The rewrite relation of the following TRS is considered.
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
|
q1#(y(x1)) |
→ |
q1#(x1) |
(17) |
|
q1#(a(x1)) |
→ |
q1#(x1) |
(15) |
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [y(x1)] |
= |
1 · x1
|
| [a(x1)] |
= |
1 · x1
|
| [q1#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
q1#(y(x1)) |
→ |
q1#(x1) |
(17) |
|
| 1 |
> |
1 |
|
q1#(a(x1)) |
→ |
q1#(x1) |
(15) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
|
q3#(y(x1)) |
→ |
q3#(x1) |
(23) |
1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [y(x1)] |
= |
1 · x1
|
| [q3#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
q3#(y(x1)) |
→ |
q3#(x1) |
(23) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
|
a#(q2(y(x1))) |
→ |
a#(y(x1)) |
(19) |
|
a#(q2(a(x1))) |
→ |
a#(a(x1)) |
(18) |
1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [a(x1)] |
= |
1 · x1
|
| [q2(x1)] |
= |
1 · x1
|
| [y(x1)] |
= |
1 · x1
|
| [a#(x1)] |
= |
1 · x1
|
together with the usable
rules
|
a(q2(a(x1))) |
→ |
q2(a(a(x1))) |
(5) |
|
a(q2(y(x1))) |
→ |
q2(a(y(x1))) |
(6) |
|
y(q2(a(x1))) |
→ |
q2(y(a(x1))) |
(8) |
|
y(q2(y(x1))) |
→ |
q2(y(y(x1))) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
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.
|
a#(q2(y(x1))) |
→ |
a#(y(x1)) |
(19) |
|
| 1 |
> |
1 |
|
a#(q2(a(x1))) |
→ |
a#(a(x1)) |
(18) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
|
y#(q2(y(x1))) |
→ |
y#(y(x1)) |
(21) |
|
y#(q2(a(x1))) |
→ |
y#(a(x1)) |
(20) |
1.1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [a(x1)] |
= |
1 · x1
|
| [q2(x1)] |
= |
1 · x1
|
| [y(x1)] |
= |
1 · x1
|
| [y#(x1)] |
= |
1 · x1
|
together with the usable
rules
|
a(q2(a(x1))) |
→ |
q2(a(a(x1))) |
(5) |
|
a(q2(y(x1))) |
→ |
q2(a(y(x1))) |
(6) |
|
y(q2(a(x1))) |
→ |
q2(y(a(x1))) |
(8) |
|
y(q2(y(x1))) |
→ |
q2(y(y(x1))) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
y#(q2(y(x1))) |
→ |
y#(y(x1)) |
(21) |
|
| 1 |
> |
1 |
|
y#(q2(a(x1))) |
→ |
y#(a(x1)) |
(20) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.