Certification Problem
Input (TPDB TRS_Standard/Rubio_04/logarquot)
The rewrite relation of the following TRS is considered.
|
min(X,0) |
→ |
X |
(1) |
|
min(s(X),s(Y)) |
→ |
min(X,Y) |
(2) |
|
quot(0,s(Y)) |
→ |
0 |
(3) |
|
quot(s(X),s(Y)) |
→ |
s(quot(min(X,Y),s(Y))) |
(4) |
|
log(s(0)) |
→ |
0 |
(5) |
|
log(s(s(X))) |
→ |
s(log(s(quot(X,s(s(0)))))) |
(6) |
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.
|
log#(s(s(X))) |
→ |
log#(s(quot(X,s(s(0))))) |
(7) |
|
quot#(s(X),s(Y)) |
→ |
min#(X,Y) |
(8) |
|
log#(s(s(X))) |
→ |
quot#(X,s(s(0))) |
(9) |
|
min#(s(X),s(Y)) |
→ |
min#(X,Y) |
(10) |
|
quot#(s(X),s(Y)) |
→ |
quot#(min(X,Y),s(Y)) |
(11) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
log#(s(s(X))) |
→ |
log#(s(quot(X,s(s(0))))) |
(7) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 2 |
| [min#(x1, x2)] |
=
|
0 |
| [log#(x1)] |
=
|
x1 + 0 |
| [log(x1)] |
=
|
0 |
| [0] |
=
|
1 |
| [quot(x1, x2)] |
=
|
x1 + 1 |
| [min(x1, x2)] |
=
|
x1 + 0 |
| [quot#(x1, x2)] |
=
|
0 |
together with the usable
rules
|
quot(s(X),s(Y)) |
→ |
s(quot(min(X,Y),s(Y))) |
(4) |
|
min(X,0) |
→ |
X |
(1) |
|
quot(0,s(Y)) |
→ |
0 |
(3) |
|
min(s(X),s(Y)) |
→ |
min(X,Y) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
log#(s(s(X))) |
→ |
log#(s(quot(X,s(s(0))))) |
(7) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
|
quot#(s(X),s(Y)) |
→ |
quot#(min(X,Y),s(Y)) |
(11) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 2 |
| [min#(x1, x2)] |
=
|
0 |
| [log#(x1)] |
=
|
0 |
| [log(x1)] |
=
|
0 |
| [0] |
=
|
42737 |
| [quot(x1, x2)] |
=
|
42736 |
| [min(x1, x2)] |
=
|
x1 + 1 |
| [quot#(x1, x2)] |
=
|
x1 + 0 |
together with the usable
rules
|
min(X,0) |
→ |
X |
(1) |
|
min(s(X),s(Y)) |
→ |
min(X,Y) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
quot#(s(X),s(Y)) |
→ |
quot#(min(X,Y),s(Y)) |
(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
|
min#(s(X),s(Y)) |
→ |
min#(X,Y) |
(10) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [min#(x1, x2)] |
=
|
x2 + 0 |
| [log#(x1)] |
=
|
0 |
| [log(x1)] |
=
|
0 |
| [0] |
=
|
42737 |
| [quot(x1, x2)] |
=
|
1143 |
| [min(x1, x2)] |
=
|
x1 + 1 |
| [quot#(x1, x2)] |
=
|
x1 + 0 |
together with the usable
rules
|
min(X,0) |
→ |
X |
(1) |
|
min(s(X),s(Y)) |
→ |
min(X,Y) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
min#(s(X),s(Y)) |
→ |
min#(X,Y) |
(10) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.