Certification Problem
Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-561)
The rewrite relation of the following TRS is considered.
a(b(x1)) |
→ |
x1 |
(1) |
a(c(x1)) |
→ |
c(c(x1)) |
(2) |
b(c(x1)) |
→ |
a(b(a(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
{a(☐), b(☐), c(☐)}
We obtain the transformed TRS
a(a(b(x1))) |
→ |
a(x1) |
(4) |
b(a(b(x1))) |
→ |
b(x1) |
(5) |
c(a(b(x1))) |
→ |
c(x1) |
(6) |
a(a(c(x1))) |
→ |
a(c(c(x1))) |
(7) |
b(a(c(x1))) |
→ |
b(c(c(x1))) |
(8) |
c(a(c(x1))) |
→ |
c(c(c(x1))) |
(9) |
a(b(c(x1))) |
→ |
a(a(b(a(b(x1))))) |
(10) |
b(b(c(x1))) |
→ |
b(a(b(a(b(x1))))) |
(11) |
c(b(c(x1))) |
→ |
c(a(b(a(b(x1))))) |
(12) |
1.1 Semantic Labeling
Root-labeling is applied.
We obtain the labeled TRS
aa(ab(ba(x1))) |
→ |
aa(x1) |
(13) |
aa(ab(bb(x1))) |
→ |
ab(x1) |
(14) |
aa(ab(bc(x1))) |
→ |
ac(x1) |
(15) |
ba(ab(ba(x1))) |
→ |
ba(x1) |
(16) |
ba(ab(bb(x1))) |
→ |
bb(x1) |
(17) |
ba(ab(bc(x1))) |
→ |
bc(x1) |
(18) |
ca(ab(ba(x1))) |
→ |
ca(x1) |
(19) |
ca(ab(bb(x1))) |
→ |
cb(x1) |
(20) |
ca(ab(bc(x1))) |
→ |
cc(x1) |
(21) |
aa(ac(ca(x1))) |
→ |
ac(cc(ca(x1))) |
(22) |
aa(ac(cb(x1))) |
→ |
ac(cc(cb(x1))) |
(23) |
aa(ac(cc(x1))) |
→ |
ac(cc(cc(x1))) |
(24) |
ba(ac(ca(x1))) |
→ |
bc(cc(ca(x1))) |
(25) |
ba(ac(cb(x1))) |
→ |
bc(cc(cb(x1))) |
(26) |
ba(ac(cc(x1))) |
→ |
bc(cc(cc(x1))) |
(27) |
ca(ac(ca(x1))) |
→ |
cc(cc(ca(x1))) |
(28) |
ca(ac(cb(x1))) |
→ |
cc(cc(cb(x1))) |
(29) |
ca(ac(cc(x1))) |
→ |
cc(cc(cc(x1))) |
(30) |
ab(bc(ca(x1))) |
→ |
aa(ab(ba(ab(ba(x1))))) |
(31) |
ab(bc(cb(x1))) |
→ |
aa(ab(ba(ab(bb(x1))))) |
(32) |
ab(bc(cc(x1))) |
→ |
aa(ab(ba(ab(bc(x1))))) |
(33) |
bb(bc(ca(x1))) |
→ |
ba(ab(ba(ab(ba(x1))))) |
(34) |
bb(bc(cb(x1))) |
→ |
ba(ab(ba(ab(bb(x1))))) |
(35) |
bb(bc(cc(x1))) |
→ |
ba(ab(ba(ab(bc(x1))))) |
(36) |
cb(bc(ca(x1))) |
→ |
ca(ab(ba(ab(ba(x1))))) |
(37) |
cb(bc(cb(x1))) |
→ |
ca(ab(ba(ab(bb(x1))))) |
(38) |
cb(bc(cc(x1))) |
→ |
ca(ab(ba(ab(bc(x1))))) |
(39) |
1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
[aa(x1)] |
= |
1 · x1
|
[ab(x1)] |
= |
1 · x1
|
[ba(x1)] |
= |
1 · x1
|
[bb(x1)] |
= |
1 · x1 + 2 |
[bc(x1)] |
= |
1 · x1
|
[ac(x1)] |
= |
1 · x1
|
[ca(x1)] |
= |
1 · x1 + 2 |
[cb(x1)] |
= |
1 · x1 + 3 |
[cc(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
aa(ab(bb(x1))) |
→ |
ab(x1) |
(14) |
ca(ab(bb(x1))) |
→ |
cb(x1) |
(20) |
ca(ab(bc(x1))) |
→ |
cc(x1) |
(21) |
ca(ac(ca(x1))) |
→ |
cc(cc(ca(x1))) |
(28) |
ca(ac(cb(x1))) |
→ |
cc(cc(cb(x1))) |
(29) |
ca(ac(cc(x1))) |
→ |
cc(cc(cc(x1))) |
(30) |
ab(bc(ca(x1))) |
→ |
aa(ab(ba(ab(ba(x1))))) |
(31) |
ab(bc(cb(x1))) |
→ |
aa(ab(ba(ab(bb(x1))))) |
(32) |
bb(bc(ca(x1))) |
→ |
ba(ab(ba(ab(ba(x1))))) |
(34) |
bb(bc(cb(x1))) |
→ |
ba(ab(ba(ab(bb(x1))))) |
(35) |
bb(bc(cc(x1))) |
→ |
ba(ab(ba(ab(bc(x1))))) |
(36) |
cb(bc(ca(x1))) |
→ |
ca(ab(ba(ab(ba(x1))))) |
(37) |
cb(bc(cb(x1))) |
→ |
ca(ab(ba(ab(bb(x1))))) |
(38) |
cb(bc(cc(x1))) |
→ |
ca(ab(ba(ab(bc(x1))))) |
(39) |
1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
aa#(ab(ba(x1))) |
→ |
aa#(x1) |
(40) |
ca#(ab(ba(x1))) |
→ |
ca#(x1) |
(41) |
ab#(bc(cc(x1))) |
→ |
aa#(ab(ba(ab(bc(x1))))) |
(42) |
ab#(bc(cc(x1))) |
→ |
ab#(ba(ab(bc(x1)))) |
(43) |
ab#(bc(cc(x1))) |
→ |
ba#(ab(bc(x1))) |
(44) |
ab#(bc(cc(x1))) |
→ |
ab#(bc(x1)) |
(45) |
1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
ab#(bc(cc(x1))) |
→ |
ab#(bc(x1)) |
(45) |
ab#(bc(cc(x1))) |
→ |
ab#(ba(ab(bc(x1)))) |
(43) |
1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[ab#(x1)] |
= |
1 · x1
|
[bc(x1)] |
= |
1 · x1
|
[cc(x1)] |
= |
1 + 1 · x1
|
[ba(x1)] |
= |
1 · x1
|
[ab(x1)] |
= |
1 · x1
|
[aa(x1)] |
= |
1 + 1 · x1
|
[bb(x1)] |
= |
1 + 1 · x1
|
[ac(x1)] |
= |
1 + 1 · x1
|
[ca(x1)] |
= |
1 · x1
|
[cb(x1)] |
= |
1 · x1
|
the
pairs
ab#(bc(cc(x1))) |
→ |
ab#(bc(x1)) |
(45) |
ab#(bc(cc(x1))) |
→ |
ab#(ba(ab(bc(x1)))) |
(43) |
could be deleted.
1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
aa#(ab(ba(x1))) |
→ |
aa#(x1) |
(40) |
1.1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ab(x1)] |
= |
1 · x1
|
[ba(x1)] |
= |
1 · x1
|
[aa#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
aa#(ab(ba(x1))) |
→ |
aa#(x1) |
(40) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
ca#(ab(ba(x1))) |
→ |
ca#(x1) |
(41) |
1.1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ab(x1)] |
= |
1 · x1
|
[ba(x1)] |
= |
1 · x1
|
[ca#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
ca#(ab(ba(x1))) |
→ |
ca#(x1) |
(41) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.