Certification Problem
Input (TPDB SRS_Relative/Zantema_06_relative/rel01)
The relative rewrite relation R/S is considered where R is the following TRS
|
a(a(b(x1))) |
→ |
b(a(x1)) |
(1) |
|
c(b(x1)) |
→ |
b(c(a(x1))) |
(2) |
and S is the following TRS.
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by matchbox @ termCOMP 2023)
1 Closure Under Flat Contexts
Using the flat contexts
{c(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
c(a(a(b(x1)))) |
→ |
c(b(a(x1))) |
(4) |
|
c(c(b(x1))) |
→ |
c(b(c(a(x1)))) |
(5) |
|
b(a(a(b(x1)))) |
→ |
b(b(a(x1))) |
(6) |
|
b(c(b(x1))) |
→ |
b(b(c(a(x1)))) |
(7) |
|
a(a(a(b(x1)))) |
→ |
a(b(a(x1))) |
(8) |
|
a(c(b(x1))) |
→ |
a(b(c(a(x1)))) |
(9) |
|
c(a(x1)) |
→ |
c(a(c(a(x1)))) |
(10) |
|
b(a(x1)) |
→ |
b(a(c(a(x1)))) |
(11) |
|
a(a(x1)) |
→ |
a(a(c(a(x1)))) |
(12) |
1.1 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(a2(a1(b2(x1)))) |
→ |
a1(b2(a2(x1))) |
(13) |
|
a2(a2(a1(b1(x1)))) |
→ |
a1(b2(a1(x1))) |
(14) |
|
a2(a2(a1(b0(x1)))) |
→ |
a1(b2(a0(x1))) |
(15) |
|
b2(a2(a1(b2(x1)))) |
→ |
b1(b2(a2(x1))) |
(16) |
|
b2(a2(a1(b1(x1)))) |
→ |
b1(b2(a1(x1))) |
(17) |
|
b2(a2(a1(b0(x1)))) |
→ |
b1(b2(a0(x1))) |
(18) |
|
c2(a2(a1(b2(x1)))) |
→ |
c1(b2(a2(x1))) |
(19) |
|
c2(a2(a1(b1(x1)))) |
→ |
c1(b2(a1(x1))) |
(20) |
|
c2(a2(a1(b0(x1)))) |
→ |
c1(b2(a0(x1))) |
(21) |
|
a0(c1(b2(x1))) |
→ |
a1(b0(c2(a2(x1)))) |
(22) |
|
a0(c1(b1(x1))) |
→ |
a1(b0(c2(a1(x1)))) |
(23) |
|
a0(c1(b0(x1))) |
→ |
a1(b0(c2(a0(x1)))) |
(24) |
|
b0(c1(b2(x1))) |
→ |
b1(b0(c2(a2(x1)))) |
(25) |
|
b0(c1(b1(x1))) |
→ |
b1(b0(c2(a1(x1)))) |
(26) |
|
b0(c1(b0(x1))) |
→ |
b1(b0(c2(a0(x1)))) |
(27) |
|
c0(c1(b2(x1))) |
→ |
c1(b0(c2(a2(x1)))) |
(28) |
|
c0(c1(b1(x1))) |
→ |
c1(b0(c2(a1(x1)))) |
(29) |
|
c0(c1(b0(x1))) |
→ |
c1(b0(c2(a0(x1)))) |
(30) |
|
a2(a2(x1)) |
→ |
a2(a0(c2(a2(x1)))) |
(31) |
|
a2(a1(x1)) |
→ |
a2(a0(c2(a1(x1)))) |
(32) |
|
a2(a0(x1)) |
→ |
a2(a0(c2(a0(x1)))) |
(33) |
|
b2(a2(x1)) |
→ |
b2(a0(c2(a2(x1)))) |
(34) |
|
b2(a1(x1)) |
→ |
b2(a0(c2(a1(x1)))) |
(35) |
|
b2(a0(x1)) |
→ |
b2(a0(c2(a0(x1)))) |
(36) |
|
c2(a2(x1)) |
→ |
c2(a0(c2(a2(x1)))) |
(37) |
|
c2(a1(x1)) |
→ |
c2(a0(c2(a1(x1)))) |
(38) |
|
c2(a0(x1)) |
→ |
c2(a0(c2(a0(x1)))) |
(39) |
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 +
|
| [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(a2(a1(b2(x1)))) |
→ |
a1(b2(a2(x1))) |
(13) |
|
a2(a2(a1(b1(x1)))) |
→ |
a1(b2(a1(x1))) |
(14) |
|
a2(a2(a1(b0(x1)))) |
→ |
a1(b2(a0(x1))) |
(15) |
|
b2(a2(a1(b2(x1)))) |
→ |
b1(b2(a2(x1))) |
(16) |
|
b2(a2(a1(b1(x1)))) |
→ |
b1(b2(a1(x1))) |
(17) |
|
b2(a2(a1(b0(x1)))) |
→ |
b1(b2(a0(x1))) |
(18) |
|
c0(c1(b2(x1))) |
→ |
c1(b0(c2(a2(x1)))) |
(28) |
|
c0(c1(b1(x1))) |
→ |
c1(b0(c2(a1(x1)))) |
(29) |
|
c0(c1(b0(x1))) |
→ |
c1(b0(c2(a0(x1)))) |
(30) |
1.1.1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
b2(a1(a2(c2(x1)))) |
→ |
a2(b2(c1(x1))) |
(40) |
|
b1(a1(a2(c2(x1)))) |
→ |
a1(b2(c1(x1))) |
(41) |
|
b0(a1(a2(c2(x1)))) |
→ |
a0(b2(c1(x1))) |
(42) |
|
b2(c1(a0(x1))) |
→ |
a2(c2(b0(a1(x1)))) |
(43) |
|
b1(c1(a0(x1))) |
→ |
a1(c2(b0(a1(x1)))) |
(44) |
|
b0(c1(a0(x1))) |
→ |
a0(c2(b0(a1(x1)))) |
(45) |
|
b2(c1(b0(x1))) |
→ |
a2(c2(b0(b1(x1)))) |
(46) |
|
b1(c1(b0(x1))) |
→ |
a1(c2(b0(b1(x1)))) |
(47) |
|
b0(c1(b0(x1))) |
→ |
a0(c2(b0(b1(x1)))) |
(48) |
|
a2(a2(x1)) |
→ |
a2(c2(a0(a2(x1)))) |
(49) |
|
a1(a2(x1)) |
→ |
a1(c2(a0(a2(x1)))) |
(50) |
|
a0(a2(x1)) |
→ |
a0(c2(a0(a2(x1)))) |
(51) |
|
a2(b2(x1)) |
→ |
a2(c2(a0(b2(x1)))) |
(52) |
|
a1(b2(x1)) |
→ |
a1(c2(a0(b2(x1)))) |
(53) |
|
a0(b2(x1)) |
→ |
a0(c2(a0(b2(x1)))) |
(54) |
|
a2(c2(x1)) |
→ |
a2(c2(a0(c2(x1)))) |
(55) |
|
a1(c2(x1)) |
→ |
a1(c2(a0(c2(x1)))) |
(56) |
|
a0(c2(x1)) |
→ |
a0(c2(a0(c2(x1)))) |
(57) |
1.1.1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the naturals
| [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(c2(x1)))) |
→ |
a2(b2(c1(x1))) |
(40) |
|
b1(a1(a2(c2(x1)))) |
→ |
a1(b2(c1(x1))) |
(41) |
|
b0(a1(a2(c2(x1)))) |
→ |
a0(b2(c1(x1))) |
(42) |
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
| [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(c1(a0(x1))) |
→ |
a2(c2(b0(a1(x1)))) |
(43) |
|
b1(c1(a0(x1))) |
→ |
a1(c2(b0(a1(x1)))) |
(44) |
|
b0(c1(a0(x1))) |
→ |
a0(c2(b0(a1(x1)))) |
(45) |
|
b2(c1(b0(x1))) |
→ |
a2(c2(b0(b1(x1)))) |
(46) |
|
b1(c1(b0(x1))) |
→ |
a1(c2(b0(b1(x1)))) |
(47) |
|
b0(c1(b0(x1))) |
→ |
a0(c2(b0(b1(x1)))) |
(48) |
1.1.1.1.1.1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
a2(a2(x1)) |
→ |
a2(a0(c2(a2(x1)))) |
(31) |
|
a2(a1(x1)) |
→ |
a2(a0(c2(a1(x1)))) |
(32) |
|
a2(a0(x1)) |
→ |
a2(a0(c2(a0(x1)))) |
(33) |
|
b2(a2(x1)) |
→ |
b2(a0(c2(a2(x1)))) |
(34) |
|
b2(a1(x1)) |
→ |
b2(a0(c2(a1(x1)))) |
(35) |
|
b2(a0(x1)) |
→ |
b2(a0(c2(a0(x1)))) |
(36) |
|
c2(a2(x1)) |
→ |
c2(a0(c2(a2(x1)))) |
(37) |
|
c2(a1(x1)) |
→ |
c2(a0(c2(a1(x1)))) |
(38) |
|
c2(a0(x1)) |
→ |
c2(a0(c2(a0(x1)))) |
(39) |
1.1.1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.