Certification Problem
Input (TPDB TRS_Standard/Beerendonk_07/23)
The rewrite relation of the following TRS is considered.
|
cond1(true,x,y,z) |
→ |
cond2(gr(y,z),x,y,z) |
(1) |
|
cond2(true,x,y,z) |
→ |
cond2(gr(y,z),x,p(y),z) |
(2) |
|
cond2(false,x,y,z) |
→ |
cond1(gr(x,z),p(x),y,z) |
(3) |
|
gr(0,x) |
→ |
false |
(4) |
|
gr(s(x),0) |
→ |
true |
(5) |
|
gr(s(x),s(y)) |
→ |
gr(x,y) |
(6) |
|
p(0) |
→ |
0 |
(7) |
|
p(s(x)) |
→ |
x |
(8) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by ttt2 @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
cond1#(true,x,y,z) |
→ |
gr#(y,z) |
(9) |
|
cond1#(true,x,y,z) |
→ |
cond2#(gr(y,z),x,y,z) |
(10) |
|
cond2#(true,x,y,z) |
→ |
p#(y) |
(11) |
|
cond2#(true,x,y,z) |
→ |
gr#(y,z) |
(12) |
|
cond2#(true,x,y,z) |
→ |
cond2#(gr(y,z),x,p(y),z) |
(13) |
|
cond2#(false,x,y,z) |
→ |
p#(x) |
(14) |
|
cond2#(false,x,y,z) |
→ |
gr#(x,z) |
(15) |
|
cond2#(false,x,y,z) |
→ |
cond1#(gr(x,z),p(x),y,z) |
(16) |
|
gr#(s(x),s(y)) |
→ |
gr#(x,y) |
(17) |
1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
|
cond2#(false,x,y,z) |
→ |
cond1#(gr(x,z),p(x),y,z) |
(16) |
|
cond1#(true,x,y,z) |
→ |
cond2#(gr(y,z),x,y,z) |
(10) |
|
cond2#(true,x,y,z) |
→ |
cond2#(gr(y,z),x,p(y),z) |
(13) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers
| [p(x1)] |
= |
-1 · x1 + 0 |
| [cond1#(x1,...,x4)] |
= |
-3 · x1 + 1 · x2 +
-∞ · x3 +
-∞ · x4 + 0 |
| [false] |
= |
4 |
| [cond2#(x1,...,x4)] |
= |
-∞ · x1 + 1 · x2 +
-∞ · x3 +
-∞ · x4 + 2 |
| [true] |
= |
5 |
| [s(x1)] |
= |
4 · x1 + 5 |
| [0] |
= |
0 |
| [gr(x1, x2)] |
= |
0 · x1 +
-∞ · x2 + 4 |
together with the usable
rules
|
p(0) |
→ |
0 |
(7) |
|
p(s(x)) |
→ |
x |
(8) |
|
gr(0,x) |
→ |
false |
(4) |
|
gr(s(x),0) |
→ |
true |
(5) |
|
gr(s(x),s(y)) |
→ |
gr(x,y) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
cond2#(false,x,y,z) |
→ |
cond1#(gr(x,z),p(x),y,z) |
(16) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
|
gr#(s(x),s(y)) |
→ |
gr#(x,y) |
(17) |
1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
gr#(s(x),s(y)) |
→ |
gr#(x,y) |
(17) |
|
|
| 2 |
> |
2 |
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.