Certification Problem
Input (TPDB TRS_Standard/Zantema_05/jw27)
The rewrite relation of the following TRS is considered.
|
f(f(a,x),a) |
→ |
f(a,f(f(x,f(a,a)),a)) |
(1) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Uncurrying
We uncurry the binary symbol
f
in combination with the following symbol map which also determines the applicative arities of these symbols.
| a |
is mapped to |
a, |
a1(x1), |
a2(x1, x2) |
There are no uncurry rules.
No rules have to be added for the eta-expansion.
Uncurrying the rules and adding the uncurrying rules yields the new set of rules
|
a2(x,a) |
→ |
a1(f(f(x,a1(a)),a)) |
(4) |
|
f(a,y1) |
→ |
a1(y1) |
(2) |
|
f(a1(x0),y1) |
→ |
a2(x0,y1) |
(3) |
1.1 Switch to Innermost Termination
The TRS is overlay and locally confluent:
10Hence, it suffices to show innermost termination in the following.
1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
a2#(x,a) |
→ |
f#(f(x,a1(a)),a) |
(5) |
|
a2#(x,a) |
→ |
f#(x,a1(a)) |
(6) |
|
f#(a1(x0),y1) |
→ |
a2#(x0,y1) |
(7) |
1.1.1.1 Reduction Pair Processor with Usable Rules
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(a) |
= |
1 |
|
weight(a) |
= |
1 |
|
|
|
| prec(a1) |
= |
0 |
|
weight(a1) |
= |
1 |
|
|
|
in combination with the following argument filter
| π(a2#) |
= |
2 |
| π(a) |
= |
[] |
| π(f#) |
= |
2 |
| π(a1) |
= |
[] |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
a2#(x,a) |
→ |
f#(x,a1(a)) |
(6) |
could be deleted.
1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
| [a2#(x1, x2)] |
= |
+ · x1 + · x2
|
| [a] |
= |
|
| [f#(x1, x2)] |
= |
+ · x1 + · x2
|
| [f(x1, x2)] |
= |
+ · x1 + · x2
|
| [a1(x1)] |
= |
+ · x1
|
| [a2(x1, x2)] |
= |
+ · x1 + · x2
|
the
pair
|
f#(a1(x0),y1) |
→ |
a2#(x0,y1) |
(7) |
could be deleted.
1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
a2#(x,a) |
→ |
f#(f(x,a1(a)),a) |
(5) |
|
| 2 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.