Certification Problem
Input (TPDB SRS_Standard/Secret_05_SRS/aprove2)
The rewrite relation of the following TRS is considered.
|
g(h(x1)) |
→ |
g(f(s(x1))) |
(1) |
|
f(s(s(s(x1)))) |
→ |
h(f(s(h(x1)))) |
(2) |
|
f(h(x1)) |
→ |
h(f(s(h(x1)))) |
(3) |
|
h(x1) |
→ |
x1 |
(4) |
|
f(f(s(s(x1)))) |
→ |
s(s(s(f(f(x1))))) |
(5) |
|
b(a(x1)) |
→ |
a(b(x1)) |
(6) |
|
a(a(a(x1))) |
→ |
b(a(a(b(x1)))) |
(7) |
|
b(b(b(b(x1)))) |
→ |
a(x1) |
(8) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by matchbox @ termCOMP 2023)
1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [h(x1)] |
= |
x1 +
|
| [g(x1)] |
= |
x1 +
|
| [f(x1)] |
= |
x1 +
|
| [b(x1)] |
= |
x1 +
|
| [a(x1)] |
= |
x1 +
|
| [s(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
|
a(a(a(x1))) |
→ |
b(a(a(b(x1)))) |
(7) |
|
b(b(b(b(x1)))) |
→ |
a(x1) |
(8) |
1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
h(g(x1)) |
→ |
s(f(g(x1))) |
(9) |
|
s(s(s(f(x1)))) |
→ |
h(s(f(h(x1)))) |
(10) |
|
h(f(x1)) |
→ |
h(s(f(h(x1)))) |
(11) |
|
h(x1) |
→ |
x1 |
(4) |
|
s(s(f(f(x1)))) |
→ |
f(f(s(s(s(x1))))) |
(12) |
|
a(b(x1)) |
→ |
b(a(x1)) |
(13) |
1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
h#(g(x1)) |
→ |
s#(f(g(x1))) |
(14) |
|
h#(f(x1)) |
→ |
h#(x1) |
(15) |
|
h#(f(x1)) |
→ |
h#(s(f(h(x1)))) |
(16) |
|
h#(f(x1)) |
→ |
s#(f(h(x1))) |
(17) |
|
a#(b(x1)) |
→ |
a#(x1) |
(18) |
|
s#(s(f(f(x1)))) |
→ |
s#(x1) |
(19) |
|
s#(s(f(f(x1)))) |
→ |
s#(s(x1)) |
(20) |
|
s#(s(f(f(x1)))) |
→ |
s#(s(s(x1))) |
(21) |
|
s#(s(s(f(x1)))) |
→ |
h#(x1) |
(22) |
|
s#(s(s(f(x1)))) |
→ |
h#(s(f(h(x1)))) |
(23) |
|
s#(s(s(f(x1)))) |
→ |
s#(f(h(x1))) |
(24) |
1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [h(x1)] |
= |
x1 +
|
| [g(x1)] |
= |
x1 +
|
| [f(x1)] |
= |
x1 +
|
| [b(x1)] |
= |
x1 +
|
| [a(x1)] |
= |
x1 +
|
| [s(x1)] |
= |
x1 +
|
| [h#(x1)] |
= |
x1 +
|
| [a#(x1)] |
= |
x1 +
|
| [s#(x1)] |
= |
x1 +
|
together with the usable
rules
|
h(g(x1)) |
→ |
s(f(g(x1))) |
(9) |
|
s(s(s(f(x1)))) |
→ |
h(s(f(h(x1)))) |
(10) |
|
h(f(x1)) |
→ |
h(s(f(h(x1)))) |
(11) |
|
h(x1) |
→ |
x1 |
(4) |
|
s(s(f(f(x1)))) |
→ |
f(f(s(s(s(x1))))) |
(12) |
|
a(b(x1)) |
→ |
b(a(x1)) |
(13) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
and
no rules
could be deleted.
1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.