MAYBE
* Step 1: InnermostRuleRemoval MAYBE
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),X2)
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(X)
activate(n__s(X)) -> s(X)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> activate(X)
add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
and(false(),Y) -> false()
and(true(),X) -> activate(X)
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
from(X) -> cons(activate(X),n__from(n__s(activate(X))))
from(X) -> n__from(X)
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
s(X) -> n__s(X)
- Signature:
{activate/1,add/2,and/2,first/2,from/1,if/3,s/1} / {0/0,cons/2,false/0,n__add/2,n__first/2,n__from/1,n__s/1
,nil/0,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,and,first,from,if,s} and constructors {0
,cons,false,n__add,n__first,n__from,n__s,nil,true}
+ Applied Processor:
InnermostRuleRemoval
+ Details:
Arguments of following rules are not normal-forms.
add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
All above mentioned rules can be savely removed.
* Step 2: DependencyPairs MAYBE
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),X2)
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(X)
activate(n__s(X)) -> s(X)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> activate(X)
and(false(),Y) -> false()
and(true(),X) -> activate(X)
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
from(X) -> cons(activate(X),n__from(n__s(activate(X))))
from(X) -> n__from(X)
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
s(X) -> n__s(X)
- Signature:
{activate/1,add/2,and/2,first/2,from/1,if/3,s/1} / {0/0,cons/2,false/0,n__add/2,n__first/2,n__from/1,n__s/1
,nil/0,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,and,first,from,if,s} and constructors {0
,cons,false,n__add,n__first,n__from,n__s,nil,true}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
activate#(X) -> c_1()
activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1))
activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__from(X)) -> c_4(from#(X))
activate#(n__s(X)) -> c_5(s#(X))
add#(X1,X2) -> c_6()
add#(0(),X) -> c_7(activate#(X))
and#(false(),Y) -> c_8()
and#(true(),X) -> c_9(activate#(X))
first#(X1,X2) -> c_10()
first#(0(),X) -> c_11()
from#(X) -> c_12(activate#(X),activate#(X))
from#(X) -> c_13()
if#(false(),X,Y) -> c_14(activate#(Y))
if#(true(),X,Y) -> c_15(activate#(X))
s#(X) -> c_16()
Weak DPs
and mark the set of starting terms.
* Step 3: UsableRules MAYBE
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1()
activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1))
activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__from(X)) -> c_4(from#(X))
activate#(n__s(X)) -> c_5(s#(X))
add#(X1,X2) -> c_6()
add#(0(),X) -> c_7(activate#(X))
and#(false(),Y) -> c_8()
and#(true(),X) -> c_9(activate#(X))
first#(X1,X2) -> c_10()
first#(0(),X) -> c_11()
from#(X) -> c_12(activate#(X),activate#(X))
from#(X) -> c_13()
if#(false(),X,Y) -> c_14(activate#(Y))
if#(true(),X,Y) -> c_15(activate#(X))
s#(X) -> c_16()
- Weak TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),X2)
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(X)
activate(n__s(X)) -> s(X)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> activate(X)
and(false(),Y) -> false()
and(true(),X) -> activate(X)
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
from(X) -> cons(activate(X),n__from(n__s(activate(X))))
from(X) -> n__from(X)
if(false(),X,Y) -> activate(Y)
if(true(),X,Y) -> activate(X)
s(X) -> n__s(X)
- Signature:
{activate/1,add/2,and/2,first/2,from/1,if/3,s/1,activate#/1,add#/2,and#/2,first#/2,from#/1,if#/3
,s#/1} / {0/0,cons/2,false/0,n__add/2,n__first/2,n__from/1,n__s/1,nil/0,true/0,c_1/0,c_2/2,c_3/3,c_4/1,c_5/1
,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2,c_13/0,c_14/1,c_15/1,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,and#,first#,from#,if#
,s#} and constructors {0,cons,false,n__add,n__first,n__from,n__s,nil,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),X2)
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(X)
activate(n__s(X)) -> s(X)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> activate(X)
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
from(X) -> cons(activate(X),n__from(n__s(activate(X))))
from(X) -> n__from(X)
s(X) -> n__s(X)
activate#(X) -> c_1()
activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1))
activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__from(X)) -> c_4(from#(X))
activate#(n__s(X)) -> c_5(s#(X))
add#(X1,X2) -> c_6()
add#(0(),X) -> c_7(activate#(X))
and#(false(),Y) -> c_8()
and#(true(),X) -> c_9(activate#(X))
first#(X1,X2) -> c_10()
first#(0(),X) -> c_11()
from#(X) -> c_12(activate#(X),activate#(X))
from#(X) -> c_13()
if#(false(),X,Y) -> c_14(activate#(Y))
if#(true(),X,Y) -> c_15(activate#(X))
s#(X) -> c_16()
* Step 4: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1()
activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1))
activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__from(X)) -> c_4(from#(X))
activate#(n__s(X)) -> c_5(s#(X))
add#(X1,X2) -> c_6()
add#(0(),X) -> c_7(activate#(X))
and#(false(),Y) -> c_8()
and#(true(),X) -> c_9(activate#(X))
first#(X1,X2) -> c_10()
first#(0(),X) -> c_11()
from#(X) -> c_12(activate#(X),activate#(X))
from#(X) -> c_13()
if#(false(),X,Y) -> c_14(activate#(Y))
if#(true(),X,Y) -> c_15(activate#(X))
s#(X) -> c_16()
- Weak TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),X2)
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(X)
activate(n__s(X)) -> s(X)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> activate(X)
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
from(X) -> cons(activate(X),n__from(n__s(activate(X))))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,add/2,and/2,first/2,from/1,if/3,s/1,activate#/1,add#/2,and#/2,first#/2,from#/1,if#/3
,s#/1} / {0/0,cons/2,false/0,n__add/2,n__first/2,n__from/1,n__s/1,nil/0,true/0,c_1/0,c_2/2,c_3/3,c_4/1,c_5/1
,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2,c_13/0,c_14/1,c_15/1,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,and#,first#,from#,if#
,s#} and constructors {0,cons,false,n__add,n__first,n__from,n__s,nil,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,6,8,10,11,13,16}
by application of
Pre({1,6,8,10,11,13,16}) = {2,3,4,5,7,9,12,14,15}.
Here rules are labelled as follows:
1: activate#(X) -> c_1()
2: activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1))
3: activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
4: activate#(n__from(X)) -> c_4(from#(X))
5: activate#(n__s(X)) -> c_5(s#(X))
6: add#(X1,X2) -> c_6()
7: add#(0(),X) -> c_7(activate#(X))
8: and#(false(),Y) -> c_8()
9: and#(true(),X) -> c_9(activate#(X))
10: first#(X1,X2) -> c_10()
11: first#(0(),X) -> c_11()
12: from#(X) -> c_12(activate#(X),activate#(X))
13: from#(X) -> c_13()
14: if#(false(),X,Y) -> c_14(activate#(Y))
15: if#(true(),X,Y) -> c_15(activate#(X))
16: s#(X) -> c_16()
* Step 5: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1))
activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__from(X)) -> c_4(from#(X))
activate#(n__s(X)) -> c_5(s#(X))
add#(0(),X) -> c_7(activate#(X))
and#(true(),X) -> c_9(activate#(X))
from#(X) -> c_12(activate#(X),activate#(X))
if#(false(),X,Y) -> c_14(activate#(Y))
if#(true(),X,Y) -> c_15(activate#(X))
- Weak DPs:
activate#(X) -> c_1()
add#(X1,X2) -> c_6()
and#(false(),Y) -> c_8()
first#(X1,X2) -> c_10()
first#(0(),X) -> c_11()
from#(X) -> c_13()
s#(X) -> c_16()
- Weak TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),X2)
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(X)
activate(n__s(X)) -> s(X)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> activate(X)
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
from(X) -> cons(activate(X),n__from(n__s(activate(X))))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,add/2,and/2,first/2,from/1,if/3,s/1,activate#/1,add#/2,and#/2,first#/2,from#/1,if#/3
,s#/1} / {0/0,cons/2,false/0,n__add/2,n__first/2,n__from/1,n__s/1,nil/0,true/0,c_1/0,c_2/2,c_3/3,c_4/1,c_5/1
,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2,c_13/0,c_14/1,c_15/1,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,and#,first#,from#,if#
,s#} and constructors {0,cons,false,n__add,n__first,n__from,n__s,nil,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{4}
by application of
Pre({4}) = {1,2,5,6,7,8,9}.
Here rules are labelled as follows:
1: activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1))
2: activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
3: activate#(n__from(X)) -> c_4(from#(X))
4: activate#(n__s(X)) -> c_5(s#(X))
5: add#(0(),X) -> c_7(activate#(X))
6: and#(true(),X) -> c_9(activate#(X))
7: from#(X) -> c_12(activate#(X),activate#(X))
8: if#(false(),X,Y) -> c_14(activate#(Y))
9: if#(true(),X,Y) -> c_15(activate#(X))
10: activate#(X) -> c_1()
11: add#(X1,X2) -> c_6()
12: and#(false(),Y) -> c_8()
13: first#(X1,X2) -> c_10()
14: first#(0(),X) -> c_11()
15: from#(X) -> c_13()
16: s#(X) -> c_16()
* Step 6: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1))
activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__from(X)) -> c_4(from#(X))
add#(0(),X) -> c_7(activate#(X))
and#(true(),X) -> c_9(activate#(X))
from#(X) -> c_12(activate#(X),activate#(X))
if#(false(),X,Y) -> c_14(activate#(Y))
if#(true(),X,Y) -> c_15(activate#(X))
- Weak DPs:
activate#(X) -> c_1()
activate#(n__s(X)) -> c_5(s#(X))
add#(X1,X2) -> c_6()
and#(false(),Y) -> c_8()
first#(X1,X2) -> c_10()
first#(0(),X) -> c_11()
from#(X) -> c_13()
s#(X) -> c_16()
- Weak TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),X2)
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(X)
activate(n__s(X)) -> s(X)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> activate(X)
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
from(X) -> cons(activate(X),n__from(n__s(activate(X))))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,add/2,and/2,first/2,from/1,if/3,s/1,activate#/1,add#/2,and#/2,first#/2,from#/1,if#/3
,s#/1} / {0/0,cons/2,false/0,n__add/2,n__first/2,n__from/1,n__s/1,nil/0,true/0,c_1/0,c_2/2,c_3/3,c_4/1,c_5/1
,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2,c_13/0,c_14/1,c_15/1,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,and#,first#,from#,if#
,s#} and constructors {0,cons,false,n__add,n__first,n__from,n__s,nil,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1))
-->_2 activate#(n__s(X)) -> c_5(s#(X)):10
-->_1 add#(0(),X) -> c_7(activate#(X)):4
-->_2 activate#(n__from(X)) -> c_4(from#(X)):3
-->_2 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_1 add#(X1,X2) -> c_6():11
-->_2 activate#(X) -> c_1():9
-->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
2:S:activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
-->_3 activate#(n__s(X)) -> c_5(s#(X)):10
-->_2 activate#(n__s(X)) -> c_5(s#(X)):10
-->_3 activate#(n__from(X)) -> c_4(from#(X)):3
-->_2 activate#(n__from(X)) -> c_4(from#(X)):3
-->_1 first#(0(),X) -> c_11():14
-->_1 first#(X1,X2) -> c_10():13
-->_3 activate#(X) -> c_1():9
-->_2 activate#(X) -> c_1():9
-->_3 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_2 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_3 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
-->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
3:S:activate#(n__from(X)) -> c_4(from#(X))
-->_1 from#(X) -> c_12(activate#(X),activate#(X)):6
-->_1 from#(X) -> c_13():15
4:S:add#(0(),X) -> c_7(activate#(X))
-->_1 activate#(n__s(X)) -> c_5(s#(X)):10
-->_1 activate#(X) -> c_1():9
-->_1 activate#(n__from(X)) -> c_4(from#(X)):3
-->_1 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_1 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
5:S:and#(true(),X) -> c_9(activate#(X))
-->_1 activate#(n__s(X)) -> c_5(s#(X)):10
-->_1 activate#(X) -> c_1():9
-->_1 activate#(n__from(X)) -> c_4(from#(X)):3
-->_1 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_1 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
6:S:from#(X) -> c_12(activate#(X),activate#(X))
-->_2 activate#(n__s(X)) -> c_5(s#(X)):10
-->_1 activate#(n__s(X)) -> c_5(s#(X)):10
-->_2 activate#(X) -> c_1():9
-->_1 activate#(X) -> c_1():9
-->_2 activate#(n__from(X)) -> c_4(from#(X)):3
-->_1 activate#(n__from(X)) -> c_4(from#(X)):3
-->_2 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_1 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
-->_1 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
7:S:if#(false(),X,Y) -> c_14(activate#(Y))
-->_1 activate#(n__s(X)) -> c_5(s#(X)):10
-->_1 activate#(X) -> c_1():9
-->_1 activate#(n__from(X)) -> c_4(from#(X)):3
-->_1 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_1 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
8:S:if#(true(),X,Y) -> c_15(activate#(X))
-->_1 activate#(n__s(X)) -> c_5(s#(X)):10
-->_1 activate#(X) -> c_1():9
-->_1 activate#(n__from(X)) -> c_4(from#(X)):3
-->_1 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_1 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
9:W:activate#(X) -> c_1()
10:W:activate#(n__s(X)) -> c_5(s#(X))
-->_1 s#(X) -> c_16():16
11:W:add#(X1,X2) -> c_6()
12:W:and#(false(),Y) -> c_8()
13:W:first#(X1,X2) -> c_10()
14:W:first#(0(),X) -> c_11()
15:W:from#(X) -> c_13()
16:W:s#(X) -> c_16()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
12: and#(false(),Y) -> c_8()
11: add#(X1,X2) -> c_6()
15: from#(X) -> c_13()
13: first#(X1,X2) -> c_10()
14: first#(0(),X) -> c_11()
9: activate#(X) -> c_1()
10: activate#(n__s(X)) -> c_5(s#(X))
16: s#(X) -> c_16()
* Step 7: SimplifyRHS MAYBE
+ Considered Problem:
- Strict DPs:
activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1))
activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__from(X)) -> c_4(from#(X))
add#(0(),X) -> c_7(activate#(X))
and#(true(),X) -> c_9(activate#(X))
from#(X) -> c_12(activate#(X),activate#(X))
if#(false(),X,Y) -> c_14(activate#(Y))
if#(true(),X,Y) -> c_15(activate#(X))
- Weak TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),X2)
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(X)
activate(n__s(X)) -> s(X)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> activate(X)
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
from(X) -> cons(activate(X),n__from(n__s(activate(X))))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,add/2,and/2,first/2,from/1,if/3,s/1,activate#/1,add#/2,and#/2,first#/2,from#/1,if#/3
,s#/1} / {0/0,cons/2,false/0,n__add/2,n__first/2,n__from/1,n__s/1,nil/0,true/0,c_1/0,c_2/2,c_3/3,c_4/1,c_5/1
,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2,c_13/0,c_14/1,c_15/1,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,and#,first#,from#,if#
,s#} and constructors {0,cons,false,n__add,n__first,n__from,n__s,nil,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1))
-->_1 add#(0(),X) -> c_7(activate#(X)):4
-->_2 activate#(n__from(X)) -> c_4(from#(X)):3
-->_2 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
2:S:activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
-->_3 activate#(n__from(X)) -> c_4(from#(X)):3
-->_2 activate#(n__from(X)) -> c_4(from#(X)):3
-->_3 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_2 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_3 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
-->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
3:S:activate#(n__from(X)) -> c_4(from#(X))
-->_1 from#(X) -> c_12(activate#(X),activate#(X)):6
4:S:add#(0(),X) -> c_7(activate#(X))
-->_1 activate#(n__from(X)) -> c_4(from#(X)):3
-->_1 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_1 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
5:S:and#(true(),X) -> c_9(activate#(X))
-->_1 activate#(n__from(X)) -> c_4(from#(X)):3
-->_1 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_1 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
6:S:from#(X) -> c_12(activate#(X),activate#(X))
-->_2 activate#(n__from(X)) -> c_4(from#(X)):3
-->_1 activate#(n__from(X)) -> c_4(from#(X)):3
-->_2 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_1 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
-->_1 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
7:S:if#(false(),X,Y) -> c_14(activate#(Y))
-->_1 activate#(n__from(X)) -> c_4(from#(X)):3
-->_1 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_1 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
8:S:if#(true(),X,Y) -> c_15(activate#(X))
-->_1 activate#(n__from(X)) -> c_4(from#(X)):3
-->_1 activate#(n__first(X1,X2)) -> c_3(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2
-->_1 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
activate#(n__first(X1,X2)) -> c_3(activate#(X1),activate#(X2))
* Step 8: RemoveHeads MAYBE
+ Considered Problem:
- Strict DPs:
activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1))
activate#(n__first(X1,X2)) -> c_3(activate#(X1),activate#(X2))
activate#(n__from(X)) -> c_4(from#(X))
add#(0(),X) -> c_7(activate#(X))
and#(true(),X) -> c_9(activate#(X))
from#(X) -> c_12(activate#(X),activate#(X))
if#(false(),X,Y) -> c_14(activate#(Y))
if#(true(),X,Y) -> c_15(activate#(X))
- Weak TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),X2)
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(X)
activate(n__s(X)) -> s(X)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> activate(X)
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
from(X) -> cons(activate(X),n__from(n__s(activate(X))))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,add/2,and/2,first/2,from/1,if/3,s/1,activate#/1,add#/2,and#/2,first#/2,from#/1,if#/3
,s#/1} / {0/0,cons/2,false/0,n__add/2,n__first/2,n__from/1,n__s/1,nil/0,true/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1
,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2,c_13/0,c_14/1,c_15/1,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,and#,first#,from#,if#
,s#} and constructors {0,cons,false,n__add,n__first,n__from,n__s,nil,true}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1))
-->_1 add#(0(),X) -> c_7(activate#(X)):4
-->_2 activate#(n__from(X)) -> c_4(from#(X)):3
-->_2 activate#(n__first(X1,X2)) -> c_3(activate#(X1),activate#(X2)):2
-->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
2:S:activate#(n__first(X1,X2)) -> c_3(activate#(X1),activate#(X2))
-->_2 activate#(n__from(X)) -> c_4(from#(X)):3
-->_1 activate#(n__from(X)) -> c_4(from#(X)):3
-->_2 activate#(n__first(X1,X2)) -> c_3(activate#(X1),activate#(X2)):2
-->_1 activate#(n__first(X1,X2)) -> c_3(activate#(X1),activate#(X2)):2
-->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
-->_1 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
3:S:activate#(n__from(X)) -> c_4(from#(X))
-->_1 from#(X) -> c_12(activate#(X),activate#(X)):6
4:S:add#(0(),X) -> c_7(activate#(X))
-->_1 activate#(n__from(X)) -> c_4(from#(X)):3
-->_1 activate#(n__first(X1,X2)) -> c_3(activate#(X1),activate#(X2)):2
-->_1 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
5:S:and#(true(),X) -> c_9(activate#(X))
-->_1 activate#(n__from(X)) -> c_4(from#(X)):3
-->_1 activate#(n__first(X1,X2)) -> c_3(activate#(X1),activate#(X2)):2
-->_1 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
6:S:from#(X) -> c_12(activate#(X),activate#(X))
-->_2 activate#(n__from(X)) -> c_4(from#(X)):3
-->_1 activate#(n__from(X)) -> c_4(from#(X)):3
-->_2 activate#(n__first(X1,X2)) -> c_3(activate#(X1),activate#(X2)):2
-->_1 activate#(n__first(X1,X2)) -> c_3(activate#(X1),activate#(X2)):2
-->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
-->_1 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
7:S:if#(false(),X,Y) -> c_14(activate#(Y))
-->_1 activate#(n__from(X)) -> c_4(from#(X)):3
-->_1 activate#(n__first(X1,X2)) -> c_3(activate#(X1),activate#(X2)):2
-->_1 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
8:S:if#(true(),X,Y) -> c_15(activate#(X))
-->_1 activate#(n__from(X)) -> c_4(from#(X)):3
-->_1 activate#(n__first(X1,X2)) -> c_3(activate#(X1),activate#(X2)):2
-->_1 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1)):1
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).
[(5,and#(true(),X) -> c_9(activate#(X)))
,(7,if#(false(),X,Y) -> c_14(activate#(Y)))
,(8,if#(true(),X,Y) -> c_15(activate#(X)))]
* Step 9: Failure MAYBE
+ Considered Problem:
- Strict DPs:
activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),X2),activate#(X1))
activate#(n__first(X1,X2)) -> c_3(activate#(X1),activate#(X2))
activate#(n__from(X)) -> c_4(from#(X))
add#(0(),X) -> c_7(activate#(X))
from#(X) -> c_12(activate#(X),activate#(X))
- Weak TRS:
activate(X) -> X
activate(n__add(X1,X2)) -> add(activate(X1),X2)
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(X)
activate(n__s(X)) -> s(X)
add(X1,X2) -> n__add(X1,X2)
add(0(),X) -> activate(X)
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
from(X) -> cons(activate(X),n__from(n__s(activate(X))))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Signature:
{activate/1,add/2,and/2,first/2,from/1,if/3,s/1,activate#/1,add#/2,and#/2,first#/2,from#/1,if#/3
,s#/1} / {0/0,cons/2,false/0,n__add/2,n__first/2,n__from/1,n__s/1,nil/0,true/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1
,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2,c_13/0,c_14/1,c_15/1,c_16/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,and#,first#,from#,if#
,s#} and constructors {0,cons,false,n__add,n__first,n__from,n__s,nil,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
MAYBE