Certification Problem
Input (TPDB SRS_Standard/Zantema_04/z122)
The rewrite relation of the following TRS is considered.
|
a(a(x1)) |
→ |
b(c(x1)) |
(1) |
|
b(b(x1)) |
→ |
c(d(x1)) |
(2) |
|
c(c(x1)) |
→ |
d(d(d(x1))) |
(3) |
|
d(c(x1)) |
→ |
b(f(x1)) |
(4) |
|
d(d(d(x1))) |
→ |
a(c(x1)) |
(5) |
|
f(f(x1)) |
→ |
f(b(x1)) |
(6) |
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(☐), d(☐), f(☐)}
We obtain the transformed TRS
|
f(f(x1)) |
→ |
f(b(x1)) |
(6) |
|
a(a(a(x1))) |
→ |
a(b(c(x1))) |
(7) |
|
b(a(a(x1))) |
→ |
b(b(c(x1))) |
(8) |
|
c(a(a(x1))) |
→ |
c(b(c(x1))) |
(9) |
|
d(a(a(x1))) |
→ |
d(b(c(x1))) |
(10) |
|
f(a(a(x1))) |
→ |
f(b(c(x1))) |
(11) |
|
a(b(b(x1))) |
→ |
a(c(d(x1))) |
(12) |
|
b(b(b(x1))) |
→ |
b(c(d(x1))) |
(13) |
|
c(b(b(x1))) |
→ |
c(c(d(x1))) |
(14) |
|
d(b(b(x1))) |
→ |
d(c(d(x1))) |
(15) |
|
f(b(b(x1))) |
→ |
f(c(d(x1))) |
(16) |
|
a(c(c(x1))) |
→ |
a(d(d(d(x1)))) |
(17) |
|
b(c(c(x1))) |
→ |
b(d(d(d(x1)))) |
(18) |
|
c(c(c(x1))) |
→ |
c(d(d(d(x1)))) |
(19) |
|
d(c(c(x1))) |
→ |
d(d(d(d(x1)))) |
(20) |
|
f(c(c(x1))) |
→ |
f(d(d(d(x1)))) |
(21) |
|
a(d(c(x1))) |
→ |
a(b(f(x1))) |
(22) |
|
b(d(c(x1))) |
→ |
b(b(f(x1))) |
(23) |
|
c(d(c(x1))) |
→ |
c(b(f(x1))) |
(24) |
|
d(d(c(x1))) |
→ |
d(b(f(x1))) |
(25) |
|
f(d(c(x1))) |
→ |
f(b(f(x1))) |
(26) |
|
a(d(d(d(x1)))) |
→ |
a(a(c(x1))) |
(27) |
|
b(d(d(d(x1)))) |
→ |
b(a(c(x1))) |
(28) |
|
c(d(d(d(x1)))) |
→ |
c(a(c(x1))) |
(29) |
|
d(d(d(d(x1)))) |
→ |
d(a(c(x1))) |
(30) |
|
f(d(d(d(x1)))) |
→ |
f(a(c(x1))) |
(31) |
1.1 Semantic Labeling
Root-labeling is applied.
We obtain the labeled TRS
There are 130 ruless (increase limit for explicit display).
1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [ff(x1)] |
= |
1 · x1 + 85 |
| [fb(x1)] |
= |
1 · x1
|
| [bf(x1)] |
= |
1 · x1 + 165 |
| [bb(x1)] |
= |
1 · x1 + 83 |
| [fa(x1)] |
= |
1 · x1
|
| [ba(x1)] |
= |
1 · x1 + 83 |
| [fc(x1)] |
= |
1 · x1
|
| [bc(x1)] |
= |
1 · x1 + 84 |
| [fd(x1)] |
= |
1 · x1
|
| [bd(x1)] |
= |
1 · x1 + 84 |
| [aa(x1)] |
= |
1 · x1 + 93 |
| [af(x1)] |
= |
1 · x1 + 177 |
| [ab(x1)] |
= |
1 · x1 + 91 |
| [cf(x1)] |
= |
1 · x1 + 185 |
| [cb(x1)] |
= |
1 · x1 + 99 |
| [ca(x1)] |
= |
1 · x1 + 100 |
| [ac(x1)] |
= |
1 · x1 + 92 |
| [cc(x1)] |
= |
1 · x1 + 100 |
| [ad(x1)] |
= |
1 · x1 + 92 |
| [cd(x1)] |
= |
1 · x1 + 99 |
| [da(x1)] |
= |
1 · x1 + 65 |
| [db(x1)] |
= |
1 · x1 + 65 |
| [df(x1)] |
= |
1 · x1 + 147 |
| [dc(x1)] |
= |
1 · x1 + 66 |
| [dd(x1)] |
= |
1 · x1 + 66 |
all of the following rules can be deleted.
There are 126 ruless (increase limit for explicit display).
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [cd(x1)] |
= |
1 · x1 + 1 |
| [dc(x1)] |
= |
1 · x1
|
| [cb(x1)] |
= |
1 · x1
|
| [bf(x1)] |
= |
1 · x1
|
| [fb(x1)] |
= |
1 · x1
|
| [fd(x1)] |
= |
1 · x1 + 2 |
all of the following rules can be deleted.
|
cd(dc(cb(x1))) |
→ |
cb(bf(fb(x1))) |
(123) |
|
fd(dc(cb(x1))) |
→ |
fb(bf(fb(x1))) |
(133) |
|
fd(dc(cd(x1))) |
→ |
fb(bf(fd(x1))) |
(136) |
1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [cd(x1)] |
= |
1 · x1
|
| [dc(x1)] |
= |
1 · x1 + 1 |
| [cb(x1)] |
= |
1 · x1
|
| [bf(x1)] |
= |
1 · x1
|
| [fd(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
cd(dc(cd(x1))) |
→ |
cb(bf(fd(x1))) |
(126) |
1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.