Certification Problem
Input (TPDB SRS_Standard/Secret_06_SRS/secr9)
The rewrite relation of the following TRS is considered.
|
a(c(x1)) |
→ |
c(b(c(c(a(x1))))) |
(1) |
|
b(b(b(x1))) |
→ |
c(b(x1)) |
(2) |
|
d(d(x1)) |
→ |
d(b(d(b(d(x1))))) |
(3) |
|
a(a(x1)) |
→ |
a(d(a(x1))) |
(4) |
|
a(b(x1)) |
→ |
c(c(a(x1))) |
(5) |
|
c(c(x1)) |
→ |
c(b(c(b(c(x1))))) |
(6) |
|
c(c(c(x1))) |
→ |
c(b(b(x1))) |
(7) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
a#(c(x1)) |
→ |
c#(b(c(c(a(x1))))) |
(8) |
|
a#(c(x1)) |
→ |
b#(c(c(a(x1)))) |
(9) |
|
a#(c(x1)) |
→ |
c#(c(a(x1))) |
(10) |
|
a#(c(x1)) |
→ |
c#(a(x1)) |
(11) |
|
a#(c(x1)) |
→ |
a#(x1) |
(12) |
|
b#(b(b(x1))) |
→ |
c#(b(x1)) |
(13) |
|
d#(d(x1)) |
→ |
d#(b(d(b(d(x1))))) |
(14) |
|
d#(d(x1)) |
→ |
b#(d(b(d(x1)))) |
(15) |
|
d#(d(x1)) |
→ |
d#(b(d(x1))) |
(16) |
|
d#(d(x1)) |
→ |
b#(d(x1)) |
(17) |
|
a#(a(x1)) |
→ |
a#(d(a(x1))) |
(18) |
|
a#(a(x1)) |
→ |
d#(a(x1)) |
(19) |
|
a#(b(x1)) |
→ |
c#(c(a(x1))) |
(20) |
|
a#(b(x1)) |
→ |
c#(a(x1)) |
(21) |
|
a#(b(x1)) |
→ |
a#(x1) |
(22) |
|
c#(c(x1)) |
→ |
c#(b(c(b(c(x1))))) |
(23) |
|
c#(c(x1)) |
→ |
b#(c(b(c(x1)))) |
(24) |
|
c#(c(x1)) |
→ |
c#(b(c(x1))) |
(25) |
|
c#(c(x1)) |
→ |
b#(c(x1)) |
(26) |
|
c#(c(c(x1))) |
→ |
c#(b(b(x1))) |
(27) |
|
c#(c(c(x1))) |
→ |
b#(b(x1)) |
(28) |
|
c#(c(c(x1))) |
→ |
b#(x1) |
(29) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
a#(a(x1)) |
→ |
a#(d(a(x1))) |
(18) |
|
a#(c(x1)) |
→ |
a#(x1) |
(12) |
|
a#(b(x1)) |
→ |
a#(x1) |
(22) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [a#(x1)] |
= |
1 · x1
|
| [a(x1)] |
= |
1 · x1
|
| [d(x1)] |
= |
0 |
| [c(x1)] |
= |
1 + 1 · x1
|
| [b(x1)] |
= |
1 · x1
|
together with the usable
rule
|
d(d(x1)) |
→ |
d(b(d(b(d(x1))))) |
(3) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [b(x1)] |
= |
1 · x1
|
| [a#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
a#(b(x1)) |
→ |
a#(x1) |
(22) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
|
a#(a(x1)) |
→ |
a#(d(a(x1))) |
(18) |
1.1.1.1.2 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [a#(x1)] |
= |
1 · x1
|
| [a(x1)] |
= |
1 + 1 · x1
|
| [d(x1)] |
= |
0 |
| [c(x1)] |
= |
0 |
| [b(x1)] |
= |
1 · x1
|
together with the usable
rule
|
d(d(x1)) |
→ |
d(b(d(b(d(x1))))) |
(3) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
a#(a(x1)) |
→ |
a#(d(a(x1))) |
(18) |
could be deleted.
1.1.1.1.2.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
d#(d(x1)) |
→ |
d#(b(d(x1))) |
(16) |
|
d#(d(x1)) |
→ |
d#(b(d(b(d(x1))))) |
(14) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [d(x1)] |
= |
1 · x1
|
| [b(x1)] |
= |
1 · x1
|
| [c(x1)] |
= |
1 · x1
|
| [d#(x1)] |
= |
1 · x1
|
together with the usable
rules
|
d(d(x1)) |
→ |
d(b(d(b(d(x1))))) |
(3) |
|
b(b(b(x1))) |
→ |
c(b(x1)) |
(2) |
|
c(c(c(x1))) |
→ |
c(b(b(x1))) |
(7) |
|
c(c(x1)) |
→ |
c(b(c(b(c(x1))))) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.2.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [d#(x1)] |
= |
1 · x1
|
| [d(x1)] |
= |
1 + 1 · x1
|
| [b(x1)] |
= |
0 |
| [c(x1)] |
= |
1 · x1
|
together with the usable
rules
|
b(b(b(x1))) |
→ |
c(b(x1)) |
(2) |
|
c(c(x1)) |
→ |
c(b(c(b(c(x1))))) |
(6) |
|
c(c(c(x1))) |
→ |
c(b(b(x1))) |
(7) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
d#(d(x1)) |
→ |
d#(b(d(x1))) |
(16) |
|
d#(d(x1)) |
→ |
d#(b(d(b(d(x1))))) |
(14) |
could be deleted.
1.1.2.1.1 P is empty
There are no pairs anymore.
-
The
3rd
component contains the
pair
|
c#(c(x1)) |
→ |
c#(b(c(b(c(x1))))) |
(23) |
|
c#(c(x1)) |
→ |
c#(b(c(x1))) |
(25) |
|
c#(c(c(x1))) |
→ |
c#(b(b(x1))) |
(27) |
|
c#(c(c(x1))) |
→ |
b#(b(x1)) |
(28) |
|
b#(b(b(x1))) |
→ |
c#(b(x1)) |
(13) |
|
c#(c(c(x1))) |
→ |
b#(x1) |
(29) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [b(x1)] |
= |
1 · x1
|
| [c(x1)] |
= |
1 · x1
|
| [c#(x1)] |
= |
1 · x1
|
| [b#(x1)] |
= |
1 · x1
|
together with the usable
rules
|
b(b(b(x1))) |
→ |
c(b(x1)) |
(2) |
|
c(c(c(x1))) |
→ |
c(b(b(x1))) |
(7) |
|
c(c(x1)) |
→ |
c(b(c(b(c(x1))))) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.3.1 Reduction Pair Processor
Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals
| [c#(x1)] |
= |
+ · x1
|
| [c(x1)] |
= |
+ · x1
|
| [b(x1)] |
= |
+ · x1
|
| [b#(x1)] |
= |
+ · x1
|
the
pair
|
c#(c(c(x1))) |
→ |
c#(b(b(x1))) |
(27) |
could be deleted.
1.1.3.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals
| [c#(x1)] |
= |
+ · x1
|
| [c(x1)] |
= |
+ · x1
|
| [b(x1)] |
= |
+ · x1
|
| [b#(x1)] |
= |
+ · x1
|
the
pairs
|
c#(c(c(x1))) |
→ |
b#(b(x1)) |
(28) |
|
c#(c(c(x1))) |
→ |
b#(x1) |
(29) |
could be deleted.
1.1.3.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.