MAYBE
* Step 1: DependencyPairs MAYBE
+ Considered Problem:
- Strict TRS:
a__U11(X1,X2) -> U11(X1,X2)
a__U11(tt(),N) -> mark(N)
a__U21(X1,X2,X3) -> U21(X1,X2,X3)
a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__and(X1,X2) -> and(X1,X2)
a__and(tt(),X) -> mark(X)
a__isNat(X) -> isNat(X)
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2))
a__isNat(s(V1)) -> a__isNat(V1)
a__plus(N,0()) -> a__U11(a__isNat(N),N)
a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N)
a__plus(X1,X2) -> plus(X1,X2)
mark(0()) -> 0()
mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3)
mark(and(X1,X2)) -> a__and(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(s(X)) -> s(mark(X))
mark(tt()) -> tt()
- Signature:
{a__U11/2,a__U21/3,a__and/2,a__isNat/1,a__plus/2,mark/1} / {0/0,U11/2,U21/3,and/2,isNat/1,plus/2,s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__and,a__isNat,a__plus
,mark} and constructors {0,U11,U21,and,isNat,plus,s,tt}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
a__U11#(X1,X2) -> c_1()
a__U11#(tt(),N) -> c_2(mark#(N))
a__U21#(X1,X2,X3) -> c_3()
a__U21#(tt(),M,N) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M))
a__and#(X1,X2) -> c_5()
a__and#(tt(),X) -> c_6(mark#(X))
a__isNat#(X) -> c_7()
a__isNat#(0()) -> c_8()
a__isNat#(plus(V1,V2)) -> c_9(a__and#(a__isNat(V1),isNat(V2)),a__isNat#(V1))
a__isNat#(s(V1)) -> c_10(a__isNat#(V1))
a__plus#(N,0()) -> c_11(a__U11#(a__isNat(N),N),a__isNat#(N))
a__plus#(N,s(M)) -> c_12(a__U21#(a__and(a__isNat(M),isNat(N)),M,N)
,a__and#(a__isNat(M),isNat(N))
,a__isNat#(M))
a__plus#(X1,X2) -> c_13()
mark#(0()) -> c_14()
mark#(U11(X1,X2)) -> c_15(a__U11#(mark(X1),X2),mark#(X1))
mark#(U21(X1,X2,X3)) -> c_16(a__U21#(mark(X1),X2,X3),mark#(X1))
mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1))
mark#(isNat(X)) -> c_18(a__isNat#(X))
mark#(plus(X1,X2)) -> c_19(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
mark#(s(X)) -> c_20(mark#(X))
mark#(tt()) -> c_21()
Weak DPs
and mark the set of starting terms.
* Step 2: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
a__U11#(X1,X2) -> c_1()
a__U11#(tt(),N) -> c_2(mark#(N))
a__U21#(X1,X2,X3) -> c_3()
a__U21#(tt(),M,N) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M))
a__and#(X1,X2) -> c_5()
a__and#(tt(),X) -> c_6(mark#(X))
a__isNat#(X) -> c_7()
a__isNat#(0()) -> c_8()
a__isNat#(plus(V1,V2)) -> c_9(a__and#(a__isNat(V1),isNat(V2)),a__isNat#(V1))
a__isNat#(s(V1)) -> c_10(a__isNat#(V1))
a__plus#(N,0()) -> c_11(a__U11#(a__isNat(N),N),a__isNat#(N))
a__plus#(N,s(M)) -> c_12(a__U21#(a__and(a__isNat(M),isNat(N)),M,N)
,a__and#(a__isNat(M),isNat(N))
,a__isNat#(M))
a__plus#(X1,X2) -> c_13()
mark#(0()) -> c_14()
mark#(U11(X1,X2)) -> c_15(a__U11#(mark(X1),X2),mark#(X1))
mark#(U21(X1,X2,X3)) -> c_16(a__U21#(mark(X1),X2,X3),mark#(X1))
mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1))
mark#(isNat(X)) -> c_18(a__isNat#(X))
mark#(plus(X1,X2)) -> c_19(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
mark#(s(X)) -> c_20(mark#(X))
mark#(tt()) -> c_21()
- Weak TRS:
a__U11(X1,X2) -> U11(X1,X2)
a__U11(tt(),N) -> mark(N)
a__U21(X1,X2,X3) -> U21(X1,X2,X3)
a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__and(X1,X2) -> and(X1,X2)
a__and(tt(),X) -> mark(X)
a__isNat(X) -> isNat(X)
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2))
a__isNat(s(V1)) -> a__isNat(V1)
a__plus(N,0()) -> a__U11(a__isNat(N),N)
a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N)
a__plus(X1,X2) -> plus(X1,X2)
mark(0()) -> 0()
mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3)
mark(and(X1,X2)) -> a__and(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(s(X)) -> s(mark(X))
mark(tt()) -> tt()
- Signature:
{a__U11/2,a__U21/3,a__and/2,a__isNat/1,a__plus/2,mark/1,a__U11#/2,a__U21#/3,a__and#/2,a__isNat#/1,a__plus#/2
,mark#/1} / {0/0,U11/2,U21/3,and/2,isNat/1,plus/2,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/3,c_5/0,c_6/1,c_7/0,c_8/0
,c_9/2,c_10/1,c_11/2,c_12/3,c_13/0,c_14/0,c_15/2,c_16/2,c_17/2,c_18/1,c_19/3,c_20/1,c_21/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U11#,a__U21#,a__and#,a__isNat#,a__plus#
,mark#} and constructors {0,U11,U21,and,isNat,plus,s,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3,5,7,8,13,14,21}
by application of
Pre({1,3,5,7,8,13,14,21}) = {2,4,6,9,10,11,12,15,16,17,18,19,20}.
Here rules are labelled as follows:
1: a__U11#(X1,X2) -> c_1()
2: a__U11#(tt(),N) -> c_2(mark#(N))
3: a__U21#(X1,X2,X3) -> c_3()
4: a__U21#(tt(),M,N) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M))
5: a__and#(X1,X2) -> c_5()
6: a__and#(tt(),X) -> c_6(mark#(X))
7: a__isNat#(X) -> c_7()
8: a__isNat#(0()) -> c_8()
9: a__isNat#(plus(V1,V2)) -> c_9(a__and#(a__isNat(V1),isNat(V2)),a__isNat#(V1))
10: a__isNat#(s(V1)) -> c_10(a__isNat#(V1))
11: a__plus#(N,0()) -> c_11(a__U11#(a__isNat(N),N),a__isNat#(N))
12: a__plus#(N,s(M)) -> c_12(a__U21#(a__and(a__isNat(M),isNat(N)),M,N)
,a__and#(a__isNat(M),isNat(N))
,a__isNat#(M))
13: a__plus#(X1,X2) -> c_13()
14: mark#(0()) -> c_14()
15: mark#(U11(X1,X2)) -> c_15(a__U11#(mark(X1),X2),mark#(X1))
16: mark#(U21(X1,X2,X3)) -> c_16(a__U21#(mark(X1),X2,X3),mark#(X1))
17: mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1))
18: mark#(isNat(X)) -> c_18(a__isNat#(X))
19: mark#(plus(X1,X2)) -> c_19(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
20: mark#(s(X)) -> c_20(mark#(X))
21: mark#(tt()) -> c_21()
* Step 3: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
a__U11#(tt(),N) -> c_2(mark#(N))
a__U21#(tt(),M,N) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M))
a__and#(tt(),X) -> c_6(mark#(X))
a__isNat#(plus(V1,V2)) -> c_9(a__and#(a__isNat(V1),isNat(V2)),a__isNat#(V1))
a__isNat#(s(V1)) -> c_10(a__isNat#(V1))
a__plus#(N,0()) -> c_11(a__U11#(a__isNat(N),N),a__isNat#(N))
a__plus#(N,s(M)) -> c_12(a__U21#(a__and(a__isNat(M),isNat(N)),M,N)
,a__and#(a__isNat(M),isNat(N))
,a__isNat#(M))
mark#(U11(X1,X2)) -> c_15(a__U11#(mark(X1),X2),mark#(X1))
mark#(U21(X1,X2,X3)) -> c_16(a__U21#(mark(X1),X2,X3),mark#(X1))
mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1))
mark#(isNat(X)) -> c_18(a__isNat#(X))
mark#(plus(X1,X2)) -> c_19(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
mark#(s(X)) -> c_20(mark#(X))
- Weak DPs:
a__U11#(X1,X2) -> c_1()
a__U21#(X1,X2,X3) -> c_3()
a__and#(X1,X2) -> c_5()
a__isNat#(X) -> c_7()
a__isNat#(0()) -> c_8()
a__plus#(X1,X2) -> c_13()
mark#(0()) -> c_14()
mark#(tt()) -> c_21()
- Weak TRS:
a__U11(X1,X2) -> U11(X1,X2)
a__U11(tt(),N) -> mark(N)
a__U21(X1,X2,X3) -> U21(X1,X2,X3)
a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__and(X1,X2) -> and(X1,X2)
a__and(tt(),X) -> mark(X)
a__isNat(X) -> isNat(X)
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2))
a__isNat(s(V1)) -> a__isNat(V1)
a__plus(N,0()) -> a__U11(a__isNat(N),N)
a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N)
a__plus(X1,X2) -> plus(X1,X2)
mark(0()) -> 0()
mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3)
mark(and(X1,X2)) -> a__and(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(s(X)) -> s(mark(X))
mark(tt()) -> tt()
- Signature:
{a__U11/2,a__U21/3,a__and/2,a__isNat/1,a__plus/2,mark/1,a__U11#/2,a__U21#/3,a__and#/2,a__isNat#/1,a__plus#/2
,mark#/1} / {0/0,U11/2,U21/3,and/2,isNat/1,plus/2,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/3,c_5/0,c_6/1,c_7/0,c_8/0
,c_9/2,c_10/1,c_11/2,c_12/3,c_13/0,c_14/0,c_15/2,c_16/2,c_17/2,c_18/1,c_19/3,c_20/1,c_21/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U11#,a__U21#,a__and#,a__isNat#,a__plus#
,mark#} and constructors {0,U11,U21,and,isNat,plus,s,tt}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:a__U11#(tt(),N) -> c_2(mark#(N))
-->_1 mark#(s(X)) -> c_20(mark#(X)):13
-->_1 mark#(plus(X1,X2)) -> c_19(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12
-->_1 mark#(isNat(X)) -> c_18(a__isNat#(X)):11
-->_1 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):10
-->_1 mark#(U21(X1,X2,X3)) -> c_16(a__U21#(mark(X1),X2,X3),mark#(X1)):9
-->_1 mark#(U11(X1,X2)) -> c_15(a__U11#(mark(X1),X2),mark#(X1)):8
-->_1 mark#(tt()) -> c_21():21
-->_1 mark#(0()) -> c_14():20
2:S:a__U21#(tt(),M,N) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M))
-->_3 mark#(s(X)) -> c_20(mark#(X)):13
-->_2 mark#(s(X)) -> c_20(mark#(X)):13
-->_3 mark#(plus(X1,X2)) -> c_19(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12
-->_2 mark#(plus(X1,X2)) -> c_19(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12
-->_3 mark#(isNat(X)) -> c_18(a__isNat#(X)):11
-->_2 mark#(isNat(X)) -> c_18(a__isNat#(X)):11
-->_3 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):10
-->_2 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):10
-->_3 mark#(U21(X1,X2,X3)) -> c_16(a__U21#(mark(X1),X2,X3),mark#(X1)):9
-->_2 mark#(U21(X1,X2,X3)) -> c_16(a__U21#(mark(X1),X2,X3),mark#(X1)):9
-->_3 mark#(U11(X1,X2)) -> c_15(a__U11#(mark(X1),X2),mark#(X1)):8
-->_2 mark#(U11(X1,X2)) -> c_15(a__U11#(mark(X1),X2),mark#(X1)):8
-->_1 a__plus#(N,s(M)) -> c_12(a__U21#(a__and(a__isNat(M),isNat(N)),M,N)
,a__and#(a__isNat(M),isNat(N))
,a__isNat#(M)):7
-->_1 a__plus#(N,0()) -> c_11(a__U11#(a__isNat(N),N),a__isNat#(N)):6
-->_3 mark#(tt()) -> c_21():21
-->_2 mark#(tt()) -> c_21():21
-->_3 mark#(0()) -> c_14():20
-->_2 mark#(0()) -> c_14():20
-->_1 a__plus#(X1,X2) -> c_13():19
3:S:a__and#(tt(),X) -> c_6(mark#(X))
-->_1 mark#(s(X)) -> c_20(mark#(X)):13
-->_1 mark#(plus(X1,X2)) -> c_19(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12
-->_1 mark#(isNat(X)) -> c_18(a__isNat#(X)):11
-->_1 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):10
-->_1 mark#(U21(X1,X2,X3)) -> c_16(a__U21#(mark(X1),X2,X3),mark#(X1)):9
-->_1 mark#(U11(X1,X2)) -> c_15(a__U11#(mark(X1),X2),mark#(X1)):8
-->_1 mark#(tt()) -> c_21():21
-->_1 mark#(0()) -> c_14():20
4:S:a__isNat#(plus(V1,V2)) -> c_9(a__and#(a__isNat(V1),isNat(V2)),a__isNat#(V1))
-->_2 a__isNat#(s(V1)) -> c_10(a__isNat#(V1)):5
-->_2 a__isNat#(0()) -> c_8():18
-->_2 a__isNat#(X) -> c_7():17
-->_1 a__and#(X1,X2) -> c_5():16
-->_2 a__isNat#(plus(V1,V2)) -> c_9(a__and#(a__isNat(V1),isNat(V2)),a__isNat#(V1)):4
-->_1 a__and#(tt(),X) -> c_6(mark#(X)):3
5:S:a__isNat#(s(V1)) -> c_10(a__isNat#(V1))
-->_1 a__isNat#(0()) -> c_8():18
-->_1 a__isNat#(X) -> c_7():17
-->_1 a__isNat#(s(V1)) -> c_10(a__isNat#(V1)):5
-->_1 a__isNat#(plus(V1,V2)) -> c_9(a__and#(a__isNat(V1),isNat(V2)),a__isNat#(V1)):4
6:S:a__plus#(N,0()) -> c_11(a__U11#(a__isNat(N),N),a__isNat#(N))
-->_2 a__isNat#(0()) -> c_8():18
-->_2 a__isNat#(X) -> c_7():17
-->_1 a__U11#(X1,X2) -> c_1():14
-->_2 a__isNat#(s(V1)) -> c_10(a__isNat#(V1)):5
-->_2 a__isNat#(plus(V1,V2)) -> c_9(a__and#(a__isNat(V1),isNat(V2)),a__isNat#(V1)):4
-->_1 a__U11#(tt(),N) -> c_2(mark#(N)):1
7:S:a__plus#(N,s(M)) -> c_12(a__U21#(a__and(a__isNat(M),isNat(N)),M,N)
,a__and#(a__isNat(M),isNat(N))
,a__isNat#(M))
-->_3 a__isNat#(0()) -> c_8():18
-->_3 a__isNat#(X) -> c_7():17
-->_2 a__and#(X1,X2) -> c_5():16
-->_1 a__U21#(X1,X2,X3) -> c_3():15
-->_3 a__isNat#(s(V1)) -> c_10(a__isNat#(V1)):5
-->_3 a__isNat#(plus(V1,V2)) -> c_9(a__and#(a__isNat(V1),isNat(V2)),a__isNat#(V1)):4
-->_2 a__and#(tt(),X) -> c_6(mark#(X)):3
-->_1 a__U21#(tt(),M,N) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)):2
8:S:mark#(U11(X1,X2)) -> c_15(a__U11#(mark(X1),X2),mark#(X1))
-->_2 mark#(s(X)) -> c_20(mark#(X)):13
-->_2 mark#(plus(X1,X2)) -> c_19(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12
-->_2 mark#(isNat(X)) -> c_18(a__isNat#(X)):11
-->_2 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):10
-->_2 mark#(U21(X1,X2,X3)) -> c_16(a__U21#(mark(X1),X2,X3),mark#(X1)):9
-->_2 mark#(tt()) -> c_21():21
-->_2 mark#(0()) -> c_14():20
-->_1 a__U11#(X1,X2) -> c_1():14
-->_2 mark#(U11(X1,X2)) -> c_15(a__U11#(mark(X1),X2),mark#(X1)):8
-->_1 a__U11#(tt(),N) -> c_2(mark#(N)):1
9:S:mark#(U21(X1,X2,X3)) -> c_16(a__U21#(mark(X1),X2,X3),mark#(X1))
-->_2 mark#(s(X)) -> c_20(mark#(X)):13
-->_2 mark#(plus(X1,X2)) -> c_19(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12
-->_2 mark#(isNat(X)) -> c_18(a__isNat#(X)):11
-->_2 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):10
-->_2 mark#(tt()) -> c_21():21
-->_2 mark#(0()) -> c_14():20
-->_1 a__U21#(X1,X2,X3) -> c_3():15
-->_2 mark#(U21(X1,X2,X3)) -> c_16(a__U21#(mark(X1),X2,X3),mark#(X1)):9
-->_2 mark#(U11(X1,X2)) -> c_15(a__U11#(mark(X1),X2),mark#(X1)):8
-->_1 a__U21#(tt(),M,N) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)):2
10:S:mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1))
-->_2 mark#(s(X)) -> c_20(mark#(X)):13
-->_2 mark#(plus(X1,X2)) -> c_19(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12
-->_2 mark#(isNat(X)) -> c_18(a__isNat#(X)):11
-->_2 mark#(tt()) -> c_21():21
-->_2 mark#(0()) -> c_14():20
-->_1 a__and#(X1,X2) -> c_5():16
-->_2 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):10
-->_2 mark#(U21(X1,X2,X3)) -> c_16(a__U21#(mark(X1),X2,X3),mark#(X1)):9
-->_2 mark#(U11(X1,X2)) -> c_15(a__U11#(mark(X1),X2),mark#(X1)):8
-->_1 a__and#(tt(),X) -> c_6(mark#(X)):3
11:S:mark#(isNat(X)) -> c_18(a__isNat#(X))
-->_1 a__isNat#(0()) -> c_8():18
-->_1 a__isNat#(X) -> c_7():17
-->_1 a__isNat#(s(V1)) -> c_10(a__isNat#(V1)):5
-->_1 a__isNat#(plus(V1,V2)) -> c_9(a__and#(a__isNat(V1),isNat(V2)),a__isNat#(V1)):4
12:S:mark#(plus(X1,X2)) -> c_19(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
-->_3 mark#(s(X)) -> c_20(mark#(X)):13
-->_2 mark#(s(X)) -> c_20(mark#(X)):13
-->_3 mark#(tt()) -> c_21():21
-->_2 mark#(tt()) -> c_21():21
-->_3 mark#(0()) -> c_14():20
-->_2 mark#(0()) -> c_14():20
-->_1 a__plus#(X1,X2) -> c_13():19
-->_3 mark#(plus(X1,X2)) -> c_19(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12
-->_2 mark#(plus(X1,X2)) -> c_19(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12
-->_3 mark#(isNat(X)) -> c_18(a__isNat#(X)):11
-->_2 mark#(isNat(X)) -> c_18(a__isNat#(X)):11
-->_3 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):10
-->_2 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):10
-->_3 mark#(U21(X1,X2,X3)) -> c_16(a__U21#(mark(X1),X2,X3),mark#(X1)):9
-->_2 mark#(U21(X1,X2,X3)) -> c_16(a__U21#(mark(X1),X2,X3),mark#(X1)):9
-->_3 mark#(U11(X1,X2)) -> c_15(a__U11#(mark(X1),X2),mark#(X1)):8
-->_2 mark#(U11(X1,X2)) -> c_15(a__U11#(mark(X1),X2),mark#(X1)):8
-->_1 a__plus#(N,s(M)) -> c_12(a__U21#(a__and(a__isNat(M),isNat(N)),M,N)
,a__and#(a__isNat(M),isNat(N))
,a__isNat#(M)):7
-->_1 a__plus#(N,0()) -> c_11(a__U11#(a__isNat(N),N),a__isNat#(N)):6
13:S:mark#(s(X)) -> c_20(mark#(X))
-->_1 mark#(tt()) -> c_21():21
-->_1 mark#(0()) -> c_14():20
-->_1 mark#(s(X)) -> c_20(mark#(X)):13
-->_1 mark#(plus(X1,X2)) -> c_19(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12
-->_1 mark#(isNat(X)) -> c_18(a__isNat#(X)):11
-->_1 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):10
-->_1 mark#(U21(X1,X2,X3)) -> c_16(a__U21#(mark(X1),X2,X3),mark#(X1)):9
-->_1 mark#(U11(X1,X2)) -> c_15(a__U11#(mark(X1),X2),mark#(X1)):8
14:W:a__U11#(X1,X2) -> c_1()
15:W:a__U21#(X1,X2,X3) -> c_3()
16:W:a__and#(X1,X2) -> c_5()
17:W:a__isNat#(X) -> c_7()
18:W:a__isNat#(0()) -> c_8()
19:W:a__plus#(X1,X2) -> c_13()
20:W:mark#(0()) -> c_14()
21:W:mark#(tt()) -> c_21()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
14: a__U11#(X1,X2) -> c_1()
15: a__U21#(X1,X2,X3) -> c_3()
16: a__and#(X1,X2) -> c_5()
17: a__isNat#(X) -> c_7()
18: a__isNat#(0()) -> c_8()
19: a__plus#(X1,X2) -> c_13()
20: mark#(0()) -> c_14()
21: mark#(tt()) -> c_21()
* Step 4: Failure MAYBE
+ Considered Problem:
- Strict DPs:
a__U11#(tt(),N) -> c_2(mark#(N))
a__U21#(tt(),M,N) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M))
a__and#(tt(),X) -> c_6(mark#(X))
a__isNat#(plus(V1,V2)) -> c_9(a__and#(a__isNat(V1),isNat(V2)),a__isNat#(V1))
a__isNat#(s(V1)) -> c_10(a__isNat#(V1))
a__plus#(N,0()) -> c_11(a__U11#(a__isNat(N),N),a__isNat#(N))
a__plus#(N,s(M)) -> c_12(a__U21#(a__and(a__isNat(M),isNat(N)),M,N)
,a__and#(a__isNat(M),isNat(N))
,a__isNat#(M))
mark#(U11(X1,X2)) -> c_15(a__U11#(mark(X1),X2),mark#(X1))
mark#(U21(X1,X2,X3)) -> c_16(a__U21#(mark(X1),X2,X3),mark#(X1))
mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1))
mark#(isNat(X)) -> c_18(a__isNat#(X))
mark#(plus(X1,X2)) -> c_19(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2))
mark#(s(X)) -> c_20(mark#(X))
- Weak TRS:
a__U11(X1,X2) -> U11(X1,X2)
a__U11(tt(),N) -> mark(N)
a__U21(X1,X2,X3) -> U21(X1,X2,X3)
a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__and(X1,X2) -> and(X1,X2)
a__and(tt(),X) -> mark(X)
a__isNat(X) -> isNat(X)
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2))
a__isNat(s(V1)) -> a__isNat(V1)
a__plus(N,0()) -> a__U11(a__isNat(N),N)
a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N)
a__plus(X1,X2) -> plus(X1,X2)
mark(0()) -> 0()
mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3)
mark(and(X1,X2)) -> a__and(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(s(X)) -> s(mark(X))
mark(tt()) -> tt()
- Signature:
{a__U11/2,a__U21/3,a__and/2,a__isNat/1,a__plus/2,mark/1,a__U11#/2,a__U21#/3,a__and#/2,a__isNat#/1,a__plus#/2
,mark#/1} / {0/0,U11/2,U21/3,and/2,isNat/1,plus/2,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/3,c_5/0,c_6/1,c_7/0,c_8/0
,c_9/2,c_10/1,c_11/2,c_12/3,c_13/0,c_14/0,c_15/2,c_16/2,c_17/2,c_18/1,c_19/3,c_20/1,c_21/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U11#,a__U21#,a__and#,a__isNat#,a__plus#
,mark#} and constructors {0,U11,U21,and,isNat,plus,s,tt}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
MAYBE