Certification Problem
Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-399)
The rewrite relation of the following TRS is considered.
a(x1) |
→ |
b(x1) |
(1) |
b(b(x1)) |
→ |
c(x1) |
(2) |
c(a(c(x1))) |
→ |
a(b(c(a(x1)))) |
(3) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by ttt2 @ termCOMP 2023)
1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
a(x1) |
→ |
b(x1) |
(1) |
b(b(x1)) |
→ |
c(x1) |
(2) |
c(a(c(x1))) |
→ |
a(c(b(a(x1)))) |
(4) |
1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
a#(x1) |
→ |
b#(x1) |
(5) |
b#(b(x1)) |
→ |
c#(x1) |
(6) |
c#(a(c(x1))) |
→ |
a#(x1) |
(7) |
c#(a(c(x1))) |
→ |
b#(a(x1)) |
(8) |
c#(a(c(x1))) |
→ |
c#(b(a(x1))) |
(9) |
c#(a(c(x1))) |
→ |
a#(c(b(a(x1)))) |
(10) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers
[b#(x1)] |
= |
0 · x1 +
-∞ |
[b(x1)] |
= |
1 · x1 +
-∞ |
[a#(x1)] |
= |
0 · x1 +
-∞ |
[c#(x1)] |
= |
1 · x1 +
-∞ |
[a(x1)] |
= |
1 · x1 +
-∞ |
[c(x1)] |
= |
2 · x1 +
-∞ |
together with the usable
rules
a(x1) |
→ |
b(x1) |
(1) |
b(b(x1)) |
→ |
c(x1) |
(2) |
c(a(c(x1))) |
→ |
a(c(b(a(x1)))) |
(4) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
c#(a(c(x1))) |
→ |
a#(x1) |
(7) |
c#(a(c(x1))) |
→ |
b#(a(x1)) |
(8) |
c#(a(c(x1))) |
→ |
c#(b(a(x1))) |
(9) |
could be deleted.
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
[b#(x1)] |
= |
· x1 +
|
[b(x1)] |
= |
· x1 +
|
[a#(x1)] |
= |
· x1 +
|
[c#(x1)] |
= |
· x1 +
|
[a(x1)] |
= |
· x1 +
|
[c(x1)] |
= |
· x1 +
|
together with the usable
rules
a(x1) |
→ |
b(x1) |
(1) |
b(b(x1)) |
→ |
c(x1) |
(2) |
c(a(c(x1))) |
→ |
a(c(b(a(x1)))) |
(4) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
c#(a(c(x1))) |
→ |
a#(c(b(a(x1)))) |
(10) |
could be deleted.
1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.