Certification Problem
Input (TPDB SRS_Standard/Secret_06_SRS/5-matchbox)
The rewrite relation of the following TRS is considered.
|
a(b(c(x1))) |
→ |
a(a(b(x1))) |
(1) |
|
a(b(c(x1))) |
→ |
b(c(b(c(x1)))) |
(2) |
|
a(b(c(x1))) |
→ |
c(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
|
c(b(a(x1))) |
→ |
b(a(a(x1))) |
(4) |
|
c(b(a(x1))) |
→ |
c(b(c(b(x1)))) |
(5) |
|
c(b(a(x1))) |
→ |
a(c(b(c(x1)))) |
(6) |
1.1 Closure Under Flat Contexts
Using the flat contexts
{c(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
c(b(a(x1))) |
→ |
c(b(c(b(x1)))) |
(5) |
|
c(c(b(a(x1)))) |
→ |
c(b(a(a(x1)))) |
(7) |
|
b(c(b(a(x1)))) |
→ |
b(b(a(a(x1)))) |
(8) |
|
a(c(b(a(x1)))) |
→ |
a(b(a(a(x1)))) |
(9) |
|
c(c(b(a(x1)))) |
→ |
c(a(c(b(c(x1))))) |
(10) |
|
b(c(b(a(x1)))) |
→ |
b(a(c(b(c(x1))))) |
(11) |
|
a(c(b(a(x1)))) |
→ |
a(a(c(b(c(x1))))) |
(12) |
1.1.1 Semantic Labeling
Root-labeling is applied.
We obtain the labeled TRS
|
cb(ba(ac(x1))) |
→ |
cb(bc(cb(bc(x1)))) |
(13) |
|
cb(ba(ab(x1))) |
→ |
cb(bc(cb(bb(x1)))) |
(14) |
|
cb(ba(aa(x1))) |
→ |
cb(bc(cb(ba(x1)))) |
(15) |
|
cc(cb(ba(ac(x1)))) |
→ |
cb(ba(aa(ac(x1)))) |
(16) |
|
cc(cb(ba(ab(x1)))) |
→ |
cb(ba(aa(ab(x1)))) |
(17) |
|
cc(cb(ba(aa(x1)))) |
→ |
cb(ba(aa(aa(x1)))) |
(18) |
|
bc(cb(ba(ac(x1)))) |
→ |
bb(ba(aa(ac(x1)))) |
(19) |
|
bc(cb(ba(ab(x1)))) |
→ |
bb(ba(aa(ab(x1)))) |
(20) |
|
bc(cb(ba(aa(x1)))) |
→ |
bb(ba(aa(aa(x1)))) |
(21) |
|
ac(cb(ba(ac(x1)))) |
→ |
ab(ba(aa(ac(x1)))) |
(22) |
|
ac(cb(ba(ab(x1)))) |
→ |
ab(ba(aa(ab(x1)))) |
(23) |
|
ac(cb(ba(aa(x1)))) |
→ |
ab(ba(aa(aa(x1)))) |
(24) |
|
cc(cb(ba(ac(x1)))) |
→ |
ca(ac(cb(bc(cc(x1))))) |
(25) |
|
cc(cb(ba(ab(x1)))) |
→ |
ca(ac(cb(bc(cb(x1))))) |
(26) |
|
cc(cb(ba(aa(x1)))) |
→ |
ca(ac(cb(bc(ca(x1))))) |
(27) |
|
bc(cb(ba(ac(x1)))) |
→ |
ba(ac(cb(bc(cc(x1))))) |
(28) |
|
bc(cb(ba(ab(x1)))) |
→ |
ba(ac(cb(bc(cb(x1))))) |
(29) |
|
bc(cb(ba(aa(x1)))) |
→ |
ba(ac(cb(bc(ca(x1))))) |
(30) |
|
ac(cb(ba(ac(x1)))) |
→ |
aa(ac(cb(bc(cc(x1))))) |
(31) |
|
ac(cb(ba(ab(x1)))) |
→ |
aa(ac(cb(bc(cb(x1))))) |
(32) |
|
ac(cb(ba(aa(x1)))) |
→ |
aa(ac(cb(bc(ca(x1))))) |
(33) |
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [cb(x1)] |
= |
1 · x1
|
| [ba(x1)] |
= |
1 · x1 + 1 |
| [ac(x1)] |
= |
1 · x1
|
| [bc(x1)] |
= |
1 · x1
|
| [ab(x1)] |
= |
1 · x1
|
| [bb(x1)] |
= |
1 · x1
|
| [aa(x1)] |
= |
1 · x1
|
| [cc(x1)] |
= |
1 · x1
|
| [ca(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
cb(ba(ac(x1))) |
→ |
cb(bc(cb(bc(x1)))) |
(13) |
|
cb(ba(ab(x1))) |
→ |
cb(bc(cb(bb(x1)))) |
(14) |
|
cc(cb(ba(ac(x1)))) |
→ |
ca(ac(cb(bc(cc(x1))))) |
(25) |
|
cc(cb(ba(ab(x1)))) |
→ |
ca(ac(cb(bc(cb(x1))))) |
(26) |
|
cc(cb(ba(aa(x1)))) |
→ |
ca(ac(cb(bc(ca(x1))))) |
(27) |
|
ac(cb(ba(ac(x1)))) |
→ |
aa(ac(cb(bc(cc(x1))))) |
(31) |
|
ac(cb(ba(ab(x1)))) |
→ |
aa(ac(cb(bc(cb(x1))))) |
(32) |
|
ac(cb(ba(aa(x1)))) |
→ |
aa(ac(cb(bc(ca(x1))))) |
(33) |
1.1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
cb#(ba(aa(x1))) |
→ |
cb#(bc(cb(ba(x1)))) |
(34) |
|
cb#(ba(aa(x1))) |
→ |
bc#(cb(ba(x1))) |
(35) |
|
cb#(ba(aa(x1))) |
→ |
cb#(ba(x1)) |
(36) |
|
cc#(cb(ba(ac(x1)))) |
→ |
cb#(ba(aa(ac(x1)))) |
(37) |
|
cc#(cb(ba(ab(x1)))) |
→ |
cb#(ba(aa(ab(x1)))) |
(38) |
|
cc#(cb(ba(aa(x1)))) |
→ |
cb#(ba(aa(aa(x1)))) |
(39) |
|
bc#(cb(ba(ac(x1)))) |
→ |
ac#(cb(bc(cc(x1)))) |
(40) |
|
bc#(cb(ba(ac(x1)))) |
→ |
cb#(bc(cc(x1))) |
(41) |
|
bc#(cb(ba(ac(x1)))) |
→ |
bc#(cc(x1)) |
(42) |
|
bc#(cb(ba(ac(x1)))) |
→ |
cc#(x1) |
(43) |
|
bc#(cb(ba(ab(x1)))) |
→ |
ac#(cb(bc(cb(x1)))) |
(44) |
|
bc#(cb(ba(ab(x1)))) |
→ |
cb#(bc(cb(x1))) |
(45) |
|
bc#(cb(ba(ab(x1)))) |
→ |
bc#(cb(x1)) |
(46) |
|
bc#(cb(ba(ab(x1)))) |
→ |
cb#(x1) |
(47) |
|
bc#(cb(ba(aa(x1)))) |
→ |
ac#(cb(bc(ca(x1)))) |
(48) |
|
bc#(cb(ba(aa(x1)))) |
→ |
cb#(bc(ca(x1))) |
(49) |
|
bc#(cb(ba(aa(x1)))) |
→ |
bc#(ca(x1)) |
(50) |
1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.