The rewrite relation of the following TRS is considered.
from(X) |
→ |
cons(X,n__from(n__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) |
s(X) |
→ |
n__s(X) |
(15) |
activate(n__from(X)) |
→ |
from(activate(X)) |
(16) |
activate(n__s(X)) |
→ |
s(activate(X)) |
(17) |
activate(X) |
→ |
X |
(18) |
2ndspos#(s(N),cons(X,Z)) |
→ |
2ndspos#(s(N),cons2(X,activate(Z))) |
(19) |
2ndspos#(s(N),cons(X,Z)) |
→ |
activate#(Z) |
(20) |
2ndspos#(s(N),cons2(X,cons(Y,Z))) |
→ |
2ndsneg#(N,activate(Z)) |
(21) |
2ndspos#(s(N),cons2(X,cons(Y,Z))) |
→ |
activate#(Z) |
(22) |
2ndsneg#(s(N),cons(X,Z)) |
→ |
2ndsneg#(s(N),cons2(X,activate(Z))) |
(23) |
2ndsneg#(s(N),cons(X,Z)) |
→ |
activate#(Z) |
(24) |
2ndsneg#(s(N),cons2(X,cons(Y,Z))) |
→ |
2ndspos#(N,activate(Z)) |
(25) |
2ndsneg#(s(N),cons2(X,cons(Y,Z))) |
→ |
activate#(Z) |
(26) |
pi#(X) |
→ |
2ndspos#(X,from(0)) |
(27) |
pi#(X) |
→ |
from#(0) |
(28) |
plus#(s(X),Y) |
→ |
s#(plus(X,Y)) |
(29) |
plus#(s(X),Y) |
→ |
plus#(X,Y) |
(30) |
times#(s(X),Y) |
→ |
plus#(Y,times(X,Y)) |
(31) |
times#(s(X),Y) |
→ |
times#(X,Y) |
(32) |
square#(X) |
→ |
times#(X,X) |
(33) |
activate#(n__from(X)) |
→ |
from#(activate(X)) |
(34) |
activate#(n__from(X)) |
→ |
activate#(X) |
(35) |
activate#(n__s(X)) |
→ |
s#(activate(X)) |
(36) |
activate#(n__s(X)) |
→ |
activate#(X) |
(37) |
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
times#(s(X),Y) |
→ |
times#(X,Y) |
(32) |
1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[s(x1)] |
= |
1 · x1
|
[times#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
times#(s(X),Y) |
→ |
times#(X,Y) |
(32) |
|
1 |
> |
1 |
2 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
2ndspos#(s(N),cons2(X,cons(Y,Z))) |
→ |
2ndsneg#(N,activate(Z)) |
(21) |
2ndsneg#(s(N),cons(X,Z)) |
→ |
2ndsneg#(s(N),cons2(X,activate(Z))) |
(23) |
2ndsneg#(s(N),cons2(X,cons(Y,Z))) |
→ |
2ndspos#(N,activate(Z)) |
(25) |
2ndspos#(s(N),cons(X,Z)) |
→ |
2ndspos#(s(N),cons2(X,activate(Z))) |
(19) |
1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
2ndsneg#(s(N),cons2(X,cons(Y,Z))) |
→ |
2ndspos#(N,activate(Z)) |
(25) |
|
1 |
> |
1 |
2ndsneg#(s(N),cons(X,Z)) |
→ |
2ndsneg#(s(N),cons2(X,activate(Z))) |
(23) |
|
1 |
≥ |
1 |
2ndspos#(s(N),cons(X,Z)) |
→ |
2ndspos#(s(N),cons2(X,activate(Z))) |
(19) |
|
1 |
≥ |
1 |
2ndspos#(s(N),cons2(X,cons(Y,Z))) |
→ |
2ndsneg#(N,activate(Z)) |
(21) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
plus#(s(X),Y) |
→ |
plus#(X,Y) |
(30) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[s(x1)] |
= |
1 · x1
|
[plus#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
plus#(s(X),Y) |
→ |
plus#(X,Y) |
(30) |
|
1 |
> |
1 |
2 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
activate#(n__s(X)) |
→ |
activate#(X) |
(37) |
activate#(n__from(X)) |
→ |
activate#(X) |
(35) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[n__s(x1)] |
= |
1 · x1
|
[n__from(x1)] |
= |
1 · x1
|
[activate#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
activate#(n__s(X)) |
→ |
activate#(X) |
(37) |
|
1 |
> |
1 |
activate#(n__from(X)) |
→ |
activate#(X) |
(35) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.