*** 1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
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)
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {activate,add,dbl,first,half,sqr,terms}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
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.
*** 1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
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()
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
UsableRules
Proof:
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()
*** 1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
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()
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
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()
*** 1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
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 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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()
*** 1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
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 DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
half#(s(s(X))) -> c_14(half#(X))
Weak TRS 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Problem (S)
Strict DP Rules:
half#(s(s(X))) -> c_14(half#(X))
Strict TRS Rules:
Weak DP Rules:
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 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
*** 1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
half#(s(s(X))) -> c_14(half#(X))
Weak TRS 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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__terms(X)) -> c_3(terms#(X)):2
-->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):1
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))
-->_3 dbl#(s(X)) -> c_7(dbl#(X)):4
-->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3
-->_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))
*** 1.1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} 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))
8: terms#(N) -> c_17(sqr#(N))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{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 any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
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
p(activate) = 1 + 2*x1 + x1^2
p(add) = 7*x1*x2 + 4*x2 + x2^2
p(cons) = 1 + x1 + x2
p(dbl) = x1
p(first) = 1 + 4*x1
p(half) = 0
p(n__first) = x1 + x2
p(n__terms) = x1
p(nil) = 0
p(recip) = 0
p(s) = 1 + x1
p(sqr) = 0
p(terms) = 1 + x1 + 2*x1^2
p(activate#) = 7 + 2*x1 + 4*x1^2
p(add#) = 2
p(dbl#) = 2*x1
p(first#) = 4 + x1*x2 + 4*x1^2 + 4*x2^2
p(half#) = 0
p(sqr#) = 2*x1^2
p(terms#) = 2 + 2*x1 + 2*x1^2
p(c_1) = 1
p(c_2) = x1
p(c_3) = x1
p(c_4) = 0
p(c_5) = x1
p(c_6) = 0
p(c_7) = x1
p(c_8) = 1
p(c_9) = 0
p(c_10) = x1
p(c_11) = 0
p(c_12) = 1
p(c_13) = 0
p(c_14) = 0
p(c_15) = 0
p(c_16) = x1 + x2 + x3
p(c_17) = 1 + x1
p(c_18) = 0
Following rules are strictly oriented:
activate#(n__first(X1,X2)) = 7 + 2*X1 + 8*X1*X2 + 4*X1^2 + 2*X2 + 4*X2^2
> 4 + X1*X2 + 4*X1^2 + 4*X2^2
= c_2(first#(X1,X2))
activate#(n__terms(X)) = 7 + 2*X + 4*X^2
> 2 + 2*X + 2*X^2
= c_3(terms#(X))
dbl#(s(X)) = 2 + 2*X
> 2*X
= c_7(dbl#(X))
first#(s(X),cons(Y,Z)) = 13 + 9*X + X*Y + X*Z + 4*X^2 + 9*Y + 8*Y*Z + 4*Y^2 + 9*Z + 4*Z^2
> 7 + 2*Z + 4*Z^2
= c_10(activate#(Z))
terms#(N) = 2 + 2*N + 2*N^2
> 1 + 2*N^2
= c_17(sqr#(N))
Following rules are (at-least) weakly oriented:
add#(s(X),Y) = 2
>= 2
= c_5(add#(X,Y))
sqr#(s(X)) = 2 + 4*X + 2*X^2
>= 2 + 2*X + 2*X^2
= c_16(add#(sqr(X),dbl(X))
,sqr#(X)
,dbl#(X))
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
add#(s(X),Y) -> c_5(add#(X,Y))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
Strict TRS Rules:
Weak DP Rules:
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 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.2 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
add#(s(X),Y) -> c_5(add#(X,Y))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
Strict TRS Rules:
Weak DP Rules:
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 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.1.1.2.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
add#(s(X),Y) -> c_5(add#(X,Y))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
Strict TRS Rules:
Weak DP Rules:
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 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
SimplifyRHS
Proof:
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))
*** 1.1.1.1.1.1.1.2.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
add#(s(X),Y) -> c_5(add#(X,Y))
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X))
Strict TRS Rules:
Weak DP Rules:
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 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
Proof:
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)
*** 1.1.1.1.1.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X))
Strict TRS Rules:
Weak DP Rules:
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 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} 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.
*** 1.1.1.1.1.1.1.2.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X))
Strict TRS Rules:
Weak DP Rules:
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 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{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 any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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) = [6]
p(activate) = [1]
p(add) = [0]
p(cons) = [1] x2 + [0]
p(dbl) = [8]
p(first) = [4] x2 + [0]
p(half) = [1] x1 + [1]
p(n__first) = [1] x1 + [1] x2 + [2]
p(n__terms) = [1] x1 + [2]
p(nil) = [0]
p(recip) = [1]
p(s) = [1] x1 + [8]
p(sqr) = [2] x1 + [6]
p(terms) = [1] x1 + [1]
p(activate#) = [8] x1 + [1]
p(add#) = [0]
p(dbl#) = [1] x1 + [1]
p(first#) = [8] x2 + [4]
p(half#) = [1] x1 + [0]
p(sqr#) = [1] x1 + [1]
p(terms#) = [1] x1 + [8]
p(c_1) = [0]
p(c_2) = [1] x1 + [8]
p(c_3) = [2] x1 + [0]
p(c_4) = [1]
p(c_5) = [1] x1 + [1]
p(c_6) = [0]
p(c_7) = [1] x1 + [2]
p(c_8) = [1]
p(c_9) = [1]
p(c_10) = [1] x1 + [3]
p(c_11) = [1]
p(c_12) = [1]
p(c_13) = [1]
p(c_14) = [1] x1 + [2]
p(c_15) = [2]
p(c_16) = [1] x1 + [1] x2 + [5]
p(c_17) = [1] x1 + [7]
p(c_18) = [1]
Following rules are strictly oriented:
sqr#(s(X)) = [1] X + [9]
> [1] X + [6]
= 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 + [17]
>= [8] X2 + [12]
= c_2(first#(X1,X2))
activate#(n__terms(X)) = [8] X + [17]
>= [2] X + [16]
= c_3(terms#(X))
first#(s(X),cons(Y,Z)) = [8] Z + [4]
>= [8] Z + [4]
= c_10(activate#(Z))
terms#(N) = [1] N + [8]
>= [1] N + [8]
= c_17(sqr#(N))
*** 1.1.1.1.1.1.1.2.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.2.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.1.1.2.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.1.2.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
add#(s(X),Y) -> c_5(add#(X,Y))
Strict TRS Rules:
Weak DP Rules:
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 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} 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.
*** 1.1.1.1.1.1.1.2.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
add#(s(X),Y) -> c_5(add#(X,Y))
Strict TRS Rules:
Weak DP Rules:
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 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{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 any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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) = 4*x1 + x1^2
p(add) = 1 + x1 + x2
p(cons) = 1 + x1 + x2
p(dbl) = 1 + 2*x1
p(first) = x1 + x1^2
p(half) = 1 + 2*x1 + x1^2
p(n__first) = x1 + x2
p(n__terms) = x1
p(nil) = 0
p(recip) = 1
p(s) = 1 + x1
p(sqr) = 1 + 2*x1 + x1^2
p(terms) = 1 + x1
p(activate#) = 6 + 6*x1^2
p(add#) = 2 + x1
p(dbl#) = 2
p(first#) = 5 + 6*x2^2
p(half#) = 4*x1
p(sqr#) = 4 + 2*x1^2
p(terms#) = 6 + 5*x1^2
p(c_1) = 0
p(c_2) = 1
p(c_3) = 0
p(c_4) = 0
p(c_5) = x1
p(c_6) = 0
p(c_7) = 1 + x1
p(c_8) = 1
p(c_9) = 0
p(c_10) = 0
p(c_11) = 1
p(c_12) = 0
p(c_13) = 0
p(c_14) = x1
p(c_15) = 0
p(c_16) = 1
p(c_17) = 0
p(c_18) = 1
Following rules are strictly oriented:
add#(s(X),Y) = 3 + X
> 2 + X
= c_5(add#(X,Y))
Following rules are (at-least) weakly oriented:
activate#(n__first(X1,X2)) = 6 + 12*X1*X2 + 6*X1^2 + 6*X2^2
>= 5 + 6*X2^2
= first#(X1,X2)
activate#(n__terms(X)) = 6 + 6*X^2
>= 6 + 5*X^2
= terms#(X)
first#(s(X),cons(Y,Z)) = 11 + 12*Y + 12*Y*Z + 6*Y^2 + 12*Z + 6*Z^2
>= 6 + 6*Z^2
= activate#(Z)
sqr#(s(X)) = 6 + 4*X + 2*X^2
>= 3 + 2*X + X^2
= add#(sqr(X),dbl(X))
sqr#(s(X)) = 6 + 4*X + 2*X^2
>= 4 + 2*X^2
= sqr#(X)
terms#(N) = 6 + 5*N^2
>= 4 + 2*N^2
= sqr#(N)
add(0(),X) = 1 + X
>= X
= X
add(s(X),Y) = 2 + X + Y
>= 2 + X + Y
= s(add(X,Y))
dbl(0()) = 1
>= 0
= 0()
dbl(s(X)) = 3 + 2*X
>= 3 + 2*X
= s(s(dbl(X)))
sqr(0()) = 1
>= 0
= 0()
sqr(s(X)) = 4 + 4*X + X^2
>= 4 + 4*X + X^2
= s(add(sqr(X),dbl(X)))
*** 1.1.1.1.1.1.1.2.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.2.1.1.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.1.1.2.1.1.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
half#(s(s(X))) -> c_14(half#(X))
Strict TRS Rules:
Weak DP Rules:
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 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
half#(s(s(X))) -> c_14(half#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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)))
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
half#(s(s(X))) -> c_14(half#(X))
*** 1.1.1.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
half#(s(s(X))) -> c_14(half#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: half#(s(s(X))) -> c_14(half#(X))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
half#(s(s(X))) -> c_14(half#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{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 any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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) = [0]
p(activate) = [0]
p(add) = [0]
p(cons) = [1] x1 + [1] x2 + [0]
p(dbl) = [4] x1 + [0]
p(first) = [0]
p(half) = [8] x1 + [2]
p(n__first) = [1]
p(n__terms) = [0]
p(nil) = [1]
p(recip) = [1] x1 + [0]
p(s) = [1] x1 + [4]
p(sqr) = [2]
p(terms) = [1]
p(activate#) = [4] x1 + [1]
p(add#) = [8] x1 + [0]
p(dbl#) = [2] x1 + [1]
p(first#) = [1] x1 + [1] x2 + [1]
p(half#) = [2] x1 + [1]
p(sqr#) = [0]
p(terms#) = [4] x1 + [1]
p(c_1) = [0]
p(c_2) = [1]
p(c_3) = [1] x1 + [1]
p(c_4) = [1]
p(c_5) = [4] x1 + [1]
p(c_6) = [8]
p(c_7) = [0]
p(c_8) = [1]
p(c_9) = [1]
p(c_10) = [4]
p(c_11) = [1]
p(c_12) = [2]
p(c_13) = [1]
p(c_14) = [1] x1 + [14]
p(c_15) = [1]
p(c_16) = [8] x3 + [0]
p(c_17) = [1]
p(c_18) = [0]
Following rules are strictly oriented:
half#(s(s(X))) = [2] X + [17]
> [2] X + [15]
= c_14(half#(X))
Following rules are (at-least) weakly oriented:
*** 1.1.1.1.1.2.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
half#(s(s(X))) -> c_14(half#(X))
Weak TRS Rules:
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.2.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
half#(s(s(X))) -> c_14(half#(X))
Weak TRS Rules:
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.2.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {activate#,add#,dbl#,first#,half#,sqr#,terms#}/{0,cons,n__first,n__terms,nil,recip,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).