Certification Problem
Input (TPDB TRS_Standard/Waldmann_06/jwteparla2)
The rewrite relation of the following TRS is considered.
|
f(f(f(a,f(a,a)),a),x) |
→ |
f(x,f(x,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), |
a3(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
|
a3(a1(a),a,x) |
→ |
f(x,f(x,a)) |
(5) |
|
f(a,y1) |
→ |
a1(y1) |
(2) |
|
f(a1(x0),y1) |
→ |
a2(x0,y1) |
(3) |
|
f(a2(x0,x1),y1) |
→ |
a3(x0,x1,y1) |
(4) |
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.
|
a3#(a1(a),a,x) |
→ |
f#(x,f(x,a)) |
(6) |
|
a3#(a1(a),a,x) |
→ |
f#(x,a) |
(7) |
|
f#(a2(x0,x1),y1) |
→ |
a3#(x0,x1,y1) |
(8) |
1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [f#(x1, x2)] |
= |
-1 + x1 + x2
|
| [f(x1, x2)] |
= |
2 |
| [a] |
= |
0 |
| [a1(x1)] |
= |
0 |
| [a2(x1, x2)] |
= |
2 |
| [a3(x1, x2, x3)] |
= |
2 |
| [a3#(x1, x2, x3)] |
= |
1 + x3
|
the
pair
|
a3#(a1(a),a,x) |
→ |
f#(x,a) |
(7) |
could be deleted.
1.1.1.1.1 Narrowing Processor
We consider all narrowings of the pair
below position
ε
to get the following set of pairs
|
a3#(a1(a),a,a) |
→ |
f#(a,a1(a)) |
(9) |
|
a3#(a1(a),a,a1(x0)) |
→ |
f#(a1(x0),a2(x0,a)) |
(10) |
|
a3#(a1(a),a,a2(x0,x1)) |
→ |
f#(a2(x0,x1),a3(x0,x1,a)) |
(11) |
1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
|
a3#(a1(a),a,a2(x0,x1)) |
→ |
f#(a2(x0,x1),a3(x0,x1,a)) |
(11) |
|
f#(a2(x0,x1),y1) |
→ |
a3#(x0,x1,y1) |
(8) |
1.1.1.1.1.1.1 Forward Instantiation Processor
We instantiate the pair
to the following set of pairs
|
f#(a2(a1(a),a),a2(y_0,y_1)) |
→ |
a3#(a1(a),a,a2(y_0,y_1)) |
(12) |
1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
| [a3#(x1, x2, x3)] |
= |
+ · x1 + · x2 + · x3
|
| [a1(x1)] |
= |
+ · x1
|
| [a] |
= |
|
| [a2(x1, x2)] |
= |
+ · x1 + · x2
|
| [f#(x1, x2)] |
= |
+ · x1 + · x2
|
| [a3(x1, x2, x3)] |
= |
+ · x1 + · x2 + · x3
|
| [f(x1, x2)] |
= |
+ · x1 + · x2
|
the
pair
|
a3#(a1(a),a,a2(x0,x1)) |
→ |
f#(a2(x0,x1),a3(x0,x1,a)) |
(11) |
could be deleted.
1.1.1.1.1.1.1.1.1 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
There are no lhss.
1.1.1.1.1.1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
f#(a2(a1(a),a),a2(y_0,y_1)) |
→ |
a3#(a1(a),a,a2(y_0,y_1)) |
(12) |
|
|
| 1 |
> |
1 |
| 1 |
> |
2 |
| 2 |
≥ |
3 |
As there is no critical graph in the transitive closure, there are no infinite chains.