YES
Termination Proof
Termination Proof
by ttt2 (version ttt2 1.15)
Input
The rewrite relation of the following TRS is considered.
|
a(a(b(x0))) |
→ |
b(a(x0)) |
|
b(a(a(x0))) |
→ |
a(a(a(b(x0)))) |
|
a(c(x0)) |
→ |
c(b(x0)) |
Proof
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
a#(a(b(x0))) |
→ |
a#(x0) |
|
a#(a(b(x0))) |
→ |
b#(a(x0)) |
|
b#(a(a(x0))) |
→ |
b#(x0) |
|
b#(a(a(x0))) |
→ |
a#(b(x0)) |
|
b#(a(a(x0))) |
→ |
a#(a(b(x0))) |
|
b#(a(a(x0))) |
→ |
a#(a(a(b(x0)))) |
|
a#(c(x0)) |
→ |
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 +
-∞
|
| [a(x1)] |
= |
0 ·
x1 + 2 |
| [a#(x1)] |
= |
0 ·
x1 +
-∞
|
| [b(x1)] |
= |
0 ·
x1 + 2 |
| [c(x1)] |
= |
9 ·
x1 + 11 |
together with the usable
rules
|
a(a(b(x0))) |
→ |
b(a(x0)) |
|
b(a(a(x0))) |
→ |
a(a(a(b(x0)))) |
|
a(c(x0)) |
→ |
c(b(x0)) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
a#(a(b(x0))) |
→ |
a#(x0) |
|
a#(a(b(x0))) |
→ |
b#(a(x0)) |
|
b#(a(a(x0))) |
→ |
b#(x0) |
|
b#(a(a(x0))) |
→ |
a#(b(x0)) |
|
b#(a(a(x0))) |
→ |
a#(a(b(x0))) |
|
b#(a(a(x0))) |
→ |
a#(a(a(b(x0)))) |
remain.
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers
| [b#(x1)] |
= |
2 ·
x1 + 0 |
| [a(x1)] |
= |
0 ·
x1 + 13 |
| [a#(x1)] |
= |
0 ·
x1 + 0 |
| [b(x1)] |
= |
2 ·
x1 + 15 |
| [c(x1)] |
= |
-∞
·
x1 + 13 |
together with the usable
rules
|
a(a(b(x0))) |
→ |
b(a(x0)) |
|
b(a(a(x0))) |
→ |
a(a(a(b(x0)))) |
|
a(c(x0)) |
→ |
c(b(x0)) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
a#(a(b(x0))) |
→ |
b#(a(x0)) |
|
b#(a(a(x0))) |
→ |
b#(x0) |
|
b#(a(a(x0))) |
→ |
a#(b(x0)) |
|
b#(a(a(x0))) |
→ |
a#(a(b(x0))) |
|
b#(a(a(x0))) |
→ |
a#(a(a(b(x0)))) |
remain.
1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1
over the naturals
| [b#(x1)] |
= |
| 1 |
1 |
0 |
1 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
·
x1 +
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
|
| [a(x1)] |
= |
| 0 |
1 |
0 |
1 |
| 0 |
0 |
1 |
0 |
| 0 |
0 |
0 |
0 |
| 1 |
0 |
0 |
0 |
·
x1 +
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 1 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
|
| [a#(x1)] |
= |
| 1 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
·
x1 +
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
|
| [b(x1)] |
= |
| 1 |
1 |
0 |
1 |
| 0 |
0 |
1 |
0 |
| 0 |
0 |
1 |
0 |
| 1 |
1 |
0 |
1 |
·
x1 +
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
|
| [c(x1)] |
= |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
·
x1 +
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
|
the
pair
remains.
1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.