MAYBE
* Step 1: DependencyPairs MAYBE
+ Considered Problem:
- Strict TRS:
U11(tt(),L) -> U12(tt(),activate(L))
U12(tt(),L) -> s(length(activate(L)))
U21(tt(),IL,M,N) -> U22(tt(),activate(IL),activate(M),activate(N))
U22(tt(),IL,M,N) -> U23(tt(),activate(IL),activate(M),activate(N))
U23(tt(),IL,M,N) -> cons(activate(N),n__take(activate(M),activate(IL)))
activate(X) -> X
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
activate(n__zeros()) -> zeros()
length(cons(N,L)) -> U11(tt(),activate(L))
length(nil()) -> 0()
take(X1,X2) -> n__take(X1,X2)
take(0(),IL) -> nil()
take(s(M),cons(N,IL)) -> U21(tt(),activate(IL),M,N)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{U11/2,U12/2,U21/4,U22/4,U23/4,activate/1,length/1,take/2,zeros/0} / {0/0,cons/2,n__take/2,n__zeros/0,nil/0
,s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,U21,U22,U23,activate,length,take
,zeros} and constructors {0,cons,n__take,n__zeros,nil,s,tt}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
U11#(tt(),L) -> c_1(U12#(tt(),activate(L)),activate#(L))
U12#(tt(),L) -> c_2(length#(activate(L)),activate#(L))
U21#(tt(),IL,M,N) -> c_3(U22#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
U22#(tt(),IL,M,N) -> c_4(U23#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
U23#(tt(),IL,M,N) -> c_5(activate#(N),activate#(M),activate#(IL))
activate#(X) -> c_6()
activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__zeros()) -> c_8(zeros#())
length#(cons(N,L)) -> c_9(U11#(tt(),activate(L)),activate#(L))
length#(nil()) -> c_10()
take#(X1,X2) -> c_11()
take#(0(),IL) -> c_12()
take#(s(M),cons(N,IL)) -> c_13(U21#(tt(),activate(IL),M,N),activate#(IL))
zeros#() -> c_14()
zeros#() -> c_15()
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules MAYBE
+ Considered Problem:
- Strict DPs:
U11#(tt(),L) -> c_1(U12#(tt(),activate(L)),activate#(L))
U12#(tt(),L) -> c_2(length#(activate(L)),activate#(L))
U21#(tt(),IL,M,N) -> c_3(U22#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
U22#(tt(),IL,M,N) -> c_4(U23#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
U23#(tt(),IL,M,N) -> c_5(activate#(N),activate#(M),activate#(IL))
activate#(X) -> c_6()
activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__zeros()) -> c_8(zeros#())
length#(cons(N,L)) -> c_9(U11#(tt(),activate(L)),activate#(L))
length#(nil()) -> c_10()
take#(X1,X2) -> c_11()
take#(0(),IL) -> c_12()
take#(s(M),cons(N,IL)) -> c_13(U21#(tt(),activate(IL),M,N),activate#(IL))
zeros#() -> c_14()
zeros#() -> c_15()
- Weak TRS:
U11(tt(),L) -> U12(tt(),activate(L))
U12(tt(),L) -> s(length(activate(L)))
U21(tt(),IL,M,N) -> U22(tt(),activate(IL),activate(M),activate(N))
U22(tt(),IL,M,N) -> U23(tt(),activate(IL),activate(M),activate(N))
U23(tt(),IL,M,N) -> cons(activate(N),n__take(activate(M),activate(IL)))
activate(X) -> X
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
activate(n__zeros()) -> zeros()
length(cons(N,L)) -> U11(tt(),activate(L))
length(nil()) -> 0()
take(X1,X2) -> n__take(X1,X2)
take(0(),IL) -> nil()
take(s(M),cons(N,IL)) -> U21(tt(),activate(IL),M,N)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{U11/2,U12/2,U21/4,U22/4,U23/4,activate/1,length/1,take/2,zeros/0,U11#/2,U12#/2,U21#/4,U22#/4,U23#/4
,activate#/1,length#/1,take#/2,zeros#/0} / {0/0,cons/2,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/2,c_2/2,c_3/4
,c_4/4,c_5/3,c_6/0,c_7/3,c_8/1,c_9/2,c_10/0,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,U23#,activate#,length#,take#
,zeros#} and constructors {0,cons,n__take,n__zeros,nil,s,tt}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
U21(tt(),IL,M,N) -> U22(tt(),activate(IL),activate(M),activate(N))
U22(tt(),IL,M,N) -> U23(tt(),activate(IL),activate(M),activate(N))
U23(tt(),IL,M,N) -> cons(activate(N),n__take(activate(M),activate(IL)))
activate(X) -> X
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
activate(n__zeros()) -> zeros()
take(X1,X2) -> n__take(X1,X2)
take(0(),IL) -> nil()
take(s(M),cons(N,IL)) -> U21(tt(),activate(IL),M,N)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
U11#(tt(),L) -> c_1(U12#(tt(),activate(L)),activate#(L))
U12#(tt(),L) -> c_2(length#(activate(L)),activate#(L))
U21#(tt(),IL,M,N) -> c_3(U22#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
U22#(tt(),IL,M,N) -> c_4(U23#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
U23#(tt(),IL,M,N) -> c_5(activate#(N),activate#(M),activate#(IL))
activate#(X) -> c_6()
activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__zeros()) -> c_8(zeros#())
length#(cons(N,L)) -> c_9(U11#(tt(),activate(L)),activate#(L))
length#(nil()) -> c_10()
take#(X1,X2) -> c_11()
take#(0(),IL) -> c_12()
take#(s(M),cons(N,IL)) -> c_13(U21#(tt(),activate(IL),M,N),activate#(IL))
zeros#() -> c_14()
zeros#() -> c_15()
* Step 3: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
U11#(tt(),L) -> c_1(U12#(tt(),activate(L)),activate#(L))
U12#(tt(),L) -> c_2(length#(activate(L)),activate#(L))
U21#(tt(),IL,M,N) -> c_3(U22#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
U22#(tt(),IL,M,N) -> c_4(U23#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
U23#(tt(),IL,M,N) -> c_5(activate#(N),activate#(M),activate#(IL))
activate#(X) -> c_6()
activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__zeros()) -> c_8(zeros#())
length#(cons(N,L)) -> c_9(U11#(tt(),activate(L)),activate#(L))
length#(nil()) -> c_10()
take#(X1,X2) -> c_11()
take#(0(),IL) -> c_12()
take#(s(M),cons(N,IL)) -> c_13(U21#(tt(),activate(IL),M,N),activate#(IL))
zeros#() -> c_14()
zeros#() -> c_15()
- Weak TRS:
U21(tt(),IL,M,N) -> U22(tt(),activate(IL),activate(M),activate(N))
U22(tt(),IL,M,N) -> U23(tt(),activate(IL),activate(M),activate(N))
U23(tt(),IL,M,N) -> cons(activate(N),n__take(activate(M),activate(IL)))
activate(X) -> X
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
activate(n__zeros()) -> zeros()
take(X1,X2) -> n__take(X1,X2)
take(0(),IL) -> nil()
take(s(M),cons(N,IL)) -> U21(tt(),activate(IL),M,N)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{U11/2,U12/2,U21/4,U22/4,U23/4,activate/1,length/1,take/2,zeros/0,U11#/2,U12#/2,U21#/4,U22#/4,U23#/4
,activate#/1,length#/1,take#/2,zeros#/0} / {0/0,cons/2,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/2,c_2/2,c_3/4
,c_4/4,c_5/3,c_6/0,c_7/3,c_8/1,c_9/2,c_10/0,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,U23#,activate#,length#,take#
,zeros#} and constructors {0,cons,n__take,n__zeros,nil,s,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{6,10,11,12,14,15}
by application of
Pre({6,10,11,12,14,15}) = {1,2,3,4,5,7,8,9,13}.
Here rules are labelled as follows:
1: U11#(tt(),L) -> c_1(U12#(tt(),activate(L)),activate#(L))
2: U12#(tt(),L) -> c_2(length#(activate(L)),activate#(L))
3: U21#(tt(),IL,M,N) -> c_3(U22#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
4: U22#(tt(),IL,M,N) -> c_4(U23#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
5: U23#(tt(),IL,M,N) -> c_5(activate#(N),activate#(M),activate#(IL))
6: activate#(X) -> c_6()
7: activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
8: activate#(n__zeros()) -> c_8(zeros#())
9: length#(cons(N,L)) -> c_9(U11#(tt(),activate(L)),activate#(L))
10: length#(nil()) -> c_10()
11: take#(X1,X2) -> c_11()
12: take#(0(),IL) -> c_12()
13: take#(s(M),cons(N,IL)) -> c_13(U21#(tt(),activate(IL),M,N),activate#(IL))
14: zeros#() -> c_14()
15: zeros#() -> c_15()
* Step 4: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
U11#(tt(),L) -> c_1(U12#(tt(),activate(L)),activate#(L))
U12#(tt(),L) -> c_2(length#(activate(L)),activate#(L))
U21#(tt(),IL,M,N) -> c_3(U22#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
U22#(tt(),IL,M,N) -> c_4(U23#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
U23#(tt(),IL,M,N) -> c_5(activate#(N),activate#(M),activate#(IL))
activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__zeros()) -> c_8(zeros#())
length#(cons(N,L)) -> c_9(U11#(tt(),activate(L)),activate#(L))
take#(s(M),cons(N,IL)) -> c_13(U21#(tt(),activate(IL),M,N),activate#(IL))
- Weak DPs:
activate#(X) -> c_6()
length#(nil()) -> c_10()
take#(X1,X2) -> c_11()
take#(0(),IL) -> c_12()
zeros#() -> c_14()
zeros#() -> c_15()
- Weak TRS:
U21(tt(),IL,M,N) -> U22(tt(),activate(IL),activate(M),activate(N))
U22(tt(),IL,M,N) -> U23(tt(),activate(IL),activate(M),activate(N))
U23(tt(),IL,M,N) -> cons(activate(N),n__take(activate(M),activate(IL)))
activate(X) -> X
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
activate(n__zeros()) -> zeros()
take(X1,X2) -> n__take(X1,X2)
take(0(),IL) -> nil()
take(s(M),cons(N,IL)) -> U21(tt(),activate(IL),M,N)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{U11/2,U12/2,U21/4,U22/4,U23/4,activate/1,length/1,take/2,zeros/0,U11#/2,U12#/2,U21#/4,U22#/4,U23#/4
,activate#/1,length#/1,take#/2,zeros#/0} / {0/0,cons/2,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/2,c_2/2,c_3/4
,c_4/4,c_5/3,c_6/0,c_7/3,c_8/1,c_9/2,c_10/0,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,U23#,activate#,length#,take#
,zeros#} and constructors {0,cons,n__take,n__zeros,nil,s,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{7}
by application of
Pre({7}) = {1,2,3,4,5,6,8,9}.
Here rules are labelled as follows:
1: U11#(tt(),L) -> c_1(U12#(tt(),activate(L)),activate#(L))
2: U12#(tt(),L) -> c_2(length#(activate(L)),activate#(L))
3: U21#(tt(),IL,M,N) -> c_3(U22#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
4: U22#(tt(),IL,M,N) -> c_4(U23#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
5: U23#(tt(),IL,M,N) -> c_5(activate#(N),activate#(M),activate#(IL))
6: activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
7: activate#(n__zeros()) -> c_8(zeros#())
8: length#(cons(N,L)) -> c_9(U11#(tt(),activate(L)),activate#(L))
9: take#(s(M),cons(N,IL)) -> c_13(U21#(tt(),activate(IL),M,N),activate#(IL))
10: activate#(X) -> c_6()
11: length#(nil()) -> c_10()
12: take#(X1,X2) -> c_11()
13: take#(0(),IL) -> c_12()
14: zeros#() -> c_14()
15: zeros#() -> c_15()
* Step 5: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
U11#(tt(),L) -> c_1(U12#(tt(),activate(L)),activate#(L))
U12#(tt(),L) -> c_2(length#(activate(L)),activate#(L))
U21#(tt(),IL,M,N) -> c_3(U22#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
U22#(tt(),IL,M,N) -> c_4(U23#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
U23#(tt(),IL,M,N) -> c_5(activate#(N),activate#(M),activate#(IL))
activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
length#(cons(N,L)) -> c_9(U11#(tt(),activate(L)),activate#(L))
take#(s(M),cons(N,IL)) -> c_13(U21#(tt(),activate(IL),M,N),activate#(IL))
- Weak DPs:
activate#(X) -> c_6()
activate#(n__zeros()) -> c_8(zeros#())
length#(nil()) -> c_10()
take#(X1,X2) -> c_11()
take#(0(),IL) -> c_12()
zeros#() -> c_14()
zeros#() -> c_15()
- Weak TRS:
U21(tt(),IL,M,N) -> U22(tt(),activate(IL),activate(M),activate(N))
U22(tt(),IL,M,N) -> U23(tt(),activate(IL),activate(M),activate(N))
U23(tt(),IL,M,N) -> cons(activate(N),n__take(activate(M),activate(IL)))
activate(X) -> X
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
activate(n__zeros()) -> zeros()
take(X1,X2) -> n__take(X1,X2)
take(0(),IL) -> nil()
take(s(M),cons(N,IL)) -> U21(tt(),activate(IL),M,N)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{U11/2,U12/2,U21/4,U22/4,U23/4,activate/1,length/1,take/2,zeros/0,U11#/2,U12#/2,U21#/4,U22#/4,U23#/4
,activate#/1,length#/1,take#/2,zeros#/0} / {0/0,cons/2,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/2,c_2/2,c_3/4
,c_4/4,c_5/3,c_6/0,c_7/3,c_8/1,c_9/2,c_10/0,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,U23#,activate#,length#,take#
,zeros#} and constructors {0,cons,n__take,n__zeros,nil,s,tt}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:U11#(tt(),L) -> c_1(U12#(tt(),activate(L)),activate#(L))
-->_2 activate#(n__zeros()) -> c_8(zeros#()):10
-->_2 activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):6
-->_1 U12#(tt(),L) -> c_2(length#(activate(L)),activate#(L)):2
-->_2 activate#(X) -> c_6():9
2:S:U12#(tt(),L) -> c_2(length#(activate(L)),activate#(L))
-->_2 activate#(n__zeros()) -> c_8(zeros#()):10
-->_1 length#(cons(N,L)) -> c_9(U11#(tt(),activate(L)),activate#(L)):7
-->_2 activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):6
-->_1 length#(nil()) -> c_10():11
-->_2 activate#(X) -> c_6():9
3:S:U21#(tt(),IL,M,N) -> c_3(U22#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
-->_4 activate#(n__zeros()) -> c_8(zeros#()):10
-->_3 activate#(n__zeros()) -> c_8(zeros#()):10
-->_2 activate#(n__zeros()) -> c_8(zeros#()):10
-->_4 activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):6
-->_3 activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):6
-->_2 activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):6
-->_1 U22#(tt(),IL,M,N) -> c_4(U23#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N)):4
-->_4 activate#(X) -> c_6():9
-->_3 activate#(X) -> c_6():9
-->_2 activate#(X) -> c_6():9
4:S:U22#(tt(),IL,M,N) -> c_4(U23#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
-->_4 activate#(n__zeros()) -> c_8(zeros#()):10
-->_3 activate#(n__zeros()) -> c_8(zeros#()):10
-->_2 activate#(n__zeros()) -> c_8(zeros#()):10
-->_4 activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):6
-->_3 activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):6
-->_2 activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):6
-->_1 U23#(tt(),IL,M,N) -> c_5(activate#(N),activate#(M),activate#(IL)):5
-->_4 activate#(X) -> c_6():9
-->_3 activate#(X) -> c_6():9
-->_2 activate#(X) -> c_6():9
5:S:U23#(tt(),IL,M,N) -> c_5(activate#(N),activate#(M),activate#(IL))
-->_3 activate#(n__zeros()) -> c_8(zeros#()):10
-->_2 activate#(n__zeros()) -> c_8(zeros#()):10
-->_1 activate#(n__zeros()) -> c_8(zeros#()):10
-->_3 activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):6
-->_2 activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):6
-->_1 activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):6
-->_3 activate#(X) -> c_6():9
-->_2 activate#(X) -> c_6():9
-->_1 activate#(X) -> c_6():9
6:S:activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
-->_3 activate#(n__zeros()) -> c_8(zeros#()):10
-->_2 activate#(n__zeros()) -> c_8(zeros#()):10
-->_1 take#(s(M),cons(N,IL)) -> c_13(U21#(tt(),activate(IL),M,N),activate#(IL)):8
-->_1 take#(0(),IL) -> c_12():13
-->_1 take#(X1,X2) -> c_11():12
-->_3 activate#(X) -> c_6():9
-->_2 activate#(X) -> c_6():9
-->_3 activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):6
-->_2 activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):6
7:S:length#(cons(N,L)) -> c_9(U11#(tt(),activate(L)),activate#(L))
-->_2 activate#(n__zeros()) -> c_8(zeros#()):10
-->_2 activate#(X) -> c_6():9
-->_2 activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):6
-->_1 U11#(tt(),L) -> c_1(U12#(tt(),activate(L)),activate#(L)):1
8:S:take#(s(M),cons(N,IL)) -> c_13(U21#(tt(),activate(IL),M,N),activate#(IL))
-->_2 activate#(n__zeros()) -> c_8(zeros#()):10
-->_2 activate#(X) -> c_6():9
-->_2 activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):6
-->_1 U21#(tt(),IL,M,N) -> c_3(U22#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N)):3
9:W:activate#(X) -> c_6()
10:W:activate#(n__zeros()) -> c_8(zeros#())
-->_1 zeros#() -> c_15():15
-->_1 zeros#() -> c_14():14
11:W:length#(nil()) -> c_10()
12:W:take#(X1,X2) -> c_11()
13:W:take#(0(),IL) -> c_12()
14:W:zeros#() -> c_14()
15:W:zeros#() -> c_15()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
11: length#(nil()) -> c_10()
12: take#(X1,X2) -> c_11()
13: take#(0(),IL) -> c_12()
9: activate#(X) -> c_6()
10: activate#(n__zeros()) -> c_8(zeros#())
14: zeros#() -> c_14()
15: zeros#() -> c_15()
* Step 6: Failure MAYBE
+ Considered Problem:
- Strict DPs:
U11#(tt(),L) -> c_1(U12#(tt(),activate(L)),activate#(L))
U12#(tt(),L) -> c_2(length#(activate(L)),activate#(L))
U21#(tt(),IL,M,N) -> c_3(U22#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
U22#(tt(),IL,M,N) -> c_4(U23#(tt(),activate(IL),activate(M),activate(N))
,activate#(IL)
,activate#(M)
,activate#(N))
U23#(tt(),IL,M,N) -> c_5(activate#(N),activate#(M),activate#(IL))
activate#(n__take(X1,X2)) -> c_7(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
length#(cons(N,L)) -> c_9(U11#(tt(),activate(L)),activate#(L))
take#(s(M),cons(N,IL)) -> c_13(U21#(tt(),activate(IL),M,N),activate#(IL))
- Weak TRS:
U21(tt(),IL,M,N) -> U22(tt(),activate(IL),activate(M),activate(N))
U22(tt(),IL,M,N) -> U23(tt(),activate(IL),activate(M),activate(N))
U23(tt(),IL,M,N) -> cons(activate(N),n__take(activate(M),activate(IL)))
activate(X) -> X
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
activate(n__zeros()) -> zeros()
take(X1,X2) -> n__take(X1,X2)
take(0(),IL) -> nil()
take(s(M),cons(N,IL)) -> U21(tt(),activate(IL),M,N)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{U11/2,U12/2,U21/4,U22/4,U23/4,activate/1,length/1,take/2,zeros/0,U11#/2,U12#/2,U21#/4,U22#/4,U23#/4
,activate#/1,length#/1,take#/2,zeros#/0} / {0/0,cons/2,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/2,c_2/2,c_3/4
,c_4/4,c_5/3,c_6/0,c_7/3,c_8/1,c_9/2,c_10/0,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,U23#,activate#,length#,take#
,zeros#} and constructors {0,cons,n__take,n__zeros,nil,s,tt}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
MAYBE