Certification Problem
Input (TPDB SRS_Relative/Waldmann_19/random-93)
The relative rewrite relation R/S is considered where R is the following TRS
|
c(a(b(x1))) |
→ |
c(a(c(x1))) |
(1) |
|
a(a(a(x1))) |
→ |
c(a(a(x1))) |
(2) |
|
a(b(a(x1))) |
→ |
c(a(c(x1))) |
(3) |
|
a(b(b(x1))) |
→ |
a(c(b(x1))) |
(4) |
and S is the following TRS.
|
b(c(c(x1))) |
→ |
b(a(b(x1))) |
(5) |
|
a(b(b(x1))) |
→ |
a(a(c(x1))) |
(6) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
b(a(c(x1))) |
→ |
c(a(c(x1))) |
(7) |
|
a(a(a(x1))) |
→ |
a(a(c(x1))) |
(8) |
|
a(b(a(x1))) |
→ |
c(a(c(x1))) |
(3) |
|
b(b(a(x1))) |
→ |
b(c(a(x1))) |
(9) |
|
c(c(b(x1))) |
→ |
b(a(b(x1))) |
(10) |
|
b(b(a(x1))) |
→ |
c(a(a(x1))) |
(11) |
1.1 Closure Under Flat Contexts
Using the flat contexts
{b(☐), a(☐), c(☐)}
We obtain the transformed TRS
|
a(a(a(x1))) |
→ |
a(a(c(x1))) |
(8) |
|
b(b(a(x1))) |
→ |
b(c(a(x1))) |
(9) |
|
b(b(a(c(x1)))) |
→ |
b(c(a(c(x1)))) |
(12) |
|
a(b(a(c(x1)))) |
→ |
a(c(a(c(x1)))) |
(13) |
|
c(b(a(c(x1)))) |
→ |
c(c(a(c(x1)))) |
(14) |
|
b(a(b(a(x1)))) |
→ |
b(c(a(c(x1)))) |
(15) |
|
a(a(b(a(x1)))) |
→ |
a(c(a(c(x1)))) |
(16) |
|
c(a(b(a(x1)))) |
→ |
c(c(a(c(x1)))) |
(17) |
|
b(c(c(b(x1)))) |
→ |
b(b(a(b(x1)))) |
(18) |
|
a(c(c(b(x1)))) |
→ |
a(b(a(b(x1)))) |
(19) |
|
c(c(c(b(x1)))) |
→ |
c(b(a(b(x1)))) |
(20) |
|
b(b(b(a(x1)))) |
→ |
b(c(a(a(x1)))) |
(21) |
|
a(b(b(a(x1)))) |
→ |
a(c(a(a(x1)))) |
(22) |
|
c(b(b(a(x1)))) |
→ |
c(c(a(a(x1)))) |
(23) |
1.1.1 Semantic Labeling
Root-labeling is applied.
We obtain the labeled TRS
|
aa(aa(aa(x1))) |
→ |
aa(ac(ca(x1))) |
(24) |
|
aa(aa(ac(x1))) |
→ |
aa(ac(cc(x1))) |
(25) |
|
aa(aa(ab(x1))) |
→ |
aa(ac(cb(x1))) |
(26) |
|
bb(ba(aa(x1))) |
→ |
bc(ca(aa(x1))) |
(27) |
|
bb(ba(ac(x1))) |
→ |
bc(ca(ac(x1))) |
(28) |
|
bb(ba(ab(x1))) |
→ |
bc(ca(ab(x1))) |
(29) |
|
bb(ba(ac(ca(x1)))) |
→ |
bc(ca(ac(ca(x1)))) |
(30) |
|
bb(ba(ac(cc(x1)))) |
→ |
bc(ca(ac(cc(x1)))) |
(31) |
|
bb(ba(ac(cb(x1)))) |
→ |
bc(ca(ac(cb(x1)))) |
(32) |
|
ab(ba(ac(ca(x1)))) |
→ |
ac(ca(ac(ca(x1)))) |
(33) |
|
ab(ba(ac(cc(x1)))) |
→ |
ac(ca(ac(cc(x1)))) |
(34) |
|
ab(ba(ac(cb(x1)))) |
→ |
ac(ca(ac(cb(x1)))) |
(35) |
|
cb(ba(ac(ca(x1)))) |
→ |
cc(ca(ac(ca(x1)))) |
(36) |
|
cb(ba(ac(cc(x1)))) |
→ |
cc(ca(ac(cc(x1)))) |
(37) |
|
cb(ba(ac(cb(x1)))) |
→ |
cc(ca(ac(cb(x1)))) |
(38) |
|
ba(ab(ba(aa(x1)))) |
→ |
bc(ca(ac(ca(x1)))) |
(39) |
|
ba(ab(ba(ac(x1)))) |
→ |
bc(ca(ac(cc(x1)))) |
(40) |
|
ba(ab(ba(ab(x1)))) |
→ |
bc(ca(ac(cb(x1)))) |
(41) |
|
aa(ab(ba(aa(x1)))) |
→ |
ac(ca(ac(ca(x1)))) |
(42) |
|
aa(ab(ba(ac(x1)))) |
→ |
ac(ca(ac(cc(x1)))) |
(43) |
|
aa(ab(ba(ab(x1)))) |
→ |
ac(ca(ac(cb(x1)))) |
(44) |
|
ca(ab(ba(aa(x1)))) |
→ |
cc(ca(ac(ca(x1)))) |
(45) |
|
ca(ab(ba(ac(x1)))) |
→ |
cc(ca(ac(cc(x1)))) |
(46) |
|
ca(ab(ba(ab(x1)))) |
→ |
cc(ca(ac(cb(x1)))) |
(47) |
|
bc(cc(cb(ba(x1)))) |
→ |
bb(ba(ab(ba(x1)))) |
(48) |
|
bc(cc(cb(bc(x1)))) |
→ |
bb(ba(ab(bc(x1)))) |
(49) |
|
bc(cc(cb(bb(x1)))) |
→ |
bb(ba(ab(bb(x1)))) |
(50) |
|
ac(cc(cb(ba(x1)))) |
→ |
ab(ba(ab(ba(x1)))) |
(51) |
|
ac(cc(cb(bc(x1)))) |
→ |
ab(ba(ab(bc(x1)))) |
(52) |
|
ac(cc(cb(bb(x1)))) |
→ |
ab(ba(ab(bb(x1)))) |
(53) |
|
cc(cc(cb(ba(x1)))) |
→ |
cb(ba(ab(ba(x1)))) |
(54) |
|
cc(cc(cb(bc(x1)))) |
→ |
cb(ba(ab(bc(x1)))) |
(55) |
|
cc(cc(cb(bb(x1)))) |
→ |
cb(ba(ab(bb(x1)))) |
(56) |
|
bb(bb(ba(aa(x1)))) |
→ |
bc(ca(aa(aa(x1)))) |
(57) |
|
bb(bb(ba(ac(x1)))) |
→ |
bc(ca(aa(ac(x1)))) |
(58) |
|
bb(bb(ba(ab(x1)))) |
→ |
bc(ca(aa(ab(x1)))) |
(59) |
|
ab(bb(ba(aa(x1)))) |
→ |
ac(ca(aa(aa(x1)))) |
(60) |
|
ab(bb(ba(ac(x1)))) |
→ |
ac(ca(aa(ac(x1)))) |
(61) |
|
ab(bb(ba(ab(x1)))) |
→ |
ac(ca(aa(ab(x1)))) |
(62) |
|
cb(bb(ba(aa(x1)))) |
→ |
cc(ca(aa(aa(x1)))) |
(63) |
|
cb(bb(ba(ac(x1)))) |
→ |
cc(ca(aa(ac(x1)))) |
(64) |
|
cb(bb(ba(ab(x1)))) |
→ |
cc(ca(aa(ab(x1)))) |
(65) |
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [aa(x1)] |
= |
1 + 1 · x1
|
| [ac(x1)] |
= |
1 + 1 · x1
|
| [ca(x1)] |
= |
1 · x1
|
| [cc(x1)] |
= |
1 + 1 · x1
|
| [ab(x1)] |
= |
1 + 1 · x1
|
| [cb(x1)] |
= |
1 + 1 · x1
|
| [bb(x1)] |
= |
1 · x1
|
| [ba(x1)] |
= |
1 + 1 · x1
|
| [bc(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
aa(aa(aa(x1))) |
→ |
aa(ac(ca(x1))) |
(24) |
|
bb(ba(aa(x1))) |
→ |
bc(ca(aa(x1))) |
(27) |
|
bb(ba(ac(x1))) |
→ |
bc(ca(ac(x1))) |
(28) |
|
bb(ba(ab(x1))) |
→ |
bc(ca(ab(x1))) |
(29) |
|
bb(ba(ac(ca(x1)))) |
→ |
bc(ca(ac(ca(x1)))) |
(30) |
|
bb(ba(ac(cc(x1)))) |
→ |
bc(ca(ac(cc(x1)))) |
(31) |
|
bb(ba(ac(cb(x1)))) |
→ |
bc(ca(ac(cb(x1)))) |
(32) |
|
ab(ba(ac(ca(x1)))) |
→ |
ac(ca(ac(ca(x1)))) |
(33) |
|
ab(ba(ac(cc(x1)))) |
→ |
ac(ca(ac(cc(x1)))) |
(34) |
|
ab(ba(ac(cb(x1)))) |
→ |
ac(ca(ac(cb(x1)))) |
(35) |
|
cb(ba(ac(ca(x1)))) |
→ |
cc(ca(ac(ca(x1)))) |
(36) |
|
cb(ba(ac(cc(x1)))) |
→ |
cc(ca(ac(cc(x1)))) |
(37) |
|
cb(ba(ac(cb(x1)))) |
→ |
cc(ca(ac(cb(x1)))) |
(38) |
|
ba(ab(ba(aa(x1)))) |
→ |
bc(ca(ac(ca(x1)))) |
(39) |
|
ba(ab(ba(ac(x1)))) |
→ |
bc(ca(ac(cc(x1)))) |
(40) |
|
ba(ab(ba(ab(x1)))) |
→ |
bc(ca(ac(cb(x1)))) |
(41) |
|
aa(ab(ba(aa(x1)))) |
→ |
ac(ca(ac(ca(x1)))) |
(42) |
|
aa(ab(ba(ac(x1)))) |
→ |
ac(ca(ac(cc(x1)))) |
(43) |
|
aa(ab(ba(ab(x1)))) |
→ |
ac(ca(ac(cb(x1)))) |
(44) |
|
ca(ab(ba(aa(x1)))) |
→ |
cc(ca(ac(ca(x1)))) |
(45) |
1.1.1.1.1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(aa) |
= |
1 |
|
weight(aa) |
= |
5 |
|
|
|
| prec(ac) |
= |
4 |
|
weight(ac) |
= |
5 |
|
|
|
| prec(cc) |
= |
2 |
|
weight(cc) |
= |
4 |
|
|
|
| prec(ab) |
= |
3 |
|
weight(ab) |
= |
7 |
|
|
|
| prec(cb) |
= |
0 |
|
weight(cb) |
= |
6 |
|
|
|
| prec(ca) |
= |
5 |
|
weight(ca) |
= |
5 |
|
|
|
| prec(ba) |
= |
6 |
|
weight(ba) |
= |
1 |
|
|
|
| prec(bc) |
= |
7 |
|
weight(bc) |
= |
7 |
|
|
|
| prec(bb) |
= |
8 |
|
weight(bb) |
= |
8 |
|
|
|
all of the following rules can be deleted.
|
aa(aa(ac(x1))) |
→ |
aa(ac(cc(x1))) |
(25) |
|
aa(aa(ab(x1))) |
→ |
aa(ac(cb(x1))) |
(26) |
|
ca(ab(ba(ac(x1)))) |
→ |
cc(ca(ac(cc(x1)))) |
(46) |
|
ca(ab(ba(ab(x1)))) |
→ |
cc(ca(ac(cb(x1)))) |
(47) |
|
bc(cc(cb(ba(x1)))) |
→ |
bb(ba(ab(ba(x1)))) |
(48) |
|
bc(cc(cb(bc(x1)))) |
→ |
bb(ba(ab(bc(x1)))) |
(49) |
|
bc(cc(cb(bb(x1)))) |
→ |
bb(ba(ab(bb(x1)))) |
(50) |
|
ac(cc(cb(ba(x1)))) |
→ |
ab(ba(ab(ba(x1)))) |
(51) |
|
ac(cc(cb(bc(x1)))) |
→ |
ab(ba(ab(bc(x1)))) |
(52) |
|
ac(cc(cb(bb(x1)))) |
→ |
ab(ba(ab(bb(x1)))) |
(53) |
|
cc(cc(cb(ba(x1)))) |
→ |
cb(ba(ab(ba(x1)))) |
(54) |
|
cc(cc(cb(bc(x1)))) |
→ |
cb(ba(ab(bc(x1)))) |
(55) |
|
cc(cc(cb(bb(x1)))) |
→ |
cb(ba(ab(bb(x1)))) |
(56) |
|
bb(bb(ba(aa(x1)))) |
→ |
bc(ca(aa(aa(x1)))) |
(57) |
|
bb(bb(ba(ac(x1)))) |
→ |
bc(ca(aa(ac(x1)))) |
(58) |
|
bb(bb(ba(ab(x1)))) |
→ |
bc(ca(aa(ab(x1)))) |
(59) |
|
ab(bb(ba(aa(x1)))) |
→ |
ac(ca(aa(aa(x1)))) |
(60) |
|
ab(bb(ba(ac(x1)))) |
→ |
ac(ca(aa(ac(x1)))) |
(61) |
|
ab(bb(ba(ab(x1)))) |
→ |
ac(ca(aa(ab(x1)))) |
(62) |
|
cb(bb(ba(aa(x1)))) |
→ |
cc(ca(aa(aa(x1)))) |
(63) |
|
cb(bb(ba(ac(x1)))) |
→ |
cc(ca(aa(ac(x1)))) |
(64) |
|
cb(bb(ba(ab(x1)))) |
→ |
cc(ca(aa(ab(x1)))) |
(65) |
1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.