Certification Problem
Input (TPDB SRS_Relative/Mixed_relative_SRS/zr02)
The relative rewrite relation R/S is considered where R is the following TRS
|
n(s(x1)) |
→ |
s(x1) |
(1) |
|
o(s(x1)) |
→ |
s(x1) |
(2) |
|
n(o(p(x1))) |
→ |
o(n(x1)) |
(3) |
and S is the following TRS.
|
t(x1) |
→ |
t(c(n(x1))) |
(4) |
|
p(s(x1)) |
→ |
s(x1) |
(5) |
|
o(n(x1)) |
→ |
n(o(x1)) |
(6) |
|
p(n(x1)) |
→ |
m(p(x1)) |
(7) |
|
p(m(x1)) |
→ |
m(p(x1)) |
(8) |
|
o(m(x1)) |
→ |
n(o(x1)) |
(9) |
|
n(p(x1)) |
→ |
p(n(x1)) |
(10) |
|
c(p(x1)) |
→ |
p(c(x1)) |
(11) |
|
c(m(x1)) |
→ |
m(c(x1)) |
(12) |
|
c(n(x1)) |
→ |
n(c(x1)) |
(13) |
|
c(o(x1)) |
→ |
o(c(x1)) |
(14) |
|
c(o(x1)) |
→ |
o(x1) |
(15) |
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
|
s(n(x1)) |
→ |
s(x1) |
(16) |
|
s(o(x1)) |
→ |
s(x1) |
(17) |
|
p(o(n(x1))) |
→ |
n(o(x1)) |
(18) |
|
t(x1) |
→ |
n(c(t(x1))) |
(19) |
|
s(p(x1)) |
→ |
s(x1) |
(20) |
|
n(o(x1)) |
→ |
o(n(x1)) |
(21) |
|
n(p(x1)) |
→ |
p(m(x1)) |
(22) |
|
m(p(x1)) |
→ |
p(m(x1)) |
(23) |
|
m(o(x1)) |
→ |
o(n(x1)) |
(24) |
|
p(n(x1)) |
→ |
n(p(x1)) |
(25) |
|
p(c(x1)) |
→ |
c(p(x1)) |
(26) |
|
m(c(x1)) |
→ |
c(m(x1)) |
(27) |
|
n(c(x1)) |
→ |
c(n(x1)) |
(28) |
|
o(c(x1)) |
→ |
c(o(x1)) |
(29) |
|
o(c(x1)) |
→ |
o(x1) |
(30) |
1.1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [t(x1)] |
= |
0 · x1 +
-∞ |
| [n(x1)] |
= |
0 · x1 +
-∞ |
| [p(x1)] |
= |
4 · x1 +
-∞ |
| [c(x1)] |
= |
0 · x1 +
-∞ |
| [s(x1)] |
= |
0 · x1 +
-∞ |
| [m(x1)] |
= |
0 · x1 +
-∞ |
| [o(x1)] |
= |
8 · x1 +
-∞ |
all of the following rules can be deleted.
|
s(o(x1)) |
→ |
s(x1) |
(17) |
|
p(o(n(x1))) |
→ |
n(o(x1)) |
(18) |
|
s(p(x1)) |
→ |
s(x1) |
(20) |
1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
| [t(x1)] |
= |
· x1 +
|
| [n(x1)] |
= |
· x1 +
|
| [p(x1)] |
= |
· x1 +
|
| [c(x1)] |
= |
· x1 +
|
| [s(x1)] |
= |
· x1 +
|
| [m(x1)] |
= |
· x1 +
|
| [o(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.