Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex1_Zan97_iGM)
The rewrite relation of the following TRS is considered.
active(g(X)) |
→ |
mark(h(X)) |
(1) |
active(c) |
→ |
mark(d) |
(2) |
active(h(d)) |
→ |
mark(g(c)) |
(3) |
mark(g(X)) |
→ |
active(g(X)) |
(4) |
mark(h(X)) |
→ |
active(h(X)) |
(5) |
mark(c) |
→ |
active(c) |
(6) |
mark(d) |
→ |
active(d) |
(7) |
g(mark(X)) |
→ |
g(X) |
(8) |
g(active(X)) |
→ |
g(X) |
(9) |
h(mark(X)) |
→ |
h(X) |
(10) |
h(active(X)) |
→ |
h(X) |
(11) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by NaTT @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
mark#(c) |
→ |
active#(c) |
(12) |
h#(mark(X)) |
→ |
h#(X) |
(13) |
mark#(g(X)) |
→ |
active#(g(X)) |
(14) |
active#(h(d)) |
→ |
g#(c) |
(15) |
active#(g(X)) |
→ |
h#(X) |
(16) |
g#(mark(X)) |
→ |
g#(X) |
(17) |
mark#(h(X)) |
→ |
active#(h(X)) |
(18) |
active#(g(X)) |
→ |
mark#(h(X)) |
(19) |
mark#(d) |
→ |
active#(d) |
(20) |
g#(active(X)) |
→ |
g#(X) |
(21) |
active#(c) |
→ |
mark#(d) |
(22) |
active#(h(d)) |
→ |
mark#(g(c)) |
(23) |
h#(active(X)) |
→ |
h#(X) |
(24) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
g#(mark(X)) |
→ |
g#(X) |
(17) |
g#(active(X)) |
→ |
g#(X) |
(21) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
0 |
[d] |
=
|
0 |
[c] |
=
|
0 |
[mark#(x1)] |
=
|
0 |
[h#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[g#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
g#(mark(X)) |
→ |
g#(X) |
(17) |
g#(active(X)) |
→ |
g#(X) |
(21) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
active#(h(d)) |
→ |
mark#(g(c)) |
(23) |
mark#(g(X)) |
→ |
active#(g(X)) |
(14) |
active#(g(X)) |
→ |
mark#(h(X)) |
(19) |
mark#(h(X)) |
→ |
active#(h(X)) |
(18) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
x1 + 1 |
[d] |
=
|
20657 |
[c] |
=
|
20653 |
[mark#(x1)] |
=
|
x1 + 1 |
[h#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[g#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
x1 + 3 |
together with the usable
rules
g(mark(X)) |
→ |
g(X) |
(8) |
h(mark(X)) |
→ |
h(X) |
(10) |
h(active(X)) |
→ |
h(X) |
(11) |
g(active(X)) |
→ |
g(X) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(h(d)) |
→ |
mark#(g(c)) |
(23) |
mark#(g(X)) |
→ |
active#(g(X)) |
(14) |
active#(g(X)) |
→ |
mark#(h(X)) |
(19) |
mark#(h(X)) |
→ |
active#(h(X)) |
(18) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
h#(active(X)) |
→ |
h#(X) |
(24) |
h#(mark(X)) |
→ |
h#(X) |
(13) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
x1 + 1 |
[d] |
=
|
20657 |
[c] |
=
|
20653 |
[mark#(x1)] |
=
|
x1 + 1 |
[h#(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
x1 + 1 |
[g#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
x1 + 3 |
together with the usable
rules
g(mark(X)) |
→ |
g(X) |
(8) |
h(mark(X)) |
→ |
h(X) |
(10) |
h(active(X)) |
→ |
h(X) |
(11) |
g(active(X)) |
→ |
g(X) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
h#(active(X)) |
→ |
h#(X) |
(24) |
h#(mark(X)) |
→ |
h#(X) |
(13) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.