Certification Problem
Input (TPDB TRS_Standard/Zantema_05/z04)
The rewrite relation of the following TRS is considered.
|
a(f,a(f,x)) |
→ |
a(x,x) |
(1) |
|
a(h,x) |
→ |
a(f,a(g,a(f,x))) |
(2) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Uncurrying
We uncurry the binary symbol
a
in combination with the following symbol map which also determines the applicative arities of these symbols.
| f |
is mapped to |
f, |
f1(x1) |
| h |
is mapped to |
h, |
h1(x1) |
| g |
is mapped to |
g, |
g1(x1) |
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
|
f1(f1(x)) |
→ |
a(x,x) |
(6) |
|
h1(x) |
→ |
f1(g1(f1(x))) |
(7) |
|
a(f,y1) |
→ |
f1(y1) |
(3) |
|
a(h,y1) |
→ |
h1(y1) |
(4) |
|
a(g,y1) |
→ |
g1(y1) |
(5) |
1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
f1#(f1(x)) |
→ |
a#(x,x) |
(8) |
|
h1#(x) |
→ |
f1#(g1(f1(x))) |
(9) |
|
h1#(x) |
→ |
f1#(x) |
(10) |
|
a#(f,y1) |
→ |
f1#(y1) |
(11) |
|
a#(h,y1) |
→ |
h1#(y1) |
(12) |
1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
|
a#(f,y1) |
→ |
f1#(y1) |
(11) |
|
f1#(f1(x)) |
→ |
a#(x,x) |
(8) |
|
a#(h,y1) |
→ |
h1#(y1) |
(12) |
|
h1#(x) |
→ |
f1#(x) |
(10) |
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.
|
f1#(f1(x)) |
→ |
a#(x,x) |
(8) |
|
|
| 1 |
> |
1 |
| 1 |
> |
2 |
|
a#(f,y1) |
→ |
f1#(y1) |
(11) |
|
| 2 |
≥ |
1 |
|
a#(h,y1) |
→ |
h1#(y1) |
(12) |
|
| 2 |
≥ |
1 |
|
h1#(x) |
→ |
f1#(x) |
(10) |
|
| 1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.