Certification Problem
Input (TPDB TRS_Standard/Strategy_removed_AG01/#4.37a)
The rewrite relation of the following TRS is considered.
|
f(c(s(x),y)) |
→ |
f(c(x,s(y))) |
(1) |
|
g(c(x,s(y))) |
→ |
g(c(s(x),y)) |
(2) |
|
g(s(f(x))) |
→ |
g(f(x)) |
(3) |
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.
|
g#(s(f(x))) |
→ |
g#(f(x)) |
(4) |
|
f#(c(s(x),y)) |
→ |
f#(c(x,s(y))) |
(5) |
|
g#(c(x,s(y))) |
→ |
g#(c(s(x),y)) |
(6) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
f#(c(s(x),y)) |
→ |
f#(c(x,s(y))) |
(5) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [c(x1, x2)] |
=
|
x1 + 28958 |
| [f(x1)] |
=
|
0 |
| [f#(x1)] |
=
|
x1 + 0 |
| [g#(x1)] |
=
|
0 |
| [g(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
f#(c(s(x),y)) |
→ |
f#(c(x,s(y))) |
(5) |
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#(s(f(x))) |
→ |
g#(f(x)) |
(4) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [c(x1, x2)] |
=
|
x1 + 40651 |
| [f(x1)] |
=
|
x1 + 22248 |
| [f#(x1)] |
=
|
0 |
| [g#(x1)] |
=
|
x1 + 0 |
| [g(x1)] |
=
|
0 |
together with the usable
rule
|
f(c(s(x),y)) |
→ |
f(c(x,s(y))) |
(1) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
g#(s(f(x))) |
→ |
g#(f(x)) |
(4) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
|
g#(c(x,s(y))) |
→ |
g#(c(s(x),y)) |
(6) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [c(x1, x2)] |
=
|
x2 + 1143 |
| [f(x1)] |
=
|
22248 |
| [f#(x1)] |
=
|
0 |
| [g#(x1)] |
=
|
x1 + 0 |
| [g(x1)] |
=
|
0 |
together with the usable
rule
|
f(c(s(x),y)) |
→ |
f(c(x,s(y))) |
(1) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
g#(c(x,s(y))) |
→ |
g#(c(s(x),y)) |
(6) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.