Certification Problem
Input (TPDB SRS_Standard/Secret_05_SRS/torpa3)
The rewrite relation of the following TRS is considered.
|
b(b(x1)) |
→ |
c(d(x1)) |
(1) |
|
c(c(x1)) |
→ |
d(d(d(x1))) |
(2) |
|
c(x1) |
→ |
g(x1) |
(3) |
|
d(d(x1)) |
→ |
c(f(x1)) |
(4) |
|
d(d(d(x1))) |
→ |
g(c(x1)) |
(5) |
|
f(x1) |
→ |
a(g(x1)) |
(6) |
|
g(x1) |
→ |
d(a(b(x1))) |
(7) |
|
g(g(x1)) |
→ |
b(c(x1)) |
(8) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by matchbox @ termCOMP 2023)
1 Closure Under Flat Contexts
Using the flat contexts
{g(☐), f(☐), d(☐), c(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
g(b(b(x1))) |
→ |
g(c(d(x1))) |
(9) |
|
g(c(c(x1))) |
→ |
g(d(d(d(x1)))) |
(10) |
|
g(c(x1)) |
→ |
g(g(x1)) |
(11) |
|
g(d(d(x1))) |
→ |
g(c(f(x1))) |
(12) |
|
g(d(d(d(x1)))) |
→ |
g(g(c(x1))) |
(13) |
|
g(f(x1)) |
→ |
g(a(g(x1))) |
(14) |
|
g(g(x1)) |
→ |
g(d(a(b(x1)))) |
(15) |
|
g(g(g(x1))) |
→ |
g(b(c(x1))) |
(16) |
|
f(b(b(x1))) |
→ |
f(c(d(x1))) |
(17) |
|
f(c(c(x1))) |
→ |
f(d(d(d(x1)))) |
(18) |
|
f(c(x1)) |
→ |
f(g(x1)) |
(19) |
|
f(d(d(x1))) |
→ |
f(c(f(x1))) |
(20) |
|
f(d(d(d(x1)))) |
→ |
f(g(c(x1))) |
(21) |
|
f(f(x1)) |
→ |
f(a(g(x1))) |
(22) |
|
f(g(x1)) |
→ |
f(d(a(b(x1)))) |
(23) |
|
f(g(g(x1))) |
→ |
f(b(c(x1))) |
(24) |
|
d(b(b(x1))) |
→ |
d(c(d(x1))) |
(25) |
|
d(c(c(x1))) |
→ |
d(d(d(d(x1)))) |
(26) |
|
d(c(x1)) |
→ |
d(g(x1)) |
(27) |
|
d(d(d(x1))) |
→ |
d(c(f(x1))) |
(28) |
|
d(d(d(d(x1)))) |
→ |
d(g(c(x1))) |
(29) |
|
d(f(x1)) |
→ |
d(a(g(x1))) |
(30) |
|
d(g(x1)) |
→ |
d(d(a(b(x1)))) |
(31) |
|
d(g(g(x1))) |
→ |
d(b(c(x1))) |
(32) |
|
c(b(b(x1))) |
→ |
c(c(d(x1))) |
(33) |
|
c(c(c(x1))) |
→ |
c(d(d(d(x1)))) |
(34) |
|
c(c(x1)) |
→ |
c(g(x1)) |
(35) |
|
c(d(d(x1))) |
→ |
c(c(f(x1))) |
(36) |
|
c(d(d(d(x1)))) |
→ |
c(g(c(x1))) |
(37) |
|
c(f(x1)) |
→ |
c(a(g(x1))) |
(38) |
|
c(g(x1)) |
→ |
c(d(a(b(x1)))) |
(39) |
|
c(g(g(x1))) |
→ |
c(b(c(x1))) |
(40) |
|
b(b(b(x1))) |
→ |
b(c(d(x1))) |
(41) |
|
b(c(c(x1))) |
→ |
b(d(d(d(x1)))) |
(42) |
|
b(c(x1)) |
→ |
b(g(x1)) |
(43) |
|
b(d(d(x1))) |
→ |
b(c(f(x1))) |
(44) |
|
b(d(d(d(x1)))) |
→ |
b(g(c(x1))) |
(45) |
|
b(f(x1)) |
→ |
b(a(g(x1))) |
(46) |
|
b(g(x1)) |
→ |
b(d(a(b(x1)))) |
(47) |
|
b(g(g(x1))) |
→ |
b(b(c(x1))) |
(48) |
|
a(b(b(x1))) |
→ |
a(c(d(x1))) |
(49) |
|
a(c(c(x1))) |
→ |
a(d(d(d(x1)))) |
(50) |
|
a(c(x1)) |
→ |
a(g(x1)) |
(51) |
|
a(d(d(x1))) |
→ |
a(c(f(x1))) |
(52) |
|
a(d(d(d(x1)))) |
→ |
a(g(c(x1))) |
(53) |
|
a(f(x1)) |
→ |
a(a(g(x1))) |
(54) |
|
a(g(x1)) |
→ |
a(d(a(b(x1)))) |
(55) |
|
a(g(g(x1))) |
→ |
a(b(c(x1))) |
(56) |
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):
| [g(x1)] |
= |
6x1 + 0 |
| [f(x1)] |
= |
6x1 + 1 |
| [d(x1)] |
= |
6x1 + 2 |
| [c(x1)] |
= |
6x1 + 3 |
| [b(x1)] |
= |
6x1 + 4 |
| [a(x1)] |
= |
6x1 + 5 |
We obtain the labeled TRS
There are 288 ruless (increase limit for explicit display).
1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [g0(x1)] |
= |
x1 +
|
| [g1(x1)] |
= |
x1 +
|
| [g2(x1)] |
= |
x1 +
|
| [g3(x1)] |
= |
x1 +
|
| [g4(x1)] |
= |
x1 +
|
| [g5(x1)] |
= |
x1 +
|
| [f0(x1)] |
= |
x1 +
|
| [f1(x1)] |
= |
x1 +
|
| [f2(x1)] |
= |
x1 +
|
| [f3(x1)] |
= |
x1 +
|
| [f4(x1)] |
= |
x1 +
|
| [f5(x1)] |
= |
x1 +
|
| [d0(x1)] |
= |
x1 +
|
| [d1(x1)] |
= |
x1 +
|
| [d2(x1)] |
= |
x1 +
|
| [d3(x1)] |
= |
x1 +
|
| [d4(x1)] |
= |
x1 +
|
| [d5(x1)] |
= |
x1 +
|
| [c0(x1)] |
= |
x1 +
|
| [c1(x1)] |
= |
x1 +
|
| [c2(x1)] |
= |
x1 +
|
| [c3(x1)] |
= |
x1 +
|
| [c4(x1)] |
= |
x1 +
|
| [c5(x1)] |
= |
x1 +
|
| [b0(x1)] |
= |
x1 +
|
| [b1(x1)] |
= |
x1 +
|
| [b2(x1)] |
= |
x1 +
|
| [b3(x1)] |
= |
x1 +
|
| [b4(x1)] |
= |
x1 +
|
| [b5(x1)] |
= |
x1 +
|
| [a0(x1)] |
= |
x1 +
|
| [a1(x1)] |
= |
x1 +
|
| [a2(x1)] |
= |
x1 +
|
| [a3(x1)] |
= |
x1 +
|
| [a4(x1)] |
= |
x1 +
|
| [a5(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
There are 288 ruless (increase limit for explicit display).
1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.