The rewrite relation of the following TRS is considered.
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
|
f#(x,s(y),z) |
→ |
f#(*(x,x),half(s(y)),z) |
(27) |
|
f#(x,s(y),z) |
→ |
f#(x,y,*(x,z)) |
(25) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(s) |
= |
0 |
|
weight(s) |
= |
1 |
|
|
|
| prec(0) |
= |
1 |
|
weight(0) |
= |
1 |
|
|
|
in combination with the following argument filter
| π(f#) |
= |
2 |
| π(s) |
= |
[1] |
| π(half) |
= |
1 |
| π(0) |
= |
[] |
together with the usable
rules
|
half(s(0)) |
→ |
0 |
(11) |
|
half(s(s(x))) |
→ |
s(half(x)) |
(12) |
|
half(0) |
→ |
0 |
(10) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
f#(x,s(y),z) |
→ |
f#(x,y,*(x,z)) |
(25) |
could be deleted.
1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [f#(x1, x2, x3)] |
= |
-2 + 2 · x2
|
| [*(x1, x2)] |
= |
2 · x1 + 2 · x2
|
| [0] |
= |
0 |
| [s(x1)] |
= |
2 + x1
|
| [+(x1, x2)] |
= |
-2 + 2 · x1
|
| [half(x1)] |
= |
-2 + x1
|
together with the usable
rules
|
half(s(0)) |
→ |
0 |
(11) |
|
half(s(s(x))) |
→ |
s(half(x)) |
(12) |
|
half(0) |
→ |
0 |
(10) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
f#(x,s(y),z) |
→ |
f#(*(x,x),half(s(y)),z) |
(27) |
could be deleted.
1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
-#(s(x),s(y)) |
→ |
-#(x,y) |
(18) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [s(x1)] |
= |
1 · x1
|
| [-#(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.
1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
-#(s(x),s(y)) |
→ |
-#(x,y) |
(18) |
|
|
| 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
|
*#(x,s(y)) |
→ |
*#(x,y) |
(19) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [s(x1)] |
= |
1 · x1
|
| [*#(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.
1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
*#(x,s(y)) |
→ |
*#(x,y) |
(19) |
|
|
| 1 |
≥ |
1 |
| 2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
|
odd#(s(s(x))) |
→ |
odd#(x) |
(20) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [s(x1)] |
= |
1 · x1
|
| [odd#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
odd#(s(s(x))) |
→ |
odd#(x) |
(20) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
|
half#(s(s(x))) |
→ |
half#(x) |
(21) |
1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [s(x1)] |
= |
1 · x1
|
| [half#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
half#(s(s(x))) |
→ |
half#(x) |
(21) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.