Certification Problem
Input (TPDB TRS_Standard/Rubio_04/quotminus)
The rewrite relation of the following TRS is considered.
|
plus(0,Y) |
→ |
Y |
(1) |
|
plus(s(X),Y) |
→ |
s(plus(X,Y)) |
(2) |
|
min(X,0) |
→ |
X |
(3) |
|
min(s(X),s(Y)) |
→ |
min(X,Y) |
(4) |
|
min(min(X,Y),Z) |
→ |
min(X,plus(Y,Z)) |
(5) |
|
quot(0,s(Y)) |
→ |
0 |
(6) |
|
quot(s(X),s(Y)) |
→ |
s(quot(min(X,Y),s(Y))) |
(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.
|
quot#(s(X),s(Y)) |
→ |
quot#(min(X,Y),s(Y)) |
(8) |
|
min#(s(X),s(Y)) |
→ |
min#(X,Y) |
(9) |
|
min#(min(X,Y),Z) |
→ |
plus#(Y,Z) |
(10) |
|
quot#(s(X),s(Y)) |
→ |
min#(X,Y) |
(11) |
|
plus#(s(X),Y) |
→ |
plus#(X,Y) |
(12) |
|
min#(min(X,Y),Z) |
→ |
min#(X,plus(Y,Z)) |
(13) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
quot#(s(X),s(Y)) |
→ |
quot#(min(X,Y),s(Y)) |
(8) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [Z] |
=
|
1 |
| [s(x1)] |
=
|
x1 + 30095 |
| [plus#(x1, x2)] |
=
|
0 |
| [min#(x1, x2)] |
=
|
0 |
| [0] |
=
|
1 |
| [quot(x1, x2)] |
=
|
0 |
| [plus(x1, x2)] |
=
|
x1 + 10157 |
| [min(x1, x2)] |
=
|
x1 + 30094 |
| [quot#(x1, x2)] |
=
|
x1 + 0 |
together with the usable
rules
|
min(s(X),s(Y)) |
→ |
min(X,Y) |
(4) |
|
min(X,0) |
→ |
X |
(3) |
|
min(min(X,Y),Z) |
→ |
min(X,plus(Y,Z)) |
(5) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
quot#(s(X),s(Y)) |
→ |
quot#(min(X,Y),s(Y)) |
(8) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
|
min#(min(X,Y),Z) |
→ |
min#(X,plus(Y,Z)) |
(13) |
|
min#(s(X),s(Y)) |
→ |
min#(X,Y) |
(9) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [Z] |
=
|
1 |
| [s(x1)] |
=
|
x1 + 30095 |
| [plus#(x1, x2)] |
=
|
0 |
| [min#(x1, x2)] |
=
|
x1 + 0 |
| [0] |
=
|
0 |
| [quot(x1, x2)] |
=
|
0 |
| [plus(x1, x2)] |
=
|
x1 + 0 |
| [min(x1, x2)] |
=
|
x1 + x2 + 30094 |
| [quot#(x1, x2)] |
=
|
0 |
together with the usable
rule
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
min#(min(X,Y),Z) |
→ |
min#(X,plus(Y,Z)) |
(13) |
|
min#(s(X),s(Y)) |
→ |
min#(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
|
plus#(s(X),Y) |
→ |
plus#(X,Y) |
(12) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [Z] |
=
|
1 |
| [s(x1)] |
=
|
x1 + 1 |
| [plus#(x1, x2)] |
=
|
x1 + 0 |
| [min#(x1, x2)] |
=
|
0 |
| [0] |
=
|
0 |
| [quot(x1, x2)] |
=
|
0 |
| [plus(x1, x2)] |
=
|
x1 + 0 |
| [min(x1, x2)] |
=
|
x1 + x2 + 42736 |
| [quot#(x1, x2)] |
=
|
0 |
together with the usable
rule
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
plus#(s(X),Y) |
→ |
plus#(X,Y) |
(12) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.