The rewrite relation of the following TRS is considered.
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
|
gcd#(s(x),s(y)) |
→ |
gcd#(minus(max(x,y),min(x,transform(y))),s(min(x,y))) |
(20) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [gcd#(x1, x2)] |
= |
-2 + 2 · x1 + x2
|
| [minus(x1, x2)] |
= |
x1 |
| [max(x1, x2)] |
= |
x1 + x2
|
| [0] |
= |
0 |
| [s(x1)] |
= |
2 + 2 · x1
|
| [min(x1, x2)] |
= |
x1 |
| [transform(x1)] |
= |
0 |
| [cons(x1, x2)] |
= |
2 + 2 · x1
|
together with the usable
rules
|
max(x,0) |
→ |
x |
(4) |
|
max(0,y) |
→ |
y |
(5) |
|
max(s(x),s(y)) |
→ |
s(max(x,y)) |
(6) |
|
min(x,0) |
→ |
0 |
(1) |
|
min(0,y) |
→ |
0 |
(2) |
|
min(s(x),s(y)) |
→ |
s(min(x,y)) |
(3) |
|
minus(x,0) |
→ |
x |
(7) |
|
minus(s(x),s(y)) |
→ |
s(minus(x,y)) |
(8) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
gcd#(s(x),s(y)) |
→ |
gcd#(minus(max(x,y),min(x,transform(y))),s(min(x,y))) |
(20) |
could be deleted.
1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
min#(s(x),s(y)) |
→ |
min#(x,y) |
(17) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [s(x1)] |
= |
1 · x1
|
| [min#(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.
|
min#(s(x),s(y)) |
→ |
min#(x,y) |
(17) |
|
|
| 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
|
max#(s(x),s(y)) |
→ |
max#(x,y) |
(18) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [s(x1)] |
= |
1 · x1
|
| [max#(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.
|
max#(s(x),s(y)) |
→ |
max#(x,y) |
(18) |
|
|
| 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
|
minus#(s(x),s(y)) |
→ |
minus#(x,y) |
(19) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [s(x1)] |
= |
1 · x1
|
| [minus#(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 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) |
(19) |
|
|
| 1 |
> |
1 |
| 2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
|
cons#(x,cons(y,s(z))) |
→ |
cons#(y,x) |
(29) |
|
cons#(cons(x,z),s(y)) |
→ |
transform#(x) |
(30) |
|
transform#(cons(x,y)) |
→ |
cons#(cons(x,x),x) |
(26) |
|
transform#(cons(x,y)) |
→ |
cons#(x,x) |
(27) |
|
transform#(s(x)) |
→ |
transform#(x) |
(28) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
| [cons#(x1, x2)] |
= |
+ · x1 + · x2
|
| [cons(x1, x2)] |
= |
+ · x1 + · x2
|
| [s(x1)] |
= |
+ · x1
|
| [transform#(x1)] |
= |
+ · x1
|
| [transform(x1)] |
= |
+ · x1
|
together with the usable
rules
|
cons(x,y) |
→ |
y |
(14) |
|
transform(cons(x,y)) |
→ |
cons(cons(x,x),x) |
(11) |
|
cons(x,cons(y,s(z))) |
→ |
cons(y,x) |
(15) |
|
cons(cons(x,z),s(y)) |
→ |
transform(x) |
(16) |
|
transform(x) |
→ |
s(s(x)) |
(10) |
|
transform(cons(x,y)) |
→ |
y |
(12) |
|
transform(s(x)) |
→ |
s(s(transform(x))) |
(13) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
cons#(cons(x,z),s(y)) |
→ |
transform#(x) |
(30) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
|
transform#(s(x)) |
→ |
transform#(x) |
(28) |
1.1.5.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [s(x1)] |
= |
1 · x1
|
| [transform#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.5.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
transform#(s(x)) |
→ |
transform#(x) |
(28) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
|
cons#(x,cons(y,s(z))) |
→ |
cons#(y,x) |
(29) |
1.1.5.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [s(x1)] |
= |
1 · x1
|
| [cons#(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.
1.1.5.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
cons#(x,cons(y,s(z))) |
→ |
cons#(y,x) |
(29) |
|
|
| 2 |
> |
1 |
| 1 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.