The rewrite relation of the following TRS is considered.
The dependency pairs are split into 8
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(58) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(56) |
1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[proper(x1)] |
= |
1 · x1
|
[from(x1)] |
= |
2 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[s(x1)] |
= |
1 · x1
|
[length(x1)] |
= |
1 · x1
|
[nil] |
= |
0 |
[ok(x1)] |
= |
2 · x1
|
[0] |
= |
0 |
[length1(x1)] |
= |
2 · x1
|
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
2 · x1
|
[top#(x1)] |
= |
2 · x1
|
together with the usable
rules
proper(from(X)) |
→ |
from(proper(X)) |
(11) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(12) |
proper(s(X)) |
→ |
s(proper(X)) |
(13) |
proper(length(X)) |
→ |
length(proper(X)) |
(14) |
proper(nil) |
→ |
ok(nil) |
(15) |
proper(0) |
→ |
ok(0) |
(16) |
proper(length1(X)) |
→ |
length1(proper(X)) |
(17) |
length1(ok(X)) |
→ |
ok(length1(X)) |
(22) |
length(ok(X)) |
→ |
ok(length(X)) |
(21) |
s(mark(X)) |
→ |
mark(s(X)) |
(10) |
s(ok(X)) |
→ |
ok(s(X)) |
(20) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(9) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(19) |
from(mark(X)) |
→ |
mark(from(X)) |
(8) |
from(ok(X)) |
→ |
ok(from(X)) |
(18) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(1) |
active(length(nil)) |
→ |
mark(0) |
(2) |
active(length(cons(X,Y))) |
→ |
mark(s(length1(Y))) |
(3) |
active(length1(X)) |
→ |
mark(length(X)) |
(4) |
active(from(X)) |
→ |
from(active(X)) |
(5) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(6) |
active(s(X)) |
→ |
s(active(X)) |
(7) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1 Narrowing Processor
We consider all narrowings of the pair
below position
1
to get the following set of pairs
top#(ok(from(x0))) |
→ |
top#(mark(cons(x0,from(s(x0))))) |
(60) |
top#(ok(length(nil))) |
→ |
top#(mark(0)) |
(61) |
top#(ok(length(cons(x0,x1)))) |
→ |
top#(mark(s(length1(x1)))) |
(62) |
top#(ok(length1(x0))) |
→ |
top#(mark(length(x0))) |
(63) |
top#(ok(from(x0))) |
→ |
top#(from(active(x0))) |
(64) |
top#(ok(cons(x0,x1))) |
→ |
top#(cons(active(x0),x1)) |
(65) |
top#(ok(s(x0))) |
→ |
top#(s(active(x0))) |
(66) |
1.1.1.1.1 Narrowing Processor
We consider all narrowings of the pair
below position
1
to get the following set of pairs
top#(mark(from(x0))) |
→ |
top#(from(proper(x0))) |
(67) |
top#(mark(cons(x0,x1))) |
→ |
top#(cons(proper(x0),proper(x1))) |
(68) |
top#(mark(s(x0))) |
→ |
top#(s(proper(x0))) |
(69) |
top#(mark(length(x0))) |
→ |
top#(length(proper(x0))) |
(70) |
top#(mark(nil)) |
→ |
top#(ok(nil)) |
(71) |
top#(mark(0)) |
→ |
top#(ok(0)) |
(72) |
top#(mark(length1(x0))) |
→ |
top#(length1(proper(x0))) |
(73) |
1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
active#(cons(X1,X2)) |
→ |
active#(X1) |
(34) |
active#(from(X)) |
→ |
active#(X) |
(32) |
active#(s(X)) |
→ |
active#(X) |
(36) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[from(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
active#(cons(X1,X2)) |
→ |
active#(X1) |
(34) |
|
1 |
> |
1 |
active#(from(X)) |
→ |
active#(X) |
(32) |
|
1 |
> |
1 |
active#(s(X)) |
→ |
active#(X) |
(36) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(43) |
proper#(from(X)) |
→ |
proper#(X) |
(41) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(44) |
proper#(s(X)) |
→ |
proper#(X) |
(46) |
proper#(length(X)) |
→ |
proper#(X) |
(48) |
proper#(length1(X)) |
→ |
proper#(X) |
(50) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[from(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[length(x1)] |
= |
1 · x1
|
[length1(x1)] |
= |
1 · x1
|
[proper#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(43) |
|
1 |
> |
1 |
proper#(from(X)) |
→ |
proper#(X) |
(41) |
|
1 |
> |
1 |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(44) |
|
1 |
> |
1 |
proper#(s(X)) |
→ |
proper#(X) |
(46) |
|
1 |
> |
1 |
proper#(length(X)) |
→ |
proper#(X) |
(48) |
|
1 |
> |
1 |
proper#(length1(X)) |
→ |
proper#(X) |
(50) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
from#(ok(X)) |
→ |
from#(X) |
(51) |
from#(mark(X)) |
→ |
from#(X) |
(37) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[from#(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.
from#(ok(X)) |
→ |
from#(X) |
(51) |
|
1 |
> |
1 |
from#(mark(X)) |
→ |
from#(X) |
(37) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(52) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(38) |
1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[cons#(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.
1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(52) |
|
1 |
> |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(38) |
|
1 |
> |
1 |
2 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
s#(ok(X)) |
→ |
s#(X) |
(53) |
s#(mark(X)) |
→ |
s#(X) |
(39) |
1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[s#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(ok(X)) |
→ |
s#(X) |
(53) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(39) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
length#(ok(X)) |
→ |
length#(X) |
(54) |
1.1.7 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[length#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.7.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
length#(ok(X)) |
→ |
length#(X) |
(54) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
length1#(ok(X)) |
→ |
length1#(X) |
(55) |
1.1.8 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[length1#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.8.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
length1#(ok(X)) |
→ |
length1#(X) |
(55) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.