The rewrite relation of the following TRS is considered.
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
|
log#(s(x1)) |
→ |
log#(half(s(x1))) |
(10) |
1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers
| [p(x1)] |
= |
-1 · x1 + 0 |
| [0(x1)] |
= |
-∞ · x1 + 0 |
| [log#(x1)] |
= |
0 · x1 + -16 |
| [s(x1)] |
= |
1 · x1 + 4 |
| [half(x1)] |
= |
-1 · x1 + 0 |
together with the usable
rules
|
s(s(p(s(x1)))) |
→ |
s(s(x1)) |
(7) |
|
half(0(x1)) |
→ |
0(s(s(half(x1)))) |
(2) |
|
half(s(0(x1))) |
→ |
0(x1) |
(3) |
|
half(s(s(x1))) |
→ |
s(half(p(s(s(x1))))) |
(4) |
|
half(half(s(s(s(s(x1)))))) |
→ |
s(s(half(half(x1)))) |
(5) |
|
p(s(s(s(x1)))) |
→ |
s(p(s(s(x1)))) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
log#(s(x1)) |
→ |
log#(half(s(x1))) |
(10) |
could be deleted.
1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
half#(0(x1)) |
→ |
half#(x1) |
(12) |
|
half#(half(s(s(s(s(x1)))))) |
→ |
half#(half(x1)) |
(19) |
|
half#(half(s(s(s(s(x1)))))) |
→ |
half#(x1) |
(18) |
1.1.1.2 Reduction Pair Processor with Usable Rules
Using the
| prec(half#) |
= |
0 |
|
stat(half#) |
= |
lex
|
| prec(p) |
= |
0 |
|
stat(p) |
= |
lex
|
| prec(0) |
= |
0 |
|
stat(0) |
= |
lex
|
| prec(half) |
= |
0 |
|
stat(half) |
= |
lex
|
| prec(s) |
= |
0 |
|
stat(s) |
= |
lex
|
| π(half#) |
= |
1 |
| π(p) |
= |
1 |
| π(0) |
= |
[1] |
| π(half) |
= |
1 |
| π(s) |
= |
1 |
together with the usable
rules
|
half(0(x1)) |
→ |
0(s(s(half(x1)))) |
(2) |
|
half(s(0(x1))) |
→ |
0(x1) |
(3) |
|
half(s(s(x1))) |
→ |
s(half(p(s(s(x1))))) |
(4) |
|
half(half(s(s(s(s(x1)))))) |
→ |
s(s(half(half(x1)))) |
(5) |
|
s(s(p(s(x1)))) |
→ |
s(s(x1)) |
(7) |
|
p(s(s(s(x1)))) |
→ |
s(p(s(s(x1)))) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
half#(0(x1)) |
→ |
half#(x1) |
(12) |
could be deleted.
1.1.1.2.1 Reduction Pair Processor with Usable Rules
Using the
| prec(half#) |
= |
0 |
|
stat(half#) |
= |
lex
|
| prec(p) |
= |
0 |
|
stat(p) |
= |
lex
|
| prec(0) |
= |
0 |
|
stat(0) |
= |
lex
|
| prec(half) |
= |
0 |
|
stat(half) |
= |
lex
|
| prec(s) |
= |
0 |
|
stat(s) |
= |
lex
|
| π(half#) |
= |
1 |
| π(p) |
= |
1 |
| π(0) |
= |
1 |
| π(half) |
= |
[1] |
| π(s) |
= |
1 |
together with the usable
rules
|
half(0(x1)) |
→ |
0(s(s(half(x1)))) |
(2) |
|
half(s(0(x1))) |
→ |
0(x1) |
(3) |
|
half(s(s(x1))) |
→ |
s(half(p(s(s(x1))))) |
(4) |
|
half(half(s(s(s(s(x1)))))) |
→ |
s(s(half(half(x1)))) |
(5) |
|
s(s(p(s(x1)))) |
→ |
s(s(x1)) |
(7) |
|
p(s(s(s(x1)))) |
→ |
s(p(s(s(x1)))) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
half#(half(s(s(s(s(x1)))))) |
→ |
half#(x1) |
(18) |
could be deleted.
1.1.1.2.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers
| [p(x1)] |
= |
-1 · x1 + 0 |
| [0(x1)] |
= |
-∞ · x1 + 2 |
| [s(x1)] |
= |
1 · x1 + 1 |
| [half#(x1)] |
= |
0 · x1 + -16 |
| [half(x1)] |
= |
0 · x1 +
-∞ |
together with the usable
rules
|
half(0(x1)) |
→ |
0(s(s(half(x1)))) |
(2) |
|
half(s(0(x1))) |
→ |
0(x1) |
(3) |
|
half(s(s(x1))) |
→ |
s(half(p(s(s(x1))))) |
(4) |
|
half(half(s(s(s(s(x1)))))) |
→ |
s(s(half(half(x1)))) |
(5) |
|
s(s(p(s(x1)))) |
→ |
s(s(x1)) |
(7) |
|
p(s(s(s(x1)))) |
→ |
s(p(s(s(x1)))) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
half#(half(s(s(s(s(x1)))))) |
→ |
half#(half(x1)) |
(19) |
could be deleted.
1.1.1.2.1.1.1 P is empty
There are no pairs anymore.
-
The
3rd
component contains the
pair
|
half#(s(s(x1))) |
→ |
half#(p(s(s(x1)))) |
(16) |
1.1.1.3 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers
| [p(x1)] |
= |
-1 · x1 + 0 |
| [s(x1)] |
= |
1 · x1 + 2 |
| [half#(x1)] |
= |
0 · x1 + 0 |
together with the usable
rules
|
s(s(p(s(x1)))) |
→ |
s(s(x1)) |
(7) |
|
p(s(s(s(x1)))) |
→ |
s(p(s(s(x1)))) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
half#(s(s(x1))) |
→ |
half#(p(s(s(x1)))) |
(16) |
could be deleted.
1.1.1.3.1 P is empty
There are no pairs anymore.
-
The
4th
component contains the
pair
|
p#(s(s(s(x1)))) |
→ |
p#(s(s(x1))) |
(22) |
1.1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
p#(s(s(s(x1)))) |
→ |
p#(s(s(x1))) |
(22) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
|
s#(s(p(s(x1)))) |
→ |
s#(s(x1)) |
(24) |
1.1.1.5 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
s#(s(p(s(x1)))) |
→ |
s#(s(x1)) |
(24) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.