Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex1_2_Luc02c_GM)
The rewrite relation of the following TRS is considered.
a__2nd(cons(X,cons(Y,Z))) |
→ |
mark(Y) |
(1) |
a__from(X) |
→ |
cons(mark(X),from(s(X))) |
(2) |
mark(2nd(X)) |
→ |
a__2nd(mark(X)) |
(3) |
mark(from(X)) |
→ |
a__from(mark(X)) |
(4) |
mark(cons(X1,X2)) |
→ |
cons(mark(X1),X2) |
(5) |
mark(s(X)) |
→ |
s(mark(X)) |
(6) |
a__2nd(X) |
→ |
2nd(X) |
(7) |
a__from(X) |
→ |
from(X) |
(8) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
a__2nd#(cons(X,cons(Y,Z))) |
→ |
mark#(Y) |
(9) |
a__from#(X) |
→ |
mark#(X) |
(10) |
mark#(2nd(X)) |
→ |
a__2nd#(mark(X)) |
(11) |
mark#(2nd(X)) |
→ |
mark#(X) |
(12) |
mark#(from(X)) |
→ |
a__from#(mark(X)) |
(13) |
mark#(from(X)) |
→ |
mark#(X) |
(14) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(15) |
mark#(s(X)) |
→ |
mark#(X) |
(16) |
1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
[a__2nd#(x1)] |
= |
+ · x1
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[mark#(x1)] |
= |
+ · x1
|
[a__from#(x1)] |
= |
+ · x1
|
[2nd(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[from(x1)] |
= |
+ · x1
|
[s(x1)] |
= |
+ · x1
|
[a__2nd(x1)] |
= |
+ · x1
|
[a__from(x1)] |
= |
+ · x1
|
the
pair
mark#(2nd(X)) |
→ |
a__2nd#(mark(X)) |
(11) |
could be deleted.
1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.