* Step 1: Sum WORST_CASE(Omega(n^1),O(n^3))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__first(X1,X2)) -> first(X1,X2)
activate(n__terms(X)) -> terms(X)
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(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)),n__terms(s(N)))
terms(X) -> n__terms(X)
- Signature:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,half,sqr
,terms} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__first(X1,X2)) -> first(X1,X2)
activate(n__terms(X)) -> terms(X)
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(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)),n__terms(s(N)))
terms(X) -> n__terms(X)
- Signature:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,half,sqr
,terms} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
add(x,y){x -> s(x)} =
add(s(x),y) ->^+ s(add(x,y))
= C[add(x,y) = add(x,y){}]
** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__first(X1,X2)) -> first(X1,X2)
activate(n__terms(X)) -> terms(X)
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(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)),n__terms(s(N)))
terms(X) -> n__terms(X)
- Signature:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,half,sqr
,terms} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
activate#(X) -> c_1()
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
add#(0(),X) -> c_4()
add#(s(X),Y) -> c_5(add#(X,Y))
dbl#(0()) -> c_6()
dbl#(s(X)) -> c_7(dbl#(X))
first#(X1,X2) -> c_8()
first#(0(),X) -> c_9()
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
half#(0()) -> c_11()
half#(dbl(X)) -> c_12()
half#(s(0())) -> c_13()
half#(s(s(X))) -> c_14(half#(X))
sqr#(0()) -> c_15()
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_17(sqr#(N))
terms#(X) -> c_18()
Weak DPs
and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1()
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
add#(0(),X) -> c_4()
add#(s(X),Y) -> c_5(add#(X,Y))
dbl#(0()) -> c_6()
dbl#(s(X)) -> c_7(dbl#(X))
first#(X1,X2) -> c_8()
first#(0(),X) -> c_9()
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
half#(0()) -> c_11()
half#(dbl(X)) -> c_12()
half#(s(0())) -> c_13()
half#(s(s(X))) -> c_14(half#(X))
sqr#(0()) -> c_15()
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_17(sqr#(N))
terms#(X) -> c_18()
- Weak TRS:
activate(X) -> X
activate(n__first(X1,X2)) -> first(X1,X2)
activate(n__terms(X)) -> terms(X)
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(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)),n__terms(s(N)))
terms(X) -> n__terms(X)
- Signature:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,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)))
activate#(X) -> c_1()
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
add#(0(),X) -> c_4()
add#(s(X),Y) -> c_5(add#(X,Y))
dbl#(0()) -> c_6()
dbl#(s(X)) -> c_7(dbl#(X))
first#(X1,X2) -> c_8()
first#(0(),X) -> c_9()
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
half#(0()) -> c_11()
half#(dbl(X)) -> c_12()
half#(s(0())) -> c_13()
half#(s(s(X))) -> c_14(half#(X))
sqr#(0()) -> c_15()
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_17(sqr#(N))
terms#(X) -> c_18()
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1()
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
add#(0(),X) -> c_4()
add#(s(X),Y) -> c_5(add#(X,Y))
dbl#(0()) -> c_6()
dbl#(s(X)) -> c_7(dbl#(X))
first#(X1,X2) -> c_8()
first#(0(),X) -> c_9()
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
half#(0()) -> c_11()
half#(dbl(X)) -> c_12()
half#(s(0())) -> c_13()
half#(s(s(X))) -> c_14(half#(X))
sqr#(0()) -> c_15()
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_17(sqr#(N))
terms#(X) -> c_18()
- 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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,4,6,8,9,11,12,13,15,18}
by application of
Pre({1,4,6,8,9,11,12,13,15,18}) = {2,3,5,7,10,14,16,17}.
Here rules are labelled as follows:
1: activate#(X) -> c_1()
2: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
3: activate#(n__terms(X)) -> c_3(terms#(X))
4: add#(0(),X) -> c_4()
5: add#(s(X),Y) -> c_5(add#(X,Y))
6: dbl#(0()) -> c_6()
7: dbl#(s(X)) -> c_7(dbl#(X))
8: first#(X1,X2) -> c_8()
9: first#(0(),X) -> c_9()
10: first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
11: half#(0()) -> c_11()
12: half#(dbl(X)) -> c_12()
13: half#(s(0())) -> c_13()
14: half#(s(s(X))) -> c_14(half#(X))
15: sqr#(0()) -> c_15()
16: sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
17: terms#(N) -> c_17(sqr#(N))
18: terms#(X) -> c_18()
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
add#(s(X),Y) -> c_5(add#(X,Y))
dbl#(s(X)) -> c_7(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
half#(s(s(X))) -> c_14(half#(X))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_17(sqr#(N))
- Weak DPs:
activate#(X) -> c_1()
add#(0(),X) -> c_4()
dbl#(0()) -> c_6()
first#(X1,X2) -> c_8()
first#(0(),X) -> c_9()
half#(0()) -> c_11()
half#(dbl(X)) -> c_12()
half#(s(0())) -> c_13()
sqr#(0()) -> c_15()
terms#(X) -> c_18()
- 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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
-->_1 first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)):5
-->_1 first#(0(),X) -> c_9():13
-->_1 first#(X1,X2) -> c_8():12
2:S:activate#(n__terms(X)) -> c_3(terms#(X))
-->_1 terms#(N) -> c_17(sqr#(N)):8
-->_1 terms#(X) -> c_18():18
3:S:add#(s(X),Y) -> c_5(add#(X,Y))
-->_1 add#(0(),X) -> c_4():10
-->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3
4:S:dbl#(s(X)) -> c_7(dbl#(X))
-->_1 dbl#(0()) -> c_6():11
-->_1 dbl#(s(X)) -> c_7(dbl#(X)):4
5:S:first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
-->_1 activate#(X) -> c_1():9
-->_1 activate#(n__terms(X)) -> c_3(terms#(X)):2
-->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):1
6:S:half#(s(s(X))) -> c_14(half#(X))
-->_1 half#(s(0())) -> c_13():16
-->_1 half#(dbl(X)) -> c_12():15
-->_1 half#(0()) -> c_11():14
-->_1 half#(s(s(X))) -> c_14(half#(X)):6
7:S:sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_2 sqr#(0()) -> c_15():17
-->_3 dbl#(0()) -> c_6():11
-->_1 add#(0(),X) -> c_4():10
-->_2 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):7
-->_3 dbl#(s(X)) -> c_7(dbl#(X)):4
-->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3
8:S:terms#(N) -> c_17(sqr#(N))
-->_1 sqr#(0()) -> c_15():17
-->_1 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):7
9:W:activate#(X) -> c_1()
10:W:add#(0(),X) -> c_4()
11:W:dbl#(0()) -> c_6()
12:W:first#(X1,X2) -> c_8()
13:W:first#(0(),X) -> c_9()
14:W:half#(0()) -> c_11()
15:W:half#(dbl(X)) -> c_12()
16:W:half#(s(0())) -> c_13()
17:W:sqr#(0()) -> c_15()
18:W:terms#(X) -> c_18()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
14: half#(0()) -> c_11()
15: half#(dbl(X)) -> c_12()
16: half#(s(0())) -> c_13()
12: first#(X1,X2) -> c_8()
13: first#(0(),X) -> c_9()
18: terms#(X) -> c_18()
10: add#(0(),X) -> c_4()
11: dbl#(0()) -> c_6()
17: sqr#(0()) -> c_15()
9: activate#(X) -> c_1()
** Step 1.b:5: Decompose WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
add#(s(X),Y) -> c_5(add#(X,Y))
dbl#(s(X)) -> c_7(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
half#(s(s(X))) -> c_14(half#(X))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_17(sqr#(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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,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:
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
add#(s(X),Y) -> c_5(add#(X,Y))
dbl#(s(X)) -> c_7(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_17(sqr#(N))
- Weak DPs:
half#(s(s(X))) -> c_14(half#(X))
- 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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
Problem (S)
- Strict DPs:
half#(s(s(X))) -> c_14(half#(X))
- Weak DPs:
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
add#(s(X),Y) -> c_5(add#(X,Y))
dbl#(s(X)) -> c_7(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_17(sqr#(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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
*** Step 1.b:5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
add#(s(X),Y) -> c_5(add#(X,Y))
dbl#(s(X)) -> c_7(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_17(sqr#(N))
- Weak DPs:
half#(s(s(X))) -> c_14(half#(X))
- 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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
-->_1 first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)):5
2:S:activate#(n__terms(X)) -> c_3(terms#(X))
-->_1 terms#(N) -> c_17(sqr#(N)):8
3:S:add#(s(X),Y) -> c_5(add#(X,Y))
-->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3
4:S:dbl#(s(X)) -> c_7(dbl#(X))
-->_1 dbl#(s(X)) -> c_7(dbl#(X)):4
5:S:first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
-->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):1
-->_1 activate#(n__terms(X)) -> c_3(terms#(X)):2
6:W:half#(s(s(X))) -> c_14(half#(X))
-->_1 half#(s(s(X))) -> c_14(half#(X)):6
7:S:sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3
-->_3 dbl#(s(X)) -> c_7(dbl#(X)):4
-->_2 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):7
8:S:terms#(N) -> c_17(sqr#(N))
-->_1 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):7
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: half#(s(s(X))) -> c_14(half#(X))
*** Step 1.b:5.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
add#(s(X),Y) -> c_5(add#(X,Y))
dbl#(s(X)) -> c_7(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_17(sqr#(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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
2: activate#(n__terms(X)) -> c_3(terms#(X))
4: dbl#(s(X)) -> c_7(dbl#(X))
5: first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
Consider the set of all dependency pairs
1: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
2: activate#(n__terms(X)) -> c_3(terms#(X))
3: add#(s(X),Y) -> c_5(add#(X,Y))
4: dbl#(s(X)) -> c_7(dbl#(X))
5: first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
7: sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
8: terms#(N) -> c_17(sqr#(N))
Processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2))
SPACE(?,?)on application of the dependency pairs
{1,2,4,5}
These cover all (indirect) predecessors of dependency pairs
{1,2,4,5,8}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
**** Step 1.b:5.a:2.a:1: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
add#(s(X),Y) -> c_5(add#(X,Y))
dbl#(s(X)) -> c_7(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_17(sqr#(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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 2, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_5) = {1},
uargs(c_7) = {1},
uargs(c_10) = {1},
uargs(c_16) = {1,2,3},
uargs(c_17) = {1}
Following symbols are considered usable:
{activate#,add#,dbl#,first#,half#,sqr#,terms#}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[1]
p(activate) = [0]
[0]
[0]
p(add) = [0 0 0] [0]
[0 0 0] x1 + [0]
[1 1 0] [0]
p(cons) = [1 1 1] [0]
[0 0 0] x2 + [1]
[0 0 1] [0]
p(dbl) = [0 0 0] [0]
[0 0 0] x1 + [0]
[1 0 0] [0]
p(first) = [0]
[0]
[0]
p(half) = [0]
[0]
[0]
p(n__first) = [0 1 1] [1 1 0] [0]
[0 0 1] x1 + [0 0 1] x2 + [1]
[0 0 0] [0 0 0] [1]
p(n__terms) = [1 1 1] [1]
[0 0 1] x1 + [1]
[0 0 0] [1]
p(nil) = [0]
[0]
[0]
p(recip) = [0]
[0]
[0]
p(s) = [1 1 0] [0]
[0 0 1] x1 + [0]
[0 0 1] [1]
p(sqr) = [0 0 0] [1]
[0 0 0] x1 + [1]
[0 0 1] [0]
p(terms) = [0]
[0]
[0]
p(activate#) = [1 1 1] [0]
[1 1 0] x1 + [0]
[1 0 1] [1]
p(add#) = [0]
[1]
[1]
p(dbl#) = [0 0 1] [1]
[1 0 0] x1 + [1]
[0 0 0] [1]
p(first#) = [0 0 0] [1 1 1] [1]
[0 0 1] x1 + [1 0 0] x2 + [0]
[0 1 1] [0 0 0] [0]
p(half#) = [0]
[0]
[0]
p(sqr#) = [1 1 1] [0]
[1 1 0] x1 + [0]
[0 0 1] [1]
p(terms#) = [1 1 1] [0]
[0 0 1] x1 + [0]
[1 1 1] [0]
p(c_1) = [0]
[0]
[0]
p(c_2) = [1 0 1] [0]
[0 0 1] x1 + [1]
[0 1 0] [0]
p(c_3) = [1 1 0] [0]
[0 0 1] x1 + [0]
[1 0 0] [1]
p(c_4) = [0]
[0]
[0]
p(c_5) = [1 0 0] [0]
[0 0 1] x1 + [0]
[0 0 0] [0]
p(c_6) = [0]
[0]
[0]
p(c_7) = [1 0 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [1]
p(c_8) = [0]
[0]
[0]
p(c_9) = [0]
[0]
[0]
p(c_10) = [1 0 0] [0]
[0 0 0] x1 + [1]
[0 0 0] [0]
p(c_11) = [0]
[0]
[0]
p(c_12) = [0]
[0]
[0]
p(c_13) = [0]
[0]
[0]
p(c_14) = [0]
[0]
[0]
p(c_15) = [0]
[0]
[0]
p(c_16) = [1 0 0] [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 1] [0 0 0] [0 0 0] [1]
p(c_17) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 0] [0]
p(c_18) = [0]
[0]
[0]
Following rules are strictly oriented:
activate#(n__first(X1,X2)) = [0 1 2] [1 1 1] [2]
[0 1 2] X1 + [1 1 1] X2 + [1]
[0 1 1] [1 1 0] [2]
> [0 1 1] [1 1 1] [1]
[0 1 1] X1 + [0 0 0] X2 + [1]
[0 0 1] [1 0 0] [0]
= c_2(first#(X1,X2))
activate#(n__terms(X)) = [1 1 2] [3]
[1 1 2] X + [2]
[1 1 1] [3]
> [1 1 2] [0]
[1 1 1] X + [0]
[1 1 1] [1]
= c_3(terms#(X))
dbl#(s(X)) = [0 0 1] [2]
[1 1 0] X + [1]
[0 0 0] [1]
> [0 0 1] [1]
[1 0 0] X + [1]
[0 0 0] [1]
= c_7(dbl#(X))
first#(s(X),cons(Y,Z)) = [0 0 0] [1 1 2] [2]
[0 0 1] X + [1 1 1] Z + [1]
[0 0 2] [0 0 0] [1]
> [1 1 1] [0]
[0 0 0] Z + [1]
[0 0 0] [0]
= c_10(activate#(Z))
Following rules are (at-least) weakly oriented:
add#(s(X),Y) = [0]
[1]
[1]
>= [0]
[1]
[0]
= c_5(add#(X,Y))
sqr#(s(X)) = [1 1 2] [1]
[1 1 1] X + [0]
[0 0 1] [2]
>= [1 1 2] [1]
[0 0 0] X + [0]
[0 0 0] [2]
= c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) = [1 1 1] [0]
[0 0 1] N + [0]
[1 1 1] [0]
>= [1 1 1] [0]
[0 0 0] N + [0]
[0 0 0] [0]
= c_17(sqr#(N))
**** Step 1.b:5.a:2.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_5(add#(X,Y))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_17(sqr#(N))
- Weak DPs:
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
dbl#(s(X)) -> c_7(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
- 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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 1.b:5.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_5(add#(X,Y))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
- Weak DPs:
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
dbl#(s(X)) -> c_7(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
terms#(N) -> c_17(sqr#(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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:add#(s(X),Y) -> c_5(add#(X,Y))
-->_1 add#(s(X),Y) -> c_5(add#(X,Y)):1
2:S:sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_3 dbl#(s(X)) -> c_7(dbl#(X)):5
-->_2 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2
-->_1 add#(s(X),Y) -> c_5(add#(X,Y)):1
3:W:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
-->_1 first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)):6
4:W:activate#(n__terms(X)) -> c_3(terms#(X))
-->_1 terms#(N) -> c_17(sqr#(N)):7
5:W:dbl#(s(X)) -> c_7(dbl#(X))
-->_1 dbl#(s(X)) -> c_7(dbl#(X)):5
6:W:first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
-->_1 activate#(n__terms(X)) -> c_3(terms#(X)):4
-->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):3
7:W:terms#(N) -> c_17(sqr#(N))
-->_1 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2
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_7(dbl#(X))
**** Step 1.b:5.a:2.b:2: SimplifyRHS WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_5(add#(X,Y))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
- Weak DPs:
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
terms#(N) -> c_17(sqr#(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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:add#(s(X),Y) -> c_5(add#(X,Y))
-->_1 add#(s(X),Y) -> c_5(add#(X,Y)):1
2:S:sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_2 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2
-->_1 add#(s(X),Y) -> c_5(add#(X,Y)):1
3:W:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
-->_1 first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)):6
4:W:activate#(n__terms(X)) -> c_3(terms#(X))
-->_1 terms#(N) -> c_17(sqr#(N)):7
6:W:first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
-->_1 activate#(n__terms(X)) -> c_3(terms#(X)):4
-->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):3
7:W:terms#(N) -> c_17(sqr#(N))
-->_1 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X))
**** Step 1.b:5.a:2.b:3: DecomposeDG WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_5(add#(X,Y))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X))
- Weak DPs:
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
terms#(N) -> c_17(sqr#(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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X))
terms#(N) -> c_17(sqr#(N))
and a lower component
add#(s(X),Y) -> c_5(add#(X,Y))
Further, following extension rules are added to the lower component.
activate#(n__first(X1,X2)) -> first#(X1,X2)
activate#(n__terms(X)) -> terms#(X)
first#(s(X),cons(Y,Z)) -> activate#(Z)
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> sqr#(X)
terms#(N) -> sqr#(N)
***** Step 1.b:5.a:2.b:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X))
- Weak DPs:
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
terms#(N) -> c_17(sqr#(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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,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: sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X))
The strictly oriented rules are moved into the weak component.
****** Step 1.b:5.a:2.b:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X))
- Weak DPs:
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
terms#(N) -> c_17(sqr#(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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,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_2) = {1},
uargs(c_3) = {1},
uargs(c_10) = {1},
uargs(c_16) = {1,2},
uargs(c_17) = {1}
Following symbols are considered usable:
{activate#,add#,dbl#,first#,half#,sqr#,terms#}
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [2] x1 + [1]
p(add) = [3] x1 + [5] x2 + [8]
p(cons) = [1] x2 + [1]
p(dbl) = [2] x1 + [2]
p(first) = [1] x1 + [1] x2 + [1]
p(half) = [2]
p(n__first) = [1] x1 + [1] x2 + [1]
p(n__terms) = [1] x1 + [0]
p(nil) = [1]
p(recip) = [1]
p(s) = [1] x1 + [1]
p(sqr) = [2] x1 + [1]
p(terms) = [1] x1 + [1]
p(activate#) = [8] x1 + [12]
p(add#) = [0]
p(dbl#) = [8] x1 + [1]
p(first#) = [8] x1 + [8] x2 + [3]
p(half#) = [2] x1 + [0]
p(sqr#) = [4] x1 + [0]
p(terms#) = [8] x1 + [1]
p(c_1) = [1]
p(c_2) = [1] x1 + [8]
p(c_3) = [1] x1 + [10]
p(c_4) = [1]
p(c_5) = [1] x1 + [2]
p(c_6) = [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [4]
p(c_9) = [1]
p(c_10) = [1] x1 + [1]
p(c_11) = [1]
p(c_12) = [1]
p(c_13) = [1]
p(c_14) = [0]
p(c_15) = [0]
p(c_16) = [8] x1 + [1] x2 + [2]
p(c_17) = [2] x1 + [0]
p(c_18) = [0]
Following rules are strictly oriented:
sqr#(s(X)) = [4] X + [4]
> [4] X + [2]
= c_16(add#(sqr(X),dbl(X)),sqr#(X))
Following rules are (at-least) weakly oriented:
activate#(n__first(X1,X2)) = [8] X1 + [8] X2 + [20]
>= [8] X1 + [8] X2 + [11]
= c_2(first#(X1,X2))
activate#(n__terms(X)) = [8] X + [12]
>= [8] X + [11]
= c_3(terms#(X))
first#(s(X),cons(Y,Z)) = [8] X + [8] Z + [19]
>= [8] Z + [13]
= c_10(activate#(Z))
terms#(N) = [8] N + [1]
>= [8] N + [0]
= c_17(sqr#(N))
****** Step 1.b:5.a:2.b:3.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X))
terms#(N) -> c_17(sqr#(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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
****** Step 1.b:5.a:2.b:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X))
terms#(N) -> c_17(sqr#(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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
-->_1 first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)):3
2:W:activate#(n__terms(X)) -> c_3(terms#(X))
-->_1 terms#(N) -> c_17(sqr#(N)):5
3:W:first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
-->_1 activate#(n__terms(X)) -> c_3(terms#(X)):2
-->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):1
4:W:sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X))
-->_2 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X)):4
5:W:terms#(N) -> c_17(sqr#(N))
-->_1 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X)):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
3: first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
2: activate#(n__terms(X)) -> c_3(terms#(X))
5: terms#(N) -> c_17(sqr#(N))
4: sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X))
****** Step 1.b:5.a:2.b:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- 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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
***** Step 1.b:5.a:2.b:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_5(add#(X,Y))
- Weak DPs:
activate#(n__first(X1,X2)) -> first#(X1,X2)
activate#(n__terms(X)) -> terms#(X)
first#(s(X),cons(Y,Z)) -> activate#(Z)
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> sqr#(X)
terms#(N) -> sqr#(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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: add#(s(X),Y) -> c_5(add#(X,Y))
The strictly oriented rules are moved into the weak component.
****** Step 1.b:5.a:2.b:3.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_5(add#(X,Y))
- Weak DPs:
activate#(n__first(X1,X2)) -> first#(X1,X2)
activate#(n__terms(X)) -> terms#(X)
first#(s(X),cons(Y,Z)) -> activate#(Z)
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> sqr#(X)
terms#(N) -> sqr#(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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_5) = {1}
Following symbols are considered usable:
{add,dbl,sqr,activate#,add#,dbl#,first#,half#,sqr#,terms#}
TcT has computed the following interpretation:
p(0) = 0
p(activate) = 1
p(add) = x1 + x2
p(cons) = x2
p(dbl) = 2*x1
p(first) = 1 + 4*x2^2
p(half) = 2 + 2*x1 + x1^2
p(n__first) = x1 + x2
p(n__terms) = x1
p(nil) = 0
p(recip) = 0
p(s) = 1 + x1
p(sqr) = x1^2
p(terms) = 1 + x1 + x1^2
p(activate#) = 6 + 3*x1 + 6*x1^2
p(add#) = x1
p(dbl#) = 2 + x1 + 4*x1^2
p(first#) = 6 + x1 + 6*x1*x2 + 2*x1^2 + 3*x2 + 6*x2^2
p(half#) = 4 + 4*x1 + x1^2
p(sqr#) = 4*x1^2
p(terms#) = 3 + 4*x1^2
p(c_1) = 1
p(c_2) = 0
p(c_3) = 0
p(c_4) = 0
p(c_5) = x1
p(c_6) = 1
p(c_7) = x1
p(c_8) = 0
p(c_9) = 1
p(c_10) = 1
p(c_11) = 0
p(c_12) = 0
p(c_13) = 1
p(c_14) = 0
p(c_15) = 1
p(c_16) = 1 + x1
p(c_17) = x1
p(c_18) = 1
Following rules are strictly oriented:
add#(s(X),Y) = 1 + X
> X
= c_5(add#(X,Y))
Following rules are (at-least) weakly oriented:
activate#(n__first(X1,X2)) = 6 + 3*X1 + 12*X1*X2 + 6*X1^2 + 3*X2 + 6*X2^2
>= 6 + X1 + 6*X1*X2 + 2*X1^2 + 3*X2 + 6*X2^2
= first#(X1,X2)
activate#(n__terms(X)) = 6 + 3*X + 6*X^2
>= 3 + 4*X^2
= terms#(X)
first#(s(X),cons(Y,Z)) = 9 + 5*X + 6*X*Z + 2*X^2 + 9*Z + 6*Z^2
>= 6 + 3*Z + 6*Z^2
= activate#(Z)
sqr#(s(X)) = 4 + 8*X + 4*X^2
>= X^2
= add#(sqr(X),dbl(X))
sqr#(s(X)) = 4 + 8*X + 4*X^2
>= 4*X^2
= sqr#(X)
terms#(N) = 3 + 4*N^2
>= 4*N^2
= sqr#(N)
add(0(),X) = X
>= X
= X
add(s(X),Y) = 1 + X + Y
>= 1 + X + Y
= s(add(X,Y))
dbl(0()) = 0
>= 0
= 0()
dbl(s(X)) = 2 + 2*X
>= 2 + 2*X
= s(s(dbl(X)))
sqr(0()) = 0
>= 0
= 0()
sqr(s(X)) = 1 + 2*X + X^2
>= 1 + 2*X + X^2
= s(add(sqr(X),dbl(X)))
****** Step 1.b:5.a:2.b:3.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
activate#(n__first(X1,X2)) -> first#(X1,X2)
activate#(n__terms(X)) -> terms#(X)
add#(s(X),Y) -> c_5(add#(X,Y))
first#(s(X),cons(Y,Z)) -> activate#(Z)
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> sqr#(X)
terms#(N) -> sqr#(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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
****** Step 1.b:5.a:2.b:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
activate#(n__first(X1,X2)) -> first#(X1,X2)
activate#(n__terms(X)) -> terms#(X)
add#(s(X),Y) -> c_5(add#(X,Y))
first#(s(X),cons(Y,Z)) -> activate#(Z)
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> sqr#(X)
terms#(N) -> sqr#(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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:activate#(n__first(X1,X2)) -> first#(X1,X2)
-->_1 first#(s(X),cons(Y,Z)) -> activate#(Z):4
2:W:activate#(n__terms(X)) -> terms#(X)
-->_1 terms#(N) -> sqr#(N):7
3:W:add#(s(X),Y) -> c_5(add#(X,Y))
-->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3
4:W:first#(s(X),cons(Y,Z)) -> activate#(Z)
-->_1 activate#(n__terms(X)) -> terms#(X):2
-->_1 activate#(n__first(X1,X2)) -> first#(X1,X2):1
5:W:sqr#(s(X)) -> add#(sqr(X),dbl(X))
-->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3
6:W:sqr#(s(X)) -> sqr#(X)
-->_1 sqr#(s(X)) -> sqr#(X):6
-->_1 sqr#(s(X)) -> add#(sqr(X),dbl(X)):5
7:W:terms#(N) -> sqr#(N)
-->_1 sqr#(s(X)) -> sqr#(X):6
-->_1 sqr#(s(X)) -> add#(sqr(X),dbl(X)):5
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: activate#(n__first(X1,X2)) -> first#(X1,X2)
4: first#(s(X),cons(Y,Z)) -> activate#(Z)
2: activate#(n__terms(X)) -> terms#(X)
7: terms#(N) -> sqr#(N)
6: sqr#(s(X)) -> sqr#(X)
5: sqr#(s(X)) -> add#(sqr(X),dbl(X))
3: add#(s(X),Y) -> c_5(add#(X,Y))
****** Step 1.b:5.a:2.b:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- 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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 1.b:5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(X))) -> c_14(half#(X))
- Weak DPs:
activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
activate#(n__terms(X)) -> c_3(terms#(X))
add#(s(X),Y) -> c_5(add#(X,Y))
dbl#(s(X)) -> c_7(dbl#(X))
first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_17(sqr#(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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:half#(s(s(X))) -> c_14(half#(X))
-->_1 half#(s(s(X))) -> c_14(half#(X)):1
2:W:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
-->_1 first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)):6
3:W:activate#(n__terms(X)) -> c_3(terms#(X))
-->_1 terms#(N) -> c_17(sqr#(N)):8
4:W:add#(s(X),Y) -> c_5(add#(X,Y))
-->_1 add#(s(X),Y) -> c_5(add#(X,Y)):4
5:W:dbl#(s(X)) -> c_7(dbl#(X))
-->_1 dbl#(s(X)) -> c_7(dbl#(X)):5
6:W:first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
-->_1 activate#(n__terms(X)) -> c_3(terms#(X)):3
-->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):2
7:W:sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_2 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):7
-->_3 dbl#(s(X)) -> c_7(dbl#(X)):5
-->_1 add#(s(X),Y) -> c_5(add#(X,Y)):4
8:W:terms#(N) -> c_17(sqr#(N))
-->_1 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):7
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
6: first#(s(X),cons(Y,Z)) -> c_10(activate#(Z))
3: activate#(n__terms(X)) -> c_3(terms#(X))
8: terms#(N) -> c_17(sqr#(N))
7: sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
4: add#(s(X),Y) -> c_5(add#(X,Y))
5: dbl#(s(X)) -> c_7(dbl#(X))
*** Step 1.b:5.b:2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(X))) -> c_14(half#(X))
- 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:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
half#(s(s(X))) -> c_14(half#(X))
*** Step 1.b:5.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(X))) -> c_14(half#(X))
- Signature:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,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_14(half#(X))
The strictly oriented rules are moved into the weak component.
**** Step 1.b:5.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(X))) -> c_14(half#(X))
- Signature:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,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_14) = {1}
Following symbols are considered usable:
{activate#,add#,dbl#,first#,half#,sqr#,terms#}
TcT has computed the following interpretation:
p(0) = [8]
p(activate) = [8]
p(add) = [1]
p(cons) = [1]
p(dbl) = [1]
p(first) = [1] x1 + [8] x2 + [0]
p(half) = [1] x1 + [0]
p(n__first) = [1] x2 + [0]
p(n__terms) = [1]
p(nil) = [1]
p(recip) = [1] x1 + [1]
p(s) = [1] x1 + [2]
p(sqr) = [2] x1 + [1]
p(terms) = [8] x1 + [1]
p(activate#) = [1] x1 + [1]
p(add#) = [8] x1 + [8]
p(dbl#) = [0]
p(first#) = [2] x1 + [1] x2 + [1]
p(half#) = [4] x1 + [0]
p(sqr#) = [2] x1 + [1]
p(terms#) = [1] x1 + [0]
p(c_1) = [8]
p(c_2) = [1] x1 + [1]
p(c_3) = [1]
p(c_4) = [1]
p(c_5) = [2] x1 + [1]
p(c_6) = [0]
p(c_7) = [1]
p(c_8) = [8]
p(c_9) = [0]
p(c_10) = [2] x1 + [2]
p(c_11) = [2]
p(c_12) = [2]
p(c_13) = [0]
p(c_14) = [1] x1 + [15]
p(c_15) = [0]
p(c_16) = [2]
p(c_17) = [1] x1 + [1]
p(c_18) = [4]
Following rules are strictly oriented:
half#(s(s(X))) = [4] X + [16]
> [4] X + [15]
= c_14(half#(X))
Following rules are (at-least) weakly oriented:
**** Step 1.b:5.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
half#(s(s(X))) -> c_14(half#(X))
- Signature:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 1.b:5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
half#(s(s(X))) -> c_14(half#(X))
- Signature:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:half#(s(s(X))) -> c_14(half#(X))
-->_1 half#(s(s(X))) -> c_14(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_14(half#(X))
**** Step 1.b:5.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1
,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1
,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr#
,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^3))