YES
Termination Proof
Termination Proof
by ttt2 (version ttt2 1.15)
Input
The rewrite relation of the following TRS is considered.
|
a(a(x0)) |
→ |
b(c(x0)) |
|
b(b(x0)) |
→ |
a(c(c(c(x0)))) |
|
c(c(x0)) |
→ |
a(b(x0)) |
Proof
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
a#(a(x0)) |
→ |
c#(x0) |
|
a#(a(x0)) |
→ |
b#(c(x0)) |
|
b#(b(x0)) |
→ |
c#(x0) |
|
b#(b(x0)) |
→ |
c#(c(x0)) |
|
b#(b(x0)) |
→ |
c#(c(c(x0))) |
|
b#(b(x0)) |
→ |
a#(c(c(c(x0)))) |
|
c#(c(x0)) |
→ |
b#(x0) |
|
c#(c(x0)) |
→ |
a#(b(x0)) |
1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1
over the arctic semiring over the integers
| [c#(x1)] |
= |
·
x1 +
|
| [c(x1)] |
= |
·
x1 +
|
| [a#(x1)] |
= |
·
x1 +
|
| [b#(x1)] |
= |
·
x1 +
|
| [a(x1)] |
= |
·
x1 +
|
| [b(x1)] |
= |
·
x1 +
|
together with the usable
rules
|
a(a(x0)) |
→ |
b(c(x0)) |
|
b(b(x0)) |
→ |
a(c(c(c(x0)))) |
|
c(c(x0)) |
→ |
a(b(x0)) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
a#(a(x0)) |
→ |
c#(x0) |
|
a#(a(x0)) |
→ |
b#(c(x0)) |
|
b#(b(x0)) |
→ |
c#(x0) |
|
b#(b(x0)) |
→ |
a#(c(c(c(x0)))) |
|
c#(c(x0)) |
→ |
b#(x0) |
|
c#(c(x0)) |
→ |
a#(b(x0)) |
remain.
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the arctic semiring over the integers
| [c#(x1)] |
= |
·
x1 +
| 1 |
-∞
|
-∞
|
|
-∞
|
-∞
|
-∞
|
|
-∞
|
-∞
|
-∞
|
|
| [c(x1)] |
= |
·
x1 +
|
| [a#(x1)] |
= |
·
x1 +
| 0 |
-∞
|
-∞
|
|
-∞
|
-∞
|
-∞
|
|
-∞
|
-∞
|
-∞
|
|
| [b#(x1)] |
= |
·
x1 +
| 0 |
-∞
|
-∞
|
|
-∞
|
-∞
|
-∞
|
|
-∞
|
-∞
|
-∞
|
|
| [a(x1)] |
= |
·
x1 +
|
| [b(x1)] |
= |
·
x1 +
|
together with the usable
rules
|
a(a(x0)) |
→ |
b(c(x0)) |
|
b(b(x0)) |
→ |
a(c(c(c(x0)))) |
|
c(c(x0)) |
→ |
a(b(x0)) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
a#(a(x0)) |
→ |
c#(x0) |
|
a#(a(x0)) |
→ |
b#(c(x0)) |
|
b#(b(x0)) |
→ |
c#(x0) |
|
b#(b(x0)) |
→ |
a#(c(c(c(x0)))) |
|
c#(c(x0)) |
→ |
b#(x0) |
remain.
1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1
over the arctic semiring over the integers
| [c#(x1)] |
= |
·
x1 +
|
| [c(x1)] |
= |
·
x1 +
|
| [a#(x1)] |
= |
·
x1 +
|
| [b#(x1)] |
= |
·
x1 +
|
| [a(x1)] |
= |
·
x1 +
|
| [b(x1)] |
= |
·
x1 +
|
together with the usable
rules
|
a(a(x0)) |
→ |
b(c(x0)) |
|
b(b(x0)) |
→ |
a(c(c(c(x0)))) |
|
c(c(x0)) |
→ |
a(b(x0)) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
a#(a(x0)) |
→ |
c#(x0) |
|
a#(a(x0)) |
→ |
b#(c(x0)) |
|
b#(b(x0)) |
→ |
a#(c(c(c(x0)))) |
|
c#(c(x0)) |
→ |
b#(x0) |
remain.
1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the arctic semiring over the integers
| [c#(x1)] |
= |
|
-∞
|
-∞
|
1 |
|
-∞
|
-∞
|
-∞
|
|
-∞
|
-∞
|
-∞
|
·
x1 +
| 0 |
-∞
|
-∞
|
|
-∞
|
-∞
|
-∞
|
|
-∞
|
-∞
|
-∞
|
|
| [c(x1)] |
= |
·
x1 +
|
| [a#(x1)] |
= |
·
x1 +
| 1 |
-∞
|
-∞
|
|
-∞
|
-∞
|
-∞
|
|
-∞
|
-∞
|
-∞
|
|
| [b#(x1)] |
= |
|
-∞
|
0 |
-∞
|
|
-∞
|
-∞
|
-∞
|
|
-∞
|
-∞
|
-∞
|
·
x1 +
| 0 |
-∞
|
-∞
|
|
-∞
|
-∞
|
-∞
|
|
-∞
|
-∞
|
-∞
|
|
| [a(x1)] |
= |
·
x1 +
|
| [b(x1)] |
= |
·
x1 +
|
together with the usable
rules
|
a(a(x0)) |
→ |
b(c(x0)) |
|
b(b(x0)) |
→ |
a(c(c(c(x0)))) |
|
c(c(x0)) |
→ |
a(b(x0)) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
a#(a(x0)) |
→ |
c#(x0) |
|
b#(b(x0)) |
→ |
a#(c(c(c(x0)))) |
remain.
1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.