The rewrite relation of the following TRS is considered.
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
if#(false,x,l) |
→ |
conviter#(half(x),cons(lastbit(x),l)) |
(20) |
|
conviter#(x,l) |
→ |
if#(zero(x),x,l) |
(17) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers
| [zero(x1)] |
= |
4 · x1 +
-∞ |
| [half(x1)] |
= |
-1 · x1 + 0 |
| [lastbit(x1)] |
= |
0 · x1 + 1 |
| [conviter#(x1, x2)] |
= |
2 · x1 + 1 · x2 + 0 |
| [if#(x1, x2, x3)] |
= |
-4 · x1 + 2 · x2 +
-∞ · x3 + 0 |
| [true] |
= |
0 |
| [0] |
= |
0 |
| [cons(x1, x2)] |
= |
0 · x1 +
-∞ · x2 + 1 |
| [false] |
= |
7 |
| [s(x1)] |
= |
2 · x1 + 3 |
together with the usable
rules
|
lastbit(0) |
→ |
0 |
(4) |
|
lastbit(s(0)) |
→ |
s(0) |
(5) |
|
lastbit(s(s(x))) |
→ |
lastbit(x) |
(6) |
|
half(0) |
→ |
0 |
(1) |
|
half(s(0)) |
→ |
0 |
(2) |
|
half(s(s(x))) |
→ |
s(half(x)) |
(3) |
|
zero(0) |
→ |
true |
(7) |
|
zero(s(x)) |
→ |
false |
(8) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
if#(false,x,l) |
→ |
conviter#(half(x),cons(lastbit(x),l)) |
(20) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
|
half#(s(s(x))) |
→ |
half#(x) |
(13) |
1.1.2 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) |
(13) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
|
lastbit#(s(s(x))) |
→ |
lastbit#(x) |
(14) |
1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
lastbit#(s(s(x))) |
→ |
lastbit#(x) |
(14) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.