*** 1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
fst(pair(XS,YS)) -> XS
head(cons(N,XS)) -> N
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(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)
Weak DP Rules:
Weak TRS Rules:
Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1}
Obligation:
Innermost
basic terms: {activate,afterNth,fst,head,natsFrom,sel,snd,splitAt,tail,take,u}/{0,cons,n__natsFrom,nil,pair,s}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
activate#(X) -> c_1()
activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
fst#(pair(XS,YS)) -> c_4()
head#(cons(N,XS)) -> c_5()
natsFrom#(N) -> c_6()
natsFrom#(X) -> c_7()
sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
snd#(pair(XS,YS)) -> c_9()
splitAt#(0(),XS) -> c_10()
splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS)),splitAt#(N,activate(XS)),activate#(XS),activate#(XS))
tail#(cons(N,XS)) -> c_12(activate#(XS))
take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
activate#(X) -> c_1()
activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
fst#(pair(XS,YS)) -> c_4()
head#(cons(N,XS)) -> c_5()
natsFrom#(N) -> c_6()
natsFrom#(X) -> c_7()
sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
snd#(pair(XS,YS)) -> c_9()
splitAt#(0(),XS) -> c_10()
splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS)),splitAt#(N,activate(XS)),activate#(XS),activate#(XS))
tail#(cons(N,XS)) -> c_12(activate#(XS))
take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
fst(pair(XS,YS)) -> XS
head(cons(N,XS)) -> N
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(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,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2,c_9/0,c_10/0,c_11/4,c_12/1,c_13/2,c_14/1}
Obligation:
Innermost
basic terms: {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#,u#}/{0,cons,n__natsFrom,nil,pair,s}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
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))
u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
activate#(X) -> c_1()
activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
fst#(pair(XS,YS)) -> c_4()
head#(cons(N,XS)) -> c_5()
natsFrom#(N) -> c_6()
natsFrom#(X) -> c_7()
sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
snd#(pair(XS,YS)) -> c_9()
splitAt#(0(),XS) -> c_10()
splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS)),splitAt#(N,activate(XS)),activate#(XS),activate#(XS))
tail#(cons(N,XS)) -> c_12(activate#(XS))
take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
*** 1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
activate#(X) -> c_1()
activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
fst#(pair(XS,YS)) -> c_4()
head#(cons(N,XS)) -> c_5()
natsFrom#(N) -> c_6()
natsFrom#(X) -> c_7()
sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
snd#(pair(XS,YS)) -> c_9()
splitAt#(0(),XS) -> c_10()
splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS)),splitAt#(N,activate(XS)),activate#(XS),activate#(XS))
tail#(cons(N,XS)) -> c_12(activate#(XS))
take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
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))
u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2,c_9/0,c_10/0,c_11/4,c_12/1,c_13/2,c_14/1}
Obligation:
Innermost
basic terms: {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#,u#}/{0,cons,n__natsFrom,nil,pair,s}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1,4,5,6,7,9,10}
by application of
Pre({1,4,5,6,7,9,10}) = {2,3,8,11,12,13,14}.
Here rules are labelled as follows:
1: activate#(X) -> c_1()
2: activate#(n__natsFrom(X)) ->
c_2(natsFrom#(X))
3: afterNth#(N,XS) ->
c_3(snd#(splitAt(N,XS))
,splitAt#(N,XS))
4: fst#(pair(XS,YS)) -> c_4()
5: head#(cons(N,XS)) -> c_5()
6: natsFrom#(N) -> c_6()
7: natsFrom#(X) -> c_7()
8: sel#(N,XS) ->
c_8(head#(afterNth(N,XS))
,afterNth#(N,XS))
9: snd#(pair(XS,YS)) -> c_9()
10: splitAt#(0(),XS) -> c_10()
11: splitAt#(s(N),cons(X,XS)) ->
c_11(u#(splitAt(N,activate(XS))
,N
,X
,activate(XS))
,splitAt#(N,activate(XS))
,activate#(XS)
,activate#(XS))
12: tail#(cons(N,XS)) ->
c_12(activate#(XS))
13: take#(N,XS) ->
c_13(fst#(splitAt(N,XS))
,splitAt#(N,XS))
14: u#(pair(YS,ZS),N,X,XS) ->
c_14(activate#(X))
*** 1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS)),splitAt#(N,activate(XS)),activate#(XS),activate#(XS))
tail#(cons(N,XS)) -> c_12(activate#(XS))
take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
Strict TRS Rules:
Weak DP Rules:
activate#(X) -> c_1()
fst#(pair(XS,YS)) -> c_4()
head#(cons(N,XS)) -> c_5()
natsFrom#(N) -> c_6()
natsFrom#(X) -> c_7()
snd#(pair(XS,YS)) -> c_9()
splitAt#(0(),XS) -> c_10()
Weak TRS Rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
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))
u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2,c_9/0,c_10/0,c_11/4,c_12/1,c_13/2,c_14/1}
Obligation:
Innermost
basic terms: {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#,u#}/{0,cons,n__natsFrom,nil,pair,s}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1}
by application of
Pre({1}) = {4,5,7}.
Here rules are labelled as follows:
1: activate#(n__natsFrom(X)) ->
c_2(natsFrom#(X))
2: afterNth#(N,XS) ->
c_3(snd#(splitAt(N,XS))
,splitAt#(N,XS))
3: sel#(N,XS) ->
c_8(head#(afterNth(N,XS))
,afterNth#(N,XS))
4: splitAt#(s(N),cons(X,XS)) ->
c_11(u#(splitAt(N,activate(XS))
,N
,X
,activate(XS))
,splitAt#(N,activate(XS))
,activate#(XS)
,activate#(XS))
5: tail#(cons(N,XS)) ->
c_12(activate#(XS))
6: take#(N,XS) ->
c_13(fst#(splitAt(N,XS))
,splitAt#(N,XS))
7: u#(pair(YS,ZS),N,X,XS) ->
c_14(activate#(X))
8: activate#(X) -> c_1()
9: fst#(pair(XS,YS)) -> c_4()
10: head#(cons(N,XS)) -> c_5()
11: natsFrom#(N) -> c_6()
12: natsFrom#(X) -> c_7()
13: snd#(pair(XS,YS)) -> c_9()
14: splitAt#(0(),XS) -> c_10()
*** 1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS)),splitAt#(N,activate(XS)),activate#(XS),activate#(XS))
tail#(cons(N,XS)) -> c_12(activate#(XS))
take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
Strict TRS Rules:
Weak DP Rules:
activate#(X) -> c_1()
activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
fst#(pair(XS,YS)) -> c_4()
head#(cons(N,XS)) -> c_5()
natsFrom#(N) -> c_6()
natsFrom#(X) -> c_7()
snd#(pair(XS,YS)) -> c_9()
splitAt#(0(),XS) -> c_10()
Weak TRS Rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
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))
u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2,c_9/0,c_10/0,c_11/4,c_12/1,c_13/2,c_14/1}
Obligation:
Innermost
basic terms: {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#,u#}/{0,cons,n__natsFrom,nil,pair,s}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{4,6}
by application of
Pre({4,6}) = {3}.
Here rules are labelled as follows:
1: afterNth#(N,XS) ->
c_3(snd#(splitAt(N,XS))
,splitAt#(N,XS))
2: sel#(N,XS) ->
c_8(head#(afterNth(N,XS))
,afterNth#(N,XS))
3: splitAt#(s(N),cons(X,XS)) ->
c_11(u#(splitAt(N,activate(XS))
,N
,X
,activate(XS))
,splitAt#(N,activate(XS))
,activate#(XS)
,activate#(XS))
4: tail#(cons(N,XS)) ->
c_12(activate#(XS))
5: take#(N,XS) ->
c_13(fst#(splitAt(N,XS))
,splitAt#(N,XS))
6: u#(pair(YS,ZS),N,X,XS) ->
c_14(activate#(X))
7: activate#(X) -> c_1()
8: activate#(n__natsFrom(X)) ->
c_2(natsFrom#(X))
9: fst#(pair(XS,YS)) -> c_4()
10: head#(cons(N,XS)) -> c_5()
11: natsFrom#(N) -> c_6()
12: natsFrom#(X) -> c_7()
13: snd#(pair(XS,YS)) -> c_9()
14: splitAt#(0(),XS) -> c_10()
*** 1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS)),splitAt#(N,activate(XS)),activate#(XS),activate#(XS))
take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
Strict TRS Rules:
Weak DP Rules:
activate#(X) -> c_1()
activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
fst#(pair(XS,YS)) -> c_4()
head#(cons(N,XS)) -> c_5()
natsFrom#(N) -> c_6()
natsFrom#(X) -> c_7()
snd#(pair(XS,YS)) -> c_9()
splitAt#(0(),XS) -> c_10()
tail#(cons(N,XS)) -> c_12(activate#(XS))
u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
Weak TRS Rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
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))
u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2,c_9/0,c_10/0,c_11/4,c_12/1,c_13/2,c_14/1}
Obligation:
Innermost
basic terms: {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#,u#}/{0,cons,n__natsFrom,nil,pair,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
-->_2 splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS)),splitAt#(N,activate(XS)),activate#(XS),activate#(XS)):3
-->_2 splitAt#(0(),XS) -> c_10():12
-->_1 snd#(pair(XS,YS)) -> c_9():11
2:S:sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
-->_1 head#(cons(N,XS)) -> c_5():8
-->_2 afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS)):1
3:S:splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS)),splitAt#(N,activate(XS)),activate#(XS),activate#(XS))
-->_1 u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X)):14
-->_4 activate#(n__natsFrom(X)) -> c_2(natsFrom#(X)):6
-->_3 activate#(n__natsFrom(X)) -> c_2(natsFrom#(X)):6
-->_2 splitAt#(0(),XS) -> c_10():12
-->_4 activate#(X) -> c_1():5
-->_3 activate#(X) -> c_1():5
-->_2 splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS)),splitAt#(N,activate(XS)),activate#(XS),activate#(XS)):3
4:S:take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
-->_2 splitAt#(0(),XS) -> c_10():12
-->_1 fst#(pair(XS,YS)) -> c_4():7
-->_2 splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS)),splitAt#(N,activate(XS)),activate#(XS),activate#(XS)):3
5:W:activate#(X) -> c_1()
6:W:activate#(n__natsFrom(X)) -> c_2(natsFrom#(X))
-->_1 natsFrom#(X) -> c_7():10
-->_1 natsFrom#(N) -> c_6():9
7:W:fst#(pair(XS,YS)) -> c_4()
8:W:head#(cons(N,XS)) -> c_5()
9:W:natsFrom#(N) -> c_6()
10:W:natsFrom#(X) -> c_7()
11:W:snd#(pair(XS,YS)) -> c_9()
12:W:splitAt#(0(),XS) -> c_10()
13:W:tail#(cons(N,XS)) -> c_12(activate#(XS))
-->_1 activate#(n__natsFrom(X)) -> c_2(natsFrom#(X)):6
-->_1 activate#(X) -> c_1():5
14:W:u#(pair(YS,ZS),N,X,XS) -> c_14(activate#(X))
-->_1 activate#(n__natsFrom(X)) -> c_2(natsFrom#(X)):6
-->_1 activate#(X) -> c_1():5
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
13: tail#(cons(N,XS)) ->
c_12(activate#(XS))
7: fst#(pair(XS,YS)) -> c_4()
8: head#(cons(N,XS)) -> c_5()
11: snd#(pair(XS,YS)) -> c_9()
12: splitAt#(0(),XS) -> c_10()
14: u#(pair(YS,ZS),N,X,XS) ->
c_14(activate#(X))
5: activate#(X) -> c_1()
6: activate#(n__natsFrom(X)) ->
c_2(natsFrom#(X))
9: natsFrom#(N) -> c_6()
10: natsFrom#(X) -> c_7()
*** 1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS)),splitAt#(N,activate(XS)),activate#(XS),activate#(XS))
take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
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))
u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2,c_9/0,c_10/0,c_11/4,c_12/1,c_13/2,c_14/1}
Obligation:
Innermost
basic terms: {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#,u#}/{0,cons,n__natsFrom,nil,pair,s}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS))
-->_2 splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS)),splitAt#(N,activate(XS)),activate#(XS),activate#(XS)):3
2:S:sel#(N,XS) -> c_8(head#(afterNth(N,XS)),afterNth#(N,XS))
-->_2 afterNth#(N,XS) -> c_3(snd#(splitAt(N,XS)),splitAt#(N,XS)):1
3:S:splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS)),splitAt#(N,activate(XS)),activate#(XS),activate#(XS))
-->_2 splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS)),splitAt#(N,activate(XS)),activate#(XS),activate#(XS)):3
4:S:take#(N,XS) -> c_13(fst#(splitAt(N,XS)),splitAt#(N,XS))
-->_2 splitAt#(s(N),cons(X,XS)) -> c_11(u#(splitAt(N,activate(XS)),N,X,activate(XS)),splitAt#(N,activate(XS)),activate#(XS),activate#(XS)):3
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
afterNth#(N,XS) -> c_3(splitAt#(N,XS))
sel#(N,XS) -> c_8(afterNth#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
take#(N,XS) -> c_13(splitAt#(N,XS))
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
afterNth#(N,XS) -> c_3(splitAt#(N,XS))
sel#(N,XS) -> c_8(afterNth#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
take#(N,XS) -> c_13(splitAt#(N,XS))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
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))
u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#,u#}/{0,cons,n__natsFrom,nil,pair,s}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
afterNth#(N,XS) -> c_3(splitAt#(N,XS))
sel#(N,XS) -> c_8(afterNth#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
take#(N,XS) -> c_13(splitAt#(N,XS))
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
afterNth#(N,XS) -> c_3(splitAt#(N,XS))
sel#(N,XS) -> c_8(afterNth#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
take#(N,XS) -> c_13(splitAt#(N,XS))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#,u#}/{0,cons,n__natsFrom,nil,pair,s}
Applied Processor:
RemoveHeads
Proof:
Consider the dependency graph
1:S:afterNth#(N,XS) -> c_3(splitAt#(N,XS))
-->_1 splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS))):3
2:S:sel#(N,XS) -> c_8(afterNth#(N,XS))
-->_1 afterNth#(N,XS) -> c_3(splitAt#(N,XS)):1
3:S:splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
-->_1 splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS))):3
4:S:take#(N,XS) -> c_13(splitAt#(N,XS))
-->_1 splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS))):3
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(2,sel#(N,XS) -> c_8(afterNth#(N,XS))),(4,take#(N,XS) -> c_13(splitAt#(N,XS)))]
*** 1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
afterNth#(N,XS) -> c_3(splitAt#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#,u#}/{0,cons,n__natsFrom,nil,pair,s}
Applied Processor:
RemoveHeads
Proof:
Consider the dependency graph
1:S:afterNth#(N,XS) -> c_3(splitAt#(N,XS))
-->_1 splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS))):3
3:S:splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
-->_1 splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS))):3
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(1,afterNth#(N,XS) -> c_3(splitAt#(N,XS)))]
*** 1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#,u#}/{0,cons,n__natsFrom,nil,pair,s}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
3: splitAt#(s(N),cons(X,XS)) ->
c_11(splitAt#(N,activate(XS)))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#,u#}/{0,cons,n__natsFrom,nil,pair,s}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_11) = {1}
Following symbols are considered usable:
{activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#,u#}
TcT has computed the following interpretation:
p(0) = [1]
p(activate) = [0]
p(afterNth) = [1] x1 + [1] x2 + [0]
p(cons) = [1] x1 + [0]
p(fst) = [1] x1 + [1]
p(head) = [2] x1 + [4]
p(n__natsFrom) = [0]
p(natsFrom) = [0]
p(nil) = [2]
p(pair) = [8]
p(s) = [1] x1 + [6]
p(sel) = [2] x2 + [0]
p(snd) = [1] x1 + [2]
p(splitAt) = [1] x1 + [1] x2 + [2]
p(tail) = [1] x1 + [0]
p(take) = [1] x1 + [4]
p(u) = [1] x1 + [8] x3 + [0]
p(activate#) = [1] x1 + [1]
p(afterNth#) = [1] x1 + [4]
p(fst#) = [0]
p(head#) = [4]
p(natsFrom#) = [2] x1 + [1]
p(sel#) = [1] x1 + [0]
p(snd#) = [2]
p(splitAt#) = [1] x1 + [0]
p(tail#) = [8]
p(take#) = [1]
p(u#) = [8] x1 + [1] x2 + [1]
p(c_1) = [2]
p(c_2) = [8] x1 + [1]
p(c_3) = [1]
p(c_4) = [1]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [1]
p(c_8) = [4]
p(c_9) = [2]
p(c_10) = [2]
p(c_11) = [1] x1 + [0]
p(c_12) = [0]
p(c_13) = [1]
p(c_14) = [1] x1 + [1]
Following rules are strictly oriented:
splitAt#(s(N),cons(X,XS)) = [1] N + [6]
> [1] N + [0]
= c_11(splitAt#(N,activate(XS)))
Following rules are (at-least) weakly oriented:
*** 1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
Weak TRS Rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#,u#}/{0,cons,n__natsFrom,nil,pair,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
Weak TRS Rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#,u#}/{0,cons,n__natsFrom,nil,pair,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS)))
-->_1 splitAt#(s(N),cons(X,XS)) -> c_11(splitAt#(N,activate(XS))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: splitAt#(s(N),cons(X,XS)) ->
c_11(splitAt#(N,activate(XS)))
*** 1.1.1.1.1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
Signature:
{activate/1,afterNth/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1,afterNth#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {activate#,afterNth#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#,u#}/{0,cons,n__natsFrom,nil,pair,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).