The rewrite relation of the following TRS is considered.
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
|
prod#(app(l1,l2)) |
→ |
prod#(l1) |
(51) |
|
prod#(cons(x,l)) |
→ |
prod#(l) |
(49) |
|
prod#(app(l1,l2)) |
→ |
prod#(l2) |
(52) |
1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [app(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [prod#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
prod#(app(l1,l2)) |
→ |
prod#(l1) |
(51) |
|
| 1 |
> |
1 |
|
prod#(cons(x,l)) |
→ |
prod#(l) |
(49) |
|
| 1 |
> |
1 |
|
prod#(app(l1,l2)) |
→ |
prod#(l2) |
(52) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
|
*#(1(x),y) |
→ |
*#(x,y) |
(35) |
|
*#(0(x),y) |
→ |
*#(x,y) |
(32) |
|
*#(*(x,y),z) |
→ |
*#(x,*(y,z)) |
(36) |
|
*#(*(x,y),z) |
→ |
*#(y,z) |
(37) |
|
*#(x,+(y,z)) |
→ |
*#(x,y) |
(39) |
|
*#(x,+(y,z)) |
→ |
*#(x,z) |
(40) |
1.1.2 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) |
(35) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
|
*#(0(x),y) |
→ |
*#(x,y) |
(32) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
|
*#(*(x,y),z) |
→ |
*#(x,*(y,z)) |
(36) |
|
| 1 |
> |
1 |
|
*#(*(x,y),z) |
→ |
*#(y,z) |
(37) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
|
*#(x,+(y,z)) |
→ |
*#(x,y) |
(39) |
|
|
| 1 |
≥ |
1 |
| 2 |
> |
2 |
|
*#(x,+(y,z)) |
→ |
*#(x,z) |
(40) |
|
|
| 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
|
sum#(app(l1,l2)) |
→ |
sum#(l1) |
(46) |
|
sum#(cons(x,l)) |
→ |
sum#(l) |
(44) |
|
sum#(app(l1,l2)) |
→ |
sum#(l2) |
(47) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [app(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [sum#(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.
|
sum#(app(l1,l2)) |
→ |
sum#(l1) |
(46) |
|
| 1 |
> |
1 |
|
sum#(cons(x,l)) |
→ |
sum#(l) |
(44) |
|
| 1 |
> |
1 |
|
sum#(app(l1,l2)) |
→ |
sum#(l2) |
(47) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
|
+#(0(x),1(y)) |
→ |
+#(x,y) |
(24) |
|
+#(0(x),0(y)) |
→ |
+#(x,y) |
(23) |
|
+#(1(x),0(y)) |
→ |
+#(x,y) |
(25) |
|
+#(1(x),1(y)) |
→ |
+#(+(x,y),1(#)) |
(27) |
|
+#(1(x),1(y)) |
→ |
+#(x,y) |
(28) |
|
+#(+(x,y),z) |
→ |
+#(x,+(y,z)) |
(29) |
|
+#(+(x,y),z) |
→ |
+#(y,z) |
(30) |
1.1.4 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
|
| [+#(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) |
|
+(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
rule
could be deleted.
1.1.4.1 Monotonic Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(#) |
= |
4 |
|
weight(#) |
= |
2 |
|
|
|
| prec(0) |
= |
1 |
|
weight(0) |
= |
1 |
|
|
|
| prec(1) |
= |
2 |
|
weight(1) |
= |
3 |
|
|
|
| prec(+) |
= |
3 |
|
weight(+) |
= |
0 |
|
|
|
| prec(+#) |
= |
0 |
|
weight(+#) |
= |
0 |
|
|
|
the
pairs
|
+#(0(x),1(y)) |
→ |
+#(x,y) |
(24) |
|
+#(0(x),0(y)) |
→ |
+#(x,y) |
(23) |
|
+#(1(x),0(y)) |
→ |
+#(x,y) |
(25) |
|
+#(1(x),1(y)) |
→ |
+#(+(x,y),1(#)) |
(27) |
|
+#(1(x),1(y)) |
→ |
+#(x,y) |
(28) |
|
+#(+(x,y),z) |
→ |
+#(x,+(y,z)) |
(29) |
|
+#(+(x,y),z) |
→ |
+#(y,z) |
(30) |
and
the
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) |
could be deleted.
1.1.4.1.1 P is empty
There are no pairs anymore.
-
The
5th
component contains the
pair
|
app#(cons(x,l1),l2) |
→ |
app#(l1,l2) |
(41) |
1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [app#(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.
|
app#(cons(x,l1),l2) |
→ |
app#(l1,l2) |
(41) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.