The rewrite relation of the following TRS is considered.
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
*#(1(x),y) |
→ |
*#(x,y) |
(52) |
|
*#(0(x),y) |
→ |
*#(x,y) |
(49) |
|
*#(j(x),y) |
→ |
*#(x,y) |
(55) |
|
*#(*(x,y),z) |
→ |
*#(x,*(y,z)) |
(56) |
|
*#(*(x,y),z) |
→ |
*#(y,z) |
(57) |
|
*#(+(x,y),z) |
→ |
*#(x,z) |
(59) |
|
*#(+(x,y),z) |
→ |
*#(y,z) |
(60) |
|
*#(x,+(y,z)) |
→ |
*#(x,y) |
(62) |
|
*#(x,+(y,z)) |
→ |
*#(x,z) |
(63) |
1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
*#(1(x),y) |
→ |
*#(x,y) |
(52) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
|
*#(0(x),y) |
→ |
*#(x,y) |
(49) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
|
*#(j(x),y) |
→ |
*#(x,y) |
(55) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
|
*#(*(x,y),z) |
→ |
*#(x,*(y,z)) |
(56) |
|
| 1 |
> |
1 |
|
*#(*(x,y),z) |
→ |
*#(y,z) |
(57) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
|
*#(+(x,y),z) |
→ |
*#(x,z) |
(59) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
|
*#(+(x,y),z) |
→ |
*#(y,z) |
(60) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
|
*#(x,+(y,z)) |
→ |
*#(x,y) |
(62) |
|
|
| 1 |
≥ |
1 |
| 2 |
> |
2 |
|
*#(x,+(y,z)) |
→ |
*#(x,z) |
(63) |
|
|
| 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
|
+#(0(x),1(y)) |
→ |
+#(x,y) |
(28) |
|
+#(0(x),0(y)) |
→ |
+#(x,y) |
(27) |
|
+#(1(x),0(y)) |
→ |
+#(x,y) |
(29) |
|
+#(0(x),j(y)) |
→ |
+#(x,y) |
(30) |
|
+#(j(x),0(y)) |
→ |
+#(x,y) |
(31) |
|
+#(1(x),1(y)) |
→ |
+#(+(x,y),1(#)) |
(32) |
|
+#(1(x),1(y)) |
→ |
+#(x,y) |
(33) |
|
+#(j(x),j(y)) |
→ |
+#(+(x,y),j(#)) |
(34) |
|
+#(j(x),j(y)) |
→ |
+#(x,y) |
(35) |
|
+#(1(x),j(y)) |
→ |
+#(x,y) |
(37) |
|
+#(j(x),1(y)) |
→ |
+#(x,y) |
(39) |
|
+#(+(x,y),z) |
→ |
+#(x,+(y,z)) |
(40) |
|
+#(+(x,y),z) |
→ |
+#(y,z) |
(41) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [+(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [#] |
= |
0 |
| [0(x1)] |
= |
1 · x1
|
| [1(x1)] |
= |
1 · x1
|
| [j(x1)] |
= |
1 · x1
|
| [+#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
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) |
|
+(0(x),j(y)) |
→ |
j(+(x,y)) |
(7) |
|
+(j(x),0(y)) |
→ |
j(+(x,y)) |
(8) |
|
+(1(x),1(y)) |
→ |
j(+(+(x,y),1(#))) |
(9) |
|
+(j(x),j(y)) |
→ |
1(+(+(x,y),j(#))) |
(10) |
|
+(1(x),j(y)) |
→ |
0(+(x,y)) |
(11) |
|
+(j(x),1(y)) |
→ |
0(+(x,y)) |
(12) |
|
+(+(x,y),z) |
→ |
+(x,+(y,z)) |
(13) |
|
0(#) |
→ |
# |
(1) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.2.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [+(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [#] |
= |
0 |
| [0(x1)] |
= |
1 + 1 · x1
|
| [1(x1)] |
= |
1 + 1 · x1
|
| [j(x1)] |
= |
1 + 1 · x1
|
| [+#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
the
pairs
|
+#(0(x),1(y)) |
→ |
+#(x,y) |
(28) |
|
+#(0(x),0(y)) |
→ |
+#(x,y) |
(27) |
|
+#(1(x),0(y)) |
→ |
+#(x,y) |
(29) |
|
+#(0(x),j(y)) |
→ |
+#(x,y) |
(30) |
|
+#(j(x),0(y)) |
→ |
+#(x,y) |
(31) |
|
+#(1(x),1(y)) |
→ |
+#(+(x,y),1(#)) |
(32) |
|
+#(1(x),1(y)) |
→ |
+#(x,y) |
(33) |
|
+#(j(x),j(y)) |
→ |
+#(+(x,y),j(#)) |
(34) |
|
+#(j(x),j(y)) |
→ |
+#(x,y) |
(35) |
|
+#(1(x),j(y)) |
→ |
+#(x,y) |
(37) |
|
+#(j(x),1(y)) |
→ |
+#(x,y) |
(39) |
and
the
rules
|
+(0(x),0(y)) |
→ |
0(+(x,y)) |
(4) |
|
+(0(x),1(y)) |
→ |
1(+(x,y)) |
(5) |
|
+(1(x),0(y)) |
→ |
1(+(x,y)) |
(6) |
|
+(0(x),j(y)) |
→ |
j(+(x,y)) |
(7) |
|
+(j(x),0(y)) |
→ |
j(+(x,y)) |
(8) |
|
+(1(x),j(y)) |
→ |
0(+(x,y)) |
(11) |
|
+(j(x),1(y)) |
→ |
0(+(x,y)) |
(12) |
|
0(#) |
→ |
# |
(1) |
could be deleted.
1.1.2.1.1 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,z)) |
(40) |
|
| 1 |
> |
1 |
|
+#(+(x,y),z) |
→ |
+#(y,z) |
(41) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
|
opp#(1(x)) |
→ |
opp#(x) |
(44) |
|
opp#(0(x)) |
→ |
opp#(x) |
(43) |
|
opp#(j(x)) |
→ |
opp#(x) |
(45) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [1(x1)] |
= |
1 · x1
|
| [0(x1)] |
= |
1 · x1
|
| [j(x1)] |
= |
1 · x1
|
| [opp#(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.
|
opp#(1(x)) |
→ |
opp#(x) |
(44) |
|
| 1 |
> |
1 |
|
opp#(0(x)) |
→ |
opp#(x) |
(43) |
|
| 1 |
> |
1 |
|
opp#(j(x)) |
→ |
opp#(x) |
(45) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.