* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
2ndsneg(0(),Z) -> rnil()
2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
2ndspos(0(),Z) -> rnil()
2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
pi(X) -> 2ndspos(X,from(0()))
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
square(X) -> times(X,X)
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1
,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,from,pi,plus,square
,times} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
2ndsneg(0(),Z) -> rnil()
2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
2ndspos(0(),Z) -> rnil()
2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
pi(X) -> 2ndspos(X,from(0()))
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
square(X) -> times(X,X)
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1
,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,from,pi,plus,square
,times} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
plus(x,y){x -> s(x)} =
plus(s(x),y) ->^+ s(plus(x,y))
= C[plus(x,y) = plus(x,y){}]
** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
2ndsneg(0(),Z) -> rnil()
2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
2ndspos(0(),Z) -> rnil()
2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
pi(X) -> 2ndspos(X,from(0()))
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
square(X) -> times(X,X)
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1
,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,from,pi,plus,square
,times} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
2ndsneg#(0(),Z) -> c_1()
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z))
2ndspos#(0(),Z) -> c_4()
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z))
activate#(X) -> c_7()
activate#(n__from(X)) -> c_8(from#(X))
from#(X) -> c_9()
from#(X) -> c_10()
pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0()))
plus#(0(),Y) -> c_12()
plus#(s(X),Y) -> c_13(plus#(X,Y))
square#(X) -> c_14(times#(X,X))
times#(0(),Y) -> c_15()
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
Weak DPs
and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
2ndsneg#(0(),Z) -> c_1()
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z))
2ndspos#(0(),Z) -> c_4()
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z))
activate#(X) -> c_7()
activate#(n__from(X)) -> c_8(from#(X))
from#(X) -> c_9()
from#(X) -> c_10()
pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0()))
plus#(0(),Y) -> c_12()
plus#(s(X),Y) -> c_13(plus#(X,Y))
square#(X) -> c_14(times#(X,X))
times#(0(),Y) -> c_15()
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak TRS:
2ndsneg(0(),Z) -> rnil()
2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
2ndspos(0(),Z) -> rnil()
2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
pi(X) -> 2ndspos(X,from(0()))
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
square(X) -> times(X,X)
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/2,c_7/0,c_8/1,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
2ndsneg#(0(),Z) -> c_1()
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z))
2ndspos#(0(),Z) -> c_4()
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z))
activate#(X) -> c_7()
activate#(n__from(X)) -> c_8(from#(X))
from#(X) -> c_9()
from#(X) -> c_10()
pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0()))
plus#(0(),Y) -> c_12()
plus#(s(X),Y) -> c_13(plus#(X,Y))
square#(X) -> c_14(times#(X,X))
times#(0(),Y) -> c_15()
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
2ndsneg#(0(),Z) -> c_1()
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z))
2ndspos#(0(),Z) -> c_4()
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z))
activate#(X) -> c_7()
activate#(n__from(X)) -> c_8(from#(X))
from#(X) -> c_9()
from#(X) -> c_10()
pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0()))
plus#(0(),Y) -> c_12()
plus#(s(X),Y) -> c_13(plus#(X,Y))
square#(X) -> c_14(times#(X,X))
times#(0(),Y) -> c_15()
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/2,c_7/0,c_8/1,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,4,7,9,10,12,15}
by application of
Pre({1,4,7,9,10,12,15}) = {2,3,5,6,8,11,13,14,16}.
Here rules are labelled as follows:
1: 2ndsneg#(0(),Z) -> c_1()
2: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z))
3: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z))
4: 2ndspos#(0(),Z) -> c_4()
5: 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z))
6: 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z))
7: activate#(X) -> c_7()
8: activate#(n__from(X)) -> c_8(from#(X))
9: from#(X) -> c_9()
10: from#(X) -> c_10()
11: pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0()))
12: plus#(0(),Y) -> c_12()
13: plus#(s(X),Y) -> c_13(plus#(X,Y))
14: square#(X) -> c_14(times#(X,X))
15: times#(0(),Y) -> c_15()
16: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
** Step 1.b:4: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z))
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z))
activate#(n__from(X)) -> c_8(from#(X))
pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0()))
plus#(s(X),Y) -> c_13(plus#(X,Y))
square#(X) -> c_14(times#(X,X))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak DPs:
2ndsneg#(0(),Z) -> c_1()
2ndspos#(0(),Z) -> c_4()
activate#(X) -> c_7()
from#(X) -> c_9()
from#(X) -> c_10()
plus#(0(),Y) -> c_12()
times#(0(),Y) -> c_15()
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/2,c_7/0,c_8/1,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{5}
by application of
Pre({5}) = {1,2,3,4}.
Here rules are labelled as follows:
1: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z))
2: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z))
3: 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z))
4: 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z))
5: activate#(n__from(X)) -> c_8(from#(X))
6: pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0()))
7: plus#(s(X),Y) -> c_13(plus#(X,Y))
8: square#(X) -> c_14(times#(X,X))
9: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
10: 2ndsneg#(0(),Z) -> c_1()
11: 2ndspos#(0(),Z) -> c_4()
12: activate#(X) -> c_7()
13: from#(X) -> c_9()
14: from#(X) -> c_10()
15: plus#(0(),Y) -> c_12()
16: times#(0(),Y) -> c_15()
** Step 1.b:5: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z))
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z))
pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0()))
plus#(s(X),Y) -> c_13(plus#(X,Y))
square#(X) -> c_14(times#(X,X))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak DPs:
2ndsneg#(0(),Z) -> c_1()
2ndspos#(0(),Z) -> c_4()
activate#(X) -> c_7()
activate#(n__from(X)) -> c_8(from#(X))
from#(X) -> c_9()
from#(X) -> c_10()
plus#(0(),Y) -> c_12()
times#(0(),Y) -> c_15()
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/2,c_7/0,c_8/1,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z))
-->_2 activate#(n__from(X)) -> c_8(from#(X)):12
-->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)):2
-->_2 activate#(X) -> c_7():11
2:S:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z))
-->_2 activate#(n__from(X)) -> c_8(from#(X)):12
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4
-->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)):3
-->_2 activate#(X) -> c_7():11
-->_1 2ndspos#(0(),Z) -> c_4():10
3:S:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z))
-->_2 activate#(n__from(X)) -> c_8(from#(X)):12
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4
-->_2 activate#(X) -> c_7():11
4:S:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z))
-->_2 activate#(n__from(X)) -> c_8(from#(X)):12
-->_2 activate#(X) -> c_7():11
-->_1 2ndsneg#(0(),Z) -> c_1():9
-->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)):2
-->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)):1
5:S:pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0()))
-->_2 from#(X) -> c_10():14
-->_2 from#(X) -> c_9():13
-->_1 2ndspos#(0(),Z) -> c_4():10
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4
-->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)):3
6:S:plus#(s(X),Y) -> c_13(plus#(X,Y))
-->_1 plus#(0(),Y) -> c_12():15
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6
7:S:square#(X) -> c_14(times#(X,X))
-->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8
-->_1 times#(0(),Y) -> c_15():16
8:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
-->_2 times#(0(),Y) -> c_15():16
-->_1 plus#(0(),Y) -> c_12():15
-->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6
9:W:2ndsneg#(0(),Z) -> c_1()
10:W:2ndspos#(0(),Z) -> c_4()
11:W:activate#(X) -> c_7()
12:W:activate#(n__from(X)) -> c_8(from#(X))
-->_1 from#(X) -> c_10():14
-->_1 from#(X) -> c_9():13
13:W:from#(X) -> c_9()
14:W:from#(X) -> c_10()
15:W:plus#(0(),Y) -> c_12()
16:W:times#(0(),Y) -> c_15()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
16: times#(0(),Y) -> c_15()
15: plus#(0(),Y) -> c_12()
10: 2ndspos#(0(),Z) -> c_4()
9: 2ndsneg#(0(),Z) -> c_1()
11: activate#(X) -> c_7()
12: activate#(n__from(X)) -> c_8(from#(X))
13: from#(X) -> c_9()
14: from#(X) -> c_10()
** Step 1.b:6: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z))
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z))
pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0()))
plus#(s(X),Y) -> c_13(plus#(X,Y))
square#(X) -> c_14(times#(X,X))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/2,c_7/0,c_8/1,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z))
-->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)):2
2:S:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z))
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4
-->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)):3
3:S:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z))
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4
4:S:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z))
-->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)):2
-->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)):1
5:S:pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0()))
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4
-->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)):3
6:S:plus#(s(X),Y) -> c_13(plus#(X,Y))
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6
7:S:square#(X) -> c_14(times#(X,X))
-->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8
8:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
-->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
pi#(X) -> c_11(2ndspos#(X,from(0())))
** Step 1.b:7: RemoveHeads WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(s(X),Y) -> c_13(plus#(X,Y))
square#(X) -> c_14(times#(X,X))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
-->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):2
2:S:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4
-->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):3
3:S:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4
4:S:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
-->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):2
-->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))):1
5:S:pi#(X) -> c_11(2ndspos#(X,from(0())))
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4
-->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):3
6:S:plus#(s(X),Y) -> c_13(plus#(X,Y))
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6
7:S:square#(X) -> c_14(times#(X,X))
-->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8
8:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
-->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(7,square#(X) -> c_14(times#(X,X)))]
** Step 1.b:8: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(s(X),Y) -> c_13(plus#(X,Y))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,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:
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
- Weak DPs:
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(s(X),Y) -> c_13(plus#(X,Y))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1
,c_15/0,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
Problem (S)
- Strict DPs:
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(s(X),Y) -> c_13(plus#(X,Y))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak DPs:
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1
,c_15/0,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
*** Step 1.b:8.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
- Weak DPs:
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(s(X),Y) -> c_13(plus#(X,Y))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
-->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):2
2:S:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4
-->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):3
3:S:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4
4:S:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
-->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))):1
-->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):2
5:W:pi#(X) -> c_11(2ndspos#(X,from(0())))
-->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):3
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4
6:W:plus#(s(X),Y) -> c_13(plus#(X,Y))
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6
8:W:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6
-->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
8: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
6: plus#(s(X),Y) -> c_13(plus#(X,Y))
*** Step 1.b:8.a:2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
- Weak DPs:
pi#(X) -> c_11(2ndspos#(X,from(0())))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
pi#(X) -> c_11(2ndspos#(X,from(0())))
*** Step 1.b:8.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
- Weak DPs:
pi#(X) -> c_11(2ndspos#(X,from(0())))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,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:
2: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
Consider the set of all dependency pairs
1: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
2: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
3: 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
4: 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
5: pi#(X) -> c_11(2ndspos#(X,from(0())))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{2}
These cover all (indirect) predecessors of dependency pairs
{1,2,3,4,5}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
**** Step 1.b:8.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
- Weak DPs:
pi#(X) -> c_11(2ndspos#(X,from(0())))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,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_5) = {1},
uargs(c_6) = {1},
uargs(c_11) = {1}
Following symbols are considered usable:
{2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#,times#}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [1]
p(2ndspos) = [1] x1 + [2] x2 + [1]
p(activate) = [3]
p(cons) = [1] x1 + [0]
p(cons2) = [1] x1 + [1] x2 + [0]
p(from) = [9]
p(n__from) = [3]
p(negrecip) = [1] x1 + [1]
p(pi) = [0]
p(plus) = [1] x2 + [0]
p(posrecip) = [2]
p(rcons) = [1] x2 + [2]
p(rnil) = [8]
p(s) = [1] x1 + [2]
p(square) = [2] x1 + [2]
p(times) = [2] x1 + [8]
p(2ndsneg#) = [4] x1 + [0]
p(2ndspos#) = [4] x1 + [0]
p(activate#) = [2] x1 + [0]
p(from#) = [1] x1 + [1]
p(pi#) = [4] x1 + [0]
p(plus#) = [1] x1 + [1]
p(square#) = [2] x1 + [0]
p(times#) = [1] x1 + [1] x2 + [4]
p(c_1) = [1]
p(c_2) = [1] x1 + [0]
p(c_3) = [1] x1 + [7]
p(c_4) = [1]
p(c_5) = [1] x1 + [0]
p(c_6) = [1] x1 + [8]
p(c_7) = [1]
p(c_8) = [1] x1 + [1]
p(c_9) = [1]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [0]
p(c_13) = [0]
p(c_14) = [1]
p(c_15) = [0]
p(c_16) = [1] x1 + [2]
Following rules are strictly oriented:
2ndsneg#(s(N),cons2(X,cons(Y,Z))) = [4] N + [8]
> [4] N + [7]
= c_3(2ndspos#(N,activate(Z)))
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N),cons(X,Z)) = [4] N + [8]
>= [4] N + [8]
= c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
2ndspos#(s(N),cons(X,Z)) = [4] N + [8]
>= [4] N + [8]
= c_5(2ndspos#(s(N),cons2(X,activate(Z))))
2ndspos#(s(N),cons2(X,cons(Y,Z))) = [4] N + [8]
>= [4] N + [8]
= c_6(2ndsneg#(N,activate(Z)))
pi#(X) = [4] X + [0]
>= [4] X + [0]
= c_11(2ndspos#(X,from(0())))
**** Step 1.b:8.a:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
- Weak DPs:
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
pi#(X) -> c_11(2ndspos#(X,from(0())))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 1.b:8.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
pi#(X) -> c_11(2ndspos#(X,from(0())))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
-->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):2
2:W:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4
-->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):3
3:W:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4
4:W:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
-->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):2
-->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))):1
5:W:pi#(X) -> c_11(2ndspos#(X,from(0())))
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4
-->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: pi#(X) -> c_11(2ndspos#(X,from(0())))
1: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
4: 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
3: 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
2: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
**** Step 1.b:8.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 1.b:8.b:1: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(s(X),Y) -> c_13(plus#(X,Y))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak DPs:
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1}
by application of
Pre({1}) = {}.
Here rules are labelled as follows:
1: pi#(X) -> c_11(2ndspos#(X,from(0())))
2: plus#(s(X),Y) -> c_13(plus#(X,Y))
3: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
4: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
5: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
6: 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
7: 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
*** Step 1.b:8.b:2: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(s(X),Y) -> c_13(plus#(X,Y))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak DPs:
2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
pi#(X) -> c_11(2ndspos#(X,from(0())))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:plus#(s(X),Y) -> c_13(plus#(X,Y))
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):1
2:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
-->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):2
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):1
3:W:2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
-->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):4
4:W:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):6
-->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):5
5:W:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):6
6:W:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
-->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):4
-->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))):3
7:W:pi#(X) -> c_11(2ndspos#(X,from(0())))
-->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):6
-->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):5
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
7: pi#(X) -> c_11(2ndspos#(X,from(0())))
3: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))))
6: 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)))
5: 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))))
4: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)))
*** Step 1.b:8.b:3: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(s(X),Y) -> c_13(plus#(X,Y))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,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:
plus#(s(X),Y) -> c_13(plus#(X,Y))
- Weak DPs:
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1
,c_15/0,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
Problem (S)
- Strict DPs:
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak DPs:
plus#(s(X),Y) -> c_13(plus#(X,Y))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1
,c_15/0,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
**** Step 1.b:8.b:3.a:1: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(s(X),Y) -> c_13(plus#(X,Y))
- Weak DPs:
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
plus#(s(X),Y) -> c_13(plus#(X,Y))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
**** Step 1.b:8.b:3.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(s(X),Y) -> c_13(plus#(X,Y))
- Weak DPs:
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak TRS:
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,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: plus#(s(X),Y) -> c_13(plus#(X,Y))
The strictly oriented rules are moved into the weak component.
***** Step 1.b:8.b:3.a:2.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(s(X),Y) -> c_13(plus#(X,Y))
- Weak DPs:
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak TRS:
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,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_13) = {1},
uargs(c_16) = {1,2}
Following symbols are considered usable:
{2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#,times#}
TcT has computed the following interpretation:
p(0) = 0
p(2ndsneg) = 4 + x2 + 4*x2^2
p(2ndspos) = x1*x2 + x1^2 + x2^2
p(activate) = 2
p(cons) = 1
p(cons2) = 1 + x1 + x2
p(from) = 4*x1
p(n__from) = 0
p(negrecip) = 1 + x1
p(pi) = 1 + x1 + x1^2
p(plus) = 4 + 4*x1 + 2*x1*x2 + 4*x2 + 2*x2^2
p(posrecip) = 1
p(rcons) = 1 + x1
p(rnil) = 1
p(s) = 1 + x1
p(square) = 1 + x1 + 4*x1^2
p(times) = 2*x2
p(2ndsneg#) = 4*x1 + x2 + x2^2
p(2ndspos#) = 1 + x1 + x1*x2 + x1^2 + 4*x2 + x2^2
p(activate#) = 1 + 4*x1 + 2*x1^2
p(from#) = 2*x1
p(pi#) = 0
p(plus#) = 3 + 5*x1
p(square#) = 1 + 4*x1
p(times#) = 6 + 4*x1 + 6*x1*x2 + 3*x1^2 + 6*x2 + 4*x2^2
p(c_1) = 0
p(c_2) = x1
p(c_3) = 1
p(c_4) = 0
p(c_5) = 0
p(c_6) = 0
p(c_7) = 1
p(c_8) = x1
p(c_9) = 1
p(c_10) = 0
p(c_11) = x1
p(c_12) = 1
p(c_13) = x1
p(c_14) = 1
p(c_15) = 0
p(c_16) = x1 + x2
Following rules are strictly oriented:
plus#(s(X),Y) = 8 + 5*X
> 3 + 5*X
= c_13(plus#(X,Y))
Following rules are (at-least) weakly oriented:
times#(s(X),Y) = 13 + 10*X + 6*X*Y + 3*X^2 + 12*Y + 4*Y^2
>= 9 + 4*X + 6*X*Y + 3*X^2 + 11*Y + 4*Y^2
= c_16(plus#(Y,times(X,Y)),times#(X,Y))
***** Step 1.b:8.b:3.a:2.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
plus#(s(X),Y) -> c_13(plus#(X,Y))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak TRS:
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 1.b:8.b:3.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
plus#(s(X),Y) -> c_13(plus#(X,Y))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak TRS:
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:plus#(s(X),Y) -> c_13(plus#(X,Y))
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):1
2:W:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
-->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):2
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
1: plus#(s(X),Y) -> c_13(plus#(X,Y))
***** Step 1.b:8.b:3.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
**** Step 1.b:8.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak DPs:
plus#(s(X),Y) -> c_13(plus#(X,Y))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):2
-->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):1
2:W:plus#(s(X),Y) -> c_13(plus#(X,Y))
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: plus#(s(X),Y) -> c_13(plus#(X,Y))
**** Step 1.b:8.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y))
-->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
times#(s(X),Y) -> c_16(times#(X,Y))
**** Step 1.b:8.b:3.b:3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
times#(s(X),Y) -> c_16(times#(X,Y))
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
times#(s(X),Y) -> c_16(times#(X,Y))
**** Step 1.b:8.b:3.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
times#(s(X),Y) -> c_16(times#(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,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: times#(s(X),Y) -> c_16(times#(X,Y))
The strictly oriented rules are moved into the weak component.
***** Step 1.b:8.b:3.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
times#(s(X),Y) -> c_16(times#(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,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_16) = {1}
Following symbols are considered usable:
{2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#,times#}
TcT has computed the following interpretation:
p(0) = [2]
p(2ndsneg) = [2] x1 + [1] x2 + [0]
p(2ndspos) = [1] x1 + [1] x2 + [0]
p(activate) = [2] x1 + [1]
p(cons) = [1] x1 + [1] x2 + [1]
p(cons2) = [1] x1 + [1] x2 + [0]
p(from) = [0]
p(n__from) = [1] x1 + [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [1]
p(square) = [0]
p(times) = [0]
p(2ndsneg#) = [0]
p(2ndspos#) = [0]
p(activate#) = [1]
p(from#) = [1]
p(pi#) = [8] x1 + [0]
p(plus#) = [1]
p(square#) = [2] x1 + [0]
p(times#) = [4] x1 + [4]
p(c_1) = [8]
p(c_2) = [8] x1 + [8]
p(c_3) = [8]
p(c_4) = [8]
p(c_5) = [8]
p(c_6) = [2]
p(c_7) = [1]
p(c_8) = [2] x1 + [0]
p(c_9) = [2]
p(c_10) = [4]
p(c_11) = [1] x1 + [1]
p(c_12) = [4]
p(c_13) = [1]
p(c_14) = [2] x1 + [1]
p(c_15) = [4]
p(c_16) = [1] x1 + [2]
Following rules are strictly oriented:
times#(s(X),Y) = [4] X + [8]
> [4] X + [6]
= c_16(times#(X,Y))
Following rules are (at-least) weakly oriented:
***** Step 1.b:8.b:3.b:4.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
times#(s(X),Y) -> c_16(times#(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 1.b:8.b:3.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
times#(s(X),Y) -> c_16(times#(X,Y))
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:times#(s(X),Y) -> c_16(times#(X,Y))
-->_1 times#(s(X),Y) -> c_16(times#(X,Y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: times#(s(X),Y) -> c_16(times#(X,Y))
***** Step 1.b:8.b:3.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1
,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2
,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0
,c_16/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#
,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))