The rewrite relation of the following TRS is considered.
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
|
app#(cons(x,l1),l2) |
→ |
app#(l1,l2) |
(41) |
1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
app#(cons(x,l1),l2) |
→ |
app#(l1,l2) |
(41) |
|
|
| 2 |
≥ |
2 |
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
|
sum#(cons(x,l)) |
→ |
sum#(l) |
(43) |
|
sum#(app(l1,l2)) |
→ |
sum#(l1) |
(46) |
|
sum#(app(l1,l2)) |
→ |
sum#(l2) |
(45) |
1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
sum#(cons(x,l)) |
→ |
sum#(l) |
(43) |
|
| 1 |
> |
1 |
|
sum#(app(l1,l2)) |
→ |
sum#(l1) |
(46) |
|
| 1 |
> |
1 |
|
sum#(app(l1,l2)) |
→ |
sum#(l2) |
(45) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
|
prod#(cons(x,l)) |
→ |
prod#(l) |
(48) |
|
prod#(app(l1,l2)) |
→ |
prod#(l1) |
(51) |
|
prod#(app(l1,l2)) |
→ |
prod#(l2) |
(50) |
1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
prod#(cons(x,l)) |
→ |
prod#(l) |
(48) |
|
| 1 |
> |
1 |
|
prod#(app(l1,l2)) |
→ |
prod#(l1) |
(51) |
|
| 1 |
> |
1 |
|
prod#(app(l1,l2)) |
→ |
prod#(l2) |
(50) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
|
*#(x,+(y,z)) |
→ |
*#(x,y) |
(39) |
|
*#(x,+(y,z)) |
→ |
*#(x,z) |
(38) |
|
*#(*(x,y),z) |
→ |
*#(x,*(y,z)) |
(37) |
|
*#(*(x,y),z) |
→ |
*#(y,z) |
(36) |
|
*#(1(x),y) |
→ |
*#(x,y) |
(33) |
|
*#(0(x),y) |
→ |
*#(x,y) |
(31) |
1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
*#(x,+(y,z)) |
→ |
*#(x,y) |
(39) |
|
|
| 2 |
> |
2 |
| 1 |
≥ |
1 |
|
*#(x,+(y,z)) |
→ |
*#(x,z) |
(38) |
|
|
| 2 |
> |
2 |
| 1 |
≥ |
1 |
|
*#(*(x,y),z) |
→ |
*#(x,*(y,z)) |
(37) |
|
| 1 |
> |
1 |
|
*#(*(x,y),z) |
→ |
*#(y,z) |
(36) |
|
|
| 2 |
≥ |
2 |
| 1 |
> |
1 |
|
*#(1(x),y) |
→ |
*#(x,y) |
(33) |
|
|
| 2 |
≥ |
2 |
| 1 |
> |
1 |
|
*#(0(x),y) |
→ |
*#(x,y) |
(31) |
|
|
| 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
|
+#(+(x,y),z) |
→ |
+#(x,+(y,z)) |
(30) |
|
+#(+(x,y),z) |
→ |
+#(y,z) |
(29) |
|
+#(1(x),1(y)) |
→ |
+#(+(x,y),1(#)) |
(27) |
|
+#(1(x),1(y)) |
→ |
+#(x,y) |
(26) |
|
+#(1(x),0(y)) |
→ |
+#(x,y) |
(25) |
|
+#(0(x),1(y)) |
→ |
+#(x,y) |
(24) |
|
+#(0(x),0(y)) |
→ |
+#(x,y) |
(22) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [0(x1)] |
= |
2 · x1 + 0 |
| [1(x1)] |
= |
2 · x1 + 2 |
| [#] |
= |
0 |
| [+#(x1, x2)] |
= |
4 · x1 + 4 · x2 + 0 |
| [+(x1, x2)] |
= |
1 · x1 + 1 · x2 + 0 |
together with the usable
rules
|
+(x,#) |
→ |
x |
(2) |
|
+(#,x) |
→ |
x |
(3) |
|
+(0(x),0(y)) |
→ |
0(+(x,y)) |
(4) |
|
+(0(x),1(y)) |
→ |
1(+(x,y)) |
(5) |
|
+(1(x),0(y)) |
→ |
1(+(x,y)) |
(6) |
|
+(1(x),1(y)) |
→ |
0(+(+(x,y),1(#))) |
(7) |
|
+(+(x,y),z) |
→ |
+(x,+(y,z)) |
(8) |
|
0(#) |
→ |
# |
(1) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
+#(1(x),1(y)) |
→ |
+#(+(x,y),1(#)) |
(27) |
|
+#(1(x),1(y)) |
→ |
+#(x,y) |
(26) |
|
+#(1(x),0(y)) |
→ |
+#(x,y) |
(25) |
|
+#(0(x),1(y)) |
→ |
+#(x,y) |
(24) |
could be deleted.
1.1.5.1 Subterm Criterion Processor
We use the projection
and remove the pairs:
|
+#(+(x,y),z) |
→ |
+#(x,+(y,z)) |
(30) |
|
+#(+(x,y),z) |
→ |
+#(y,z) |
(29) |
|
+#(0(x),0(y)) |
→ |
+#(x,y) |
(22) |
1.1.5.1.1 P is empty
There are no pairs anymore.