Certification Problem
Input (TPDB TRS_Standard/AotoYamada_05/017)
The rewrite relation of the following TRS is considered.
|
app(app(app(uncurry,f),x),y) |
→ |
app(app(f,x),y) |
(1) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Uncurrying
We uncurry the binary symbol
app
in combination with the following symbol map which also determines the applicative arities of these symbols.
| uncurry |
is mapped to |
uncurry, |
uncurry1(x1), |
uncurry2(x1, x2), |
uncurry3(x1, x2, x3) |
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
|
uncurry3(f,x,y) |
→ |
app(app(f,x),y) |
(5) |
|
app(uncurry,y1) |
→ |
uncurry1(y1) |
(2) |
|
app(uncurry1(x0),y1) |
→ |
uncurry2(x0,y1) |
(3) |
|
app(uncurry2(x0,x1),y1) |
→ |
uncurry3(x0,x1,y1) |
(4) |
1.1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(uncurry) |
= |
0 |
|
weight(uncurry) |
= |
1 |
|
|
|
| prec(uncurry1) |
= |
1 |
|
weight(uncurry1) |
= |
1 |
|
|
|
| prec(uncurry3) |
= |
4 |
|
weight(uncurry3) |
= |
0 |
|
|
|
| prec(app) |
= |
3 |
|
weight(app) |
= |
0 |
|
|
|
| prec(uncurry2) |
= |
2 |
|
weight(uncurry2) |
= |
1 |
|
|
|
all of the following rules can be deleted.
|
uncurry3(f,x,y) |
→ |
app(app(f,x),y) |
(5) |
|
app(uncurry,y1) |
→ |
uncurry1(y1) |
(2) |
|
app(uncurry1(x0),y1) |
→ |
uncurry2(x0,y1) |
(3) |
|
app(uncurry2(x0,x1),y1) |
→ |
uncurry3(x0,x1,y1) |
(4) |
1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.