Certification Problem
Input (TPDB SRS_Relative/Zantema_06_relative/rel05)
The relative rewrite relation R/S is considered where R is the following TRS
|
b(c(a(x1))) |
→ |
d(d(x1)) |
(1) |
|
b(x1) |
→ |
c(c(x1)) |
(2) |
|
a(a(x1)) |
→ |
a(x1) |
(3) |
and S is the following TRS.
|
a(b(x1)) |
→ |
d(x1) |
(4) |
|
d(x1) |
→ |
a(b(x1)) |
(5) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by matchbox @ termCOMP 2023)
1 Closure Under Flat Contexts
Using the flat contexts
{d(☐), c(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
d(b(c(a(x1)))) |
→ |
d(d(d(x1))) |
(6) |
|
d(b(x1)) |
→ |
d(c(c(x1))) |
(7) |
|
d(a(a(x1))) |
→ |
d(a(x1)) |
(8) |
|
c(b(c(a(x1)))) |
→ |
c(d(d(x1))) |
(9) |
|
c(b(x1)) |
→ |
c(c(c(x1))) |
(10) |
|
c(a(a(x1))) |
→ |
c(a(x1)) |
(11) |
|
b(b(c(a(x1)))) |
→ |
b(d(d(x1))) |
(12) |
|
b(b(x1)) |
→ |
b(c(c(x1))) |
(13) |
|
b(a(a(x1))) |
→ |
b(a(x1)) |
(14) |
|
a(b(c(a(x1)))) |
→ |
a(d(d(x1))) |
(15) |
|
a(b(x1)) |
→ |
a(c(c(x1))) |
(16) |
|
a(a(a(x1))) |
→ |
a(a(x1)) |
(17) |
|
d(a(b(x1))) |
→ |
d(d(x1)) |
(18) |
|
d(d(x1)) |
→ |
d(a(b(x1))) |
(19) |
|
c(a(b(x1))) |
→ |
c(d(x1)) |
(20) |
|
c(d(x1)) |
→ |
c(a(b(x1))) |
(21) |
|
b(a(b(x1))) |
→ |
b(d(x1)) |
(22) |
|
b(d(x1)) |
→ |
b(a(b(x1))) |
(23) |
|
a(a(b(x1))) |
→ |
a(d(x1)) |
(24) |
|
a(d(x1)) |
→ |
a(a(b(x1))) |
(25) |
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
|
b2(b1(c3(a2(x1)))) |
→ |
b0(d0(d2(x1))) |
(26) |
|
b2(b1(c3(a1(x1)))) |
→ |
b0(d0(d1(x1))) |
(27) |
|
b2(b1(c3(a3(x1)))) |
→ |
b0(d0(d3(x1))) |
(28) |
|
b2(b1(c3(a0(x1)))) |
→ |
b0(d0(d0(x1))) |
(29) |
|
c2(b1(c3(a2(x1)))) |
→ |
c0(d0(d2(x1))) |
(30) |
|
c2(b1(c3(a1(x1)))) |
→ |
c0(d0(d1(x1))) |
(31) |
|
c2(b1(c3(a3(x1)))) |
→ |
c0(d0(d3(x1))) |
(32) |
|
c2(b1(c3(a0(x1)))) |
→ |
c0(d0(d0(x1))) |
(33) |
|
a2(b1(c3(a2(x1)))) |
→ |
a0(d0(d2(x1))) |
(34) |
|
a2(b1(c3(a1(x1)))) |
→ |
a0(d0(d1(x1))) |
(35) |
|
a2(b1(c3(a3(x1)))) |
→ |
a0(d0(d3(x1))) |
(36) |
|
a2(b1(c3(a0(x1)))) |
→ |
a0(d0(d0(x1))) |
(37) |
|
d2(b1(c3(a2(x1)))) |
→ |
d0(d0(d2(x1))) |
(38) |
|
d2(b1(c3(a1(x1)))) |
→ |
d0(d0(d1(x1))) |
(39) |
|
d2(b1(c3(a3(x1)))) |
→ |
d0(d0(d3(x1))) |
(40) |
|
d2(b1(c3(a0(x1)))) |
→ |
d0(d0(d0(x1))) |
(41) |
|
b2(b2(x1)) |
→ |
b1(c1(c2(x1))) |
(42) |
|
b2(b1(x1)) |
→ |
b1(c1(c1(x1))) |
(43) |
|
b2(b3(x1)) |
→ |
b1(c1(c3(x1))) |
(44) |
|
b2(b0(x1)) |
→ |
b1(c1(c0(x1))) |
(45) |
|
c2(b2(x1)) |
→ |
c1(c1(c2(x1))) |
(46) |
|
c2(b1(x1)) |
→ |
c1(c1(c1(x1))) |
(47) |
|
c2(b3(x1)) |
→ |
c1(c1(c3(x1))) |
(48) |
|
c2(b0(x1)) |
→ |
c1(c1(c0(x1))) |
(49) |
|
a2(b2(x1)) |
→ |
a1(c1(c2(x1))) |
(50) |
|
a2(b1(x1)) |
→ |
a1(c1(c1(x1))) |
(51) |
|
a2(b3(x1)) |
→ |
a1(c1(c3(x1))) |
(52) |
|
a2(b0(x1)) |
→ |
a1(c1(c0(x1))) |
(53) |
|
d2(b2(x1)) |
→ |
d1(c1(c2(x1))) |
(54) |
|
d2(b1(x1)) |
→ |
d1(c1(c1(x1))) |
(55) |
|
d2(b3(x1)) |
→ |
d1(c1(c3(x1))) |
(56) |
|
d2(b0(x1)) |
→ |
d1(c1(c0(x1))) |
(57) |
|
b3(a3(a2(x1))) |
→ |
b3(a2(x1)) |
(58) |
|
b3(a3(a1(x1))) |
→ |
b3(a1(x1)) |
(59) |
|
b3(a3(a3(x1))) |
→ |
b3(a3(x1)) |
(60) |
|
b3(a3(a0(x1))) |
→ |
b3(a0(x1)) |
(61) |
|
c3(a3(a2(x1))) |
→ |
c3(a2(x1)) |
(62) |
|
c3(a3(a1(x1))) |
→ |
c3(a1(x1)) |
(63) |
|
c3(a3(a3(x1))) |
→ |
c3(a3(x1)) |
(64) |
|
c3(a3(a0(x1))) |
→ |
c3(a0(x1)) |
(65) |
|
a3(a3(a2(x1))) |
→ |
a3(a2(x1)) |
(66) |
|
a3(a3(a1(x1))) |
→ |
a3(a1(x1)) |
(67) |
|
a3(a3(a3(x1))) |
→ |
a3(a3(x1)) |
(68) |
|
a3(a3(a0(x1))) |
→ |
a3(a0(x1)) |
(69) |
|
d3(a3(a2(x1))) |
→ |
d3(a2(x1)) |
(70) |
|
d3(a3(a1(x1))) |
→ |
d3(a1(x1)) |
(71) |
|
d3(a3(a3(x1))) |
→ |
d3(a3(x1)) |
(72) |
|
d3(a3(a0(x1))) |
→ |
d3(a0(x1)) |
(73) |
|
b3(a2(b2(x1))) |
→ |
b0(d2(x1)) |
(74) |
|
b3(a2(b1(x1))) |
→ |
b0(d1(x1)) |
(75) |
|
b3(a2(b3(x1))) |
→ |
b0(d3(x1)) |
(76) |
|
b3(a2(b0(x1))) |
→ |
b0(d0(x1)) |
(77) |
|
c3(a2(b2(x1))) |
→ |
c0(d2(x1)) |
(78) |
|
c3(a2(b1(x1))) |
→ |
c0(d1(x1)) |
(79) |
|
c3(a2(b3(x1))) |
→ |
c0(d3(x1)) |
(80) |
|
c3(a2(b0(x1))) |
→ |
c0(d0(x1)) |
(81) |
|
a3(a2(b2(x1))) |
→ |
a0(d2(x1)) |
(82) |
|
a3(a2(b1(x1))) |
→ |
a0(d1(x1)) |
(83) |
|
a3(a2(b3(x1))) |
→ |
a0(d3(x1)) |
(84) |
|
a3(a2(b0(x1))) |
→ |
a0(d0(x1)) |
(85) |
|
d3(a2(b2(x1))) |
→ |
d0(d2(x1)) |
(86) |
|
d3(a2(b1(x1))) |
→ |
d0(d1(x1)) |
(87) |
|
d3(a2(b3(x1))) |
→ |
d0(d3(x1)) |
(88) |
|
d3(a2(b0(x1))) |
→ |
d0(d0(x1)) |
(89) |
|
b0(d2(x1)) |
→ |
b3(a2(b2(x1))) |
(90) |
|
b0(d1(x1)) |
→ |
b3(a2(b1(x1))) |
(91) |
|
b0(d3(x1)) |
→ |
b3(a2(b3(x1))) |
(92) |
|
b0(d0(x1)) |
→ |
b3(a2(b0(x1))) |
(93) |
|
c0(d2(x1)) |
→ |
c3(a2(b2(x1))) |
(94) |
|
c0(d1(x1)) |
→ |
c3(a2(b1(x1))) |
(95) |
|
c0(d3(x1)) |
→ |
c3(a2(b3(x1))) |
(96) |
|
c0(d0(x1)) |
→ |
c3(a2(b0(x1))) |
(97) |
|
a0(d2(x1)) |
→ |
a3(a2(b2(x1))) |
(98) |
|
a0(d1(x1)) |
→ |
a3(a2(b1(x1))) |
(99) |
|
a0(d3(x1)) |
→ |
a3(a2(b3(x1))) |
(100) |
|
a0(d0(x1)) |
→ |
a3(a2(b0(x1))) |
(101) |
|
d0(d2(x1)) |
→ |
d3(a2(b2(x1))) |
(102) |
|
d0(d1(x1)) |
→ |
d3(a2(b1(x1))) |
(103) |
|
d0(d3(x1)) |
→ |
d3(a2(b3(x1))) |
(104) |
|
d0(d0(x1)) |
→ |
d3(a2(b0(x1))) |
(105) |
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.
|
b2(b1(c3(a2(x1)))) |
→ |
b0(d0(d2(x1))) |
(26) |
|
b2(b1(c3(a1(x1)))) |
→ |
b0(d0(d1(x1))) |
(27) |
|
b2(b1(c3(a3(x1)))) |
→ |
b0(d0(d3(x1))) |
(28) |
|
b2(b1(c3(a0(x1)))) |
→ |
b0(d0(d0(x1))) |
(29) |
|
c2(b1(c3(a2(x1)))) |
→ |
c0(d0(d2(x1))) |
(30) |
|
c2(b1(c3(a1(x1)))) |
→ |
c0(d0(d1(x1))) |
(31) |
|
c2(b1(c3(a3(x1)))) |
→ |
c0(d0(d3(x1))) |
(32) |
|
c2(b1(c3(a0(x1)))) |
→ |
c0(d0(d0(x1))) |
(33) |
|
a2(b1(c3(a3(x1)))) |
→ |
a0(d0(d3(x1))) |
(36) |
|
a2(b1(c3(a0(x1)))) |
→ |
a0(d0(d0(x1))) |
(37) |
|
d2(b1(c3(a2(x1)))) |
→ |
d0(d0(d2(x1))) |
(38) |
|
d2(b1(c3(a1(x1)))) |
→ |
d0(d0(d1(x1))) |
(39) |
|
d2(b1(c3(a3(x1)))) |
→ |
d0(d0(d3(x1))) |
(40) |
|
d2(b1(c3(a0(x1)))) |
→ |
d0(d0(d0(x1))) |
(41) |
|
b2(b2(x1)) |
→ |
b1(c1(c2(x1))) |
(42) |
|
b2(b1(x1)) |
→ |
b1(c1(c1(x1))) |
(43) |
|
c2(b2(x1)) |
→ |
c1(c1(c2(x1))) |
(46) |
|
c2(b1(x1)) |
→ |
c1(c1(c1(x1))) |
(47) |
|
c2(b3(x1)) |
→ |
c1(c1(c3(x1))) |
(48) |
|
c2(b0(x1)) |
→ |
c1(c1(c0(x1))) |
(49) |
|
a2(b2(x1)) |
→ |
a1(c1(c2(x1))) |
(50) |
|
a2(b1(x1)) |
→ |
a1(c1(c1(x1))) |
(51) |
|
d2(b2(x1)) |
→ |
d1(c1(c2(x1))) |
(54) |
|
d2(b1(x1)) |
→ |
d1(c1(c1(x1))) |
(55) |
1.1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the naturals
| [d0(x1)] |
= |
· x1 +
|
| [d1(x1)] |
= |
· x1 +
|
| [d2(x1)] |
= |
· x1 +
|
| [d3(x1)] |
= |
· x1 +
|
| [c0(x1)] |
= |
· x1 +
|
| [c1(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.
|
a2(b1(c3(a2(x1)))) |
→ |
a0(d0(d2(x1))) |
(34) |
|
a2(b1(c3(a1(x1)))) |
→ |
a0(d0(d1(x1))) |
(35) |
1.1.1.1.1 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 +
|
| [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.
|
b2(b3(x1)) |
→ |
b1(c1(c3(x1))) |
(44) |
|
b2(b0(x1)) |
→ |
b1(c1(c0(x1))) |
(45) |
|
a2(b3(x1)) |
→ |
a1(c1(c3(x1))) |
(52) |
|
a2(b0(x1)) |
→ |
a1(c1(c0(x1))) |
(53) |
|
d2(b3(x1)) |
→ |
d1(c1(c3(x1))) |
(56) |
|
d2(b0(x1)) |
→ |
d1(c1(c0(x1))) |
(57) |
|
b3(a3(a2(x1))) |
→ |
b3(a2(x1)) |
(58) |
|
b3(a3(a1(x1))) |
→ |
b3(a1(x1)) |
(59) |
|
b3(a3(a3(x1))) |
→ |
b3(a3(x1)) |
(60) |
|
b3(a3(a0(x1))) |
→ |
b3(a0(x1)) |
(61) |
|
c3(a3(a2(x1))) |
→ |
c3(a2(x1)) |
(62) |
|
c3(a3(a1(x1))) |
→ |
c3(a1(x1)) |
(63) |
|
c3(a3(a3(x1))) |
→ |
c3(a3(x1)) |
(64) |
|
c3(a3(a0(x1))) |
→ |
c3(a0(x1)) |
(65) |
|
a3(a3(a2(x1))) |
→ |
a3(a2(x1)) |
(66) |
|
a3(a3(a1(x1))) |
→ |
a3(a1(x1)) |
(67) |
|
a3(a3(a3(x1))) |
→ |
a3(a3(x1)) |
(68) |
|
a3(a3(a0(x1))) |
→ |
a3(a0(x1)) |
(69) |
|
d3(a3(a2(x1))) |
→ |
d3(a2(x1)) |
(70) |
|
d3(a3(a1(x1))) |
→ |
d3(a1(x1)) |
(71) |
|
d3(a3(a3(x1))) |
→ |
d3(a3(x1)) |
(72) |
|
d3(a3(a0(x1))) |
→ |
d3(a0(x1)) |
(73) |
1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.