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.