Certification Problem
Input (TPDB SRS_Relative/Zantema_06_relative/rel13)
The relative rewrite relation R/S is considered where R is the following TRS
|
o(l(x1)) |
→ |
r(x1) |
(1) |
|
n(l(o(x1))) |
→ |
r(o(x1)) |
(2) |
|
L(l(o(x1))) |
→ |
L(r(o(x1))) |
(3) |
|
r(o(x1)) |
→ |
l(x1) |
(4) |
|
o(r(n(x1))) |
→ |
o(l(x1)) |
(5) |
|
o(r(R(x1))) |
→ |
o(l(R(x1))) |
(6) |
and S is the following TRS.
|
L(x1) |
→ |
L(n(x1)) |
(7) |
|
R(x1) |
→ |
n(R(x1)) |
(8) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by matchbox @ termCOMP 2023)
1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [L(x1)] |
= |
x1 +
|
| [R(x1)] |
= |
x1 +
|
| [o(x1)] |
= |
x1 +
|
| [n(x1)] |
= |
x1 +
|
| [l(x1)] |
= |
x1 +
|
| [r(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
|
o(l(x1)) |
→ |
r(x1) |
(1) |
|
r(o(x1)) |
→ |
l(x1) |
(4) |
1.1 Closure Under Flat Contexts
Using the flat contexts
{L(☐), R(☐), o(☐), n(☐), l(☐), r(☐)}
We obtain the transformed TRS
|
L(n(l(o(x1)))) |
→ |
L(r(o(x1))) |
(9) |
|
L(L(l(o(x1)))) |
→ |
L(L(r(o(x1)))) |
(10) |
|
L(o(r(n(x1)))) |
→ |
L(o(l(x1))) |
(11) |
|
L(o(r(R(x1)))) |
→ |
L(o(l(R(x1)))) |
(12) |
|
R(n(l(o(x1)))) |
→ |
R(r(o(x1))) |
(13) |
|
R(L(l(o(x1)))) |
→ |
R(L(r(o(x1)))) |
(14) |
|
R(o(r(n(x1)))) |
→ |
R(o(l(x1))) |
(15) |
|
R(o(r(R(x1)))) |
→ |
R(o(l(R(x1)))) |
(16) |
|
o(n(l(o(x1)))) |
→ |
o(r(o(x1))) |
(17) |
|
o(L(l(o(x1)))) |
→ |
o(L(r(o(x1)))) |
(18) |
|
o(o(r(n(x1)))) |
→ |
o(o(l(x1))) |
(19) |
|
o(o(r(R(x1)))) |
→ |
o(o(l(R(x1)))) |
(20) |
|
n(n(l(o(x1)))) |
→ |
n(r(o(x1))) |
(21) |
|
n(L(l(o(x1)))) |
→ |
n(L(r(o(x1)))) |
(22) |
|
n(o(r(n(x1)))) |
→ |
n(o(l(x1))) |
(23) |
|
n(o(r(R(x1)))) |
→ |
n(o(l(R(x1)))) |
(24) |
|
l(n(l(o(x1)))) |
→ |
l(r(o(x1))) |
(25) |
|
l(L(l(o(x1)))) |
→ |
l(L(r(o(x1)))) |
(26) |
|
l(o(r(n(x1)))) |
→ |
l(o(l(x1))) |
(27) |
|
l(o(r(R(x1)))) |
→ |
l(o(l(R(x1)))) |
(28) |
|
r(n(l(o(x1)))) |
→ |
r(r(o(x1))) |
(29) |
|
r(L(l(o(x1)))) |
→ |
r(L(r(o(x1)))) |
(30) |
|
r(o(r(n(x1)))) |
→ |
r(o(l(x1))) |
(31) |
|
r(o(r(R(x1)))) |
→ |
r(o(l(R(x1)))) |
(32) |
|
L(L(x1)) |
→ |
L(L(n(x1))) |
(33) |
|
L(R(x1)) |
→ |
L(n(R(x1))) |
(34) |
|
R(L(x1)) |
→ |
R(L(n(x1))) |
(35) |
|
R(R(x1)) |
→ |
R(n(R(x1))) |
(36) |
|
o(L(x1)) |
→ |
o(L(n(x1))) |
(37) |
|
o(R(x1)) |
→ |
o(n(R(x1))) |
(38) |
|
n(L(x1)) |
→ |
n(L(n(x1))) |
(39) |
|
n(R(x1)) |
→ |
n(n(R(x1))) |
(40) |
|
l(L(x1)) |
→ |
l(L(n(x1))) |
(41) |
|
l(R(x1)) |
→ |
l(n(R(x1))) |
(42) |
|
r(L(x1)) |
→ |
r(L(n(x1))) |
(43) |
|
r(R(x1)) |
→ |
r(n(R(x1))) |
(44) |
1.1.1 Semantic Labeling
The following interpretations form a
model
of the rules.
As carrier we take the set
{0,...,5}.
Symbols are labeled by the interpretation of their arguments using the interpretations
(modulo 6):
| [L(x1)] |
= |
6x1 + 0 |
| [R(x1)] |
= |
6x1 + 1 |
| [o(x1)] |
= |
6x1 + 2 |
| [n(x1)] |
= |
6x1 + 3 |
| [l(x1)] |
= |
6x1 + 4 |
| [r(x1)] |
= |
6x1 + 5 |
We obtain the labeled TRS
There are 216 ruless (increase limit for explicit display).
1.1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [L0(x1)] |
= |
x1 +
|
| [L1(x1)] |
= |
x1 +
|
| [L2(x1)] |
= |
x1 +
|
| [L3(x1)] |
= |
x1 +
|
| [L4(x1)] |
= |
x1 +
|
| [L5(x1)] |
= |
x1 +
|
| [R0(x1)] |
= |
x1 +
|
| [R1(x1)] |
= |
x1 +
|
| [R2(x1)] |
= |
x1 +
|
| [R3(x1)] |
= |
x1 +
|
| [R4(x1)] |
= |
x1 +
|
| [R5(x1)] |
= |
x1 +
|
| [o0(x1)] |
= |
x1 +
|
| [o1(x1)] |
= |
x1 +
|
| [o2(x1)] |
= |
x1 +
|
| [o3(x1)] |
= |
x1 +
|
| [o4(x1)] |
= |
x1 +
|
| [o5(x1)] |
= |
x1 +
|
| [n0(x1)] |
= |
x1 +
|
| [n1(x1)] |
= |
x1 +
|
| [n2(x1)] |
= |
x1 +
|
| [n3(x1)] |
= |
x1 +
|
| [n4(x1)] |
= |
x1 +
|
| [n5(x1)] |
= |
x1 +
|
| [l0(x1)] |
= |
x1 +
|
| [l1(x1)] |
= |
x1 +
|
| [l2(x1)] |
= |
x1 +
|
| [l3(x1)] |
= |
x1 +
|
| [l4(x1)] |
= |
x1 +
|
| [l5(x1)] |
= |
x1 +
|
| [r0(x1)] |
= |
x1 +
|
| [r1(x1)] |
= |
x1 +
|
| [r2(x1)] |
= |
x1 +
|
| [r3(x1)] |
= |
x1 +
|
| [r5(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
There are 204 ruless (increase limit for explicit display).
1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.