Certification Problem
Input (TPDB SRS_Relative/Waldmann_19/random-187)
The rewrite relation of the following TRS is considered.
|
c(c(a(x1))) |
→ |
b(b(c(x1))) |
(1) |
|
a(b(c(x1))) |
→ |
a(b(a(x1))) |
(2) |
|
c(c(c(x1))) |
→ |
b(c(a(x1))) |
(3) |
|
a(b(b(x1))) |
→ |
a(c(a(x1))) |
(4) |
|
c(c(a(x1))) |
→ |
b(b(b(x1))) |
(5) |
|
c(b(a(x1))) |
→ |
c(c(b(x1))) |
(6) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by matchbox @ termCOMP 2023)
1 Split
We split R in the relative problem D/R-D and R-D, where the rules D
|
a(b(c(x1))) |
→ |
a(b(a(x1))) |
(2) |
|
c(c(c(x1))) |
→ |
b(c(a(x1))) |
(3) |
|
c(c(a(x1))) |
→ |
b(b(b(x1))) |
(5) |
|
c(b(a(x1))) |
→ |
c(c(b(x1))) |
(6) |
are deleted.
1.1 Closure Under Flat Contexts
Using the flat contexts
{c(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
c(a(b(c(x1)))) |
→ |
c(a(b(a(x1)))) |
(7) |
|
c(c(c(c(x1)))) |
→ |
c(b(c(a(x1)))) |
(8) |
|
c(c(c(a(x1)))) |
→ |
c(b(b(b(x1)))) |
(9) |
|
c(c(b(a(x1)))) |
→ |
c(c(c(b(x1)))) |
(10) |
|
b(a(b(c(x1)))) |
→ |
b(a(b(a(x1)))) |
(11) |
|
b(c(c(c(x1)))) |
→ |
b(b(c(a(x1)))) |
(12) |
|
b(c(c(a(x1)))) |
→ |
b(b(b(b(x1)))) |
(13) |
|
b(c(b(a(x1)))) |
→ |
b(c(c(b(x1)))) |
(14) |
|
a(a(b(c(x1)))) |
→ |
a(a(b(a(x1)))) |
(15) |
|
a(c(c(c(x1)))) |
→ |
a(b(c(a(x1)))) |
(16) |
|
a(c(c(a(x1)))) |
→ |
a(b(b(b(x1)))) |
(17) |
|
a(c(b(a(x1)))) |
→ |
a(c(c(b(x1)))) |
(18) |
|
c(c(c(a(x1)))) |
→ |
c(b(b(c(x1)))) |
(19) |
|
c(a(b(b(x1)))) |
→ |
c(a(c(a(x1)))) |
(20) |
|
b(c(c(a(x1)))) |
→ |
b(b(b(c(x1)))) |
(21) |
|
b(a(b(b(x1)))) |
→ |
b(a(c(a(x1)))) |
(22) |
|
a(c(c(a(x1)))) |
→ |
a(b(b(c(x1)))) |
(23) |
|
a(a(b(b(x1)))) |
→ |
a(a(c(a(x1)))) |
(24) |
1.1.1 Closure Under Flat Contexts
Using the flat contexts
{c(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
c(c(a(b(c(x1))))) |
→ |
c(c(a(b(a(x1))))) |
(25) |
|
c(c(c(c(c(x1))))) |
→ |
c(c(b(c(a(x1))))) |
(26) |
|
c(c(c(c(a(x1))))) |
→ |
c(c(b(b(b(x1))))) |
(27) |
|
c(c(c(b(a(x1))))) |
→ |
c(c(c(c(b(x1))))) |
(28) |
|
c(b(a(b(c(x1))))) |
→ |
c(b(a(b(a(x1))))) |
(29) |
|
c(b(c(c(c(x1))))) |
→ |
c(b(b(c(a(x1))))) |
(30) |
|
c(b(c(c(a(x1))))) |
→ |
c(b(b(b(b(x1))))) |
(31) |
|
c(b(c(b(a(x1))))) |
→ |
c(b(c(c(b(x1))))) |
(32) |
|
c(a(a(b(c(x1))))) |
→ |
c(a(a(b(a(x1))))) |
(33) |
|
c(a(c(c(c(x1))))) |
→ |
c(a(b(c(a(x1))))) |
(34) |
|
c(a(c(c(a(x1))))) |
→ |
c(a(b(b(b(x1))))) |
(35) |
|
c(a(c(b(a(x1))))) |
→ |
c(a(c(c(b(x1))))) |
(36) |
|
b(c(a(b(c(x1))))) |
→ |
b(c(a(b(a(x1))))) |
(37) |
|
b(c(c(c(c(x1))))) |
→ |
b(c(b(c(a(x1))))) |
(38) |
|
b(c(c(c(a(x1))))) |
→ |
b(c(b(b(b(x1))))) |
(39) |
|
b(c(c(b(a(x1))))) |
→ |
b(c(c(c(b(x1))))) |
(40) |
|
b(b(a(b(c(x1))))) |
→ |
b(b(a(b(a(x1))))) |
(41) |
|
b(b(c(c(c(x1))))) |
→ |
b(b(b(c(a(x1))))) |
(42) |
|
b(b(c(c(a(x1))))) |
→ |
b(b(b(b(b(x1))))) |
(43) |
|
b(b(c(b(a(x1))))) |
→ |
b(b(c(c(b(x1))))) |
(44) |
|
b(a(a(b(c(x1))))) |
→ |
b(a(a(b(a(x1))))) |
(45) |
|
b(a(c(c(c(x1))))) |
→ |
b(a(b(c(a(x1))))) |
(46) |
|
b(a(c(c(a(x1))))) |
→ |
b(a(b(b(b(x1))))) |
(47) |
|
b(a(c(b(a(x1))))) |
→ |
b(a(c(c(b(x1))))) |
(48) |
|
a(c(a(b(c(x1))))) |
→ |
a(c(a(b(a(x1))))) |
(49) |
|
a(c(c(c(c(x1))))) |
→ |
a(c(b(c(a(x1))))) |
(50) |
|
a(c(c(c(a(x1))))) |
→ |
a(c(b(b(b(x1))))) |
(51) |
|
a(c(c(b(a(x1))))) |
→ |
a(c(c(c(b(x1))))) |
(52) |
|
a(b(a(b(c(x1))))) |
→ |
a(b(a(b(a(x1))))) |
(53) |
|
a(b(c(c(c(x1))))) |
→ |
a(b(b(c(a(x1))))) |
(54) |
|
a(b(c(c(a(x1))))) |
→ |
a(b(b(b(b(x1))))) |
(55) |
|
a(b(c(b(a(x1))))) |
→ |
a(b(c(c(b(x1))))) |
(56) |
|
a(a(a(b(c(x1))))) |
→ |
a(a(a(b(a(x1))))) |
(57) |
|
a(a(c(c(c(x1))))) |
→ |
a(a(b(c(a(x1))))) |
(58) |
|
a(a(c(c(a(x1))))) |
→ |
a(a(b(b(b(x1))))) |
(59) |
|
a(a(c(b(a(x1))))) |
→ |
a(a(c(c(b(x1))))) |
(60) |
|
c(c(c(c(a(x1))))) |
→ |
c(c(b(b(c(x1))))) |
(61) |
|
c(c(a(b(b(x1))))) |
→ |
c(c(a(c(a(x1))))) |
(62) |
|
c(b(c(c(a(x1))))) |
→ |
c(b(b(b(c(x1))))) |
(63) |
|
c(b(a(b(b(x1))))) |
→ |
c(b(a(c(a(x1))))) |
(64) |
|
c(a(c(c(a(x1))))) |
→ |
c(a(b(b(c(x1))))) |
(65) |
|
c(a(a(b(b(x1))))) |
→ |
c(a(a(c(a(x1))))) |
(66) |
|
b(c(c(c(a(x1))))) |
→ |
b(c(b(b(c(x1))))) |
(67) |
|
b(c(a(b(b(x1))))) |
→ |
b(c(a(c(a(x1))))) |
(68) |
|
b(b(c(c(a(x1))))) |
→ |
b(b(b(b(c(x1))))) |
(69) |
|
b(b(a(b(b(x1))))) |
→ |
b(b(a(c(a(x1))))) |
(70) |
|
b(a(c(c(a(x1))))) |
→ |
b(a(b(b(c(x1))))) |
(71) |
|
b(a(a(b(b(x1))))) |
→ |
b(a(a(c(a(x1))))) |
(72) |
|
a(c(c(c(a(x1))))) |
→ |
a(c(b(b(c(x1))))) |
(73) |
|
a(c(a(b(b(x1))))) |
→ |
a(c(a(c(a(x1))))) |
(74) |
|
a(b(c(c(a(x1))))) |
→ |
a(b(b(b(c(x1))))) |
(75) |
|
a(b(a(b(b(x1))))) |
→ |
a(b(a(c(a(x1))))) |
(76) |
|
a(a(c(c(a(x1))))) |
→ |
a(a(b(b(c(x1))))) |
(77) |
|
a(a(a(b(b(x1))))) |
→ |
a(a(a(c(a(x1))))) |
(78) |
1.1.1.1 Semantic Labeling
The following interpretations form a
model
of the rules.
As carrier we take the set
{0,...,8}.
Symbols are labeled by the interpretation of their arguments using the interpretations
(modulo 9):
| [c(x1)] |
= |
3x1 + 0 |
| [b(x1)] |
= |
3x1 + 1 |
| [a(x1)] |
= |
3x1 + 2 |
We obtain the labeled TRS
There are 486 ruless (increase limit for explicit display).
1.1.1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [c0(x1)] |
= |
x1 +
|
| [c3(x1)] |
= |
x1 +
|
| [c6(x1)] |
= |
x1 +
|
| [c1(x1)] |
= |
x1 +
|
| [c4(x1)] |
= |
x1 +
|
| [c7(x1)] |
= |
x1 +
|
| [c2(x1)] |
= |
x1 +
|
| [c5(x1)] |
= |
x1 +
|
| [c8(x1)] |
= |
x1 +
|
| [b0(x1)] |
= |
x1 +
|
| [b3(x1)] |
= |
x1 +
|
| [b6(x1)] |
= |
x1 +
|
| [b1(x1)] |
= |
x1 +
|
| [b4(x1)] |
= |
x1 +
|
| [b7(x1)] |
= |
x1 +
|
| [b2(x1)] |
= |
x1 +
|
| [b5(x1)] |
= |
x1 +
|
| [b8(x1)] |
= |
x1 +
|
| [a0(x1)] |
= |
x1 +
|
| [a3(x1)] |
= |
x1 +
|
| [a6(x1)] |
= |
x1 +
|
| [a1(x1)] |
= |
x1 +
|
| [a4(x1)] |
= |
x1 +
|
| [a7(x1)] |
= |
x1 +
|
| [a2(x1)] |
= |
x1 +
|
| [a5(x1)] |
= |
x1 +
|
| [a8(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
There are 454 ruless (increase limit for explicit display).
1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.
1.2 Closure Under Flat Contexts
Using the flat contexts
{c(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
c(c(c(a(x1)))) |
→ |
c(b(b(c(x1)))) |
(19) |
|
c(a(b(b(x1)))) |
→ |
c(a(c(a(x1)))) |
(20) |
|
b(c(c(a(x1)))) |
→ |
b(b(b(c(x1)))) |
(21) |
|
b(a(b(b(x1)))) |
→ |
b(a(c(a(x1)))) |
(22) |
|
a(c(c(a(x1)))) |
→ |
a(b(b(c(x1)))) |
(23) |
|
a(a(b(b(x1)))) |
→ |
a(a(c(a(x1)))) |
(24) |
1.2.1 Semantic Labeling
The following interpretations form a
model
of the rules.
As carrier we take the set
{0,1,2}.
Symbols are labeled by the interpretation of their arguments using the interpretations
(modulo 3):
| [c(x1)] |
= |
3x1 + 0 |
| [b(x1)] |
= |
3x1 + 1 |
| [a(x1)] |
= |
3x1 + 2 |
We obtain the labeled TRS
|
c0(c0(c2(a0(x1)))) |
→ |
c1(b1(b0(c0(x1)))) |
(565) |
|
c0(c0(c2(a2(x1)))) |
→ |
c1(b1(b0(c2(x1)))) |
(566) |
|
c0(c0(c2(a1(x1)))) |
→ |
c1(b1(b0(c1(x1)))) |
(567) |
|
a0(c0(c2(a0(x1)))) |
→ |
a1(b1(b0(c0(x1)))) |
(568) |
|
a0(c0(c2(a2(x1)))) |
→ |
a1(b1(b0(c2(x1)))) |
(569) |
|
a0(c0(c2(a1(x1)))) |
→ |
a1(b1(b0(c1(x1)))) |
(570) |
|
b0(c0(c2(a0(x1)))) |
→ |
b1(b1(b0(c0(x1)))) |
(571) |
|
b0(c0(c2(a2(x1)))) |
→ |
b1(b1(b0(c2(x1)))) |
(572) |
|
b0(c0(c2(a1(x1)))) |
→ |
b1(b1(b0(c1(x1)))) |
(573) |
|
c2(a1(b1(b0(x1)))) |
→ |
c2(a0(c2(a0(x1)))) |
(574) |
|
c2(a1(b1(b2(x1)))) |
→ |
c2(a0(c2(a2(x1)))) |
(575) |
|
c2(a1(b1(b1(x1)))) |
→ |
c2(a0(c2(a1(x1)))) |
(576) |
|
a2(a1(b1(b0(x1)))) |
→ |
a2(a0(c2(a0(x1)))) |
(577) |
|
a2(a1(b1(b2(x1)))) |
→ |
a2(a0(c2(a2(x1)))) |
(578) |
|
a2(a1(b1(b1(x1)))) |
→ |
a2(a0(c2(a1(x1)))) |
(579) |
|
b2(a1(b1(b0(x1)))) |
→ |
b2(a0(c2(a0(x1)))) |
(580) |
|
b2(a1(b1(b2(x1)))) |
→ |
b2(a0(c2(a2(x1)))) |
(581) |
|
b2(a1(b1(b1(x1)))) |
→ |
b2(a0(c2(a1(x1)))) |
(582) |
1.2.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [c0(x1)] |
= |
x1 +
|
| [c1(x1)] |
= |
x1 +
|
| [c2(x1)] |
= |
x1 +
|
| [b0(x1)] |
= |
x1 +
|
| [b1(x1)] |
= |
x1 +
|
| [b2(x1)] |
= |
x1 +
|
| [a0(x1)] |
= |
x1 +
|
| [a1(x1)] |
= |
x1 +
|
| [a2(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
|
c0(c0(c2(a0(x1)))) |
→ |
c1(b1(b0(c0(x1)))) |
(565) |
|
c0(c0(c2(a2(x1)))) |
→ |
c1(b1(b0(c2(x1)))) |
(566) |
|
c0(c0(c2(a1(x1)))) |
→ |
c1(b1(b0(c1(x1)))) |
(567) |
|
a0(c0(c2(a2(x1)))) |
→ |
a1(b1(b0(c2(x1)))) |
(569) |
|
a0(c0(c2(a1(x1)))) |
→ |
a1(b1(b0(c1(x1)))) |
(570) |
|
b0(c0(c2(a2(x1)))) |
→ |
b1(b1(b0(c2(x1)))) |
(572) |
|
b0(c0(c2(a1(x1)))) |
→ |
b1(b1(b0(c1(x1)))) |
(573) |
|
c2(a1(b1(b2(x1)))) |
→ |
c2(a0(c2(a2(x1)))) |
(575) |
|
a2(a1(b1(b2(x1)))) |
→ |
a2(a0(c2(a2(x1)))) |
(578) |
|
b2(a1(b1(b2(x1)))) |
→ |
b2(a0(c2(a2(x1)))) |
(581) |
1.2.1.1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
a0(c2(c0(a0(x1)))) |
→ |
c0(b0(b1(a1(x1)))) |
(583) |
|
a0(c2(c0(b0(x1)))) |
→ |
c0(b0(b1(b1(x1)))) |
(584) |
|
b0(b1(a1(c2(x1)))) |
→ |
a0(c2(a0(c2(x1)))) |
(585) |
|
b1(b1(a1(c2(x1)))) |
→ |
a1(c2(a0(c2(x1)))) |
(586) |
|
b0(b1(a1(a2(x1)))) |
→ |
a0(c2(a0(a2(x1)))) |
(587) |
|
b1(b1(a1(a2(x1)))) |
→ |
a1(c2(a0(a2(x1)))) |
(588) |
|
b0(b1(a1(b2(x1)))) |
→ |
a0(c2(a0(b2(x1)))) |
(589) |
|
b1(b1(a1(b2(x1)))) |
→ |
a1(c2(a0(b2(x1)))) |
(590) |
1.2.1.1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the naturals
| [c0(x1)] |
= |
· x1 +
|
| [c2(x1)] |
= |
· x1 +
|
| [b0(x1)] |
= |
· x1 +
|
| [b1(x1)] |
= |
· x1 +
|
| [b2(x1)] |
= |
· x1 +
|
| [a0(x1)] |
= |
· x1 +
|
| [a1(x1)] |
= |
· x1 +
|
| [a2(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
|
a0(c2(c0(a0(x1)))) |
→ |
c0(b0(b1(a1(x1)))) |
(583) |
|
a0(c2(c0(b0(x1)))) |
→ |
c0(b0(b1(b1(x1)))) |
(584) |
|
b0(b1(a1(c2(x1)))) |
→ |
a0(c2(a0(c2(x1)))) |
(585) |
|
b1(b1(a1(c2(x1)))) |
→ |
a1(c2(a0(c2(x1)))) |
(586) |
|
b0(b1(a1(a2(x1)))) |
→ |
a0(c2(a0(a2(x1)))) |
(587) |
|
b1(b1(a1(a2(x1)))) |
→ |
a1(c2(a0(a2(x1)))) |
(588) |
|
b0(b1(a1(b2(x1)))) |
→ |
a0(c2(a0(b2(x1)))) |
(589) |
|
b1(b1(a1(b2(x1)))) |
→ |
a1(c2(a0(b2(x1)))) |
(590) |
1.2.1.1.1.1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
There are no rules.
1.2.1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.