Certification Problem
Input (TPDB SRS_Relative/Mixed_relative_SRS/zr09)
The relative rewrite relation R/S is considered where R is the following TRS
|
a(b(x1)) |
→ |
b(a(x1)) |
(1) |
|
d(c(x1)) |
→ |
d(a(x1)) |
(2) |
and S is the following TRS.
|
a(x1) |
→ |
b(c(x1)) |
(3) |
|
b(c(x1)) |
→ |
a(c(x1)) |
(4) |
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
{d(☐), c(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
d(d(c(x1))) |
→ |
d(d(a(x1))) |
(5) |
|
c(d(c(x1))) |
→ |
c(d(a(x1))) |
(6) |
|
b(d(c(x1))) |
→ |
b(d(a(x1))) |
(7) |
|
a(d(c(x1))) |
→ |
a(d(a(x1))) |
(8) |
|
d(a(b(x1))) |
→ |
d(b(a(x1))) |
(9) |
|
d(a(x1)) |
→ |
d(b(c(x1))) |
(10) |
|
d(b(c(x1))) |
→ |
d(a(c(x1))) |
(11) |
|
c(a(b(x1))) |
→ |
c(b(a(x1))) |
(12) |
|
c(a(x1)) |
→ |
c(b(c(x1))) |
(13) |
|
c(b(c(x1))) |
→ |
c(a(c(x1))) |
(14) |
|
b(a(b(x1))) |
→ |
b(b(a(x1))) |
(15) |
|
b(a(x1)) |
→ |
b(b(c(x1))) |
(16) |
|
b(b(c(x1))) |
→ |
b(a(c(x1))) |
(17) |
|
a(a(b(x1))) |
→ |
a(b(a(x1))) |
(18) |
|
a(a(x1)) |
→ |
a(b(c(x1))) |
(19) |
|
a(b(c(x1))) |
→ |
a(a(c(x1))) |
(20) |
1.1.1 Semantic Labeling
The following interpretations form a
model
of the rules.
As carrier we take the set
{0,...,3}.
Symbols are labeled by the interpretation of their arguments using the interpretations
(modulo 4):
| [d(x1)] |
= |
4x1 + 0 |
| [c(x1)] |
= |
4x1 + 1 |
| [b(x1)] |
= |
4x1 + 2 |
| [a(x1)] |
= |
4x1 + 3 |
We obtain the labeled TRS
|
a0(d1(c3(x1))) |
→ |
a0(d3(a3(x1))) |
(21) |
|
a0(d1(c2(x1))) |
→ |
a0(d3(a2(x1))) |
(22) |
|
a0(d1(c0(x1))) |
→ |
a0(d3(a0(x1))) |
(23) |
|
a0(d1(c1(x1))) |
→ |
a0(d3(a1(x1))) |
(24) |
|
b0(d1(c3(x1))) |
→ |
b0(d3(a3(x1))) |
(25) |
|
b0(d1(c2(x1))) |
→ |
b0(d3(a2(x1))) |
(26) |
|
b0(d1(c0(x1))) |
→ |
b0(d3(a0(x1))) |
(27) |
|
b0(d1(c1(x1))) |
→ |
b0(d3(a1(x1))) |
(28) |
|
d0(d1(c3(x1))) |
→ |
d0(d3(a3(x1))) |
(29) |
|
d0(d1(c2(x1))) |
→ |
d0(d3(a2(x1))) |
(30) |
|
d0(d1(c0(x1))) |
→ |
d0(d3(a0(x1))) |
(31) |
|
d0(d1(c1(x1))) |
→ |
d0(d3(a1(x1))) |
(32) |
|
c0(d1(c3(x1))) |
→ |
c0(d3(a3(x1))) |
(33) |
|
c0(d1(c2(x1))) |
→ |
c0(d3(a2(x1))) |
(34) |
|
c0(d1(c0(x1))) |
→ |
c0(d3(a0(x1))) |
(35) |
|
c0(d1(c1(x1))) |
→ |
c0(d3(a1(x1))) |
(36) |
|
a3(a2(b3(x1))) |
→ |
a2(b3(a3(x1))) |
(37) |
|
a3(a2(b2(x1))) |
→ |
a2(b3(a2(x1))) |
(38) |
|
a3(a2(b0(x1))) |
→ |
a2(b3(a0(x1))) |
(39) |
|
a3(a2(b1(x1))) |
→ |
a2(b3(a1(x1))) |
(40) |
|
b3(a2(b3(x1))) |
→ |
b2(b3(a3(x1))) |
(41) |
|
b3(a2(b2(x1))) |
→ |
b2(b3(a2(x1))) |
(42) |
|
b3(a2(b0(x1))) |
→ |
b2(b3(a0(x1))) |
(43) |
|
b3(a2(b1(x1))) |
→ |
b2(b3(a1(x1))) |
(44) |
|
d3(a2(b3(x1))) |
→ |
d2(b3(a3(x1))) |
(45) |
|
d3(a2(b2(x1))) |
→ |
d2(b3(a2(x1))) |
(46) |
|
d3(a2(b0(x1))) |
→ |
d2(b3(a0(x1))) |
(47) |
|
d3(a2(b1(x1))) |
→ |
d2(b3(a1(x1))) |
(48) |
|
c3(a2(b3(x1))) |
→ |
c2(b3(a3(x1))) |
(49) |
|
c3(a2(b2(x1))) |
→ |
c2(b3(a2(x1))) |
(50) |
|
c3(a2(b0(x1))) |
→ |
c2(b3(a0(x1))) |
(51) |
|
c3(a2(b1(x1))) |
→ |
c2(b3(a1(x1))) |
(52) |
|
a3(a3(x1)) |
→ |
a2(b1(c3(x1))) |
(53) |
|
a3(a2(x1)) |
→ |
a2(b1(c2(x1))) |
(54) |
|
a3(a0(x1)) |
→ |
a2(b1(c0(x1))) |
(55) |
|
a3(a1(x1)) |
→ |
a2(b1(c1(x1))) |
(56) |
|
b3(a3(x1)) |
→ |
b2(b1(c3(x1))) |
(57) |
|
b3(a2(x1)) |
→ |
b2(b1(c2(x1))) |
(58) |
|
b3(a0(x1)) |
→ |
b2(b1(c0(x1))) |
(59) |
|
b3(a1(x1)) |
→ |
b2(b1(c1(x1))) |
(60) |
|
d3(a3(x1)) |
→ |
d2(b1(c3(x1))) |
(61) |
|
d3(a2(x1)) |
→ |
d2(b1(c2(x1))) |
(62) |
|
d3(a0(x1)) |
→ |
d2(b1(c0(x1))) |
(63) |
|
d3(a1(x1)) |
→ |
d2(b1(c1(x1))) |
(64) |
|
c3(a3(x1)) |
→ |
c2(b1(c3(x1))) |
(65) |
|
c3(a2(x1)) |
→ |
c2(b1(c2(x1))) |
(66) |
|
c3(a0(x1)) |
→ |
c2(b1(c0(x1))) |
(67) |
|
c3(a1(x1)) |
→ |
c2(b1(c1(x1))) |
(68) |
|
a2(b1(c3(x1))) |
→ |
a3(a1(c3(x1))) |
(69) |
|
a2(b1(c2(x1))) |
→ |
a3(a1(c2(x1))) |
(70) |
|
a2(b1(c0(x1))) |
→ |
a3(a1(c0(x1))) |
(71) |
|
a2(b1(c1(x1))) |
→ |
a3(a1(c1(x1))) |
(72) |
|
b2(b1(c3(x1))) |
→ |
b3(a1(c3(x1))) |
(73) |
|
b2(b1(c2(x1))) |
→ |
b3(a1(c2(x1))) |
(74) |
|
b2(b1(c0(x1))) |
→ |
b3(a1(c0(x1))) |
(75) |
|
b2(b1(c1(x1))) |
→ |
b3(a1(c1(x1))) |
(76) |
|
d2(b1(c3(x1))) |
→ |
d3(a1(c3(x1))) |
(77) |
|
d2(b1(c2(x1))) |
→ |
d3(a1(c2(x1))) |
(78) |
|
d2(b1(c0(x1))) |
→ |
d3(a1(c0(x1))) |
(79) |
|
d2(b1(c1(x1))) |
→ |
d3(a1(c1(x1))) |
(80) |
|
c2(b1(c3(x1))) |
→ |
c3(a1(c3(x1))) |
(81) |
|
c2(b1(c2(x1))) |
→ |
c3(a1(c2(x1))) |
(82) |
|
c2(b1(c0(x1))) |
→ |
c3(a1(c0(x1))) |
(83) |
|
c2(b1(c1(x1))) |
→ |
c3(a1(c1(x1))) |
(84) |
1.1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [d0(x1)] |
= |
x1 +
|
| [d1(x1)] |
= |
x1 +
|
| [d2(x1)] |
= |
x1 +
|
| [d3(x1)] |
= |
x1 +
|
| [c0(x1)] |
= |
x1 +
|
| [c1(x1)] |
= |
x1 +
|
| [c2(x1)] |
= |
x1 +
|
| [c3(x1)] |
= |
x1 +
|
| [b0(x1)] |
= |
x1 +
|
| [b1(x1)] |
= |
x1 +
|
| [b2(x1)] |
= |
x1 +
|
| [b3(x1)] |
= |
x1 +
|
| [a0(x1)] |
= |
x1 +
|
| [a1(x1)] |
= |
x1 +
|
| [a2(x1)] |
= |
x1 +
|
| [a3(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
|
a0(d1(c3(x1))) |
→ |
a0(d3(a3(x1))) |
(21) |
|
a0(d1(c2(x1))) |
→ |
a0(d3(a2(x1))) |
(22) |
|
a0(d1(c0(x1))) |
→ |
a0(d3(a0(x1))) |
(23) |
|
a0(d1(c1(x1))) |
→ |
a0(d3(a1(x1))) |
(24) |
|
b0(d1(c3(x1))) |
→ |
b0(d3(a3(x1))) |
(25) |
|
b0(d1(c2(x1))) |
→ |
b0(d3(a2(x1))) |
(26) |
|
b0(d1(c0(x1))) |
→ |
b0(d3(a0(x1))) |
(27) |
|
b0(d1(c1(x1))) |
→ |
b0(d3(a1(x1))) |
(28) |
|
d0(d1(c3(x1))) |
→ |
d0(d3(a3(x1))) |
(29) |
|
d0(d1(c2(x1))) |
→ |
d0(d3(a2(x1))) |
(30) |
|
d0(d1(c0(x1))) |
→ |
d0(d3(a0(x1))) |
(31) |
|
d0(d1(c1(x1))) |
→ |
d0(d3(a1(x1))) |
(32) |
|
c0(d1(c3(x1))) |
→ |
c0(d3(a3(x1))) |
(33) |
|
c0(d1(c2(x1))) |
→ |
c0(d3(a2(x1))) |
(34) |
|
c0(d1(c0(x1))) |
→ |
c0(d3(a0(x1))) |
(35) |
|
c0(d1(c1(x1))) |
→ |
c0(d3(a1(x1))) |
(36) |
|
a3(a2(b0(x1))) |
→ |
a2(b3(a0(x1))) |
(39) |
|
a3(a2(b1(x1))) |
→ |
a2(b3(a1(x1))) |
(40) |
|
b3(a2(b0(x1))) |
→ |
b2(b3(a0(x1))) |
(43) |
|
b3(a2(b1(x1))) |
→ |
b2(b3(a1(x1))) |
(44) |
|
d3(a2(b0(x1))) |
→ |
d2(b3(a0(x1))) |
(47) |
|
d3(a2(b1(x1))) |
→ |
d2(b3(a1(x1))) |
(48) |
|
c3(a2(b0(x1))) |
→ |
c2(b3(a0(x1))) |
(51) |
|
c3(a2(b1(x1))) |
→ |
c2(b3(a1(x1))) |
(52) |
|
a3(a3(x1)) |
→ |
a2(b1(c3(x1))) |
(53) |
|
a3(a2(x1)) |
→ |
a2(b1(c2(x1))) |
(54) |
|
a3(a0(x1)) |
→ |
a2(b1(c0(x1))) |
(55) |
|
b3(a3(x1)) |
→ |
b2(b1(c3(x1))) |
(57) |
|
b3(a2(x1)) |
→ |
b2(b1(c2(x1))) |
(58) |
|
b3(a0(x1)) |
→ |
b2(b1(c0(x1))) |
(59) |
|
d3(a3(x1)) |
→ |
d2(b1(c3(x1))) |
(61) |
|
d3(a2(x1)) |
→ |
d2(b1(c2(x1))) |
(62) |
|
d3(a0(x1)) |
→ |
d2(b1(c0(x1))) |
(63) |
|
c3(a3(x1)) |
→ |
c2(b1(c3(x1))) |
(65) |
|
c3(a2(x1)) |
→ |
c2(b1(c2(x1))) |
(66) |
|
c3(a0(x1)) |
→ |
c2(b1(c0(x1))) |
(67) |
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(a(b(x1))) |
→ |
c(b(a(x1))) |
(12) |
|
b(a(b(x1))) |
→ |
b(b(a(x1))) |
(15) |
|
a(a(b(x1))) |
→ |
a(b(a(x1))) |
(18) |
|
c(a(x1)) |
→ |
c(b(c(x1))) |
(13) |
|
c(b(c(x1))) |
→ |
c(a(c(x1))) |
(14) |
|
b(a(x1)) |
→ |
b(b(c(x1))) |
(16) |
|
b(b(c(x1))) |
→ |
b(a(c(x1))) |
(17) |
|
a(a(x1)) |
→ |
a(b(c(x1))) |
(19) |
|
a(b(c(x1))) |
→ |
a(a(c(x1))) |
(20) |
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
|
a2(a1(b2(x1))) |
→ |
a1(b2(a2(x1))) |
(85) |
|
a2(a1(b1(x1))) |
→ |
a1(b2(a1(x1))) |
(86) |
|
a2(a1(b0(x1))) |
→ |
a1(b2(a0(x1))) |
(87) |
|
b2(a1(b2(x1))) |
→ |
b1(b2(a2(x1))) |
(88) |
|
b2(a1(b1(x1))) |
→ |
b1(b2(a1(x1))) |
(89) |
|
b2(a1(b0(x1))) |
→ |
b1(b2(a0(x1))) |
(90) |
|
c2(a1(b2(x1))) |
→ |
c1(b2(a2(x1))) |
(91) |
|
c2(a1(b1(x1))) |
→ |
c1(b2(a1(x1))) |
(92) |
|
c2(a1(b0(x1))) |
→ |
c1(b2(a0(x1))) |
(93) |
|
a2(a2(x1)) |
→ |
a1(b0(c2(x1))) |
(94) |
|
a2(a1(x1)) |
→ |
a1(b0(c1(x1))) |
(95) |
|
a2(a0(x1)) |
→ |
a1(b0(c0(x1))) |
(96) |
|
b2(a2(x1)) |
→ |
b1(b0(c2(x1))) |
(97) |
|
b2(a1(x1)) |
→ |
b1(b0(c1(x1))) |
(98) |
|
b2(a0(x1)) |
→ |
b1(b0(c0(x1))) |
(99) |
|
c2(a2(x1)) |
→ |
c1(b0(c2(x1))) |
(100) |
|
c2(a1(x1)) |
→ |
c1(b0(c1(x1))) |
(101) |
|
c2(a0(x1)) |
→ |
c1(b0(c0(x1))) |
(102) |
|
a1(b0(c2(x1))) |
→ |
a2(a0(c2(x1))) |
(103) |
|
a1(b0(c1(x1))) |
→ |
a2(a0(c1(x1))) |
(104) |
|
a1(b0(c0(x1))) |
→ |
a2(a0(c0(x1))) |
(105) |
|
b1(b0(c2(x1))) |
→ |
b2(a0(c2(x1))) |
(106) |
|
b1(b0(c1(x1))) |
→ |
b2(a0(c1(x1))) |
(107) |
|
b1(b0(c0(x1))) |
→ |
b2(a0(c0(x1))) |
(108) |
|
c1(b0(c2(x1))) |
→ |
c2(a0(c2(x1))) |
(109) |
|
c1(b0(c1(x1))) |
→ |
c2(a0(c1(x1))) |
(110) |
|
c1(b0(c0(x1))) |
→ |
c2(a0(c0(x1))) |
(111) |
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.
|
a2(a1(b0(x1))) |
→ |
a1(b2(a0(x1))) |
(87) |
|
b2(a1(b0(x1))) |
→ |
b1(b2(a0(x1))) |
(90) |
|
c2(a1(b0(x1))) |
→ |
c1(b2(a0(x1))) |
(93) |
|
a2(a2(x1)) |
→ |
a1(b0(c2(x1))) |
(94) |
|
a2(a1(x1)) |
→ |
a1(b0(c1(x1))) |
(95) |
|
b2(a2(x1)) |
→ |
b1(b0(c2(x1))) |
(97) |
|
b2(a1(x1)) |
→ |
b1(b0(c1(x1))) |
(98) |
|
c2(a2(x1)) |
→ |
c1(b0(c2(x1))) |
(100) |
|
c2(a1(x1)) |
→ |
c1(b0(c1(x1))) |
(101) |
1.2.1.1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
b2(a1(a2(x1))) |
→ |
a2(b2(a1(x1))) |
(112) |
|
b1(a1(a2(x1))) |
→ |
a1(b2(a1(x1))) |
(113) |
|
b2(a1(b2(x1))) |
→ |
a2(b2(b1(x1))) |
(114) |
|
b1(a1(b2(x1))) |
→ |
a1(b2(b1(x1))) |
(115) |
|
b2(a1(c2(x1))) |
→ |
a2(b2(c1(x1))) |
(116) |
|
b1(a1(c2(x1))) |
→ |
a1(b2(c1(x1))) |
(117) |
|
a0(a2(x1)) |
→ |
c0(b0(a1(x1))) |
(118) |
|
a0(b2(x1)) |
→ |
c0(b0(b1(x1))) |
(119) |
|
a0(c2(x1)) |
→ |
c0(b0(c1(x1))) |
(120) |
|
c2(b0(a1(x1))) |
→ |
c2(a0(a2(x1))) |
(121) |
|
c1(b0(a1(x1))) |
→ |
c1(a0(a2(x1))) |
(122) |
|
c0(b0(a1(x1))) |
→ |
c0(a0(a2(x1))) |
(123) |
|
c2(b0(b1(x1))) |
→ |
c2(a0(b2(x1))) |
(124) |
|
c1(b0(b1(x1))) |
→ |
c1(a0(b2(x1))) |
(125) |
|
c0(b0(b1(x1))) |
→ |
c0(a0(b2(x1))) |
(126) |
|
c2(b0(c1(x1))) |
→ |
c2(a0(c2(x1))) |
(127) |
|
c1(b0(c1(x1))) |
→ |
c1(a0(c2(x1))) |
(128) |
|
c0(b0(c1(x1))) |
→ |
c0(a0(c2(x1))) |
(129) |
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 +
|
| [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.
|
b2(a1(a2(x1))) |
→ |
a2(b2(a1(x1))) |
(112) |
|
b1(a1(a2(x1))) |
→ |
a1(b2(a1(x1))) |
(113) |
|
b2(a1(b2(x1))) |
→ |
a2(b2(b1(x1))) |
(114) |
|
b1(a1(b2(x1))) |
→ |
a1(b2(b1(x1))) |
(115) |
|
b2(a1(c2(x1))) |
→ |
a2(b2(c1(x1))) |
(116) |
|
b1(a1(c2(x1))) |
→ |
a1(b2(c1(x1))) |
(117) |
1.2.1.1.1.1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
a2(a0(x1)) |
→ |
a1(b0(c0(x1))) |
(96) |
|
b2(a0(x1)) |
→ |
b1(b0(c0(x1))) |
(99) |
|
c2(a0(x1)) |
→ |
c1(b0(c0(x1))) |
(102) |
|
a1(b0(c2(x1))) |
→ |
a2(a0(c2(x1))) |
(103) |
|
a1(b0(c1(x1))) |
→ |
a2(a0(c1(x1))) |
(104) |
|
a1(b0(c0(x1))) |
→ |
a2(a0(c0(x1))) |
(105) |
|
b1(b0(c2(x1))) |
→ |
b2(a0(c2(x1))) |
(106) |
|
b1(b0(c1(x1))) |
→ |
b2(a0(c1(x1))) |
(107) |
|
b1(b0(c0(x1))) |
→ |
b2(a0(c0(x1))) |
(108) |
|
c1(b0(c2(x1))) |
→ |
c2(a0(c2(x1))) |
(109) |
|
c1(b0(c1(x1))) |
→ |
c2(a0(c1(x1))) |
(110) |
|
c1(b0(c0(x1))) |
→ |
c2(a0(c0(x1))) |
(111) |
1.2.1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.