Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex4_Zan97_FR)
The rewrite relation of the following TRS is considered.
|
f(X) |
→ |
cons(X,n__f(n__g(X))) |
(1) |
|
g(0) |
→ |
s(0) |
(2) |
|
g(s(X)) |
→ |
s(s(g(X))) |
(3) |
|
sel(0,cons(X,Y)) |
→ |
X |
(4) |
|
sel(s(X),cons(Y,Z)) |
→ |
sel(X,activate(Z)) |
(5) |
|
f(X) |
→ |
n__f(X) |
(6) |
|
g(X) |
→ |
n__g(X) |
(7) |
|
activate(n__f(X)) |
→ |
f(activate(X)) |
(8) |
|
activate(n__g(X)) |
→ |
g(activate(X)) |
(9) |
|
activate(X) |
→ |
X |
(10) |
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.
|
activate#(n__g(X)) |
→ |
activate#(X) |
(11) |
|
activate#(n__f(X)) |
→ |
f#(activate(X)) |
(12) |
|
activate#(n__f(X)) |
→ |
activate#(X) |
(13) |
|
g#(s(X)) |
→ |
g#(X) |
(14) |
|
sel#(s(X),cons(Y,Z)) |
→ |
sel#(X,activate(Z)) |
(15) |
|
activate#(n__g(X)) |
→ |
g#(activate(X)) |
(16) |
|
sel#(s(X),cons(Y,Z)) |
→ |
activate#(Z) |
(17) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
sel#(s(X),cons(Y,Z)) |
→ |
sel#(X,activate(Z)) |
(15) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [activate(x1)] |
=
|
1 |
| [activate#(x1)] |
=
|
0 |
| [f(x1)] |
=
|
2 |
| [0] |
=
|
11798 |
| [sel#(x1, x2)] |
=
|
x1 + 0 |
| [sel(x1, x2)] |
=
|
0 |
| [n__f(x1)] |
=
|
x1 + 3 |
| [f#(x1)] |
=
|
0 |
| [g#(x1)] |
=
|
0 |
| [n__g(x1)] |
=
|
x1 + 8366 |
| [cons(x1, x2)] |
=
|
3 |
| [g(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
sel#(s(X),cons(Y,Z)) |
→ |
sel#(X,activate(Z)) |
(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
|
activate#(n__f(X)) |
→ |
activate#(X) |
(13) |
|
activate#(n__g(X)) |
→ |
activate#(X) |
(11) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [activate(x1)] |
=
|
1 |
| [activate#(x1)] |
=
|
x1 + 0 |
| [f(x1)] |
=
|
2 |
| [0] |
=
|
11798 |
| [sel#(x1, x2)] |
=
|
0 |
| [sel(x1, x2)] |
=
|
0 |
| [n__f(x1)] |
=
|
x1 + 3 |
| [f#(x1)] |
=
|
0 |
| [g#(x1)] |
=
|
0 |
| [n__g(x1)] |
=
|
x1 + 8366 |
| [cons(x1, x2)] |
=
|
3 |
| [g(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
|
activate#(n__f(X)) |
→ |
activate#(X) |
(13) |
|
activate#(n__g(X)) |
→ |
activate#(X) |
(11) |
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
| [s(x1)] |
=
|
x1 + 1 |
| [activate(x1)] |
=
|
1 |
| [activate#(x1)] |
=
|
0 |
| [f(x1)] |
=
|
2 |
| [0] |
=
|
11798 |
| [sel#(x1, x2)] |
=
|
0 |
| [sel(x1, x2)] |
=
|
0 |
| [n__f(x1)] |
=
|
x1 + 3 |
| [f#(x1)] |
=
|
0 |
| [g#(x1)] |
=
|
x1 + 0 |
| [n__g(x1)] |
=
|
x1 + 1 |
| [cons(x1, x2)] |
=
|
3 |
| [g(x1)] |
=
|
0 |
having no usable rules (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.