Certification Problem
Input (TPDB TRS_Standard/Der95/21)
The rewrite relation of the following TRS is considered.
|
p(s(x)) |
→ |
x |
(1) |
|
fact(0) |
→ |
s(0) |
(2) |
|
fact(s(x)) |
→ |
*(s(x),fact(p(s(x)))) |
(3) |
|
*(0,y) |
→ |
0 |
(4) |
|
*(s(x),y) |
→ |
+(*(x,y),y) |
(5) |
|
+(x,0) |
→ |
x |
(6) |
|
+(x,s(y)) |
→ |
s(+(x,y)) |
(7) |
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.
|
*#(s(x),y) |
→ |
+#(*(x,y),y) |
(8) |
|
fact#(s(x)) |
→ |
p#(s(x)) |
(9) |
|
fact#(s(x)) |
→ |
fact#(p(s(x))) |
(10) |
|
*#(s(x),y) |
→ |
*#(x,y) |
(11) |
|
+#(x,s(y)) |
→ |
+#(x,y) |
(12) |
|
fact#(s(x)) |
→ |
*#(s(x),fact(p(s(x)))) |
(13) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
fact#(s(x)) |
→ |
fact#(p(s(x))) |
(10) |
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(s) |
= |
1 |
|
status(s) |
= |
[1] |
|
list-extension(s) |
= |
Lex |
| prec(*#) |
= |
0 |
|
status(*#) |
= |
[] |
|
list-extension(*#) |
= |
Lex |
| prec(fact#) |
= |
0 |
|
status(fact#) |
= |
[1] |
|
list-extension(fact#) |
= |
Lex |
| prec(p#) |
= |
0 |
|
status(p#) |
= |
[] |
|
list-extension(p#) |
= |
Lex |
| prec(p) |
= |
0 |
|
status(p) |
= |
[] |
|
list-extension(p) |
= |
Lex |
| prec(0) |
= |
0 |
|
status(0) |
= |
[] |
|
list-extension(0) |
= |
Lex |
| prec(fact) |
= |
0 |
|
status(fact) |
= |
[] |
|
list-extension(fact) |
= |
Lex |
| prec(+) |
= |
0 |
|
status(+) |
= |
[] |
|
list-extension(+) |
= |
Lex |
| prec(+#) |
= |
0 |
|
status(+#) |
= |
[1, 2] |
|
list-extension(+#) |
= |
Lex |
| prec(*) |
= |
0 |
|
status(*) |
= |
[2, 1] |
|
list-extension(*) |
= |
Lex |
and the following
Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 7720 |
| [*#(x1, x2)] |
=
|
max(x1 + 1, 0) |
| [fact#(x1)] |
=
|
x1 + 1 |
| [p#(x1)] |
=
|
1 |
| [p(x1)] |
=
|
x1 + 0 |
| [0] |
=
|
0 |
| [fact(x1)] |
=
|
1 |
| [+(x1, x2)] |
=
|
1 |
| [+#(x1, x2)] |
=
|
max(x1 + 1, x2 + 1, 0) |
| [*(x1, x2)] |
=
|
max(x1 + 1, x2 + 1, 0) |
together with the usable
rule
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
fact#(s(x)) |
→ |
fact#(p(s(x))) |
(10) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
|
*#(s(x),y) |
→ |
*#(x,y) |
(11) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [*#(x1, x2)] |
=
|
x1 + 0 |
| [fact#(x1)] |
=
|
0 |
| [p#(x1)] |
=
|
0 |
| [p(x1)] |
=
|
x1 + 0 |
| [0] |
=
|
0 |
| [fact(x1)] |
=
|
0 |
| [+(x1, x2)] |
=
|
0 |
| [+#(x1, x2)] |
=
|
0 |
| [*(x1, x2)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
*#(s(x),y) |
→ |
*#(x,y) |
(11) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
|
+#(x,s(y)) |
→ |
+#(x,y) |
(12) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [*#(x1, x2)] |
=
|
0 |
| [fact#(x1)] |
=
|
0 |
| [p#(x1)] |
=
|
0 |
| [p(x1)] |
=
|
x1 + 0 |
| [0] |
=
|
0 |
| [fact(x1)] |
=
|
0 |
| [+(x1, x2)] |
=
|
0 |
| [+#(x1, x2)] |
=
|
x2 + 0 |
| [*(x1, x2)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
+#(x,s(y)) |
→ |
+#(x,y) |
(12) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.