Certification Problem
Input (TPDB SRS_Relative/Zantema_06_relative/rel09)
The relative rewrite relation R/S is considered where R is the following TRS
|
b(q(b(x1))) |
→ |
b(p(b(x1))) |
(1) |
and S is the following TRS.
|
0(p(0(x1))) |
→ |
q(x1) |
(2) |
|
1(p(1(x1))) |
→ |
q(x1) |
(3) |
|
0(q(0(x1))) |
→ |
q(x1) |
(4) |
|
1(q(1(x1))) |
→ |
q(x1) |
(5) |
|
p(x1) |
→ |
1(p(1(0(1(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
are deleted.
1.1 Closure Under Flat Contexts
Using the flat contexts
{1(☐), 0(☐), b(☐), q(☐), p(☐)}
We obtain the transformed TRS
|
1(0(p(0(x1)))) |
→ |
1(q(x1)) |
(7) |
|
0(0(p(0(x1)))) |
→ |
0(q(x1)) |
(8) |
|
b(0(p(0(x1)))) |
→ |
b(q(x1)) |
(9) |
|
q(0(p(0(x1)))) |
→ |
q(q(x1)) |
(10) |
|
p(0(p(0(x1)))) |
→ |
p(q(x1)) |
(11) |
|
1(b(q(b(x1)))) |
→ |
1(b(p(b(x1)))) |
(12) |
|
1(1(p(1(x1)))) |
→ |
1(q(x1)) |
(13) |
|
1(0(q(0(x1)))) |
→ |
1(q(x1)) |
(14) |
|
1(1(q(1(x1)))) |
→ |
1(q(x1)) |
(15) |
|
1(p(x1)) |
→ |
1(1(p(1(0(1(x1)))))) |
(16) |
|
0(b(q(b(x1)))) |
→ |
0(b(p(b(x1)))) |
(17) |
|
0(1(p(1(x1)))) |
→ |
0(q(x1)) |
(18) |
|
0(0(q(0(x1)))) |
→ |
0(q(x1)) |
(19) |
|
0(1(q(1(x1)))) |
→ |
0(q(x1)) |
(20) |
|
0(p(x1)) |
→ |
0(1(p(1(0(1(x1)))))) |
(21) |
|
b(b(q(b(x1)))) |
→ |
b(b(p(b(x1)))) |
(22) |
|
b(1(p(1(x1)))) |
→ |
b(q(x1)) |
(23) |
|
b(0(q(0(x1)))) |
→ |
b(q(x1)) |
(24) |
|
b(1(q(1(x1)))) |
→ |
b(q(x1)) |
(25) |
|
b(p(x1)) |
→ |
b(1(p(1(0(1(x1)))))) |
(26) |
|
q(b(q(b(x1)))) |
→ |
q(b(p(b(x1)))) |
(27) |
|
q(1(p(1(x1)))) |
→ |
q(q(x1)) |
(28) |
|
q(0(q(0(x1)))) |
→ |
q(q(x1)) |
(29) |
|
q(1(q(1(x1)))) |
→ |
q(q(x1)) |
(30) |
|
q(p(x1)) |
→ |
q(1(p(1(0(1(x1)))))) |
(31) |
|
p(b(q(b(x1)))) |
→ |
p(b(p(b(x1)))) |
(32) |
|
p(1(p(1(x1)))) |
→ |
p(q(x1)) |
(33) |
|
p(0(q(0(x1)))) |
→ |
p(q(x1)) |
(34) |
|
p(1(q(1(x1)))) |
→ |
p(q(x1)) |
(35) |
|
p(p(x1)) |
→ |
p(1(p(1(0(1(x1)))))) |
(36) |
1.1.1 Semantic Labeling
The following interpretations form a
model
of the rules.
As carrier we take the set
{0,...,4}.
Symbols are labeled by the interpretation of their arguments using the interpretations
(modulo 5):
| [1(x1)] |
= |
5x1 + 0 |
| [0(x1)] |
= |
5x1 + 1 |
| [b(x1)] |
= |
5x1 + 2 |
| [q(x1)] |
= |
5x1 + 3 |
| [p(x1)] |
= |
5x1 + 4 |
We obtain the labeled TRS
There are 150 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
| [10(x1)] |
= |
x1 +
|
| [11(x1)] |
= |
x1 +
|
| [12(x1)] |
= |
x1 +
|
| [13(x1)] |
= |
x1 +
|
| [14(x1)] |
= |
x1 +
|
| [00(x1)] |
= |
x1 +
|
| [01(x1)] |
= |
x1 +
|
| [02(x1)] |
= |
x1 +
|
| [03(x1)] |
= |
x1 +
|
| [04(x1)] |
= |
x1 +
|
| [b0(x1)] |
= |
x1 +
|
| [b1(x1)] |
= |
x1 +
|
| [b2(x1)] |
= |
x1 +
|
| [b3(x1)] |
= |
x1 +
|
| [b4(x1)] |
= |
x1 +
|
| [q0(x1)] |
= |
x1 +
|
| [q1(x1)] |
= |
x1 +
|
| [q2(x1)] |
= |
x1 +
|
| [q3(x1)] |
= |
x1 +
|
| [q4(x1)] |
= |
x1 +
|
| [p0(x1)] |
= |
x1 +
|
| [p1(x1)] |
= |
x1 +
|
| [p2(x1)] |
= |
x1 +
|
| [p3(x1)] |
= |
x1 +
|
| [p4(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
|
b1(04(p1(02(x1)))) |
→ |
b3(q2(x1)) |
(37) |
|
b1(04(p1(03(x1)))) |
→ |
b3(q3(x1)) |
(38) |
|
b1(04(p1(04(x1)))) |
→ |
b3(q4(x1)) |
(39) |
|
b1(04(p1(01(x1)))) |
→ |
b3(q1(x1)) |
(40) |
|
b1(04(p1(00(x1)))) |
→ |
b3(q0(x1)) |
(41) |
|
q1(04(p1(02(x1)))) |
→ |
q3(q2(x1)) |
(42) |
|
q1(04(p1(03(x1)))) |
→ |
q3(q3(x1)) |
(43) |
|
q1(04(p1(04(x1)))) |
→ |
q3(q4(x1)) |
(44) |
|
q1(04(p1(01(x1)))) |
→ |
q3(q1(x1)) |
(45) |
|
q1(04(p1(00(x1)))) |
→ |
q3(q0(x1)) |
(46) |
|
p1(04(p1(02(x1)))) |
→ |
p3(q2(x1)) |
(47) |
|
p1(04(p1(03(x1)))) |
→ |
p3(q3(x1)) |
(48) |
|
p1(04(p1(04(x1)))) |
→ |
p3(q4(x1)) |
(49) |
|
p1(04(p1(01(x1)))) |
→ |
p3(q1(x1)) |
(50) |
|
p1(04(p1(00(x1)))) |
→ |
p3(q0(x1)) |
(51) |
|
01(04(p1(02(x1)))) |
→ |
03(q2(x1)) |
(52) |
|
01(04(p1(03(x1)))) |
→ |
03(q3(x1)) |
(53) |
|
01(04(p1(04(x1)))) |
→ |
03(q4(x1)) |
(54) |
|
01(04(p1(01(x1)))) |
→ |
03(q1(x1)) |
(55) |
|
01(04(p1(00(x1)))) |
→ |
03(q0(x1)) |
(56) |
|
11(04(p1(02(x1)))) |
→ |
13(q2(x1)) |
(57) |
|
11(04(p1(03(x1)))) |
→ |
13(q3(x1)) |
(58) |
|
11(04(p1(04(x1)))) |
→ |
13(q4(x1)) |
(59) |
|
11(04(p1(01(x1)))) |
→ |
13(q1(x1)) |
(60) |
|
11(04(p1(00(x1)))) |
→ |
13(q0(x1)) |
(61) |
|
b0(14(p0(13(x1)))) |
→ |
b3(q3(x1)) |
(88) |
|
q0(14(p0(12(x1)))) |
→ |
q3(q2(x1)) |
(92) |
|
q0(14(p0(13(x1)))) |
→ |
q3(q3(x1)) |
(93) |
|
q0(14(p0(14(x1)))) |
→ |
q3(q4(x1)) |
(94) |
|
q0(14(p0(11(x1)))) |
→ |
q3(q1(x1)) |
(95) |
|
q0(14(p0(10(x1)))) |
→ |
q3(q0(x1)) |
(96) |
|
p0(14(p0(13(x1)))) |
→ |
p3(q3(x1)) |
(98) |
|
00(14(p0(13(x1)))) |
→ |
03(q3(x1)) |
(103) |
|
10(14(p0(13(x1)))) |
→ |
13(q3(x1)) |
(108) |
|
b1(03(q1(02(x1)))) |
→ |
b3(q2(x1)) |
(112) |
|
b1(03(q1(03(x1)))) |
→ |
b3(q3(x1)) |
(113) |
|
b1(03(q1(04(x1)))) |
→ |
b3(q4(x1)) |
(114) |
|
b1(03(q1(01(x1)))) |
→ |
b3(q1(x1)) |
(115) |
|
b1(03(q1(00(x1)))) |
→ |
b3(q0(x1)) |
(116) |
|
q1(03(q1(02(x1)))) |
→ |
q3(q2(x1)) |
(117) |
|
q1(03(q1(03(x1)))) |
→ |
q3(q3(x1)) |
(118) |
|
q1(03(q1(04(x1)))) |
→ |
q3(q4(x1)) |
(119) |
|
q1(03(q1(01(x1)))) |
→ |
q3(q1(x1)) |
(120) |
|
q1(03(q1(00(x1)))) |
→ |
q3(q0(x1)) |
(121) |
|
p1(03(q1(02(x1)))) |
→ |
p3(q2(x1)) |
(122) |
|
p1(03(q1(03(x1)))) |
→ |
p3(q3(x1)) |
(123) |
|
p1(03(q1(04(x1)))) |
→ |
p3(q4(x1)) |
(124) |
|
p1(03(q1(01(x1)))) |
→ |
p3(q1(x1)) |
(125) |
|
p1(03(q1(00(x1)))) |
→ |
p3(q0(x1)) |
(126) |
|
01(03(q1(02(x1)))) |
→ |
03(q2(x1)) |
(127) |
|
01(03(q1(03(x1)))) |
→ |
03(q3(x1)) |
(128) |
|
01(03(q1(04(x1)))) |
→ |
03(q4(x1)) |
(129) |
|
01(03(q1(01(x1)))) |
→ |
03(q1(x1)) |
(130) |
|
01(03(q1(00(x1)))) |
→ |
03(q0(x1)) |
(131) |
|
11(03(q1(02(x1)))) |
→ |
13(q2(x1)) |
(132) |
|
11(03(q1(03(x1)))) |
→ |
13(q3(x1)) |
(133) |
|
11(03(q1(04(x1)))) |
→ |
13(q4(x1)) |
(134) |
|
11(03(q1(01(x1)))) |
→ |
13(q1(x1)) |
(135) |
|
b0(13(q0(13(x1)))) |
→ |
b3(q3(x1)) |
(138) |
|
q0(13(q0(12(x1)))) |
→ |
q3(q2(x1)) |
(142) |
|
q0(13(q0(13(x1)))) |
→ |
q3(q3(x1)) |
(143) |
|
q0(13(q0(14(x1)))) |
→ |
q3(q4(x1)) |
(144) |
|
q0(13(q0(11(x1)))) |
→ |
q3(q1(x1)) |
(145) |
|
q0(13(q0(10(x1)))) |
→ |
q3(q0(x1)) |
(146) |
|
p0(13(q0(13(x1)))) |
→ |
p3(q3(x1)) |
(148) |
|
00(13(q0(13(x1)))) |
→ |
03(q3(x1)) |
(153) |
|
10(13(q0(13(x1)))) |
→ |
13(q3(x1)) |
(158) |
|
b4(p4(x1)) |
→ |
b0(14(p0(11(00(14(x1)))))) |
(164) |
|
b4(p1(x1)) |
→ |
b0(14(p0(11(00(11(x1)))))) |
(165) |
|
q4(p4(x1)) |
→ |
q0(14(p0(11(00(14(x1)))))) |
(169) |
|
q4(p1(x1)) |
→ |
q0(14(p0(11(00(11(x1)))))) |
(170) |
|
p4(p2(x1)) |
→ |
p0(14(p0(11(00(12(x1)))))) |
(172) |
|
p4(p3(x1)) |
→ |
p0(14(p0(11(00(13(x1)))))) |
(173) |
|
p4(p4(x1)) |
→ |
p0(14(p0(11(00(14(x1)))))) |
(174) |
|
p4(p1(x1)) |
→ |
p0(14(p0(11(00(11(x1)))))) |
(175) |
|
p4(p0(x1)) |
→ |
p0(14(p0(11(00(10(x1)))))) |
(176) |
|
04(p2(x1)) |
→ |
00(14(p0(11(00(12(x1)))))) |
(177) |
|
04(p3(x1)) |
→ |
00(14(p0(11(00(13(x1)))))) |
(178) |
|
04(p4(x1)) |
→ |
00(14(p0(11(00(14(x1)))))) |
(179) |
|
04(p1(x1)) |
→ |
00(14(p0(11(00(11(x1)))))) |
(180) |
|
04(p0(x1)) |
→ |
00(14(p0(11(00(10(x1)))))) |
(181) |
|
14(p4(x1)) |
→ |
10(14(p0(11(00(14(x1)))))) |
(184) |
|
14(p1(x1)) |
→ |
10(14(p0(11(00(11(x1)))))) |
(185) |
1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.
1.2 Rule Removal
Using the
matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals
| [1(x1)] |
= |
· x1
|
| [0(x1)] |
= |
· x1
|
| [b(x1)] |
= |
· x1
|
| [q(x1)] |
= |
· x1
|
| [p(x1)] |
= |
· x1
|
all of the following rules can be deleted.
1.2.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [1(x1)] |
= |
x1 +
|
| [0(x1)] |
= |
x1 +
|
| [b(x1)] |
= |
x1 +
|
| [q(x1)] |
= |
x1 +
|
| [p(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
|
b(q(b(x1))) |
→ |
b(p(b(x1))) |
(1) |
1.2.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.