Certification Problem
Input (TPDB SRS_Standard/Waldmann_19/random-147)
The rewrite relation of the following TRS is considered.
|
a(b(a(a(x1)))) |
→ |
b(b(b(b(x1)))) |
(1) |
|
b(b(a(b(x1)))) |
→ |
a(b(a(a(x1)))) |
(2) |
|
b(a(b(b(x1)))) |
→ |
a(a(b(a(x1)))) |
(3) |
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
|
a(a(b(a(x1)))) |
→ |
b(b(b(b(x1)))) |
(4) |
|
b(a(b(b(x1)))) |
→ |
a(a(b(a(x1)))) |
(3) |
|
b(b(a(b(x1)))) |
→ |
a(b(a(a(x1)))) |
(2) |
1.1 Closure Under Flat Contexts
Using the flat contexts
{a(☐), b(☐)}
We obtain the transformed TRS
|
a(a(a(b(a(x1))))) |
→ |
a(b(b(b(b(x1))))) |
(5) |
|
b(a(a(b(a(x1))))) |
→ |
b(b(b(b(b(x1))))) |
(6) |
|
a(b(a(b(b(x1))))) |
→ |
a(a(a(b(a(x1))))) |
(7) |
|
b(b(a(b(b(x1))))) |
→ |
b(a(a(b(a(x1))))) |
(8) |
|
a(b(b(a(b(x1))))) |
→ |
a(a(b(a(a(x1))))) |
(9) |
|
b(b(b(a(b(x1))))) |
→ |
b(a(b(a(a(x1))))) |
(10) |
1.1.1 Semantic Labeling
Root-labeling is applied.
We obtain the labeled TRS
|
aa(aa(ab(ba(aa(x1))))) |
→ |
ab(bb(bb(bb(ba(x1))))) |
(11) |
|
aa(aa(ab(ba(ab(x1))))) |
→ |
ab(bb(bb(bb(bb(x1))))) |
(12) |
|
ba(aa(ab(ba(aa(x1))))) |
→ |
bb(bb(bb(bb(ba(x1))))) |
(13) |
|
ba(aa(ab(ba(ab(x1))))) |
→ |
bb(bb(bb(bb(bb(x1))))) |
(14) |
|
ab(ba(ab(bb(ba(x1))))) |
→ |
aa(aa(ab(ba(aa(x1))))) |
(15) |
|
ab(ba(ab(bb(bb(x1))))) |
→ |
aa(aa(ab(ba(ab(x1))))) |
(16) |
|
bb(ba(ab(bb(ba(x1))))) |
→ |
ba(aa(ab(ba(aa(x1))))) |
(17) |
|
bb(ba(ab(bb(bb(x1))))) |
→ |
ba(aa(ab(ba(ab(x1))))) |
(18) |
|
ab(bb(ba(ab(ba(x1))))) |
→ |
aa(ab(ba(aa(aa(x1))))) |
(19) |
|
ab(bb(ba(ab(bb(x1))))) |
→ |
aa(ab(ba(aa(ab(x1))))) |
(20) |
|
bb(bb(ba(ab(ba(x1))))) |
→ |
ba(ab(ba(aa(aa(x1))))) |
(21) |
|
bb(bb(ba(ab(bb(x1))))) |
→ |
ba(ab(ba(aa(ab(x1))))) |
(22) |
1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
aa#(aa(ab(ba(aa(x1))))) |
→ |
ab#(bb(bb(bb(ba(x1))))) |
(23) |
|
aa#(aa(ab(ba(aa(x1))))) |
→ |
bb#(bb(bb(ba(x1)))) |
(24) |
|
aa#(aa(ab(ba(aa(x1))))) |
→ |
bb#(bb(ba(x1))) |
(25) |
|
aa#(aa(ab(ba(aa(x1))))) |
→ |
bb#(ba(x1)) |
(26) |
|
aa#(aa(ab(ba(aa(x1))))) |
→ |
ba#(x1) |
(27) |
|
aa#(aa(ab(ba(ab(x1))))) |
→ |
ab#(bb(bb(bb(bb(x1))))) |
(28) |
|
aa#(aa(ab(ba(ab(x1))))) |
→ |
bb#(bb(bb(bb(x1)))) |
(29) |
|
aa#(aa(ab(ba(ab(x1))))) |
→ |
bb#(bb(bb(x1))) |
(30) |
|
aa#(aa(ab(ba(ab(x1))))) |
→ |
bb#(bb(x1)) |
(31) |
|
aa#(aa(ab(ba(ab(x1))))) |
→ |
bb#(x1) |
(32) |
|
ba#(aa(ab(ba(aa(x1))))) |
→ |
bb#(bb(bb(bb(ba(x1))))) |
(33) |
|
ba#(aa(ab(ba(aa(x1))))) |
→ |
bb#(bb(bb(ba(x1)))) |
(34) |
|
ba#(aa(ab(ba(aa(x1))))) |
→ |
bb#(bb(ba(x1))) |
(35) |
|
ba#(aa(ab(ba(aa(x1))))) |
→ |
bb#(ba(x1)) |
(36) |
|
ba#(aa(ab(ba(aa(x1))))) |
→ |
ba#(x1) |
(37) |
|
ba#(aa(ab(ba(ab(x1))))) |
→ |
bb#(bb(bb(bb(bb(x1))))) |
(38) |
|
ba#(aa(ab(ba(ab(x1))))) |
→ |
bb#(bb(bb(bb(x1)))) |
(39) |
|
ba#(aa(ab(ba(ab(x1))))) |
→ |
bb#(bb(bb(x1))) |
(40) |
|
ba#(aa(ab(ba(ab(x1))))) |
→ |
bb#(bb(x1)) |
(41) |
|
ba#(aa(ab(ba(ab(x1))))) |
→ |
bb#(x1) |
(42) |
|
ab#(ba(ab(bb(ba(x1))))) |
→ |
aa#(aa(ab(ba(aa(x1))))) |
(43) |
|
ab#(ba(ab(bb(ba(x1))))) |
→ |
aa#(ab(ba(aa(x1)))) |
(44) |
|
ab#(ba(ab(bb(ba(x1))))) |
→ |
ab#(ba(aa(x1))) |
(45) |
|
ab#(ba(ab(bb(ba(x1))))) |
→ |
ba#(aa(x1)) |
(46) |
|
ab#(ba(ab(bb(ba(x1))))) |
→ |
aa#(x1) |
(47) |
|
ab#(ba(ab(bb(bb(x1))))) |
→ |
aa#(aa(ab(ba(ab(x1))))) |
(48) |
|
ab#(ba(ab(bb(bb(x1))))) |
→ |
aa#(ab(ba(ab(x1)))) |
(49) |
|
ab#(ba(ab(bb(bb(x1))))) |
→ |
ab#(ba(ab(x1))) |
(50) |
|
ab#(ba(ab(bb(bb(x1))))) |
→ |
ba#(ab(x1)) |
(51) |
|
ab#(ba(ab(bb(bb(x1))))) |
→ |
ab#(x1) |
(52) |
|
bb#(ba(ab(bb(ba(x1))))) |
→ |
ba#(aa(ab(ba(aa(x1))))) |
(53) |
|
bb#(ba(ab(bb(ba(x1))))) |
→ |
aa#(ab(ba(aa(x1)))) |
(54) |
|
bb#(ba(ab(bb(ba(x1))))) |
→ |
ab#(ba(aa(x1))) |
(55) |
|
bb#(ba(ab(bb(ba(x1))))) |
→ |
ba#(aa(x1)) |
(56) |
|
bb#(ba(ab(bb(ba(x1))))) |
→ |
aa#(x1) |
(57) |
|
bb#(ba(ab(bb(bb(x1))))) |
→ |
ba#(aa(ab(ba(ab(x1))))) |
(58) |
|
bb#(ba(ab(bb(bb(x1))))) |
→ |
aa#(ab(ba(ab(x1)))) |
(59) |
|
bb#(ba(ab(bb(bb(x1))))) |
→ |
ab#(ba(ab(x1))) |
(60) |
|
bb#(ba(ab(bb(bb(x1))))) |
→ |
ba#(ab(x1)) |
(61) |
|
bb#(ba(ab(bb(bb(x1))))) |
→ |
ab#(x1) |
(62) |
|
ab#(bb(ba(ab(ba(x1))))) |
→ |
aa#(ab(ba(aa(aa(x1))))) |
(63) |
|
ab#(bb(ba(ab(ba(x1))))) |
→ |
ab#(ba(aa(aa(x1)))) |
(64) |
|
ab#(bb(ba(ab(ba(x1))))) |
→ |
ba#(aa(aa(x1))) |
(65) |
|
ab#(bb(ba(ab(ba(x1))))) |
→ |
aa#(aa(x1)) |
(66) |
|
ab#(bb(ba(ab(ba(x1))))) |
→ |
aa#(x1) |
(67) |
|
ab#(bb(ba(ab(bb(x1))))) |
→ |
aa#(ab(ba(aa(ab(x1))))) |
(68) |
|
ab#(bb(ba(ab(bb(x1))))) |
→ |
ab#(ba(aa(ab(x1)))) |
(69) |
|
ab#(bb(ba(ab(bb(x1))))) |
→ |
ba#(aa(ab(x1))) |
(70) |
|
ab#(bb(ba(ab(bb(x1))))) |
→ |
aa#(ab(x1)) |
(71) |
|
ab#(bb(ba(ab(bb(x1))))) |
→ |
ab#(x1) |
(72) |
|
bb#(bb(ba(ab(ba(x1))))) |
→ |
ba#(ab(ba(aa(aa(x1))))) |
(73) |
|
bb#(bb(ba(ab(ba(x1))))) |
→ |
ab#(ba(aa(aa(x1)))) |
(74) |
|
bb#(bb(ba(ab(ba(x1))))) |
→ |
ba#(aa(aa(x1))) |
(75) |
|
bb#(bb(ba(ab(ba(x1))))) |
→ |
aa#(aa(x1)) |
(76) |
|
bb#(bb(ba(ab(ba(x1))))) |
→ |
aa#(x1) |
(77) |
|
bb#(bb(ba(ab(bb(x1))))) |
→ |
ba#(ab(ba(aa(ab(x1))))) |
(78) |
|
bb#(bb(ba(ab(bb(x1))))) |
→ |
ab#(ba(aa(ab(x1)))) |
(79) |
|
bb#(bb(ba(ab(bb(x1))))) |
→ |
ba#(aa(ab(x1))) |
(80) |
|
bb#(bb(ba(ab(bb(x1))))) |
→ |
aa#(ab(x1)) |
(81) |
|
bb#(bb(ba(ab(bb(x1))))) |
→ |
ab#(x1) |
(82) |
1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [aa#(x1)] |
= |
1 + 1 · x1
|
| [aa(x1)] |
= |
1 + 1 · x1
|
| [ab(x1)] |
= |
1 + 1 · x1
|
| [ba(x1)] |
= |
1 + 1 · x1
|
| [ab#(x1)] |
= |
1 + 1 · x1
|
| [bb(x1)] |
= |
1 + 1 · x1
|
| [bb#(x1)] |
= |
1 · x1
|
| [ba#(x1)] |
= |
1 · x1
|
the
pairs
|
aa#(aa(ab(ba(aa(x1))))) |
→ |
bb#(bb(bb(ba(x1)))) |
(24) |
|
aa#(aa(ab(ba(aa(x1))))) |
→ |
bb#(bb(ba(x1))) |
(25) |
|
aa#(aa(ab(ba(aa(x1))))) |
→ |
bb#(ba(x1)) |
(26) |
|
aa#(aa(ab(ba(aa(x1))))) |
→ |
ba#(x1) |
(27) |
|
aa#(aa(ab(ba(ab(x1))))) |
→ |
bb#(bb(bb(bb(x1)))) |
(29) |
|
aa#(aa(ab(ba(ab(x1))))) |
→ |
bb#(bb(bb(x1))) |
(30) |
|
aa#(aa(ab(ba(ab(x1))))) |
→ |
bb#(bb(x1)) |
(31) |
|
aa#(aa(ab(ba(ab(x1))))) |
→ |
bb#(x1) |
(32) |
|
ba#(aa(ab(ba(aa(x1))))) |
→ |
bb#(bb(bb(ba(x1)))) |
(34) |
|
ba#(aa(ab(ba(aa(x1))))) |
→ |
bb#(bb(ba(x1))) |
(35) |
|
ba#(aa(ab(ba(aa(x1))))) |
→ |
bb#(ba(x1)) |
(36) |
|
ba#(aa(ab(ba(aa(x1))))) |
→ |
ba#(x1) |
(37) |
|
ba#(aa(ab(ba(ab(x1))))) |
→ |
bb#(bb(bb(bb(x1)))) |
(39) |
|
ba#(aa(ab(ba(ab(x1))))) |
→ |
bb#(bb(bb(x1))) |
(40) |
|
ba#(aa(ab(ba(ab(x1))))) |
→ |
bb#(bb(x1)) |
(41) |
|
ba#(aa(ab(ba(ab(x1))))) |
→ |
bb#(x1) |
(42) |
|
ab#(ba(ab(bb(ba(x1))))) |
→ |
aa#(ab(ba(aa(x1)))) |
(44) |
|
ab#(ba(ab(bb(ba(x1))))) |
→ |
ab#(ba(aa(x1))) |
(45) |
|
ab#(ba(ab(bb(ba(x1))))) |
→ |
ba#(aa(x1)) |
(46) |
|
ab#(ba(ab(bb(ba(x1))))) |
→ |
aa#(x1) |
(47) |
|
ab#(ba(ab(bb(bb(x1))))) |
→ |
aa#(ab(ba(ab(x1)))) |
(49) |
|
ab#(ba(ab(bb(bb(x1))))) |
→ |
ab#(ba(ab(x1))) |
(50) |
|
ab#(ba(ab(bb(bb(x1))))) |
→ |
ba#(ab(x1)) |
(51) |
|
ab#(ba(ab(bb(bb(x1))))) |
→ |
ab#(x1) |
(52) |
|
bb#(ba(ab(bb(ba(x1))))) |
→ |
ab#(ba(aa(x1))) |
(55) |
|
bb#(ba(ab(bb(ba(x1))))) |
→ |
ba#(aa(x1)) |
(56) |
|
bb#(ba(ab(bb(ba(x1))))) |
→ |
aa#(x1) |
(57) |
|
bb#(ba(ab(bb(bb(x1))))) |
→ |
ab#(ba(ab(x1))) |
(60) |
|
bb#(ba(ab(bb(bb(x1))))) |
→ |
ba#(ab(x1)) |
(61) |
|
bb#(ba(ab(bb(bb(x1))))) |
→ |
ab#(x1) |
(62) |
|
ab#(bb(ba(ab(ba(x1))))) |
→ |
ab#(ba(aa(aa(x1)))) |
(64) |
|
ab#(bb(ba(ab(ba(x1))))) |
→ |
ba#(aa(aa(x1))) |
(65) |
|
ab#(bb(ba(ab(ba(x1))))) |
→ |
aa#(aa(x1)) |
(66) |
|
ab#(bb(ba(ab(ba(x1))))) |
→ |
aa#(x1) |
(67) |
|
ab#(bb(ba(ab(bb(x1))))) |
→ |
ab#(ba(aa(ab(x1)))) |
(69) |
|
ab#(bb(ba(ab(bb(x1))))) |
→ |
ba#(aa(ab(x1))) |
(70) |
|
ab#(bb(ba(ab(bb(x1))))) |
→ |
aa#(ab(x1)) |
(71) |
|
ab#(bb(ba(ab(bb(x1))))) |
→ |
ab#(x1) |
(72) |
|
bb#(bb(ba(ab(ba(x1))))) |
→ |
ba#(aa(aa(x1))) |
(75) |
|
bb#(bb(ba(ab(ba(x1))))) |
→ |
aa#(aa(x1)) |
(76) |
|
bb#(bb(ba(ab(ba(x1))))) |
→ |
aa#(x1) |
(77) |
|
bb#(bb(ba(ab(bb(x1))))) |
→ |
ba#(aa(ab(x1))) |
(80) |
|
bb#(bb(ba(ab(bb(x1))))) |
→ |
aa#(ab(x1)) |
(81) |
|
bb#(bb(ba(ab(bb(x1))))) |
→ |
ab#(x1) |
(82) |
could be deleted.
1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.