* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(activate(X))
activate(n__s(X)) -> s(activate(X))
afterNth(N,XS) -> snd(splitAt(N,XS))
fst(pair(XS,YS)) -> XS
head(cons(N,XS)) -> N
natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
natsFrom(X) -> n__natsFrom(X)
s(X) -> n__s(X)
sel(N,XS) -> head(afterNth(N,XS))
snd(pair(XS,YS)) -> YS
splitAt(0(),XS) -> pair(nil(),XS)
splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
tail(cons(N,XS)) -> activate(XS)
take(N,XS) -> fst(splitAt(N,XS))
u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
- Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4} / {0/0,cons/2
,n__natsFrom/1,n__s/1,nil/0,pair/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,afterNth,fst,head,natsFrom,s,sel,snd,splitAt
,tail,take,u} and constructors {0,cons,n__natsFrom,n__s,nil,pair}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(activate(X))
activate(n__s(X)) -> s(activate(X))
afterNth(N,XS) -> snd(splitAt(N,XS))
fst(pair(XS,YS)) -> XS
head(cons(N,XS)) -> N
natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
natsFrom(X) -> n__natsFrom(X)
s(X) -> n__s(X)
sel(N,XS) -> head(afterNth(N,XS))
snd(pair(XS,YS)) -> YS
splitAt(0(),XS) -> pair(nil(),XS)
splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
tail(cons(N,XS)) -> activate(XS)
take(N,XS) -> fst(splitAt(N,XS))
u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
- Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4} / {0/0,cons/2
,n__natsFrom/1,n__s/1,nil/0,pair/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,afterNth,fst,head,natsFrom,s,sel,snd,splitAt
,tail,take,u} and constructors {0,cons,n__natsFrom,n__s,nil,pair}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
activate(x){x -> n__natsFrom(x)} =
activate(n__natsFrom(x)) ->^+ natsFrom(activate(x))
= C[activate(x) = activate(x){}]
** Step 1.b:1: InnermostRuleRemoval WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(activate(X))
activate(n__s(X)) -> s(activate(X))
afterNth(N,XS) -> snd(splitAt(N,XS))
fst(pair(XS,YS)) -> XS
head(cons(N,XS)) -> N
natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
natsFrom(X) -> n__natsFrom(X)
s(X) -> n__s(X)
sel(N,XS) -> head(afterNth(N,XS))
snd(pair(XS,YS)) -> YS
splitAt(0(),XS) -> pair(nil(),XS)
splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
tail(cons(N,XS)) -> activate(XS)
take(N,XS) -> fst(splitAt(N,XS))
u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
- Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4} / {0/0,cons/2
,n__natsFrom/1,n__s/1,nil/0,pair/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,afterNth,fst,head,natsFrom,s,sel,snd,splitAt
,tail,take,u} and constructors {0,cons,n__natsFrom,n__s,nil,pair}
+ Applied Processor:
InnermostRuleRemoval
+ Details:
Arguments of following rules are not normal-forms.
splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
All above mentioned rules can be savely removed.
** Step 1.b:2: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(activate(X))
activate(n__s(X)) -> s(activate(X))
afterNth(N,XS) -> snd(splitAt(N,XS))
fst(pair(XS,YS)) -> XS
head(cons(N,XS)) -> N
natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
natsFrom(X) -> n__natsFrom(X)
s(X) -> n__s(X)
sel(N,XS) -> head(afterNth(N,XS))
snd(pair(XS,YS)) -> YS
splitAt(0(),XS) -> pair(nil(),XS)
tail(cons(N,XS)) -> activate(XS)
take(N,XS) -> fst(splitAt(N,XS))
u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
- Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4} / {0/0,cons/2
,n__natsFrom/1,n__s/1,nil/0,pair/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,afterNth,fst,head,natsFrom,s,sel,snd,splitAt
,tail,take,u} and constructors {0,cons,n__natsFrom,n__s,nil,pair}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:
Strict DPs
activate#(X) -> c_1()
activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)))
activate#(n__s(X)) -> c_3(s#(activate(X)))
afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)))
fst#(pair(XS,YS)) -> c_5()
head#(cons(N,XS)) -> c_6()
natsFrom#(N) -> c_7()
natsFrom#(X) -> c_8()
s#(X) -> c_9()
sel#(N,XS) -> c_10(head#(afterNth(N,XS)))
snd#(pair(XS,YS)) -> c_11()
splitAt#(0(),XS) -> c_12()
tail#(cons(N,XS)) -> c_13(activate#(XS))
take#(N,XS) -> c_14(fst#(splitAt(N,XS)))
u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X))
Weak DPs
and mark the set of starting terms.
** Step 1.b:3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1()
activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)))
activate#(n__s(X)) -> c_3(s#(activate(X)))
afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)))
fst#(pair(XS,YS)) -> c_5()
head#(cons(N,XS)) -> c_6()
natsFrom#(N) -> c_7()
natsFrom#(X) -> c_8()
s#(X) -> c_9()
sel#(N,XS) -> c_10(head#(afterNth(N,XS)))
snd#(pair(XS,YS)) -> c_11()
splitAt#(0(),XS) -> c_12()
tail#(cons(N,XS)) -> c_13(activate#(XS))
take#(N,XS) -> c_14(fst#(splitAt(N,XS)))
u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X))
- Strict TRS:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(activate(X))
activate(n__s(X)) -> s(activate(X))
afterNth(N,XS) -> snd(splitAt(N,XS))
fst(pair(XS,YS)) -> XS
head(cons(N,XS)) -> N
natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
natsFrom(X) -> n__natsFrom(X)
s(X) -> n__s(X)
sel(N,XS) -> head(afterNth(N,XS))
snd(pair(XS,YS)) -> YS
splitAt(0(),XS) -> pair(nil(),XS)
tail(cons(N,XS)) -> activate(XS)
take(N,XS) -> fst(splitAt(N,XS))
u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
- Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0
,c_12/0,c_13/1,c_14/1,c_15/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd#
,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(activate(X))
activate(n__s(X)) -> s(activate(X))
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
natsFrom(X) -> n__natsFrom(X)
s(X) -> n__s(X)
snd(pair(XS,YS)) -> YS
splitAt(0(),XS) -> pair(nil(),XS)
activate#(X) -> c_1()
activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)))
activate#(n__s(X)) -> c_3(s#(activate(X)))
afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)))
fst#(pair(XS,YS)) -> c_5()
head#(cons(N,XS)) -> c_6()
natsFrom#(N) -> c_7()
natsFrom#(X) -> c_8()
s#(X) -> c_9()
sel#(N,XS) -> c_10(head#(afterNth(N,XS)))
snd#(pair(XS,YS)) -> c_11()
splitAt#(0(),XS) -> c_12()
tail#(cons(N,XS)) -> c_13(activate#(XS))
take#(N,XS) -> c_14(fst#(splitAt(N,XS)))
u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X))
** Step 1.b:4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1()
activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)))
activate#(n__s(X)) -> c_3(s#(activate(X)))
afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)))
fst#(pair(XS,YS)) -> c_5()
head#(cons(N,XS)) -> c_6()
natsFrom#(N) -> c_7()
natsFrom#(X) -> c_8()
s#(X) -> c_9()
sel#(N,XS) -> c_10(head#(afterNth(N,XS)))
snd#(pair(XS,YS)) -> c_11()
splitAt#(0(),XS) -> c_12()
tail#(cons(N,XS)) -> c_13(activate#(XS))
take#(N,XS) -> c_14(fst#(splitAt(N,XS)))
u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X))
- Strict TRS:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(activate(X))
activate(n__s(X)) -> s(activate(X))
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
natsFrom(X) -> n__natsFrom(X)
s(X) -> n__s(X)
snd(pair(XS,YS)) -> YS
splitAt(0(),XS) -> pair(nil(),XS)
- Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0
,c_12/0,c_13/1,c_14/1,c_15/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd#
,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(natsFrom) = {1},
uargs(s) = {1},
uargs(snd) = {1},
uargs(fst#) = {1},
uargs(head#) = {1},
uargs(natsFrom#) = {1},
uargs(s#) = {1},
uargs(snd#) = {1},
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_10) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_15) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [5]
p(activate) = [5] x1 + [1]
p(afterNth) = [3] x1 + [3] x2 + [6]
p(cons) = [1] x2 + [0]
p(fst) = [0]
p(head) = [0]
p(n__natsFrom) = [1] x1 + [1]
p(n__s) = [1] x1 + [1]
p(natsFrom) = [1] x1 + [4]
p(nil) = [4]
p(pair) = [1] x1 + [1] x2 + [1]
p(s) = [1] x1 + [4]
p(sel) = [1] x2 + [1]
p(snd) = [1] x1 + [2]
p(splitAt) = [2] x1 + [2] x2 + [3]
p(tail) = [0]
p(take) = [0]
p(u) = [0]
p(activate#) = [5] x1 + [0]
p(afterNth#) = [2] x1 + [4] x2 + [0]
p(fst#) = [1] x1 + [0]
p(head#) = [1] x1 + [0]
p(natsFrom#) = [1] x1 + [4]
p(s#) = [1] x1 + [1]
p(sel#) = [3] x1 + [3] x2 + [0]
p(snd#) = [1] x1 + [0]
p(splitAt#) = [1] x2 + [1]
p(tail#) = [6] x1 + [0]
p(take#) = [5] x1 + [2] x2 + [1]
p(u#) = [1] x2 + [7] x3 + [1] x4 + [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [4]
p(c_3) = [1] x1 + [4]
p(c_4) = [1] x1 + [5]
p(c_5) = [4]
p(c_6) = [0]
p(c_7) = [1]
p(c_8) = [1]
p(c_9) = [0]
p(c_10) = [1] x1 + [0]
p(c_11) = [1]
p(c_12) = [4]
p(c_13) = [1] x1 + [1]
p(c_14) = [1] x1 + [0]
p(c_15) = [1] x1 + [0]
Following rules are strictly oriented:
natsFrom#(N) = [1] N + [4]
> [1]
= c_7()
natsFrom#(X) = [1] X + [4]
> [1]
= c_8()
s#(X) = [1] X + [1]
> [0]
= c_9()
activate(X) = [5] X + [1]
> [1] X + [0]
= X
activate(n__natsFrom(X)) = [5] X + [6]
> [5] X + [5]
= natsFrom(activate(X))
activate(n__s(X)) = [5] X + [6]
> [5] X + [5]
= s(activate(X))
afterNth(N,XS) = [3] N + [3] XS + [6]
> [2] N + [2] XS + [5]
= snd(splitAt(N,XS))
natsFrom(N) = [1] N + [4]
> [1] N + [2]
= cons(N,n__natsFrom(n__s(N)))
natsFrom(X) = [1] X + [4]
> [1] X + [1]
= n__natsFrom(X)
s(X) = [1] X + [4]
> [1] X + [1]
= n__s(X)
snd(pair(XS,YS)) = [1] XS + [1] YS + [3]
> [1] YS + [0]
= YS
splitAt(0(),XS) = [2] XS + [13]
> [1] XS + [5]
= pair(nil(),XS)
Following rules are (at-least) weakly oriented:
activate#(X) = [5] X + [0]
>= [0]
= c_1()
activate#(n__natsFrom(X)) = [5] X + [5]
>= [5] X + [9]
= c_2(natsFrom#(activate(X)))
activate#(n__s(X)) = [5] X + [5]
>= [5] X + [6]
= c_3(s#(activate(X)))
afterNth#(N,XS) = [2] N + [4] XS + [0]
>= [2] N + [2] XS + [8]
= c_4(snd#(splitAt(N,XS)))
fst#(pair(XS,YS)) = [1] XS + [1] YS + [1]
>= [4]
= c_5()
head#(cons(N,XS)) = [1] XS + [0]
>= [0]
= c_6()
sel#(N,XS) = [3] N + [3] XS + [0]
>= [3] N + [3] XS + [6]
= c_10(head#(afterNth(N,XS)))
snd#(pair(XS,YS)) = [1] XS + [1] YS + [1]
>= [1]
= c_11()
splitAt#(0(),XS) = [1] XS + [1]
>= [4]
= c_12()
tail#(cons(N,XS)) = [6] XS + [0]
>= [5] XS + [1]
= c_13(activate#(XS))
take#(N,XS) = [5] N + [2] XS + [1]
>= [2] N + [2] XS + [3]
= c_14(fst#(splitAt(N,XS)))
u#(pair(YS,ZS),N,X,XS) = [1] N + [7] X + [1] XS + [0]
>= [5] X + [0]
= c_15(activate#(X))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:5: PredecessorEstimation WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1()
activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)))
activate#(n__s(X)) -> c_3(s#(activate(X)))
afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)))
fst#(pair(XS,YS)) -> c_5()
head#(cons(N,XS)) -> c_6()
sel#(N,XS) -> c_10(head#(afterNth(N,XS)))
snd#(pair(XS,YS)) -> c_11()
splitAt#(0(),XS) -> c_12()
tail#(cons(N,XS)) -> c_13(activate#(XS))
take#(N,XS) -> c_14(fst#(splitAt(N,XS)))
u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X))
- Weak DPs:
natsFrom#(N) -> c_7()
natsFrom#(X) -> c_8()
s#(X) -> c_9()
- Weak TRS:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(activate(X))
activate(n__s(X)) -> s(activate(X))
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
natsFrom(X) -> n__natsFrom(X)
s(X) -> n__s(X)
snd(pair(XS,YS)) -> YS
splitAt(0(),XS) -> pair(nil(),XS)
- Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0
,c_12/0,c_13/1,c_14/1,c_15/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd#
,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,3,5,6,8,9}
by application of
Pre({1,2,3,5,6,8,9}) = {4,7,10,11,12}.
Here rules are labelled as follows:
1: activate#(X) -> c_1()
2: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)))
3: activate#(n__s(X)) -> c_3(s#(activate(X)))
4: afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)))
5: fst#(pair(XS,YS)) -> c_5()
6: head#(cons(N,XS)) -> c_6()
7: sel#(N,XS) -> c_10(head#(afterNth(N,XS)))
8: snd#(pair(XS,YS)) -> c_11()
9: splitAt#(0(),XS) -> c_12()
10: tail#(cons(N,XS)) -> c_13(activate#(XS))
11: take#(N,XS) -> c_14(fst#(splitAt(N,XS)))
12: u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X))
13: natsFrom#(N) -> c_7()
14: natsFrom#(X) -> c_8()
15: s#(X) -> c_9()
** Step 1.b:6: PredecessorEstimation WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)))
sel#(N,XS) -> c_10(head#(afterNth(N,XS)))
tail#(cons(N,XS)) -> c_13(activate#(XS))
take#(N,XS) -> c_14(fst#(splitAt(N,XS)))
u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X))
- Weak DPs:
activate#(X) -> c_1()
activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)))
activate#(n__s(X)) -> c_3(s#(activate(X)))
fst#(pair(XS,YS)) -> c_5()
head#(cons(N,XS)) -> c_6()
natsFrom#(N) -> c_7()
natsFrom#(X) -> c_8()
s#(X) -> c_9()
snd#(pair(XS,YS)) -> c_11()
splitAt#(0(),XS) -> c_12()
- Weak TRS:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(activate(X))
activate(n__s(X)) -> s(activate(X))
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
natsFrom(X) -> n__natsFrom(X)
s(X) -> n__s(X)
snd(pair(XS,YS)) -> YS
splitAt(0(),XS) -> pair(nil(),XS)
- Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0
,c_12/0,c_13/1,c_14/1,c_15/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd#
,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,3,4,5}
by application of
Pre({1,2,3,4,5}) = {}.
Here rules are labelled as follows:
1: afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)))
2: sel#(N,XS) -> c_10(head#(afterNth(N,XS)))
3: tail#(cons(N,XS)) -> c_13(activate#(XS))
4: take#(N,XS) -> c_14(fst#(splitAt(N,XS)))
5: u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X))
6: activate#(X) -> c_1()
7: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)))
8: activate#(n__s(X)) -> c_3(s#(activate(X)))
9: fst#(pair(XS,YS)) -> c_5()
10: head#(cons(N,XS)) -> c_6()
11: natsFrom#(N) -> c_7()
12: natsFrom#(X) -> c_8()
13: s#(X) -> c_9()
14: snd#(pair(XS,YS)) -> c_11()
15: splitAt#(0(),XS) -> c_12()
** Step 1.b:7: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
activate#(X) -> c_1()
activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)))
activate#(n__s(X)) -> c_3(s#(activate(X)))
afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)))
fst#(pair(XS,YS)) -> c_5()
head#(cons(N,XS)) -> c_6()
natsFrom#(N) -> c_7()
natsFrom#(X) -> c_8()
s#(X) -> c_9()
sel#(N,XS) -> c_10(head#(afterNth(N,XS)))
snd#(pair(XS,YS)) -> c_11()
splitAt#(0(),XS) -> c_12()
tail#(cons(N,XS)) -> c_13(activate#(XS))
take#(N,XS) -> c_14(fst#(splitAt(N,XS)))
u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X))
- Weak TRS:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(activate(X))
activate(n__s(X)) -> s(activate(X))
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
natsFrom(X) -> n__natsFrom(X)
s(X) -> n__s(X)
snd(pair(XS,YS)) -> YS
splitAt(0(),XS) -> pair(nil(),XS)
- Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0
,c_12/0,c_13/1,c_14/1,c_15/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd#
,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:activate#(X) -> c_1()
2:W:activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)))
-->_1 natsFrom#(X) -> c_8():8
-->_1 natsFrom#(N) -> c_7():7
3:W:activate#(n__s(X)) -> c_3(s#(activate(X)))
-->_1 s#(X) -> c_9():9
4:W:afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)))
-->_1 snd#(pair(XS,YS)) -> c_11():11
5:W:fst#(pair(XS,YS)) -> c_5()
6:W:head#(cons(N,XS)) -> c_6()
7:W:natsFrom#(N) -> c_7()
8:W:natsFrom#(X) -> c_8()
9:W:s#(X) -> c_9()
10:W:sel#(N,XS) -> c_10(head#(afterNth(N,XS)))
-->_1 head#(cons(N,XS)) -> c_6():6
11:W:snd#(pair(XS,YS)) -> c_11()
12:W:splitAt#(0(),XS) -> c_12()
13:W:tail#(cons(N,XS)) -> c_13(activate#(XS))
-->_1 activate#(n__s(X)) -> c_3(s#(activate(X))):3
-->_1 activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))):2
-->_1 activate#(X) -> c_1():1
14:W:take#(N,XS) -> c_14(fst#(splitAt(N,XS)))
-->_1 fst#(pair(XS,YS)) -> c_5():5
15:W:u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X))
-->_1 activate#(n__s(X)) -> c_3(s#(activate(X))):3
-->_1 activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X))):2
-->_1 activate#(X) -> c_1():1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
15: u#(pair(YS,ZS),N,X,XS) -> c_15(activate#(X))
14: take#(N,XS) -> c_14(fst#(splitAt(N,XS)))
13: tail#(cons(N,XS)) -> c_13(activate#(XS))
12: splitAt#(0(),XS) -> c_12()
10: sel#(N,XS) -> c_10(head#(afterNth(N,XS)))
6: head#(cons(N,XS)) -> c_6()
5: fst#(pair(XS,YS)) -> c_5()
4: afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)))
11: snd#(pair(XS,YS)) -> c_11()
3: activate#(n__s(X)) -> c_3(s#(activate(X)))
9: s#(X) -> c_9()
2: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)))
7: natsFrom#(N) -> c_7()
8: natsFrom#(X) -> c_8()
1: activate#(X) -> c_1()
** Step 1.b:8: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(activate(X))
activate(n__s(X)) -> s(activate(X))
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
natsFrom(X) -> n__natsFrom(X)
s(X) -> n__s(X)
snd(pair(XS,YS)) -> YS
splitAt(0(),XS) -> pair(nil(),XS)
- Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1
,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2
,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0
,c_12/0,c_13/1,c_14/1,c_15/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd#
,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))