Certification Problem
Input (TPDB SRS_Standard/Waldmann_06_SRS/uni-1)
The rewrite relation of the following TRS is considered.
|
a(a(a(x1))) |
→ |
a(c(a(a(x1)))) |
(1) |
|
c(c(c(x1))) |
→ |
a(x1) |
(2) |
|
a(x1) |
→ |
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
|
a(a(a(x1))) |
→ |
a(a(c(a(x1)))) |
(4) |
|
c(c(c(x1))) |
→ |
a(x1) |
(2) |
|
a(x1) |
→ |
x1 |
(3) |
1.1 Closure Under Flat Contexts
Using the flat contexts
{a(☐), c(☐)}
We obtain the transformed TRS
|
a(a(a(x1))) |
→ |
a(a(c(a(x1)))) |
(4) |
|
a(c(c(c(x1)))) |
→ |
a(a(x1)) |
(5) |
|
c(c(c(c(x1)))) |
→ |
c(a(x1)) |
(6) |
|
a(a(x1)) |
→ |
a(x1) |
(7) |
|
c(a(x1)) |
→ |
c(x1) |
(8) |
1.1.1 Semantic Labeling
Root-labeling is applied.
We obtain the labeled TRS
|
aa(aa(aa(x1))) |
→ |
aa(ac(ca(aa(x1)))) |
(9) |
|
aa(aa(ac(x1))) |
→ |
aa(ac(ca(ac(x1)))) |
(10) |
|
ac(cc(cc(ca(x1)))) |
→ |
aa(aa(x1)) |
(11) |
|
ac(cc(cc(cc(x1)))) |
→ |
aa(ac(x1)) |
(12) |
|
cc(cc(cc(ca(x1)))) |
→ |
ca(aa(x1)) |
(13) |
|
cc(cc(cc(cc(x1)))) |
→ |
ca(ac(x1)) |
(14) |
|
aa(aa(x1)) |
→ |
aa(x1) |
(15) |
|
aa(ac(x1)) |
→ |
ac(x1) |
(16) |
|
ca(aa(x1)) |
→ |
ca(x1) |
(17) |
|
ca(ac(x1)) |
→ |
cc(x1) |
(18) |
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [aa(x1)] |
= |
1 · x1 + 3 |
| [ac(x1)] |
= |
1 · x1
|
| [ca(x1)] |
= |
1 · x1 + 3 |
| [cc(x1)] |
= |
1 · x1 + 2 |
all of the following rules can be deleted.
|
ac(cc(cc(ca(x1)))) |
→ |
aa(aa(x1)) |
(11) |
|
ac(cc(cc(cc(x1)))) |
→ |
aa(ac(x1)) |
(12) |
|
cc(cc(cc(ca(x1)))) |
→ |
ca(aa(x1)) |
(13) |
|
cc(cc(cc(cc(x1)))) |
→ |
ca(ac(x1)) |
(14) |
|
aa(aa(x1)) |
→ |
aa(x1) |
(15) |
|
aa(ac(x1)) |
→ |
ac(x1) |
(16) |
|
ca(aa(x1)) |
→ |
ca(x1) |
(17) |
|
ca(ac(x1)) |
→ |
cc(x1) |
(18) |
1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [aa(x1)] |
= |
1 · x1 + 1 |
| [ac(x1)] |
= |
1 · x1
|
| [ca(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
aa(aa(aa(x1))) |
→ |
aa(ac(ca(aa(x1)))) |
(9) |
|
aa(aa(ac(x1))) |
→ |
aa(ac(ca(ac(x1)))) |
(10) |
1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.