Certification Problem
Input (TPDB TRS_Standard/Strategy_removed_mixed_05/ex3)
The rewrite relation of the following TRS is considered.
|
f(g(a)) |
→ |
f(s(g(b))) |
(1) |
|
f(f(x)) |
→ |
b |
(2) |
|
g(x) |
→ |
f(g(x)) |
(3) |
Property / Task
Prove or disprove termination.Answer / Result
No.Proof (by AProVE @ termCOMP 2023)
1 Constant to Unary
Every constant is turned into a unary function symbol to obtain the TRS
|
f(g(a'(x))) |
→ |
f(s(g(b'(x)))) |
(4) |
|
f(f(x)) |
→ |
b'(x) |
(5) |
|
g(x) |
→ |
f(g(x)) |
(3) |
1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
a'(g(f(x))) |
→ |
b'(g(s(f(x)))) |
(6) |
|
f(f(x)) |
→ |
b'(x) |
(5) |
|
g(x) |
→ |
g(f(x)) |
(7) |
1.1.1 Rule Removal
The following rules have been removed.
|
a'(g(f(x))) |
→ |
b'(g(s(f(x)))) |
(6) |
1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
g#(x) |
→ |
g#(f(x)) |
(8) |
|
g#(x) |
→ |
f#(x) |
(9) |
It remains to prove infiniteness of the resulting DP problem.
1.1.1.1.1 Pair and Rule Removal
Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem.
The following pairs have been deleted.
and the following rules have been deleted.
1.1.1.1.1.1 Pair and Rule Removal
Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem.
The following pairs have been deleted.
and the following rules have been deleted.
1.1.1.1.1.1.1 Loop
The following loop proves infiniteness of the DP problem.
| t0
|
= |
g#(x) |
|
→P
|
g#(f(x)) |
|
= |
t1
|
where t1 =
t0σ
and
σ =
{x/f(x)}