Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex18_Luc06_iGM)
The rewrite relation of the following TRS is considered.
active(f(f(a))) |
→ |
mark(f(g(f(a)))) |
(1) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(2) |
mark(a) |
→ |
active(a) |
(3) |
mark(g(X)) |
→ |
active(g(X)) |
(4) |
f(mark(X)) |
→ |
f(X) |
(5) |
f(active(X)) |
→ |
f(X) |
(6) |
g(mark(X)) |
→ |
g(X) |
(7) |
g(active(X)) |
→ |
g(X) |
(8) |
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#(f(X)) |
→ |
f#(mark(X)) |
(9) |
f#(mark(X)) |
→ |
f#(X) |
(10) |
active#(f(f(a))) |
→ |
f#(g(f(a))) |
(11) |
g#(active(X)) |
→ |
g#(X) |
(12) |
mark#(g(X)) |
→ |
active#(g(X)) |
(13) |
mark#(a) |
→ |
active#(a) |
(14) |
active#(f(f(a))) |
→ |
g#(f(a)) |
(15) |
g#(mark(X)) |
→ |
g#(X) |
(16) |
mark#(f(X)) |
→ |
mark#(X) |
(17) |
mark#(f(X)) |
→ |
active#(f(mark(X))) |
(18) |
active#(f(f(a))) |
→ |
mark#(f(g(f(a)))) |
(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
mark#(g(X)) |
→ |
active#(g(X)) |
(13) |
active#(f(f(a))) |
→ |
mark#(f(g(f(a)))) |
(19) |
mark#(f(X)) |
→ |
active#(f(mark(X))) |
(18) |
mark#(f(X)) |
→ |
mark#(X) |
(17) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[a] |
=
|
1 |
[f(x1)] |
=
|
x1 + 28960 |
[mark#(x1)] |
=
|
x1 + 28959 |
[mark(x1)] |
=
|
x1 + 28958 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 0 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
1 |
together with the usable
rules
mark(g(X)) |
→ |
active(g(X)) |
(4) |
g(active(X)) |
→ |
g(X) |
(8) |
active(f(f(a))) |
→ |
mark(f(g(f(a)))) |
(1) |
mark(a) |
→ |
active(a) |
(3) |
f(mark(X)) |
→ |
f(X) |
(5) |
g(mark(X)) |
→ |
g(X) |
(7) |
f(active(X)) |
→ |
f(X) |
(6) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(g(X)) |
→ |
active#(g(X)) |
(13) |
active#(f(f(a))) |
→ |
mark#(f(g(f(a)))) |
(19) |
mark#(f(X)) |
→ |
active#(f(mark(X))) |
(18) |
mark#(f(X)) |
→ |
mark#(X) |
(17) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
g#(active(X)) |
→ |
g#(X) |
(12) |
g#(mark(X)) |
→ |
g#(X) |
(16) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[a] |
=
|
1 |
[f(x1)] |
=
|
x1 + 1 |
[mark#(x1)] |
=
|
x1 + 28959 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
x1 + 0 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
1 |
together with the usable
rules
mark(g(X)) |
→ |
active(g(X)) |
(4) |
g(active(X)) |
→ |
g(X) |
(8) |
active(f(f(a))) |
→ |
mark(f(g(f(a)))) |
(1) |
mark(a) |
→ |
active(a) |
(3) |
f(mark(X)) |
→ |
f(X) |
(5) |
g(mark(X)) |
→ |
g(X) |
(7) |
f(active(X)) |
→ |
f(X) |
(6) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
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
[a] |
=
|
1 |
[f(x1)] |
=
|
41063 |
[mark#(x1)] |
=
|
x1 + 28959 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
x1 + 0 |
[g#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
0 |
together with the usable
rules
g(active(X)) |
→ |
g(X) |
(8) |
f(mark(X)) |
→ |
f(X) |
(5) |
g(mark(X)) |
→ |
g(X) |
(7) |
f(active(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.