Certification Problem
Input (TPDB SRS_Standard/Trafo_06/dup09)
The rewrite relation of the following TRS is considered.
|
0(0(*(*(x1)))) |
→ |
*(*(1(1(x1)))) |
(1) |
|
1(1(*(*(x1)))) |
→ |
0(0(#(#(x1)))) |
(2) |
|
#(#(0(0(x1)))) |
→ |
0(0(#(#(x1)))) |
(3) |
|
#(#(1(1(x1)))) |
→ |
1(1(#(#(x1)))) |
(4) |
|
#(#($($(x1)))) |
→ |
*(*($($(x1)))) |
(5) |
|
#(#(#(#(x1)))) |
→ |
#(#(x1)) |
(6) |
|
#(#(*(*(x1)))) |
→ |
*(*(x1)) |
(7) |
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
| [*(x1)] |
= |
x1 +
|
| [$(x1)] |
= |
x1 +
|
| [#(x1)] |
= |
x1 +
|
| [1(x1)] |
= |
x1 +
|
| [0(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
|
#(#(#(#(x1)))) |
→ |
#(#(x1)) |
(6) |
|
#(#(*(*(x1)))) |
→ |
*(*(x1)) |
(7) |
1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
##(#(1(1(x1)))) |
→ |
##(x1) |
(8) |
|
##(#(1(1(x1)))) |
→ |
##(#(x1)) |
(9) |
|
##(#(1(1(x1)))) |
→ |
1#(#(#(x1))) |
(10) |
|
##(#(1(1(x1)))) |
→ |
1#(1(#(#(x1)))) |
(11) |
|
##(#(0(0(x1)))) |
→ |
##(x1) |
(12) |
|
##(#(0(0(x1)))) |
→ |
##(#(x1)) |
(13) |
|
##(#(0(0(x1)))) |
→ |
0#(#(#(x1))) |
(14) |
|
##(#(0(0(x1)))) |
→ |
0#(0(#(#(x1)))) |
(15) |
|
1#(1(*(*(x1)))) |
→ |
##(x1) |
(16) |
|
1#(1(*(*(x1)))) |
→ |
##(#(x1)) |
(17) |
|
1#(1(*(*(x1)))) |
→ |
0#(#(#(x1))) |
(18) |
|
1#(1(*(*(x1)))) |
→ |
0#(0(#(#(x1)))) |
(19) |
|
0#(0(*(*(x1)))) |
→ |
1#(x1) |
(20) |
|
0#(0(*(*(x1)))) |
→ |
1#(1(x1)) |
(21) |
1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [*(x1)] |
= |
x1 +
|
| [$(x1)] |
= |
x1 +
|
| [#(x1)] |
= |
x1 +
|
| [1(x1)] |
= |
x1 +
|
| [0(x1)] |
= |
x1 +
|
| [##(x1)] |
= |
x1 +
|
| [1#(x1)] |
= |
x1 +
|
| [0#(x1)] |
= |
x1 +
|
together with the usable
rules
|
0(0(*(*(x1)))) |
→ |
*(*(1(1(x1)))) |
(1) |
|
1(1(*(*(x1)))) |
→ |
0(0(#(#(x1)))) |
(2) |
|
#(#(0(0(x1)))) |
→ |
0(0(#(#(x1)))) |
(3) |
|
#(#(1(1(x1)))) |
→ |
1(1(#(#(x1)))) |
(4) |
|
#(#($($(x1)))) |
→ |
*(*($($(x1)))) |
(5) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
##(#(1(1(x1)))) |
→ |
##(x1) |
(8) |
|
##(#(1(1(x1)))) |
→ |
##(#(x1)) |
(9) |
|
##(#(1(1(x1)))) |
→ |
1#(#(#(x1))) |
(10) |
|
##(#(1(1(x1)))) |
→ |
1#(1(#(#(x1)))) |
(11) |
|
##(#(0(0(x1)))) |
→ |
##(x1) |
(12) |
|
##(#(0(0(x1)))) |
→ |
##(#(x1)) |
(13) |
|
##(#(0(0(x1)))) |
→ |
0#(#(#(x1))) |
(14) |
|
##(#(0(0(x1)))) |
→ |
0#(0(#(#(x1)))) |
(15) |
|
1#(1(*(*(x1)))) |
→ |
##(x1) |
(16) |
|
1#(1(*(*(x1)))) |
→ |
##(#(x1)) |
(17) |
|
1#(1(*(*(x1)))) |
→ |
0#(#(#(x1))) |
(18) |
|
1#(1(*(*(x1)))) |
→ |
0#(0(#(#(x1)))) |
(19) |
|
0#(0(*(*(x1)))) |
→ |
1#(x1) |
(20) |
|
0#(0(*(*(x1)))) |
→ |
1#(1(x1)) |
(21) |
and
no rules
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.