Certification Problem
Input (TPDB TRS_Standard/Strategy_removed_mixed_05/test830)
The rewrite relation of the following TRS is considered.
|
f(s(X)) |
→ |
f(X) |
(1) |
|
g(cons(0,Y)) |
→ |
g(Y) |
(2) |
|
g(cons(s(X),Y)) |
→ |
s(X) |
(3) |
|
h(cons(X,Y)) |
→ |
h(g(cons(X,Y))) |
(4) |
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#(s(X)) |
→ |
f#(X) |
(5) |
|
h#(cons(X,Y)) |
→ |
h#(g(cons(X,Y))) |
(6) |
|
g#(cons(0,Y)) |
→ |
g#(Y) |
(7) |
|
h#(cons(X,Y)) |
→ |
g#(cons(X,Y)) |
(8) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
h#(cons(X,Y)) |
→ |
h#(g(cons(X,Y))) |
(6) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [h(x1)] |
=
|
0 |
| [s(x1)] |
=
|
8856 |
| [f(x1)] |
=
|
0 |
| [0] |
=
|
23676 |
| [h#(x1)] |
=
|
x1 + 0 |
| [f#(x1)] |
=
|
0 |
| [g#(x1)] |
=
|
0 |
| [cons(x1, x2)] |
=
|
x1 + 8857 |
| [g(x1)] |
=
|
8856 |
together with the usable
rules
|
g(cons(s(X),Y)) |
→ |
s(X) |
(3) |
|
g(cons(0,Y)) |
→ |
g(Y) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
h#(cons(X,Y)) |
→ |
h#(g(cons(X,Y))) |
(6) |
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#(cons(0,Y)) |
→ |
g#(Y) |
(7) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [h(x1)] |
=
|
0 |
| [s(x1)] |
=
|
8856 |
| [f(x1)] |
=
|
0 |
| [0] |
=
|
23676 |
| [h#(x1)] |
=
|
x1 + 0 |
| [f#(x1)] |
=
|
0 |
| [g#(x1)] |
=
|
x1 + 0 |
| [cons(x1, x2)] |
=
|
x1 + x2 + 8857 |
| [g(x1)] |
=
|
8856 |
together with the usable
rules
|
g(cons(s(X),Y)) |
→ |
s(X) |
(3) |
|
g(cons(0,Y)) |
→ |
g(Y) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
g#(cons(0,Y)) |
→ |
g#(Y) |
(7) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [h(x1)] |
=
|
0 |
| [s(x1)] |
=
|
x1 + 30613 |
| [f(x1)] |
=
|
0 |
| [0] |
=
|
16304 |
| [h#(x1)] |
=
|
x1 + 0 |
| [f#(x1)] |
=
|
x1 + 0 |
| [g#(x1)] |
=
|
0 |
| [cons(x1, x2)] |
=
|
x1 + x2 + 0 |
| [g(x1)] |
=
|
x1 + 0 |
together with the usable
rules
|
g(cons(s(X),Y)) |
→ |
s(X) |
(3) |
|
g(cons(0,Y)) |
→ |
g(Y) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.