MAYBE
* Step 1: InnermostRuleRemoval MAYBE
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U11(tt()) -> tt()
U21(tt()) -> tt()
U31(tt()) -> tt()
U41(tt(),V2) -> U42(isNatIList(activate(V2)))
U42(tt()) -> tt()
U51(tt(),V2) -> U52(isNatList(activate(V2)))
U52(tt()) -> tt()
U61(tt(),L,N) -> U62(isNat(activate(N)),activate(L))
U62(tt(),L) -> s(length(activate(L)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__length(X)) -> length(X)
activate(n__nil()) -> nil()
activate(n__s(X)) -> s(X)
activate(n__zeros()) -> zeros()
cons(X1,X2) -> n__cons(X1,X2)
isNat(n__0()) -> tt()
isNat(n__length(V1)) -> U11(isNatList(activate(V1)))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNatIList(V) -> U31(isNatList(activate(V)))
isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2))
isNatIList(n__zeros()) -> tt()
isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2))
isNatList(n__nil()) -> tt()
length(X) -> n__length(X)
length(cons(N,L)) -> U61(isNatList(activate(L)),activate(L),N)
length(nil()) -> 0()
nil() -> n__nil()
s(X) -> n__s(X)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1
,isNatList/1,length/1,nil/0,s/1,zeros/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0,n__s/1,n__zeros/0,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U21,U31,U41,U42,U51,U52,U61,U62,activate,cons,isNat
,isNatIList,isNatList,length,nil,s,zeros} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__zeros,tt}
+ Applied Processor:
InnermostRuleRemoval
+ Details:
Arguments of following rules are not normal-forms.
length(cons(N,L)) -> U61(isNatList(activate(L)),activate(L),N)
length(nil()) -> 0()
All above mentioned rules can be savely removed.
* Step 2: DependencyPairs MAYBE
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U11(tt()) -> tt()
U21(tt()) -> tt()
U31(tt()) -> tt()
U41(tt(),V2) -> U42(isNatIList(activate(V2)))
U42(tt()) -> tt()
U51(tt(),V2) -> U52(isNatList(activate(V2)))
U52(tt()) -> tt()
U61(tt(),L,N) -> U62(isNat(activate(N)),activate(L))
U62(tt(),L) -> s(length(activate(L)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__length(X)) -> length(X)
activate(n__nil()) -> nil()
activate(n__s(X)) -> s(X)
activate(n__zeros()) -> zeros()
cons(X1,X2) -> n__cons(X1,X2)
isNat(n__0()) -> tt()
isNat(n__length(V1)) -> U11(isNatList(activate(V1)))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNatIList(V) -> U31(isNatList(activate(V)))
isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2))
isNatIList(n__zeros()) -> tt()
isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2))
isNatList(n__nil()) -> tt()
length(X) -> n__length(X)
nil() -> n__nil()
s(X) -> n__s(X)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1
,isNatList/1,length/1,nil/0,s/1,zeros/0} / {n__0/0,n__cons/2,n__length/1,n__nil/0,n__s/1,n__zeros/0,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U21,U31,U41,U42,U51,U52,U61,U62,activate,cons,isNat
,isNatIList,isNatList,length,nil,s,zeros} and constructors {n__0,n__cons,n__length,n__nil,n__s,n__zeros,tt}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
0#() -> c_1()
U11#(tt()) -> c_2()
U21#(tt()) -> c_3()
U31#(tt()) -> c_4()
U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2))
U42#(tt()) -> c_6()
U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2))
U52#(tt()) -> c_8()
U61#(tt(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L))
U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L))
activate#(X) -> c_11()
activate#(n__0()) -> c_12(0#())
activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2))
activate#(n__length(X)) -> c_14(length#(X))
activate#(n__nil()) -> c_15(nil#())
activate#(n__s(X)) -> c_16(s#(X))
activate#(n__zeros()) -> c_17(zeros#())
cons#(X1,X2) -> c_18()
isNat#(n__0()) -> c_19()
isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1))
isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V))
isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatIList#(n__zeros()) -> c_24()
isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatList#(n__nil()) -> c_26()
length#(X) -> c_27()
nil#() -> c_28()
s#(X) -> c_29()
zeros#() -> c_30(cons#(0(),n__zeros()),0#())
zeros#() -> c_31()
Weak DPs
and mark the set of starting terms.
* Step 3: UsableRules MAYBE
+ Considered Problem:
- Strict DPs:
0#() -> c_1()
U11#(tt()) -> c_2()
U21#(tt()) -> c_3()
U31#(tt()) -> c_4()
U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2))
U42#(tt()) -> c_6()
U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2))
U52#(tt()) -> c_8()
U61#(tt(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L))
U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L))
activate#(X) -> c_11()
activate#(n__0()) -> c_12(0#())
activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2))
activate#(n__length(X)) -> c_14(length#(X))
activate#(n__nil()) -> c_15(nil#())
activate#(n__s(X)) -> c_16(s#(X))
activate#(n__zeros()) -> c_17(zeros#())
cons#(X1,X2) -> c_18()
isNat#(n__0()) -> c_19()
isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1))
isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V))
isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatIList#(n__zeros()) -> c_24()
isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatList#(n__nil()) -> c_26()
length#(X) -> c_27()
nil#() -> c_28()
s#(X) -> c_29()
zeros#() -> c_30(cons#(0(),n__zeros()),0#())
zeros#() -> c_31()
- Weak TRS:
0() -> n__0()
U11(tt()) -> tt()
U21(tt()) -> tt()
U31(tt()) -> tt()
U41(tt(),V2) -> U42(isNatIList(activate(V2)))
U42(tt()) -> tt()
U51(tt(),V2) -> U52(isNatList(activate(V2)))
U52(tt()) -> tt()
U61(tt(),L,N) -> U62(isNat(activate(N)),activate(L))
U62(tt(),L) -> s(length(activate(L)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__length(X)) -> length(X)
activate(n__nil()) -> nil()
activate(n__s(X)) -> s(X)
activate(n__zeros()) -> zeros()
cons(X1,X2) -> n__cons(X1,X2)
isNat(n__0()) -> tt()
isNat(n__length(V1)) -> U11(isNatList(activate(V1)))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNatIList(V) -> U31(isNatList(activate(V)))
isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2))
isNatIList(n__zeros()) -> tt()
isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2))
isNatList(n__nil()) -> tt()
length(X) -> n__length(X)
nil() -> n__nil()
s(X) -> n__s(X)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1
,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2
,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2
,n__length/1,n__nil/0,n__s/1,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/3,c_8/0,c_9/4,c_10/3
,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/3,c_21/3,c_22/3,c_23/4,c_24/0,c_25/4
,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate#
,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil
,n__s,n__zeros,tt}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
0() -> n__0()
U11(tt()) -> tt()
U21(tt()) -> tt()
U31(tt()) -> tt()
U41(tt(),V2) -> U42(isNatIList(activate(V2)))
U42(tt()) -> tt()
U51(tt(),V2) -> U52(isNatList(activate(V2)))
U52(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__length(X)) -> length(X)
activate(n__nil()) -> nil()
activate(n__s(X)) -> s(X)
activate(n__zeros()) -> zeros()
cons(X1,X2) -> n__cons(X1,X2)
isNat(n__0()) -> tt()
isNat(n__length(V1)) -> U11(isNatList(activate(V1)))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNatIList(V) -> U31(isNatList(activate(V)))
isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2))
isNatIList(n__zeros()) -> tt()
isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2))
isNatList(n__nil()) -> tt()
length(X) -> n__length(X)
nil() -> n__nil()
s(X) -> n__s(X)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
0#() -> c_1()
U11#(tt()) -> c_2()
U21#(tt()) -> c_3()
U31#(tt()) -> c_4()
U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2))
U42#(tt()) -> c_6()
U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2))
U52#(tt()) -> c_8()
U61#(tt(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L))
U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L))
activate#(X) -> c_11()
activate#(n__0()) -> c_12(0#())
activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2))
activate#(n__length(X)) -> c_14(length#(X))
activate#(n__nil()) -> c_15(nil#())
activate#(n__s(X)) -> c_16(s#(X))
activate#(n__zeros()) -> c_17(zeros#())
cons#(X1,X2) -> c_18()
isNat#(n__0()) -> c_19()
isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1))
isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V))
isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatIList#(n__zeros()) -> c_24()
isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatList#(n__nil()) -> c_26()
length#(X) -> c_27()
nil#() -> c_28()
s#(X) -> c_29()
zeros#() -> c_30(cons#(0(),n__zeros()),0#())
zeros#() -> c_31()
* Step 4: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
0#() -> c_1()
U11#(tt()) -> c_2()
U21#(tt()) -> c_3()
U31#(tt()) -> c_4()
U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2))
U42#(tt()) -> c_6()
U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2))
U52#(tt()) -> c_8()
U61#(tt(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L))
U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L))
activate#(X) -> c_11()
activate#(n__0()) -> c_12(0#())
activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2))
activate#(n__length(X)) -> c_14(length#(X))
activate#(n__nil()) -> c_15(nil#())
activate#(n__s(X)) -> c_16(s#(X))
activate#(n__zeros()) -> c_17(zeros#())
cons#(X1,X2) -> c_18()
isNat#(n__0()) -> c_19()
isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1))
isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V))
isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatIList#(n__zeros()) -> c_24()
isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatList#(n__nil()) -> c_26()
length#(X) -> c_27()
nil#() -> c_28()
s#(X) -> c_29()
zeros#() -> c_30(cons#(0(),n__zeros()),0#())
zeros#() -> c_31()
- Weak TRS:
0() -> n__0()
U11(tt()) -> tt()
U21(tt()) -> tt()
U31(tt()) -> tt()
U41(tt(),V2) -> U42(isNatIList(activate(V2)))
U42(tt()) -> tt()
U51(tt(),V2) -> U52(isNatList(activate(V2)))
U52(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__length(X)) -> length(X)
activate(n__nil()) -> nil()
activate(n__s(X)) -> s(X)
activate(n__zeros()) -> zeros()
cons(X1,X2) -> n__cons(X1,X2)
isNat(n__0()) -> tt()
isNat(n__length(V1)) -> U11(isNatList(activate(V1)))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNatIList(V) -> U31(isNatList(activate(V)))
isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2))
isNatIList(n__zeros()) -> tt()
isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2))
isNatList(n__nil()) -> tt()
length(X) -> n__length(X)
nil() -> n__nil()
s(X) -> n__s(X)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1
,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2
,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2
,n__length/1,n__nil/0,n__s/1,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/3,c_8/0,c_9/4,c_10/3
,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/3,c_21/3,c_22/3,c_23/4,c_24/0,c_25/4
,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate#
,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil
,n__s,n__zeros,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,3,4,6,8,11,18,19,24,26,27,28,29,31}
by application of
Pre({1,2,3,4,6,8,11,18,19,24,26,27,28,29,31}) = {5,7,9,10,12,13,14,15,16,17,20,21,22,23,25,30}.
Here rules are labelled as follows:
1: 0#() -> c_1()
2: U11#(tt()) -> c_2()
3: U21#(tt()) -> c_3()
4: U31#(tt()) -> c_4()
5: U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2))
6: U42#(tt()) -> c_6()
7: U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2))
8: U52#(tt()) -> c_8()
9: U61#(tt(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L))
10: U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L))
11: activate#(X) -> c_11()
12: activate#(n__0()) -> c_12(0#())
13: activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2))
14: activate#(n__length(X)) -> c_14(length#(X))
15: activate#(n__nil()) -> c_15(nil#())
16: activate#(n__s(X)) -> c_16(s#(X))
17: activate#(n__zeros()) -> c_17(zeros#())
18: cons#(X1,X2) -> c_18()
19: isNat#(n__0()) -> c_19()
20: isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1))
21: isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
22: isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V))
23: isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
24: isNatIList#(n__zeros()) -> c_24()
25: isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
26: isNatList#(n__nil()) -> c_26()
27: length#(X) -> c_27()
28: nil#() -> c_28()
29: s#(X) -> c_29()
30: zeros#() -> c_30(cons#(0(),n__zeros()),0#())
31: zeros#() -> c_31()
* Step 5: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2))
U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2))
U61#(tt(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L))
U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L))
activate#(n__0()) -> c_12(0#())
activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2))
activate#(n__length(X)) -> c_14(length#(X))
activate#(n__nil()) -> c_15(nil#())
activate#(n__s(X)) -> c_16(s#(X))
activate#(n__zeros()) -> c_17(zeros#())
isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1))
isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V))
isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
zeros#() -> c_30(cons#(0(),n__zeros()),0#())
- Weak DPs:
0#() -> c_1()
U11#(tt()) -> c_2()
U21#(tt()) -> c_3()
U31#(tt()) -> c_4()
U42#(tt()) -> c_6()
U52#(tt()) -> c_8()
activate#(X) -> c_11()
cons#(X1,X2) -> c_18()
isNat#(n__0()) -> c_19()
isNatIList#(n__zeros()) -> c_24()
isNatList#(n__nil()) -> c_26()
length#(X) -> c_27()
nil#() -> c_28()
s#(X) -> c_29()
zeros#() -> c_31()
- Weak TRS:
0() -> n__0()
U11(tt()) -> tt()
U21(tt()) -> tt()
U31(tt()) -> tt()
U41(tt(),V2) -> U42(isNatIList(activate(V2)))
U42(tt()) -> tt()
U51(tt(),V2) -> U52(isNatList(activate(V2)))
U52(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__length(X)) -> length(X)
activate(n__nil()) -> nil()
activate(n__s(X)) -> s(X)
activate(n__zeros()) -> zeros()
cons(X1,X2) -> n__cons(X1,X2)
isNat(n__0()) -> tt()
isNat(n__length(V1)) -> U11(isNatList(activate(V1)))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNatIList(V) -> U31(isNatList(activate(V)))
isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2))
isNatIList(n__zeros()) -> tt()
isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2))
isNatList(n__nil()) -> tt()
length(X) -> n__length(X)
nil() -> n__nil()
s(X) -> n__s(X)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1
,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2
,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2
,n__length/1,n__nil/0,n__s/1,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/3,c_8/0,c_9/4,c_10/3
,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/3,c_21/3,c_22/3,c_23/4,c_24/0,c_25/4
,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate#
,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil
,n__s,n__zeros,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{5,6,7,8,9,16}
by application of
Pre({5,6,7,8,9,16}) = {1,2,3,4,10,11,12,13,14,15}.
Here rules are labelled as follows:
1: U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2))
2: U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2))
3: U61#(tt(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L))
4: U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L))
5: activate#(n__0()) -> c_12(0#())
6: activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2))
7: activate#(n__length(X)) -> c_14(length#(X))
8: activate#(n__nil()) -> c_15(nil#())
9: activate#(n__s(X)) -> c_16(s#(X))
10: activate#(n__zeros()) -> c_17(zeros#())
11: isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1))
12: isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
13: isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V))
14: isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
15: isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
16: zeros#() -> c_30(cons#(0(),n__zeros()),0#())
17: 0#() -> c_1()
18: U11#(tt()) -> c_2()
19: U21#(tt()) -> c_3()
20: U31#(tt()) -> c_4()
21: U42#(tt()) -> c_6()
22: U52#(tt()) -> c_8()
23: activate#(X) -> c_11()
24: cons#(X1,X2) -> c_18()
25: isNat#(n__0()) -> c_19()
26: isNatIList#(n__zeros()) -> c_24()
27: isNatList#(n__nil()) -> c_26()
28: length#(X) -> c_27()
29: nil#() -> c_28()
30: s#(X) -> c_29()
31: zeros#() -> c_31()
* Step 6: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2))
U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2))
U61#(tt(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L))
U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L))
activate#(n__zeros()) -> c_17(zeros#())
isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1))
isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V))
isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
- Weak DPs:
0#() -> c_1()
U11#(tt()) -> c_2()
U21#(tt()) -> c_3()
U31#(tt()) -> c_4()
U42#(tt()) -> c_6()
U52#(tt()) -> c_8()
activate#(X) -> c_11()
activate#(n__0()) -> c_12(0#())
activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2))
activate#(n__length(X)) -> c_14(length#(X))
activate#(n__nil()) -> c_15(nil#())
activate#(n__s(X)) -> c_16(s#(X))
cons#(X1,X2) -> c_18()
isNat#(n__0()) -> c_19()
isNatIList#(n__zeros()) -> c_24()
isNatList#(n__nil()) -> c_26()
length#(X) -> c_27()
nil#() -> c_28()
s#(X) -> c_29()
zeros#() -> c_30(cons#(0(),n__zeros()),0#())
zeros#() -> c_31()
- Weak TRS:
0() -> n__0()
U11(tt()) -> tt()
U21(tt()) -> tt()
U31(tt()) -> tt()
U41(tt(),V2) -> U42(isNatIList(activate(V2)))
U42(tt()) -> tt()
U51(tt(),V2) -> U52(isNatList(activate(V2)))
U52(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__length(X)) -> length(X)
activate(n__nil()) -> nil()
activate(n__s(X)) -> s(X)
activate(n__zeros()) -> zeros()
cons(X1,X2) -> n__cons(X1,X2)
isNat(n__0()) -> tt()
isNat(n__length(V1)) -> U11(isNatList(activate(V1)))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNatIList(V) -> U31(isNatList(activate(V)))
isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2))
isNatIList(n__zeros()) -> tt()
isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2))
isNatList(n__nil()) -> tt()
length(X) -> n__length(X)
nil() -> n__nil()
s(X) -> n__s(X)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1
,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2
,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2
,n__length/1,n__nil/0,n__s/1,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/3,c_8/0,c_9/4,c_10/3
,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/3,c_21/3,c_22/3,c_23/4,c_24/0,c_25/4
,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate#
,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil
,n__s,n__zeros,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{5}
by application of
Pre({5}) = {1,2,3,4,6,7,8,9,10}.
Here rules are labelled as follows:
1: U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2))
2: U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2))
3: U61#(tt(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L))
4: U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L))
5: activate#(n__zeros()) -> c_17(zeros#())
6: isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1))
7: isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
8: isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V))
9: isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
10: isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
11: 0#() -> c_1()
12: U11#(tt()) -> c_2()
13: U21#(tt()) -> c_3()
14: U31#(tt()) -> c_4()
15: U42#(tt()) -> c_6()
16: U52#(tt()) -> c_8()
17: activate#(X) -> c_11()
18: activate#(n__0()) -> c_12(0#())
19: activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2))
20: activate#(n__length(X)) -> c_14(length#(X))
21: activate#(n__nil()) -> c_15(nil#())
22: activate#(n__s(X)) -> c_16(s#(X))
23: cons#(X1,X2) -> c_18()
24: isNat#(n__0()) -> c_19()
25: isNatIList#(n__zeros()) -> c_24()
26: isNatList#(n__nil()) -> c_26()
27: length#(X) -> c_27()
28: nil#() -> c_28()
29: s#(X) -> c_29()
30: zeros#() -> c_30(cons#(0(),n__zeros()),0#())
31: zeros#() -> c_31()
* Step 7: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2))
U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2))
U61#(tt(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L))
U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L))
isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1))
isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V))
isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
- Weak DPs:
0#() -> c_1()
U11#(tt()) -> c_2()
U21#(tt()) -> c_3()
U31#(tt()) -> c_4()
U42#(tt()) -> c_6()
U52#(tt()) -> c_8()
activate#(X) -> c_11()
activate#(n__0()) -> c_12(0#())
activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2))
activate#(n__length(X)) -> c_14(length#(X))
activate#(n__nil()) -> c_15(nil#())
activate#(n__s(X)) -> c_16(s#(X))
activate#(n__zeros()) -> c_17(zeros#())
cons#(X1,X2) -> c_18()
isNat#(n__0()) -> c_19()
isNatIList#(n__zeros()) -> c_24()
isNatList#(n__nil()) -> c_26()
length#(X) -> c_27()
nil#() -> c_28()
s#(X) -> c_29()
zeros#() -> c_30(cons#(0(),n__zeros()),0#())
zeros#() -> c_31()
- Weak TRS:
0() -> n__0()
U11(tt()) -> tt()
U21(tt()) -> tt()
U31(tt()) -> tt()
U41(tt(),V2) -> U42(isNatIList(activate(V2)))
U42(tt()) -> tt()
U51(tt(),V2) -> U52(isNatList(activate(V2)))
U52(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__length(X)) -> length(X)
activate(n__nil()) -> nil()
activate(n__s(X)) -> s(X)
activate(n__zeros()) -> zeros()
cons(X1,X2) -> n__cons(X1,X2)
isNat(n__0()) -> tt()
isNat(n__length(V1)) -> U11(isNatList(activate(V1)))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNatIList(V) -> U31(isNatList(activate(V)))
isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2))
isNatIList(n__zeros()) -> tt()
isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2))
isNatList(n__nil()) -> tt()
length(X) -> n__length(X)
nil() -> n__nil()
s(X) -> n__s(X)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1
,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2
,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2
,n__length/1,n__nil/0,n__s/1,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/3,c_8/0,c_9/4,c_10/3
,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/3,c_21/3,c_22/3,c_23/4,c_24/0,c_25/4
,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate#
,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil
,n__s,n__zeros,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{4}
by application of
Pre({4}) = {3}.
Here rules are labelled as follows:
1: U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2))
2: U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2))
3: U61#(tt(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L))
4: U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L))
5: isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1))
6: isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
7: isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V))
8: isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
9: isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
10: 0#() -> c_1()
11: U11#(tt()) -> c_2()
12: U21#(tt()) -> c_3()
13: U31#(tt()) -> c_4()
14: U42#(tt()) -> c_6()
15: U52#(tt()) -> c_8()
16: activate#(X) -> c_11()
17: activate#(n__0()) -> c_12(0#())
18: activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2))
19: activate#(n__length(X)) -> c_14(length#(X))
20: activate#(n__nil()) -> c_15(nil#())
21: activate#(n__s(X)) -> c_16(s#(X))
22: activate#(n__zeros()) -> c_17(zeros#())
23: cons#(X1,X2) -> c_18()
24: isNat#(n__0()) -> c_19()
25: isNatIList#(n__zeros()) -> c_24()
26: isNatList#(n__nil()) -> c_26()
27: length#(X) -> c_27()
28: nil#() -> c_28()
29: s#(X) -> c_29()
30: zeros#() -> c_30(cons#(0(),n__zeros()),0#())
31: zeros#() -> c_31()
* Step 8: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2))
U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2))
U61#(tt(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L))
isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1))
isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V))
isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
- Weak DPs:
0#() -> c_1()
U11#(tt()) -> c_2()
U21#(tt()) -> c_3()
U31#(tt()) -> c_4()
U42#(tt()) -> c_6()
U52#(tt()) -> c_8()
U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L))
activate#(X) -> c_11()
activate#(n__0()) -> c_12(0#())
activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2))
activate#(n__length(X)) -> c_14(length#(X))
activate#(n__nil()) -> c_15(nil#())
activate#(n__s(X)) -> c_16(s#(X))
activate#(n__zeros()) -> c_17(zeros#())
cons#(X1,X2) -> c_18()
isNat#(n__0()) -> c_19()
isNatIList#(n__zeros()) -> c_24()
isNatList#(n__nil()) -> c_26()
length#(X) -> c_27()
nil#() -> c_28()
s#(X) -> c_29()
zeros#() -> c_30(cons#(0(),n__zeros()),0#())
zeros#() -> c_31()
- Weak TRS:
0() -> n__0()
U11(tt()) -> tt()
U21(tt()) -> tt()
U31(tt()) -> tt()
U41(tt(),V2) -> U42(isNatIList(activate(V2)))
U42(tt()) -> tt()
U51(tt(),V2) -> U52(isNatList(activate(V2)))
U52(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__length(X)) -> length(X)
activate(n__nil()) -> nil()
activate(n__s(X)) -> s(X)
activate(n__zeros()) -> zeros()
cons(X1,X2) -> n__cons(X1,X2)
isNat(n__0()) -> tt()
isNat(n__length(V1)) -> U11(isNatList(activate(V1)))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNatIList(V) -> U31(isNatList(activate(V)))
isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2))
isNatIList(n__zeros()) -> tt()
isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2))
isNatList(n__nil()) -> tt()
length(X) -> n__length(X)
nil() -> n__nil()
s(X) -> n__s(X)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1
,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2
,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2
,n__length/1,n__nil/0,n__s/1,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/3,c_8/0,c_9/4,c_10/3
,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/3,c_21/3,c_22/3,c_23/4,c_24/0,c_25/4
,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate#
,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil
,n__s,n__zeros,tt}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2))
-->_3 activate#(n__zeros()) -> c_17(zeros#()):22
-->_3 activate#(n__s(X)) -> c_16(s#(X)):21
-->_3 activate#(n__nil()) -> c_15(nil#()):20
-->_3 activate#(n__length(X)) -> c_14(length#(X)):19
-->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18
-->_3 activate#(n__0()) -> c_12(0#()):17
-->_2 isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):7
-->_2 isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)):6
-->_2 isNatIList#(n__zeros()) -> c_24():25
-->_3 activate#(X) -> c_11():16
-->_1 U42#(tt()) -> c_6():13
2:S:U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2))
-->_3 activate#(n__zeros()) -> c_17(zeros#()):22
-->_3 activate#(n__s(X)) -> c_16(s#(X)):21
-->_3 activate#(n__nil()) -> c_15(nil#()):20
-->_3 activate#(n__length(X)) -> c_14(length#(X)):19
-->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18
-->_3 activate#(n__0()) -> c_12(0#()):17
-->_2 isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):8
-->_2 isNatList#(n__nil()) -> c_26():26
-->_3 activate#(X) -> c_11():16
-->_1 U52#(tt()) -> c_8():14
3:S:U61#(tt(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L))
,isNat#(activate(N))
,activate#(N)
,activate#(L))
-->_4 activate#(n__zeros()) -> c_17(zeros#()):22
-->_3 activate#(n__zeros()) -> c_17(zeros#()):22
-->_4 activate#(n__s(X)) -> c_16(s#(X)):21
-->_3 activate#(n__s(X)) -> c_16(s#(X)):21
-->_4 activate#(n__nil()) -> c_15(nil#()):20
-->_3 activate#(n__nil()) -> c_15(nil#()):20
-->_4 activate#(n__length(X)) -> c_14(length#(X)):19
-->_3 activate#(n__length(X)) -> c_14(length#(X)):19
-->_4 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18
-->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18
-->_4 activate#(n__0()) -> c_12(0#()):17
-->_3 activate#(n__0()) -> c_12(0#()):17
-->_1 U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L)):15
-->_2 isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):5
-->_2 isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):4
-->_2 isNat#(n__0()) -> c_19():24
-->_4 activate#(X) -> c_11():16
-->_3 activate#(X) -> c_11():16
4:S:isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1))
-->_3 activate#(n__zeros()) -> c_17(zeros#()):22
-->_3 activate#(n__s(X)) -> c_16(s#(X)):21
-->_3 activate#(n__nil()) -> c_15(nil#()):20
-->_3 activate#(n__length(X)) -> c_14(length#(X)):19
-->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18
-->_3 activate#(n__0()) -> c_12(0#()):17
-->_2 isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):8
-->_2 isNatList#(n__nil()) -> c_26():26
-->_3 activate#(X) -> c_11():16
-->_1 U11#(tt()) -> c_2():10
5:S:isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
-->_3 activate#(n__zeros()) -> c_17(zeros#()):22
-->_3 activate#(n__s(X)) -> c_16(s#(X)):21
-->_3 activate#(n__nil()) -> c_15(nil#()):20
-->_3 activate#(n__length(X)) -> c_14(length#(X)):19
-->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18
-->_3 activate#(n__0()) -> c_12(0#()):17
-->_2 isNat#(n__0()) -> c_19():24
-->_3 activate#(X) -> c_11():16
-->_1 U21#(tt()) -> c_3():11
-->_2 isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):5
-->_2 isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):4
6:S:isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V))
-->_3 activate#(n__zeros()) -> c_17(zeros#()):22
-->_3 activate#(n__s(X)) -> c_16(s#(X)):21
-->_3 activate#(n__nil()) -> c_15(nil#()):20
-->_3 activate#(n__length(X)) -> c_14(length#(X)):19
-->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18
-->_3 activate#(n__0()) -> c_12(0#()):17
-->_2 isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):8
-->_2 isNatList#(n__nil()) -> c_26():26
-->_3 activate#(X) -> c_11():16
-->_1 U31#(tt()) -> c_4():12
7:S:isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_4 activate#(n__zeros()) -> c_17(zeros#()):22
-->_3 activate#(n__zeros()) -> c_17(zeros#()):22
-->_4 activate#(n__s(X)) -> c_16(s#(X)):21
-->_3 activate#(n__s(X)) -> c_16(s#(X)):21
-->_4 activate#(n__nil()) -> c_15(nil#()):20
-->_3 activate#(n__nil()) -> c_15(nil#()):20
-->_4 activate#(n__length(X)) -> c_14(length#(X)):19
-->_3 activate#(n__length(X)) -> c_14(length#(X)):19
-->_4 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18
-->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18
-->_4 activate#(n__0()) -> c_12(0#()):17
-->_3 activate#(n__0()) -> c_12(0#()):17
-->_2 isNat#(n__0()) -> c_19():24
-->_4 activate#(X) -> c_11():16
-->_3 activate#(X) -> c_11():16
-->_2 isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):5
-->_2 isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):4
-->_1 U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)):1
8:S:isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_4 activate#(n__zeros()) -> c_17(zeros#()):22
-->_3 activate#(n__zeros()) -> c_17(zeros#()):22
-->_4 activate#(n__s(X)) -> c_16(s#(X)):21
-->_3 activate#(n__s(X)) -> c_16(s#(X)):21
-->_4 activate#(n__nil()) -> c_15(nil#()):20
-->_3 activate#(n__nil()) -> c_15(nil#()):20
-->_4 activate#(n__length(X)) -> c_14(length#(X)):19
-->_3 activate#(n__length(X)) -> c_14(length#(X)):19
-->_4 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18
-->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18
-->_4 activate#(n__0()) -> c_12(0#()):17
-->_3 activate#(n__0()) -> c_12(0#()):17
-->_2 isNat#(n__0()) -> c_19():24
-->_4 activate#(X) -> c_11():16
-->_3 activate#(X) -> c_11():16
-->_2 isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):5
-->_2 isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):4
-->_1 U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)):2
9:W:0#() -> c_1()
10:W:U11#(tt()) -> c_2()
11:W:U21#(tt()) -> c_3()
12:W:U31#(tt()) -> c_4()
13:W:U42#(tt()) -> c_6()
14:W:U52#(tt()) -> c_8()
15:W:U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L))
-->_3 activate#(n__zeros()) -> c_17(zeros#()):22
-->_3 activate#(n__s(X)) -> c_16(s#(X)):21
-->_3 activate#(n__nil()) -> c_15(nil#()):20
-->_3 activate#(n__length(X)) -> c_14(length#(X)):19
-->_3 activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2)):18
-->_3 activate#(n__0()) -> c_12(0#()):17
-->_1 s#(X) -> c_29():29
-->_2 length#(X) -> c_27():27
-->_3 activate#(X) -> c_11():16
16:W:activate#(X) -> c_11()
17:W:activate#(n__0()) -> c_12(0#())
-->_1 0#() -> c_1():9
18:W:activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2))
-->_1 cons#(X1,X2) -> c_18():23
19:W:activate#(n__length(X)) -> c_14(length#(X))
-->_1 length#(X) -> c_27():27
20:W:activate#(n__nil()) -> c_15(nil#())
-->_1 nil#() -> c_28():28
21:W:activate#(n__s(X)) -> c_16(s#(X))
-->_1 s#(X) -> c_29():29
22:W:activate#(n__zeros()) -> c_17(zeros#())
-->_1 zeros#() -> c_30(cons#(0(),n__zeros()),0#()):30
-->_1 zeros#() -> c_31():31
23:W:cons#(X1,X2) -> c_18()
24:W:isNat#(n__0()) -> c_19()
25:W:isNatIList#(n__zeros()) -> c_24()
26:W:isNatList#(n__nil()) -> c_26()
27:W:length#(X) -> c_27()
28:W:nil#() -> c_28()
29:W:s#(X) -> c_29()
30:W:zeros#() -> c_30(cons#(0(),n__zeros()),0#())
-->_1 cons#(X1,X2) -> c_18():23
-->_2 0#() -> c_1():9
31:W:zeros#() -> c_31()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
15: U62#(tt(),L) -> c_10(s#(length(activate(L))),length#(activate(L)),activate#(L))
13: U42#(tt()) -> c_6()
25: isNatIList#(n__zeros()) -> c_24()
12: U31#(tt()) -> c_4()
10: U11#(tt()) -> c_2()
14: U52#(tt()) -> c_8()
26: isNatList#(n__nil()) -> c_26()
11: U21#(tt()) -> c_3()
16: activate#(X) -> c_11()
24: isNat#(n__0()) -> c_19()
17: activate#(n__0()) -> c_12(0#())
18: activate#(n__cons(X1,X2)) -> c_13(cons#(X1,X2))
19: activate#(n__length(X)) -> c_14(length#(X))
27: length#(X) -> c_27()
20: activate#(n__nil()) -> c_15(nil#())
28: nil#() -> c_28()
21: activate#(n__s(X)) -> c_16(s#(X))
29: s#(X) -> c_29()
22: activate#(n__zeros()) -> c_17(zeros#())
31: zeros#() -> c_31()
30: zeros#() -> c_30(cons#(0(),n__zeros()),0#())
9: 0#() -> c_1()
23: cons#(X1,X2) -> c_18()
* Step 9: SimplifyRHS MAYBE
+ Considered Problem:
- Strict DPs:
U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2))
U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2))
U61#(tt(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L)),isNat#(activate(N)),activate#(N),activate#(L))
isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1))
isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V))
isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
- Weak TRS:
0() -> n__0()
U11(tt()) -> tt()
U21(tt()) -> tt()
U31(tt()) -> tt()
U41(tt(),V2) -> U42(isNatIList(activate(V2)))
U42(tt()) -> tt()
U51(tt(),V2) -> U52(isNatList(activate(V2)))
U52(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__length(X)) -> length(X)
activate(n__nil()) -> nil()
activate(n__s(X)) -> s(X)
activate(n__zeros()) -> zeros()
cons(X1,X2) -> n__cons(X1,X2)
isNat(n__0()) -> tt()
isNat(n__length(V1)) -> U11(isNatList(activate(V1)))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNatIList(V) -> U31(isNatList(activate(V)))
isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2))
isNatIList(n__zeros()) -> tt()
isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2))
isNatList(n__nil()) -> tt()
length(X) -> n__length(X)
nil() -> n__nil()
s(X) -> n__s(X)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1
,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2
,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2
,n__length/1,n__nil/0,n__s/1,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/3,c_8/0,c_9/4,c_10/3
,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/3,c_21/3,c_22/3,c_23/4,c_24/0,c_25/4
,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate#
,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil
,n__s,n__zeros,tt}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2))
-->_2 isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):7
-->_2 isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V)):6
2:S:U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2))
-->_2 isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):8
3:S:U61#(tt(),L,N) -> c_9(U62#(isNat(activate(N)),activate(L))
,isNat#(activate(N))
,activate#(N)
,activate#(L))
-->_2 isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):5
-->_2 isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):4
4:S:isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1))
-->_2 isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):8
5:S:isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
-->_2 isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):5
-->_2 isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):4
6:S:isNatIList#(V) -> c_22(U31#(isNatList(activate(V))),isNatList#(activate(V)),activate#(V))
-->_2 isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):8
7:S:isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_2 isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):5
-->_2 isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):4
-->_1 U41#(tt(),V2) -> c_5(U42#(isNatIList(activate(V2))),isNatIList#(activate(V2)),activate#(V2)):1
8:S:isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_2 isNat#(n__s(V1)) -> c_21(U21#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):5
-->_2 isNat#(n__length(V1)) -> c_20(U11#(isNatList(activate(V1))),isNatList#(activate(V1)),activate#(V1)):4
-->_1 U51#(tt(),V2) -> c_7(U52#(isNatList(activate(V2))),isNatList#(activate(V2)),activate#(V2)):2
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
U41#(tt(),V2) -> c_5(isNatIList#(activate(V2)))
U51#(tt(),V2) -> c_7(isNatList#(activate(V2)))
U61#(tt(),L,N) -> c_9(isNat#(activate(N)))
isNat#(n__length(V1)) -> c_20(isNatList#(activate(V1)))
isNat#(n__s(V1)) -> c_21(isNat#(activate(V1)))
isNatIList#(V) -> c_22(isNatList#(activate(V)))
isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
* Step 10: UsableRules MAYBE
+ Considered Problem:
- Strict DPs:
U41#(tt(),V2) -> c_5(isNatIList#(activate(V2)))
U51#(tt(),V2) -> c_7(isNatList#(activate(V2)))
U61#(tt(),L,N) -> c_9(isNat#(activate(N)))
isNat#(n__length(V1)) -> c_20(isNatList#(activate(V1)))
isNat#(n__s(V1)) -> c_21(isNat#(activate(V1)))
isNatIList#(V) -> c_22(isNatList#(activate(V)))
isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
- Weak TRS:
0() -> n__0()
U11(tt()) -> tt()
U21(tt()) -> tt()
U31(tt()) -> tt()
U41(tt(),V2) -> U42(isNatIList(activate(V2)))
U42(tt()) -> tt()
U51(tt(),V2) -> U52(isNatList(activate(V2)))
U52(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__length(X)) -> length(X)
activate(n__nil()) -> nil()
activate(n__s(X)) -> s(X)
activate(n__zeros()) -> zeros()
cons(X1,X2) -> n__cons(X1,X2)
isNat(n__0()) -> tt()
isNat(n__length(V1)) -> U11(isNatList(activate(V1)))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNatIList(V) -> U31(isNatList(activate(V)))
isNatIList(n__cons(V1,V2)) -> U41(isNat(activate(V1)),activate(V2))
isNatIList(n__zeros()) -> tt()
isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2))
isNatList(n__nil()) -> tt()
length(X) -> n__length(X)
nil() -> n__nil()
s(X) -> n__s(X)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1
,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2
,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2
,n__length/1,n__nil/0,n__s/1,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/3
,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/1,c_22/1,c_23/2,c_24/0,c_25/2
,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate#
,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil
,n__s,n__zeros,tt}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
0() -> n__0()
U11(tt()) -> tt()
U21(tt()) -> tt()
U51(tt(),V2) -> U52(isNatList(activate(V2)))
U52(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__length(X)) -> length(X)
activate(n__nil()) -> nil()
activate(n__s(X)) -> s(X)
activate(n__zeros()) -> zeros()
cons(X1,X2) -> n__cons(X1,X2)
isNat(n__0()) -> tt()
isNat(n__length(V1)) -> U11(isNatList(activate(V1)))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2))
isNatList(n__nil()) -> tt()
length(X) -> n__length(X)
nil() -> n__nil()
s(X) -> n__s(X)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
U41#(tt(),V2) -> c_5(isNatIList#(activate(V2)))
U51#(tt(),V2) -> c_7(isNatList#(activate(V2)))
U61#(tt(),L,N) -> c_9(isNat#(activate(N)))
isNat#(n__length(V1)) -> c_20(isNatList#(activate(V1)))
isNat#(n__s(V1)) -> c_21(isNat#(activate(V1)))
isNatIList#(V) -> c_22(isNatList#(activate(V)))
isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
* Step 11: NaturalMI MAYBE
+ Considered Problem:
- Strict DPs:
U41#(tt(),V2) -> c_5(isNatIList#(activate(V2)))
U51#(tt(),V2) -> c_7(isNatList#(activate(V2)))
U61#(tt(),L,N) -> c_9(isNat#(activate(N)))
isNat#(n__length(V1)) -> c_20(isNatList#(activate(V1)))
isNat#(n__s(V1)) -> c_21(isNat#(activate(V1)))
isNatIList#(V) -> c_22(isNatList#(activate(V)))
isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
- Weak TRS:
0() -> n__0()
U11(tt()) -> tt()
U21(tt()) -> tt()
U51(tt(),V2) -> U52(isNatList(activate(V2)))
U52(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__length(X)) -> length(X)
activate(n__nil()) -> nil()
activate(n__s(X)) -> s(X)
activate(n__zeros()) -> zeros()
cons(X1,X2) -> n__cons(X1,X2)
isNat(n__0()) -> tt()
isNat(n__length(V1)) -> U11(isNatList(activate(V1)))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2))
isNatList(n__nil()) -> tt()
length(X) -> n__length(X)
nil() -> n__nil()
s(X) -> n__s(X)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1
,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2
,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2
,n__length/1,n__nil/0,n__s/1,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/3
,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/1,c_22/1,c_23/2,c_24/0,c_25/2
,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate#
,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil
,n__s,n__zeros,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
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(c_5) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_20) = {1},
uargs(c_21) = {1},
uargs(c_22) = {1},
uargs(c_23) = {1,2},
uargs(c_25) = {1,2}
Following symbols are considered usable:
{0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#
,nil#,s#,zeros#}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [0]
p(U21) = [0]
p(U31) = [0]
p(U41) = [0]
p(U42) = [0]
p(U51) = [0]
p(U52) = [0]
p(U61) = [0]
p(U62) = [0]
p(activate) = [4]
p(cons) = [0]
p(isNat) = [4]
p(isNatIList) = [0]
p(isNatList) = [0]
p(length) = [0]
p(n__0) = [0]
p(n__cons) = [0]
p(n__length) = [0]
p(n__nil) = [0]
p(n__s) = [1]
p(n__zeros) = [4]
p(nil) = [4]
p(s) = [1]
p(tt) = [0]
p(zeros) = [1]
p(0#) = [4]
p(U11#) = [1]
p(U21#) = [1]
p(U31#) = [1]
p(U41#) = [0]
p(U42#) = [2] x1 + [1]
p(U51#) = [0]
p(U52#) = [1]
p(U61#) = [1] x2 + [1] x3 + [6]
p(U62#) = [2] x2 + [1]
p(activate#) = [0]
p(cons#) = [0]
p(isNat#) = [0]
p(isNatIList#) = [0]
p(isNatList#) = [0]
p(length#) = [0]
p(nil#) = [0]
p(s#) = [2] x1 + [4]
p(zeros#) = [0]
p(c_1) = [1]
p(c_2) = [1]
p(c_3) = [2]
p(c_4) = [0]
p(c_5) = [4] x1 + [0]
p(c_6) = [1]
p(c_7) = [4] x1 + [0]
p(c_8) = [1]
p(c_9) = [2] x1 + [2]
p(c_10) = [4] x3 + [1]
p(c_11) = [1]
p(c_12) = [0]
p(c_13) = [1] x1 + [2]
p(c_14) = [1]
p(c_15) = [0]
p(c_16) = [2] x1 + [0]
p(c_17) = [1]
p(c_18) = [0]
p(c_19) = [1]
p(c_20) = [1] x1 + [0]
p(c_21) = [1] x1 + [0]
p(c_22) = [1] x1 + [0]
p(c_23) = [4] x1 + [1] x2 + [0]
p(c_24) = [0]
p(c_25) = [4] x1 + [2] x2 + [0]
p(c_26) = [1]
p(c_27) = [1]
p(c_28) = [1]
p(c_29) = [0]
p(c_30) = [1] x2 + [0]
p(c_31) = [1]
Following rules are strictly oriented:
U61#(tt(),L,N) = [1] L + [1] N + [6]
> [2]
= c_9(isNat#(activate(N)))
Following rules are (at-least) weakly oriented:
U41#(tt(),V2) = [0]
>= [0]
= c_5(isNatIList#(activate(V2)))
U51#(tt(),V2) = [0]
>= [0]
= c_7(isNatList#(activate(V2)))
isNat#(n__length(V1)) = [0]
>= [0]
= c_20(isNatList#(activate(V1)))
isNat#(n__s(V1)) = [0]
>= [0]
= c_21(isNat#(activate(V1)))
isNatIList#(V) = [0]
>= [0]
= c_22(isNatList#(activate(V)))
isNatIList#(n__cons(V1,V2)) = [0]
>= [0]
= c_23(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNatList#(n__cons(V1,V2)) = [0]
>= [0]
= c_25(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
* Step 12: NaturalMI MAYBE
+ Considered Problem:
- Strict DPs:
U41#(tt(),V2) -> c_5(isNatIList#(activate(V2)))
U51#(tt(),V2) -> c_7(isNatList#(activate(V2)))
isNat#(n__length(V1)) -> c_20(isNatList#(activate(V1)))
isNat#(n__s(V1)) -> c_21(isNat#(activate(V1)))
isNatIList#(V) -> c_22(isNatList#(activate(V)))
isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
- Weak DPs:
U61#(tt(),L,N) -> c_9(isNat#(activate(N)))
- Weak TRS:
0() -> n__0()
U11(tt()) -> tt()
U21(tt()) -> tt()
U51(tt(),V2) -> U52(isNatList(activate(V2)))
U52(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__length(X)) -> length(X)
activate(n__nil()) -> nil()
activate(n__s(X)) -> s(X)
activate(n__zeros()) -> zeros()
cons(X1,X2) -> n__cons(X1,X2)
isNat(n__0()) -> tt()
isNat(n__length(V1)) -> U11(isNatList(activate(V1)))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2))
isNatList(n__nil()) -> tt()
length(X) -> n__length(X)
nil() -> n__nil()
s(X) -> n__s(X)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1
,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2
,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2
,n__length/1,n__nil/0,n__s/1,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/3
,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/1,c_22/1,c_23/2,c_24/0,c_25/2
,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate#
,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil
,n__s,n__zeros,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
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(c_5) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_20) = {1},
uargs(c_21) = {1},
uargs(c_22) = {1},
uargs(c_23) = {1,2},
uargs(c_25) = {1,2}
Following symbols are considered usable:
{0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate#,cons#,isNat#,isNatIList#,isNatList#,length#
,nil#,s#,zeros#}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [4] x1 + [4]
p(U21) = [0]
p(U31) = [4]
p(U41) = [1] x2 + [2]
p(U42) = [1]
p(U51) = [0]
p(U52) = [1] x1 + [0]
p(U61) = [0]
p(U62) = [4] x1 + [1]
p(activate) = [3] x1 + [0]
p(cons) = [2] x1 + [1] x2 + [5]
p(isNat) = [4] x1 + [4]
p(isNatIList) = [2] x1 + [1]
p(isNatList) = [1] x1 + [2]
p(length) = [1] x1 + [1]
p(n__0) = [0]
p(n__cons) = [0]
p(n__length) = [1]
p(n__nil) = [2]
p(n__s) = [1]
p(n__zeros) = [1]
p(nil) = [4]
p(s) = [0]
p(tt) = [0]
p(zeros) = [0]
p(0#) = [4]
p(U11#) = [2] x1 + [0]
p(U21#) = [1] x1 + [1]
p(U31#) = [2]
p(U41#) = [4]
p(U42#) = [1] x1 + [4]
p(U51#) = [0]
p(U52#) = [1] x1 + [1]
p(U61#) = [1] x1 + [4] x3 + [4]
p(U62#) = [1] x1 + [0]
p(activate#) = [0]
p(cons#) = [4] x1 + [4]
p(isNat#) = [0]
p(isNatIList#) = [4]
p(isNatList#) = [0]
p(length#) = [1]
p(nil#) = [0]
p(s#) = [2] x1 + [2]
p(zeros#) = [0]
p(c_1) = [2]
p(c_2) = [1]
p(c_3) = [2]
p(c_4) = [0]
p(c_5) = [1] x1 + [0]
p(c_6) = [1]
p(c_7) = [1] x1 + [0]
p(c_8) = [1]
p(c_9) = [1] x1 + [4]
p(c_10) = [1] x1 + [1]
p(c_11) = [0]
p(c_12) = [1] x1 + [1]
p(c_13) = [1] x1 + [4]
p(c_14) = [2]
p(c_15) = [0]
p(c_16) = [4] x1 + [0]
p(c_17) = [4] x1 + [0]
p(c_18) = [1]
p(c_19) = [1]
p(c_20) = [4] x1 + [0]
p(c_21) = [4] x1 + [0]
p(c_22) = [1] x1 + [3]
p(c_23) = [1] x1 + [1] x2 + [0]
p(c_24) = [0]
p(c_25) = [4] x1 + [2] x2 + [0]
p(c_26) = [0]
p(c_27) = [2]
p(c_28) = [0]
p(c_29) = [0]
p(c_30) = [4] x1 + [4] x2 + [0]
p(c_31) = [1]
Following rules are strictly oriented:
isNatIList#(V) = [4]
> [3]
= c_22(isNatList#(activate(V)))
Following rules are (at-least) weakly oriented:
U41#(tt(),V2) = [4]
>= [4]
= c_5(isNatIList#(activate(V2)))
U51#(tt(),V2) = [0]
>= [0]
= c_7(isNatList#(activate(V2)))
U61#(tt(),L,N) = [4] N + [4]
>= [4]
= c_9(isNat#(activate(N)))
isNat#(n__length(V1)) = [0]
>= [0]
= c_20(isNatList#(activate(V1)))
isNat#(n__s(V1)) = [0]
>= [0]
= c_21(isNat#(activate(V1)))
isNatIList#(n__cons(V1,V2)) = [4]
>= [4]
= c_23(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNatList#(n__cons(V1,V2)) = [0]
>= [0]
= c_25(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
* Step 13: Failure MAYBE
+ Considered Problem:
- Strict DPs:
U41#(tt(),V2) -> c_5(isNatIList#(activate(V2)))
U51#(tt(),V2) -> c_7(isNatList#(activate(V2)))
isNat#(n__length(V1)) -> c_20(isNatList#(activate(V1)))
isNat#(n__s(V1)) -> c_21(isNat#(activate(V1)))
isNatIList#(n__cons(V1,V2)) -> c_23(U41#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
isNatList#(n__cons(V1,V2)) -> c_25(U51#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
- Weak DPs:
U61#(tt(),L,N) -> c_9(isNat#(activate(N)))
isNatIList#(V) -> c_22(isNatList#(activate(V)))
- Weak TRS:
0() -> n__0()
U11(tt()) -> tt()
U21(tt()) -> tt()
U51(tt(),V2) -> U52(isNatList(activate(V2)))
U52(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__length(X)) -> length(X)
activate(n__nil()) -> nil()
activate(n__s(X)) -> s(X)
activate(n__zeros()) -> zeros()
cons(X1,X2) -> n__cons(X1,X2)
isNat(n__0()) -> tt()
isNat(n__length(V1)) -> U11(isNatList(activate(V1)))
isNat(n__s(V1)) -> U21(isNat(activate(V1)))
isNatList(n__cons(V1,V2)) -> U51(isNat(activate(V1)),activate(V2))
isNatList(n__nil()) -> tt()
length(X) -> n__length(X)
nil() -> n__nil()
s(X) -> n__s(X)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{0/0,U11/1,U21/1,U31/1,U41/2,U42/1,U51/2,U52/1,U61/3,U62/2,activate/1,cons/2,isNat/1,isNatIList/1
,isNatList/1,length/1,nil/0,s/1,zeros/0,0#/0,U11#/1,U21#/1,U31#/1,U41#/2,U42#/1,U51#/2,U52#/1,U61#/3,U62#/2
,activate#/1,cons#/2,isNat#/1,isNatIList#/1,isNatList#/1,length#/1,nil#/0,s#/1,zeros#/0} / {n__0/0,n__cons/2
,n__length/1,n__nil/0,n__s/1,n__zeros/0,tt/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/1,c_10/3
,c_11/0,c_12/1,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/0,c_19/0,c_20/1,c_21/1,c_22/1,c_23/2,c_24/0,c_25/2
,c_26/0,c_27/0,c_28/0,c_29/0,c_30/2,c_31/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,U42#,U51#,U52#,U61#,U62#,activate#
,cons#,isNat#,isNatIList#,isNatList#,length#,nil#,s#,zeros#} and constructors {n__0,n__cons,n__length,n__nil
,n__s,n__zeros,tt}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
MAYBE