Certification Problem
Input (TPDB TRS_Standard/SK90/4.28)
The rewrite relation of the following TRS is considered.
|
f(x,nil) |
→ |
g(nil,x) |
(1) |
|
f(x,g(y,z)) |
→ |
g(f(x,y),z) |
(2) |
|
++(x,nil) |
→ |
x |
(3) |
|
++(x,g(y,z)) |
→ |
g(++(x,y),z) |
(4) |
|
null(nil) |
→ |
true |
(5) |
|
null(g(x,y)) |
→ |
false |
(6) |
|
mem(nil,y) |
→ |
false |
(7) |
|
mem(g(x,y),z) |
→ |
or(=(y,z),mem(x,z)) |
(8) |
|
mem(x,max(x)) |
→ |
not(null(x)) |
(9) |
|
max(g(g(nil,x),y)) |
→ |
max'(x,y) |
(10) |
|
max(g(g(g(x,y),z),u)) |
→ |
max'(max(g(g(x,y),z)),u) |
(11) |
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.
|
mem#(x,max(x)) |
→ |
null#(x) |
(12) |
|
++#(x,g(y,z)) |
→ |
++#(x,y) |
(13) |
|
max#(g(g(g(x,y),z),u)) |
→ |
max#(g(g(x,y),z)) |
(14) |
|
f#(x,g(y,z)) |
→ |
f#(x,y) |
(15) |
|
mem#(g(x,y),z) |
→ |
mem#(x,z) |
(16) |
1.1 Dependency Graph Processor
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
|
mem#(g(x,y),z) |
→ |
mem#(x,z) |
(16) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [mem(x1, x2)] |
=
|
0 |
| [max'(x1, x2)] |
=
|
0 |
| [u] |
=
|
0 |
| [++(x1, x2)] |
=
|
0 |
| [false] |
=
|
0 |
| [mem#(x1, x2)] |
=
|
x1 + 0 |
| [true] |
=
|
0 |
| [f(x1, x2)] |
=
|
0 |
| [null(x1)] |
=
|
0 |
| [max(x1)] |
=
|
0 |
| [=(x1, x2)] |
=
|
0 |
| [++#(x1, x2)] |
=
|
0 |
| [nil] |
=
|
0 |
| [max#(x1)] |
=
|
0 |
| [or(x1, x2)] |
=
|
0 |
| [f#(x1, x2)] |
=
|
0 |
| [null#(x1)] |
=
|
0 |
| [g(x1, x2)] |
=
|
x1 + 1 |
| [not(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
mem#(g(x,y),z) |
→ |
mem#(x,z) |
(16) |
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) |
(15) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [mem(x1, x2)] |
=
|
0 |
| [max'(x1, x2)] |
=
|
0 |
| [u] |
=
|
0 |
| [++(x1, x2)] |
=
|
0 |
| [false] |
=
|
0 |
| [mem#(x1, x2)] |
=
|
0 |
| [true] |
=
|
0 |
| [f(x1, x2)] |
=
|
0 |
| [null(x1)] |
=
|
0 |
| [max(x1)] |
=
|
0 |
| [=(x1, x2)] |
=
|
0 |
| [++#(x1, x2)] |
=
|
0 |
| [nil] |
=
|
0 |
| [max#(x1)] |
=
|
0 |
| [or(x1, x2)] |
=
|
0 |
| [f#(x1, x2)] |
=
|
x2 + 0 |
| [null#(x1)] |
=
|
0 |
| [g(x1, x2)] |
=
|
x1 + 1 |
| [not(x1)] |
=
|
0 |
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) |
(15) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
|
max#(g(g(g(x,y),z),u)) |
→ |
max#(g(g(x,y),z)) |
(14) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [mem(x1, x2)] |
=
|
0 |
| [max'(x1, x2)] |
=
|
0 |
| [u] |
=
|
1 |
| [++(x1, x2)] |
=
|
0 |
| [false] |
=
|
0 |
| [mem#(x1, x2)] |
=
|
0 |
| [true] |
=
|
0 |
| [f(x1, x2)] |
=
|
0 |
| [null(x1)] |
=
|
0 |
| [max(x1)] |
=
|
0 |
| [=(x1, x2)] |
=
|
0 |
| [++#(x1, x2)] |
=
|
0 |
| [nil] |
=
|
0 |
| [max#(x1)] |
=
|
x1 + 0 |
| [or(x1, x2)] |
=
|
0 |
| [f#(x1, x2)] |
=
|
0 |
| [null#(x1)] |
=
|
0 |
| [g(x1, x2)] |
=
|
x1 + x2 + 28958 |
| [not(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
max#(g(g(g(x,y),z),u)) |
→ |
max#(g(g(x,y),z)) |
(14) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
|
++#(x,g(y,z)) |
→ |
++#(x,y) |
(13) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [mem(x1, x2)] |
=
|
0 |
| [max'(x1, x2)] |
=
|
0 |
| [u] |
=
|
1 |
| [++(x1, x2)] |
=
|
0 |
| [false] |
=
|
0 |
| [mem#(x1, x2)] |
=
|
0 |
| [true] |
=
|
0 |
| [f(x1, x2)] |
=
|
0 |
| [null(x1)] |
=
|
0 |
| [max(x1)] |
=
|
0 |
| [=(x1, x2)] |
=
|
0 |
| [++#(x1, x2)] |
=
|
x2 + 0 |
| [nil] |
=
|
0 |
| [max#(x1)] |
=
|
0 |
| [or(x1, x2)] |
=
|
0 |
| [f#(x1, x2)] |
=
|
0 |
| [null#(x1)] |
=
|
0 |
| [g(x1, x2)] |
=
|
x1 + 28958 |
| [not(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
++#(x,g(y,z)) |
→ |
++#(x,y) |
(13) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 0
components.