Certification Problem
Input (TPDB TRS_Standard/Zantema_05/z03)
The rewrite relation of the following TRS is considered.
|
a(f,a(g,a(f,x))) |
→ |
a(f,a(g,a(g,a(f,x)))) |
(1) |
|
a(g,a(f,a(g,x))) |
→ |
a(g,a(f,a(f,a(g,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) |
| 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(g1(f1(x))) |
→ |
f1(g1(g1(f1(x)))) |
(5) |
|
g1(f1(g1(x))) |
→ |
g1(f1(f1(g1(x)))) |
(6) |
|
a(f,y1) |
→ |
f1(y1) |
(3) |
|
a(g,y1) |
→ |
g1(y1) |
(4) |
1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [f1(x1)] |
= |
1 · x1
|
| [g1(x1)] |
= |
1 · x1
|
| [a(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [f] |
= |
0 |
| [g] |
= |
1 |
all of the following rules can be deleted.
1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [f1(x1)] |
= |
1 · x1
|
| [g1(x1)] |
= |
1 · x1
|
| [a(x1, x2)] |
= |
1 + 2 · x1 + 2 · x2
|
| [f] |
= |
0 |
all of the following rules can be deleted.
1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
f1#(g1(f1(x))) |
→ |
f1#(g1(g1(f1(x)))) |
(7) |
|
f1#(g1(f1(x))) |
→ |
g1#(g1(f1(x))) |
(8) |
|
g1#(f1(g1(x))) |
→ |
g1#(f1(f1(g1(x)))) |
(9) |
|
g1#(f1(g1(x))) |
→ |
f1#(f1(g1(x))) |
(10) |
1.1.1.1.1 Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(f1#) |
= |
1 |
|
weight(f1#) |
= |
1 |
|
|
|
| prec(g1) |
= |
0 |
|
weight(g1) |
= |
1 |
|
|
|
| prec(f1) |
= |
2 |
|
weight(f1) |
= |
1 |
|
|
|
in combination with the following argument filter
| π(f1#) |
= |
[] |
| π(g1#) |
= |
1 |
| π(g1) |
= |
[] |
| π(f1) |
= |
[] |
the
pairs
|
f1#(g1(f1(x))) |
→ |
g1#(g1(f1(x))) |
(8) |
|
g1#(f1(g1(x))) |
→ |
f1#(f1(g1(x))) |
(10) |
could be deleted.
1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.