Certification Problem
Input (TPDB TRS_Standard/AG01/#3.8a)
The rewrite relation of the following TRS is considered.
|
pred(s(x)) |
→ |
x |
(1) |
|
minus(x,0) |
→ |
x |
(2) |
|
minus(x,s(y)) |
→ |
pred(minus(x,y)) |
(3) |
|
quot(0,s(y)) |
→ |
0 |
(4) |
|
quot(s(x),s(y)) |
→ |
s(quot(minus(x,y),s(y))) |
(5) |
|
log(s(0)) |
→ |
0 |
(6) |
|
log(s(s(x))) |
→ |
s(log(s(quot(x,s(s(0)))))) |
(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.
|
log#(s(s(x))) |
→ |
quot#(x,s(s(0))) |
(8) |
|
minus#(x,s(y)) |
→ |
minus#(x,y) |
(9) |
|
minus#(x,s(y)) |
→ |
pred#(minus(x,y)) |
(10) |
|
quot#(s(x),s(y)) |
→ |
quot#(minus(x,y),s(y)) |
(11) |
|
log#(s(s(x))) |
→ |
log#(s(quot(x,s(s(0))))) |
(12) |
|
quot#(s(x),s(y)) |
→ |
minus#(x,y) |
(13) |
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))))) |
(12) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 2 |
| [minus(x1, x2)] |
=
|
x1 + 0 |
| [log#(x1)] |
=
|
x1 + 0 |
| [pred(x1)] |
=
|
x1 + 0 |
| [log(x1)] |
=
|
0 |
| [0] |
=
|
1 |
| [quot(x1, x2)] |
=
|
x1 + 1 |
| [minus#(x1, x2)] |
=
|
0 |
| [pred#(x1)] |
=
|
0 |
| [quot#(x1, x2)] |
=
|
0 |
together with the usable
rules
|
quot(0,s(y)) |
→ |
0 |
(4) |
|
pred(s(x)) |
→ |
x |
(1) |
|
minus(x,s(y)) |
→ |
pred(minus(x,y)) |
(3) |
|
quot(s(x),s(y)) |
→ |
s(quot(minus(x,y),s(y))) |
(5) |
|
minus(x,0) |
→ |
x |
(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))))) |
(12) |
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#(minus(x,y),s(y)) |
(11) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 2 |
| [minus(x1, x2)] |
=
|
x1 + 0 |
| [log#(x1)] |
=
|
x1 + 0 |
| [pred(x1)] |
=
|
x1 + 0 |
| [log(x1)] |
=
|
0 |
| [0] |
=
|
1 |
| [quot(x1, x2)] |
=
|
x1 + 40651 |
| [minus#(x1, x2)] |
=
|
0 |
| [pred#(x1)] |
=
|
0 |
| [quot#(x1, x2)] |
=
|
x1 + 0 |
together with the usable
rules
|
quot(0,s(y)) |
→ |
0 |
(4) |
|
pred(s(x)) |
→ |
x |
(1) |
|
minus(x,s(y)) |
→ |
pred(minus(x,y)) |
(3) |
|
quot(s(x),s(y)) |
→ |
s(quot(minus(x,y),s(y))) |
(5) |
|
minus(x,0) |
→ |
x |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
quot#(s(x),s(y)) |
→ |
quot#(minus(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
|
minus#(x,s(y)) |
→ |
minus#(x,y) |
(9) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 2 |
| [minus(x1, x2)] |
=
|
x1 + 0 |
| [log#(x1)] |
=
|
x1 + 0 |
| [pred(x1)] |
=
|
x1 + 0 |
| [log(x1)] |
=
|
0 |
| [0] |
=
|
1 |
| [quot(x1, x2)] |
=
|
x1 + 40651 |
| [minus#(x1, x2)] |
=
|
x2 + 0 |
| [pred#(x1)] |
=
|
0 |
| [quot#(x1, x2)] |
=
|
x1 + 0 |
together with the usable
rules
|
quot(0,s(y)) |
→ |
0 |
(4) |
|
pred(s(x)) |
→ |
x |
(1) |
|
minus(x,s(y)) |
→ |
pred(minus(x,y)) |
(3) |
|
quot(s(x),s(y)) |
→ |
s(quot(minus(x,y),s(y))) |
(5) |
|
minus(x,0) |
→ |
x |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
minus#(x,s(y)) |
→ |
minus#(x,y) |
(9) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.