Certification Problem
Input (TPDB SRS_Standard/Zantema_06/beans2)
The rewrite relation of the following TRS is considered.
|
b(a(a(x1))) |
→ |
a(b(c(x1))) |
(1) |
|
c(a(x1)) |
→ |
a(c(x1)) |
(2) |
|
c(b(x1)) |
→ |
b(a(x1)) |
(3) |
|
L(a(a(x1))) |
→ |
L(a(b(c(x1)))) |
(4) |
|
c(R(x1)) |
→ |
b(a(R(x1))) |
(5) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
b#(a(a(x1))) |
→ |
b#(c(x1)) |
(6) |
|
b#(a(a(x1))) |
→ |
c#(x1) |
(7) |
|
c#(a(x1)) |
→ |
c#(x1) |
(8) |
|
c#(b(x1)) |
→ |
b#(a(x1)) |
(9) |
|
L#(a(a(x1))) |
→ |
L#(a(b(c(x1)))) |
(10) |
|
L#(a(a(x1))) |
→ |
b#(c(x1)) |
(11) |
|
L#(a(a(x1))) |
→ |
c#(x1) |
(12) |
|
c#(R(x1)) |
→ |
b#(a(R(x1))) |
(13) |
1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
|
L#(a(a(x1))) |
→ |
L#(a(b(c(x1)))) |
(10) |
1.1.1 Switch to Innermost Termination
The TRS does not have overlaps with the pairs and is locally confluent:
20
Hence, it suffices to show innermost termination in the following.
1.1.1.1 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
|
c(a(x1)) |
→ |
a(c(x1)) |
(2) |
|
c(b(x1)) |
→ |
b(a(x1)) |
(3) |
|
c(R(x1)) |
→ |
b(a(R(x1))) |
(5) |
|
b(a(a(x1))) |
→ |
a(b(c(x1))) |
(1) |
1.1.1.1.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
|
b(a(a(x0))) |
|
c(a(x0)) |
|
c(b(x0)) |
|
c(R(x0)) |
1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [L#(x1)] |
= |
2 + 2 · x1
|
| [a(x1)] |
= |
1 + x1
|
| [b(x1)] |
= |
-1 + x1
|
| [c(x1)] |
= |
1 + x1
|
| [R(x1)] |
= |
0 |
the
pair
|
L#(a(a(x1))) |
→ |
L#(a(b(c(x1)))) |
(10) |
could be deleted.
1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
b#(a(a(x1))) |
→ |
c#(x1) |
(7) |
|
c#(a(x1)) |
→ |
c#(x1) |
(8) |
|
c#(b(x1)) |
→ |
b#(a(x1)) |
(9) |
|
b#(a(a(x1))) |
→ |
b#(c(x1)) |
(6) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [c(x1)] |
= |
1 · x1
|
| [a(x1)] |
= |
1 · x1
|
| [b(x1)] |
= |
1 · x1
|
| [R(x1)] |
= |
1 · x1
|
| [c#(x1)] |
= |
1 · x1
|
| [b#(x1)] |
= |
1 · x1
|
together with the usable
rules
|
c(a(x1)) |
→ |
a(c(x1)) |
(2) |
|
c(b(x1)) |
→ |
b(a(x1)) |
(3) |
|
c(R(x1)) |
→ |
b(a(R(x1))) |
(5) |
|
b(a(a(x1))) |
→ |
a(b(c(x1))) |
(1) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.2.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [c(x1)] |
= |
2 + 2 · x1
|
| [a(x1)] |
= |
2 + 2 · x1
|
| [b(x1)] |
= |
1 · x1
|
| [R(x1)] |
= |
1 · x1
|
| [b#(x1)] |
= |
1 · x1
|
| [c#(x1)] |
= |
2 + 2 · x1
|
the
pairs
|
b#(a(a(x1))) |
→ |
c#(x1) |
(7) |
|
c#(a(x1)) |
→ |
c#(x1) |
(8) |
|
b#(a(a(x1))) |
→ |
b#(c(x1)) |
(6) |
and
no rules
could be deleted.
1.1.2.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [b(x1)] |
= |
1 · x1
|
| [a(x1)] |
= |
1 · x1
|
| [b#(x1)] |
= |
1 · x1
|
| [c#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.2.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [c#(x1)] |
= |
2 + 2 · x1
|
| [b(x1)] |
= |
2 + x1
|
| [b#(x1)] |
= |
1 + 2 · x1
|
| [a(x1)] |
= |
2 + x1
|
the
pair
|
c#(b(x1)) |
→ |
b#(a(x1)) |
(9) |
could be deleted.
1.1.2.1.1.1.1 P is empty
There are no pairs anymore.