The rewrite relation of the following TRS is considered.
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
prod#(cons(x,l)) |
→ |
prod#(l) |
(37) |
1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[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#(cons(x,l)) |
→ |
prod#(l) |
(37) |
|
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) |
(30) |
*#(0(x),y) |
→ |
*#(x,y) |
(27) |
*#(*(x,y),z) |
→ |
*#(x,*(y,z)) |
(31) |
*#(*(x,y),z) |
→ |
*#(y,z) |
(32) |
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) |
(30) |
|
1 |
> |
1 |
2 |
≥ |
2 |
*#(0(x),y) |
→ |
*#(x,y) |
(27) |
|
1 |
> |
1 |
2 |
≥ |
2 |
*#(*(x,y),z) |
→ |
*#(x,*(y,z)) |
(31) |
|
1 |
> |
1 |
*#(*(x,y),z) |
→ |
*#(y,z) |
(32) |
|
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#(cons(x,l)) |
→ |
sum#(l) |
(35) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[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#(cons(x,l)) |
→ |
sum#(l) |
(35) |
|
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) |
(19) |
+#(0(x),0(y)) |
→ |
+#(x,y) |
(18) |
+#(1(x),0(y)) |
→ |
+#(x,y) |
(20) |
+#(1(x),1(y)) |
→ |
+#(+(x,y),1(#)) |
(22) |
+#(1(x),1(y)) |
→ |
+#(x,y) |
(23) |
+#(+(x,y),z) |
→ |
+#(x,+(y,z)) |
(24) |
+#(+(x,y),z) |
→ |
+#(y,z) |
(25) |
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) |
(19) |
+#(0(x),0(y)) |
→ |
+#(x,y) |
(18) |
+#(1(x),0(y)) |
→ |
+#(x,y) |
(20) |
+#(1(x),1(y)) |
→ |
+#(+(x,y),1(#)) |
(22) |
+#(1(x),1(y)) |
→ |
+#(x,y) |
(23) |
+#(+(x,y),z) |
→ |
+#(x,+(y,z)) |
(24) |
+#(+(x,y),z) |
→ |
+#(y,z) |
(25) |
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.