Certification Problem
Input (TPDB SRS_Standard/Waldmann_06_SRS/sym-4)
The rewrite relation of the following TRS is considered.
|
b(b(x1)) |
→ |
c(c(c(c(x1)))) |
(1) |
|
c(x1) |
→ |
x1 |
(2) |
|
b(c(b(x1))) |
→ |
b(b(b(x1))) |
(3) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Closure Under Flat Contexts
Using the flat contexts
{b(☐), c(☐)}
We obtain the transformed TRS
|
b(c(b(x1))) |
→ |
b(b(b(x1))) |
(3) |
|
b(b(b(x1))) |
→ |
b(c(c(c(c(x1))))) |
(4) |
|
c(b(b(x1))) |
→ |
c(c(c(c(c(x1))))) |
(5) |
|
b(c(x1)) |
→ |
b(x1) |
(6) |
|
c(c(x1)) |
→ |
c(x1) |
(7) |
1.1 Semantic Labeling
Root-labeling is applied.
We obtain the labeled TRS
|
bc(cb(bb(x1))) |
→ |
bb(bb(bb(x1))) |
(8) |
|
bc(cb(bc(x1))) |
→ |
bb(bb(bc(x1))) |
(9) |
|
bb(bb(bb(x1))) |
→ |
bc(cc(cc(cc(cb(x1))))) |
(10) |
|
bb(bb(bc(x1))) |
→ |
bc(cc(cc(cc(cc(x1))))) |
(11) |
|
cb(bb(bb(x1))) |
→ |
cc(cc(cc(cc(cb(x1))))) |
(12) |
|
cb(bb(bc(x1))) |
→ |
cc(cc(cc(cc(cc(x1))))) |
(13) |
|
bc(cb(x1)) |
→ |
bb(x1) |
(14) |
|
bc(cc(x1)) |
→ |
bc(x1) |
(15) |
|
cc(cb(x1)) |
→ |
cb(x1) |
(16) |
|
cc(cc(x1)) |
→ |
cc(x1) |
(17) |
1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [bc(x1)] |
= |
1 · x1
|
| [cb(x1)] |
= |
1 · x1 + 2 |
| [bb(x1)] |
= |
1 · x1 + 1 |
| [cc(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
bb(bb(bb(x1))) |
→ |
bc(cc(cc(cc(cb(x1))))) |
(10) |
|
bb(bb(bc(x1))) |
→ |
bc(cc(cc(cc(cc(x1))))) |
(11) |
|
cb(bb(bb(x1))) |
→ |
cc(cc(cc(cc(cb(x1))))) |
(12) |
|
cb(bb(bc(x1))) |
→ |
cc(cc(cc(cc(cc(x1))))) |
(13) |
|
bc(cb(x1)) |
→ |
bb(x1) |
(14) |
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [bc(x1)] |
= |
1 · x1
|
| [cb(x1)] |
= |
1 · x1 + 1 |
| [bb(x1)] |
= |
1 · x1
|
| [cc(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
bc(cb(bb(x1))) |
→ |
bb(bb(bb(x1))) |
(8) |
|
bc(cb(bc(x1))) |
→ |
bb(bb(bc(x1))) |
(9) |
1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [bc(x1)] |
= |
1 · x1
|
| [cc(x1)] |
= |
1 · x1 + 1 |
| [cb(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
bc(cc(x1)) |
→ |
bc(x1) |
(15) |
|
cc(cb(x1)) |
→ |
cb(x1) |
(16) |
|
cc(cc(x1)) |
→ |
cc(x1) |
(17) |
1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.