Certification Problem
Input (TPDB TRS_Standard/Zantema_05/z29)
The rewrite relation of the following TRS is considered.
|
a(lambda(x),y) |
→ |
lambda(a(x,1)) |
(1) |
|
a(lambda(x),y) |
→ |
lambda(a(x,a(y,t))) |
(2) |
|
a(a(x,y),z) |
→ |
a(x,a(y,z)) |
(3) |
|
lambda(x) |
→ |
x |
(4) |
|
a(x,y) |
→ |
x |
(5) |
|
a(x,y) |
→ |
y |
(6) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [a(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [lambda(x1)] |
= |
1 + 1 · x1
|
| [1] |
= |
0 |
| [t] |
= |
0 |
all of the following rules can be deleted.
1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
a#(lambda(x),y) |
→ |
a#(x,1) |
(7) |
|
a#(lambda(x),y) |
→ |
a#(x,a(y,t)) |
(8) |
|
a#(lambda(x),y) |
→ |
a#(y,t) |
(9) |
|
a#(a(x,y),z) |
→ |
a#(x,a(y,z)) |
(10) |
|
a#(a(x,y),z) |
→ |
a#(y,z) |
(11) |
1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [a(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [lambda(x1)] |
= |
1 + 1 · x1
|
| [1] |
= |
0 |
| [t] |
= |
0 |
| [a#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
the
pairs
|
a#(lambda(x),y) |
→ |
a#(x,1) |
(7) |
|
a#(lambda(x),y) |
→ |
a#(x,a(y,t)) |
(8) |
|
a#(lambda(x),y) |
→ |
a#(y,t) |
(9) |
and
no rules
could be deleted.
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.
|
a#(a(x,y),z) |
→ |
a#(x,a(y,z)) |
(10) |
|
| 1 |
> |
1 |
|
a#(a(x,y),z) |
→ |
a#(y,z) |
(11) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.