MAYBE
* Step 1: DependencyPairs MAYBE
+ Considered Problem:
- Strict TRS:
main(0()) -> 0()
main(S(x1)) -> sum#1(map#2(Cons(S(x1),unfoldr#2(S(x1)))))
map#2(Cons(x2,x5)) -> Cons(mult#2(x2,x2),map#2(x5))
map#2(Nil()) -> Nil()
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> plus#2(x2,mult#2(x4,x2))
plus#2(0(),x8) -> x8
plus#2(S(x4),x2) -> S(plus#2(x4,x2))
sum#1(Cons(x2,x1)) -> plus#2(x2,sum#1(x1))
sum#1(Nil()) -> 0()
unfoldr#2(0()) -> Nil()
unfoldr#2(S(x2)) -> Cons(x2,unfoldr#2(x2))
- Signature:
{main/1,map#2/1,mult#2/2,plus#2/2,sum#1/1,unfoldr#2/1} / {0/0,Cons/2,Nil/0,S/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {main,map#2,mult#2,plus#2,sum#1
,unfoldr#2} and constructors {0,Cons,Nil,S}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
main#(0()) -> c_1()
main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1)))))
,map#2#(Cons(S(x1),unfoldr#2(S(x1))))
,unfoldr#2#(S(x1)))
map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5))
map#2#(Nil()) -> c_4()
mult#2#(0(),x2) -> c_5()
mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2))
plus#2#(0(),x8) -> c_7()
plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2))
sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1))
sum#1#(Nil()) -> c_10()
unfoldr#2#(0()) -> c_11()
unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2))
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules MAYBE
+ Considered Problem:
- Strict DPs:
main#(0()) -> c_1()
main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1)))))
,map#2#(Cons(S(x1),unfoldr#2(S(x1))))
,unfoldr#2#(S(x1)))
map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5))
map#2#(Nil()) -> c_4()
mult#2#(0(),x2) -> c_5()
mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2))
plus#2#(0(),x8) -> c_7()
plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2))
sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1))
sum#1#(Nil()) -> c_10()
unfoldr#2#(0()) -> c_11()
unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2))
- Weak TRS:
main(0()) -> 0()
main(S(x1)) -> sum#1(map#2(Cons(S(x1),unfoldr#2(S(x1)))))
map#2(Cons(x2,x5)) -> Cons(mult#2(x2,x2),map#2(x5))
map#2(Nil()) -> Nil()
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> plus#2(x2,mult#2(x4,x2))
plus#2(0(),x8) -> x8
plus#2(S(x4),x2) -> S(plus#2(x4,x2))
sum#1(Cons(x2,x1)) -> plus#2(x2,sum#1(x1))
sum#1(Nil()) -> 0()
unfoldr#2(0()) -> Nil()
unfoldr#2(S(x2)) -> Cons(x2,unfoldr#2(x2))
- Signature:
{main/1,map#2/1,mult#2/2,plus#2/2,sum#1/1,unfoldr#2/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2,sum#1#/1
,unfoldr#2#/1} / {0/0,Cons/2,Nil/0,S/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/2,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#,map#2#,mult#2#,plus#2#,sum#1#
,unfoldr#2#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
map#2(Cons(x2,x5)) -> Cons(mult#2(x2,x2),map#2(x5))
map#2(Nil()) -> Nil()
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> plus#2(x2,mult#2(x4,x2))
plus#2(0(),x8) -> x8
plus#2(S(x4),x2) -> S(plus#2(x4,x2))
sum#1(Cons(x2,x1)) -> plus#2(x2,sum#1(x1))
sum#1(Nil()) -> 0()
unfoldr#2(0()) -> Nil()
unfoldr#2(S(x2)) -> Cons(x2,unfoldr#2(x2))
main#(0()) -> c_1()
main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1)))))
,map#2#(Cons(S(x1),unfoldr#2(S(x1))))
,unfoldr#2#(S(x1)))
map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5))
map#2#(Nil()) -> c_4()
mult#2#(0(),x2) -> c_5()
mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2))
plus#2#(0(),x8) -> c_7()
plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2))
sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1))
sum#1#(Nil()) -> c_10()
unfoldr#2#(0()) -> c_11()
unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2))
* Step 3: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
main#(0()) -> c_1()
main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1)))))
,map#2#(Cons(S(x1),unfoldr#2(S(x1))))
,unfoldr#2#(S(x1)))
map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5))
map#2#(Nil()) -> c_4()
mult#2#(0(),x2) -> c_5()
mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2))
plus#2#(0(),x8) -> c_7()
plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2))
sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1))
sum#1#(Nil()) -> c_10()
unfoldr#2#(0()) -> c_11()
unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2))
- Weak TRS:
map#2(Cons(x2,x5)) -> Cons(mult#2(x2,x2),map#2(x5))
map#2(Nil()) -> Nil()
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> plus#2(x2,mult#2(x4,x2))
plus#2(0(),x8) -> x8
plus#2(S(x4),x2) -> S(plus#2(x4,x2))
sum#1(Cons(x2,x1)) -> plus#2(x2,sum#1(x1))
sum#1(Nil()) -> 0()
unfoldr#2(0()) -> Nil()
unfoldr#2(S(x2)) -> Cons(x2,unfoldr#2(x2))
- Signature:
{main/1,map#2/1,mult#2/2,plus#2/2,sum#1/1,unfoldr#2/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2,sum#1#/1
,unfoldr#2#/1} / {0/0,Cons/2,Nil/0,S/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/2,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#,map#2#,mult#2#,plus#2#,sum#1#
,unfoldr#2#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,4,5,7,10,11}
by application of
Pre({1,4,5,7,10,11}) = {2,3,6,8,9,12}.
Here rules are labelled as follows:
1: main#(0()) -> c_1()
2: main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1)))))
,map#2#(Cons(S(x1),unfoldr#2(S(x1))))
,unfoldr#2#(S(x1)))
3: map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5))
4: map#2#(Nil()) -> c_4()
5: mult#2#(0(),x2) -> c_5()
6: mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2))
7: plus#2#(0(),x8) -> c_7()
8: plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2))
9: sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1))
10: sum#1#(Nil()) -> c_10()
11: unfoldr#2#(0()) -> c_11()
12: unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2))
* Step 4: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1)))))
,map#2#(Cons(S(x1),unfoldr#2(S(x1))))
,unfoldr#2#(S(x1)))
map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5))
mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2))
plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2))
sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1))
unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2))
- Weak DPs:
main#(0()) -> c_1()
map#2#(Nil()) -> c_4()
mult#2#(0(),x2) -> c_5()
plus#2#(0(),x8) -> c_7()
sum#1#(Nil()) -> c_10()
unfoldr#2#(0()) -> c_11()
- Weak TRS:
map#2(Cons(x2,x5)) -> Cons(mult#2(x2,x2),map#2(x5))
map#2(Nil()) -> Nil()
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> plus#2(x2,mult#2(x4,x2))
plus#2(0(),x8) -> x8
plus#2(S(x4),x2) -> S(plus#2(x4,x2))
sum#1(Cons(x2,x1)) -> plus#2(x2,sum#1(x1))
sum#1(Nil()) -> 0()
unfoldr#2(0()) -> Nil()
unfoldr#2(S(x2)) -> Cons(x2,unfoldr#2(x2))
- Signature:
{main/1,map#2/1,mult#2/2,plus#2/2,sum#1/1,unfoldr#2/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2,sum#1#/1
,unfoldr#2#/1} / {0/0,Cons/2,Nil/0,S/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/2,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#,map#2#,mult#2#,plus#2#,sum#1#
,unfoldr#2#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1)))))
,map#2#(Cons(S(x1),unfoldr#2(S(x1))))
,unfoldr#2#(S(x1)))
-->_3 unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2)):6
-->_1 sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1)):5
-->_2 map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5)):2
-->_1 sum#1#(Nil()) -> c_10():11
2:S:map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5))
-->_1 mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2)):3
-->_1 mult#2#(0(),x2) -> c_5():9
-->_2 map#2#(Nil()) -> c_4():8
-->_2 map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5)):2
3:S:mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2))
-->_1 plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2)):4
-->_1 plus#2#(0(),x8) -> c_7():10
-->_2 mult#2#(0(),x2) -> c_5():9
-->_2 mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2)):3
4:S:plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2))
-->_1 plus#2#(0(),x8) -> c_7():10
-->_1 plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2)):4
5:S:sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1))
-->_2 sum#1#(Nil()) -> c_10():11
-->_1 plus#2#(0(),x8) -> c_7():10
-->_2 sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1)):5
-->_1 plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2)):4
6:S:unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2))
-->_1 unfoldr#2#(0()) -> c_11():12
-->_1 unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2)):6
7:W:main#(0()) -> c_1()
8:W:map#2#(Nil()) -> c_4()
9:W:mult#2#(0(),x2) -> c_5()
10:W:plus#2#(0(),x8) -> c_7()
11:W:sum#1#(Nil()) -> c_10()
12:W:unfoldr#2#(0()) -> c_11()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
7: main#(0()) -> c_1()
8: map#2#(Nil()) -> c_4()
9: mult#2#(0(),x2) -> c_5()
10: plus#2#(0(),x8) -> c_7()
11: sum#1#(Nil()) -> c_10()
12: unfoldr#2#(0()) -> c_11()
* Step 5: NaturalMI MAYBE
+ Considered Problem:
- Strict DPs:
main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1)))))
,map#2#(Cons(S(x1),unfoldr#2(S(x1))))
,unfoldr#2#(S(x1)))
map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5))
mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2))
plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2))
sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1))
unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2))
- Weak TRS:
map#2(Cons(x2,x5)) -> Cons(mult#2(x2,x2),map#2(x5))
map#2(Nil()) -> Nil()
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> plus#2(x2,mult#2(x4,x2))
plus#2(0(),x8) -> x8
plus#2(S(x4),x2) -> S(plus#2(x4,x2))
sum#1(Cons(x2,x1)) -> plus#2(x2,sum#1(x1))
sum#1(Nil()) -> 0()
unfoldr#2(0()) -> Nil()
unfoldr#2(S(x2)) -> Cons(x2,unfoldr#2(x2))
- Signature:
{main/1,map#2/1,mult#2/2,plus#2/2,sum#1/1,unfoldr#2/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2,sum#1#/1
,unfoldr#2#/1} / {0/0,Cons/2,Nil/0,S/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/2,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#,map#2#,mult#2#,plus#2#,sum#1#
,unfoldr#2#} and constructors {0,Cons,Nil,S}
+ 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_2) = {1,2,3},
uargs(c_3) = {1,2},
uargs(c_6) = {1,2},
uargs(c_8) = {1},
uargs(c_9) = {1,2},
uargs(c_12) = {1}
Following symbols are considered usable:
{main#,map#2#,mult#2#,plus#2#,sum#1#,unfoldr#2#}
TcT has computed the following interpretation:
p(0) = [0]
p(Cons) = [0]
p(Nil) = [0]
p(S) = [9]
p(main) = [0]
p(map#2) = [0]
p(mult#2) = [0]
p(plus#2) = [0]
p(sum#1) = [0]
p(unfoldr#2) = [1] x1 + [15]
p(main#) = [13]
p(map#2#) = [0]
p(mult#2#) = [0]
p(plus#2#) = [0]
p(sum#1#) = [0]
p(unfoldr#2#) = [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [8] x3 + [0]
p(c_3) = [1] x1 + [2] x2 + [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [4] x1 + [4] x2 + [0]
p(c_7) = [1]
p(c_8) = [8] x1 + [0]
p(c_9) = [2] x1 + [1] x2 + [0]
p(c_10) = [2]
p(c_11) = [0]
p(c_12) = [1] x1 + [0]
Following rules are strictly oriented:
main#(S(x1)) = [13]
> [0]
= c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1))))),map#2#(Cons(S(x1),unfoldr#2(S(x1)))),unfoldr#2#(S(x1)))
Following rules are (at-least) weakly oriented:
map#2#(Cons(x2,x5)) = [0]
>= [0]
= c_3(mult#2#(x2,x2),map#2#(x5))
mult#2#(S(x4),x2) = [0]
>= [0]
= c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2))
plus#2#(S(x4),x2) = [0]
>= [0]
= c_8(plus#2#(x4,x2))
sum#1#(Cons(x2,x1)) = [0]
>= [0]
= c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1))
unfoldr#2#(S(x2)) = [0]
>= [0]
= c_12(unfoldr#2#(x2))
* Step 6: Failure MAYBE
+ Considered Problem:
- Strict DPs:
map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5))
mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2))
plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2))
sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1))
unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2))
- Weak DPs:
main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1)))))
,map#2#(Cons(S(x1),unfoldr#2(S(x1))))
,unfoldr#2#(S(x1)))
- Weak TRS:
map#2(Cons(x2,x5)) -> Cons(mult#2(x2,x2),map#2(x5))
map#2(Nil()) -> Nil()
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> plus#2(x2,mult#2(x4,x2))
plus#2(0(),x8) -> x8
plus#2(S(x4),x2) -> S(plus#2(x4,x2))
sum#1(Cons(x2,x1)) -> plus#2(x2,sum#1(x1))
sum#1(Nil()) -> 0()
unfoldr#2(0()) -> Nil()
unfoldr#2(S(x2)) -> Cons(x2,unfoldr#2(x2))
- Signature:
{main/1,map#2/1,mult#2/2,plus#2/2,sum#1/1,unfoldr#2/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2,sum#1#/1
,unfoldr#2#/1} / {0/0,Cons/2,Nil/0,S/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/2,c_10/0,c_11/0
,c_12/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#,map#2#,mult#2#,plus#2#,sum#1#
,unfoldr#2#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
MAYBE