Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/ExSec11_1_Luc02a_L)
The rewrite relation of the following TRS is considered.
|
terms(N) |
→ |
cons(recip(sqr(N))) |
(1) |
|
sqr(0) |
→ |
0 |
(2) |
|
sqr(s(X)) |
→ |
s(add(sqr(X),dbl(X))) |
(3) |
|
dbl(0) |
→ |
0 |
(4) |
|
dbl(s(X)) |
→ |
s(s(dbl(X))) |
(5) |
|
add(0,X) |
→ |
X |
(6) |
|
add(s(X),Y) |
→ |
s(add(X,Y)) |
(7) |
|
first(0,X) |
→ |
nil |
(8) |
|
first(s(X),cons(Y)) |
→ |
cons(Y) |
(9) |
|
half(0) |
→ |
0 |
(10) |
|
half(s(0)) |
→ |
0 |
(11) |
|
half(s(s(X))) |
→ |
s(half(X)) |
(12) |
|
half(dbl(X)) |
→ |
X |
(13) |
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.
|
add#(s(X),Y) |
→ |
add#(X,Y) |
(14) |
|
sqr#(s(X)) |
→ |
dbl#(X) |
(15) |
|
terms#(N) |
→ |
sqr#(N) |
(16) |
|
sqr#(s(X)) |
→ |
sqr#(X) |
(17) |
|
dbl#(s(X)) |
→ |
dbl#(X) |
(18) |
|
half#(s(s(X))) |
→ |
half#(X) |
(19) |
|
sqr#(s(X)) |
→ |
add#(sqr(X),dbl(X)) |
(20) |
1.1 Dependency Graph Processor
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
|
half#(s(s(X))) |
→ |
half#(X) |
(19) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [recip(x1)] |
=
|
0 |
| [dbl(x1)] |
=
|
0 |
| [dbl#(x1)] |
=
|
0 |
| [terms#(x1)] |
=
|
0 |
| [half#(x1)] |
=
|
x1 + 0 |
| [half(x1)] |
=
|
0 |
| [sqr#(x1)] |
=
|
0 |
| [0] |
=
|
0 |
| [first#(x1, x2)] |
=
|
0 |
| [nil] |
=
|
0 |
| [first(x1, x2)] |
=
|
0 |
| [cons(x1)] |
=
|
0 |
| [add#(x1, x2)] |
=
|
0 |
| [add(x1, x2)] |
=
|
0 |
| [sqr(x1)] |
=
|
0 |
| [terms(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
half#(s(s(X))) |
→ |
half#(X) |
(19) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
|
sqr#(s(X)) |
→ |
sqr#(X) |
(17) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [recip(x1)] |
=
|
0 |
| [dbl(x1)] |
=
|
0 |
| [dbl#(x1)] |
=
|
0 |
| [terms#(x1)] |
=
|
0 |
| [half#(x1)] |
=
|
0 |
| [half(x1)] |
=
|
0 |
| [sqr#(x1)] |
=
|
x1 + 0 |
| [0] |
=
|
0 |
| [first#(x1, x2)] |
=
|
0 |
| [nil] |
=
|
0 |
| [first(x1, x2)] |
=
|
0 |
| [cons(x1)] |
=
|
0 |
| [add#(x1, x2)] |
=
|
0 |
| [add(x1, x2)] |
=
|
0 |
| [sqr(x1)] |
=
|
0 |
| [terms(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
sqr#(s(X)) |
→ |
sqr#(X) |
(17) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
|
dbl#(s(X)) |
→ |
dbl#(X) |
(18) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [recip(x1)] |
=
|
0 |
| [dbl(x1)] |
=
|
0 |
| [dbl#(x1)] |
=
|
x1 + 0 |
| [terms#(x1)] |
=
|
0 |
| [half#(x1)] |
=
|
0 |
| [half(x1)] |
=
|
0 |
| [sqr#(x1)] |
=
|
0 |
| [0] |
=
|
0 |
| [first#(x1, x2)] |
=
|
0 |
| [nil] |
=
|
0 |
| [first(x1, x2)] |
=
|
0 |
| [cons(x1)] |
=
|
0 |
| [add#(x1, x2)] |
=
|
0 |
| [add(x1, x2)] |
=
|
0 |
| [sqr(x1)] |
=
|
0 |
| [terms(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
dbl#(s(X)) |
→ |
dbl#(X) |
(18) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
|
add#(s(X),Y) |
→ |
add#(X,Y) |
(14) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [recip(x1)] |
=
|
0 |
| [dbl(x1)] |
=
|
0 |
| [dbl#(x1)] |
=
|
0 |
| [terms#(x1)] |
=
|
0 |
| [half#(x1)] |
=
|
0 |
| [half(x1)] |
=
|
0 |
| [sqr#(x1)] |
=
|
0 |
| [0] |
=
|
0 |
| [first#(x1, x2)] |
=
|
0 |
| [nil] |
=
|
0 |
| [first(x1, x2)] |
=
|
0 |
| [cons(x1)] |
=
|
0 |
| [add#(x1, x2)] |
=
|
x1 + 0 |
| [add(x1, x2)] |
=
|
0 |
| [sqr(x1)] |
=
|
0 |
| [terms(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
add#(s(X),Y) |
→ |
add#(X,Y) |
(14) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 0
components.