Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex14_AEGL02_GM)
The rewrite relation of the following TRS is considered.
a__from(X) |
→ |
cons(mark(X),from(s(X))) |
(1) |
a__length(nil) |
→ |
0 |
(2) |
a__length(cons(X,Y)) |
→ |
s(a__length1(Y)) |
(3) |
a__length1(X) |
→ |
a__length(X) |
(4) |
mark(from(X)) |
→ |
a__from(mark(X)) |
(5) |
mark(length(X)) |
→ |
a__length(X) |
(6) |
mark(length1(X)) |
→ |
a__length1(X) |
(7) |
mark(cons(X1,X2)) |
→ |
cons(mark(X1),X2) |
(8) |
mark(s(X)) |
→ |
s(mark(X)) |
(9) |
mark(nil) |
→ |
nil |
(10) |
mark(0) |
→ |
0 |
(11) |
a__from(X) |
→ |
from(X) |
(12) |
a__length(X) |
→ |
length(X) |
(13) |
a__length1(X) |
→ |
length1(X) |
(14) |
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__from#(X) |
→ |
mark#(X) |
(15) |
a__length#(cons(X,Y)) |
→ |
a__length1#(Y) |
(16) |
a__length1#(X) |
→ |
a__length#(X) |
(17) |
mark#(from(X)) |
→ |
a__from#(mark(X)) |
(18) |
mark#(from(X)) |
→ |
mark#(X) |
(19) |
mark#(length(X)) |
→ |
a__length#(X) |
(20) |
mark#(length1(X)) |
→ |
a__length1#(X) |
(21) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(22) |
mark#(s(X)) |
→ |
mark#(X) |
(23) |
1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
mark#(from(X)) |
→ |
a__from#(mark(X)) |
(18) |
a__from#(X) |
→ |
mark#(X) |
(15) |
mark#(from(X)) |
→ |
mark#(X) |
(19) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(22) |
mark#(s(X)) |
→ |
mark#(X) |
(23) |
1.1.1 Reduction Pair Processor
Using the
prec(from) |
= |
4 |
|
stat(from) |
= |
lex
|
prec(mark) |
= |
4 |
|
stat(mark) |
= |
lex
|
prec(a__from) |
= |
4 |
|
stat(a__from) |
= |
lex
|
prec(length) |
= |
0 |
|
stat(length) |
= |
lex
|
prec(a__length) |
= |
3 |
|
stat(a__length) |
= |
lex
|
prec(length1) |
= |
1 |
|
stat(length1) |
= |
lex
|
prec(a__length1) |
= |
3 |
|
stat(a__length1) |
= |
lex
|
prec(nil) |
= |
4 |
|
stat(nil) |
= |
lex
|
prec(0) |
= |
2 |
|
stat(0) |
= |
lex
|
π(mark#) |
= |
1 |
π(from) |
= |
[1] |
π(a__from#) |
= |
1 |
π(mark) |
= |
[1] |
π(cons) |
= |
1 |
π(s) |
= |
1 |
π(a__from) |
= |
[1] |
π(length) |
= |
[] |
π(a__length) |
= |
[] |
π(length1) |
= |
[] |
π(a__length1) |
= |
[] |
π(nil) |
= |
[] |
π(0) |
= |
[] |
the
pair
mark#(from(X)) |
→ |
mark#(X) |
(19) |
could be deleted.
1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[a__from#(x1)] |
= |
2 · x1
|
[mark(x1)] |
= |
2 · x1
|
[from(x1)] |
= |
2 + 2 · x1
|
[a__from(x1)] |
= |
2 + 2 · x1
|
[length(x1)] |
= |
0 |
[a__length(x1)] |
= |
0 |
[length1(x1)] |
= |
0 |
[a__length1(x1)] |
= |
0 |
[cons(x1, x2)] |
= |
1 + x1
|
[s(x1)] |
= |
2 · x1
|
[nil] |
= |
1 |
[0] |
= |
0 |
[mark#(x1)] |
= |
2 · x1
|
the
pairs
mark#(from(X)) |
→ |
a__from#(mark(X)) |
(18) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(22) |
could be deleted.
1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
mark#(s(X)) |
→ |
mark#(X) |
(23) |
1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[s(x1)] |
= |
1 · x1
|
[mark#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
mark#(s(X)) |
→ |
mark#(X) |
(23) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
a__length1#(X) |
→ |
a__length#(X) |
(17) |
a__length#(cons(X,Y)) |
→ |
a__length1#(Y) |
(16) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[a__length#(x1)] |
= |
1 · x1
|
[a__length1#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
a__length#(cons(X,Y)) |
→ |
a__length1#(Y) |
(16) |
|
1 |
> |
1 |
a__length1#(X) |
→ |
a__length#(X) |
(17) |
|
1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.