Certification Problem
Input (TPDB SRS_Standard/Waldmann_06_SRS/jw4)
The rewrite relation of the following TRS is considered.
|
b(a(b(b(x1)))) |
→ |
b(b(b(a(b(x1))))) |
(1) |
|
b(a(a(b(b(x1))))) |
→ |
b(a(b(b(a(a(b(x1))))))) |
(2) |
|
b(a(a(a(b(b(x1)))))) |
→ |
b(a(a(b(b(a(a(a(b(x1))))))))) |
(3) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
b(b(a(b(x1)))) |
→ |
b(a(b(b(b(x1))))) |
(4) |
|
b(b(a(a(b(x1))))) |
→ |
b(a(a(b(b(a(b(x1))))))) |
(5) |
|
b(b(a(a(a(b(x1)))))) |
→ |
b(a(a(a(b(b(a(a(b(x1))))))))) |
(6) |
1.1 Semantic Labeling
Root-labeling is applied.
We obtain the labeled TRS
|
bb(ba(ab(bb(x1)))) |
→ |
ba(ab(bb(bb(bb(x1))))) |
(7) |
|
bb(ba(ab(ba(x1)))) |
→ |
ba(ab(bb(bb(ba(x1))))) |
(8) |
|
bb(ba(aa(ab(bb(x1))))) |
→ |
ba(aa(ab(bb(ba(ab(bb(x1))))))) |
(9) |
|
bb(ba(aa(ab(ba(x1))))) |
→ |
ba(aa(ab(bb(ba(ab(ba(x1))))))) |
(10) |
|
bb(ba(aa(aa(ab(bb(x1)))))) |
→ |
ba(aa(aa(ab(bb(ba(aa(ab(bb(x1))))))))) |
(11) |
|
bb(ba(aa(aa(ab(ba(x1)))))) |
→ |
ba(aa(aa(ab(bb(ba(aa(ab(ba(x1))))))))) |
(12) |
1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
bb#(ba(ab(bb(x1)))) |
→ |
bb#(bb(bb(x1))) |
(13) |
|
bb#(ba(ab(bb(x1)))) |
→ |
bb#(bb(x1)) |
(14) |
|
bb#(ba(ab(ba(x1)))) |
→ |
bb#(bb(ba(x1))) |
(15) |
|
bb#(ba(ab(ba(x1)))) |
→ |
bb#(ba(x1)) |
(16) |
|
bb#(ba(aa(ab(bb(x1))))) |
→ |
bb#(ba(ab(bb(x1)))) |
(17) |
|
bb#(ba(aa(ab(ba(x1))))) |
→ |
bb#(ba(ab(ba(x1)))) |
(18) |
|
bb#(ba(aa(aa(ab(bb(x1)))))) |
→ |
bb#(ba(aa(ab(bb(x1))))) |
(19) |
|
bb#(ba(aa(aa(ab(ba(x1)))))) |
→ |
bb#(ba(aa(ab(ba(x1))))) |
(20) |
1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals
| [bb#(x1)] |
= |
+ · x1
|
| [ba(x1)] |
= |
+ · x1
|
| [ab(x1)] |
= |
+ · x1
|
| [bb(x1)] |
= |
+ · x1
|
| [aa(x1)] |
= |
+ · x1
|
the
pairs
|
bb#(ba(aa(aa(ab(bb(x1)))))) |
→ |
bb#(ba(aa(ab(bb(x1))))) |
(19) |
|
bb#(ba(aa(aa(ab(ba(x1)))))) |
→ |
bb#(ba(aa(ab(ba(x1))))) |
(20) |
could be deleted.
1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals
| [bb#(x1)] |
= |
+ · x1
|
| [ba(x1)] |
= |
+ · x1
|
| [ab(x1)] |
= |
+ · x1
|
| [bb(x1)] |
= |
+ · x1
|
| [aa(x1)] |
= |
+ · x1
|
the
pairs
|
bb#(ba(aa(ab(bb(x1))))) |
→ |
bb#(ba(ab(bb(x1)))) |
(17) |
|
bb#(ba(aa(ab(ba(x1))))) |
→ |
bb#(ba(ab(ba(x1)))) |
(18) |
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [bb#(x1)] |
= |
1 · x1
|
| [ba(x1)] |
= |
1 + 1 · x1
|
| [ab(x1)] |
= |
1 · x1
|
| [bb(x1)] |
= |
1 · x1
|
| [aa(x1)] |
= |
0 |
the
pairs
|
bb#(ba(ab(bb(x1)))) |
→ |
bb#(bb(bb(x1))) |
(13) |
|
bb#(ba(ab(bb(x1)))) |
→ |
bb#(bb(x1)) |
(14) |
|
bb#(ba(ab(ba(x1)))) |
→ |
bb#(bb(ba(x1))) |
(15) |
|
bb#(ba(ab(ba(x1)))) |
→ |
bb#(ba(x1)) |
(16) |
could be deleted.
1.1.1.1.1.1.1 P is empty
There are no pairs anymore.