Certification Problem
Input (TPDB SRS_Standard/Mixed_SRS/3)
The rewrite relation of the following TRS is considered.
|
a(a(x1)) |
→ |
a(b(a(x1))) |
(1) |
|
b(a(b(x1))) |
→ |
a(c(a(x1))) |
(2) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Closure Under Flat Contexts
Using the flat contexts
{a(☐), b(☐), c(☐)}
We obtain the transformed TRS
|
a(a(x1)) |
→ |
a(b(a(x1))) |
(1) |
|
a(b(a(b(x1)))) |
→ |
a(a(c(a(x1)))) |
(3) |
|
b(b(a(b(x1)))) |
→ |
b(a(c(a(x1)))) |
(4) |
|
c(b(a(b(x1)))) |
→ |
c(a(c(a(x1)))) |
(5) |
1.1 Semantic Labeling
Root-labeling is applied.
We obtain the labeled TRS
|
aa(aa(x1)) |
→ |
ab(ba(aa(x1))) |
(6) |
|
aa(ab(x1)) |
→ |
ab(ba(ab(x1))) |
(7) |
|
aa(ac(x1)) |
→ |
ab(ba(ac(x1))) |
(8) |
|
ab(ba(ab(ba(x1)))) |
→ |
aa(ac(ca(aa(x1)))) |
(9) |
|
ab(ba(ab(bb(x1)))) |
→ |
aa(ac(ca(ab(x1)))) |
(10) |
|
ab(ba(ab(bc(x1)))) |
→ |
aa(ac(ca(ac(x1)))) |
(11) |
|
bb(ba(ab(ba(x1)))) |
→ |
ba(ac(ca(aa(x1)))) |
(12) |
|
bb(ba(ab(bb(x1)))) |
→ |
ba(ac(ca(ab(x1)))) |
(13) |
|
bb(ba(ab(bc(x1)))) |
→ |
ba(ac(ca(ac(x1)))) |
(14) |
|
cb(ba(ab(ba(x1)))) |
→ |
ca(ac(ca(aa(x1)))) |
(15) |
|
cb(ba(ab(bb(x1)))) |
→ |
ca(ac(ca(ab(x1)))) |
(16) |
|
cb(ba(ab(bc(x1)))) |
→ |
ca(ac(ca(ac(x1)))) |
(17) |
1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [aa(x1)] |
= |
1 · x1
|
| [ab(x1)] |
= |
1 · x1
|
| [ba(x1)] |
= |
1 · x1
|
| [ac(x1)] |
= |
1 · x1
|
| [ca(x1)] |
= |
1 · x1
|
| [bb(x1)] |
= |
1 · x1 + 1 |
| [bc(x1)] |
= |
1 · x1 + 1 |
| [cb(x1)] |
= |
1 · x1 + 1 |
all of the following rules can be deleted.
|
ab(ba(ab(bb(x1)))) |
→ |
aa(ac(ca(ab(x1)))) |
(10) |
|
ab(ba(ab(bc(x1)))) |
→ |
aa(ac(ca(ac(x1)))) |
(11) |
|
bb(ba(ab(ba(x1)))) |
→ |
ba(ac(ca(aa(x1)))) |
(12) |
|
bb(ba(ab(bb(x1)))) |
→ |
ba(ac(ca(ab(x1)))) |
(13) |
|
bb(ba(ab(bc(x1)))) |
→ |
ba(ac(ca(ac(x1)))) |
(14) |
|
cb(ba(ab(ba(x1)))) |
→ |
ca(ac(ca(aa(x1)))) |
(15) |
|
cb(ba(ab(bb(x1)))) |
→ |
ca(ac(ca(ab(x1)))) |
(16) |
|
cb(ba(ab(bc(x1)))) |
→ |
ca(ac(ca(ac(x1)))) |
(17) |
1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
aa#(aa(x1)) |
→ |
ab#(ba(aa(x1))) |
(18) |
|
aa#(ab(x1)) |
→ |
ab#(ba(ab(x1))) |
(19) |
|
aa#(ac(x1)) |
→ |
ab#(ba(ac(x1))) |
(20) |
|
ab#(ba(ab(ba(x1)))) |
→ |
aa#(ac(ca(aa(x1)))) |
(21) |
|
ab#(ba(ab(ba(x1)))) |
→ |
aa#(x1) |
(22) |
1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.