Certification Problem
Input (TPDB TRS_Standard/AG01/#3.42)
The rewrite relation of the following TRS is considered.
|
half(0) |
→ |
0 |
(1) |
|
half(s(0)) |
→ |
0 |
(2) |
|
half(s(s(x))) |
→ |
s(half(x)) |
(3) |
|
lastbit(0) |
→ |
0 |
(4) |
|
lastbit(s(0)) |
→ |
s(0) |
(5) |
|
lastbit(s(s(x))) |
→ |
lastbit(x) |
(6) |
|
conv(0) |
→ |
cons(nil,0) |
(7) |
|
conv(s(x)) |
→ |
cons(conv(half(s(x))),lastbit(s(x))) |
(8) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by NaTT @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
half#(s(s(x))) |
→ |
half#(x) |
(9) |
|
conv#(s(x)) |
→ |
lastbit#(s(x)) |
(10) |
|
conv#(s(x)) |
→ |
conv#(half(s(x))) |
(11) |
|
lastbit#(s(s(x))) |
→ |
lastbit#(x) |
(12) |
|
conv#(s(x)) |
→ |
half#(s(x)) |
(13) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
conv#(s(x)) |
→ |
conv#(half(s(x))) |
(11) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the argument filter in combination with the following Weighted Path Order with the following precedence and status
| prec(conv) |
= |
0 |
|
status(conv) |
= |
[] |
|
list-extension(conv) |
= |
Lex |
| prec(s) |
= |
0 |
|
status(s) |
= |
[1] |
|
list-extension(s) |
= |
Lex |
| prec(conv#) |
= |
0 |
|
status(conv#) |
= |
[1] |
|
list-extension(conv#) |
= |
Lex |
| prec(half#) |
= |
0 |
|
status(half#) |
= |
[] |
|
list-extension(half#) |
= |
Lex |
| prec(lastbit#) |
= |
0 |
|
status(lastbit#) |
= |
[] |
|
list-extension(lastbit#) |
= |
Lex |
| prec(half) |
= |
0 |
|
status(half) |
= |
[] |
|
list-extension(half) |
= |
Lex |
| prec(lastbit) |
= |
0 |
|
status(lastbit) |
= |
[] |
|
list-extension(lastbit) |
= |
Lex |
| prec(0) |
= |
0 |
|
status(0) |
= |
[] |
|
list-extension(0) |
= |
Lex |
| prec(nil) |
= |
0 |
|
status(nil) |
= |
[] |
|
list-extension(nil) |
= |
Lex |
| prec(cons) |
= |
0 |
|
status(cons) |
= |
[1, 2] |
|
list-extension(cons) |
= |
Lex |
and the following
Max-polynomial interpretation
| [conv(x1)] |
=
|
1 |
| [s(x1)] |
=
|
x1 + 21239 |
| [conv#(x1)] |
=
|
x1 + 1 |
| [half#(x1)] |
=
|
1 |
| [lastbit#(x1)] |
=
|
1 |
| [half(x1)] |
=
|
x1 + 0 |
| [lastbit(x1)] |
=
|
1 |
| [0] |
=
|
1 |
| [nil] |
=
|
0 |
| [cons(x1, x2)] |
=
|
x1 + x2 + 1 |
together with the usable
rules
|
half(0) |
→ |
0 |
(1) |
|
half(s(s(x))) |
→ |
s(half(x)) |
(3) |
|
half(s(0)) |
→ |
0 |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
conv#(s(x)) |
→ |
conv#(half(s(x))) |
(11) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
|
lastbit#(s(s(x))) |
→ |
lastbit#(x) |
(12) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [conv(x1)] |
=
|
0 |
| [s(x1)] |
=
|
x1 + 1 |
| [conv#(x1)] |
=
|
0 |
| [half#(x1)] |
=
|
0 |
| [lastbit#(x1)] |
=
|
x1 + 0 |
| [half(x1)] |
=
|
11293 |
| [lastbit(x1)] |
=
|
0 |
| [0] |
=
|
11294 |
| [nil] |
=
|
0 |
| [cons(x1, x2)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
lastbit#(s(s(x))) |
→ |
lastbit#(x) |
(12) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
|
half#(s(s(x))) |
→ |
half#(x) |
(9) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [conv(x1)] |
=
|
0 |
| [s(x1)] |
=
|
x1 + 1 |
| [conv#(x1)] |
=
|
0 |
| [half#(x1)] |
=
|
x1 + 0 |
| [lastbit#(x1)] |
=
|
0 |
| [half(x1)] |
=
|
20163 |
| [lastbit(x1)] |
=
|
0 |
| [0] |
=
|
20164 |
| [nil] |
=
|
0 |
| [cons(x1, x2)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
half#(s(s(x))) |
→ |
half#(x) |
(9) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.