*** 1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
2ndsneg(0(),Z) -> rnil()
2ndsneg(s(N),cons(X,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z)))
2ndspos(0(),Z) -> rnil()
2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z)))
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
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))
Weak DP Rules:
Weak TRS Rules:
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1}
Obligation:
Full
basic terms: {2ndsneg,2ndspos,activate,cons,from,pi,plus,square,times}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
DependencyPairs {dpKind_ = WIDP}
Proof:
We add the following weak dependency pairs:
Strict DPs
2ndsneg#(0(),Z) -> c_1()
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) -> c_3()
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
2ndsneg#(0(),Z) -> c_1()
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) -> c_3()
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Strict TRS Rules:
2ndsneg(0(),Z) -> rnil()
2ndsneg(s(N),cons(X,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z)))
2ndspos(0(),Z) -> rnil()
2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z)))
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
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))
Weak DP Rules:
Weak TRS Rules:
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
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,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) -> c_3()
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
*** 1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
2ndsneg#(0(),Z) -> c_1()
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) -> c_3()
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Strict TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
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))
Weak DP Rules:
Weak TRS Rules:
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1,3,15}
by application of
Pre({1,3,15}) = {2,4,5,8,10,11,12,14}.
Here rules are labelled as follows:
1: 2ndsneg#(0(),Z) -> c_1()
2: 2ndsneg#(s(N)
,cons(X,n__cons(Y,Z))) ->
c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
3: 2ndspos#(0(),Z) -> c_3()
4: 2ndspos#(s(N)
,cons(X,n__cons(Y,Z))) ->
c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
5: activate#(X) -> c_5(X)
6: activate#(n__cons(X1,X2)) ->
c_6(cons#(X1,X2))
7: activate#(n__from(X)) ->
c_7(from#(X))
8: cons#(X1,X2) -> c_8(X1,X2)
9: from#(X) -> c_9(cons#(X
,n__from(s(X))))
10: from#(X) -> c_10(X)
11: pi#(X) -> c_11(2ndspos#(X
,from(0())))
12: plus#(0(),Y) -> c_12(Y)
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)))
*** 1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Strict TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
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))
Weak DP Rules:
2ndsneg#(0(),Z) -> c_1()
2ndspos#(0(),Z) -> c_3()
times#(0(),Y) -> c_15()
Weak TRS Rules:
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
PathAnalysis {onlyLinear = True}
Proof:
We employ 'linear path analysis' using the following approximated dependency graph:
1:S:2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
-->_1 activate#(n__from(X)) -> c_7(from#(X)):5
-->_1 activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)):4
-->_1 activate#(X) -> c_5(X):3
-->_2 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))):2
-->_2 2ndspos#(0(),Z) -> c_3():15
2:S:2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
-->_1 activate#(n__from(X)) -> c_7(from#(X)):5
-->_1 activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)):4
-->_1 activate#(X) -> c_5(X):3
-->_2 2ndsneg#(0(),Z) -> c_1():14
-->_2 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))):1
3:S:activate#(X) -> c_5(X)
-->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))):13
-->_1 square#(X) -> c_14(times#(X,X)):12
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):11
-->_1 plus#(0(),Y) -> c_12(Y):10
-->_1 pi#(X) -> c_11(2ndspos#(X,from(0()))):9
-->_1 from#(X) -> c_10(X):8
-->_1 from#(X) -> c_9(cons#(X,n__from(s(X)))):7
-->_1 cons#(X1,X2) -> c_8(X1,X2):6
-->_1 activate#(n__from(X)) -> c_7(from#(X)):5
-->_1 activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)):4
-->_1 times#(0(),Y) -> c_15():16
-->_1 2ndspos#(0(),Z) -> c_3():15
-->_1 2ndsneg#(0(),Z) -> c_1():14
-->_1 activate#(X) -> c_5(X):3
-->_1 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))):2
-->_1 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))):1
4:S:activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
-->_1 cons#(X1,X2) -> c_8(X1,X2):6
5:S:activate#(n__from(X)) -> c_7(from#(X))
-->_1 from#(X) -> c_10(X):8
-->_1 from#(X) -> c_9(cons#(X,n__from(s(X)))):7
6:S:cons#(X1,X2) -> c_8(X1,X2)
-->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))):13
-->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))):13
-->_2 square#(X) -> c_14(times#(X,X)):12
-->_1 square#(X) -> c_14(times#(X,X)):12
-->_2 plus#(s(X),Y) -> c_13(plus#(X,Y)):11
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):11
-->_2 plus#(0(),Y) -> c_12(Y):10
-->_1 plus#(0(),Y) -> c_12(Y):10
-->_2 pi#(X) -> c_11(2ndspos#(X,from(0()))):9
-->_1 pi#(X) -> c_11(2ndspos#(X,from(0()))):9
-->_2 from#(X) -> c_10(X):8
-->_1 from#(X) -> c_10(X):8
-->_2 from#(X) -> c_9(cons#(X,n__from(s(X)))):7
-->_1 from#(X) -> c_9(cons#(X,n__from(s(X)))):7
-->_2 times#(0(),Y) -> c_15():16
-->_1 times#(0(),Y) -> c_15():16
-->_2 2ndspos#(0(),Z) -> c_3():15
-->_1 2ndspos#(0(),Z) -> c_3():15
-->_2 2ndsneg#(0(),Z) -> c_1():14
-->_1 2ndsneg#(0(),Z) -> c_1():14
-->_2 cons#(X1,X2) -> c_8(X1,X2):6
-->_1 cons#(X1,X2) -> c_8(X1,X2):6
-->_2 activate#(n__from(X)) -> c_7(from#(X)):5
-->_1 activate#(n__from(X)) -> c_7(from#(X)):5
-->_2 activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)):4
-->_1 activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)):4
-->_2 activate#(X) -> c_5(X):3
-->_1 activate#(X) -> c_5(X):3
-->_2 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))):2
-->_1 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))):2
-->_2 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))):1
-->_1 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))):1
7:S:from#(X) -> c_9(cons#(X,n__from(s(X))))
-->_1 cons#(X1,X2) -> c_8(X1,X2):6
8:S:from#(X) -> c_10(X)
-->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))):13
-->_1 square#(X) -> c_14(times#(X,X)):12
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):11
-->_1 plus#(0(),Y) -> c_12(Y):10
-->_1 pi#(X) -> c_11(2ndspos#(X,from(0()))):9
-->_1 times#(0(),Y) -> c_15():16
-->_1 2ndspos#(0(),Z) -> c_3():15
-->_1 2ndsneg#(0(),Z) -> c_1():14
-->_1 from#(X) -> c_10(X):8
-->_1 from#(X) -> c_9(cons#(X,n__from(s(X)))):7
-->_1 cons#(X1,X2) -> c_8(X1,X2):6
-->_1 activate#(n__from(X)) -> c_7(from#(X)):5
-->_1 activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)):4
-->_1 activate#(X) -> c_5(X):3
-->_1 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))):2
-->_1 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))):1
9:S:pi#(X) -> c_11(2ndspos#(X,from(0())))
-->_1 2ndspos#(0(),Z) -> c_3():15
-->_1 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))):2
10:S:plus#(0(),Y) -> c_12(Y)
-->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))):13
-->_1 square#(X) -> c_14(times#(X,X)):12
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):11
-->_1 times#(0(),Y) -> c_15():16
-->_1 2ndspos#(0(),Z) -> c_3():15
-->_1 2ndsneg#(0(),Z) -> c_1():14
-->_1 plus#(0(),Y) -> c_12(Y):10
-->_1 pi#(X) -> c_11(2ndspos#(X,from(0()))):9
-->_1 from#(X) -> c_10(X):8
-->_1 from#(X) -> c_9(cons#(X,n__from(s(X)))):7
-->_1 cons#(X1,X2) -> c_8(X1,X2):6
-->_1 activate#(n__from(X)) -> c_7(from#(X)):5
-->_1 activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)):4
-->_1 activate#(X) -> c_5(X):3
-->_1 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))):2
-->_1 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))):1
11:S:plus#(s(X),Y) -> c_13(plus#(X,Y))
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):11
-->_1 plus#(0(),Y) -> c_12(Y):10
12:S:square#(X) -> c_14(times#(X,X))
-->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))):13
-->_1 times#(0(),Y) -> c_15():16
13:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)))
-->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):11
-->_1 plus#(0(),Y) -> c_12(Y):10
14:W:2ndsneg#(0(),Z) -> c_1()
15:W:2ndspos#(0(),Z) -> c_3()
16:W:times#(0(),Y) -> c_15()
Obtaining following paths:
1.) Path: [1,4]
2.) Path: [1,3]
3.) Path: [1,2]
*** 1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Strict TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
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))
Weak DP Rules:
times#(0(),Y) -> c_15()
Weak TRS Rules:
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [0]
p(2ndspos) = [0]
p(activate) = [1] x1 + [2]
p(cons) = [1] x2 + [0]
p(from) = [1] x1 + [0]
p(n__cons) = [1] x2 + [0]
p(n__from) = [1] x1 + [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [1] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [0]
p(square) = [0]
p(times) = [0]
p(2ndsneg#) = [1] x2 + [0]
p(2ndspos#) = [1] x2 + [0]
p(activate#) = [0]
p(cons#) = [0]
p(from#) = [0]
p(pi#) = [0]
p(plus#) = [1] x2 + [0]
p(square#) = [0]
p(times#) = [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [0]
p(c_15) = [0]
p(c_16) = [1] x1 + [0]
Following rules are strictly oriented:
activate(X) = [1] X + [2]
> [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [2]
> [1] X2 + [0]
= cons(X1,X2)
activate(n__from(X)) = [1] X + [2]
> [1] X + [0]
= from(X)
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N) = [1] Z + [0]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [2]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = [1] Z + [0]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [2]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [0]
>= [0]
= c_5(X)
activate#(n__cons(X1,X2)) = [0]
>= [0]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [0]
>= [0]
= c_7(from#(X))
cons#(X1,X2) = [0]
>= [0]
= c_8(X1,X2)
from#(X) = [0]
>= [0]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [0]
>= [0]
= c_10(X)
pi#(X) = [0]
>= [0]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_12(Y)
plus#(s(X),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_13(plus#(X,Y))
square#(X) = [0]
>= [0]
= c_14(times#(X,X))
times#(0(),Y) = [0]
>= [0]
= c_15()
times#(s(X),Y) = [0]
>= [0]
= c_16(plus#(Y,times(X,Y)))
cons(X1,X2) = [1] X2 + [0]
>= [1] X2 + [0]
= n__cons(X1,X2)
from(X) = [1] X + [0]
>= [1] X + [0]
= cons(X,n__from(s(X)))
from(X) = [1] X + [0]
>= [1] X + [0]
= n__from(X)
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [0]
= s(plus(X,Y))
times(0(),Y) = [0]
>= [0]
= 0()
times(s(X),Y) = [0]
>= [0]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Strict TRS Rules:
cons(X1,X2) -> n__cons(X1,X2)
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))
Weak DP Rules:
times#(0(),Y) -> c_15()
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [0]
p(2ndspos) = [0]
p(activate) = [1] x1 + [6]
p(cons) = [1] x2 + [0]
p(from) = [0]
p(n__cons) = [1] x2 + [4]
p(n__from) = [6]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [1] x2 + [3]
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) = [5]
p(2ndsneg#) = [1] x2 + [0]
p(2ndspos#) = [1] x2 + [4]
p(activate#) = [3]
p(cons#) = [0]
p(from#) = [0]
p(pi#) = [0]
p(plus#) = [4] x1 + [1] x2 + [0]
p(square#) = [7] x1 + [0]
p(times#) = [7] x2 + [6]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [4]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [1]
p(c_9) = [1] x1 + [0]
p(c_10) = [1]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [2]
p(c_15) = [6]
p(c_16) = [1] x1 + [0]
Following rules are strictly oriented:
activate#(X) = [3]
> [0]
= c_5(X)
activate#(n__cons(X1,X2)) = [3]
> [0]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [3]
> [0]
= c_7(from#(X))
plus#(s(X),Y) = [4] X + [1] Y + [4]
> [4] X + [1] Y + [0]
= c_13(plus#(X,Y))
times#(s(X),Y) = [7] Y + [6]
> [4] Y + [5]
= c_16(plus#(Y,times(X,Y)))
plus(0(),Y) = [1] Y + [3]
> [1] Y + [0]
= Y
times(0(),Y) = [5]
> [0]
= 0()
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N) = [1] Z + [4]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [13]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = [1] Z + [8]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [13]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
cons#(X1,X2) = [0]
>= [1]
= c_8(X1,X2)
from#(X) = [0]
>= [0]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [0]
>= [1]
= c_10(X)
pi#(X) = [0]
>= [4]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_12(Y)
square#(X) = [7] X + [0]
>= [7] X + [8]
= c_14(times#(X,X))
times#(0(),Y) = [7] Y + [6]
>= [6]
= c_15()
activate(X) = [1] X + [6]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [10]
>= [1] X2 + [0]
= cons(X1,X2)
activate(n__from(X)) = [12]
>= [0]
= from(X)
cons(X1,X2) = [1] X2 + [0]
>= [1] X2 + [4]
= n__cons(X1,X2)
from(X) = [0]
>= [6]
= cons(X,n__from(s(X)))
from(X) = [0]
>= [6]
= n__from(X)
plus(s(X),Y) = [1] Y + [3]
>= [1] Y + [4]
= s(plus(X,Y))
times(s(X),Y) = [5]
>= [8]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
square#(X) -> c_14(times#(X,X))
Strict TRS Rules:
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(s(X),Y) -> s(plus(X,Y))
times(s(X),Y) -> plus(Y,times(X,Y))
Weak DP Rules:
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
plus#(s(X),Y) -> c_13(plus#(X,Y))
times#(0(),Y) -> c_15()
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [1] x1 + [0]
p(2ndspos) = [1] x2 + [0]
p(activate) = [1] x1 + [0]
p(cons) = [1] x2 + [0]
p(from) = [0]
p(n__cons) = [1] x2 + [0]
p(n__from) = [0]
p(negrecip) = [0]
p(pi) = [0]
p(plus) = [1] x2 + [2]
p(posrecip) = [1]
p(rcons) = [1] x1 + [1] x2 + [1]
p(rnil) = [0]
p(s) = [1] x1 + [0]
p(square) = [1] x1 + [2]
p(times) = [4] x2 + [0]
p(2ndsneg#) = [1] x2 + [4]
p(2ndspos#) = [1] x2 + [2]
p(activate#) = [3]
p(cons#) = [2]
p(from#) = [1]
p(pi#) = [1]
p(plus#) = [1] x2 + [4]
p(square#) = [6] x1 + [1]
p(times#) = [4] x2 + [6]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [1]
p(c_5) = [1]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [3]
p(c_10) = [0]
p(c_11) = [1] x1 + [2]
p(c_12) = [1] x1 + [1]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [1]
p(c_15) = [0]
p(c_16) = [1] x1 + [2]
Following rules are strictly oriented:
cons#(X1,X2) = [2]
> [0]
= c_8(X1,X2)
from#(X) = [1]
> [0]
= c_10(X)
plus#(0(),Y) = [1] Y + [4]
> [1] Y + [1]
= c_12(Y)
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N) = [1] Z + [4]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [5]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = [1] Z + [2]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [8]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [3]
>= [1]
= c_5(X)
activate#(n__cons(X1,X2)) = [3]
>= [2]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [3]
>= [1]
= c_7(from#(X))
from#(X) = [1]
>= [5]
= c_9(cons#(X,n__from(s(X))))
pi#(X) = [1]
>= [4]
= c_11(2ndspos#(X,from(0())))
plus#(s(X),Y) = [1] Y + [4]
>= [1] Y + [4]
= c_13(plus#(X,Y))
square#(X) = [6] X + [1]
>= [4] X + [7]
= c_14(times#(X,X))
times#(0(),Y) = [4] Y + [6]
>= [0]
= c_15()
times#(s(X),Y) = [4] Y + [6]
>= [4] Y + [6]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [0]
>= [1] X2 + [0]
= cons(X1,X2)
activate(n__from(X)) = [0]
>= [0]
= from(X)
cons(X1,X2) = [1] X2 + [0]
>= [1] X2 + [0]
= n__cons(X1,X2)
from(X) = [0]
>= [0]
= cons(X,n__from(s(X)))
from(X) = [0]
>= [0]
= n__from(X)
plus(0(),Y) = [1] Y + [2]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [2]
>= [1] Y + [2]
= s(plus(X,Y))
times(0(),Y) = [4] Y + [0]
>= [0]
= 0()
times(s(X),Y) = [4] Y + [0]
>= [4] Y + [2]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
from#(X) -> c_9(cons#(X,n__from(s(X))))
pi#(X) -> c_11(2ndspos#(X,from(0())))
square#(X) -> c_14(times#(X,X))
Strict TRS Rules:
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(s(X),Y) -> s(plus(X,Y))
times(s(X),Y) -> plus(Y,times(X,Y))
Weak DP Rules:
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_10(X)
plus#(0(),Y) -> c_12(Y)
plus#(s(X),Y) -> c_13(plus#(X,Y))
times#(0(),Y) -> c_15()
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [2]
p(2ndsneg) = [0]
p(2ndspos) = [0]
p(activate) = [1] x1 + [5]
p(cons) = [1] x2 + [0]
p(from) = [1] x1 + [5]
p(n__cons) = [1] x2 + [4]
p(n__from) = [1] x1 + [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [1] x2 + [7]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [0]
p(square) = [0]
p(times) = [4]
p(2ndsneg#) = [1] x2 + [0]
p(2ndspos#) = [1] x2 + [2]
p(activate#) = [0]
p(cons#) = [0]
p(from#) = [0]
p(pi#) = [0]
p(plus#) = [1] x2 + [1]
p(square#) = [5] x1 + [2]
p(times#) = [5] x1 + [5]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [3]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [1]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [0]
p(c_15) = [0]
p(c_16) = [1] x1 + [0]
Following rules are strictly oriented:
from(X) = [1] X + [5]
> [1] X + [0]
= cons(X,n__from(s(X)))
from(X) = [1] X + [5]
> [1] X + [0]
= n__from(X)
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N) = [1] Z + [4]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [7]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = [1] Z + [6]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [8]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [0]
>= [0]
= c_5(X)
activate#(n__cons(X1,X2)) = [0]
>= [0]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [0]
>= [0]
= c_7(from#(X))
cons#(X1,X2) = [0]
>= [0]
= c_8(X1,X2)
from#(X) = [0]
>= [0]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [0]
>= [0]
= c_10(X)
pi#(X) = [0]
>= [9]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [1]
>= [1] Y + [1]
= c_12(Y)
plus#(s(X),Y) = [1] Y + [1]
>= [1] Y + [1]
= c_13(plus#(X,Y))
square#(X) = [5] X + [2]
>= [5] X + [5]
= c_14(times#(X,X))
times#(0(),Y) = [15]
>= [0]
= c_15()
times#(s(X),Y) = [5] X + [5]
>= [5]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [1] X + [5]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [9]
>= [1] X2 + [0]
= cons(X1,X2)
activate(n__from(X)) = [1] X + [5]
>= [1] X + [5]
= from(X)
cons(X1,X2) = [1] X2 + [0]
>= [1] X2 + [4]
= n__cons(X1,X2)
plus(0(),Y) = [1] Y + [7]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [7]
>= [1] Y + [7]
= s(plus(X,Y))
times(0(),Y) = [4]
>= [2]
= 0()
times(s(X),Y) = [4]
>= [11]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
from#(X) -> c_9(cons#(X,n__from(s(X))))
pi#(X) -> c_11(2ndspos#(X,from(0())))
square#(X) -> c_14(times#(X,X))
Strict TRS Rules:
cons(X1,X2) -> n__cons(X1,X2)
plus(s(X),Y) -> s(plus(X,Y))
times(s(X),Y) -> plus(Y,times(X,Y))
Weak DP Rules:
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_10(X)
plus#(0(),Y) -> c_12(Y)
plus#(s(X),Y) -> c_13(plus#(X,Y))
times#(0(),Y) -> c_15()
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [1]
p(2ndsneg) = [0]
p(2ndspos) = [0]
p(activate) = [4] x1 + [7]
p(cons) = [4] x2 + [0]
p(from) = [7]
p(n__cons) = [1] x2 + [1]
p(n__from) = [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [1] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [2]
p(square) = [0]
p(times) = [4] x1 + [1] x2 + [4]
p(2ndsneg#) = [1] x2 + [0]
p(2ndspos#) = [1] x2 + [0]
p(activate#) = [0]
p(cons#) = [0]
p(from#) = [0]
p(pi#) = [0]
p(plus#) = [2] x1 + [1] x2 + [6]
p(square#) = [7] x1 + [0]
p(times#) = [4] x1 + [3] x2 + [7]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [4]
p(c_14) = [1] x1 + [0]
p(c_15) = [1]
p(c_16) = [1] x1 + [4]
Following rules are strictly oriented:
times(s(X),Y) = [4] X + [1] Y + [12]
> [4] X + [1] Y + [4]
= plus(Y,times(X,Y))
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N) = [4] Z + [4]
,cons(X,n__cons(Y,Z)))
>= [4] Z + [7]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = [4] Z + [4]
,cons(X,n__cons(Y,Z)))
>= [4] Z + [7]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [0]
>= [0]
= c_5(X)
activate#(n__cons(X1,X2)) = [0]
>= [0]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [0]
>= [0]
= c_7(from#(X))
cons#(X1,X2) = [0]
>= [0]
= c_8(X1,X2)
from#(X) = [0]
>= [0]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [0]
>= [0]
= c_10(X)
pi#(X) = [0]
>= [7]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [8]
>= [1] Y + [0]
= c_12(Y)
plus#(s(X),Y) = [2] X + [1] Y + [10]
>= [2] X + [1] Y + [10]
= c_13(plus#(X,Y))
square#(X) = [7] X + [0]
>= [7] X + [7]
= c_14(times#(X,X))
times#(0(),Y) = [3] Y + [11]
>= [1]
= c_15()
times#(s(X),Y) = [4] X + [3] Y + [15]
>= [4] X + [3] Y + [14]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [4] X + [7]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [4] X2 + [11]
>= [4] X2 + [0]
= cons(X1,X2)
activate(n__from(X)) = [7]
>= [7]
= from(X)
cons(X1,X2) = [4] X2 + [0]
>= [1] X2 + [1]
= n__cons(X1,X2)
from(X) = [7]
>= [0]
= cons(X,n__from(s(X)))
from(X) = [7]
>= [0]
= n__from(X)
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [2]
= s(plus(X,Y))
times(0(),Y) = [1] Y + [8]
>= [1]
= 0()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
from#(X) -> c_9(cons#(X,n__from(s(X))))
pi#(X) -> c_11(2ndspos#(X,from(0())))
square#(X) -> c_14(times#(X,X))
Strict TRS Rules:
cons(X1,X2) -> n__cons(X1,X2)
plus(s(X),Y) -> s(plus(X,Y))
Weak DP Rules:
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_10(X)
plus#(0(),Y) -> c_12(Y)
plus#(s(X),Y) -> c_13(plus#(X,Y))
times#(0(),Y) -> c_15()
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [0]
p(2ndspos) = [0]
p(activate) = [1] x1 + [0]
p(cons) = [1] x2 + [0]
p(from) = [0]
p(n__cons) = [1] x2 + [2]
p(n__from) = [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [1] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [4]
p(square) = [0]
p(times) = [0]
p(2ndsneg#) = [1] x1 + [1] x2 + [0]
p(2ndspos#) = [1] x1 + [1] x2 + [3]
p(activate#) = [2]
p(cons#) = [2]
p(from#) = [2]
p(pi#) = [4] x1 + [2]
p(plus#) = [1] x1 + [1] x2 + [0]
p(square#) = [1] x1 + [4]
p(times#) = [1] x2 + [1]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [5]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [5]
p(c_5) = [2]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [2]
p(c_9) = [1] x1 + [0]
p(c_10) = [1]
p(c_11) = [1] x1 + [7]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [2]
p(c_14) = [1] x1 + [0]
p(c_15) = [1]
p(c_16) = [1] x1 + [0]
Following rules are strictly oriented:
2ndspos#(s(N) = [1] N + [1] Z + [9]
,cons(X,n__cons(Y,Z)))
> [1] N + [1] Z + [7]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
square#(X) = [1] X + [4]
> [1] X + [1]
= c_14(times#(X,X))
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N) = [1] N + [1] Z + [6]
,cons(X,n__cons(Y,Z)))
>= [1] N + [1] Z + [10]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
activate#(X) = [2]
>= [2]
= c_5(X)
activate#(n__cons(X1,X2)) = [2]
>= [2]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [2]
>= [2]
= c_7(from#(X))
cons#(X1,X2) = [2]
>= [2]
= c_8(X1,X2)
from#(X) = [2]
>= [2]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [2]
>= [1]
= c_10(X)
pi#(X) = [4] X + [2]
>= [1] X + [10]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_12(Y)
plus#(s(X),Y) = [1] X + [1] Y + [4]
>= [1] X + [1] Y + [2]
= c_13(plus#(X,Y))
times#(0(),Y) = [1] Y + [1]
>= [1]
= c_15()
times#(s(X),Y) = [1] Y + [1]
>= [1] Y + [0]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [2]
>= [1] X2 + [0]
= cons(X1,X2)
activate(n__from(X)) = [0]
>= [0]
= from(X)
cons(X1,X2) = [1] X2 + [0]
>= [1] X2 + [2]
= n__cons(X1,X2)
from(X) = [0]
>= [0]
= cons(X,n__from(s(X)))
from(X) = [0]
>= [0]
= n__from(X)
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [4]
= s(plus(X,Y))
times(0(),Y) = [0]
>= [0]
= 0()
times(s(X),Y) = [0]
>= [0]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
from#(X) -> c_9(cons#(X,n__from(s(X))))
pi#(X) -> c_11(2ndspos#(X,from(0())))
Strict TRS Rules:
cons(X1,X2) -> n__cons(X1,X2)
plus(s(X),Y) -> s(plus(X,Y))
Weak DP Rules:
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_10(X)
plus#(0(),Y) -> c_12(Y)
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)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [0]
p(2ndspos) = [0]
p(activate) = [4] x1 + [0]
p(cons) = [4] x2 + [0]
p(from) = [4]
p(n__cons) = [1] x2 + [0]
p(n__from) = [1]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [1] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [0]
p(square) = [0]
p(times) = [0]
p(2ndsneg#) = [1] x2 + [0]
p(2ndspos#) = [1] x2 + [0]
p(activate#) = [0]
p(cons#) = [0]
p(from#) = [0]
p(pi#) = [5]
p(plus#) = [1] x2 + [0]
p(square#) = [0]
p(times#) = [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [0]
p(c_15) = [0]
p(c_16) = [1] x1 + [0]
Following rules are strictly oriented:
pi#(X) = [5]
> [4]
= c_11(2ndspos#(X,from(0())))
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N) = [4] Z + [0]
,cons(X,n__cons(Y,Z)))
>= [4] Z + [0]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = [4] Z + [0]
,cons(X,n__cons(Y,Z)))
>= [4] Z + [0]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [0]
>= [0]
= c_5(X)
activate#(n__cons(X1,X2)) = [0]
>= [0]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [0]
>= [0]
= c_7(from#(X))
cons#(X1,X2) = [0]
>= [0]
= c_8(X1,X2)
from#(X) = [0]
>= [0]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [0]
>= [0]
= c_10(X)
plus#(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_12(Y)
plus#(s(X),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_13(plus#(X,Y))
square#(X) = [0]
>= [0]
= c_14(times#(X,X))
times#(0(),Y) = [0]
>= [0]
= c_15()
times#(s(X),Y) = [0]
>= [0]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [4] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [4] X2 + [0]
>= [4] X2 + [0]
= cons(X1,X2)
activate(n__from(X)) = [4]
>= [4]
= from(X)
cons(X1,X2) = [4] X2 + [0]
>= [1] X2 + [0]
= n__cons(X1,X2)
from(X) = [4]
>= [4]
= cons(X,n__from(s(X)))
from(X) = [4]
>= [1]
= n__from(X)
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [0]
= s(plus(X,Y))
times(0(),Y) = [0]
>= [0]
= 0()
times(s(X),Y) = [0]
>= [0]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
from#(X) -> c_9(cons#(X,n__from(s(X))))
Strict TRS Rules:
cons(X1,X2) -> n__cons(X1,X2)
plus(s(X),Y) -> s(plus(X,Y))
Weak DP Rules:
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [4]
p(2ndsneg) = [0]
p(2ndspos) = [0]
p(activate) = [1] x1 + [4]
p(cons) = [1] x2 + [3]
p(from) = [4]
p(n__cons) = [1] x2 + [7]
p(n__from) = [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [1] x2 + [1]
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) = [1] x1 + [4]
p(2ndsneg#) = [1] x2 + [0]
p(2ndspos#) = [1] x2 + [3]
p(activate#) = [2]
p(cons#) = [2]
p(from#) = [1]
p(pi#) = [1] x1 + [7]
p(plus#) = [1] x2 + [0]
p(square#) = [1] x1 + [7]
p(times#) = [1] x1 + [7]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [5]
p(c_5) = [2]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [4]
p(c_10) = [1]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [0]
p(c_15) = [1]
p(c_16) = [1] x1 + [4]
Following rules are strictly oriented:
2ndsneg#(s(N) = [1] Z + [10]
,cons(X,n__cons(Y,Z)))
> [1] Z + [9]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
Following rules are (at-least) weakly oriented:
2ndspos#(s(N) = [1] Z + [13]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [11]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [2]
>= [2]
= c_5(X)
activate#(n__cons(X1,X2)) = [2]
>= [2]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [2]
>= [1]
= c_7(from#(X))
cons#(X1,X2) = [2]
>= [0]
= c_8(X1,X2)
from#(X) = [1]
>= [6]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [1]
>= [1]
= c_10(X)
pi#(X) = [1] X + [7]
>= [7]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_12(Y)
plus#(s(X),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_13(plus#(X,Y))
square#(X) = [1] X + [7]
>= [1] X + [7]
= c_14(times#(X,X))
times#(0(),Y) = [11]
>= [1]
= c_15()
times#(s(X),Y) = [1] X + [8]
>= [1] X + [8]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [1] X + [4]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [11]
>= [1] X2 + [3]
= cons(X1,X2)
activate(n__from(X)) = [4]
>= [4]
= from(X)
cons(X1,X2) = [1] X2 + [3]
>= [1] X2 + [7]
= n__cons(X1,X2)
from(X) = [4]
>= [3]
= cons(X,n__from(s(X)))
from(X) = [4]
>= [0]
= n__from(X)
plus(0(),Y) = [1] Y + [1]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [1]
>= [1] Y + [2]
= s(plus(X,Y))
times(0(),Y) = [8]
>= [4]
= 0()
times(s(X),Y) = [1] X + [5]
>= [1] X + [5]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
from#(X) -> c_9(cons#(X,n__from(s(X))))
Strict TRS Rules:
cons(X1,X2) -> n__cons(X1,X2)
plus(s(X),Y) -> s(plus(X,Y))
Weak DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [4] x2 + [1]
p(2ndspos) = [1] x1 + [0]
p(activate) = [1] x1 + [2]
p(cons) = [1] x2 + [1]
p(from) = [1]
p(n__cons) = [1] x2 + [7]
p(n__from) = [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [1] x1 + [0]
p(plus) = [1] x2 + [0]
p(posrecip) = [1]
p(rcons) = [1] x2 + [4]
p(rnil) = [0]
p(s) = [1] x1 + [0]
p(square) = [1] x1 + [0]
p(times) = [1] x2 + [1]
p(2ndsneg#) = [1] x2 + [2]
p(2ndspos#) = [1] x2 + [4]
p(activate#) = [3]
p(cons#) = [0]
p(from#) = [1]
p(pi#) = [6]
p(plus#) = [1] x2 + [0]
p(square#) = [3] x1 + [5]
p(times#) = [2] x1 + [1] x2 + [1]
p(c_1) = [1]
p(c_2) = [1] x1 + [1] x2 + [1]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [5]
p(c_5) = [1]
p(c_6) = [1] x1 + [3]
p(c_7) = [1] x1 + [1]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [1]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [4]
p(c_15) = [1]
p(c_16) = [1] x1 + [0]
Following rules are strictly oriented:
from#(X) = [1]
> [0]
= c_9(cons#(X,n__from(s(X))))
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N) = [1] Z + [10]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [10]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = [1] Z + [12]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [12]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [3]
>= [1]
= c_5(X)
activate#(n__cons(X1,X2)) = [3]
>= [3]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [3]
>= [2]
= c_7(from#(X))
cons#(X1,X2) = [0]
>= [0]
= c_8(X1,X2)
from#(X) = [1]
>= [0]
= c_10(X)
pi#(X) = [6]
>= [6]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_12(Y)
plus#(s(X),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_13(plus#(X,Y))
square#(X) = [3] X + [5]
>= [3] X + [5]
= c_14(times#(X,X))
times#(0(),Y) = [1] Y + [1]
>= [1]
= c_15()
times#(s(X),Y) = [2] X + [1] Y + [1]
>= [1] Y + [1]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [1] X + [2]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [9]
>= [1] X2 + [1]
= cons(X1,X2)
activate(n__from(X)) = [2]
>= [1]
= from(X)
cons(X1,X2) = [1] X2 + [1]
>= [1] X2 + [7]
= n__cons(X1,X2)
from(X) = [1]
>= [1]
= cons(X,n__from(s(X)))
from(X) = [1]
>= [0]
= n__from(X)
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [0]
= s(plus(X,Y))
times(0(),Y) = [1] Y + [1]
>= [0]
= 0()
times(s(X),Y) = [1] Y + [1]
>= [1] Y + [1]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
cons(X1,X2) -> n__cons(X1,X2)
plus(s(X),Y) -> s(plus(X,Y))
Weak DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [4]
p(2ndsneg) = [2] x1 + [4]
p(2ndspos) = [4]
p(activate) = [1] x1 + [2]
p(cons) = [1] x2 + [2]
p(from) = [2]
p(n__cons) = [1] x2 + [0]
p(n__from) = [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [1]
p(plus) = [1] x2 + [0]
p(posrecip) = [1]
p(rcons) = [0]
p(rnil) = [0]
p(s) = [1] x1 + [1]
p(square) = [1]
p(times) = [4]
p(2ndsneg#) = [4] x1 + [1] x2 + [0]
p(2ndspos#) = [4] x1 + [1] x2 + [4]
p(activate#) = [0]
p(cons#) = [0]
p(from#) = [0]
p(pi#) = [6] x1 + [7]
p(plus#) = [1] x1 + [1] x2 + [0]
p(square#) = [7] x1 + [5]
p(times#) = [2] x1 + [4] x2 + [2]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [1]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [1]
p(c_13) = [1] x1 + [1]
p(c_14) = [1] x1 + [2]
p(c_15) = [0]
p(c_16) = [1] x1 + [0]
Following rules are strictly oriented:
cons(X1,X2) = [1] X2 + [2]
> [1] X2 + [0]
= n__cons(X1,X2)
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N) = [4] N + [1] Z + [6]
,cons(X,n__cons(Y,Z)))
>= [4] N + [1] Z + [6]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = [4] N + [1] Z + [10]
,cons(X,n__cons(Y,Z)))
>= [4] N + [1] Z + [2]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [0]
>= [0]
= c_5(X)
activate#(n__cons(X1,X2)) = [0]
>= [0]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [0]
>= [0]
= c_7(from#(X))
cons#(X1,X2) = [0]
>= [0]
= c_8(X1,X2)
from#(X) = [0]
>= [0]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [0]
>= [0]
= c_10(X)
pi#(X) = [6] X + [7]
>= [4] X + [6]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [4]
>= [1] Y + [1]
= c_12(Y)
plus#(s(X),Y) = [1] X + [1] Y + [1]
>= [1] X + [1] Y + [1]
= c_13(plus#(X,Y))
square#(X) = [7] X + [5]
>= [6] X + [4]
= c_14(times#(X,X))
times#(0(),Y) = [4] Y + [10]
>= [0]
= c_15()
times#(s(X),Y) = [2] X + [4] Y + [4]
>= [1] Y + [4]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [1] X + [2]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [2]
>= [1] X2 + [2]
= cons(X1,X2)
activate(n__from(X)) = [2]
>= [2]
= from(X)
from(X) = [2]
>= [2]
= cons(X,n__from(s(X)))
from(X) = [2]
>= [0]
= n__from(X)
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [1]
= s(plus(X,Y))
times(0(),Y) = [4]
>= [4]
= 0()
times(s(X),Y) = [4]
>= [4]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
plus(s(X),Y) -> s(plus(X,Y))
Weak DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
NaturalPI {shape = Mixed 3, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any intersect of strict-rules and rewrite-rules, greedy = NoGreedy}
Proof:
We apply a polynomial interpretation of kind constructor-based(mixed(3)):
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = 0
p(2ndsneg) = 0
p(2ndspos) = 0
p(activate) = x1
p(cons) = x2
p(from) = 0
p(n__cons) = x2
p(n__from) = 0
p(negrecip) = 0
p(pi) = 0
p(plus) = x1 + x1^2 + x2
p(posrecip) = 0
p(rcons) = 0
p(rnil) = 0
p(s) = 1 + x1
p(square) = 0
p(times) = x1*x2 + x1*x2^2
p(2ndsneg#) = 1 + x1 + x1*x2 + x1^2*x2 + x2
p(2ndspos#) = 1 + x1 + x1^2*x2 + x2
p(activate#) = 1
p(cons#) = 0
p(from#) = 0
p(pi#) = 1 + x1
p(plus#) = x2
p(square#) = 1 + x1 + x1^2 + x1^3
p(times#) = 1 + x1 + x1*x2 + x1*x2^2
p(c_1) = 0
p(c_2) = x1 + x2
p(c_3) = 0
p(c_4) = x1 + x2
p(c_5) = 0
p(c_6) = x1
p(c_7) = 1 + x1
p(c_8) = 0
p(c_9) = x1
p(c_10) = 0
p(c_11) = x1
p(c_12) = x1
p(c_13) = x1
p(c_14) = x1
p(c_15) = 0
p(c_16) = x1
Following rules are strictly oriented:
plus(s(X),Y) = 2 + 3*X + X^2 + Y
> 1 + X + X^2 + Y
= s(plus(X,Y))
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N) = 2 + N + 3*N*Z + N^2*Z + 3*Z
,cons(X,n__cons(Y,Z)))
>= 2 + N + N^2*Z + Z
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = 2 + N + 2*N*Z + N^2*Z + 2*Z
,cons(X,n__cons(Y,Z)))
>= 2 + N + N*Z + N^2*Z + Z
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = 1
>= 0
= c_5(X)
activate#(n__cons(X1,X2)) = 1
>= 0
= c_6(cons#(X1,X2))
activate#(n__from(X)) = 1
>= 1
= c_7(from#(X))
cons#(X1,X2) = 0
>= 0
= c_8(X1,X2)
from#(X) = 0
>= 0
= c_9(cons#(X,n__from(s(X))))
from#(X) = 0
>= 0
= c_10(X)
pi#(X) = 1 + X
>= 1 + X
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = Y
>= Y
= c_12(Y)
plus#(s(X),Y) = Y
>= Y
= c_13(plus#(X,Y))
square#(X) = 1 + X + X^2 + X^3
>= 1 + X + X^2 + X^3
= c_14(times#(X,X))
times#(0(),Y) = 1
>= 0
= c_15()
times#(s(X),Y) = 2 + X + X*Y + X*Y^2 + Y + Y^2
>= X*Y + X*Y^2
= c_16(plus#(Y,times(X,Y)))
activate(X) = X
>= X
= X
activate(n__cons(X1,X2)) = X2
>= X2
= cons(X1,X2)
activate(n__from(X)) = 0
>= 0
= from(X)
cons(X1,X2) = X2
>= X2
= n__cons(X1,X2)
from(X) = 0
>= 0
= cons(X,n__from(s(X)))
from(X) = 0
>= 0
= n__from(X)
plus(0(),Y) = Y
>= Y
= Y
times(0(),Y) = 0
>= 0
= 0()
times(s(X),Y) = X*Y + X*Y^2 + Y + Y^2
>= X*Y + X*Y^2 + Y + Y^2
= plus(Y,times(X,Y))
*** 1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
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,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.2 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Strict TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
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))
Weak DP Rules:
2ndspos#(0(),Z) -> c_3()
Weak TRS Rules:
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [0]
p(2ndspos) = [0]
p(activate) = [1] x1 + [2]
p(cons) = [1] x2 + [0]
p(from) = [1] x1 + [0]
p(n__cons) = [1] x2 + [0]
p(n__from) = [1] x1 + [1]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [1] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [3]
p(square) = [0]
p(times) = [3] x2 + [0]
p(2ndsneg#) = [1] x2 + [6]
p(2ndspos#) = [1] x2 + [0]
p(activate#) = [0]
p(cons#) = [7]
p(from#) = [0]
p(pi#) = [0]
p(plus#) = [2] x1 + [1] x2 + [3]
p(square#) = [5] x1 + [0]
p(times#) = [5] x2 + [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [4]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [0]
p(c_15) = [0]
p(c_16) = [1] x1 + [0]
Following rules are strictly oriented:
2ndsneg#(s(N) = [1] Z + [6]
,cons(X,n__cons(Y,Z)))
> [1] Z + [2]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
cons#(X1,X2) = [7]
> [0]
= c_8(X1,X2)
plus#(s(X),Y) = [2] X + [1] Y + [9]
> [2] X + [1] Y + [3]
= c_13(plus#(X,Y))
activate(X) = [1] X + [2]
> [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [2]
> [1] X2 + [0]
= cons(X1,X2)
activate(n__from(X)) = [1] X + [3]
> [1] X + [0]
= from(X)
Following rules are (at-least) weakly oriented:
2ndspos#(0(),Z) = [1] Z + [0]
>= [0]
= c_3()
2ndspos#(s(N) = [1] Z + [0]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [8]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [0]
>= [0]
= c_5(X)
activate#(n__cons(X1,X2)) = [0]
>= [7]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [0]
>= [0]
= c_7(from#(X))
from#(X) = [0]
>= [7]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [0]
>= [0]
= c_10(X)
pi#(X) = [0]
>= [0]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [3]
>= [1] Y + [4]
= c_12(Y)
square#(X) = [5] X + [0]
>= [5] X + [0]
= c_14(times#(X,X))
times#(s(X),Y) = [5] Y + [0]
>= [5] Y + [3]
= c_16(plus#(Y,times(X,Y)))
cons(X1,X2) = [1] X2 + [0]
>= [1] X2 + [0]
= n__cons(X1,X2)
from(X) = [1] X + [0]
>= [1] X + [4]
= cons(X,n__from(s(X)))
from(X) = [1] X + [0]
>= [1] X + [1]
= n__from(X)
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [3]
= s(plus(X,Y))
times(0(),Y) = [3] Y + [0]
>= [0]
= 0()
times(s(X),Y) = [3] Y + [0]
>= [3] Y + [0]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.2.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
square#(X) -> c_14(times#(X,X))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)))
Strict TRS Rules:
cons(X1,X2) -> n__cons(X1,X2)
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))
Weak DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) -> c_3()
cons#(X1,X2) -> c_8(X1,X2)
plus#(s(X),Y) -> c_13(plus#(X,Y))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [5]
p(2ndsneg) = [0]
p(2ndspos) = [0]
p(activate) = [1] x1 + [2]
p(cons) = [1] x2 + [3]
p(from) = [2]
p(n__cons) = [1] x2 + [2]
p(n__from) = [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [1] x2 + [2]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [4]
p(square) = [1] x1 + [0]
p(times) = [5] x2 + [3]
p(2ndsneg#) = [2] x1 + [1] x2 + [0]
p(2ndspos#) = [2] x1 + [1] x2 + [0]
p(activate#) = [0]
p(cons#) = [5]
p(from#) = [0]
p(pi#) = [4] x1 + [4]
p(plus#) = [1] x2 + [1]
p(square#) = [5] x1 + [0]
p(times#) = [5] x2 + [2]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [4]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [4]
p(c_8) = [4]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [0]
p(c_15) = [0]
p(c_16) = [1] x1 + [0]
Following rules are strictly oriented:
2ndspos#(s(N) = [2] N + [1] Z + [13]
,cons(X,n__cons(Y,Z)))
> [2] N + [1] Z + [2]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
pi#(X) = [4] X + [4]
> [2] X + [2]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [1]
> [1] Y + [0]
= c_12(Y)
cons(X1,X2) = [1] X2 + [3]
> [1] X2 + [2]
= n__cons(X1,X2)
from(X) = [2]
> [0]
= n__from(X)
plus(0(),Y) = [1] Y + [2]
> [1] Y + [0]
= Y
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N) = [2] N + [1] Z + [13]
,cons(X,n__cons(Y,Z)))
>= [2] N + [1] Z + [6]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) = [1] Z + [10]
>= [0]
= c_3()
activate#(X) = [0]
>= [0]
= c_5(X)
activate#(n__cons(X1,X2)) = [0]
>= [5]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [0]
>= [4]
= c_7(from#(X))
cons#(X1,X2) = [5]
>= [4]
= c_8(X1,X2)
from#(X) = [0]
>= [5]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [0]
>= [0]
= c_10(X)
plus#(s(X),Y) = [1] Y + [1]
>= [1] Y + [1]
= c_13(plus#(X,Y))
square#(X) = [5] X + [0]
>= [5] X + [2]
= c_14(times#(X,X))
times#(s(X),Y) = [5] Y + [2]
>= [5] Y + [4]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [1] X + [2]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [4]
>= [1] X2 + [3]
= cons(X1,X2)
activate(n__from(X)) = [2]
>= [2]
= from(X)
from(X) = [2]
>= [3]
= cons(X,n__from(s(X)))
plus(s(X),Y) = [1] Y + [2]
>= [1] Y + [6]
= s(plus(X,Y))
times(0(),Y) = [5] Y + [3]
>= [5]
= 0()
times(s(X),Y) = [5] Y + [3]
>= [5] Y + [5]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.2.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
square#(X) -> c_14(times#(X,X))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)))
Strict TRS Rules:
from(X) -> cons(X,n__from(s(X)))
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
Weak DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) -> c_3()
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
cons#(X1,X2) -> c_8(X1,X2)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
plus#(s(X),Y) -> c_13(plus#(X,Y))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> n__from(X)
plus(0(),Y) -> Y
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [3]
p(2ndsneg) = [0]
p(2ndspos) = [0]
p(activate) = [1] x1 + [1]
p(cons) = [1] x2 + [5]
p(from) = [0]
p(n__cons) = [1] x2 + [4]
p(n__from) = [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [1] x2 + [5]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [3]
p(square) = [0]
p(times) = [1] x2 + [0]
p(2ndsneg#) = [1] x2 + [0]
p(2ndspos#) = [1] x2 + [0]
p(activate#) = [4]
p(cons#) = [2]
p(from#) = [1]
p(pi#) = [1]
p(plus#) = [3] x1 + [1] x2 + [1]
p(square#) = [7] x1 + [0]
p(times#) = [3] x1 + [4] x2 + [5]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [4]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [4]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [5]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [1]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [3]
p(c_14) = [1] x1 + [0]
p(c_15) = [2]
p(c_16) = [1] x1 + [0]
Following rules are strictly oriented:
activate#(X) = [4]
> [0]
= c_5(X)
activate#(n__cons(X1,X2)) = [4]
> [2]
= c_6(cons#(X1,X2))
from#(X) = [1]
> [0]
= c_10(X)
times#(s(X),Y) = [3] X + [4] Y + [14]
> [4] Y + [1]
= c_16(plus#(Y,times(X,Y)))
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N) = [1] Z + [9]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [9]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) = [1] Z + [0]
>= [0]
= c_3()
2ndspos#(s(N) = [1] Z + [9]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [9]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(n__from(X)) = [4]
>= [6]
= c_7(from#(X))
cons#(X1,X2) = [2]
>= [0]
= c_8(X1,X2)
from#(X) = [1]
>= [2]
= c_9(cons#(X,n__from(s(X))))
pi#(X) = [1]
>= [1]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [10]
>= [1] Y + [0]
= c_12(Y)
plus#(s(X),Y) = [3] X + [1] Y + [10]
>= [3] X + [1] Y + [4]
= c_13(plus#(X,Y))
square#(X) = [7] X + [0]
>= [7] X + [5]
= c_14(times#(X,X))
activate(X) = [1] X + [1]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [5]
>= [1] X2 + [5]
= cons(X1,X2)
activate(n__from(X)) = [1]
>= [0]
= from(X)
cons(X1,X2) = [1] X2 + [5]
>= [1] X2 + [4]
= n__cons(X1,X2)
from(X) = [0]
>= [5]
= cons(X,n__from(s(X)))
from(X) = [0]
>= [0]
= n__from(X)
plus(0(),Y) = [1] Y + [5]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [5]
>= [1] Y + [8]
= s(plus(X,Y))
times(0(),Y) = [1] Y + [0]
>= [3]
= 0()
times(s(X),Y) = [1] Y + [0]
>= [1] Y + [5]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.2.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
activate#(n__from(X)) -> c_7(from#(X))
from#(X) -> c_9(cons#(X,n__from(s(X))))
square#(X) -> c_14(times#(X,X))
Strict TRS Rules:
from(X) -> cons(X,n__from(s(X)))
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
Weak DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) -> c_3()
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
plus#(s(X),Y) -> c_13(plus#(X,Y))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> n__from(X)
plus(0(),Y) -> Y
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [2]
p(2ndsneg) = [1] x1 + [4]
p(2ndspos) = [4] x1 + [0]
p(activate) = [3] x1 + [0]
p(cons) = [3] x2 + [0]
p(from) = [3]
p(n__cons) = [1] x2 + [0]
p(n__from) = [1]
p(negrecip) = [1] x1 + [0]
p(pi) = [4] x1 + [1]
p(plus) = [1] x2 + [4]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [0]
p(rnil) = [1]
p(s) = [1] x1 + [4]
p(square) = [2]
p(times) = [2] x1 + [1] x2 + [4]
p(2ndsneg#) = [1] x2 + [1]
p(2ndspos#) = [1] x2 + [1]
p(activate#) = [0]
p(cons#) = [0]
p(from#) = [0]
p(pi#) = [4]
p(plus#) = [1] x1 + [1] x2 + [4]
p(square#) = [4] x1 + [1]
p(times#) = [2] x1 + [2] x2 + [0]
p(c_1) = [4]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [1]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [2]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [1]
p(c_13) = [1] x1 + [2]
p(c_14) = [1] x1 + [0]
p(c_15) = [0]
p(c_16) = [1] x1 + [0]
Following rules are strictly oriented:
square#(X) = [4] X + [1]
> [4] X + [0]
= c_14(times#(X,X))
times(0(),Y) = [1] Y + [8]
> [2]
= 0()
times(s(X),Y) = [2] X + [1] Y + [12]
> [2] X + [1] Y + [8]
= plus(Y,times(X,Y))
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N) = [3] Z + [1]
,cons(X,n__cons(Y,Z)))
>= [3] Z + [1]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) = [1] Z + [1]
>= [1]
= c_3()
2ndspos#(s(N) = [3] Z + [1]
,cons(X,n__cons(Y,Z)))
>= [3] Z + [1]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [0]
>= [0]
= c_5(X)
activate#(n__cons(X1,X2)) = [0]
>= [0]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [0]
>= [0]
= c_7(from#(X))
cons#(X1,X2) = [0]
>= [0]
= c_8(X1,X2)
from#(X) = [0]
>= [2]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [0]
>= [0]
= c_10(X)
pi#(X) = [4]
>= [4]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [6]
>= [1] Y + [1]
= c_12(Y)
plus#(s(X),Y) = [1] X + [1] Y + [8]
>= [1] X + [1] Y + [6]
= c_13(plus#(X,Y))
times#(s(X),Y) = [2] X + [2] Y + [8]
>= [2] X + [2] Y + [8]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [3] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [3] X2 + [0]
>= [3] X2 + [0]
= cons(X1,X2)
activate(n__from(X)) = [3]
>= [3]
= from(X)
cons(X1,X2) = [3] X2 + [0]
>= [1] X2 + [0]
= n__cons(X1,X2)
from(X) = [3]
>= [3]
= cons(X,n__from(s(X)))
from(X) = [3]
>= [1]
= n__from(X)
plus(0(),Y) = [1] Y + [4]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [4]
>= [1] Y + [8]
= s(plus(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.2.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
activate#(n__from(X)) -> c_7(from#(X))
from#(X) -> c_9(cons#(X,n__from(s(X))))
Strict TRS Rules:
from(X) -> cons(X,n__from(s(X)))
plus(s(X),Y) -> s(plus(X,Y))
Weak DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) -> c_3()
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [1]
p(2ndsneg) = [0]
p(2ndspos) = [0]
p(activate) = [1] x1 + [0]
p(cons) = [1] x2 + [4]
p(from) = [0]
p(n__cons) = [1] x2 + [4]
p(n__from) = [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [1] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [0]
p(square) = [0]
p(times) = [2] x2 + [1]
p(2ndsneg#) = [1] x2 + [0]
p(2ndspos#) = [1] x2 + [4]
p(activate#) = [3]
p(cons#) = [0]
p(from#) = [0]
p(pi#) = [2] x1 + [5]
p(plus#) = [4] x1 + [1] x2 + [1]
p(square#) = [7] x1 + [6]
p(times#) = [6] x2 + [6]
p(c_1) = [1]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [3]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [1]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [0]
p(c_15) = [0]
p(c_16) = [1] x1 + [4]
Following rules are strictly oriented:
activate#(n__from(X)) = [3]
> [0]
= c_7(from#(X))
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N) = [1] Z + [8]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [7]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) = [1] Z + [4]
>= [0]
= c_3()
2ndspos#(s(N) = [1] Z + [12]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [3]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [3]
>= [3]
= c_5(X)
activate#(n__cons(X1,X2)) = [3]
>= [0]
= c_6(cons#(X1,X2))
cons#(X1,X2) = [0]
>= [0]
= c_8(X1,X2)
from#(X) = [0]
>= [0]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [0]
>= [0]
= c_10(X)
pi#(X) = [2] X + [5]
>= [4]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [5]
>= [1] Y + [1]
= c_12(Y)
plus#(s(X),Y) = [4] X + [1] Y + [1]
>= [4] X + [1] Y + [1]
= c_13(plus#(X,Y))
square#(X) = [7] X + [6]
>= [6] X + [6]
= c_14(times#(X,X))
times#(s(X),Y) = [6] Y + [6]
>= [6] Y + [6]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [4]
>= [1] X2 + [4]
= cons(X1,X2)
activate(n__from(X)) = [0]
>= [0]
= from(X)
cons(X1,X2) = [1] X2 + [4]
>= [1] X2 + [4]
= n__cons(X1,X2)
from(X) = [0]
>= [4]
= cons(X,n__from(s(X)))
from(X) = [0]
>= [0]
= n__from(X)
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [0]
= s(plus(X,Y))
times(0(),Y) = [2] Y + [1]
>= [1]
= 0()
times(s(X),Y) = [2] Y + [1]
>= [2] Y + [1]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.2.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
from#(X) -> c_9(cons#(X,n__from(s(X))))
Strict TRS Rules:
from(X) -> cons(X,n__from(s(X)))
plus(s(X),Y) -> s(plus(X,Y))
Weak DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) -> c_3()
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [4]
p(2ndspos) = [1] x1 + [1] x2 + [4]
p(activate) = [1] x1 + [0]
p(cons) = [1] x2 + [0]
p(from) = [1] x1 + [0]
p(n__cons) = [1] x2 + [0]
p(n__from) = [1] x1 + [0]
p(negrecip) = [0]
p(pi) = [1]
p(plus) = [1] x2 + [4]
p(posrecip) = [1]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [4]
p(square) = [0]
p(times) = [2] x1 + [4]
p(2ndsneg#) = [2] x1 + [1] x2 + [1]
p(2ndspos#) = [2] x1 + [1] x2 + [1]
p(activate#) = [2]
p(cons#) = [0]
p(from#) = [1]
p(pi#) = [4] x1 + [6]
p(plus#) = [1] x2 + [4]
p(square#) = [5] x1 + [7]
p(times#) = [2] x1 + [2] x2 + [7]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [6]
p(c_3) = [1]
p(c_4) = [1] x1 + [1] x2 + [6]
p(c_5) = [2]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [5]
p(c_12) = [1] x1 + [1]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [0]
p(c_15) = [1]
p(c_16) = [1] x1 + [3]
Following rules are strictly oriented:
from#(X) = [1]
> [0]
= c_9(cons#(X,n__from(s(X))))
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N) = [2] N + [1] Z + [9]
,cons(X,n__cons(Y,Z)))
>= [2] N + [1] Z + [9]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) = [1] Z + [1]
>= [1]
= c_3()
2ndspos#(s(N) = [2] N + [1] Z + [9]
,cons(X,n__cons(Y,Z)))
>= [2] N + [1] Z + [9]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [2]
>= [2]
= c_5(X)
activate#(n__cons(X1,X2)) = [2]
>= [0]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [2]
>= [1]
= c_7(from#(X))
cons#(X1,X2) = [0]
>= [0]
= c_8(X1,X2)
from#(X) = [1]
>= [0]
= c_10(X)
pi#(X) = [4] X + [6]
>= [2] X + [6]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [4]
>= [1] Y + [1]
= c_12(Y)
plus#(s(X),Y) = [1] Y + [4]
>= [1] Y + [4]
= c_13(plus#(X,Y))
square#(X) = [5] X + [7]
>= [4] X + [7]
= c_14(times#(X,X))
times#(s(X),Y) = [2] X + [2] Y + [15]
>= [2] X + [11]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [0]
>= [1] X2 + [0]
= cons(X1,X2)
activate(n__from(X)) = [1] X + [0]
>= [1] X + [0]
= from(X)
cons(X1,X2) = [1] X2 + [0]
>= [1] X2 + [0]
= n__cons(X1,X2)
from(X) = [1] X + [0]
>= [1] X + [4]
= cons(X,n__from(s(X)))
from(X) = [1] X + [0]
>= [1] X + [0]
= n__from(X)
plus(0(),Y) = [1] Y + [4]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [4]
>= [1] Y + [8]
= s(plus(X,Y))
times(0(),Y) = [4]
>= [0]
= 0()
times(s(X),Y) = [2] X + [12]
>= [2] X + [8]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.2.1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
from(X) -> cons(X,n__from(s(X)))
plus(s(X),Y) -> s(plus(X,Y))
Weak DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) -> c_3()
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [4] x1 + [1]
p(2ndspos) = [4] x1 + [0]
p(activate) = [7] x1 + [3]
p(cons) = [7] x2 + [2]
p(from) = [3]
p(n__cons) = [1] x2 + [1]
p(n__from) = [0]
p(negrecip) = [1] x1 + [2]
p(pi) = [1] x1 + [1]
p(plus) = [1] x2 + [0]
p(posrecip) = [1] x1 + [1]
p(rcons) = [1] x2 + [0]
p(rnil) = [4]
p(s) = [1] x1 + [1]
p(square) = [2]
p(times) = [2] x2 + [0]
p(2ndsneg#) = [2] x1 + [1] x2 + [4]
p(2ndspos#) = [2] x1 + [1] x2 + [0]
p(activate#) = [3]
p(cons#) = [2]
p(from#) = [2]
p(pi#) = [2] x1 + [3]
p(plus#) = [1] x2 + [0]
p(square#) = [6] x1 + [7]
p(times#) = [1] x1 + [5] x2 + [5]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [5]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [3]
p(c_6) = [1] x1 + [1]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [2]
p(c_15) = [1]
p(c_16) = [1] x1 + [2]
Following rules are strictly oriented:
from(X) = [3]
> [2]
= cons(X,n__from(s(X)))
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N) = [2] N + [7] Z + [15]
,cons(X,n__cons(Y,Z)))
>= [2] N + [7] Z + [11]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) = [1] Z + [0]
>= [0]
= c_3()
2ndspos#(s(N) = [2] N + [7] Z + [11]
,cons(X,n__cons(Y,Z)))
>= [2] N + [7] Z + [10]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [3]
>= [3]
= c_5(X)
activate#(n__cons(X1,X2)) = [3]
>= [3]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [3]
>= [2]
= c_7(from#(X))
cons#(X1,X2) = [2]
>= [0]
= c_8(X1,X2)
from#(X) = [2]
>= [2]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [2]
>= [0]
= c_10(X)
pi#(X) = [2] X + [3]
>= [2] X + [3]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_12(Y)
plus#(s(X),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_13(plus#(X,Y))
square#(X) = [6] X + [7]
>= [6] X + [7]
= c_14(times#(X,X))
times#(s(X),Y) = [1] X + [5] Y + [6]
>= [2] Y + [2]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [7] X + [3]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [7] X2 + [10]
>= [7] X2 + [2]
= cons(X1,X2)
activate(n__from(X)) = [3]
>= [3]
= from(X)
cons(X1,X2) = [7] X2 + [2]
>= [1] X2 + [1]
= n__cons(X1,X2)
from(X) = [3]
>= [0]
= n__from(X)
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [1]
= s(plus(X,Y))
times(0(),Y) = [2] Y + [0]
>= [0]
= 0()
times(s(X),Y) = [2] Y + [0]
>= [2] Y + [0]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.2.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
plus(s(X),Y) -> s(plus(X,Y))
Weak DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) -> c_3()
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
NaturalPI {shape = Mixed 3, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any intersect of strict-rules and rewrite-rules, greedy = NoGreedy}
Proof:
We apply a polynomial interpretation of kind constructor-based(mixed(3)):
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = 0
p(2ndsneg) = 0
p(2ndspos) = 0
p(activate) = 1 + x1
p(cons) = 1 + x2
p(from) = 1
p(n__cons) = x2
p(n__from) = 0
p(negrecip) = 0
p(pi) = 0
p(plus) = x1 + x1^2 + x2
p(posrecip) = 0
p(rcons) = 0
p(rnil) = 0
p(s) = 1 + x1
p(square) = 0
p(times) = x1*x2 + x1*x2^2
p(2ndsneg#) = x1*x2^2 + x1^2 + x2
p(2ndspos#) = x1*x2^2 + x1^2 + x2
p(activate#) = 1
p(cons#) = 0
p(from#) = 0
p(pi#) = 1 + x1 + x1^2 + x1^3
p(plus#) = x1 + x1^2 + x2
p(square#) = x1^2 + x1^3
p(times#) = x1*x2 + x1*x2^2
p(c_1) = 0
p(c_2) = x1 + x2
p(c_3) = 0
p(c_4) = 1 + x1 + x2
p(c_5) = 1
p(c_6) = 1 + x1
p(c_7) = x1
p(c_8) = 0
p(c_9) = x1
p(c_10) = 0
p(c_11) = x1
p(c_12) = x1
p(c_13) = x1
p(c_14) = x1
p(c_15) = 0
p(c_16) = x1
Following rules are strictly oriented:
plus(s(X),Y) = 2 + 3*X + X^2 + Y
> 1 + X + X^2 + Y
= s(plus(X,Y))
Following rules are (at-least) weakly oriented:
2ndsneg#(s(N) = 3 + 3*N + 2*N*Z + N*Z^2 + N^2 + 3*Z + Z^2
,cons(X,n__cons(Y,Z)))
>= 2 + N + 2*N*Z + N*Z^2 + N^2 + Z
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) = Z
>= 0
= c_3()
2ndspos#(s(N) = 3 + 3*N + 2*N*Z + N*Z^2 + N^2 + 3*Z + Z^2
,cons(X,n__cons(Y,Z)))
>= 3 + N + 2*N*Z + N*Z^2 + N^2 + Z
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = 1
>= 1
= c_5(X)
activate#(n__cons(X1,X2)) = 1
>= 1
= c_6(cons#(X1,X2))
activate#(n__from(X)) = 1
>= 0
= c_7(from#(X))
cons#(X1,X2) = 0
>= 0
= c_8(X1,X2)
from#(X) = 0
>= 0
= c_9(cons#(X,n__from(s(X))))
from#(X) = 0
>= 0
= c_10(X)
pi#(X) = 1 + X + X^2 + X^3
>= 1 + X + X^2
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = Y
>= Y
= c_12(Y)
plus#(s(X),Y) = 2 + 3*X + X^2 + Y
>= X + X^2 + Y
= c_13(plus#(X,Y))
square#(X) = X^2 + X^3
>= X^2 + X^3
= c_14(times#(X,X))
times#(s(X),Y) = X*Y + X*Y^2 + Y + Y^2
>= X*Y + X*Y^2 + Y + Y^2
= c_16(plus#(Y,times(X,Y)))
activate(X) = 1 + X
>= X
= X
activate(n__cons(X1,X2)) = 1 + X2
>= 1 + X2
= cons(X1,X2)
activate(n__from(X)) = 1
>= 1
= from(X)
cons(X1,X2) = 1 + X2
>= X2
= n__cons(X1,X2)
from(X) = 1
>= 1
= cons(X,n__from(s(X)))
from(X) = 1
>= 0
= n__from(X)
plus(0(),Y) = Y
>= Y
= Y
times(0(),Y) = 0
>= 0
= 0()
times(s(X),Y) = X*Y + X*Y^2 + Y + Y^2
>= X*Y + X*Y^2 + Y + Y^2
= plus(Y,times(X,Y))
*** 1.1.1.1.2.1.1.1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(0(),Z) -> c_3()
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
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,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.3 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Strict TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
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))
Weak DP Rules:
2ndsneg#(0(),Z) -> c_1()
Weak TRS Rules:
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [0]
p(2ndspos) = [0]
p(activate) = [1] x1 + [2]
p(cons) = [1] x2 + [0]
p(from) = [1] x1 + [0]
p(n__cons) = [1] x2 + [7]
p(n__from) = [1] x1 + [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [1] x1 + [0]
p(plus) = [1] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [3]
p(square) = [0]
p(times) = [1] x2 + [0]
p(2ndsneg#) = [1] x2 + [0]
p(2ndspos#) = [1] x2 + [0]
p(activate#) = [3]
p(cons#) = [0]
p(from#) = [0]
p(pi#) = [0]
p(plus#) = [4] x1 + [1] x2 + [0]
p(square#) = [5] x1 + [0]
p(times#) = [5] x2 + [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [0]
p(c_15) = [0]
p(c_16) = [1] x1 + [0]
Following rules are strictly oriented:
2ndsneg#(s(N) = [1] Z + [7]
,cons(X,n__cons(Y,Z)))
> [1] Z + [5]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = [1] Z + [7]
,cons(X,n__cons(Y,Z)))
> [1] Z + [5]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [3]
> [0]
= c_5(X)
activate#(n__cons(X1,X2)) = [3]
> [0]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [3]
> [0]
= c_7(from#(X))
plus#(s(X),Y) = [4] X + [1] Y + [12]
> [4] X + [1] Y + [0]
= c_13(plus#(X,Y))
activate(X) = [1] X + [2]
> [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [9]
> [1] X2 + [0]
= cons(X1,X2)
activate(n__from(X)) = [1] X + [2]
> [1] X + [0]
= from(X)
Following rules are (at-least) weakly oriented:
2ndsneg#(0(),Z) = [1] Z + [0]
>= [0]
= c_1()
cons#(X1,X2) = [0]
>= [0]
= c_8(X1,X2)
from#(X) = [0]
>= [0]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [0]
>= [0]
= c_10(X)
pi#(X) = [0]
>= [0]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_12(Y)
square#(X) = [5] X + [0]
>= [5] X + [0]
= c_14(times#(X,X))
times#(s(X),Y) = [5] Y + [0]
>= [5] Y + [0]
= c_16(plus#(Y,times(X,Y)))
cons(X1,X2) = [1] X2 + [0]
>= [1] X2 + [7]
= n__cons(X1,X2)
from(X) = [1] X + [0]
>= [1] X + [3]
= cons(X,n__from(s(X)))
from(X) = [1] X + [0]
>= [1] X + [0]
= n__from(X)
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [3]
= s(plus(X,Y))
times(0(),Y) = [1] Y + [0]
>= [0]
= 0()
times(s(X),Y) = [1] Y + [0]
>= [1] Y + [0]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.3.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
square#(X) -> c_14(times#(X,X))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)))
Strict TRS Rules:
cons(X1,X2) -> n__cons(X1,X2)
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))
Weak DP Rules:
2ndsneg#(0(),Z) -> c_1()
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
plus#(s(X),Y) -> c_13(plus#(X,Y))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [0]
p(2ndspos) = [1] x2 + [0]
p(activate) = [1] x1 + [5]
p(cons) = [1] x2 + [4]
p(from) = [3]
p(n__cons) = [1] x2 + [2]
p(n__from) = [2]
p(negrecip) = [1] x1 + [1]
p(pi) = [2]
p(plus) = [1] x2 + [4]
p(posrecip) = [1] x1 + [1]
p(rcons) = [1] x2 + [4]
p(rnil) = [1]
p(s) = [1] x1 + [0]
p(square) = [0]
p(times) = [2]
p(2ndsneg#) = [1] x2 + [0]
p(2ndspos#) = [1] x2 + [0]
p(activate#) = [1]
p(cons#) = [1]
p(from#) = [1]
p(pi#) = [0]
p(plus#) = [1] x2 + [0]
p(square#) = [1] x1 + [0]
p(times#) = [1] x2 + [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [1]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [0]
p(c_15) = [0]
p(c_16) = [1] x1 + [0]
Following rules are strictly oriented:
cons#(X1,X2) = [1]
> [0]
= c_8(X1,X2)
from#(X) = [1]
> [0]
= c_10(X)
cons(X1,X2) = [1] X2 + [4]
> [1] X2 + [2]
= n__cons(X1,X2)
from(X) = [3]
> [2]
= n__from(X)
plus(0(),Y) = [1] Y + [4]
> [1] Y + [0]
= Y
times(0(),Y) = [2]
> [0]
= 0()
Following rules are (at-least) weakly oriented:
2ndsneg#(0(),Z) = [1] Z + [0]
>= [0]
= c_1()
2ndsneg#(s(N) = [1] Z + [6]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [6]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = [1] Z + [6]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [6]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [1]
>= [1]
= c_5(X)
activate#(n__cons(X1,X2)) = [1]
>= [1]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [1]
>= [1]
= c_7(from#(X))
from#(X) = [1]
>= [1]
= c_9(cons#(X,n__from(s(X))))
pi#(X) = [0]
>= [3]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_12(Y)
plus#(s(X),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_13(plus#(X,Y))
square#(X) = [1] X + [0]
>= [1] X + [0]
= c_14(times#(X,X))
times#(s(X),Y) = [1] Y + [0]
>= [2]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [1] X + [5]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [7]
>= [1] X2 + [4]
= cons(X1,X2)
activate(n__from(X)) = [7]
>= [3]
= from(X)
from(X) = [3]
>= [6]
= cons(X,n__from(s(X)))
plus(s(X),Y) = [1] Y + [4]
>= [1] Y + [4]
= s(plus(X,Y))
times(s(X),Y) = [2]
>= [6]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.3.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
from#(X) -> c_9(cons#(X,n__from(s(X))))
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
square#(X) -> c_14(times#(X,X))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)))
Strict TRS Rules:
from(X) -> cons(X,n__from(s(X)))
plus(s(X),Y) -> s(plus(X,Y))
times(s(X),Y) -> plus(Y,times(X,Y))
Weak DP Rules:
2ndsneg#(0(),Z) -> c_1()
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_10(X)
plus#(s(X),Y) -> c_13(plus#(X,Y))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [0]
p(2ndspos) = [0]
p(activate) = [1] x1 + [2]
p(cons) = [1] x2 + [1]
p(from) = [5]
p(n__cons) = [1] x2 + [0]
p(n__from) = [4]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [1] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [5]
p(square) = [0]
p(times) = [0]
p(2ndsneg#) = [1] x1 + [1] x2 + [5]
p(2ndspos#) = [1] x1 + [1] x2 + [3]
p(activate#) = [0]
p(cons#) = [0]
p(from#) = [0]
p(pi#) = [4] x1 + [0]
p(plus#) = [1] x2 + [1]
p(square#) = [1]
p(times#) = [5]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [6]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [0]
p(c_15) = [0]
p(c_16) = [1] x1 + [0]
Following rules are strictly oriented:
plus#(0(),Y) = [1] Y + [1]
> [1] Y + [0]
= c_12(Y)
times#(s(X),Y) = [5]
> [1]
= c_16(plus#(Y,times(X,Y)))
Following rules are (at-least) weakly oriented:
2ndsneg#(0(),Z) = [1] Z + [5]
>= [0]
= c_1()
2ndsneg#(s(N) = [1] N + [1] Z + [11]
,cons(X,n__cons(Y,Z)))
>= [1] N + [1] Z + [11]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = [1] N + [1] Z + [9]
,cons(X,n__cons(Y,Z)))
>= [1] N + [1] Z + [7]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [0]
>= [0]
= c_5(X)
activate#(n__cons(X1,X2)) = [0]
>= [0]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [0]
>= [0]
= c_7(from#(X))
cons#(X1,X2) = [0]
>= [0]
= c_8(X1,X2)
from#(X) = [0]
>= [0]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [0]
>= [0]
= c_10(X)
pi#(X) = [4] X + [0]
>= [1] X + [8]
= c_11(2ndspos#(X,from(0())))
plus#(s(X),Y) = [1] Y + [1]
>= [1] Y + [1]
= c_13(plus#(X,Y))
square#(X) = [1]
>= [5]
= c_14(times#(X,X))
activate(X) = [1] X + [2]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [2]
>= [1] X2 + [1]
= cons(X1,X2)
activate(n__from(X)) = [6]
>= [5]
= from(X)
cons(X1,X2) = [1] X2 + [1]
>= [1] X2 + [0]
= n__cons(X1,X2)
from(X) = [5]
>= [5]
= cons(X,n__from(s(X)))
from(X) = [5]
>= [4]
= n__from(X)
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [5]
= s(plus(X,Y))
times(0(),Y) = [0]
>= [0]
= 0()
times(s(X),Y) = [0]
>= [0]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.3.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
from#(X) -> c_9(cons#(X,n__from(s(X))))
pi#(X) -> c_11(2ndspos#(X,from(0())))
square#(X) -> c_14(times#(X,X))
Strict TRS Rules:
from(X) -> cons(X,n__from(s(X)))
plus(s(X),Y) -> s(plus(X,Y))
times(s(X),Y) -> plus(Y,times(X,Y))
Weak DP Rules:
2ndsneg#(0(),Z) -> c_1()
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_10(X)
plus#(0(),Y) -> c_12(Y)
plus#(s(X),Y) -> c_13(plus#(X,Y))
times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [0]
p(2ndspos) = [0]
p(activate) = [3] x1 + [0]
p(cons) = [3] x2 + [2]
p(from) = [6]
p(n__cons) = [1] x2 + [2]
p(n__from) = [2]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [1] x2 + [0]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [0]
p(square) = [0]
p(times) = [1] x2 + [0]
p(2ndsneg#) = [1] x2 + [2]
p(2ndspos#) = [1] x2 + [0]
p(activate#) = [0]
p(cons#) = [0]
p(from#) = [0]
p(pi#) = [0]
p(plus#) = [1] x1 + [1] x2 + [0]
p(square#) = [3] x1 + [5]
p(times#) = [1] x1 + [2] x2 + [4]
p(c_1) = [2]
p(c_2) = [1] x1 + [1] x2 + [4]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [6]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [4]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [0]
p(c_15) = [0]
p(c_16) = [1] x1 + [0]
Following rules are strictly oriented:
square#(X) = [3] X + [5]
> [3] X + [4]
= c_14(times#(X,X))
Following rules are (at-least) weakly oriented:
2ndsneg#(0(),Z) = [1] Z + [2]
>= [2]
= c_1()
2ndsneg#(s(N) = [3] Z + [10]
,cons(X,n__cons(Y,Z)))
>= [3] Z + [4]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = [3] Z + [8]
,cons(X,n__cons(Y,Z)))
>= [3] Z + [8]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [0]
>= [0]
= c_5(X)
activate#(n__cons(X1,X2)) = [0]
>= [0]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [0]
>= [0]
= c_7(from#(X))
cons#(X1,X2) = [0]
>= [0]
= c_8(X1,X2)
from#(X) = [0]
>= [0]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [0]
>= [0]
= c_10(X)
pi#(X) = [0]
>= [10]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_12(Y)
plus#(s(X),Y) = [1] X + [1] Y + [0]
>= [1] X + [1] Y + [0]
= c_13(plus#(X,Y))
times#(s(X),Y) = [1] X + [2] Y + [4]
>= [2] Y + [0]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [3] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [3] X2 + [6]
>= [3] X2 + [2]
= cons(X1,X2)
activate(n__from(X)) = [6]
>= [6]
= from(X)
cons(X1,X2) = [3] X2 + [2]
>= [1] X2 + [2]
= n__cons(X1,X2)
from(X) = [6]
>= [8]
= cons(X,n__from(s(X)))
from(X) = [6]
>= [2]
= n__from(X)
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [0]
= s(plus(X,Y))
times(0(),Y) = [1] Y + [0]
>= [0]
= 0()
times(s(X),Y) = [1] Y + [0]
>= [1] Y + [0]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.3.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
from#(X) -> c_9(cons#(X,n__from(s(X))))
pi#(X) -> c_11(2ndspos#(X,from(0())))
Strict TRS Rules:
from(X) -> cons(X,n__from(s(X)))
plus(s(X),Y) -> s(plus(X,Y))
times(s(X),Y) -> plus(Y,times(X,Y))
Weak DP Rules:
2ndsneg#(0(),Z) -> c_1()
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_10(X)
plus#(0(),Y) -> c_12(Y)
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)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [0]
p(2ndspos) = [0]
p(activate) = [1] x1 + [1]
p(cons) = [1] x2 + [0]
p(from) = [1]
p(n__cons) = [1] x2 + [0]
p(n__from) = [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [1] x2 + [0]
p(posrecip) = [1]
p(rcons) = [1] x1 + [4]
p(rnil) = [0]
p(s) = [1] x1 + [1]
p(square) = [4] x1 + [0]
p(times) = [1] x2 + [0]
p(2ndsneg#) = [1] x1 + [1] x2 + [0]
p(2ndspos#) = [1] x1 + [1] x2 + [0]
p(activate#) = [0]
p(cons#) = [0]
p(from#) = [0]
p(pi#) = [1] x1 + [1]
p(plus#) = [5] x1 + [1] x2 + [0]
p(square#) = [7] x1 + [0]
p(times#) = [7] x2 + [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [2]
p(c_10) = [0]
p(c_11) = [1] x1 + [1]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [5]
p(c_14) = [1] x1 + [0]
p(c_15) = [4]
p(c_16) = [1] x1 + [0]
Following rules are strictly oriented:
from(X) = [1]
> [0]
= cons(X,n__from(s(X)))
Following rules are (at-least) weakly oriented:
2ndsneg#(0(),Z) = [1] Z + [0]
>= [0]
= c_1()
2ndsneg#(s(N) = [1] N + [1] Z + [1]
,cons(X,n__cons(Y,Z)))
>= [1] N + [1] Z + [1]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = [1] N + [1] Z + [1]
,cons(X,n__cons(Y,Z)))
>= [1] N + [1] Z + [1]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [0]
>= [0]
= c_5(X)
activate#(n__cons(X1,X2)) = [0]
>= [0]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [0]
>= [0]
= c_7(from#(X))
cons#(X1,X2) = [0]
>= [0]
= c_8(X1,X2)
from#(X) = [0]
>= [2]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [0]
>= [0]
= c_10(X)
pi#(X) = [1] X + [1]
>= [1] X + [2]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_12(Y)
plus#(s(X),Y) = [5] X + [1] Y + [5]
>= [5] X + [1] Y + [5]
= c_13(plus#(X,Y))
square#(X) = [7] X + [0]
>= [7] X + [0]
= c_14(times#(X,X))
times#(s(X),Y) = [7] Y + [0]
>= [6] Y + [0]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [1] X + [1]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [1]
>= [1] X2 + [0]
= cons(X1,X2)
activate(n__from(X)) = [1]
>= [1]
= from(X)
cons(X1,X2) = [1] X2 + [0]
>= [1] X2 + [0]
= n__cons(X1,X2)
from(X) = [1]
>= [0]
= n__from(X)
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [1]
= s(plus(X,Y))
times(0(),Y) = [1] Y + [0]
>= [0]
= 0()
times(s(X),Y) = [1] Y + [0]
>= [1] Y + [0]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.3.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
from#(X) -> c_9(cons#(X,n__from(s(X))))
pi#(X) -> c_11(2ndspos#(X,from(0())))
Strict TRS Rules:
plus(s(X),Y) -> s(plus(X,Y))
times(s(X),Y) -> plus(Y,times(X,Y))
Weak DP Rules:
2ndsneg#(0(),Z) -> c_1()
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_10(X)
plus#(0(),Y) -> c_12(Y)
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)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [1] x1 + [1]
p(2ndspos) = [1]
p(activate) = [1] x1 + [0]
p(cons) = [1] x2 + [0]
p(from) = [0]
p(n__cons) = [1] x2 + [0]
p(n__from) = [0]
p(negrecip) = [1]
p(pi) = [1]
p(plus) = [1] x2 + [0]
p(posrecip) = [1] x1 + [2]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [2]
p(s) = [1] x1 + [0]
p(square) = [1] x1 + [1]
p(times) = [0]
p(2ndsneg#) = [1] x2 + [2]
p(2ndspos#) = [1] x2 + [2]
p(activate#) = [0]
p(cons#) = [0]
p(from#) = [0]
p(pi#) = [4] x1 + [6]
p(plus#) = [3] x1 + [1] x2 + [0]
p(square#) = [3] x1 + [4]
p(times#) = [3] x2 + [4]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [1]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [3]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [0]
p(c_15) = [1]
p(c_16) = [1] x1 + [4]
Following rules are strictly oriented:
pi#(X) = [4] X + [6]
> [5]
= c_11(2ndspos#(X,from(0())))
Following rules are (at-least) weakly oriented:
2ndsneg#(0(),Z) = [1] Z + [2]
>= [0]
= c_1()
2ndsneg#(s(N) = [1] Z + [2]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [2]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = [1] Z + [2]
,cons(X,n__cons(Y,Z)))
>= [1] Z + [2]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [0]
>= [0]
= c_5(X)
activate#(n__cons(X1,X2)) = [0]
>= [0]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [0]
>= [0]
= c_7(from#(X))
cons#(X1,X2) = [0]
>= [0]
= c_8(X1,X2)
from#(X) = [0]
>= [0]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [0]
>= [0]
= c_10(X)
plus#(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_12(Y)
plus#(s(X),Y) = [3] X + [1] Y + [0]
>= [3] X + [1] Y + [0]
= c_13(plus#(X,Y))
square#(X) = [3] X + [4]
>= [3] X + [4]
= c_14(times#(X,X))
times#(s(X),Y) = [3] Y + [4]
>= [3] Y + [4]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [1] X2 + [0]
>= [1] X2 + [0]
= cons(X1,X2)
activate(n__from(X)) = [0]
>= [0]
= from(X)
cons(X1,X2) = [1] X2 + [0]
>= [1] X2 + [0]
= n__cons(X1,X2)
from(X) = [0]
>= [0]
= cons(X,n__from(s(X)))
from(X) = [0]
>= [0]
= n__from(X)
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [0]
= s(plus(X,Y))
times(0(),Y) = [0]
>= [0]
= 0()
times(s(X),Y) = [0]
>= [0]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.3.1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
from#(X) -> c_9(cons#(X,n__from(s(X))))
Strict TRS Rules:
plus(s(X),Y) -> s(plus(X,Y))
times(s(X),Y) -> plus(Y,times(X,Y))
Weak DP Rules:
2ndsneg#(0(),Z) -> c_1()
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(2ndsneg) = [0]
p(2ndspos) = [0]
p(activate) = [4] x1 + [3]
p(cons) = [4] x2 + [0]
p(from) = [1]
p(n__cons) = [1] x2 + [0]
p(n__from) = [0]
p(negrecip) = [1] x1 + [0]
p(pi) = [0]
p(plus) = [1] x2 + [3]
p(posrecip) = [1] x1 + [0]
p(rcons) = [1] x1 + [1] x2 + [0]
p(rnil) = [2]
p(s) = [1] x1 + [1]
p(square) = [0]
p(times) = [2] x1 + [1] x2 + [0]
p(2ndsneg#) = [4] x1 + [1] x2 + [1]
p(2ndspos#) = [4] x1 + [1] x2 + [1]
p(activate#) = [1]
p(cons#) = [0]
p(from#) = [1]
p(pi#) = [4] x1 + [2]
p(plus#) = [1] x2 + [4]
p(square#) = [3] x1 + [4]
p(times#) = [2] x1 + [1] x2 + [3]
p(c_1) = [1]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [0]
p(c_4) = [1] x1 + [1] x2 + [0]
p(c_5) = [1]
p(c_6) = [1] x1 + [1]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [1]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [1]
p(c_15) = [0]
p(c_16) = [1] x1 + [1]
Following rules are strictly oriented:
from#(X) = [1]
> [0]
= c_9(cons#(X,n__from(s(X))))
Following rules are (at-least) weakly oriented:
2ndsneg#(0(),Z) = [1] Z + [1]
>= [1]
= c_1()
2ndsneg#(s(N) = [4] N + [4] Z + [5]
,cons(X,n__cons(Y,Z)))
>= [4] N + [4] Z + [5]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = [4] N + [4] Z + [5]
,cons(X,n__cons(Y,Z)))
>= [4] N + [4] Z + [5]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [1]
>= [1]
= c_5(X)
activate#(n__cons(X1,X2)) = [1]
>= [1]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [1]
>= [1]
= c_7(from#(X))
cons#(X1,X2) = [0]
>= [0]
= c_8(X1,X2)
from#(X) = [1]
>= [1]
= c_10(X)
pi#(X) = [4] X + [2]
>= [4] X + [2]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [4]
>= [1] Y + [0]
= c_12(Y)
plus#(s(X),Y) = [1] Y + [4]
>= [1] Y + [4]
= c_13(plus#(X,Y))
square#(X) = [3] X + [4]
>= [3] X + [4]
= c_14(times#(X,X))
times#(s(X),Y) = [2] X + [1] Y + [5]
>= [2] X + [1] Y + [5]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [4] X + [3]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [4] X2 + [3]
>= [4] X2 + [0]
= cons(X1,X2)
activate(n__from(X)) = [3]
>= [1]
= from(X)
cons(X1,X2) = [4] X2 + [0]
>= [1] X2 + [0]
= n__cons(X1,X2)
from(X) = [1]
>= [0]
= cons(X,n__from(s(X)))
from(X) = [1]
>= [0]
= n__from(X)
plus(0(),Y) = [1] Y + [3]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [3]
>= [1] Y + [4]
= s(plus(X,Y))
times(0(),Y) = [1] Y + [0]
>= [0]
= 0()
times(s(X),Y) = [2] X + [1] Y + [2]
>= [2] X + [1] Y + [3]
= plus(Y,times(X,Y))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.3.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
plus(s(X),Y) -> s(plus(X,Y))
times(s(X),Y) -> plus(Y,times(X,Y))
Weak DP Rules:
2ndsneg#(0(),Z) -> c_1()
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [4]
p(2ndsneg) = [4] x2 + [2]
p(2ndspos) = [2] x2 + [2]
p(activate) = [4] x1 + [0]
p(cons) = [4] x2 + [0]
p(from) = [0]
p(n__cons) = [1] x2 + [0]
p(n__from) = [0]
p(negrecip) = [0]
p(pi) = [2] x1 + [1]
p(plus) = [1] x2 + [0]
p(posrecip) = [1]
p(rcons) = [1] x2 + [0]
p(rnil) = [0]
p(s) = [1] x1 + [1]
p(square) = [4] x1 + [0]
p(times) = [1] x1 + [4]
p(2ndsneg#) = [1] x1 + [1] x2 + [2]
p(2ndspos#) = [1] x1 + [1] x2 + [2]
p(activate#) = [0]
p(cons#) = [0]
p(from#) = [0]
p(pi#) = [4] x1 + [4]
p(plus#) = [1] x2 + [0]
p(square#) = [6] x1 + [0]
p(times#) = [4] x1 + [1] x2 + [0]
p(c_1) = [1]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [1]
p(c_4) = [1] x1 + [1] x2 + [1]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [1]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [0]
p(c_15) = [0]
p(c_16) = [1] x1 + [0]
Following rules are strictly oriented:
times(s(X),Y) = [1] X + [5]
> [1] X + [4]
= plus(Y,times(X,Y))
Following rules are (at-least) weakly oriented:
2ndsneg#(0(),Z) = [1] Z + [6]
>= [1]
= c_1()
2ndsneg#(s(N) = [1] N + [4] Z + [3]
,cons(X,n__cons(Y,Z)))
>= [1] N + [4] Z + [2]
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = [1] N + [4] Z + [3]
,cons(X,n__cons(Y,Z)))
>= [1] N + [4] Z + [3]
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = [0]
>= [0]
= c_5(X)
activate#(n__cons(X1,X2)) = [0]
>= [0]
= c_6(cons#(X1,X2))
activate#(n__from(X)) = [0]
>= [0]
= c_7(from#(X))
cons#(X1,X2) = [0]
>= [0]
= c_8(X1,X2)
from#(X) = [0]
>= [0]
= c_9(cons#(X,n__from(s(X))))
from#(X) = [0]
>= [0]
= c_10(X)
pi#(X) = [4] X + [4]
>= [1] X + [3]
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_12(Y)
plus#(s(X),Y) = [1] Y + [0]
>= [1] Y + [0]
= c_13(plus#(X,Y))
square#(X) = [6] X + [0]
>= [5] X + [0]
= c_14(times#(X,X))
times#(s(X),Y) = [4] X + [1] Y + [4]
>= [1] X + [4]
= c_16(plus#(Y,times(X,Y)))
activate(X) = [4] X + [0]
>= [1] X + [0]
= X
activate(n__cons(X1,X2)) = [4] X2 + [0]
>= [4] X2 + [0]
= cons(X1,X2)
activate(n__from(X)) = [0]
>= [0]
= from(X)
cons(X1,X2) = [4] X2 + [0]
>= [1] X2 + [0]
= n__cons(X1,X2)
from(X) = [0]
>= [0]
= cons(X,n__from(s(X)))
from(X) = [0]
>= [0]
= n__from(X)
plus(0(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
plus(s(X),Y) = [1] Y + [0]
>= [1] Y + [1]
= s(plus(X,Y))
times(0(),Y) = [8]
>= [4]
= 0()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.3.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
plus(s(X),Y) -> s(plus(X,Y))
Weak DP Rules:
2ndsneg#(0(),Z) -> c_1()
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
plus(0(),Y) -> Y
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
Signature:
{2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
NaturalPI {shape = Mixed 3, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any intersect of strict-rules and rewrite-rules, greedy = NoGreedy}
Proof:
We apply a polynomial interpretation of kind constructor-based(mixed(3)):
The following argument positions are considered usable:
uargs(plus) = {2},
uargs(s) = {1},
uargs(2ndsneg#) = {2},
uargs(2ndspos#) = {2},
uargs(plus#) = {2},
uargs(c_2) = {1,2},
uargs(c_4) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_9) = {1},
uargs(c_11) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1},
uargs(c_16) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = 0
p(2ndsneg) = 0
p(2ndspos) = 0
p(activate) = x1
p(cons) = x2
p(from) = 0
p(n__cons) = x2
p(n__from) = 0
p(negrecip) = 0
p(pi) = 0
p(plus) = x1 + x1^2 + x2
p(posrecip) = 0
p(rcons) = 0
p(rnil) = 0
p(s) = 1 + x1
p(square) = 0
p(times) = x1*x2 + x1*x2^2
p(2ndsneg#) = x2
p(2ndspos#) = x2
p(activate#) = 0
p(cons#) = 0
p(from#) = 0
p(pi#) = 0
p(plus#) = x1 + x2
p(square#) = x1^2 + x1^3
p(times#) = x1*x2 + x1*x2^2
p(c_1) = 0
p(c_2) = x1 + x2
p(c_3) = 0
p(c_4) = x1 + x2
p(c_5) = 0
p(c_6) = x1
p(c_7) = x1
p(c_8) = 0
p(c_9) = x1
p(c_10) = 0
p(c_11) = x1
p(c_12) = x1
p(c_13) = x1
p(c_14) = x1
p(c_15) = 0
p(c_16) = x1
Following rules are strictly oriented:
plus(s(X),Y) = 2 + 3*X + X^2 + Y
> 1 + X + X^2 + Y
= s(plus(X,Y))
Following rules are (at-least) weakly oriented:
2ndsneg#(0(),Z) = Z
>= 0
= c_1()
2ndsneg#(s(N) = Z
,cons(X,n__cons(Y,Z)))
>= Z
= c_2(activate#(Y)
,2ndspos#(N,activate(Z)))
2ndspos#(s(N) = Z
,cons(X,n__cons(Y,Z)))
>= Z
= c_4(activate#(Y)
,2ndsneg#(N,activate(Z)))
activate#(X) = 0
>= 0
= c_5(X)
activate#(n__cons(X1,X2)) = 0
>= 0
= c_6(cons#(X1,X2))
activate#(n__from(X)) = 0
>= 0
= c_7(from#(X))
cons#(X1,X2) = 0
>= 0
= c_8(X1,X2)
from#(X) = 0
>= 0
= c_9(cons#(X,n__from(s(X))))
from#(X) = 0
>= 0
= c_10(X)
pi#(X) = 0
>= 0
= c_11(2ndspos#(X,from(0())))
plus#(0(),Y) = Y
>= Y
= c_12(Y)
plus#(s(X),Y) = 1 + X + Y
>= X + Y
= c_13(plus#(X,Y))
square#(X) = X^2 + X^3
>= X^2 + X^3
= c_14(times#(X,X))
times#(s(X),Y) = X*Y + X*Y^2 + Y + Y^2
>= X*Y + X*Y^2 + Y
= c_16(plus#(Y,times(X,Y)))
activate(X) = X
>= X
= X
activate(n__cons(X1,X2)) = X2
>= X2
= cons(X1,X2)
activate(n__from(X)) = 0
>= 0
= from(X)
cons(X1,X2) = X2
>= X2
= n__cons(X1,X2)
from(X) = 0
>= 0
= cons(X,n__from(s(X)))
from(X) = 0
>= 0
= n__from(X)
plus(0(),Y) = Y
>= Y
= Y
times(0(),Y) = 0
>= 0
= 0()
times(s(X),Y) = X*Y + X*Y^2 + Y + Y^2
>= X*Y + X*Y^2 + Y + Y^2
= plus(Y,times(X,Y))
*** 1.1.1.1.3.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
2ndsneg#(0(),Z) -> c_1()
2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)))
2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)))
activate#(X) -> c_5(X)
activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2))
activate#(n__from(X)) -> c_7(from#(X))
cons#(X1,X2) -> c_8(X1,X2)
from#(X) -> c_9(cons#(X,n__from(s(X))))
from#(X) -> c_10(X)
pi#(X) -> c_11(2ndspos#(X,from(0())))
plus#(0(),Y) -> c_12(Y)
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)))
Weak TRS Rules:
activate(X) -> X
activate(n__cons(X1,X2)) -> cons(X1,X2)
activate(n__from(X)) -> from(X)
cons(X1,X2) -> n__cons(X1,X2)
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,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1}
Obligation:
Full
basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).