Certification Problem
Input (TPDB SRS_Standard/Zantema_04/z005)
The rewrite relation of the following TRS is considered.
|
a(c(b(x1))) |
→ |
b(a(b(a(x1)))) |
(1) |
|
b(x1) |
→ |
c(a(c(x1))) |
(2) |
|
a(a(x1)) |
→ |
a(b(c(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
|
b(c(a(x1))) |
→ |
a(b(a(b(x1)))) |
(4) |
|
b(x1) |
→ |
c(a(c(x1))) |
(2) |
|
a(a(x1)) |
→ |
a(c(b(a(x1)))) |
(5) |
1.1 Closure Under Flat Contexts
Using the flat contexts
{b(☐), c(☐), a(☐)}
We obtain the transformed TRS
|
a(a(x1)) |
→ |
a(c(b(a(x1)))) |
(5) |
|
b(b(c(a(x1)))) |
→ |
b(a(b(a(b(x1))))) |
(6) |
|
c(b(c(a(x1)))) |
→ |
c(a(b(a(b(x1))))) |
(7) |
|
a(b(c(a(x1)))) |
→ |
a(a(b(a(b(x1))))) |
(8) |
|
b(b(x1)) |
→ |
b(c(a(c(x1)))) |
(9) |
|
c(b(x1)) |
→ |
c(c(a(c(x1)))) |
(10) |
|
a(b(x1)) |
→ |
a(c(a(c(x1)))) |
(11) |
1.1.1 Semantic Labeling
Root-labeling is applied.
We obtain the labeled TRS
|
aa(aa(x1)) |
→ |
ac(cb(ba(aa(x1)))) |
(12) |
|
aa(ac(x1)) |
→ |
ac(cb(ba(ac(x1)))) |
(13) |
|
aa(ab(x1)) |
→ |
ac(cb(ba(ab(x1)))) |
(14) |
|
bb(bc(ca(aa(x1)))) |
→ |
ba(ab(ba(ab(ba(x1))))) |
(15) |
|
bb(bc(ca(ac(x1)))) |
→ |
ba(ab(ba(ab(bc(x1))))) |
(16) |
|
bb(bc(ca(ab(x1)))) |
→ |
ba(ab(ba(ab(bb(x1))))) |
(17) |
|
cb(bc(ca(aa(x1)))) |
→ |
ca(ab(ba(ab(ba(x1))))) |
(18) |
|
cb(bc(ca(ac(x1)))) |
→ |
ca(ab(ba(ab(bc(x1))))) |
(19) |
|
cb(bc(ca(ab(x1)))) |
→ |
ca(ab(ba(ab(bb(x1))))) |
(20) |
|
ab(bc(ca(aa(x1)))) |
→ |
aa(ab(ba(ab(ba(x1))))) |
(21) |
|
ab(bc(ca(ac(x1)))) |
→ |
aa(ab(ba(ab(bc(x1))))) |
(22) |
|
ab(bc(ca(ab(x1)))) |
→ |
aa(ab(ba(ab(bb(x1))))) |
(23) |
|
bb(ba(x1)) |
→ |
bc(ca(ac(ca(x1)))) |
(24) |
|
bb(bc(x1)) |
→ |
bc(ca(ac(cc(x1)))) |
(25) |
|
bb(bb(x1)) |
→ |
bc(ca(ac(cb(x1)))) |
(26) |
|
cb(ba(x1)) |
→ |
cc(ca(ac(ca(x1)))) |
(27) |
|
cb(bc(x1)) |
→ |
cc(ca(ac(cc(x1)))) |
(28) |
|
cb(bb(x1)) |
→ |
cc(ca(ac(cb(x1)))) |
(29) |
|
ab(ba(x1)) |
→ |
ac(ca(ac(ca(x1)))) |
(30) |
|
ab(bc(x1)) |
→ |
ac(ca(ac(cc(x1)))) |
(31) |
|
ab(bb(x1)) |
→ |
ac(ca(ac(cb(x1)))) |
(32) |
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [aa(x1)] |
= |
1 · x1
|
| [ac(x1)] |
= |
1 · x1
|
| [cb(x1)] |
= |
1 · x1
|
| [ba(x1)] |
= |
1 · x1
|
| [ab(x1)] |
= |
1 · x1
|
| [bb(x1)] |
= |
1 · x1 + 1 |
| [bc(x1)] |
= |
1 · x1 + 1 |
| [ca(x1)] |
= |
1 · x1
|
| [cc(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
bb(bc(ca(aa(x1)))) |
→ |
ba(ab(ba(ab(ba(x1))))) |
(15) |
|
bb(bc(ca(ac(x1)))) |
→ |
ba(ab(ba(ab(bc(x1))))) |
(16) |
|
bb(bc(ca(ab(x1)))) |
→ |
ba(ab(ba(ab(bb(x1))))) |
(17) |
|
cb(bc(ca(aa(x1)))) |
→ |
ca(ab(ba(ab(ba(x1))))) |
(18) |
|
ab(bc(ca(aa(x1)))) |
→ |
aa(ab(ba(ab(ba(x1))))) |
(21) |
|
bb(bc(x1)) |
→ |
bc(ca(ac(cc(x1)))) |
(25) |
|
bb(bb(x1)) |
→ |
bc(ca(ac(cb(x1)))) |
(26) |
|
cb(bc(x1)) |
→ |
cc(ca(ac(cc(x1)))) |
(28) |
|
cb(bb(x1)) |
→ |
cc(ca(ac(cb(x1)))) |
(29) |
|
ab(bc(x1)) |
→ |
ac(ca(ac(cc(x1)))) |
(31) |
|
ab(bb(x1)) |
→ |
ac(ca(ac(cb(x1)))) |
(32) |
1.1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
aa#(aa(x1)) |
→ |
cb#(ba(aa(x1))) |
(33) |
|
aa#(ac(x1)) |
→ |
cb#(ba(ac(x1))) |
(34) |
|
aa#(ab(x1)) |
→ |
cb#(ba(ab(x1))) |
(35) |
|
cb#(bc(ca(ac(x1)))) |
→ |
ab#(ba(ab(bc(x1)))) |
(36) |
|
cb#(bc(ca(ac(x1)))) |
→ |
ab#(bc(x1)) |
(37) |
|
cb#(bc(ca(ab(x1)))) |
→ |
ab#(ba(ab(bb(x1)))) |
(38) |
|
cb#(bc(ca(ab(x1)))) |
→ |
ab#(bb(x1)) |
(39) |
|
cb#(bc(ca(ab(x1)))) |
→ |
bb#(x1) |
(40) |
|
ab#(bc(ca(ac(x1)))) |
→ |
aa#(ab(ba(ab(bc(x1))))) |
(41) |
|
ab#(bc(ca(ac(x1)))) |
→ |
ab#(ba(ab(bc(x1)))) |
(42) |
|
ab#(bc(ca(ac(x1)))) |
→ |
ab#(bc(x1)) |
(43) |
|
ab#(bc(ca(ab(x1)))) |
→ |
aa#(ab(ba(ab(bb(x1))))) |
(44) |
|
ab#(bc(ca(ab(x1)))) |
→ |
ab#(ba(ab(bb(x1)))) |
(45) |
|
ab#(bc(ca(ab(x1)))) |
→ |
ab#(bb(x1)) |
(46) |
|
ab#(bc(ca(ab(x1)))) |
→ |
bb#(x1) |
(47) |
1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.