Certification Problem
Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-543)
The rewrite relation of the following TRS is considered.
|
a(b(x1)) |
→ |
x1 |
(1) |
|
a(b(c(x1))) |
→ |
b(c(b(c(a(a(b(x1))))))) |
(2) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by ttt2 @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
a#(b(c(x1))) |
→ |
a#(b(x1)) |
(3) |
|
a#(b(c(x1))) |
→ |
a#(a(b(x1))) |
(4) |
1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers
| [a(x1)] |
= |
1 · x1 + 0 |
| [a#(x1)] |
= |
0 · x1 + 0 |
| [b(x1)] |
= |
-1 · x1 + 1 |
| [c(x1)] |
= |
1 · x1 + 3 |
together with the usable
rules
|
a(b(x1)) |
→ |
x1 |
(1) |
|
a(b(c(x1))) |
→ |
b(c(b(c(a(a(b(x1))))))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
a#(b(c(x1))) |
→ |
a#(b(x1)) |
(3) |
could be deleted.
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the rationals with delta = 1/64
| [a(x1)] |
= |
2 · x1 + 0 |
| [a#(x1)] |
= |
1 · x1 + 0 |
| [b(x1)] |
= |
1/2 · x1 + 0 |
| [c(x1)] |
= |
2 · x1 + 1 |
together with the usable
rules
|
a(b(x1)) |
→ |
x1 |
(1) |
|
a(b(c(x1))) |
→ |
b(c(b(c(a(a(b(x1))))))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
a#(b(c(x1))) |
→ |
a#(a(b(x1))) |
(4) |
could be deleted.
1.1.1.1 P is empty
There are no pairs anymore.