YES
Termination Proof
Termination Proof
by ttt2 (version ttt2 1.15)
Input
The rewrite relation of the following TRS is considered.
|
a(b(x0)) |
→ |
x0 |
|
a(c(x0)) |
→ |
c(c(b(a(x0)))) |
|
b(c(x0)) |
→ |
a(b(x0)) |
Proof
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
a#(c(x0)) |
→ |
a#(x0) |
|
a#(c(x0)) |
→ |
b#(a(x0)) |
|
b#(c(x0)) |
→ |
b#(x0) |
|
b#(c(x0)) |
→ |
a#(b(x0)) |
1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers
| [b#(x1)] |
= |
0 ·
x1 + -16 |
| [a(x1)] |
= |
3 ·
x1 + 0 |
| [a#(x1)] |
= |
0 ·
x1 + 0 |
| [b(x1)] |
= |
-3 ·
x1 + 0 |
| [c(x1)] |
= |
3 ·
x1 + 12 |
together with the usable
rules
|
a(b(x0)) |
→ |
x0 |
|
a(c(x0)) |
→ |
c(c(b(a(x0)))) |
|
b(c(x0)) |
→ |
a(b(x0)) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
remains.
1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.