Certification Problem
Input (TPDB SRS_Standard/Secret_05_SRS/torpa4)
The rewrite relation of the following TRS is considered.
|
a(b(c(a(x1)))) |
→ |
b(a(c(b(a(b(x1)))))) |
(1) |
|
a(d(x1)) |
→ |
c(x1) |
(2) |
|
a(f(f(x1))) |
→ |
g(x1) |
(3) |
|
b(g(x1)) |
→ |
g(b(x1)) |
(4) |
|
c(x1) |
→ |
f(f(x1)) |
(5) |
|
c(a(c(x1))) |
→ |
b(c(a(b(c(x1))))) |
(6) |
|
c(d(x1)) |
→ |
a(a(x1)) |
(7) |
|
g(x1) |
→ |
c(a(x1)) |
(8) |
|
g(x1) |
→ |
d(d(d(d(x1)))) |
(9) |
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
|
a(c(b(a(x1)))) |
→ |
b(a(b(c(a(b(x1)))))) |
(10) |
|
d(a(x1)) |
→ |
c(x1) |
(11) |
|
f(f(a(x1))) |
→ |
g(x1) |
(12) |
|
g(b(x1)) |
→ |
b(g(x1)) |
(13) |
|
c(x1) |
→ |
f(f(x1)) |
(5) |
|
c(a(c(x1))) |
→ |
c(b(a(c(b(x1))))) |
(14) |
|
d(c(x1)) |
→ |
a(a(x1)) |
(15) |
|
g(x1) |
→ |
a(c(x1)) |
(16) |
|
g(x1) |
→ |
d(d(d(d(x1)))) |
(9) |
1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [a(x1)] |
= |
1 · x1 + 12 |
| [c(x1)] |
= |
1 · x1 + 18 |
| [b(x1)] |
= |
1 · x1
|
| [d(x1)] |
= |
1 · x1 + 7 |
| [f(x1)] |
= |
1 · x1 + 9 |
| [g(x1)] |
= |
1 · x1 + 30 |
all of the following rules can be deleted.
|
d(a(x1)) |
→ |
c(x1) |
(11) |
|
d(c(x1)) |
→ |
a(a(x1)) |
(15) |
|
g(x1) |
→ |
d(d(d(d(x1)))) |
(9) |
1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
a#(c(b(a(x1)))) |
→ |
a#(b(c(a(b(x1))))) |
(17) |
|
a#(c(b(a(x1)))) |
→ |
c#(a(b(x1))) |
(18) |
|
a#(c(b(a(x1)))) |
→ |
a#(b(x1)) |
(19) |
|
f#(f(a(x1))) |
→ |
g#(x1) |
(20) |
|
g#(b(x1)) |
→ |
g#(x1) |
(21) |
|
c#(x1) |
→ |
f#(f(x1)) |
(22) |
|
c#(x1) |
→ |
f#(x1) |
(23) |
|
c#(a(c(x1))) |
→ |
c#(b(a(c(b(x1))))) |
(24) |
|
c#(a(c(x1))) |
→ |
a#(c(b(x1))) |
(25) |
|
c#(a(c(x1))) |
→ |
c#(b(x1)) |
(26) |
|
g#(x1) |
→ |
a#(c(x1)) |
(27) |
|
g#(x1) |
→ |
c#(x1) |
(28) |
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.