Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex4_7_77_Bor03_iGM)
The rewrite relation of the following TRS is considered.
active(zeros) |
→ |
mark(cons(0,zeros)) |
(1) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(2) |
mark(zeros) |
→ |
active(zeros) |
(3) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(4) |
mark(0) |
→ |
active(0) |
(5) |
mark(tail(X)) |
→ |
active(tail(mark(X))) |
(6) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(7) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(8) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(9) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(10) |
tail(mark(X)) |
→ |
tail(X) |
(11) |
tail(active(X)) |
→ |
tail(X) |
(12) |
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.
mark#(tail(X)) |
→ |
active#(tail(mark(X))) |
(13) |
tail#(mark(X)) |
→ |
tail#(X) |
(14) |
active#(zeros) |
→ |
cons#(0,zeros) |
(15) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(16) |
mark#(zeros) |
→ |
active#(zeros) |
(17) |
active#(zeros) |
→ |
mark#(cons(0,zeros)) |
(18) |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(19) |
tail#(active(X)) |
→ |
tail#(X) |
(20) |
mark#(0) |
→ |
active#(0) |
(21) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(22) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(23) |
mark#(cons(X1,X2)) |
→ |
cons#(mark(X1),X2) |
(24) |
mark#(tail(X)) |
→ |
tail#(mark(X)) |
(25) |
active#(tail(cons(X,XS))) |
→ |
mark#(XS) |
(26) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(27) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(28) |
mark#(tail(X)) |
→ |
mark#(X) |
(29) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
mark#(tail(X)) |
→ |
mark#(X) |
(29) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(28) |
active#(zeros) |
→ |
mark#(cons(0,zeros)) |
(18) |
active#(tail(cons(X,XS))) |
→ |
mark#(XS) |
(26) |
mark#(zeros) |
→ |
active#(zeros) |
(17) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(23) |
mark#(tail(X)) |
→ |
active#(tail(mark(X))) |
(13) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[zeros] |
=
|
8857 |
[tail(x1)] |
=
|
8857 |
[mark#(x1)] |
=
|
8857 |
[0] |
=
|
1 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
1 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
1 |
[active#(x1)] |
=
|
x1 + 0 |
together with the usable
rules
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(8) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(10) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(7) |
tail(active(X)) |
→ |
tail(X) |
(12) |
tail(mark(X)) |
→ |
tail(X) |
(11) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(28) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
tail#(mark(X)) |
→ |
tail#(X) |
(14) |
tail#(active(X)) |
→ |
tail#(X) |
(20) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[zeros] |
=
|
1 |
[tail(x1)] |
=
|
0 |
[mark#(x1)] |
=
|
x1 + 6534 |
[0] |
=
|
0 |
[tail#(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
x1 + 4 |
[active(x1)] |
=
|
x1 + 3 |
[cons(x1, x2)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
8855 |
together with the usable
rules
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(8) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(10) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(7) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
tail#(mark(X)) |
→ |
tail#(X) |
(14) |
tail#(active(X)) |
→ |
tail#(X) |
(20) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(19) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(27) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(16) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(22) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
x1 + x2 + 0 |
[zeros] |
=
|
1 |
[tail(x1)] |
=
|
0 |
[mark#(x1)] |
=
|
x1 + 6534 |
[0] |
=
|
0 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 2 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
8855 |
together with the usable
rules
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(8) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(10) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(7) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(19) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(27) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(16) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(22) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.