Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex6_GM04_iGM)
The rewrite relation of the following TRS is considered.
active(c) |
→ |
mark(f(g(c))) |
(1) |
active(f(g(X))) |
→ |
mark(g(X)) |
(2) |
mark(c) |
→ |
active(c) |
(3) |
mark(f(X)) |
→ |
active(f(X)) |
(4) |
mark(g(X)) |
→ |
active(g(X)) |
(5) |
f(mark(X)) |
→ |
f(X) |
(6) |
f(active(X)) |
→ |
f(X) |
(7) |
g(mark(X)) |
→ |
g(X) |
(8) |
g(active(X)) |
→ |
g(X) |
(9) |
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.
f#(mark(X)) |
→ |
f#(X) |
(10) |
mark#(c) |
→ |
active#(c) |
(11) |
active#(c) |
→ |
g#(c) |
(12) |
mark#(f(X)) |
→ |
active#(f(X)) |
(13) |
active#(c) |
→ |
mark#(f(g(c))) |
(14) |
g#(mark(X)) |
→ |
g#(X) |
(15) |
mark#(g(X)) |
→ |
active#(g(X)) |
(16) |
g#(active(X)) |
→ |
g#(X) |
(17) |
active#(f(g(X))) |
→ |
mark#(g(X)) |
(18) |
active#(c) |
→ |
f#(g(c)) |
(19) |
f#(active(X)) |
→ |
f#(X) |
(20) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
g#(active(X)) |
→ |
g#(X) |
(17) |
g#(mark(X)) |
→ |
g#(X) |
(15) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[c] |
=
|
0 |
[f(x1)] |
=
|
0 |
[mark#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[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#(active(X)) |
→ |
g#(X) |
(17) |
g#(mark(X)) |
→ |
g#(X) |
(15) |
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#(f(g(X))) |
→ |
mark#(g(X)) |
(18) |
mark#(f(X)) |
→ |
active#(f(X)) |
(13) |
mark#(c) |
→ |
active#(c) |
(11) |
mark#(g(X)) |
→ |
active#(g(X)) |
(16) |
active#(c) |
→ |
mark#(f(g(c))) |
(14) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[c] |
=
|
15443 |
[f(x1)] |
=
|
7722 |
[mark#(x1)] |
=
|
x1 + 7720 |
[mark(x1)] |
=
|
1 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
1 |
together with the usable
rules
g(mark(X)) |
→ |
g(X) |
(8) |
f(active(X)) |
→ |
f(X) |
(7) |
g(active(X)) |
→ |
g(X) |
(9) |
f(mark(X)) |
→ |
f(X) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(f(g(X))) |
→ |
mark#(g(X)) |
(18) |
mark#(f(X)) |
→ |
active#(f(X)) |
(13) |
mark#(c) |
→ |
active#(c) |
(11) |
mark#(g(X)) |
→ |
active#(g(X)) |
(16) |
active#(c) |
→ |
mark#(f(g(c))) |
(14) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
f#(active(X)) |
→ |
f#(X) |
(20) |
f#(mark(X)) |
→ |
f#(X) |
(10) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[c] |
=
|
15443 |
[f(x1)] |
=
|
7722 |
[mark#(x1)] |
=
|
x1 + 7720 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
x1 + 0 |
[g#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
1 |
together with the usable
rules
g(mark(X)) |
→ |
g(X) |
(8) |
f(active(X)) |
→ |
f(X) |
(7) |
g(active(X)) |
→ |
g(X) |
(9) |
f(mark(X)) |
→ |
f(X) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
f#(active(X)) |
→ |
f#(X) |
(20) |
f#(mark(X)) |
→ |
f#(X) |
(10) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.