Certification Problem
Input (TPDB SRS_Relative/Waldmann_19/random-186)
The rewrite relation of the following TRS is considered.
|
b(c(b(x1))) |
→ |
a(b(b(x1))) |
(1) |
|
a(c(c(x1))) |
→ |
c(a(a(x1))) |
(2) |
|
b(c(a(x1))) |
→ |
a(b(a(x1))) |
(3) |
|
b(c(b(x1))) |
→ |
b(c(c(x1))) |
(4) |
|
a(b(c(x1))) |
→ |
b(b(b(x1))) |
(5) |
|
c(a(c(x1))) |
→ |
c(b(a(x1))) |
(6) |
|
a(a(a(x1))) |
→ |
a(c(c(x1))) |
(7) |
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
|
b(c(b(x1))) |
→ |
a(b(b(x1))) |
(1) |
are deleted.
1.1 Closure Under Flat Contexts
Using the flat contexts
{c(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
c(b(c(b(x1)))) |
→ |
c(a(b(b(x1)))) |
(8) |
|
b(b(c(b(x1)))) |
→ |
b(a(b(b(x1)))) |
(9) |
|
a(b(c(b(x1)))) |
→ |
a(a(b(b(x1)))) |
(10) |
|
c(a(c(c(x1)))) |
→ |
c(c(a(a(x1)))) |
(11) |
|
c(b(c(a(x1)))) |
→ |
c(a(b(a(x1)))) |
(12) |
|
c(b(c(b(x1)))) |
→ |
c(b(c(c(x1)))) |
(13) |
|
c(a(b(c(x1)))) |
→ |
c(b(b(b(x1)))) |
(14) |
|
c(c(a(c(x1)))) |
→ |
c(c(b(a(x1)))) |
(15) |
|
c(a(a(a(x1)))) |
→ |
c(a(c(c(x1)))) |
(16) |
|
b(a(c(c(x1)))) |
→ |
b(c(a(a(x1)))) |
(17) |
|
b(b(c(a(x1)))) |
→ |
b(a(b(a(x1)))) |
(18) |
|
b(b(c(b(x1)))) |
→ |
b(b(c(c(x1)))) |
(19) |
|
b(a(b(c(x1)))) |
→ |
b(b(b(b(x1)))) |
(20) |
|
b(c(a(c(x1)))) |
→ |
b(c(b(a(x1)))) |
(21) |
|
b(a(a(a(x1)))) |
→ |
b(a(c(c(x1)))) |
(22) |
|
a(a(c(c(x1)))) |
→ |
a(c(a(a(x1)))) |
(23) |
|
a(b(c(a(x1)))) |
→ |
a(a(b(a(x1)))) |
(24) |
|
a(b(c(b(x1)))) |
→ |
a(b(c(c(x1)))) |
(25) |
|
a(a(b(c(x1)))) |
→ |
a(b(b(b(x1)))) |
(26) |
|
a(c(a(c(x1)))) |
→ |
a(c(b(a(x1)))) |
(27) |
|
a(a(a(a(x1)))) |
→ |
a(a(c(c(x1)))) |
(28) |
1.1.1 Closure Under Flat Contexts
Using the flat contexts
{c(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
c(c(b(c(b(x1))))) |
→ |
c(c(a(b(b(x1))))) |
(29) |
|
c(b(b(c(b(x1))))) |
→ |
c(b(a(b(b(x1))))) |
(30) |
|
c(a(b(c(b(x1))))) |
→ |
c(a(a(b(b(x1))))) |
(31) |
|
b(c(b(c(b(x1))))) |
→ |
b(c(a(b(b(x1))))) |
(32) |
|
b(b(b(c(b(x1))))) |
→ |
b(b(a(b(b(x1))))) |
(33) |
|
b(a(b(c(b(x1))))) |
→ |
b(a(a(b(b(x1))))) |
(34) |
|
a(c(b(c(b(x1))))) |
→ |
a(c(a(b(b(x1))))) |
(35) |
|
a(b(b(c(b(x1))))) |
→ |
a(b(a(b(b(x1))))) |
(36) |
|
a(a(b(c(b(x1))))) |
→ |
a(a(a(b(b(x1))))) |
(37) |
|
c(c(a(c(c(x1))))) |
→ |
c(c(c(a(a(x1))))) |
(38) |
|
c(c(b(c(a(x1))))) |
→ |
c(c(a(b(a(x1))))) |
(39) |
|
c(c(b(c(b(x1))))) |
→ |
c(c(b(c(c(x1))))) |
(40) |
|
c(c(a(b(c(x1))))) |
→ |
c(c(b(b(b(x1))))) |
(41) |
|
c(c(c(a(c(x1))))) |
→ |
c(c(c(b(a(x1))))) |
(42) |
|
c(c(a(a(a(x1))))) |
→ |
c(c(a(c(c(x1))))) |
(43) |
|
c(b(a(c(c(x1))))) |
→ |
c(b(c(a(a(x1))))) |
(44) |
|
c(b(b(c(a(x1))))) |
→ |
c(b(a(b(a(x1))))) |
(45) |
|
c(b(b(c(b(x1))))) |
→ |
c(b(b(c(c(x1))))) |
(46) |
|
c(b(a(b(c(x1))))) |
→ |
c(b(b(b(b(x1))))) |
(47) |
|
c(b(c(a(c(x1))))) |
→ |
c(b(c(b(a(x1))))) |
(48) |
|
c(b(a(a(a(x1))))) |
→ |
c(b(a(c(c(x1))))) |
(49) |
|
c(a(a(c(c(x1))))) |
→ |
c(a(c(a(a(x1))))) |
(50) |
|
c(a(b(c(a(x1))))) |
→ |
c(a(a(b(a(x1))))) |
(51) |
|
c(a(b(c(b(x1))))) |
→ |
c(a(b(c(c(x1))))) |
(52) |
|
c(a(a(b(c(x1))))) |
→ |
c(a(b(b(b(x1))))) |
(53) |
|
c(a(c(a(c(x1))))) |
→ |
c(a(c(b(a(x1))))) |
(54) |
|
c(a(a(a(a(x1))))) |
→ |
c(a(a(c(c(x1))))) |
(55) |
|
b(c(a(c(c(x1))))) |
→ |
b(c(c(a(a(x1))))) |
(56) |
|
b(c(b(c(a(x1))))) |
→ |
b(c(a(b(a(x1))))) |
(57) |
|
b(c(b(c(b(x1))))) |
→ |
b(c(b(c(c(x1))))) |
(58) |
|
b(c(a(b(c(x1))))) |
→ |
b(c(b(b(b(x1))))) |
(59) |
|
b(c(c(a(c(x1))))) |
→ |
b(c(c(b(a(x1))))) |
(60) |
|
b(c(a(a(a(x1))))) |
→ |
b(c(a(c(c(x1))))) |
(61) |
|
b(b(a(c(c(x1))))) |
→ |
b(b(c(a(a(x1))))) |
(62) |
|
b(b(b(c(a(x1))))) |
→ |
b(b(a(b(a(x1))))) |
(63) |
|
b(b(b(c(b(x1))))) |
→ |
b(b(b(c(c(x1))))) |
(64) |
|
b(b(a(b(c(x1))))) |
→ |
b(b(b(b(b(x1))))) |
(65) |
|
b(b(c(a(c(x1))))) |
→ |
b(b(c(b(a(x1))))) |
(66) |
|
b(b(a(a(a(x1))))) |
→ |
b(b(a(c(c(x1))))) |
(67) |
|
b(a(a(c(c(x1))))) |
→ |
b(a(c(a(a(x1))))) |
(68) |
|
b(a(b(c(a(x1))))) |
→ |
b(a(a(b(a(x1))))) |
(69) |
|
b(a(b(c(b(x1))))) |
→ |
b(a(b(c(c(x1))))) |
(70) |
|
b(a(a(b(c(x1))))) |
→ |
b(a(b(b(b(x1))))) |
(71) |
|
b(a(c(a(c(x1))))) |
→ |
b(a(c(b(a(x1))))) |
(72) |
|
b(a(a(a(a(x1))))) |
→ |
b(a(a(c(c(x1))))) |
(73) |
|
a(c(a(c(c(x1))))) |
→ |
a(c(c(a(a(x1))))) |
(74) |
|
a(c(b(c(a(x1))))) |
→ |
a(c(a(b(a(x1))))) |
(75) |
|
a(c(b(c(b(x1))))) |
→ |
a(c(b(c(c(x1))))) |
(76) |
|
a(c(a(b(c(x1))))) |
→ |
a(c(b(b(b(x1))))) |
(77) |
|
a(c(c(a(c(x1))))) |
→ |
a(c(c(b(a(x1))))) |
(78) |
|
a(c(a(a(a(x1))))) |
→ |
a(c(a(c(c(x1))))) |
(79) |
|
a(b(a(c(c(x1))))) |
→ |
a(b(c(a(a(x1))))) |
(80) |
|
a(b(b(c(a(x1))))) |
→ |
a(b(a(b(a(x1))))) |
(81) |
|
a(b(b(c(b(x1))))) |
→ |
a(b(b(c(c(x1))))) |
(82) |
|
a(b(a(b(c(x1))))) |
→ |
a(b(b(b(b(x1))))) |
(83) |
|
a(b(c(a(c(x1))))) |
→ |
a(b(c(b(a(x1))))) |
(84) |
|
a(b(a(a(a(x1))))) |
→ |
a(b(a(c(c(x1))))) |
(85) |
|
a(a(a(c(c(x1))))) |
→ |
a(a(c(a(a(x1))))) |
(86) |
|
a(a(b(c(a(x1))))) |
→ |
a(a(a(b(a(x1))))) |
(87) |
|
a(a(b(c(b(x1))))) |
→ |
a(a(b(c(c(x1))))) |
(88) |
|
a(a(a(b(c(x1))))) |
→ |
a(a(b(b(b(x1))))) |
(89) |
|
a(a(c(a(c(x1))))) |
→ |
a(a(c(b(a(x1))))) |
(90) |
|
a(a(a(a(a(x1))))) |
→ |
a(a(a(c(c(x1))))) |
(91) |
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 567 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 233 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 Split
We split R in the relative problem D/R-D and R-D, where the rules D
|
b(c(a(x1))) |
→ |
a(b(a(x1))) |
(3) |
|
b(c(b(x1))) |
→ |
b(c(c(x1))) |
(4) |
|
a(b(c(x1))) |
→ |
b(b(b(x1))) |
(5) |
|
c(a(c(x1))) |
→ |
c(b(a(x1))) |
(6) |
are deleted.
1.2.1 Closure Under Flat Contexts
Using the flat contexts
{c(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
c(b(c(a(x1)))) |
→ |
c(a(b(a(x1)))) |
(12) |
|
c(b(c(b(x1)))) |
→ |
c(b(c(c(x1)))) |
(13) |
|
c(a(b(c(x1)))) |
→ |
c(b(b(b(x1)))) |
(14) |
|
c(c(a(c(x1)))) |
→ |
c(c(b(a(x1)))) |
(15) |
|
b(b(c(a(x1)))) |
→ |
b(a(b(a(x1)))) |
(18) |
|
b(b(c(b(x1)))) |
→ |
b(b(c(c(x1)))) |
(19) |
|
b(a(b(c(x1)))) |
→ |
b(b(b(b(x1)))) |
(20) |
|
b(c(a(c(x1)))) |
→ |
b(c(b(a(x1)))) |
(21) |
|
a(b(c(a(x1)))) |
→ |
a(a(b(a(x1)))) |
(24) |
|
a(b(c(b(x1)))) |
→ |
a(b(c(c(x1)))) |
(25) |
|
a(a(b(c(x1)))) |
→ |
a(b(b(b(x1)))) |
(26) |
|
a(c(a(c(x1)))) |
→ |
a(c(b(a(x1)))) |
(27) |
|
c(a(c(c(x1)))) |
→ |
c(c(a(a(x1)))) |
(11) |
|
c(a(a(a(x1)))) |
→ |
c(a(c(c(x1)))) |
(16) |
|
b(a(c(c(x1)))) |
→ |
b(c(a(a(x1)))) |
(17) |
|
b(a(a(a(x1)))) |
→ |
b(a(c(c(x1)))) |
(22) |
|
a(a(c(c(x1)))) |
→ |
a(c(a(a(x1)))) |
(23) |
|
a(a(a(a(x1)))) |
→ |
a(a(c(c(x1)))) |
(28) |
1.2.1.1 Closure Under Flat Contexts
Using the flat contexts
{c(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
c(c(b(c(a(x1))))) |
→ |
c(c(a(b(a(x1))))) |
(39) |
|
c(c(b(c(b(x1))))) |
→ |
c(c(b(c(c(x1))))) |
(40) |
|
c(c(a(b(c(x1))))) |
→ |
c(c(b(b(b(x1))))) |
(41) |
|
c(c(c(a(c(x1))))) |
→ |
c(c(c(b(a(x1))))) |
(42) |
|
c(b(b(c(a(x1))))) |
→ |
c(b(a(b(a(x1))))) |
(45) |
|
c(b(b(c(b(x1))))) |
→ |
c(b(b(c(c(x1))))) |
(46) |
|
c(b(a(b(c(x1))))) |
→ |
c(b(b(b(b(x1))))) |
(47) |
|
c(b(c(a(c(x1))))) |
→ |
c(b(c(b(a(x1))))) |
(48) |
|
c(a(b(c(a(x1))))) |
→ |
c(a(a(b(a(x1))))) |
(51) |
|
c(a(b(c(b(x1))))) |
→ |
c(a(b(c(c(x1))))) |
(52) |
|
c(a(a(b(c(x1))))) |
→ |
c(a(b(b(b(x1))))) |
(53) |
|
c(a(c(a(c(x1))))) |
→ |
c(a(c(b(a(x1))))) |
(54) |
|
b(c(b(c(a(x1))))) |
→ |
b(c(a(b(a(x1))))) |
(57) |
|
b(c(b(c(b(x1))))) |
→ |
b(c(b(c(c(x1))))) |
(58) |
|
b(c(a(b(c(x1))))) |
→ |
b(c(b(b(b(x1))))) |
(59) |
|
b(c(c(a(c(x1))))) |
→ |
b(c(c(b(a(x1))))) |
(60) |
|
b(b(b(c(a(x1))))) |
→ |
b(b(a(b(a(x1))))) |
(63) |
|
b(b(b(c(b(x1))))) |
→ |
b(b(b(c(c(x1))))) |
(64) |
|
b(b(a(b(c(x1))))) |
→ |
b(b(b(b(b(x1))))) |
(65) |
|
b(b(c(a(c(x1))))) |
→ |
b(b(c(b(a(x1))))) |
(66) |
|
b(a(b(c(a(x1))))) |
→ |
b(a(a(b(a(x1))))) |
(69) |
|
b(a(b(c(b(x1))))) |
→ |
b(a(b(c(c(x1))))) |
(70) |
|
b(a(a(b(c(x1))))) |
→ |
b(a(b(b(b(x1))))) |
(71) |
|
b(a(c(a(c(x1))))) |
→ |
b(a(c(b(a(x1))))) |
(72) |
|
a(c(b(c(a(x1))))) |
→ |
a(c(a(b(a(x1))))) |
(75) |
|
a(c(b(c(b(x1))))) |
→ |
a(c(b(c(c(x1))))) |
(76) |
|
a(c(a(b(c(x1))))) |
→ |
a(c(b(b(b(x1))))) |
(77) |
|
a(c(c(a(c(x1))))) |
→ |
a(c(c(b(a(x1))))) |
(78) |
|
a(b(b(c(a(x1))))) |
→ |
a(b(a(b(a(x1))))) |
(81) |
|
a(b(b(c(b(x1))))) |
→ |
a(b(b(c(c(x1))))) |
(82) |
|
a(b(a(b(c(x1))))) |
→ |
a(b(b(b(b(x1))))) |
(83) |
|
a(b(c(a(c(x1))))) |
→ |
a(b(c(b(a(x1))))) |
(84) |
|
a(a(b(c(a(x1))))) |
→ |
a(a(a(b(a(x1))))) |
(87) |
|
a(a(b(c(b(x1))))) |
→ |
a(a(b(c(c(x1))))) |
(88) |
|
a(a(a(b(c(x1))))) |
→ |
a(a(b(b(b(x1))))) |
(89) |
|
a(a(c(a(c(x1))))) |
→ |
a(a(c(b(a(x1))))) |
(90) |
|
c(c(a(c(c(x1))))) |
→ |
c(c(c(a(a(x1))))) |
(38) |
|
c(c(a(a(a(x1))))) |
→ |
c(c(a(c(c(x1))))) |
(43) |
|
c(b(a(c(c(x1))))) |
→ |
c(b(c(a(a(x1))))) |
(44) |
|
c(b(a(a(a(x1))))) |
→ |
c(b(a(c(c(x1))))) |
(49) |
|
c(a(a(c(c(x1))))) |
→ |
c(a(c(a(a(x1))))) |
(50) |
|
c(a(a(a(a(x1))))) |
→ |
c(a(a(c(c(x1))))) |
(55) |
|
b(c(a(c(c(x1))))) |
→ |
b(c(c(a(a(x1))))) |
(56) |
|
b(c(a(a(a(x1))))) |
→ |
b(c(a(c(c(x1))))) |
(61) |
|
b(b(a(c(c(x1))))) |
→ |
b(b(c(a(a(x1))))) |
(62) |
|
b(b(a(a(a(x1))))) |
→ |
b(b(a(c(c(x1))))) |
(67) |
|
b(a(a(c(c(x1))))) |
→ |
b(a(c(a(a(x1))))) |
(68) |
|
b(a(a(a(a(x1))))) |
→ |
b(a(a(c(c(x1))))) |
(73) |
|
a(c(a(c(c(x1))))) |
→ |
a(c(c(a(a(x1))))) |
(74) |
|
a(c(a(a(a(x1))))) |
→ |
a(c(a(c(c(x1))))) |
(79) |
|
a(b(a(c(c(x1))))) |
→ |
a(b(c(a(a(x1))))) |
(80) |
|
a(b(a(a(a(x1))))) |
→ |
a(b(a(c(c(x1))))) |
(85) |
|
a(a(a(c(c(x1))))) |
→ |
a(a(c(a(a(x1))))) |
(86) |
|
a(a(a(a(a(x1))))) |
→ |
a(a(a(c(c(x1))))) |
(91) |
1.2.1.1.1 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).
1.2.1.1.1.1 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 152 ruless (increase limit for explicit display).
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 334 ruless (increase limit for explicit display).
1.2.1.1.1.1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the naturals
| [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 159 ruless (increase limit for explicit display).
1.2.1.1.1.1.1.1.1 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.
|
a8(c8(b6(a1(a5(x1))))) |
→ |
a8(b8(a7(a5(a8(x1))))) |
(659) |
|
a2(c8(b6(a1(a5(x1))))) |
→ |
a2(b8(a7(a5(a8(x1))))) |
(660) |
|
a5(c8(b6(a1(a5(x1))))) |
→ |
a5(b8(a7(a5(a8(x1))))) |
(661) |
|
a7(c5(b6(a1(a5(x1))))) |
→ |
a7(b5(a7(a5(a8(x1))))) |
(665) |
|
a1(c5(b6(a1(a5(x1))))) |
→ |
a1(b5(a7(a5(a8(x1))))) |
(666) |
|
a4(c5(b6(a1(a5(x1))))) |
→ |
a4(b5(a7(a5(a8(x1))))) |
(667) |
|
a8(c8(b6(c1(a3(x1))))) |
→ |
a8(b8(a7(c5(a6(x1))))) |
(668) |
|
a2(c8(b6(c1(a3(x1))))) |
→ |
a2(b8(a7(c5(a6(x1))))) |
(669) |
|
a5(c8(b6(c1(a3(x1))))) |
→ |
a5(b8(a7(c5(a6(x1))))) |
(670) |
|
a7(c5(b6(c1(a3(x1))))) |
→ |
a7(b5(a7(c5(a6(x1))))) |
(674) |
|
a1(c5(b6(c1(a3(x1))))) |
→ |
a1(b5(a7(c5(a6(x1))))) |
(675) |
|
a4(c5(b6(c1(a3(x1))))) |
→ |
a4(b5(a7(c5(a6(x1))))) |
(676) |
|
a8(c8(b6(b1(a4(x1))))) |
→ |
a8(b8(a7(b5(a7(x1))))) |
(677) |
|
a2(c8(b6(b1(a4(x1))))) |
→ |
a2(b8(a7(b5(a7(x1))))) |
(678) |
|
a5(c8(b6(b1(a4(x1))))) |
→ |
a5(b8(a7(b5(a7(x1))))) |
(679) |
|
a7(c5(b6(b1(a4(x1))))) |
→ |
a7(b5(a7(b5(a7(x1))))) |
(683) |
|
a1(c5(b6(b1(a4(x1))))) |
→ |
a1(b5(a7(b5(a7(x1))))) |
(684) |
|
a4(c5(b6(b1(a4(x1))))) |
→ |
a4(b5(a7(b5(a7(x1))))) |
(685) |
|
a8(c8(b6(a1(c5(x1))))) |
→ |
a8(b8(a7(a5(c8(x1))))) |
(686) |
|
a2(c8(b6(a1(c5(x1))))) |
→ |
a2(b8(a7(a5(c8(x1))))) |
(687) |
|
a5(c8(b6(a1(c5(x1))))) |
→ |
a5(b8(a7(a5(c8(x1))))) |
(688) |
|
a7(c5(b6(a1(c5(x1))))) |
→ |
a7(b5(a7(a5(c8(x1))))) |
(692) |
|
a1(c5(b6(a1(c5(x1))))) |
→ |
a1(b5(a7(a5(c8(x1))))) |
(693) |
|
a4(c5(b6(a1(c5(x1))))) |
→ |
a4(b5(a7(a5(c8(x1))))) |
(694) |
|
a8(c8(b6(c1(c3(x1))))) |
→ |
a8(b8(a7(c5(c6(x1))))) |
(695) |
|
a2(c8(b6(c1(c3(x1))))) |
→ |
a2(b8(a7(c5(c6(x1))))) |
(696) |
|
a5(c8(b6(c1(c3(x1))))) |
→ |
a5(b8(a7(c5(c6(x1))))) |
(697) |
|
a7(c5(b6(c1(c3(x1))))) |
→ |
a7(b5(a7(c5(c6(x1))))) |
(701) |
|
a1(c5(b6(c1(c3(x1))))) |
→ |
a1(b5(a7(c5(c6(x1))))) |
(702) |
|
a4(c5(b6(c1(c3(x1))))) |
→ |
a4(b5(a7(c5(c6(x1))))) |
(703) |
|
a8(c8(b6(b1(c4(x1))))) |
→ |
a8(b8(a7(b5(c7(x1))))) |
(704) |
|
a2(c8(b6(b1(c4(x1))))) |
→ |
a2(b8(a7(b5(c7(x1))))) |
(705) |
|
a5(c8(b6(b1(c4(x1))))) |
→ |
a5(b8(a7(b5(c7(x1))))) |
(706) |
|
a7(c5(b6(b1(c4(x1))))) |
→ |
a7(b5(a7(b5(c7(x1))))) |
(710) |
|
a1(c5(b6(b1(c4(x1))))) |
→ |
a1(b5(a7(b5(c7(x1))))) |
(711) |
|
a4(c5(b6(b1(c4(x1))))) |
→ |
a4(b5(a7(b5(c7(x1))))) |
(712) |
|
a8(c8(b6(a1(b5(x1))))) |
→ |
a8(b8(a7(a5(b8(x1))))) |
(713) |
|
a2(c8(b6(a1(b5(x1))))) |
→ |
a2(b8(a7(a5(b8(x1))))) |
(714) |
|
a5(c8(b6(a1(b5(x1))))) |
→ |
a5(b8(a7(a5(b8(x1))))) |
(715) |
|
a7(c5(b6(a1(b5(x1))))) |
→ |
a7(b5(a7(a5(b8(x1))))) |
(719) |
|
a1(c5(b6(a1(b5(x1))))) |
→ |
a1(b5(a7(a5(b8(x1))))) |
(720) |
|
a4(c5(b6(a1(b5(x1))))) |
→ |
a4(b5(a7(a5(b8(x1))))) |
(721) |
|
a8(c8(b6(b1(b4(x1))))) |
→ |
a8(b8(a7(b5(b7(x1))))) |
(722) |
|
a2(c8(b6(b1(b4(x1))))) |
→ |
a2(b8(a7(b5(b7(x1))))) |
(723) |
|
a5(c8(b6(b1(b4(x1))))) |
→ |
a5(b8(a7(b5(b7(x1))))) |
(724) |
|
a7(c5(b6(b1(b4(x1))))) |
→ |
a7(b5(a7(b5(b7(x1))))) |
(728) |
|
a1(c5(b6(b1(b4(x1))))) |
→ |
a1(b5(a7(b5(b7(x1))))) |
(729) |
|
a4(c5(b6(b1(b4(x1))))) |
→ |
a4(b5(a7(b5(b7(x1))))) |
(730) |
|
b0(c1(b3(a1(a5(x1))))) |
→ |
c0(c0(b0(a1(a5(x1))))) |
(735) |
|
b0(c1(b3(c1(a3(x1))))) |
→ |
c0(c0(b0(c1(a3(x1))))) |
(743) |
|
b0(c1(b3(b1(a4(x1))))) |
→ |
c0(c0(b0(b1(a4(x1))))) |
(751) |
|
b0(c1(b3(a1(c5(x1))))) |
→ |
c0(c0(b0(a1(c5(x1))))) |
(759) |
|
b0(c1(b3(c1(c3(x1))))) |
→ |
c0(c0(b0(c1(c3(x1))))) |
(767) |
|
b0(c1(b3(b1(c4(x1))))) |
→ |
c0(c0(b0(b1(c4(x1))))) |
(775) |
|
b0(c1(b3(a1(b5(x1))))) |
→ |
c0(c0(b0(a1(b5(x1))))) |
(783) |
|
b0(c1(b3(c1(b3(x1))))) |
→ |
c0(c0(b0(c1(b3(x1))))) |
(791) |
|
b0(c1(b3(b1(b4(x1))))) |
→ |
c0(c0(b0(b1(b4(x1))))) |
(799) |
|
c3(b0(a1(c5(b6(x1))))) |
→ |
b3(b1(b4(c4(b3(x1))))) |
(803) |
|
c8(c6(a0(b2(a7(x1))))) |
→ |
a8(a8(c8(b6(a1(x1))))) |
(849) |
|
c2(c6(a0(b2(a7(x1))))) |
→ |
a2(a8(c8(b6(a1(x1))))) |
(850) |
|
c5(c6(a0(b2(a7(x1))))) |
→ |
a5(a8(c8(b6(a1(x1))))) |
(851) |
|
c6(c0(a0(b2(a7(x1))))) |
→ |
a6(a2(c8(b6(a1(x1))))) |
(852) |
|
c0(c0(a0(b2(a7(x1))))) |
→ |
a0(a2(c8(b6(a1(x1))))) |
(853) |
|
c3(c0(a0(b2(a7(x1))))) |
→ |
a3(a2(c8(b6(a1(x1))))) |
(854) |
|
c7(c3(a0(b2(a7(x1))))) |
→ |
a7(a5(c8(b6(a1(x1))))) |
(855) |
|
c1(c3(a0(b2(a7(x1))))) |
→ |
a1(a5(c8(b6(a1(x1))))) |
(856) |
|
c4(c3(a0(b2(a7(x1))))) |
→ |
a4(a5(c8(b6(a1(x1))))) |
(857) |
|
c8(c6(a0(b2(c7(x1))))) |
→ |
a8(a8(c8(b6(c1(x1))))) |
(876) |
|
c2(c6(a0(b2(c7(x1))))) |
→ |
a2(a8(c8(b6(c1(x1))))) |
(877) |
|
c5(c6(a0(b2(c7(x1))))) |
→ |
a5(a8(c8(b6(c1(x1))))) |
(878) |
|
c6(c0(a0(b2(c7(x1))))) |
→ |
a6(a2(c8(b6(c1(x1))))) |
(879) |
|
c0(c0(a0(b2(c7(x1))))) |
→ |
a0(a2(c8(b6(c1(x1))))) |
(880) |
|
c3(c0(a0(b2(c7(x1))))) |
→ |
a3(a2(c8(b6(c1(x1))))) |
(881) |
|
c7(c3(a0(b2(c7(x1))))) |
→ |
a7(a5(c8(b6(c1(x1))))) |
(882) |
|
c1(c3(a0(b2(c7(x1))))) |
→ |
a1(a5(c8(b6(c1(x1))))) |
(883) |
|
c4(c3(a0(b2(c7(x1))))) |
→ |
a4(a5(c8(b6(c1(x1))))) |
(884) |
|
c8(c6(a0(b2(b7(x1))))) |
→ |
a8(a8(c8(b6(b1(x1))))) |
(903) |
|
c2(c6(a0(b2(b7(x1))))) |
→ |
a2(a8(c8(b6(b1(x1))))) |
(904) |
|
c5(c6(a0(b2(b7(x1))))) |
→ |
a5(a8(c8(b6(b1(x1))))) |
(905) |
|
c6(c0(a0(b2(b7(x1))))) |
→ |
a6(a2(c8(b6(b1(x1))))) |
(906) |
|
c0(c0(a0(b2(b7(x1))))) |
→ |
a0(a2(c8(b6(b1(x1))))) |
(907) |
|
c3(c0(a0(b2(b7(x1))))) |
→ |
a3(a2(c8(b6(b1(x1))))) |
(908) |
|
c7(c3(a0(b2(b7(x1))))) |
→ |
a7(a5(c8(b6(b1(x1))))) |
(909) |
|
c1(c3(a0(b2(b7(x1))))) |
→ |
a1(a5(c8(b6(b1(x1))))) |
(910) |
|
c4(c3(a0(b2(b7(x1))))) |
→ |
a4(a5(c8(b6(b1(x1))))) |
(911) |
1.2.1.1.1.1.1.1.1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
a8(a2(a0(c6(c8(x1))))) |
→ |
a2(a6(c8(a8(a8(x1))))) |
(253) |
|
a8(a2(a0(c6(c2(x1))))) |
→ |
a2(a6(c8(a8(a2(x1))))) |
(252) |
|
a8(a2(a0(c6(c5(x1))))) |
→ |
a2(a6(c8(a8(a5(x1))))) |
(251) |
|
a8(a2(a0(c0(c6(x1))))) |
→ |
a2(a6(c8(a2(a6(x1))))) |
(250) |
|
a8(a2(a0(c0(c0(x1))))) |
→ |
a2(a6(c8(a2(a0(x1))))) |
(249) |
|
a8(a2(a0(c0(c3(x1))))) |
→ |
a2(a6(c8(a2(a3(x1))))) |
(248) |
|
a8(a2(a0(c3(c7(x1))))) |
→ |
a2(a6(c8(a5(a7(x1))))) |
(247) |
|
a8(a2(a0(c3(c1(x1))))) |
→ |
a2(a6(c8(a5(a1(x1))))) |
(246) |
|
a8(a2(a0(c3(c4(x1))))) |
→ |
a2(a6(c8(a5(a4(x1))))) |
(245) |
|
a6(c2(a0(c6(c8(x1))))) |
→ |
a0(c6(c8(a8(a8(x1))))) |
(244) |
|
a6(c2(a0(c6(c2(x1))))) |
→ |
a0(c6(c8(a8(a2(x1))))) |
(243) |
|
a6(c2(a0(c6(c5(x1))))) |
→ |
a0(c6(c8(a8(a5(x1))))) |
(242) |
|
a6(c2(a0(c0(c6(x1))))) |
→ |
a0(c6(c8(a2(a6(x1))))) |
(241) |
|
a6(c2(a0(c0(c0(x1))))) |
→ |
a0(c6(c8(a2(a0(x1))))) |
(240) |
|
a6(c2(a0(c0(c3(x1))))) |
→ |
a0(c6(c8(a2(a3(x1))))) |
(239) |
|
a6(c2(a0(c3(c7(x1))))) |
→ |
a0(c6(c8(a5(a7(x1))))) |
(238) |
|
a6(c2(a0(c3(c1(x1))))) |
→ |
a0(c6(c8(a5(a1(x1))))) |
(237) |
|
a6(c2(a0(c3(c4(x1))))) |
→ |
a0(c6(c8(a5(a4(x1))))) |
(236) |
|
c8(a2(a0(c6(c8(x1))))) |
→ |
c2(a6(c8(a8(a8(x1))))) |
(226) |
|
c8(a2(a0(c6(c2(x1))))) |
→ |
c2(a6(c8(a8(a2(x1))))) |
(225) |
|
c8(a2(a0(c6(c5(x1))))) |
→ |
c2(a6(c8(a8(a5(x1))))) |
(224) |
|
c8(a2(a0(c0(c6(x1))))) |
→ |
c2(a6(c8(a2(a6(x1))))) |
(223) |
|
c8(a2(a0(c0(c0(x1))))) |
→ |
c2(a6(c8(a2(a0(x1))))) |
(222) |
|
c8(a2(a0(c0(c3(x1))))) |
→ |
c2(a6(c8(a2(a3(x1))))) |
(221) |
|
c8(a2(a0(c3(c7(x1))))) |
→ |
c2(a6(c8(a5(a7(x1))))) |
(220) |
|
c8(a2(a0(c3(c1(x1))))) |
→ |
c2(a6(c8(a5(a1(x1))))) |
(219) |
|
c8(a2(a0(c3(c4(x1))))) |
→ |
c2(a6(c8(a5(a4(x1))))) |
(218) |
|
c6(c2(a0(c6(c8(x1))))) |
→ |
c0(c6(c8(a8(a8(x1))))) |
(217) |
|
c6(c2(a0(c6(c2(x1))))) |
→ |
c0(c6(c8(a8(a2(x1))))) |
(216) |
|
c6(c2(a0(c6(c5(x1))))) |
→ |
c0(c6(c8(a8(a5(x1))))) |
(215) |
|
c6(c2(a0(c0(c6(x1))))) |
→ |
c0(c6(c8(a2(a6(x1))))) |
(214) |
|
c6(c2(a0(c0(c0(x1))))) |
→ |
c0(c6(c8(a2(a0(x1))))) |
(213) |
|
c6(c2(a0(c0(c3(x1))))) |
→ |
c0(c6(c8(a2(a3(x1))))) |
(212) |
|
c6(c2(a0(c3(c7(x1))))) |
→ |
c0(c6(c8(a5(a7(x1))))) |
(211) |
|
c6(c2(a0(c3(c1(x1))))) |
→ |
c0(c6(c8(a5(a1(x1))))) |
(210) |
|
c6(c2(a0(c3(c4(x1))))) |
→ |
c0(c6(c8(a5(a4(x1))))) |
(209) |
|
a8(a8(a8(a8(a8(x1))))) |
→ |
a8(a2(a0(c6(c8(x1))))) |
(658) |
|
a8(a8(a8(a8(a2(x1))))) |
→ |
a8(a2(a0(c6(c2(x1))))) |
(657) |
|
a8(a8(a8(a8(a5(x1))))) |
→ |
a8(a2(a0(c6(c5(x1))))) |
(656) |
|
a8(a8(a8(a2(a6(x1))))) |
→ |
a8(a2(a0(c0(c6(x1))))) |
(655) |
|
a8(a8(a8(a2(a0(x1))))) |
→ |
a8(a2(a0(c0(c0(x1))))) |
(654) |
|
a8(a8(a8(a2(a3(x1))))) |
→ |
a8(a2(a0(c0(c3(x1))))) |
(653) |
|
a8(a8(a8(a5(a7(x1))))) |
→ |
a8(a2(a0(c3(c7(x1))))) |
(652) |
|
a8(a8(a8(a5(a1(x1))))) |
→ |
a8(a2(a0(c3(c1(x1))))) |
(651) |
|
a8(a8(a8(a5(a4(x1))))) |
→ |
a8(a2(a0(c3(c4(x1))))) |
(650) |
|
a6(c8(a8(a8(a8(x1))))) |
→ |
a6(c2(a0(c6(c8(x1))))) |
(649) |
|
a6(c8(a8(a8(a2(x1))))) |
→ |
a6(c2(a0(c6(c2(x1))))) |
(648) |
|
a6(c8(a8(a8(a5(x1))))) |
→ |
a6(c2(a0(c6(c5(x1))))) |
(647) |
|
a6(c8(a8(a2(a6(x1))))) |
→ |
a6(c2(a0(c0(c6(x1))))) |
(646) |
|
a6(c8(a8(a2(a0(x1))))) |
→ |
a6(c2(a0(c0(c0(x1))))) |
(645) |
|
a6(c8(a8(a2(a3(x1))))) |
→ |
a6(c2(a0(c0(c3(x1))))) |
(644) |
|
a6(c8(a8(a5(a7(x1))))) |
→ |
a6(c2(a0(c3(c7(x1))))) |
(643) |
|
a6(c8(a8(a5(a1(x1))))) |
→ |
a6(c2(a0(c3(c1(x1))))) |
(642) |
|
a6(c8(a8(a5(a4(x1))))) |
→ |
a6(c2(a0(c3(c4(x1))))) |
(641) |
|
c8(a8(a8(a8(a8(x1))))) |
→ |
c8(a2(a0(c6(c8(x1))))) |
(631) |
|
c8(a8(a8(a8(a2(x1))))) |
→ |
c8(a2(a0(c6(c2(x1))))) |
(630) |
|
c8(a8(a8(a8(a5(x1))))) |
→ |
c8(a2(a0(c6(c5(x1))))) |
(629) |
|
c8(a8(a8(a2(a6(x1))))) |
→ |
c8(a2(a0(c0(c6(x1))))) |
(628) |
|
c8(a8(a8(a2(a0(x1))))) |
→ |
c8(a2(a0(c0(c0(x1))))) |
(627) |
|
c8(a8(a8(a2(a3(x1))))) |
→ |
c8(a2(a0(c0(c3(x1))))) |
(626) |
|
c8(a8(a8(a5(a7(x1))))) |
→ |
c8(a2(a0(c3(c7(x1))))) |
(625) |
|
c8(a8(a8(a5(a1(x1))))) |
→ |
c8(a2(a0(c3(c1(x1))))) |
(624) |
|
c8(a8(a8(a5(a4(x1))))) |
→ |
c8(a2(a0(c3(c4(x1))))) |
(623) |
|
c6(c8(a8(a8(a8(x1))))) |
→ |
c6(c2(a0(c6(c8(x1))))) |
(622) |
|
c6(c8(a8(a8(a2(x1))))) |
→ |
c6(c2(a0(c6(c2(x1))))) |
(621) |
|
c6(c8(a8(a8(a5(x1))))) |
→ |
c6(c2(a0(c6(c5(x1))))) |
(620) |
|
c6(c8(a8(a2(a6(x1))))) |
→ |
c6(c2(a0(c0(c6(x1))))) |
(619) |
|
c6(c8(a8(a2(a0(x1))))) |
→ |
c6(c2(a0(c0(c0(x1))))) |
(618) |
|
c6(c8(a8(a2(a3(x1))))) |
→ |
c6(c2(a0(c0(c3(x1))))) |
(617) |
|
c6(c8(a8(a5(a7(x1))))) |
→ |
c6(c2(a0(c3(c7(x1))))) |
(616) |
|
c6(c8(a8(a5(a1(x1))))) |
→ |
c6(c2(a0(c3(c1(x1))))) |
(615) |
|
c6(c8(a8(a5(a4(x1))))) |
→ |
c6(c2(a0(c3(c4(x1))))) |
(614) |
|
b8(a8(a8(a8(a8(x1))))) |
→ |
b8(a2(a0(c6(c8(x1))))) |
(604) |
|
b8(a8(a8(a8(a2(x1))))) |
→ |
b8(a2(a0(c6(c2(x1))))) |
(603) |
|
b8(a8(a8(a8(a5(x1))))) |
→ |
b8(a2(a0(c6(c5(x1))))) |
(602) |
|
b8(a8(a8(a2(a6(x1))))) |
→ |
b8(a2(a0(c0(c6(x1))))) |
(601) |
|
b8(a8(a8(a2(a0(x1))))) |
→ |
b8(a2(a0(c0(c0(x1))))) |
(600) |
|
b8(a8(a8(a2(a3(x1))))) |
→ |
b8(a2(a0(c0(c3(x1))))) |
(599) |
|
b8(a8(a8(a5(a7(x1))))) |
→ |
b8(a2(a0(c3(c7(x1))))) |
(598) |
|
b8(a8(a8(a5(a1(x1))))) |
→ |
b8(a2(a0(c3(c1(x1))))) |
(597) |
|
b8(a8(a8(a5(a4(x1))))) |
→ |
b8(a2(a0(c3(c4(x1))))) |
(596) |
|
b6(c8(a8(a8(a8(x1))))) |
→ |
b6(c2(a0(c6(c8(x1))))) |
(595) |
|
b6(c8(a8(a8(a2(x1))))) |
→ |
b6(c2(a0(c6(c2(x1))))) |
(594) |
|
b6(c8(a8(a8(a5(x1))))) |
→ |
b6(c2(a0(c6(c5(x1))))) |
(593) |
|
b6(c8(a8(a2(a6(x1))))) |
→ |
b6(c2(a0(c0(c6(x1))))) |
(592) |
|
b6(c8(a8(a2(a0(x1))))) |
→ |
b6(c2(a0(c0(c0(x1))))) |
(591) |
|
b6(c8(a8(a2(a3(x1))))) |
→ |
b6(c2(a0(c0(c3(x1))))) |
(590) |
|
b6(c8(a8(a5(a7(x1))))) |
→ |
b6(c2(a0(c3(c7(x1))))) |
(589) |
|
b6(c8(a8(a5(a1(x1))))) |
→ |
b6(c2(a0(c3(c1(x1))))) |
(588) |
|
b6(c8(a8(a5(a4(x1))))) |
→ |
b6(c2(a0(c3(c4(x1))))) |
(587) |
1.2.1.1.1.1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.
1.2.2 Closure Under Flat Contexts
Using the flat contexts
{c(☐), a(☐)}
We obtain the transformed TRS
|
c(a(c(c(x1)))) |
→ |
c(c(a(a(x1)))) |
(11) |
|
c(a(a(a(x1)))) |
→ |
c(a(c(c(x1)))) |
(16) |
|
a(a(c(c(x1)))) |
→ |
a(c(a(a(x1)))) |
(23) |
|
a(a(a(a(x1)))) |
→ |
a(a(c(c(x1)))) |
(28) |
1.2.2.1 Semantic Labeling
The following interpretations form a
model
of the rules.
As carrier we take the set
{0,1}.
Symbols are labeled by the interpretation of their arguments using the interpretations
(modulo 2):
| [c(x1)] |
= |
2x1 + 0 |
| [a(x1)] |
= |
2x1 + 1 |
We obtain the labeled TRS
|
a1(a0(c0(c1(x1)))) |
→ |
a0(c1(a1(a1(x1)))) |
(993) |
|
a1(a0(c0(c0(x1)))) |
→ |
a0(c1(a1(a0(x1)))) |
(994) |
|
c1(a0(c0(c1(x1)))) |
→ |
c0(c1(a1(a1(x1)))) |
(995) |
|
c1(a0(c0(c0(x1)))) |
→ |
c0(c1(a1(a0(x1)))) |
(996) |
|
a1(a1(a1(a1(x1)))) |
→ |
a1(a0(c0(c1(x1)))) |
(997) |
|
a1(a1(a1(a0(x1)))) |
→ |
a1(a0(c0(c0(x1)))) |
(998) |
|
c1(a1(a1(a1(x1)))) |
→ |
c1(a0(c0(c1(x1)))) |
(999) |
|
c1(a1(a1(a0(x1)))) |
→ |
c1(a0(c0(c0(x1)))) |
(1000) |
1.2.2.1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
c1(c0(a0(a1(x1)))) |
→ |
a1(a1(c1(a0(x1)))) |
(1001) |
|
c0(c0(a0(a1(x1)))) |
→ |
a0(a1(c1(a0(x1)))) |
(1002) |
|
c1(c0(a0(c1(x1)))) |
→ |
a1(a1(c1(c0(x1)))) |
(1003) |
|
c0(c0(a0(c1(x1)))) |
→ |
a0(a1(c1(c0(x1)))) |
(1004) |
|
a1(a1(a1(a1(x1)))) |
→ |
c1(c0(a0(a1(x1)))) |
(1005) |
|
a0(a1(a1(a1(x1)))) |
→ |
c0(c0(a0(a1(x1)))) |
(1006) |
|
a1(a1(a1(c1(x1)))) |
→ |
c1(c0(a0(c1(x1)))) |
(1007) |
|
a0(a1(a1(c1(x1)))) |
→ |
c0(c0(a0(c1(x1)))) |
(1008) |
1.2.2.1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the naturals
| [c0(x1)] |
= |
· x1 +
|
| [c1(x1)] |
= |
· x1 +
|
| [a0(x1)] |
= |
· x1 +
|
| [a1(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
|
c1(c0(a0(a1(x1)))) |
→ |
a1(a1(c1(a0(x1)))) |
(1001) |
|
c0(c0(a0(a1(x1)))) |
→ |
a0(a1(c1(a0(x1)))) |
(1002) |
|
c1(c0(a0(c1(x1)))) |
→ |
a1(a1(c1(c0(x1)))) |
(1003) |
|
c0(c0(a0(c1(x1)))) |
→ |
a0(a1(c1(c0(x1)))) |
(1004) |
|
a1(a1(a1(a1(x1)))) |
→ |
c1(c0(a0(a1(x1)))) |
(1005) |
|
a0(a1(a1(a1(x1)))) |
→ |
c0(c0(a0(a1(x1)))) |
(1006) |
|
a1(a1(a1(c1(x1)))) |
→ |
c1(c0(a0(c1(x1)))) |
(1007) |
|
a0(a1(a1(c1(x1)))) |
→ |
c0(c0(a0(c1(x1)))) |
(1008) |
1.2.2.1.1.1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
There are no rules.
1.2.2.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.