The rewrite relation of the following TRS is considered.
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
gcd#(s(x),s(y)) |
→ |
gcd#(-(max(x,y),min(x,y)),s(min(x,y))) |
(18) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers
[-(x1, x2)] |
= |
1 · x1 +
-∞ · x2 +
-∞ |
[min(x1, x2)] |
= |
0 · x1 +
-∞ · x2 + 0 |
[max(x1, x2)] |
= |
0 · x1 + 0 · x2 +
-∞ |
[0] |
= |
0 |
[gcd#(x1, x2)] |
= |
1 · x1 + 0 · x2 + 0 |
[s(x1)] |
= |
3 · x1 + 4 |
together with the usable
rules
min(x,0) |
→ |
0 |
(1) |
min(0,y) |
→ |
0 |
(2) |
min(s(x),s(y)) |
→ |
s(min(x,y)) |
(3) |
max(x,0) |
→ |
x |
(4) |
max(0,y) |
→ |
y |
(5) |
max(s(x),s(y)) |
→ |
s(max(x,y)) |
(6) |
-(x,0) |
→ |
x |
(7) |
-(s(x),s(y)) |
→ |
-(x,y) |
(8) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
gcd#(s(x),s(y)) |
→ |
gcd#(-(max(x,y),min(x,y)),s(min(x,y))) |
(18) |
could be deleted.
1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
-#(s(x),s(y)) |
→ |
-#(x,y) |
(14) |
1.1.2 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) |
(14) |
|
2 |
> |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
max#(s(x),s(y)) |
→ |
max#(x,y) |
(13) |
1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
max#(s(x),s(y)) |
→ |
max#(x,y) |
(13) |
|
2 |
> |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
min#(s(x),s(y)) |
→ |
min#(x,y) |
(12) |
1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
min#(s(x),s(y)) |
→ |
min#(x,y) |
(12) |
|
2 |
> |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.