The rewrite relation of the following TRS is considered.
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
|
quot#(s(x),s(y),z) |
→ |
quot#(minus(p(ack(0,x)),y),s(y),s(z)) |
(25) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [quot#(x1, x2, x3)] |
= |
-1 + x1
|
| [minus(x1, x2)] |
= |
x1 |
| [p(x1)] |
= |
-2 + x1
|
| [ack(x1, x2)] |
= |
2 + 2 · x1 + x2
|
| [0] |
= |
0 |
| [s(x1)] |
= |
2 + x1
|
| [plus(x1, x2)] |
= |
x1 + x2
|
together with the usable
rules
|
ack(0,x) |
→ |
s(x) |
(15) |
|
ack(0,x) |
→ |
plus(x,s(0)) |
(16) |
|
p(s(x)) |
→ |
x |
(10) |
|
p(0) |
→ |
0 |
(11) |
|
minus(s(x),s(y)) |
→ |
minus(x,y) |
(4) |
|
minus(minus(x,y),z) |
→ |
minus(x,plus(y,z)) |
(1) |
|
minus(0,y) |
→ |
0 |
(2) |
|
minus(x,0) |
→ |
x |
(3) |
|
plus(0,y) |
→ |
y |
(5) |
|
plus(s(x),y) |
→ |
plus(x,s(y)) |
(6) |
|
plus(s(x),y) |
→ |
s(plus(y,x)) |
(7) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
quot#(s(x),s(y),z) |
→ |
quot#(minus(p(ack(0,x)),y),s(y),s(z)) |
(25) |
could be deleted.
1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
minus#(s(x),s(y)) |
→ |
minus#(x,y) |
(21) |
|
minus#(minus(x,y),z) |
→ |
minus#(x,plus(y,z)) |
(19) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [plus(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [0] |
= |
0 |
| [s(x1)] |
= |
1 · x1
|
| [minus(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [minus#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
together with the usable
rules
|
plus(0,y) |
→ |
y |
(5) |
|
plus(s(x),y) |
→ |
plus(x,s(y)) |
(6) |
|
plus(s(x),y) |
→ |
s(plus(y,x)) |
(7) |
(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.
|
minus#(s(x),s(y)) |
→ |
minus#(x,y) |
(21) |
|
|
| 1 |
> |
1 |
| 2 |
> |
2 |
|
minus#(minus(x,y),z) |
→ |
minus#(x,plus(y,z)) |
(19) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
|
ack#(s(x),s(y)) |
→ |
ack#(x,ack(s(x),y)) |
(31) |
|
ack#(s(x),0) |
→ |
ack#(x,s(0)) |
(30) |
|
ack#(s(x),s(y)) |
→ |
ack#(s(x),y) |
(32) |
1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
ack#(s(x),0) |
→ |
ack#(x,s(0)) |
(30) |
|
| 1 |
> |
1 |
|
ack#(s(x),s(y)) |
→ |
ack#(x,ack(s(x),y)) |
(31) |
|
| 1 |
> |
1 |
|
ack#(s(x),s(y)) |
→ |
ack#(s(x),y) |
(32) |
|
|
| 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
|
plus#(s(x),y) |
→ |
plus#(y,x) |
(23) |
|
plus#(s(x),y) |
→ |
plus#(x,s(y)) |
(22) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [s(x1)] |
= |
1 · x1
|
| [plus#(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.
1.1.4.1 Monotonic Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(s) |
= |
1 |
|
weight(s) |
= |
1 |
|
|
|
| prec(plus#) |
= |
0 |
|
weight(plus#) |
= |
0 |
|
|
|
the
pairs
|
plus#(s(x),y) |
→ |
plus#(y,x) |
(23) |
|
plus#(s(x),y) |
→ |
plus#(x,s(y)) |
(22) |
and
no rules
could be deleted.
1.1.4.1.1 P is empty
There are no pairs anymore.