MAYBE
* Step 1: DependencyPairs MAYBE
+ Considered Problem:
- Strict TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
half(0()) -> 0()
half(dbl(X)) -> X
half(s(0())) -> 0()
half(s(s(X))) -> s(half(X))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
terms(N) -> cons(recip(sqr(N)),terms(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1} / {0/0,cons/2,nil/0,recip/1,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add,dbl,first,half,sqr,terms} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
add#(0(),X) -> c_1()
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(0()) -> c_7()
half#(dbl(X)) -> c_8()
half#(s(0())) -> c_9()
half#(s(s(X))) -> c_10(half#(X))
sqr#(0()) -> c_11()
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules MAYBE
+ Considered Problem:
- Strict DPs:
add#(0(),X) -> c_1()
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(0()) -> c_7()
half#(dbl(X)) -> c_8()
half#(s(0())) -> c_9()
half#(s(s(X))) -> c_10(half#(X))
sqr#(0()) -> c_11()
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
half(0()) -> 0()
half(dbl(X)) -> X
half(s(0())) -> 0()
half(s(s(X))) -> s(half(X))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
terms(N) -> cons(recip(sqr(N)),terms(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
add#(0(),X) -> c_1()
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(0()) -> c_7()
half#(dbl(X)) -> c_8()
half#(s(0())) -> c_9()
half#(s(s(X))) -> c_10(half#(X))
sqr#(0()) -> c_11()
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
* Step 3: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
add#(0(),X) -> c_1()
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(0()) -> c_7()
half#(dbl(X)) -> c_8()
half#(s(0())) -> c_9()
half#(s(s(X))) -> c_10(half#(X))
sqr#(0()) -> c_11()
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3,5,7,8,9,11}
by application of
Pre({1,3,5,7,8,9,11}) = {2,4,6,10,12,13}.
Here rules are labelled as follows:
1: add#(0(),X) -> c_1()
2: add#(s(X),Y) -> c_2(add#(X,Y))
3: dbl#(0()) -> c_3()
4: dbl#(s(X)) -> c_4(dbl#(X))
5: first#(0(),X) -> c_5()
6: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
7: half#(0()) -> c_7()
8: half#(dbl(X)) -> c_8()
9: half#(s(0())) -> c_9()
10: half#(s(s(X))) -> c_10(half#(X))
11: sqr#(0()) -> c_11()
12: sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
13: terms#(N) -> c_13(sqr#(N),terms#(s(N)))
* Step 4: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(s(X)) -> c_4(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak DPs:
add#(0(),X) -> c_1()
dbl#(0()) -> c_3()
first#(0(),X) -> c_5()
half#(0()) -> c_7()
half#(dbl(X)) -> c_8()
half#(s(0())) -> c_9()
sqr#(0()) -> c_11()
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(0(),X) -> c_1():7
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
2:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(0()) -> c_3():8
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
3:S:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
-->_1 first#(0(),X) -> c_5():9
-->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):3
4:S:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(0())) -> c_9():12
-->_1 half#(dbl(X)) -> c_8():11
-->_1 half#(0()) -> c_7():10
-->_1 half#(s(s(X))) -> c_10(half#(X)):4
5:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_2 sqr#(0()) -> c_11():13
-->_3 dbl#(0()) -> c_3():8
-->_1 add#(0(),X) -> c_1():7
-->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):5
-->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
6:S:terms#(N) -> c_13(sqr#(N),terms#(s(N)))
-->_1 sqr#(0()) -> c_11():13
-->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):6
-->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):5
7:W:add#(0(),X) -> c_1()
8:W:dbl#(0()) -> c_3()
9:W:first#(0(),X) -> c_5()
10:W:half#(0()) -> c_7()
11:W:half#(dbl(X)) -> c_8()
12:W:half#(s(0())) -> c_9()
13:W:sqr#(0()) -> c_11()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
13: sqr#(0()) -> c_11()
10: half#(0()) -> c_7()
11: half#(dbl(X)) -> c_8()
12: half#(s(0())) -> c_9()
9: first#(0(),X) -> c_5()
8: dbl#(0()) -> c_3()
7: add#(0(),X) -> c_1()
* Step 5: Decompose MAYBE
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(s(X)) -> c_4(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
- Weak DPs:
dbl#(s(X)) -> c_4(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
Problem (S)
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
** Step 5.a:1: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
- Weak DPs:
dbl#(s(X)) -> c_4(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
2:W:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
3:W:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
-->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):3
4:W:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):4
5:W:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
-->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
-->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):5
6:W:terms#(N) -> c_13(sqr#(N),terms#(s(N)))
-->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):5
-->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):6
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: half#(s(s(X))) -> c_10(half#(X))
3: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
2: dbl#(s(X)) -> c_4(dbl#(X))
** Step 5.a:2: SimplifyRHS MAYBE
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
- Weak DPs:
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
5:W:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
-->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):5
6:W:terms#(N) -> c_13(sqr#(N),terms#(s(N)))
-->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):5
-->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):6
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
** Step 5.a:3: Failure MAYBE
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
- Weak DPs:
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
** Step 5.b:1: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):1
2:S:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
-->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):2
3:S:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):3
4:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):6
-->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
-->_3 dbl#(s(X)) -> c_4(dbl#(X)):1
5:S:terms#(N) -> c_13(sqr#(N),terms#(s(N)))
-->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):5
-->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
6:W:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):6
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: add#(s(X),Y) -> c_2(add#(X,Y))
** Step 5.b:2: SimplifyRHS MAYBE
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):1
2:S:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
-->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):2
3:S:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):3
4:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
-->_3 dbl#(s(X)) -> c_4(dbl#(X)):1
5:S:terms#(N) -> c_13(sqr#(N),terms#(s(N)))
-->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):5
-->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
** Step 5.b:3: UsableRules MAYBE
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
dbl#(s(X)) -> c_4(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
** Step 5.b:4: Decompose MAYBE
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
- Weak DPs:
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
Problem (S)
- Strict DPs:
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak DPs:
dbl#(s(X)) -> c_4(dbl#(X))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
*** Step 5.b:4.a:1: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
- Weak DPs:
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):1
2:W:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
-->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):2
3:W:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):3
4:W:sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
-->_2 dbl#(s(X)) -> c_4(dbl#(X)):1
-->_1 sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)):4
5:W:terms#(N) -> c_13(sqr#(N),terms#(s(N)))
-->_1 sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)):4
-->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):5
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: half#(s(s(X))) -> c_10(half#(X))
2: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
*** Step 5.b:4.a:2: Failure MAYBE
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
- Weak DPs:
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
*** Step 5.b:4.b:1: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak DPs:
dbl#(s(X)) -> c_4(dbl#(X))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
-->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):1
2:S:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):2
3:S:sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
-->_2 dbl#(s(X)) -> c_4(dbl#(X)):5
-->_1 sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)):3
4:S:terms#(N) -> c_13(sqr#(N),terms#(s(N)))
-->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):4
-->_1 sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)):3
5:W:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):5
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: dbl#(s(X)) -> c_4(dbl#(X))
*** Step 5.b:4.b:2: SimplifyRHS MAYBE
+ Considered Problem:
- Strict DPs:
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
-->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):1
2:S:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):2
3:S:sqr#(s(X)) -> c_12(sqr#(X),dbl#(X))
-->_1 sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)):3
4:S:terms#(N) -> c_13(sqr#(N),terms#(s(N)))
-->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):4
-->_1 sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)):3
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
sqr#(s(X)) -> c_12(sqr#(X))
*** Step 5.b:4.b:3: Decompose MAYBE
+ Considered Problem:
- Strict DPs:
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
- Strict DPs:
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
- Weak DPs:
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
Problem (S)
- Strict DPs:
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak DPs:
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
**** Step 5.b:4.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
- Weak DPs:
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
-->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):1
2:W:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):2
3:W:sqr#(s(X)) -> c_12(sqr#(X))
-->_1 sqr#(s(X)) -> c_12(sqr#(X)):3
4:W:terms#(N) -> c_13(sqr#(N),terms#(s(N)))
-->_1 sqr#(s(X)) -> c_12(sqr#(X)):3
-->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: terms#(N) -> c_13(sqr#(N),terms#(s(N)))
3: sqr#(s(X)) -> c_12(sqr#(X))
2: half#(s(s(X))) -> c_10(half#(X))
**** Step 5.b:4.b:3.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
The strictly oriented rules are moved into the weak component.
***** Step 5.b:4.b:3.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_6) = {1}
Following symbols are considered usable:
{add#,dbl#,first#,half#,sqr#,terms#}
TcT has computed the following interpretation:
p(0) = [0]
p(add) = [1] x1 + [2] x2 + [0]
p(cons) = [1] x2 + [1]
p(dbl) = [1]
p(first) = [2]
p(half) = [0]
p(nil) = [0]
p(recip) = [1]
p(s) = [1] x1 + [9]
p(sqr) = [2]
p(terms) = [1]
p(add#) = [1] x1 + [4] x2 + [0]
p(dbl#) = [2]
p(first#) = [1] x1 + [12] x2 + [8]
p(half#) = [2]
p(sqr#) = [2] x1 + [1]
p(terms#) = [2] x1 + [1]
p(c_1) = [1]
p(c_2) = [8] x1 + [1]
p(c_3) = [8]
p(c_4) = [1] x1 + [0]
p(c_5) = [2]
p(c_6) = [1] x1 + [13]
p(c_7) = [4]
p(c_8) = [0]
p(c_9) = [2]
p(c_10) = [1] x1 + [1]
p(c_11) = [0]
p(c_12) = [4] x1 + [2]
p(c_13) = [4] x1 + [2]
Following rules are strictly oriented:
first#(s(X),cons(Y,Z)) = [1] X + [12] Z + [29]
> [1] X + [12] Z + [21]
= c_6(first#(X,Z))
Following rules are (at-least) weakly oriented:
***** Step 5.b:4.b:3.a:2.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 5.b:4.b:3.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
-->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
***** Step 5.b:4.b:3.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
**** Step 5.b:4.b:3.b:1: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak DPs:
first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):1
2:S:sqr#(s(X)) -> c_12(sqr#(X))
-->_1 sqr#(s(X)) -> c_12(sqr#(X)):2
3:S:terms#(N) -> c_13(sqr#(N),terms#(s(N)))
-->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):3
-->_1 sqr#(s(X)) -> c_12(sqr#(X)):2
4:W:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
-->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
**** Step 5.b:4.b:3.b:2: Decompose MAYBE
+ Considered Problem:
- Strict DPs:
half#(s(s(X))) -> c_10(half#(X))
sqr#(s(X)) -> c_12(sqr#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
- Strict DPs:
half#(s(s(X))) -> c_10(half#(X))
- Weak DPs:
sqr#(s(X)) -> c_12(sqr#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
Problem (S)
- Strict DPs:
sqr#(s(X)) -> c_12(sqr#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak DPs:
half#(s(s(X))) -> c_10(half#(X))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
***** Step 5.b:4.b:3.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(X))) -> c_10(half#(X))
- Weak DPs:
sqr#(s(X)) -> c_12(sqr#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):1
2:W:sqr#(s(X)) -> c_12(sqr#(X))
-->_1 sqr#(s(X)) -> c_12(sqr#(X)):2
3:W:terms#(N) -> c_13(sqr#(N),terms#(s(N)))
-->_1 sqr#(s(X)) -> c_12(sqr#(X)):2
-->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: terms#(N) -> c_13(sqr#(N),terms#(s(N)))
2: sqr#(s(X)) -> c_12(sqr#(X))
***** Step 5.b:4.b:3.b:2.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(X))) -> c_10(half#(X))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: half#(s(s(X))) -> c_10(half#(X))
The strictly oriented rules are moved into the weak component.
****** Step 5.b:4.b:3.b:2.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(X))) -> c_10(half#(X))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_10) = {1}
Following symbols are considered usable:
{add#,dbl#,first#,half#,sqr#,terms#}
TcT has computed the following interpretation:
p(0) = [1]
p(add) = [2]
p(cons) = [0]
p(dbl) = [2] x1 + [8]
p(first) = [1] x1 + [0]
p(half) = [1] x1 + [2]
p(nil) = [1]
p(recip) = [1] x1 + [0]
p(s) = [1] x1 + [1]
p(sqr) = [2]
p(terms) = [1]
p(add#) = [1] x1 + [1]
p(dbl#) = [8]
p(first#) = [2] x1 + [2] x2 + [2]
p(half#) = [8] x1 + [3]
p(sqr#) = [2]
p(terms#) = [1] x1 + [1]
p(c_1) = [0]
p(c_2) = [1] x1 + [1]
p(c_3) = [4]
p(c_4) = [1]
p(c_5) = [1]
p(c_6) = [2] x1 + [0]
p(c_7) = [1]
p(c_8) = [0]
p(c_9) = [1]
p(c_10) = [1] x1 + [11]
p(c_11) = [2]
p(c_12) = [4] x1 + [0]
p(c_13) = [0]
Following rules are strictly oriented:
half#(s(s(X))) = [8] X + [19]
> [8] X + [14]
= c_10(half#(X))
Following rules are (at-least) weakly oriented:
****** Step 5.b:4.b:3.b:2.a:2.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
half#(s(s(X))) -> c_10(half#(X))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
****** Step 5.b:4.b:3.b:2.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
half#(s(s(X))) -> c_10(half#(X))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: half#(s(s(X))) -> c_10(half#(X))
****** Step 5.b:4.b:3.b:2.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
***** Step 5.b:4.b:3.b:2.b:1: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
sqr#(s(X)) -> c_12(sqr#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak DPs:
half#(s(s(X))) -> c_10(half#(X))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:sqr#(s(X)) -> c_12(sqr#(X))
-->_1 sqr#(s(X)) -> c_12(sqr#(X)):1
2:S:terms#(N) -> c_13(sqr#(N),terms#(s(N)))
-->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):2
-->_1 sqr#(s(X)) -> c_12(sqr#(X)):1
3:W:half#(s(s(X))) -> c_10(half#(X))
-->_1 half#(s(s(X))) -> c_10(half#(X)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: half#(s(s(X))) -> c_10(half#(X))
***** Step 5.b:4.b:3.b:2.b:2: Decompose MAYBE
+ Considered Problem:
- Strict DPs:
sqr#(s(X)) -> c_12(sqr#(X))
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
- Strict DPs:
sqr#(s(X)) -> c_12(sqr#(X))
- Weak DPs:
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
Problem (S)
- Strict DPs:
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak DPs:
sqr#(s(X)) -> c_12(sqr#(X))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
****** Step 5.b:4.b:3.b:2.b:2.a:1: Failure MAYBE
+ Considered Problem:
- Strict DPs:
sqr#(s(X)) -> c_12(sqr#(X))
- Weak DPs:
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
****** Step 5.b:4.b:3.b:2.b:2.b:1: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Weak DPs:
sqr#(s(X)) -> c_12(sqr#(X))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:terms#(N) -> c_13(sqr#(N),terms#(s(N)))
-->_1 sqr#(s(X)) -> c_12(sqr#(X)):2
-->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):1
2:W:sqr#(s(X)) -> c_12(sqr#(X))
-->_1 sqr#(s(X)) -> c_12(sqr#(X)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: sqr#(s(X)) -> c_12(sqr#(X))
****** Step 5.b:4.b:3.b:2.b:2.b:2: SimplifyRHS MAYBE
+ Considered Problem:
- Strict DPs:
terms#(N) -> c_13(sqr#(N),terms#(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:terms#(N) -> c_13(sqr#(N),terms#(s(N)))
-->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
terms#(N) -> c_13(terms#(s(N)))
****** Step 5.b:4.b:3.b:2.b:2.b:3: Failure MAYBE
+ Considered Problem:
- Strict DPs:
terms#(N) -> c_13(terms#(s(N)))
- Signature:
{add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1,terms#/1} / {0/0,cons/2
,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
,cons,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
MAYBE