Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Transformed_CSR_04/ExAppendixB_AEL03_Z)

The rewrite relation of the following TRS is considered.

from(X) → cons(X,n__from(s(X))) (1)
2ndspos(0,Z) → rnil (2)
2ndspos(s(N),cons(X,Z)) → 2ndspos(s(N),cons2(X,activate(Z))) (3)
2ndspos(s(N),cons2(X,cons(Y,Z))) → rcons(posrecip(Y),2ndsneg(N,activate(Z))) (4)
2ndsneg(0,Z) → rnil (5)
2ndsneg(s(N),cons(X,Z)) → 2ndsneg(s(N),cons2(X,activate(Z))) (6)
2ndsneg(s(N),cons2(X,cons(Y,Z))) → rcons(negrecip(Y),2ndspos(N,activate(Z))) (7)
pi(X) → 2ndspos(X,from(0)) (8)
plus(0,Y) → Y (9)
plus(s(X),Y) → s(plus(X,Y)) (10)
times(0,Y) → 0 (11)
times(s(X),Y) → plus(Y,times(X,Y)) (12)
square(X) → times(X,X) (13)
from(X) → n__from(X) (14)
activate(n__from(X)) → from(X) (15)
activate(X) → X (16)
The evaluation strategy is innermost.

Property / Task

Determine bounds on the runtime complexity.

Answer / Result

An upperbound for the complexity is O(n2).

Proof (by AProVE @ termCOMP 2023)

1 Dependency Tuples

We get the following set of dependency tuples:
from#(z0) → c (18)
originates from
from(z0) → cons(z0,n__from(s(z0))) (17)
from#(z0) → c1 (20)
originates from
from(z0) → n__from(z0) (19)
2ndspos#(0,z0) → c2 (22)
originates from
2ndspos(0,z0) → rnil (21)
2ndspos#(s(z0),cons(z1,z2)) → c3(2ndspos#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (24)
originates from
2ndspos(s(z0),cons(z1,z2)) → 2ndspos(s(z0),cons2(z1,activate(z2))) (23)
2ndspos#(s(z0),cons2(z1,cons(z2,z3))) → c4(2ndsneg#(z0,activate(z3)),activate#(z3)) (26)
originates from
2ndspos(s(z0),cons2(z1,cons(z2,z3))) → rcons(posrecip(z2),2ndsneg(z0,activate(z3))) (25)
2ndsneg#(0,z0) → c5 (28)
originates from
2ndsneg(0,z0) → rnil (27)
2ndsneg#(s(z0),cons(z1,z2)) → c6(2ndsneg#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (30)
originates from
2ndsneg(s(z0),cons(z1,z2)) → 2ndsneg(s(z0),cons2(z1,activate(z2))) (29)
2ndsneg#(s(z0),cons2(z1,cons(z2,z3))) → c7(2ndspos#(z0,activate(z3)),activate#(z3)) (32)
originates from
2ndsneg(s(z0),cons2(z1,cons(z2,z3))) → rcons(negrecip(z2),2ndspos(z0,activate(z3))) (31)
pi#(z0) → c8(2ndspos#(z0,from(0)),from#(0)) (34)
originates from
pi(z0) → 2ndspos(z0,from(0)) (33)
plus#(0,z0) → c9 (36)
originates from
plus(0,z0) → z0 (35)
plus#(s(z0),z1) → c10(plus#(z0,z1)) (38)
originates from
plus(s(z0),z1) → s(plus(z0,z1)) (37)
times#(0,z0) → c11 (40)
originates from
times(0,z0) → 0 (39)
times#(s(z0),z1) → c12(plus#(z1,times(z0,z1)),times#(z0,z1)) (42)
originates from
times(s(z0),z1) → plus(z1,times(z0,z1)) (41)
square#(z0) → c13(times#(z0,z0)) (44)
originates from
square(z0) → times(z0,z0) (43)
activate#(n__from(z0)) → c14(from#(z0)) (46)
originates from
activate(n__from(z0)) → from(z0) (45)
activate#(z0) → c15 (48)
originates from
activate(z0) → z0 (47)
Moreover, we add the following terms to the innermost strategy.
from#(z0)
from#(z0)
2ndspos#(0,z0)
2ndspos#(s(z0),cons(z1,z2))
2ndspos#(s(z0),cons2(z1,cons(z2,z3)))
2ndsneg#(0,z0)
2ndsneg#(s(z0),cons(z1,z2))
2ndsneg#(s(z0),cons2(z1,cons(z2,z3)))
pi#(z0)
plus#(0,z0)
plus#(s(z0),z1)
times#(0,z0)
times#(s(z0),z1)
square#(z0)
activate#(n__from(z0))
activate#(z0)

1.1 Usable Rules

We remove the following rules since they are not usable.
2ndspos(0,z0) → rnil (21)
2ndspos(s(z0),cons(z1,z2)) → 2ndspos(s(z0),cons2(z1,activate(z2))) (23)
2ndspos(s(z0),cons2(z1,cons(z2,z3))) → rcons(posrecip(z2),2ndsneg(z0,activate(z3))) (25)
2ndsneg(0,z0) → rnil (27)
2ndsneg(s(z0),cons(z1,z2)) → 2ndsneg(s(z0),cons2(z1,activate(z2))) (29)
2ndsneg(s(z0),cons2(z1,cons(z2,z3))) → rcons(negrecip(z2),2ndspos(z0,activate(z3))) (31)
pi(z0) → 2ndspos(z0,from(0)) (33)
square(z0) → times(z0,z0) (43)

1.1.1 Rule Shifting

The rules
2ndspos#(0,z0) → c2 (22)
2ndsneg#(0,z0) → c5 (28)
times#(0,z0) → c11 (40)
square#(z0) → c13(times#(z0,z0)) (44)
are strictly oriented by the following linear polynomial interpretation over the naturals
[c] = 0
[c1] = 0
[c2] = 0
[c3(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c4(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c5] = 0
[c6(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c7(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c8(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c9] = 0
[c10(x1)] = 1 · x1 + 0
[c11] = 0
[c12(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c13(x1)] = 1 · x1 + 0
[c14(x1)] = 1 · x1 + 0
[c15] = 0
[activate(x1)] = 1 + 1 · x1
[from(x1)] = 1 + 1 · x1
[times(x1, x2)] = 1 + 1 · x1 + 1 · x2
[plus(x1, x2)] = 1 · x2 + 0
[from#(x1)] = 0
[2ndspos#(x1, x2)] = 1 · x1 + 0
[2ndsneg#(x1, x2)] = 1 · x1 + 0
[pi#(x1)] = 1 · x1 + 0
[plus#(x1, x2)] = 0
[times#(x1, x2)] = 1 · x1 + 0
[square#(x1)] = 1 + 1 · x1
[activate#(x1)] = 0
[0] = 1
[s(x1)] = 1 · x1 + 0
[cons(x1, x2)] = 1 + 1 · x1 + 1 · x2
[n__from(x1)] = 1 + 1 · x1
[cons2(x1, x2)] = 1 + 1 · x1
which has the intended complexity. Here, only the following usable rules have been considered:
from#(z0) → c (18)
from#(z0) → c1 (20)
2ndspos#(0,z0) → c2 (22)
2ndspos#(s(z0),cons(z1,z2)) → c3(2ndspos#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (24)
2ndspos#(s(z0),cons2(z1,cons(z2,z3))) → c4(2ndsneg#(z0,activate(z3)),activate#(z3)) (26)
2ndsneg#(0,z0) → c5 (28)
2ndsneg#(s(z0),cons(z1,z2)) → c6(2ndsneg#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (30)
2ndsneg#(s(z0),cons2(z1,cons(z2,z3))) → c7(2ndspos#(z0,activate(z3)),activate#(z3)) (32)
pi#(z0) → c8(2ndspos#(z0,from(0)),from#(0)) (34)
plus#(0,z0) → c9 (36)
plus#(s(z0),z1) → c10(plus#(z0,z1)) (38)
times#(0,z0) → c11 (40)
times#(s(z0),z1) → c12(plus#(z1,times(z0,z1)),times#(z0,z1)) (42)
square#(z0) → c13(times#(z0,z0)) (44)
activate#(n__from(z0)) → c14(from#(z0)) (46)
activate#(z0) → c15 (48)

1.1.1.1 Rule Shifting

The rules
2ndspos#(s(z0),cons2(z1,cons(z2,z3))) → c4(2ndsneg#(z0,activate(z3)),activate#(z3)) (26)
2ndsneg#(s(z0),cons2(z1,cons(z2,z3))) → c7(2ndspos#(z0,activate(z3)),activate#(z3)) (32)
plus#(0,z0) → c9 (36)
are strictly oriented by the following linear polynomial interpretation over the naturals
[c] = 0
[c1] = 0
[c2] = 0
[c3(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c4(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c5] = 0
[c6(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c7(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c8(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c9] = 0
[c10(x1)] = 1 · x1 + 0
[c11] = 0
[c12(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c13(x1)] = 1 · x1 + 0
[c14(x1)] = 1 · x1 + 0
[c15] = 0
[activate(x1)] = 1 + 1 · x1
[from(x1)] = 1 + 1 · x1
[times(x1, x2)] = 1 + 1 · x1 + 1 · x2
[plus(x1, x2)] = 1 + 1 · x1 + 1 · x2
[from#(x1)] = 0
[2ndspos#(x1, x2)] = 1 + 1 · x1
[2ndsneg#(x1, x2)] = 1 + 1 · x1
[pi#(x1)] = 1 + 1 · x1
[plus#(x1, x2)] = 1
[times#(x1, x2)] = 1 + 1 · x1
[square#(x1)] = 1 + 1 · x1
[activate#(x1)] = 0
[0] = 1
[s(x1)] = 1 + 1 · x1
[cons(x1, x2)] = 1 + 1 · x1 + 1 · x2
[n__from(x1)] = 1 + 1 · x1
[cons2(x1, x2)] = 1 + 1 · x1
which has the intended complexity. Here, only the following usable rules have been considered:
from#(z0) → c (18)
from#(z0) → c1 (20)
2ndspos#(0,z0) → c2 (22)
2ndspos#(s(z0),cons(z1,z2)) → c3(2ndspos#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (24)
2ndspos#(s(z0),cons2(z1,cons(z2,z3))) → c4(2ndsneg#(z0,activate(z3)),activate#(z3)) (26)
2ndsneg#(0,z0) → c5 (28)
2ndsneg#(s(z0),cons(z1,z2)) → c6(2ndsneg#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (30)
2ndsneg#(s(z0),cons2(z1,cons(z2,z3))) → c7(2ndspos#(z0,activate(z3)),activate#(z3)) (32)
pi#(z0) → c8(2ndspos#(z0,from(0)),from#(0)) (34)
plus#(0,z0) → c9 (36)
plus#(s(z0),z1) → c10(plus#(z0,z1)) (38)
times#(0,z0) → c11 (40)
times#(s(z0),z1) → c12(plus#(z1,times(z0,z1)),times#(z0,z1)) (42)
square#(z0) → c13(times#(z0,z0)) (44)
activate#(n__from(z0)) → c14(from#(z0)) (46)
activate#(z0) → c15 (48)

1.1.1.1.1 Rule Shifting

The rules
times#(s(z0),z1) → c12(plus#(z1,times(z0,z1)),times#(z0,z1)) (42)
are strictly oriented by the following linear polynomial interpretation over the naturals
[c] = 0
[c1] = 0
[c2] = 0
[c3(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c4(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c5] = 0
[c6(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c7(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c8(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c9] = 0
[c10(x1)] = 1 · x1 + 0
[c11] = 0
[c12(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c13(x1)] = 1 · x1 + 0
[c14(x1)] = 1 · x1 + 0
[c15] = 0
[activate(x1)] = 1
[from(x1)] = 1 + 1 · x1
[times(x1, x2)] = 1 + 1 · x1 + 1 · x2
[plus(x1, x2)] = 1 + 1 · x1 + 1 · x2
[from#(x1)] = 0
[2ndspos#(x1, x2)] = 0
[2ndsneg#(x1, x2)] = 0
[pi#(x1)] = 0
[plus#(x1, x2)] = 0
[times#(x1, x2)] = 1 + 1 · x1
[square#(x1)] = 1 + 1 · x1
[activate#(x1)] = 0
[0] = 1
[s(x1)] = 1 + 1 · x1
[cons(x1, x2)] = 1 + 1 · x1
[n__from(x1)] = 1 · x1 + 0
[cons2(x1, x2)] = 1 + 1 · x1
which has the intended complexity. Here, only the following usable rules have been considered:
from#(z0) → c (18)
from#(z0) → c1 (20)
2ndspos#(0,z0) → c2 (22)
2ndspos#(s(z0),cons(z1,z2)) → c3(2ndspos#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (24)
2ndspos#(s(z0),cons2(z1,cons(z2,z3))) → c4(2ndsneg#(z0,activate(z3)),activate#(z3)) (26)
2ndsneg#(0,z0) → c5 (28)
2ndsneg#(s(z0),cons(z1,z2)) → c6(2ndsneg#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (30)
2ndsneg#(s(z0),cons2(z1,cons(z2,z3))) → c7(2ndspos#(z0,activate(z3)),activate#(z3)) (32)
pi#(z0) → c8(2ndspos#(z0,from(0)),from#(0)) (34)
plus#(0,z0) → c9 (36)
plus#(s(z0),z1) → c10(plus#(z0,z1)) (38)
times#(0,z0) → c11 (40)
times#(s(z0),z1) → c12(plus#(z1,times(z0,z1)),times#(z0,z1)) (42)
square#(z0) → c13(times#(z0,z0)) (44)
activate#(n__from(z0)) → c14(from#(z0)) (46)
activate#(z0) → c15 (48)

1.1.1.1.1.1 Rule Shifting

The rules
pi#(z0) → c8(2ndspos#(z0,from(0)),from#(0)) (34)
are strictly oriented by the following linear polynomial interpretation over the naturals
[c] = 0
[c1] = 0
[c2] = 0
[c3(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c4(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c5] = 0
[c6(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c7(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c8(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c9] = 0
[c10(x1)] = 1 · x1 + 0
[c11] = 0
[c12(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c13(x1)] = 1 · x1 + 0
[c14(x1)] = 1 · x1 + 0
[c15] = 0
[activate(x1)] = 1
[from(x1)] = 1 + 1 · x1
[times(x1, x2)] = 1 + 1 · x1 + 1 · x2
[plus(x1, x2)] = 1 + 1 · x1 + 1 · x2
[from#(x1)] = 0
[2ndspos#(x1, x2)] = 0
[2ndsneg#(x1, x2)] = 0
[pi#(x1)] = 1
[plus#(x1, x2)] = 1
[times#(x1, x2)] = 1 + 1 · x1
[square#(x1)] = 1 + 1 · x1
[activate#(x1)] = 0
[0] = 1
[s(x1)] = 1 + 1 · x1
[cons(x1, x2)] = 1 + 1 · x1
[n__from(x1)] = 1 · x1 + 0
[cons2(x1, x2)] = 1 + 1 · x1
which has the intended complexity. Here, only the following usable rules have been considered:
from#(z0) → c (18)
from#(z0) → c1 (20)
2ndspos#(0,z0) → c2 (22)
2ndspos#(s(z0),cons(z1,z2)) → c3(2ndspos#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (24)
2ndspos#(s(z0),cons2(z1,cons(z2,z3))) → c4(2ndsneg#(z0,activate(z3)),activate#(z3)) (26)
2ndsneg#(0,z0) → c5 (28)
2ndsneg#(s(z0),cons(z1,z2)) → c6(2ndsneg#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (30)
2ndsneg#(s(z0),cons2(z1,cons(z2,z3))) → c7(2ndspos#(z0,activate(z3)),activate#(z3)) (32)
pi#(z0) → c8(2ndspos#(z0,from(0)),from#(0)) (34)
plus#(0,z0) → c9 (36)
plus#(s(z0),z1) → c10(plus#(z0,z1)) (38)
times#(0,z0) → c11 (40)
times#(s(z0),z1) → c12(plus#(z1,times(z0,z1)),times#(z0,z1)) (42)
square#(z0) → c13(times#(z0,z0)) (44)
activate#(n__from(z0)) → c14(from#(z0)) (46)
activate#(z0) → c15 (48)

1.1.1.1.1.1.1 Rule Shifting

The rules
plus#(s(z0),z1) → c10(plus#(z0,z1)) (38)
are strictly oriented by the following non-linear polynomial interpretation over the naturals
[c] = 0
[c1] = 0
[c2] = 0
[c3(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c4(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c5] = 0
[c6(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c7(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c8(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c9] = 0
[c10(x1)] = 1 · x1 + 0
[c11] = 0
[c12(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c13(x1)] = 1 · x1 + 0
[c14(x1)] = 1 · x1 + 0
[c15] = 0
[activate(x1)] = 0
[from(x1)] = 0
[times(x1, x2)] = 2 · x2 · x2 + 0
[plus(x1, x2)] = 1 + 1 · x1 + 1 · x2 · x2 + 2 · x1 · x2 + 1 · x1 · x1
[from#(x1)] = 0
[2ndspos#(x1, x2)] = 0
[2ndsneg#(x1, x2)] = 0
[pi#(x1)] = 2 + 1 · x1 + 2 · x1 · x1
[plus#(x1, x2)] = 1 · x1 + 0
[times#(x1, x2)] = 2 + 1 · x2 + 1 · x1 · x2
[square#(x1)] = 2 + 2 · x1 + 2 · x1 · x1
[activate#(x1)] = 0
[0] = 0
[s(x1)] = 1 + 1 · x1
[cons(x1, x2)] = 0
[n__from(x1)] = 0
[cons2(x1, x2)] = 0
which has the intended complexity. Here, only the following usable rules have been considered:
from#(z0) → c (18)
from#(z0) → c1 (20)
2ndspos#(0,z0) → c2 (22)
2ndspos#(s(z0),cons(z1,z2)) → c3(2ndspos#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (24)
2ndspos#(s(z0),cons2(z1,cons(z2,z3))) → c4(2ndsneg#(z0,activate(z3)),activate#(z3)) (26)
2ndsneg#(0,z0) → c5 (28)
2ndsneg#(s(z0),cons(z1,z2)) → c6(2ndsneg#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (30)
2ndsneg#(s(z0),cons2(z1,cons(z2,z3))) → c7(2ndspos#(z0,activate(z3)),activate#(z3)) (32)
pi#(z0) → c8(2ndspos#(z0,from(0)),from#(0)) (34)
plus#(0,z0) → c9 (36)
plus#(s(z0),z1) → c10(plus#(z0,z1)) (38)
times#(0,z0) → c11 (40)
times#(s(z0),z1) → c12(plus#(z1,times(z0,z1)),times#(z0,z1)) (42)
square#(z0) → c13(times#(z0,z0)) (44)
activate#(n__from(z0)) → c14(from#(z0)) (46)
activate#(z0) → c15 (48)

1.1.1.1.1.1.1.1 Rule Shifting

The rules
2ndspos#(s(z0),cons(z1,z2)) → c3(2ndspos#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (24)
are strictly oriented by the following linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the rationals with delta = 1
[square#(x1)] =
1 0
4 0
+
1 4
0 0
· x1
[pi#(x1)] =
4 0
1 0
+
1 3
0 1
· x1
[c] =
0 0
0 0
[c1] =
0 0
0 0
[c12(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[times#(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
1 0
0 0
· x2
[c8(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[c14(x1)] =
0 0
0 0
+
1 0
0 0
· x1
[c5] =
4 0
0 0
[c6(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[activate#(x1)] =
0 0
0 0
+
0 0
0 0
· x1
[c2] =
0 0
0 0
[c9] =
0 0
0 0
[c10(x1)] =
0 0
0 0
+
1 0
0 0
· x1
[c13(x1)] =
0 0
3 0
+
1 0
0 1
· x1
[2ndspos#(x1, x2)] =
0 0
0 0
+
1 2
0 0
· x1 +
1 2
0 0
· x2
[c11] =
0 0
0 0
[plus#(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[from#(x1)] =
0 0
0 0
+
0 0
0 0
· x1
[c3(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[c4(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[2ndsneg#(x1, x2)] =
4 0
0 0
+
1 1
0 0
· x1 +
1 0
0 0
· x2
[c7(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[c15] =
0 0
0 0
[s(x1)] =
4 0
0 0
+
1 1
0 1
· x1
[plus(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[0] =
0 0
0 0
[cons(x1, x2)] =
0 0
2 0
+
0 0
0 0
· x1 +
1 2
0 0
· x2
[n__from(x1)] =
0 0
0 0
+
0 0
0 0
· x1
[from(x1)] =
0 0
2 0
+
0 0
0 0
· x1
[activate(x1)] =
0 0
4 0
+
1 0
0 1
· x1
[times(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[cons2(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
1 0
0 0
· x2
which has the intended complexity. Here, only the following usable rules have been considered:
from#(z0) → c (18)
from#(z0) → c1 (20)
2ndspos#(0,z0) → c2 (22)
2ndspos#(s(z0),cons(z1,z2)) → c3(2ndspos#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (24)
2ndspos#(s(z0),cons2(z1,cons(z2,z3))) → c4(2ndsneg#(z0,activate(z3)),activate#(z3)) (26)
2ndsneg#(0,z0) → c5 (28)
2ndsneg#(s(z0),cons(z1,z2)) → c6(2ndsneg#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (30)
2ndsneg#(s(z0),cons2(z1,cons(z2,z3))) → c7(2ndspos#(z0,activate(z3)),activate#(z3)) (32)
pi#(z0) → c8(2ndspos#(z0,from(0)),from#(0)) (34)
plus#(0,z0) → c9 (36)
plus#(s(z0),z1) → c10(plus#(z0,z1)) (38)
times#(0,z0) → c11 (40)
times#(s(z0),z1) → c12(plus#(z1,times(z0,z1)),times#(z0,z1)) (42)
square#(z0) → c13(times#(z0,z0)) (44)
activate#(n__from(z0)) → c14(from#(z0)) (46)
activate#(z0) → c15 (48)
from(z0) → cons(z0,n__from(s(z0))) (17)
from(z0) → n__from(z0) (19)
activate(n__from(z0)) → from(z0) (45)
activate(z0) → z0 (47)

1.1.1.1.1.1.1.1.1 Rule Shifting

The rules
2ndsneg#(s(z0),cons(z1,z2)) → c6(2ndsneg#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (30)
are strictly oriented by the following linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the rationals with delta = 1
[square#(x1)] =
3 0
4 0
+
1 4
0 0
· x1
[pi#(x1)] =
1 0
4 0
+
0 3
0 1
· x1
[c] =
0 0
0 0
[c1] =
0 0
0 0
[c12(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[times#(x1, x2)] =
0 0
0 0
+
0 4
0 0
· x1 +
1 0
0 0
· x2
[c8(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[c14(x1)] =
0 0
0 0
+
1 0
0 0
· x1
[c5] =
0 0
0 0
[c6(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 2
0 0
· x2
[activate#(x1)] =
0 0
2 0
+
0 0
0 0
· x1
[c2] =
0 0
0 0
[c9] =
0 0
0 0
[c10(x1)] =
0 0
0 0
+
1 0
0 0
· x1
[c13(x1)] =
0 0
4 0
+
1 0
0 1
· x1
[2ndspos#(x1, x2)] =
0 0
0 0
+
0 2
0 0
· x1 +
1 0
0 0
· x2
[c11] =
0 0
0 0
[plus#(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[from#(x1)] =
0 0
0 0
+
0 0
0 0
· x1
[c3(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[c4(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[2ndsneg#(x1, x2)] =
0 0
0 0
+
0 2
0 0
· x1 +
1 2
0 0
· x2
[c7(x1, x2)] =
0 0
0 0
+
1 3
0 0
· x1 +
1 0
0 0
· x2
[c15] =
0 0
2 0
[s(x1)] =
0 0
4 0
+
0 0
0 1
· x1
[plus(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[0] =
0 0
0 0
[cons(x1, x2)] =
0 0
4 0
+
0 0
0 0
· x1 +
1 2
0 0
· x2
[n__from(x1)] =
0 0
0 0
+
0 0
0 0
· x1
[from(x1)] =
0 0
4 0
+
0 0
0 0
· x1
[activate(x1)] =
0 0
4 0
+
1 0
0 1
· x1
[times(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[cons2(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
1 0
0 0
· x2
which has the intended complexity. Here, only the following usable rules have been considered:
from#(z0) → c (18)
from#(z0) → c1 (20)
2ndspos#(0,z0) → c2 (22)
2ndspos#(s(z0),cons(z1,z2)) → c3(2ndspos#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (24)
2ndspos#(s(z0),cons2(z1,cons(z2,z3))) → c4(2ndsneg#(z0,activate(z3)),activate#(z3)) (26)
2ndsneg#(0,z0) → c5 (28)
2ndsneg#(s(z0),cons(z1,z2)) → c6(2ndsneg#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (30)
2ndsneg#(s(z0),cons2(z1,cons(z2,z3))) → c7(2ndspos#(z0,activate(z3)),activate#(z3)) (32)
pi#(z0) → c8(2ndspos#(z0,from(0)),from#(0)) (34)
plus#(0,z0) → c9 (36)
plus#(s(z0),z1) → c10(plus#(z0,z1)) (38)
times#(0,z0) → c11 (40)
times#(s(z0),z1) → c12(plus#(z1,times(z0,z1)),times#(z0,z1)) (42)
square#(z0) → c13(times#(z0,z0)) (44)
activate#(n__from(z0)) → c14(from#(z0)) (46)
activate#(z0) → c15 (48)
from(z0) → cons(z0,n__from(s(z0))) (17)
from(z0) → n__from(z0) (19)
activate(n__from(z0)) → from(z0) (45)
activate(z0) → z0 (47)

1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules
activate#(n__from(z0)) → c14(from#(z0)) (46)
are strictly oriented by the following linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the rationals with delta = 1
[square#(x1)] =
4 0
4 0
+
1 4
0 0
· x1
[pi#(x1)] =
4 0
4 0
+
1 4
0 1
· x1
[c] =
0 0
0 0
[c1] =
0 0
0 0
[c12(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[times#(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
1 0
0 0
· x2
[c8(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[c14(x1)] =
0 0
0 0
+
1 0
0 0
· x1
[c5] =
4 0
1 0
[c6(x1, x2)] =
0 0
1 0
+
1 3
0 0
· x1 +
1 0
0 0
· x2
[activate#(x1)] =
1 0
0 0
+
0 0
0 0
· x1
[c2] =
0 0
0 0
[c9] =
0 0
4 0
[c10(x1)] =
0 0
2 0
+
1 0
0 0
· x1
[c13(x1)] =
4 0
4 0
+
1 0
0 1
· x1
[2ndspos#(x1, x2)] =
0 0
0 0
+
1 2
0 0
· x1 +
1 2
0 0
· x2
[c11] =
0 0
0 0
[plus#(x1, x2)] =
0 0
4 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[from#(x1)] =
0 0
0 0
+
0 0
0 0
· x1
[c3(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[c4(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[2ndsneg#(x1, x2)] =
4 0
1 0
+
1 0
0 0
· x1 +
1 2
0 0
· x2
[c7(x1, x2)] =
0 0
1 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[c15] =
1 0
0 0
[s(x1)] =
1 0
4 0
+
1 4
0 0
· x1
[plus(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[0] =
0 0
0 0
[cons(x1, x2)] =
0 0
2 0
+
0 0
0 0
· x1 +
1 2
0 0
· x2
[n__from(x1)] =
0 0
0 0
+
0 0
0 0
· x1
[from(x1)] =
0 0
2 0
+
0 0
0 0
· x1
[activate(x1)] =
0 0
2 0
+
1 0
0 1
· x1
[times(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[cons2(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
1 0
0 0
· x2
which has the intended complexity. Here, only the following usable rules have been considered:
from#(z0) → c (18)
from#(z0) → c1 (20)
2ndspos#(0,z0) → c2 (22)
2ndspos#(s(z0),cons(z1,z2)) → c3(2ndspos#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (24)
2ndspos#(s(z0),cons2(z1,cons(z2,z3))) → c4(2ndsneg#(z0,activate(z3)),activate#(z3)) (26)
2ndsneg#(0,z0) → c5 (28)
2ndsneg#(s(z0),cons(z1,z2)) → c6(2ndsneg#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (30)
2ndsneg#(s(z0),cons2(z1,cons(z2,z3))) → c7(2ndspos#(z0,activate(z3)),activate#(z3)) (32)
pi#(z0) → c8(2ndspos#(z0,from(0)),from#(0)) (34)
plus#(0,z0) → c9 (36)
plus#(s(z0),z1) → c10(plus#(z0,z1)) (38)
times#(0,z0) → c11 (40)
times#(s(z0),z1) → c12(plus#(z1,times(z0,z1)),times#(z0,z1)) (42)
square#(z0) → c13(times#(z0,z0)) (44)
activate#(n__from(z0)) → c14(from#(z0)) (46)
activate#(z0) → c15 (48)
from(z0) → cons(z0,n__from(s(z0))) (17)
from(z0) → n__from(z0) (19)
activate(n__from(z0)) → from(z0) (45)
activate(z0) → z0 (47)

1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules
activate#(z0) → c15 (48)
are strictly oriented by the following linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the rationals with delta = 1
[square#(x1)] =
4 0
4 0
+
1 4
0 1
· x1
[pi#(x1)] =
4 0
4 0
+
1 4
0 1
· x1
[c] =
0 0
0 0
[c1] =
0 0
0 0
[c12(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[times#(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
1 0
0 0
· x2
[c8(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[c14(x1)] =
0 0
0 0
+
1 0
0 0
· x1
[c5] =
0 0
0 0
[c6(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[activate#(x1)] =
1 0
0 0
+
0 0
0 0
· x1
[c2] =
0 0
0 0
[c9] =
0 0
0 0
[c10(x1)] =
0 0
0 0
+
1 0
0 0
· x1
[c13(x1)] =
0 0
4 0
+
1 0
0 1
· x1
[2ndspos#(x1, x2)] =
0 0
0 0
+
1 2
0 1
· x1 +
1 2
0 0
· x2
[c11] =
0 0
0 0
[plus#(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[from#(x1)] =
0 0
0 0
+
0 0
0 0
· x1
[c3(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[c4(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[2ndsneg#(x1, x2)] =
0 0
0 0
+
1 4
0 0
· x1 +
1 2
0 0
· x2
[c7(x1, x2)] =
1 0
0 0
+
1 2
0 0
· x1 +
1 0
0 0
· x2
[c15] =
0 0
0 0
[s(x1)] =
0 0
2 0
+
1 4
0 0
· x1
[plus(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[0] =
0 0
0 0
[cons(x1, x2)] =
0 0
1 0
+
0 0
0 0
· x1 +
1 2
0 0
· x2
[n__from(x1)] =
0 0
0 0
+
0 0
0 0
· x1
[from(x1)] =
0 0
1 0
+
0 0
0 0
· x1
[activate(x1)] =
0 0
1 0
+
1 0
0 1
· x1
[times(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[cons2(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
1 0
0 0
· x2
which has the intended complexity. Here, only the following usable rules have been considered:
from#(z0) → c (18)
from#(z0) → c1 (20)
2ndspos#(0,z0) → c2 (22)
2ndspos#(s(z0),cons(z1,z2)) → c3(2ndspos#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (24)
2ndspos#(s(z0),cons2(z1,cons(z2,z3))) → c4(2ndsneg#(z0,activate(z3)),activate#(z3)) (26)
2ndsneg#(0,z0) → c5 (28)
2ndsneg#(s(z0),cons(z1,z2)) → c6(2ndsneg#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (30)
2ndsneg#(s(z0),cons2(z1,cons(z2,z3))) → c7(2ndspos#(z0,activate(z3)),activate#(z3)) (32)
pi#(z0) → c8(2ndspos#(z0,from(0)),from#(0)) (34)
plus#(0,z0) → c9 (36)
plus#(s(z0),z1) → c10(plus#(z0,z1)) (38)
times#(0,z0) → c11 (40)
times#(s(z0),z1) → c12(plus#(z1,times(z0,z1)),times#(z0,z1)) (42)
square#(z0) → c13(times#(z0,z0)) (44)
activate#(n__from(z0)) → c14(from#(z0)) (46)
activate#(z0) → c15 (48)
from(z0) → cons(z0,n__from(s(z0))) (17)
from(z0) → n__from(z0) (19)
activate(n__from(z0)) → from(z0) (45)
activate(z0) → z0 (47)

1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules
from#(z0) → c1 (20)
are strictly oriented by the following linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the rationals with delta = 1
[square#(x1)] =
4 0
4 0
+
1 4
0 1
· x1
[pi#(x1)] =
4 0
4 0
+
1 3
0 1
· x1
[c] =
1 0
0 0
[c1] =
0 0
0 0
[c12(x1, x2)] =
0 0
2 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[times#(x1, x2)] =
0 0
4 0
+
0 0
0 0
· x1 +
1 0
0 0
· x2
[c8(x1, x2)] =
0 0
0 0
+
1 3
0 0
· x1 +
1 0
0 0
· x2
[c14(x1)] =
0 0
0 0
+
1 0
0 0
· x1
[c5] =
0 0
0 0
[c6(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[activate#(x1)] =
1 0
0 0
+
0 0
0 0
· x1
[c2] =
0 0
0 0
[c9] =
0 0
4 0
[c10(x1)] =
0 0
0 0
+
1 0
0 0
· x1
[c13(x1)] =
0 0
0 0
+
1 0
0 1
· x1
[2ndspos#(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 1
0 0
· x2
[c11] =
0 0
4 0
[plus#(x1, x2)] =
0 0
4 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[from#(x1)] =
1 0
0 0
+
0 0
0 0
· x1
[c3(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[c4(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[2ndsneg#(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 1
0 0
· x2
[c7(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[c15] =
1 0
0 0
[s(x1)] =
2 0
0 0
+
1 1
0 0
· x1
[plus(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[0] =
0 0
0 0
[cons(x1, x2)] =
0 0
1 0
+
0 0
0 0
· x1 +
1 4
0 0
· x2
[n__from(x1)] =
0 0
0 0
+
0 0
0 0
· x1
[from(x1)] =
0 0
1 0
+
0 0
0 0
· x1
[activate(x1)] =
0 0
1 0
+
1 0
0 4
· x1
[times(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[cons2(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
1 0
0 0
· x2
which has the intended complexity. Here, only the following usable rules have been considered:
from#(z0) → c (18)
from#(z0) → c1 (20)
2ndspos#(0,z0) → c2 (22)
2ndspos#(s(z0),cons(z1,z2)) → c3(2ndspos#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (24)
2ndspos#(s(z0),cons2(z1,cons(z2,z3))) → c4(2ndsneg#(z0,activate(z3)),activate#(z3)) (26)
2ndsneg#(0,z0) → c5 (28)
2ndsneg#(s(z0),cons(z1,z2)) → c6(2ndsneg#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (30)
2ndsneg#(s(z0),cons2(z1,cons(z2,z3))) → c7(2ndspos#(z0,activate(z3)),activate#(z3)) (32)
pi#(z0) → c8(2ndspos#(z0,from(0)),from#(0)) (34)
plus#(0,z0) → c9 (36)
plus#(s(z0),z1) → c10(plus#(z0,z1)) (38)
times#(0,z0) → c11 (40)
times#(s(z0),z1) → c12(plus#(z1,times(z0,z1)),times#(z0,z1)) (42)
square#(z0) → c13(times#(z0,z0)) (44)
activate#(n__from(z0)) → c14(from#(z0)) (46)
activate#(z0) → c15 (48)
from(z0) → cons(z0,n__from(s(z0))) (17)
from(z0) → n__from(z0) (19)
activate(n__from(z0)) → from(z0) (45)
activate(z0) → z0 (47)

1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules
from#(z0) → c (18)
are strictly oriented by the following linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the rationals with delta = 1
[square#(x1)] =
1 0
4 0
+
1 4
0 0
· x1
[pi#(x1)] =
4 0
4 0
+
1 3
0 0
· x1
[c] =
0 0
0 0
[c1] =
0 0
0 0
[c12(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[times#(x1, x2)] =
0 0
1 0
+
0 0
0 0
· x1 +
1 0
0 0
· x2
[c8(x1, x2)] =
0 0
0 0
+
1 3
0 0
· x1 +
1 0
0 0
· x2
[c14(x1)] =
0 0
0 0
+
1 0
0 0
· x1
[c5] =
0 0
0 0
[c6(x1, x2)] =
0 0
0 0
+
1 0
0 1
· x1 +
1 0
0 0
· x2
[activate#(x1)] =
1 0
0 0
+
0 0
0 0
· x1
[c2] =
0 0
0 0
[c9] =
0 0
4 0
[c10(x1)] =
0 0
0 0
+
1 0
0 0
· x1
[c13(x1)] =
0 0
3 0
+
1 0
0 1
· x1
[2ndspos#(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 2
0 0
· x2
[c11] =
0 0
1 0
[plus#(x1, x2)] =
0 0
4 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[from#(x1)] =
1 0
0 0
+
0 0
0 0
· x1
[c3(x1, x2)] =
0 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[c4(x1, x2)] =
0 0
0 0
+
1 0
0 1
· x1 +
1 0
0 0
· x2
[2ndsneg#(x1, x2)] =
0 0
0 0
+
1 4
0 0
· x1 +
1 1
0 0
· x2
[c7(x1, x2)] =
3 0
0 0
+
1 0
0 0
· x1 +
1 0
0 0
· x2
[c15] =
0 0
0 0
[s(x1)] =
4 0
4 0
+
1 4
0 0
· x1
[plus(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[0] =
0 0
0 0
[cons(x1, x2)] =
0 0
1 0
+
0 0
0 0
· x1 +
1 2
0 0
· x2
[n__from(x1)] =
0 0
0 0
+
0 0
0 0
· x1
[from(x1)] =
0 0
1 0
+
0 0
0 0
· x1
[activate(x1)] =
0 0
2 0
+
1 0
0 1
· x1
[times(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
[cons2(x1, x2)] =
0 0
0 0
+
0 0
0 0
· x1 +
1 0
0 0
· x2
which has the intended complexity. Here, only the following usable rules have been considered:
from#(z0) → c (18)
from#(z0) → c1 (20)
2ndspos#(0,z0) → c2 (22)
2ndspos#(s(z0),cons(z1,z2)) → c3(2ndspos#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (24)
2ndspos#(s(z0),cons2(z1,cons(z2,z3))) → c4(2ndsneg#(z0,activate(z3)),activate#(z3)) (26)
2ndsneg#(0,z0) → c5 (28)
2ndsneg#(s(z0),cons(z1,z2)) → c6(2ndsneg#(s(z0),cons2(z1,activate(z2))),activate#(z2)) (30)
2ndsneg#(s(z0),cons2(z1,cons(z2,z3))) → c7(2ndspos#(z0,activate(z3)),activate#(z3)) (32)
pi#(z0) → c8(2ndspos#(z0,from(0)),from#(0)) (34)
plus#(0,z0) → c9 (36)
plus#(s(z0),z1) → c10(plus#(z0,z1)) (38)
times#(0,z0) → c11 (40)
times#(s(z0),z1) → c12(plus#(z1,times(z0,z1)),times#(z0,z1)) (42)
square#(z0) → c13(times#(z0,z0)) (44)
activate#(n__from(z0)) → c14(from#(z0)) (46)
activate#(z0) → c15 (48)
from(z0) → cons(z0,n__from(s(z0))) (17)
from(z0) → n__from(z0) (19)
activate(n__from(z0)) → from(z0) (45)
activate(z0) → z0 (47)

1.1.1.1.1.1.1.1.1.1.1.1.1.1 R is empty

There are no rules in the TRS R. Hence, R/S has complexity O(1).