieve) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__filter,a__nats,a__sieve,a__zprimes,mark}
TcT has computed the following interpretation:
p(0) = [3]
[0]
p(a__filter) = [1 4] x_1 + [1 0] x_2 + [1 0] x_3 + [0]
[0 1] [0 1] [0 1] [0]
p(a__nats) = [1 6] x_1 + [2]
[0 1] [0]
p(a__sieve) = [1 4] x_1 + [2]
[0 1] [1]
p(a__zprimes) = [7]
[2]
p(cons) = [1 2] x_1 + [2]
[0 1] [0]
p(filter) = [1 4] x_1 + [1 0] x_2 + [1 0] x_3 + [0]
[0 1] [0 1] [0 1] [0]
p(mark) = [1 4] x_1 + [0]
[0 1] [0]
p(nats) = [1 6] x_1 + [2]
[0 1] [0]
p(s) = [1 4] x_1 + [0]
[0 1] [0]
p(sieve) = [1 4] x_1 + [0]
[0 1] [1]
p(zprimes) = [4]
[2]
Following rules are strictly oriented:
a__sieve(X) = [1 4] X + [2]
[0 1] [1]
> [1 4] X + [0]
[0 1] [1]
= sieve(X)
Following rules are (at-least) weakly oriented:
a__filter(X1,X2,X3) = [1 4] X1 + [1 0] X2 + [1 0] X3 + [0]
[0 1] [0 1] [0 1] [0]
>= [1 4] X1 + [1 0] X2 + [1 0] X3 + [0]
[0 1] [0 1] [0 1] [0]
= filter(X1,X2,X3)
a__filter(cons(X,Y),0(),M) = [1 0] M + [1 6] X + [5]
[0 1] [0 1] [0]
>= [5]
[0]
= cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) = [1 0] M + [1 4] N + [1 6] X + [2]
[0 1] [0 1] [0 1] [0]
>= [1 6] X + [2]
[0 1] [0]
= cons(mark(X),filter(Y,N,M))
a__nats(N) = [1 6] N + [2]
[0 1] [0]
>= [1 6] N + [2]
[0 1] [0]
= cons(mark(N),nats(s(N)))
a__nats(X) = [1 6] X + [2]
[0 1] [0]
>= [1 6] X + [2]
[0 1] [0]
= nats(X)
a__sieve(cons(0(),Y)) = [7]
[1]
>= [5]
[0]
= cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) = [1 10] N + [4]
[0 1] [1]
>= [1 10] N + [2]
[0 1] [0]
= cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() = [7]
[2]
>= [7]
[1]
= a__sieve(a__nats(s(s(0()))))
a__zprimes() = [7]
[2]
>= [4]
[2]
= zprimes()
mark(0()) = [3]
[0]
>= [3]
[0]
= 0()
mark(cons(X1,X2)) = [1 6] X1 + [2]
[0 1] [0]
>= [1 6] X1 + [2]
[0 1] [0]
= cons(mark(X1),X2)
mark(filter(X1,X2,X3)) = [1 8] X1 + [1 4] X2 + [1 4] X3 + [0]
[0 1] [0 1] [0 1] [0]
>= [1 8] X1 + [1 4] X2 + [1 4] X3 + [0]
[0 1] [0 1] [0 1] [0]
= a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) = [1 10] X + [2]
[0 1] [0]
>= [1 10] X + [2]
[0 1] [0]
= a__nats(mark(X))
mark(s(X)) = [1 8] X + [0]
[0 1] [0]
>= [1 8] X + [0]
[0 1] [0]
= s(mark(X))
mark(sieve(X)) = [1 8] X + [4]
[0 1] [1]
>= [1 8] X + [2]
[0 1] [1]
= a__sieve(mark(X))
mark(zprimes()) = [12]
[2]
>= [7]
[2]
= a__zprimes()
** Step 1.b:9: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__filter(X1,X2,X3) -> filter(X1,X2,X3)
a__nats(X) -> nats(X)
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
a__nats(N) -> cons(mark(N),nats(s(N)))
a__sieve(X) -> sieve(X)
a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() -> a__sieve(a__nats(s(s(0()))))
a__zprimes() -> zprimes()
mark(0()) -> 0()
mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) -> a__nats(mark(X))
mark(sieve(X)) -> a__sieve(mark(X))
mark(zprimes()) -> a__zprimes()
- Signature:
{a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes
,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes}
+ 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__filter) = {1,2,3},
uargs(a__nats) = {1},
uargs(a__sieve) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__filter,a__nats,a__sieve,a__zprimes,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
p(a__filter) = [1 4] x_1 + [1 0] x_2 + [1 0] x_3 + [0]
[0 1] [0 1] [0 1] [0]
p(a__nats) = [1 6] x_1 + [3]
[0 1] [1]
p(a__sieve) = [1 4] x_1 + [0]
[0 1] [0]
p(a__zprimes) = [7]
[2]
p(cons) = [1 1] x_1 + [0]
[0 1] [0]
p(filter) = [1 4] x_1 + [1 0] x_2 + [1 0] x_3 + [0]
[0 1] [0 1] [0 1] [0]
p(mark) = [1 4] x_1 + [0]
[0 1] [0]
p(nats) = [1 6] x_1 + [0]
[0 1] [1]
p(s) = [1 4] x_1 + [0]
[0 1] [0]
p(sieve) = [1 4] x_1 + [0]
[0 1] [0]
p(zprimes) = [4]
[2]
Following rules are strictly oriented:
a__nats(X) = [1 6] X + [3]
[0 1] [1]
> [1 6] X + [0]
[0 1] [1]
= nats(X)
Following rules are (at-least) weakly oriented:
a__filter(X1,X2,X3) = [1 4] X1 + [1 0] X2 + [1 0] X3 + [0]
[0 1] [0 1] [0 1] [0]
>= [1 4] X1 + [1 0] X2 + [1 0] X3 + [0]
[0 1] [0 1] [0 1] [0]
= filter(X1,X2,X3)
a__filter(cons(X,Y),0(),M) = [1 0] M + [1 5] X + [0]
[0 1] [0 1] [0]
>= [0]
[0]
= cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) = [1 0] M + [1 4] N + [1 5] X + [0]
[0 1] [0 1] [0 1] [0]
>= [1 5] X + [0]
[0 1] [0]
= cons(mark(X),filter(Y,N,M))
a__nats(N) = [1 6] N + [3]
[0 1] [1]
>= [1 5] N + [0]
[0 1] [0]
= cons(mark(N),nats(s(N)))
a__sieve(X) = [1 4] X + [0]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= sieve(X)
a__sieve(cons(0(),Y)) = [0]
[0]
>= [0]
[0]
= cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) = [1 9] N + [0]
[0 1] [0]
>= [1 9] N + [0]
[0 1] [0]
= cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() = [7]
[2]
>= [7]
[1]
= a__sieve(a__nats(s(s(0()))))
a__zprimes() = [7]
[2]
>= [4]
[2]
= zprimes()
mark(0()) = [0]
[0]
>= [0]
[0]
= 0()
mark(cons(X1,X2)) = [1 5] X1 + [0]
[0 1] [0]
>= [1 5] X1 + [0]
[0 1] [0]
= cons(mark(X1),X2)
mark(filter(X1,X2,X3)) = [1 8] X1 + [1 4] X2 + [1 4] X3 + [0]
[0 1] [0 1] [0 1] [0]
>= [1 8] X1 + [1 4] X2 + [1 4] X3 + [0]
[0 1] [0 1] [0 1] [0]
= a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) = [1 10] X + [4]
[0 1] [1]
>= [1 10] X + [3]
[0 1] [1]
= a__nats(mark(X))
mark(s(X)) = [1 8] X + [0]
[0 1] [0]
>= [1 8] X + [0]
[0 1] [0]
= s(mark(X))
mark(sieve(X)) = [1 8] X + [0]
[0 1] [0]
>= [1 8] X + [0]
[0 1] [0]
= a__sieve(mark(X))
mark(zprimes()) = [12]
[2]
>= [7]
[2]
= a__zprimes()
** Step 1.b:10: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__filter(X1,X2,X3) -> filter(X1,X2,X3)
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
a__nats(N) -> cons(mark(N),nats(s(N)))
a__nats(X) -> nats(X)
a__sieve(X) -> sieve(X)
a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() -> a__sieve(a__nats(s(s(0()))))
a__zprimes() -> zprimes()
mark(0()) -> 0()
mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) -> a__nats(mark(X))
mark(sieve(X)) -> a__sieve(mark(X))
mark(zprimes()) -> a__zprimes()
- Signature:
{a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes
,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes}
+ 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__filter) = {1,2,3},
uargs(a__nats) = {1},
uargs(a__sieve) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__filter,a__nats,a__sieve,a__zprimes,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
p(a__filter) = [1 1] x_1 + [1 2] x_2 + [1 0] x_3 + [0]
[0 1] [0 1] [0 1] [0]
p(a__nats) = [1 2] x_1 + [3]
[0 1] [4]
p(a__sieve) = [1 1] x_1 + [0]
[0 1] [3]
p(a__zprimes) = [7]
[7]
p(cons) = [1 1] x_1 + [3]
[0 1] [4]
p(filter) = [1 1] x_1 + [1 2] x_2 + [1 0] x_3 + [0]
[0 1] [0 1] [0 1] [0]
p(mark) = [1 1] x_1 + [0]
[0 1] [0]
p(nats) = [1 2] x_1 + [3]
[0 1] [4]
p(s) = [1 0] x_1 + [0]
[0 1] [0]
p(sieve) = [1 1] x_1 + [0]
[0 1] [3]
p(zprimes) = [0]
[7]
Following rules are strictly oriented:
mark(cons(X1,X2)) = [1 2] X1 + [7]
[0 1] [4]
> [1 2] X1 + [3]
[0 1] [4]
= cons(mark(X1),X2)
Following rules are (at-least) weakly oriented:
a__filter(X1,X2,X3) = [1 1] X1 + [1 2] X2 + [1 0] X3 + [0]
[0 1] [0 1] [0 1] [0]
>= [1 1] X1 + [1 2] X2 + [1 0] X3 + [0]
[0 1] [0 1] [0 1] [0]
= filter(X1,X2,X3)
a__filter(cons(X,Y),0(),M) = [1 0] M + [1 2] X + [7]
[0 1] [0 1] [4]
>= [3]
[4]
= cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) = [1 0] M + [1 2] N + [1 2] X + [7]
[0 1] [0 1] [0 1] [4]
>= [1 2] X + [3]
[0 1] [4]
= cons(mark(X),filter(Y,N,M))
a__nats(N) = [1 2] N + [3]
[0 1] [4]
>= [1 2] N + [3]
[0 1] [4]
= cons(mark(N),nats(s(N)))
a__nats(X) = [1 2] X + [3]
[0 1] [4]
>= [1 2] X + [3]
[0 1] [4]
= nats(X)
a__sieve(X) = [1 1] X + [0]
[0 1] [3]
>= [1 1] X + [0]
[0 1] [3]
= sieve(X)
a__sieve(cons(0(),Y)) = [7]
[7]
>= [3]
[4]
= cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) = [1 2] N + [7]
[0 1] [7]
>= [1 2] N + [3]
[0 1] [4]
= cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() = [7]
[7]
>= [7]
[7]
= a__sieve(a__nats(s(s(0()))))
a__zprimes() = [7]
[7]
>= [0]
[7]
= zprimes()
mark(0()) = [0]
[0]
>= [0]
[0]
= 0()
mark(filter(X1,X2,X3)) = [1 2] X1 + [1 3] X2 + [1 1] X3 + [0]
[0 1] [0 1] [0 1] [0]
>= [1 2] X1 + [1 3] X2 + [1 1] X3 + [0]
[0 1] [0 1] [0 1] [0]
= a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) = [1 3] X + [7]
[0 1] [4]
>= [1 3] X + [3]
[0 1] [4]
= a__nats(mark(X))
mark(s(X)) = [1 1] X + [0]
[0 1] [0]
>= [1 1] X + [0]
[0 1] [0]
= s(mark(X))
mark(sieve(X)) = [1 2] X + [3]
[0 1] [3]
>= [1 2] X + [0]
[0 1] [3]
= a__sieve(mark(X))
mark(zprimes()) = [7]
[7]
>= [7]
[7]
= a__zprimes()
** Step 1.b:11: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__filter(X1,X2,X3) -> filter(X1,X2,X3)
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
a__nats(N) -> cons(mark(N),nats(s(N)))
a__nats(X) -> nats(X)
a__sieve(X) -> sieve(X)
a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() -> a__sieve(a__nats(s(s(0()))))
a__zprimes() -> zprimes()
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) -> a__nats(mark(X))
mark(sieve(X)) -> a__sieve(mark(X))
mark(zprimes()) -> a__zprimes()
- Signature:
{a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes
,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes}
+ 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__filter) = {1,2,3},
uargs(a__nats) = {1},
uargs(a__sieve) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__filter,a__nats,a__sieve,a__zprimes,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
p(a__filter) = [1 1] x_1 + [1 0] x_2 + [1 0] x_3 + [1]
[0 1] [0 1] [0 1] [1]
p(a__nats) = [1 4] x_1 + [0]
[0 1] [2]
p(a__sieve) = [1 2] x_1 + [0]
[0 1] [3]
p(a__zprimes) = [4]
[5]
p(cons) = [1 0] x_1 + [0]
[0 1] [0]
p(filter) = [1 1] x_1 + [1 0] x_2 + [1 0] x_3 + [0]
[0 1] [0 1] [0 1] [1]
p(mark) = [1 1] x_1 + [0]
[0 1] [0]
p(nats) = [1 4] x_1 + [0]
[0 1] [2]
p(s) = [1 3] x_1 + [0]
[0 1] [0]
p(sieve) = [1 2] x_1 + [0]
[0 1] [3]
p(zprimes) = [3]
[5]
Following rules are strictly oriented:
a__filter(X1,X2,X3) = [1 1] X1 + [1 0] X2 + [1 0] X3 + [1]
[0 1] [0 1] [0 1] [1]
> [1 1] X1 + [1 0] X2 + [1 0] X3 + [0]
[0 1] [0 1] [0 1] [1]
= filter(X1,X2,X3)
Following rules are (at-least) weakly oriented:
a__filter(cons(X,Y),0(),M) = [1 0] M + [1 1] X + [1]
[0 1] [0 1] [1]
>= [0]
[0]
= cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) = [1 0] M + [1 3] N + [1 1] X + [1]
[0 1] [0 1] [0 1] [1]
>= [1 1] X + [0]
[0 1] [0]
= cons(mark(X),filter(Y,N,M))
a__nats(N) = [1 4] N + [0]
[0 1] [2]
>= [1 1] N + [0]
[0 1] [0]
= cons(mark(N),nats(s(N)))
a__nats(X) = [1 4] X + [0]
[0 1] [2]
>= [1 4] X + [0]
[0 1] [2]
= nats(X)
a__sieve(X) = [1 2] X + [0]
[0 1] [3]
>= [1 2] X + [0]
[0 1] [3]
= sieve(X)
a__sieve(cons(0(),Y)) = [0]
[3]
>= [0]
[0]
= cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) = [1 5] N + [0]
[0 1] [3]
>= [1 4] N + [0]
[0 1] [0]
= cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() = [4]
[5]
>= [4]
[5]
= a__sieve(a__nats(s(s(0()))))
a__zprimes() = [4]
[5]
>= [3]
[5]
= zprimes()
mark(0()) = [0]
[0]
>= [0]
[0]
= 0()
mark(cons(X1,X2)) = [1 1] X1 + [0]
[0 1] [0]
>= [1 1] X1 + [0]
[0 1] [0]
= cons(mark(X1),X2)
mark(filter(X1,X2,X3)) = [1 2] X1 + [1 1] X2 + [1 1] X3 + [1]
[0 1] [0 1] [0 1] [1]
>= [1 2] X1 + [1 1] X2 + [1 1] X3 + [1]
[0 1] [0 1] [0 1] [1]
= a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) = [1 5] X + [2]
[0 1] [2]
>= [1 5] X + [0]
[0 1] [2]
= a__nats(mark(X))
mark(s(X)) = [1 4] X + [0]
[0 1] [0]
>= [1 4] X + [0]
[0 1] [0]
= s(mark(X))
mark(sieve(X)) = [1 3] X + [3]
[0 1] [3]
>= [1 3] X + [0]
[0 1] [3]
= a__sieve(mark(X))
mark(zprimes()) = [8]
[5]
>= [4]
[5]
= a__zprimes()
** Step 1.b:12: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
mark(s(X)) -> s(mark(X))
- Weak TRS:
a__filter(X1,X2,X3) -> filter(X1,X2,X3)
a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
a__nats(N) -> cons(mark(N),nats(s(N)))
a__nats(X) -> nats(X)
a__sieve(X) -> sieve(X)
a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() -> a__sieve(a__nats(s(s(0()))))
a__zprimes() -> zprimes()
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) -> a__nats(mark(X))
mark(sieve(X)) -> a__sieve(mark(X))
mark(zprimes()) -> a__zprimes()
- Signature:
{a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes
,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes}
+ 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__filter) = {1,2,3},
uargs(a__nats) = {1},
uargs(a__sieve) = {1},
uargs(cons) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{a__filter,a__nats,a__sieve,a__zprimes,mark}
TcT has computed the following interpretation:
p(0) = [0]
[0]
p(a__filter) = [1 1] x_1 + [1 2] x_2 + [1 0] x_3 + [7]
[0 1] [0 1] [0 1] [4]
p(a__nats) = [1 1] x_1 + [0]
[0 1] [0]
p(a__sieve) = [1 2] x_1 + [0]
[0 1] [4]
p(a__zprimes) = [7]
[7]
p(cons) = [1 0] x_1 + [0]
[0 1] [0]
p(filter) = [1 1] x_1 + [1 2] x_2 + [1 0] x_3 + [4]
[0 1] [0 1] [0 1] [4]
p(mark) = [1 1] x_1 + [0]
[0 1] [0]
p(nats) = [1 1] x_1 + [0]
[0 1] [0]
p(s) = [1 0] x_1 + [0]
[0 1] [1]
p(sieve) = [1 2] x_1 + [0]
[0 1] [4]
p(zprimes) = [2]
[7]
Following rules are strictly oriented:
mark(s(X)) = [1 1] X + [1]
[0 1] [1]
> [1 1] X + [0]
[0 1] [1]
= s(mark(X))
Following rules are (at-least) weakly oriented:
a__filter(X1,X2,X3) = [1 1] X1 + [1 2] X2 + [1 0] X3 + [7]
[0 1] [0 1] [0 1] [4]
>= [1 1] X1 + [1 2] X2 + [1 0] X3 + [4]
[0 1] [0 1] [0 1] [4]
= filter(X1,X2,X3)
a__filter(cons(X,Y),0(),M) = [1 0] M + [1 1] X + [7]
[0 1] [0 1] [4]
>= [0]
[0]
= cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) = [1 0] M + [1 2] N + [1 1] X + [9]
[0 1] [0 1] [0 1] [5]
>= [1 1] X + [0]
[0 1] [0]
= cons(mark(X),filter(Y,N,M))
a__nats(N) = [1 1] N + [0]
[0 1] [0]
>= [1 1] N + [0]
[0 1] [0]
= cons(mark(N),nats(s(N)))
a__nats(X) = [1 1] X + [0]
[0 1] [0]
>= [1 1] X + [0]
[0 1] [0]
= nats(X)
a__sieve(X) = [1 2] X + [0]
[0 1] [4]
>= [1 2] X + [0]
[0 1] [4]
= sieve(X)
a__sieve(cons(0(),Y)) = [0]
[4]
>= [0]
[0]
= cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) = [1 2] N + [2]
[0 1] [5]
>= [1 1] N + [0]
[0 1] [1]
= cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() = [7]
[7]
>= [6]
[6]
= a__sieve(a__nats(s(s(0()))))
a__zprimes() = [7]
[7]
>= [2]
[7]
= zprimes()
mark(0()) = [0]
[0]
>= [0]
[0]
= 0()
mark(cons(X1,X2)) = [1 1] X1 + [0]
[0 1] [0]
>= [1 1] X1 + [0]
[0 1] [0]
= cons(mark(X1),X2)
mark(filter(X1,X2,X3)) = [1 2] X1 + [1 3] X2 + [1 1] X3 + [8]
[0 1] [0 1] [0 1] [4]
>= [1 2] X1 + [1 3] X2 + [1 1] X3 + [7]
[0 1] [0 1] [0 1] [4]
= a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) = [1 2] X + [0]
[0 1] [0]
>= [1 2] X + [0]
[0 1] [0]
= a__nats(mark(X))
mark(sieve(X)) = [1 3] X + [4]
[0 1] [4]
>= [1 3] X + [0]
[0 1] [4]
= a__sieve(mark(X))
mark(zprimes()) = [9]
[7]
>= [7]
[7]
= a__zprimes()
** Step 1.b:13: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a__filter(X1,X2,X3) -> filter(X1,X2,X3)
a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
a__nats(N) -> cons(mark(N),nats(s(N)))
a__nats(X) -> nats(X)
a__sieve(X) -> sieve(X)
a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
a__zprimes() -> a__sieve(a__nats(s(s(0()))))
a__zprimes() -> zprimes()
mark(0()) -> 0()
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
mark(nats(X)) -> a__nats(mark(X))
mark(s(X)) -> s(mark(X))
mark(sieve(X)) -> a__sieve(mark(X))
mark(zprimes()) -> a__zprimes()
- Signature:
{a__filter/3,a__nats/1,a__sieve/1,a__zprimes/0,mark/1} / {0/0,cons/2,filter/3,nats/1,s/1,sieve/1,zprimes/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__filter,a__nats,a__sieve,a__zprimes
,mark} and constructors {0,cons,filter,nats,s,sieve,zprimes}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))