The rewrite relation of the following TRS is considered.
[false] |
= |
|
[s(x1)] |
= |
· x1 +
|
[and(x1, x2)] |
= |
· x1 + · x2 +
|
[mark(x1)] |
= |
· x1 +
|
[if(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
[nil] |
= |
|
[true] |
= |
|
[add(x1, x2)] |
= |
· x1 + · x2 +
|
[from(x1)] |
= |
· x1 +
|
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[first(x1, x2)] |
= |
· x1 + · x2 +
|
[0] |
= |
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
[false] |
= |
|
[s(x1)] |
= |
· x1 +
|
[and(x1, x2)] |
= |
· x1 + · x2 +
|
[mark(x1)] |
= |
· x1 +
|
[if(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
[nil] |
= |
|
[true] |
= |
|
[add(x1, x2)] |
= |
· x1 + · x2 +
|
[from(x1)] |
= |
· x1 +
|
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[first(x1, x2)] |
= |
· x1 + · x2 +
|
[0] |
= |
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
[false] |
= |
|
[s(x1)] |
= |
· x1 +
|
[and(x1, x2)] |
= |
· x1 + · x2 +
|
[mark(x1)] |
= |
· x1 +
|
[if(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
[nil] |
= |
|
[add(x1, x2)] |
= |
· x1 + · x2 +
|
[from(x1)] |
= |
· x1 +
|
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[first(x1, x2)] |
= |
· x1 + · x2 +
|
[0] |
= |
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
The dependency pairs are split into 8
components.
-
The
1st
component contains the
pair
mark#(add(X1,X2)) |
→ |
mark#(X1) |
(58) |
mark#(if(X1,X2,X3)) |
→ |
mark#(X1) |
(55) |
1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
mark#(add(X1,X2)) |
→ |
mark#(X1) |
(58) |
|
1 |
> |
1 |
mark#(if(X1,X2,X3)) |
→ |
mark#(X1) |
(55) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
add#(X1,active(X2)) |
→ |
add#(X1,X2) |
(78) |
add#(active(X1),X2) |
→ |
add#(X1,X2) |
(77) |
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(76) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(75) |
1.1.1.1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
add#(X1,active(X2)) |
→ |
add#(X1,X2) |
(78) |
|
2 |
> |
2 |
1 |
≥ |
1 |
add#(active(X1),X2) |
→ |
add#(X1,X2) |
(77) |
|
2 |
≥ |
2 |
1 |
> |
1 |
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(76) |
|
2 |
> |
2 |
1 |
≥ |
1 |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(75) |
|
2 |
≥ |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
s#(active(X)) |
→ |
s#(X) |
(80) |
s#(mark(X)) |
→ |
s#(X) |
(79) |
1.1.1.1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(active(X)) |
→ |
s#(X) |
(80) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(79) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
from#(mark(X)) |
→ |
from#(X) |
(89) |
from#(active(X)) |
→ |
from#(X) |
(90) |
1.1.1.1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
from#(mark(X)) |
→ |
from#(X) |
(89) |
|
1 |
> |
1 |
from#(active(X)) |
→ |
from#(X) |
(90) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(85) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(88) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(87) |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(86) |
1.1.1.1.1.5 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(85) |
|
2 |
≥ |
2 |
1 |
> |
1 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(88) |
|
2 |
> |
2 |
1 |
≥ |
1 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(87) |
|
2 |
≥ |
2 |
1 |
> |
1 |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(86) |
|
2 |
> |
2 |
1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
if#(mark(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(69) |
if#(X1,X2,active(X3)) |
→ |
if#(X1,X2,X3) |
(74) |
if#(X1,active(X2),X3) |
→ |
if#(X1,X2,X3) |
(73) |
if#(active(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(72) |
if#(X1,X2,mark(X3)) |
→ |
if#(X1,X2,X3) |
(71) |
if#(X1,mark(X2),X3) |
→ |
if#(X1,X2,X3) |
(70) |
1.1.1.1.1.6 Subterm Criterion Processor
We use the projection
and remove the pairs:
if#(X1,active(X2),X3) |
→ |
if#(X1,X2,X3) |
(73) |
if#(X1,mark(X2),X3) |
→ |
if#(X1,X2,X3) |
(70) |
1.1.1.1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
if#(mark(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(69) |
|
3 |
≥ |
3 |
2 |
≥ |
2 |
1 |
> |
1 |
if#(X1,X2,active(X3)) |
→ |
if#(X1,X2,X3) |
(74) |
|
3 |
> |
3 |
2 |
≥ |
2 |
1 |
≥ |
1 |
if#(active(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(72) |
|
3 |
≥ |
3 |
2 |
≥ |
2 |
1 |
> |
1 |
if#(X1,X2,mark(X3)) |
→ |
if#(X1,X2,X3) |
(71) |
|
3 |
> |
3 |
2 |
≥ |
2 |
1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(65) |
and#(X1,active(X2)) |
→ |
and#(X1,X2) |
(68) |
and#(active(X1),X2) |
→ |
and#(X1,X2) |
(67) |
and#(X1,mark(X2)) |
→ |
and#(X1,X2) |
(66) |
1.1.1.1.1.7 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(65) |
|
2 |
≥ |
2 |
1 |
> |
1 |
and#(X1,active(X2)) |
→ |
and#(X1,X2) |
(68) |
|
2 |
> |
2 |
1 |
≥ |
1 |
and#(active(X1),X2) |
→ |
and#(X1,X2) |
(67) |
|
2 |
≥ |
2 |
1 |
> |
1 |
and#(X1,mark(X2)) |
→ |
and#(X1,X2) |
(66) |
|
2 |
> |
2 |
1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(81) |
first#(X1,active(X2)) |
→ |
first#(X1,X2) |
(84) |
first#(active(X1),X2) |
→ |
first#(X1,X2) |
(83) |
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(82) |
1.1.1.1.1.8 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(81) |
|
2 |
≥ |
2 |
1 |
> |
1 |
first#(X1,active(X2)) |
→ |
first#(X1,X2) |
(84) |
|
2 |
> |
2 |
1 |
≥ |
1 |
first#(active(X1),X2) |
→ |
first#(X1,X2) |
(83) |
|
2 |
≥ |
2 |
1 |
> |
1 |
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(82) |
|
2 |
> |
2 |
1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.