MAYBE
* Step 1: DependencyPairs MAYBE
+ Considered Problem:
- Strict TRS:
a__U11(X) -> U11(X)
a__U11(tt()) -> a__U12(tt())
a__U12(X) -> U12(X)
a__U12(tt()) -> tt()
a____(X,nil()) -> mark(X)
a____(X1,X2) -> __(X1,X2)
a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z)))
a____(nil(),X) -> mark(X)
a__isNePal(X) -> isNePal(X)
a__isNePal(__(I,__(P,I))) -> a__U11(tt())
mark(U11(X)) -> a__U11(mark(X))
mark(U12(X)) -> a__U12(mark(X))
mark(__(X1,X2)) -> a____(mark(X1),mark(X2))
mark(isNePal(X)) -> a__isNePal(mark(X))
mark(nil()) -> nil()
mark(tt()) -> tt()
- Signature:
{a__U11/1,a__U12/1,a____/2,a__isNePal/1,mark/1} / {U11/1,U12/1,__/2,isNePal/1,nil/0,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U11,a__U12,a____,a__isNePal
,mark} and constructors {U11,U12,__,isNePal,nil,tt}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
a__U11#(X) -> c_1()
a__U11#(tt()) -> c_2(a__U12#(tt()))
a__U12#(X) -> c_3()
a__U12#(tt()) -> c_4()
a____#(X,nil()) -> c_5(mark#(X))
a____#(X1,X2) -> c_6()
a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z))
a____#(nil(),X) -> c_8(mark#(X))
a__isNePal#(X) -> c_9()
a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt()))
mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X))
mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X))
mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X))
mark#(nil()) -> c_15()
mark#(tt()) -> c_16()
Weak DPs
and mark the set of starting terms.
* Step 2: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
a__U11#(X) -> c_1()
a__U11#(tt()) -> c_2(a__U12#(tt()))
a__U12#(X) -> c_3()
a__U12#(tt()) -> c_4()
a____#(X,nil()) -> c_5(mark#(X))
a____#(X1,X2) -> c_6()
a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z))
a____#(nil(),X) -> c_8(mark#(X))
a__isNePal#(X) -> c_9()
a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt()))
mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X))
mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X))
mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X))
mark#(nil()) -> c_15()
mark#(tt()) -> c_16()
- Weak TRS:
a__U11(X) -> U11(X)
a__U11(tt()) -> a__U12(tt())
a__U12(X) -> U12(X)
a__U12(tt()) -> tt()
a____(X,nil()) -> mark(X)
a____(X1,X2) -> __(X1,X2)
a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z)))
a____(nil(),X) -> mark(X)
a__isNePal(X) -> isNePal(X)
a__isNePal(__(I,__(P,I))) -> a__U11(tt())
mark(U11(X)) -> a__U11(mark(X))
mark(U12(X)) -> a__U12(mark(X))
mark(__(X1,X2)) -> a____(mark(X1),mark(X2))
mark(isNePal(X)) -> a__isNePal(mark(X))
mark(nil()) -> nil()
mark(tt()) -> tt()
- Signature:
{a__U11/1,a__U12/1,a____/2,a__isNePal/1,mark/1,a__U11#/1,a__U12#/1,a____#/2,a__isNePal#/1,mark#/1} / {U11/1
,U12/1,__/2,isNePal/1,nil/0,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/5,c_8/1,c_9/0,c_10/1,c_11/2,c_12/2
,c_13/3,c_14/2,c_15/0,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a____#,a__isNePal#
,mark#} and constructors {U11,U12,__,isNePal,nil,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3,4,6,9,15,16}
by application of
Pre({1,3,4,6,9,15,16}) = {2,5,7,8,10,11,12,13,14}.
Here rules are labelled as follows:
1: a__U11#(X) -> c_1()
2: a__U11#(tt()) -> c_2(a__U12#(tt()))
3: a__U12#(X) -> c_3()
4: a__U12#(tt()) -> c_4()
5: a____#(X,nil()) -> c_5(mark#(X))
6: a____#(X1,X2) -> c_6()
7: a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z))
8: a____#(nil(),X) -> c_8(mark#(X))
9: a__isNePal#(X) -> c_9()
10: a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt()))
11: mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X))
12: mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X))
13: mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
14: mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X))
15: mark#(nil()) -> c_15()
16: mark#(tt()) -> c_16()
* Step 3: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
a__U11#(tt()) -> c_2(a__U12#(tt()))
a____#(X,nil()) -> c_5(mark#(X))
a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z))
a____#(nil(),X) -> c_8(mark#(X))
a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt()))
mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X))
mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X))
mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X))
- Weak DPs:
a__U11#(X) -> c_1()
a__U12#(X) -> c_3()
a__U12#(tt()) -> c_4()
a____#(X1,X2) -> c_6()
a__isNePal#(X) -> c_9()
mark#(nil()) -> c_15()
mark#(tt()) -> c_16()
- Weak TRS:
a__U11(X) -> U11(X)
a__U11(tt()) -> a__U12(tt())
a__U12(X) -> U12(X)
a__U12(tt()) -> tt()
a____(X,nil()) -> mark(X)
a____(X1,X2) -> __(X1,X2)
a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z)))
a____(nil(),X) -> mark(X)
a__isNePal(X) -> isNePal(X)
a__isNePal(__(I,__(P,I))) -> a__U11(tt())
mark(U11(X)) -> a__U11(mark(X))
mark(U12(X)) -> a__U12(mark(X))
mark(__(X1,X2)) -> a____(mark(X1),mark(X2))
mark(isNePal(X)) -> a__isNePal(mark(X))
mark(nil()) -> nil()
mark(tt()) -> tt()
- Signature:
{a__U11/1,a__U12/1,a____/2,a__isNePal/1,mark/1,a__U11#/1,a__U12#/1,a____#/2,a__isNePal#/1,mark#/1} / {U11/1
,U12/1,__/2,isNePal/1,nil/0,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/5,c_8/1,c_9/0,c_10/1,c_11/2,c_12/2
,c_13/3,c_14/2,c_15/0,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a____#,a__isNePal#
,mark#} and constructors {U11,U12,__,isNePal,nil,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1}
by application of
Pre({1}) = {5,6}.
Here rules are labelled as follows:
1: a__U11#(tt()) -> c_2(a__U12#(tt()))
2: a____#(X,nil()) -> c_5(mark#(X))
3: a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z))
4: a____#(nil(),X) -> c_8(mark#(X))
5: a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt()))
6: mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X))
7: mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X))
8: mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
9: mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X))
10: a__U11#(X) -> c_1()
11: a__U12#(X) -> c_3()
12: a__U12#(tt()) -> c_4()
13: a____#(X1,X2) -> c_6()
14: a__isNePal#(X) -> c_9()
15: mark#(nil()) -> c_15()
16: mark#(tt()) -> c_16()
* Step 4: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
a____#(X,nil()) -> c_5(mark#(X))
a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z))
a____#(nil(),X) -> c_8(mark#(X))
a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt()))
mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X))
mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X))
mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X))
- Weak DPs:
a__U11#(X) -> c_1()
a__U11#(tt()) -> c_2(a__U12#(tt()))
a__U12#(X) -> c_3()
a__U12#(tt()) -> c_4()
a____#(X1,X2) -> c_6()
a__isNePal#(X) -> c_9()
mark#(nil()) -> c_15()
mark#(tt()) -> c_16()
- Weak TRS:
a__U11(X) -> U11(X)
a__U11(tt()) -> a__U12(tt())
a__U12(X) -> U12(X)
a__U12(tt()) -> tt()
a____(X,nil()) -> mark(X)
a____(X1,X2) -> __(X1,X2)
a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z)))
a____(nil(),X) -> mark(X)
a__isNePal(X) -> isNePal(X)
a__isNePal(__(I,__(P,I))) -> a__U11(tt())
mark(U11(X)) -> a__U11(mark(X))
mark(U12(X)) -> a__U12(mark(X))
mark(__(X1,X2)) -> a____(mark(X1),mark(X2))
mark(isNePal(X)) -> a__isNePal(mark(X))
mark(nil()) -> nil()
mark(tt()) -> tt()
- Signature:
{a__U11/1,a__U12/1,a____/2,a__isNePal/1,mark/1,a__U11#/1,a__U12#/1,a____#/2,a__isNePal#/1,mark#/1} / {U11/1
,U12/1,__/2,isNePal/1,nil/0,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/5,c_8/1,c_9/0,c_10/1,c_11/2,c_12/2
,c_13/3,c_14/2,c_15/0,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a____#,a__isNePal#
,mark#} and constructors {U11,U12,__,isNePal,nil,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{4}
by application of
Pre({4}) = {8}.
Here rules are labelled as follows:
1: a____#(X,nil()) -> c_5(mark#(X))
2: a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z))
3: a____#(nil(),X) -> c_8(mark#(X))
4: a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt()))
5: mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X))
6: mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X))
7: mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
8: mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X))
9: a__U11#(X) -> c_1()
10: a__U11#(tt()) -> c_2(a__U12#(tt()))
11: a__U12#(X) -> c_3()
12: a__U12#(tt()) -> c_4()
13: a____#(X1,X2) -> c_6()
14: a__isNePal#(X) -> c_9()
15: mark#(nil()) -> c_15()
16: mark#(tt()) -> c_16()
* Step 5: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
a____#(X,nil()) -> c_5(mark#(X))
a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z))
a____#(nil(),X) -> c_8(mark#(X))
mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X))
mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X))
mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X))
- Weak DPs:
a__U11#(X) -> c_1()
a__U11#(tt()) -> c_2(a__U12#(tt()))
a__U12#(X) -> c_3()
a__U12#(tt()) -> c_4()
a____#(X1,X2) -> c_6()
a__isNePal#(X) -> c_9()
a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt()))
mark#(nil()) -> c_15()
mark#(tt()) -> c_16()
- Weak TRS:
a__U11(X) -> U11(X)
a__U11(tt()) -> a__U12(tt())
a__U12(X) -> U12(X)
a__U12(tt()) -> tt()
a____(X,nil()) -> mark(X)
a____(X1,X2) -> __(X1,X2)
a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z)))
a____(nil(),X) -> mark(X)
a__isNePal(X) -> isNePal(X)
a__isNePal(__(I,__(P,I))) -> a__U11(tt())
mark(U11(X)) -> a__U11(mark(X))
mark(U12(X)) -> a__U12(mark(X))
mark(__(X1,X2)) -> a____(mark(X1),mark(X2))
mark(isNePal(X)) -> a__isNePal(mark(X))
mark(nil()) -> nil()
mark(tt()) -> tt()
- Signature:
{a__U11/1,a__U12/1,a____/2,a__isNePal/1,mark/1,a__U11#/1,a__U12#/1,a____#/2,a__isNePal#/1,mark#/1} / {U11/1
,U12/1,__/2,isNePal/1,nil/0,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/5,c_8/1,c_9/0,c_10/1,c_11/2,c_12/2
,c_13/3,c_14/2,c_15/0,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a____#,a__isNePal#
,mark#} and constructors {U11,U12,__,isNePal,nil,tt}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:a____#(X,nil()) -> c_5(mark#(X))
-->_1 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_1 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_1 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_1 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
-->_1 mark#(tt()) -> c_16():16
-->_1 mark#(nil()) -> c_15():15
2:S:a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z))
-->_5 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_4 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_5 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_4 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_5 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_4 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_5 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
-->_4 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
-->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
-->_3 a____#(nil(),X) -> c_8(mark#(X)):3
-->_1 a____#(nil(),X) -> c_8(mark#(X)):3
-->_5 mark#(tt()) -> c_16():16
-->_4 mark#(tt()) -> c_16():16
-->_2 mark#(tt()) -> c_16():16
-->_5 mark#(nil()) -> c_15():15
-->_4 mark#(nil()) -> c_15():15
-->_2 mark#(nil()) -> c_15():15
-->_3 a____#(X1,X2) -> c_6():12
-->_1 a____#(X1,X2) -> c_6():12
-->_3 a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z)):2
-->_1 a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z)):2
-->_3 a____#(X,nil()) -> c_5(mark#(X)):1
-->_1 a____#(X,nil()) -> c_5(mark#(X)):1
3:S:a____#(nil(),X) -> c_8(mark#(X))
-->_1 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_1 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_1 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_1 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
-->_1 mark#(tt()) -> c_16():16
-->_1 mark#(nil()) -> c_15():15
4:S:mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X))
-->_1 a__U11#(tt()) -> c_2(a__U12#(tt())):9
-->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_2 mark#(tt()) -> c_16():16
-->_2 mark#(nil()) -> c_15():15
-->_1 a__U11#(X) -> c_1():8
-->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
5:S:mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X))
-->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_2 mark#(tt()) -> c_16():16
-->_2 mark#(nil()) -> c_15():15
-->_1 a__U12#(tt()) -> c_4():11
-->_1 a__U12#(X) -> c_3():10
-->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
6:S:mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
-->_3 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_3 mark#(tt()) -> c_16():16
-->_2 mark#(tt()) -> c_16():16
-->_3 mark#(nil()) -> c_15():15
-->_2 mark#(nil()) -> c_15():15
-->_1 a____#(X1,X2) -> c_6():12
-->_3 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_3 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_3 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
-->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
-->_1 a____#(nil(),X) -> c_8(mark#(X)):3
-->_1 a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z)):2
-->_1 a____#(X,nil()) -> c_5(mark#(X)):1
7:S:mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X))
-->_1 a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt())):14
-->_2 mark#(tt()) -> c_16():16
-->_2 mark#(nil()) -> c_15():15
-->_1 a__isNePal#(X) -> c_9():13
-->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
8:W:a__U11#(X) -> c_1()
9:W:a__U11#(tt()) -> c_2(a__U12#(tt()))
-->_1 a__U12#(tt()) -> c_4():11
-->_1 a__U12#(X) -> c_3():10
10:W:a__U12#(X) -> c_3()
11:W:a__U12#(tt()) -> c_4()
12:W:a____#(X1,X2) -> c_6()
13:W:a__isNePal#(X) -> c_9()
14:W:a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt()))
-->_1 a__U11#(tt()) -> c_2(a__U12#(tt())):9
-->_1 a__U11#(X) -> c_1():8
15:W:mark#(nil()) -> c_15()
16:W:mark#(tt()) -> c_16()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
12: a____#(X1,X2) -> c_6()
13: a__isNePal#(X) -> c_9()
15: mark#(nil()) -> c_15()
16: mark#(tt()) -> c_16()
14: a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt()))
8: a__U11#(X) -> c_1()
9: a__U11#(tt()) -> c_2(a__U12#(tt()))
10: a__U12#(X) -> c_3()
11: a__U12#(tt()) -> c_4()
* Step 6: SimplifyRHS MAYBE
+ Considered Problem:
- Strict DPs:
a____#(X,nil()) -> c_5(mark#(X))
a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z))
a____#(nil(),X) -> c_8(mark#(X))
mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X))
mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X))
mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X))
- Weak TRS:
a__U11(X) -> U11(X)
a__U11(tt()) -> a__U12(tt())
a__U12(X) -> U12(X)
a__U12(tt()) -> tt()
a____(X,nil()) -> mark(X)
a____(X1,X2) -> __(X1,X2)
a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z)))
a____(nil(),X) -> mark(X)
a__isNePal(X) -> isNePal(X)
a__isNePal(__(I,__(P,I))) -> a__U11(tt())
mark(U11(X)) -> a__U11(mark(X))
mark(U12(X)) -> a__U12(mark(X))
mark(__(X1,X2)) -> a____(mark(X1),mark(X2))
mark(isNePal(X)) -> a__isNePal(mark(X))
mark(nil()) -> nil()
mark(tt()) -> tt()
- Signature:
{a__U11/1,a__U12/1,a____/2,a__isNePal/1,mark/1,a__U11#/1,a__U12#/1,a____#/2,a__isNePal#/1,mark#/1} / {U11/1
,U12/1,__/2,isNePal/1,nil/0,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/5,c_8/1,c_9/0,c_10/1,c_11/2,c_12/2
,c_13/3,c_14/2,c_15/0,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a____#,a__isNePal#
,mark#} and constructors {U11,U12,__,isNePal,nil,tt}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:a____#(X,nil()) -> c_5(mark#(X))
-->_1 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_1 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_1 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_1 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
2:S:a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z))
-->_5 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_4 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_5 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_4 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_5 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_4 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_5 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
-->_4 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
-->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
-->_3 a____#(nil(),X) -> c_8(mark#(X)):3
-->_1 a____#(nil(),X) -> c_8(mark#(X)):3
-->_3 a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z)):2
-->_1 a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z)):2
-->_3 a____#(X,nil()) -> c_5(mark#(X)):1
-->_1 a____#(X,nil()) -> c_5(mark#(X)):1
3:S:a____#(nil(),X) -> c_8(mark#(X))
-->_1 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_1 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_1 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_1 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
4:S:mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X))
-->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
5:S:mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X))
-->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
6:S:mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
-->_3 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_3 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_3 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_3 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
-->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
-->_1 a____#(nil(),X) -> c_8(mark#(X)):3
-->_1 a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z)):2
-->_1 a____#(X,nil()) -> c_5(mark#(X)):1
7:S:mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X))
-->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7
-->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6
-->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5
-->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
mark#(U11(X)) -> c_11(mark#(X))
mark#(U12(X)) -> c_12(mark#(X))
mark#(isNePal(X)) -> c_14(mark#(X))
* Step 7: WeightGap MAYBE
+ Considered Problem:
- Strict DPs:
a____#(X,nil()) -> c_5(mark#(X))
a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z))
a____#(nil(),X) -> c_8(mark#(X))
mark#(U11(X)) -> c_11(mark#(X))
mark#(U12(X)) -> c_12(mark#(X))
mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
mark#(isNePal(X)) -> c_14(mark#(X))
- Weak TRS:
a__U11(X) -> U11(X)
a__U11(tt()) -> a__U12(tt())
a__U12(X) -> U12(X)
a__U12(tt()) -> tt()
a____(X,nil()) -> mark(X)
a____(X1,X2) -> __(X1,X2)
a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z)))
a____(nil(),X) -> mark(X)
a__isNePal(X) -> isNePal(X)
a__isNePal(__(I,__(P,I))) -> a__U11(tt())
mark(U11(X)) -> a__U11(mark(X))
mark(U12(X)) -> a__U12(mark(X))
mark(__(X1,X2)) -> a____(mark(X1),mark(X2))
mark(isNePal(X)) -> a__isNePal(mark(X))
mark(nil()) -> nil()
mark(tt()) -> tt()
- Signature:
{a__U11/1,a__U12/1,a____/2,a__isNePal/1,mark/1,a__U11#/1,a__U12#/1,a____#/2,a__isNePal#/1,mark#/1} / {U11/1
,U12/1,__/2,isNePal/1,nil/0,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/5,c_8/1,c_9/0,c_10/1,c_11/1,c_12/1
,c_13/3,c_14/1,c_15/0,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a____#,a__isNePal#
,mark#} and constructors {U11,U12,__,isNePal,nil,tt}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(a__U11) = {1},
uargs(a__U12) = {1},
uargs(a____) = {1,2},
uargs(a__isNePal) = {1},
uargs(a____#) = {1,2},
uargs(c_5) = {1},
uargs(c_7) = {1,2,3,4,5},
uargs(c_8) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1,2,3},
uargs(c_14) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(U11) = [0]
p(U12) = [0]
p(__) = [0]
p(a__U11) = [1] x1 + [0]
p(a__U12) = [1] x1 + [0]
p(a____) = [1] x1 + [1] x2 + [0]
p(a__isNePal) = [1] x1 + [0]
p(isNePal) = [0]
p(mark) = [0]
p(nil) = [0]
p(tt) = [0]
p(a__U11#) = [0]
p(a__U12#) = [0]
p(a____#) = [1] x1 + [1] x2 + [4]
p(a__isNePal#) = [0]
p(mark#) = [2]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [1] x1 + [0]
p(c_6) = [0]
p(c_7) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0]
p(c_8) = [1] x1 + [0]
p(c_9) = [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [1] x2 + [1] x3 + [2]
p(c_14) = [1] x1 + [4]
p(c_15) = [0]
p(c_16) = [1]
Following rules are strictly oriented:
a____#(X,nil()) = [1] X + [4]
> [2]
= c_5(mark#(X))
a____#(nil(),X) = [1] X + [4]
> [2]
= c_8(mark#(X))
Following rules are (at-least) weakly oriented:
a____#(__(X,Y),Z) = [1] Z + [4]
>= [14]
= c_7(a____#(mark(X),a____(mark(Y),mark(Z))),mark#(X),a____#(mark(Y),mark(Z)),mark#(Y),mark#(Z))
mark#(U11(X)) = [2]
>= [2]
= c_11(mark#(X))
mark#(U12(X)) = [2]
>= [2]
= c_12(mark#(X))
mark#(__(X1,X2)) = [2]
>= [10]
= c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
mark#(isNePal(X)) = [2]
>= [6]
= c_14(mark#(X))
a__U11(X) = [1] X + [0]
>= [0]
= U11(X)
a__U11(tt()) = [0]
>= [0]
= a__U12(tt())
a__U12(X) = [1] X + [0]
>= [0]
= U12(X)
a__U12(tt()) = [0]
>= [0]
= tt()
a____(X,nil()) = [1] X + [0]
>= [0]
= mark(X)
a____(X1,X2) = [1] X1 + [1] X2 + [0]
>= [0]
= __(X1,X2)
a____(__(X,Y),Z) = [1] Z + [0]
>= [0]
= a____(mark(X),a____(mark(Y),mark(Z)))
a____(nil(),X) = [1] X + [0]
>= [0]
= mark(X)
a__isNePal(X) = [1] X + [0]
>= [0]
= isNePal(X)
a__isNePal(__(I,__(P,I))) = [0]
>= [0]
= a__U11(tt())
mark(U11(X)) = [0]
>= [0]
= a__U11(mark(X))
mark(U12(X)) = [0]
>= [0]
= a__U12(mark(X))
mark(__(X1,X2)) = [0]
>= [0]
= a____(mark(X1),mark(X2))
mark(isNePal(X)) = [0]
>= [0]
= a__isNePal(mark(X))
mark(nil()) = [0]
>= [0]
= nil()
mark(tt()) = [0]
>= [0]
= tt()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 8: Failure MAYBE
+ Considered Problem:
- Strict DPs:
a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z)))
,mark#(X)
,a____#(mark(Y),mark(Z))
,mark#(Y)
,mark#(Z))
mark#(U11(X)) -> c_11(mark#(X))
mark#(U12(X)) -> c_12(mark#(X))
mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
mark#(isNePal(X)) -> c_14(mark#(X))
- Weak DPs:
a____#(X,nil()) -> c_5(mark#(X))
a____#(nil(),X) -> c_8(mark#(X))
- Weak TRS:
a__U11(X) -> U11(X)
a__U11(tt()) -> a__U12(tt())
a__U12(X) -> U12(X)
a__U12(tt()) -> tt()
a____(X,nil()) -> mark(X)
a____(X1,X2) -> __(X1,X2)
a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z)))
a____(nil(),X) -> mark(X)
a__isNePal(X) -> isNePal(X)
a__isNePal(__(I,__(P,I))) -> a__U11(tt())
mark(U11(X)) -> a__U11(mark(X))
mark(U12(X)) -> a__U12(mark(X))
mark(__(X1,X2)) -> a____(mark(X1),mark(X2))
mark(isNePal(X)) -> a__isNePal(mark(X))
mark(nil()) -> nil()
mark(tt()) -> tt()
- Signature:
{a__U11/1,a__U12/1,a____/2,a__isNePal/1,mark/1,a__U11#/1,a__U12#/1,a____#/2,a__isNePal#/1,mark#/1} / {U11/1
,U12/1,__/2,isNePal/1,nil/0,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/5,c_8/1,c_9/0,c_10/1,c_11/1,c_12/1
,c_13/3,c_14/1,c_15/0,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a____#,a__isNePal#
,mark#} and constructors {U11,U12,__,isNePal,nil,tt}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
MAYBE