MAYBE
* Step 1: InnermostRuleRemoval MAYBE
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
U31(tt()) -> 0()
U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(activate(X1),activate(X2))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(N,0()) -> U11(isNat(N),N)
plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(N,0()) -> U31(isNat(N))
x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N)
x(X1,X2) -> n__x(X1,X2)
- Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1
,n__x/2,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U21,U31,U41,activate,and,isNat,plus,s
,x} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt}
+ Applied Processor:
InnermostRuleRemoval
+ Details:
Arguments of following rules are not normal-forms.
plus(N,0()) -> U11(isNat(N),N)
plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N)
x(N,0()) -> U31(isNat(N))
x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N)
All above mentioned rules can be savely removed.
* Step 2: DependencyPairs MAYBE
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
U31(tt()) -> 0()
U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(activate(X1),activate(X2))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
- Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1
,n__x/2,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U21,U31,U41,activate,and,isNat,plus,s
,x} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
0#() -> c_1()
U11#(tt(),N) -> c_2(activate#(N))
U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
U31#(tt()) -> c_4(0#())
U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))
,x#(activate(N),activate(M))
,activate#(N)
,activate#(M)
,activate#(N))
activate#(X) -> c_6()
activate#(n__0()) -> c_7(0#())
activate#(n__isNat(X)) -> c_8(isNat#(X))
activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X))
activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
and#(tt(),X) -> c_12(activate#(X))
isNat#(X) -> c_13()
isNat#(n__0()) -> c_14()
isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
plus#(X1,X2) -> c_18()
s#(X) -> c_19()
x#(X1,X2) -> c_20()
Weak DPs
and mark the set of starting terms.
* Step 3: UsableRules MAYBE
+ Considered Problem:
- Strict DPs:
0#() -> c_1()
U11#(tt(),N) -> c_2(activate#(N))
U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
U31#(tt()) -> c_4(0#())
U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))
,x#(activate(N),activate(M))
,activate#(N)
,activate#(M)
,activate#(N))
activate#(X) -> c_6()
activate#(n__0()) -> c_7(0#())
activate#(n__isNat(X)) -> c_8(isNat#(X))
activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X))
activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
and#(tt(),X) -> c_12(activate#(X))
isNat#(X) -> c_13()
isNat#(n__0()) -> c_14()
isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
plus#(X1,X2) -> c_18()
s#(X) -> c_19()
x#(X1,X2) -> c_20()
- Weak TRS:
0() -> n__0()
U11(tt(),N) -> activate(N)
U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
U31(tt()) -> 0()
U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(activate(X1),activate(X2))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
- Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3
,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0
,c_2/1,c_3/4,c_4/1,c_5/5,c_6/0,c_7/1,c_8/1,c_9/3,c_10/2,c_11/3,c_12/1,c_13/0,c_14/0,c_15/4,c_16/2,c_17/4
,c_18/0,c_19/0,c_20/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s#
,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
0() -> n__0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(activate(X1),activate(X2))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
0#() -> c_1()
U11#(tt(),N) -> c_2(activate#(N))
U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
U31#(tt()) -> c_4(0#())
U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))
,x#(activate(N),activate(M))
,activate#(N)
,activate#(M)
,activate#(N))
activate#(X) -> c_6()
activate#(n__0()) -> c_7(0#())
activate#(n__isNat(X)) -> c_8(isNat#(X))
activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X))
activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
and#(tt(),X) -> c_12(activate#(X))
isNat#(X) -> c_13()
isNat#(n__0()) -> c_14()
isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
plus#(X1,X2) -> c_18()
s#(X) -> c_19()
x#(X1,X2) -> c_20()
* Step 4: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
0#() -> c_1()
U11#(tt(),N) -> c_2(activate#(N))
U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
U31#(tt()) -> c_4(0#())
U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))
,x#(activate(N),activate(M))
,activate#(N)
,activate#(M)
,activate#(N))
activate#(X) -> c_6()
activate#(n__0()) -> c_7(0#())
activate#(n__isNat(X)) -> c_8(isNat#(X))
activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X))
activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
and#(tt(),X) -> c_12(activate#(X))
isNat#(X) -> c_13()
isNat#(n__0()) -> c_14()
isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
plus#(X1,X2) -> c_18()
s#(X) -> c_19()
x#(X1,X2) -> c_20()
- Weak TRS:
0() -> n__0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(activate(X1),activate(X2))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
- Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3
,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0
,c_2/1,c_3/4,c_4/1,c_5/5,c_6/0,c_7/1,c_8/1,c_9/3,c_10/2,c_11/3,c_12/1,c_13/0,c_14/0,c_15/4,c_16/2,c_17/4
,c_18/0,c_19/0,c_20/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s#
,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,6,13,14,18,19,20}
by application of
Pre({1,6,13,14,18,19,20}) = {2,3,4,5,7,8,9,10,11,12,15,16,17}.
Here rules are labelled as follows:
1: 0#() -> c_1()
2: U11#(tt(),N) -> c_2(activate#(N))
3: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
4: U31#(tt()) -> c_4(0#())
5: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))
,x#(activate(N),activate(M))
,activate#(N)
,activate#(M)
,activate#(N))
6: activate#(X) -> c_6()
7: activate#(n__0()) -> c_7(0#())
8: activate#(n__isNat(X)) -> c_8(isNat#(X))
9: activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
10: activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X))
11: activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
12: and#(tt(),X) -> c_12(activate#(X))
13: isNat#(X) -> c_13()
14: isNat#(n__0()) -> c_14()
15: isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
16: isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1))
17: isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
18: plus#(X1,X2) -> c_18()
19: s#(X) -> c_19()
20: x#(X1,X2) -> c_20()
* Step 5: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
U11#(tt(),N) -> c_2(activate#(N))
U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
U31#(tt()) -> c_4(0#())
U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))
,x#(activate(N),activate(M))
,activate#(N)
,activate#(M)
,activate#(N))
activate#(n__0()) -> c_7(0#())
activate#(n__isNat(X)) -> c_8(isNat#(X))
activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X))
activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
and#(tt(),X) -> c_12(activate#(X))
isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
- Weak DPs:
0#() -> c_1()
activate#(X) -> c_6()
isNat#(X) -> c_13()
isNat#(n__0()) -> c_14()
plus#(X1,X2) -> c_18()
s#(X) -> c_19()
x#(X1,X2) -> c_20()
- Weak TRS:
0() -> n__0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(activate(X1),activate(X2))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
- Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3
,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0
,c_2/1,c_3/4,c_4/1,c_5/5,c_6/0,c_7/1,c_8/1,c_9/3,c_10/2,c_11/3,c_12/1,c_13/0,c_14/0,c_15/4,c_16/2,c_17/4
,c_18/0,c_19/0,c_20/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s#
,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{3,5}
by application of
Pre({3,5}) = {1,2,4,7,8,9,10,11,12,13}.
Here rules are labelled as follows:
1: U11#(tt(),N) -> c_2(activate#(N))
2: U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
3: U31#(tt()) -> c_4(0#())
4: U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))
,x#(activate(N),activate(M))
,activate#(N)
,activate#(M)
,activate#(N))
5: activate#(n__0()) -> c_7(0#())
6: activate#(n__isNat(X)) -> c_8(isNat#(X))
7: activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
8: activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X))
9: activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
10: and#(tt(),X) -> c_12(activate#(X))
11: isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
12: isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1))
13: isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
14: 0#() -> c_1()
15: activate#(X) -> c_6()
16: isNat#(X) -> c_13()
17: isNat#(n__0()) -> c_14()
18: plus#(X1,X2) -> c_18()
19: s#(X) -> c_19()
20: x#(X1,X2) -> c_20()
* Step 6: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
U11#(tt(),N) -> c_2(activate#(N))
U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))
,x#(activate(N),activate(M))
,activate#(N)
,activate#(M)
,activate#(N))
activate#(n__isNat(X)) -> c_8(isNat#(X))
activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X))
activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
and#(tt(),X) -> c_12(activate#(X))
isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
- Weak DPs:
0#() -> c_1()
U31#(tt()) -> c_4(0#())
activate#(X) -> c_6()
activate#(n__0()) -> c_7(0#())
isNat#(X) -> c_13()
isNat#(n__0()) -> c_14()
plus#(X1,X2) -> c_18()
s#(X) -> c_19()
x#(X1,X2) -> c_20()
- Weak TRS:
0() -> n__0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(activate(X1),activate(X2))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
- Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3
,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0
,c_2/1,c_3/4,c_4/1,c_5/5,c_6/0,c_7/1,c_8/1,c_9/3,c_10/2,c_11/3,c_12/1,c_13/0,c_14/0,c_15/4,c_16/2,c_17/4
,c_18/0,c_19/0,c_20/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s#
,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:U11#(tt(),N) -> c_2(activate#(N))
-->_1 activate#(n__0()) -> c_7(0#()):15
-->_1 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_1 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_1 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_1 activate#(X) -> c_6():14
2:S:U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
-->_4 activate#(n__0()) -> c_7(0#()):15
-->_3 activate#(n__0()) -> c_7(0#()):15
-->_4 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_3 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_4 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_3 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_4 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_3 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_4 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_3 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_1 s#(X) -> c_19():19
-->_2 plus#(X1,X2) -> c_18():18
-->_4 activate#(X) -> c_6():14
-->_3 activate#(X) -> c_6():14
3:S:U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))
,x#(activate(N),activate(M))
,activate#(N)
,activate#(M)
,activate#(N))
-->_5 activate#(n__0()) -> c_7(0#()):15
-->_4 activate#(n__0()) -> c_7(0#()):15
-->_3 activate#(n__0()) -> c_7(0#()):15
-->_5 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_4 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_3 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_5 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_4 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_3 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_5 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_4 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_3 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_5 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_4 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_3 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_2 x#(X1,X2) -> c_20():20
-->_1 plus#(X1,X2) -> c_18():18
-->_5 activate#(X) -> c_6():14
-->_4 activate#(X) -> c_6():14
-->_3 activate#(X) -> c_6():14
4:S:activate#(n__isNat(X)) -> c_8(isNat#(X))
-->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):11
-->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1)):10
-->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):9
-->_1 isNat#(n__0()) -> c_14():17
-->_1 isNat#(X) -> c_13():16
5:S:activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
-->_3 activate#(n__0()) -> c_7(0#()):15
-->_2 activate#(n__0()) -> c_7(0#()):15
-->_3 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_2 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_3 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_2 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_1 plus#(X1,X2) -> c_18():18
-->_3 activate#(X) -> c_6():14
-->_2 activate#(X) -> c_6():14
-->_3 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_2 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_3 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_2 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
6:S:activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X))
-->_2 activate#(n__0()) -> c_7(0#()):15
-->_2 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_1 s#(X) -> c_19():19
-->_2 activate#(X) -> c_6():14
-->_2 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_2 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_2 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
7:S:activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
-->_3 activate#(n__0()) -> c_7(0#()):15
-->_2 activate#(n__0()) -> c_7(0#()):15
-->_1 x#(X1,X2) -> c_20():20
-->_3 activate#(X) -> c_6():14
-->_2 activate#(X) -> c_6():14
-->_3 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_2 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_3 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_2 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_3 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_2 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_3 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_2 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
8:S:and#(tt(),X) -> c_12(activate#(X))
-->_1 activate#(n__0()) -> c_7(0#()):15
-->_1 activate#(X) -> c_6():14
-->_1 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_1 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_1 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
9:S:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_4 activate#(n__0()) -> c_7(0#()):15
-->_3 activate#(n__0()) -> c_7(0#()):15
-->_2 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):11
-->_2 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1)):10
-->_2 isNat#(n__0()) -> c_14():17
-->_2 isNat#(X) -> c_13():16
-->_4 activate#(X) -> c_6():14
-->_3 activate#(X) -> c_6():14
-->_2 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):9
-->_1 and#(tt(),X) -> c_12(activate#(X)):8
-->_4 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_3 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_4 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_3 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_4 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_3 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_4 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_3 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
10:S:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1))
-->_2 activate#(n__0()) -> c_7(0#()):15
-->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):11
-->_1 isNat#(n__0()) -> c_14():17
-->_1 isNat#(X) -> c_13():16
-->_2 activate#(X) -> c_6():14
-->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1)):10
-->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):9
-->_2 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_2 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_2 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_2 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
11:S:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_4 activate#(n__0()) -> c_7(0#()):15
-->_3 activate#(n__0()) -> c_7(0#()):15
-->_2 isNat#(n__0()) -> c_14():17
-->_2 isNat#(X) -> c_13():16
-->_4 activate#(X) -> c_6():14
-->_3 activate#(X) -> c_6():14
-->_2 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):11
-->_2 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1)):10
-->_2 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):9
-->_1 and#(tt(),X) -> c_12(activate#(X)):8
-->_4 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_3 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_4 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_3 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_4 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_3 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_4 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_3 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
12:W:0#() -> c_1()
13:W:U31#(tt()) -> c_4(0#())
-->_1 0#() -> c_1():12
14:W:activate#(X) -> c_6()
15:W:activate#(n__0()) -> c_7(0#())
-->_1 0#() -> c_1():12
16:W:isNat#(X) -> c_13()
17:W:isNat#(n__0()) -> c_14()
18:W:plus#(X1,X2) -> c_18()
19:W:s#(X) -> c_19()
20:W:x#(X1,X2) -> c_20()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
13: U31#(tt()) -> c_4(0#())
16: isNat#(X) -> c_13()
17: isNat#(n__0()) -> c_14()
18: plus#(X1,X2) -> c_18()
19: s#(X) -> c_19()
14: activate#(X) -> c_6()
20: x#(X1,X2) -> c_20()
15: activate#(n__0()) -> c_7(0#())
12: 0#() -> c_1()
* Step 7: SimplifyRHS MAYBE
+ Considered Problem:
- Strict DPs:
U11#(tt(),N) -> c_2(activate#(N))
U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))
,x#(activate(N),activate(M))
,activate#(N)
,activate#(M)
,activate#(N))
activate#(n__isNat(X)) -> c_8(isNat#(X))
activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X))
activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
and#(tt(),X) -> c_12(activate#(X))
isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
- Weak TRS:
0() -> n__0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(activate(X1),activate(X2))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
- Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3
,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0
,c_2/1,c_3/4,c_4/1,c_5/5,c_6/0,c_7/1,c_8/1,c_9/3,c_10/2,c_11/3,c_12/1,c_13/0,c_14/0,c_15/4,c_16/2,c_17/4
,c_18/0,c_19/0,c_20/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s#
,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:U11#(tt(),N) -> c_2(activate#(N))
-->_1 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_1 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_1 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
2:S:U21#(tt(),M,N) -> c_3(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
-->_4 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_3 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_4 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_3 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_4 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_3 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_4 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_3 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
3:S:U41#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N))
,x#(activate(N),activate(M))
,activate#(N)
,activate#(M)
,activate#(N))
-->_5 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_4 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_3 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_5 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_4 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_3 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_5 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_4 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_3 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_5 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_4 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_3 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
4:S:activate#(n__isNat(X)) -> c_8(isNat#(X))
-->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):11
-->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1)):10
-->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):9
5:S:activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
-->_3 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_2 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_3 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_2 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_3 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_2 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_3 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_2 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
6:S:activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X))
-->_2 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_2 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_2 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_2 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
7:S:activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
-->_3 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_2 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_3 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_2 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_3 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_2 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_3 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_2 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
8:S:and#(tt(),X) -> c_12(activate#(X))
-->_1 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_1 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_1 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
9:S:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):11
-->_2 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1)):10
-->_2 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):9
-->_1 and#(tt(),X) -> c_12(activate#(X)):8
-->_4 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_3 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_4 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_3 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_4 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_3 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_4 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_3 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
10:S:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1))
-->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):11
-->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1)):10
-->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):9
-->_2 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_2 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_2 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_2 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
11:S:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):11
-->_2 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1)):10
-->_2 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):9
-->_1 and#(tt(),X) -> c_12(activate#(X)):8
-->_4 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_3 activate#(n__x(X1,X2)) -> c_11(x#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):7
-->_4 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_3 activate#(n__s(X)) -> c_10(s#(activate(X)),activate#(X)):6
-->_4 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_3 activate#(n__plus(X1,X2)) -> c_9(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):5
-->_4 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_3 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
U21#(tt(),M,N) -> c_3(activate#(N),activate#(M))
U41#(tt(),M,N) -> c_5(activate#(N),activate#(M),activate#(N))
activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_10(activate#(X))
activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2))
* Step 8: RemoveHeads MAYBE
+ Considered Problem:
- Strict DPs:
U11#(tt(),N) -> c_2(activate#(N))
U21#(tt(),M,N) -> c_3(activate#(N),activate#(M))
U41#(tt(),M,N) -> c_5(activate#(N),activate#(M),activate#(N))
activate#(n__isNat(X)) -> c_8(isNat#(X))
activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_10(activate#(X))
activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2))
and#(tt(),X) -> c_12(activate#(X))
isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
- Weak TRS:
0() -> n__0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(activate(X1),activate(X2))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
- Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3
,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0
,c_2/1,c_3/2,c_4/1,c_5/3,c_6/0,c_7/1,c_8/1,c_9/2,c_10/1,c_11/2,c_12/1,c_13/0,c_14/0,c_15/4,c_16/2,c_17/4
,c_18/0,c_19/0,c_20/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s#
,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:U11#(tt(),N) -> c_2(activate#(N))
-->_1 activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2)):7
-->_1 activate#(n__s(X)) -> c_10(activate#(X)):6
-->_1 activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2)):5
-->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
2:S:U21#(tt(),M,N) -> c_3(activate#(N),activate#(M))
-->_2 activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2)):7
-->_1 activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2)):7
-->_2 activate#(n__s(X)) -> c_10(activate#(X)):6
-->_1 activate#(n__s(X)) -> c_10(activate#(X)):6
-->_2 activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2)):5
-->_1 activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2)):5
-->_2 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
3:S:U41#(tt(),M,N) -> c_5(activate#(N),activate#(M),activate#(N))
-->_3 activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2)):7
-->_2 activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2)):7
-->_1 activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2)):7
-->_3 activate#(n__s(X)) -> c_10(activate#(X)):6
-->_2 activate#(n__s(X)) -> c_10(activate#(X)):6
-->_1 activate#(n__s(X)) -> c_10(activate#(X)):6
-->_3 activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2)):5
-->_2 activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2)):5
-->_1 activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2)):5
-->_3 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_2 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
4:S:activate#(n__isNat(X)) -> c_8(isNat#(X))
-->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):11
-->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1)):10
-->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):9
5:S:activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2))
-->_2 activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2)):7
-->_1 activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2)):7
-->_2 activate#(n__s(X)) -> c_10(activate#(X)):6
-->_1 activate#(n__s(X)) -> c_10(activate#(X)):6
-->_2 activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2)):5
-->_1 activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2)):5
-->_2 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
6:S:activate#(n__s(X)) -> c_10(activate#(X))
-->_1 activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2)):7
-->_1 activate#(n__s(X)) -> c_10(activate#(X)):6
-->_1 activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2)):5
-->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
7:S:activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2))
-->_2 activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2)):7
-->_1 activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2)):7
-->_2 activate#(n__s(X)) -> c_10(activate#(X)):6
-->_1 activate#(n__s(X)) -> c_10(activate#(X)):6
-->_2 activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2)):5
-->_1 activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2)):5
-->_2 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
8:S:and#(tt(),X) -> c_12(activate#(X))
-->_1 activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2)):7
-->_1 activate#(n__s(X)) -> c_10(activate#(X)):6
-->_1 activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2)):5
-->_1 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
9:S:isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):11
-->_2 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1)):10
-->_2 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):9
-->_1 and#(tt(),X) -> c_12(activate#(X)):8
-->_4 activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2)):7
-->_3 activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2)):7
-->_4 activate#(n__s(X)) -> c_10(activate#(X)):6
-->_3 activate#(n__s(X)) -> c_10(activate#(X)):6
-->_4 activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2)):5
-->_3 activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2)):5
-->_4 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_3 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
10:S:isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1))
-->_1 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):11
-->_1 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1)):10
-->_1 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):9
-->_2 activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2)):7
-->_2 activate#(n__s(X)) -> c_10(activate#(X)):6
-->_2 activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2)):5
-->_2 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
11:S:isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):11
-->_2 isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1)):10
-->_2 isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):9
-->_1 and#(tt(),X) -> c_12(activate#(X)):8
-->_4 activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2)):7
-->_3 activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2)):7
-->_4 activate#(n__s(X)) -> c_10(activate#(X)):6
-->_3 activate#(n__s(X)) -> c_10(activate#(X)):6
-->_4 activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2)):5
-->_3 activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2)):5
-->_4 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
-->_3 activate#(n__isNat(X)) -> c_8(isNat#(X)):4
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(1,U11#(tt(),N) -> c_2(activate#(N)))
,(2,U21#(tt(),M,N) -> c_3(activate#(N),activate#(M)))
,(3,U41#(tt(),M,N) -> c_5(activate#(N),activate#(M),activate#(N)))]
* Step 9: Failure MAYBE
+ Considered Problem:
- Strict DPs:
activate#(n__isNat(X)) -> c_8(isNat#(X))
activate#(n__plus(X1,X2)) -> c_9(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_10(activate#(X))
activate#(n__x(X1,X2)) -> c_11(activate#(X1),activate#(X2))
and#(tt(),X) -> c_12(activate#(X))
isNat#(n__plus(V1,V2)) -> c_15(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_16(isNat#(activate(V1)),activate#(V1))
isNat#(n__x(V1,V2)) -> c_17(and#(isNat(activate(V1)),n__isNat(activate(V2)))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
- Weak TRS:
0() -> n__0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__isNat(X)) -> isNat(X)
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__x(X1,X2)) -> x(activate(X1),activate(X2))
and(tt(),X) -> activate(X)
isNat(X) -> n__isNat(X)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
isNat(n__s(V1)) -> isNat(activate(V1))
isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
- Signature:
{0/0,U11/2,U21/3,U31/1,U41/3,activate/1,and/2,isNat/1,plus/2,s/1,x/2,0#/0,U11#/2,U21#/3,U31#/1,U41#/3
,activate#/1,and#/2,isNat#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__isNat/1,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0
,c_2/1,c_3/2,c_4/1,c_5/3,c_6/0,c_7/1,c_8/1,c_9/2,c_10/1,c_11/2,c_12/1,c_13/0,c_14/0,c_15/4,c_16/2,c_17/4
,c_18/0,c_19/0,c_20/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U21#,U31#,U41#,activate#,and#,isNat#,plus#,s#
,x#} and constructors {n__0,n__isNat,n__plus,n__s,n__x,tt}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
MAYBE