Certification Problem
Input (TPDB TRS_Standard/SK90/2.41)
The rewrite relation of the following TRS is considered.
|
norm(nil) |
→ |
0 |
(1) |
|
norm(g(x,y)) |
→ |
s(norm(x)) |
(2) |
|
f(x,nil) |
→ |
g(nil,x) |
(3) |
|
f(x,g(y,z)) |
→ |
g(f(x,y),z) |
(4) |
|
rem(nil,y) |
→ |
nil |
(5) |
|
rem(g(x,y),0) |
→ |
g(x,y) |
(6) |
|
rem(g(x,y),s(z)) |
→ |
rem(x,z) |
(7) |
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.
|
rem#(g(x,y),s(z)) |
→ |
rem#(x,z) |
(8) |
|
f#(x,g(y,z)) |
→ |
f#(x,y) |
(9) |
|
norm#(g(x,y)) |
→ |
norm#(x) |
(10) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
norm#(g(x,y)) |
→ |
norm#(x) |
(10) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
0 |
| [norm#(x1)] |
=
|
x1 + 0 |
| [norm(x1)] |
=
|
0 |
| [rem#(x1, x2)] |
=
|
0 |
| [f(x1, x2)] |
=
|
0 |
| [0] |
=
|
0 |
| [nil] |
=
|
0 |
| [rem(x1, x2)] |
=
|
0 |
| [f#(x1, x2)] |
=
|
0 |
| [g(x1, x2)] |
=
|
x1 + 1 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
norm#(g(x,y)) |
→ |
norm#(x) |
(10) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
|
f#(x,g(y,z)) |
→ |
f#(x,y) |
(9) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
0 |
| [norm#(x1)] |
=
|
0 |
| [norm(x1)] |
=
|
0 |
| [rem#(x1, x2)] |
=
|
0 |
| [f(x1, x2)] |
=
|
0 |
| [0] |
=
|
0 |
| [nil] |
=
|
0 |
| [rem(x1, x2)] |
=
|
0 |
| [f#(x1, x2)] |
=
|
x2 + 0 |
| [g(x1, x2)] |
=
|
x1 + 1 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
f#(x,g(y,z)) |
→ |
f#(x,y) |
(9) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
|
rem#(g(x,y),s(z)) |
→ |
rem#(x,z) |
(8) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [norm#(x1)] |
=
|
0 |
| [norm(x1)] |
=
|
0 |
| [rem#(x1, x2)] |
=
|
x2 + 0 |
| [f(x1, x2)] |
=
|
0 |
| [0] |
=
|
0 |
| [nil] |
=
|
0 |
| [rem(x1, x2)] |
=
|
0 |
| [f#(x1, x2)] |
=
|
0 |
| [g(x1, x2)] |
=
|
1 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
rem#(g(x,y),s(z)) |
→ |
rem#(x,z) |
(8) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.