Certification Problem
Input (TPDB SRS_Standard/Secret_05_SRS/aprove4)
The rewrite relation of the following TRS is considered.
|
log(s(x1)) |
→ |
s(log(half(s(x1)))) |
(1) |
|
half(0(x1)) |
→ |
0(s(s(half(x1)))) |
(2) |
|
half(s(0(x1))) |
→ |
0(x1) |
(3) |
|
half(s(s(x1))) |
→ |
s(half(p(s(s(x1))))) |
(4) |
|
half(half(s(s(s(s(x1)))))) |
→ |
s(s(half(half(x1)))) |
(5) |
|
p(s(s(s(x1)))) |
→ |
s(p(s(s(x1)))) |
(6) |
|
s(s(p(s(x1)))) |
→ |
s(s(x1)) |
(7) |
|
0(x1) |
→ |
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
| [0(x1)] |
= |
x1 +
|
| [s(x1)] |
= |
x1 +
|
| [p(x1)] |
= |
x1 +
|
| [log(x1)] |
= |
x1 +
|
| [half(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
s(log(x1)) |
→ |
s(half(log(s(x1)))) |
(9) |
|
0(half(x1)) |
→ |
half(s(s(0(x1)))) |
(10) |
|
0(s(half(x1))) |
→ |
0(x1) |
(11) |
|
s(s(half(x1))) |
→ |
s(s(p(half(s(x1))))) |
(12) |
|
s(s(s(s(half(half(x1)))))) |
→ |
half(half(s(s(x1)))) |
(13) |
|
s(s(s(p(x1)))) |
→ |
s(s(p(s(x1)))) |
(14) |
|
s(p(s(s(x1)))) |
→ |
s(s(x1)) |
(15) |
1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
0#(s(half(x1))) |
→ |
0#(x1) |
(16) |
|
0#(half(x1)) |
→ |
0#(x1) |
(17) |
|
0#(half(x1)) |
→ |
s#(0(x1)) |
(18) |
|
0#(half(x1)) |
→ |
s#(s(0(x1))) |
(19) |
|
s#(s(s(s(half(half(x1)))))) |
→ |
s#(x1) |
(20) |
|
s#(s(s(s(half(half(x1)))))) |
→ |
s#(s(x1)) |
(21) |
|
s#(s(s(p(x1)))) |
→ |
s#(x1) |
(22) |
|
s#(s(s(p(x1)))) |
→ |
s#(s(p(s(x1)))) |
(23) |
|
s#(s(s(p(x1)))) |
→ |
s#(p(s(x1))) |
(24) |
|
s#(s(half(x1))) |
→ |
s#(x1) |
(25) |
|
s#(s(half(x1))) |
→ |
s#(s(p(half(s(x1))))) |
(26) |
|
s#(s(half(x1))) |
→ |
s#(p(half(s(x1)))) |
(27) |
|
s#(log(x1)) |
→ |
s#(x1) |
(28) |
|
s#(log(x1)) |
→ |
s#(half(log(s(x1)))) |
(29) |
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
| [0(x1)] |
= |
x1 +
|
| [s(x1)] |
= |
x1 +
|
| [p(x1)] |
= |
x1 +
|
| [log(x1)] |
= |
x1 +
|
| [half(x1)] |
= |
x1 +
|
| [0#(x1)] |
= |
x1 +
|
| [s#(x1)] |
= |
x1 +
|
together with the usable
rules
|
s(log(x1)) |
→ |
s(half(log(s(x1)))) |
(9) |
|
0(half(x1)) |
→ |
half(s(s(0(x1)))) |
(10) |
|
0(s(half(x1))) |
→ |
0(x1) |
(11) |
|
s(s(half(x1))) |
→ |
s(s(p(half(s(x1))))) |
(12) |
|
s(s(s(s(half(half(x1)))))) |
→ |
half(half(s(s(x1)))) |
(13) |
|
s(s(s(p(x1)))) |
→ |
s(s(p(s(x1)))) |
(14) |
|
s(p(s(s(x1)))) |
→ |
s(s(x1)) |
(15) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
0#(half(x1)) |
→ |
s#(0(x1)) |
(18) |
|
0#(half(x1)) |
→ |
s#(s(0(x1))) |
(19) |
|
s#(log(x1)) |
→ |
s#(x1) |
(28) |
and
no rules
could be deleted.
1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.