The rewrite relation of the following TRS is considered.
[mark(x1)] |
= |
· x1 +
|
[add(x1, x2)] |
= |
· x1 + · x2 +
|
[fst(x1, x2)] |
= |
· x1 + · x2 +
|
[nil] |
= |
|
[s(x1)] |
= |
· x1 +
|
[0] |
= |
|
[from(x1)] |
= |
· x1 +
|
[len(x1)] |
= |
· x1 +
|
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
[mark(x1)] |
= |
· x1 +
|
[add(x1, x2)] |
= |
· x1 + · x2 +
|
[fst(x1, x2)] |
= |
· x1 + · x2 +
|
[nil] |
= |
|
[s(x1)] |
= |
· x1 +
|
[0] |
= |
|
[from(x1)] |
= |
· x1 +
|
[len(x1)] |
= |
· x1 +
|
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
[mark(x1)] |
= |
· x1 +
|
[add(x1, x2)] |
= |
· x1 + · x2 +
|
[fst(x1, x2)] |
= |
· x1 + · x2 +
|
[nil] |
= |
|
[s(x1)] |
= |
· x1 +
|
[0] |
= |
|
[from(x1)] |
= |
· x1 +
|
[len(x1)] |
= |
· x1 +
|
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
The dependency pairs are split into 7
components.
-
The
1st
component contains the
pair
s#(mark(X)) |
→ |
s#(X) |
(61) |
s#(active(X)) |
→ |
s#(X) |
(62) |
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.
s#(mark(X)) |
→ |
s#(X) |
(61) |
|
1 |
> |
1 |
s#(active(X)) |
→ |
s#(X) |
(62) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
mark#(len(X)) |
→ |
mark#(X) |
(54) |
mark#(add(X1,X2)) |
→ |
mark#(X1) |
(51) |
mark#(add(X1,X2)) |
→ |
mark#(X2) |
(50) |
mark#(from(X)) |
→ |
mark#(X) |
(47) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(44) |
mark#(fst(X1,X2)) |
→ |
mark#(X1) |
(38) |
mark#(fst(X1,X2)) |
→ |
mark#(X2) |
(37) |
1.1.1.1.1.2 Subterm Criterion Processor
We use the projection
and remove the pairs:
mark#(len(X)) |
→ |
mark#(X) |
(54) |
mark#(add(X1,X2)) |
→ |
mark#(X1) |
(51) |
mark#(add(X1,X2)) |
→ |
mark#(X2) |
(50) |
mark#(from(X)) |
→ |
mark#(X) |
(47) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(44) |
mark#(fst(X1,X2)) |
→ |
mark#(X1) |
(38) |
mark#(fst(X1,X2)) |
→ |
mark#(X2) |
(37) |
1.1.1.1.1.2.1 P is empty
There are no pairs anymore.
-
The
3rd
component contains the
pair
add#(X1,active(X2)) |
→ |
add#(X1,X2) |
(72) |
add#(active(X1),X2) |
→ |
add#(X1,X2) |
(71) |
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(70) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(69) |
1.1.1.1.1.3 Subterm Criterion Processor
We use the projection
and remove the pairs:
add#(active(X1),X2) |
→ |
add#(X1,X2) |
(71) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(69) |
1.1.1.1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
add#(X1,active(X2)) |
→ |
add#(X1,X2) |
(72) |
|
2 |
> |
2 |
1 |
≥ |
1 |
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(70) |
|
2 |
> |
2 |
1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
fst#(mark(X1),X2) |
→ |
fst#(X1,X2) |
(57) |
fst#(X1,active(X2)) |
→ |
fst#(X1,X2) |
(60) |
fst#(active(X1),X2) |
→ |
fst#(X1,X2) |
(59) |
fst#(X1,mark(X2)) |
→ |
fst#(X1,X2) |
(58) |
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.
fst#(mark(X1),X2) |
→ |
fst#(X1,X2) |
(57) |
|
2 |
≥ |
2 |
1 |
> |
1 |
fst#(X1,active(X2)) |
→ |
fst#(X1,X2) |
(60) |
|
2 |
> |
2 |
1 |
≥ |
1 |
fst#(active(X1),X2) |
→ |
fst#(X1,X2) |
(59) |
|
2 |
≥ |
2 |
1 |
> |
1 |
fst#(X1,mark(X2)) |
→ |
fst#(X1,X2) |
(58) |
|
2 |
> |
2 |
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) |
(63) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(66) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(65) |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(64) |
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) |
(63) |
|
2 |
≥ |
2 |
1 |
> |
1 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(66) |
|
2 |
> |
2 |
1 |
≥ |
1 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(65) |
|
2 |
≥ |
2 |
1 |
> |
1 |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(64) |
|
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
from#(mark(X)) |
→ |
from#(X) |
(67) |
from#(active(X)) |
→ |
from#(X) |
(68) |
1.1.1.1.1.6 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) |
(67) |
|
1 |
> |
1 |
from#(active(X)) |
→ |
from#(X) |
(68) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
len#(mark(X)) |
→ |
len#(X) |
(73) |
len#(active(X)) |
→ |
len#(X) |
(74) |
1.1.1.1.1.7 Subterm Criterion Processor
We use the projection
and remove the pairs:
len#(mark(X)) |
→ |
len#(X) |
(73) |
len#(active(X)) |
→ |
len#(X) |
(74) |
1.1.1.1.1.7.1 P is empty
There are no pairs anymore.