Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex4_7_77_Bor03_C)
The rewrite relation of the following TRS is considered.
active(zeros) |
→ |
mark(cons(0,zeros)) |
(1) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(2) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(3) |
active(tail(X)) |
→ |
tail(active(X)) |
(4) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(5) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(6) |
proper(zeros) |
→ |
ok(zeros) |
(7) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(8) |
proper(0) |
→ |
ok(0) |
(9) |
proper(tail(X)) |
→ |
tail(proper(X)) |
(10) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(11) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(12) |
top(mark(X)) |
→ |
top(proper(X)) |
(13) |
top(ok(X)) |
→ |
top(active(X)) |
(14) |
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.
top#(mark(X)) |
→ |
top#(proper(X)) |
(15) |
top#(ok(X)) |
→ |
top#(active(X)) |
(16) |
active#(zeros) |
→ |
cons#(0,zeros) |
(17) |
proper#(tail(X)) |
→ |
proper#(X) |
(18) |
active#(cons(X1,X2)) |
→ |
cons#(active(X1),X2) |
(19) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(20) |
proper#(cons(X1,X2)) |
→ |
cons#(proper(X1),proper(X2)) |
(21) |
top#(ok(X)) |
→ |
active#(X) |
(22) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(23) |
tail#(ok(X)) |
→ |
tail#(X) |
(24) |
active#(tail(X)) |
→ |
tail#(active(X)) |
(25) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(26) |
top#(mark(X)) |
→ |
proper#(X) |
(27) |
tail#(mark(X)) |
→ |
tail#(X) |
(28) |
proper#(tail(X)) |
→ |
tail#(proper(X)) |
(29) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(30) |
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(31) |
active#(tail(X)) |
→ |
active#(X) |
(32) |
1.1 Dependency Graph Processor
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(16) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(15) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the argument filter
π(proper) |
= |
1 |
π(ok) |
= |
1 |
π(active) |
= |
1 |
in combination with the following Weighted Path Order with the following precedence and status
prec(cons#) |
= |
0 |
|
status(cons#) |
= |
[] |
|
list-extension(cons#) |
= |
Lex |
prec(top) |
= |
0 |
|
status(top) |
= |
[] |
|
list-extension(top) |
= |
Lex |
prec(top#) |
= |
0 |
|
status(top#) |
= |
[1] |
|
list-extension(top#) |
= |
Lex |
prec(zeros) |
= |
6 |
|
status(zeros) |
= |
[] |
|
list-extension(zeros) |
= |
Lex |
prec(tail) |
= |
3 |
|
status(tail) |
= |
[1] |
|
list-extension(tail) |
= |
Lex |
prec(0) |
= |
2 |
|
status(0) |
= |
[] |
|
list-extension(0) |
= |
Lex |
prec(tail#) |
= |
0 |
|
status(tail#) |
= |
[] |
|
list-extension(tail#) |
= |
Lex |
prec(mark) |
= |
1 |
|
status(mark) |
= |
[1] |
|
list-extension(mark) |
= |
Lex |
prec(proper#) |
= |
0 |
|
status(proper#) |
= |
[] |
|
list-extension(proper#) |
= |
Lex |
prec(cons) |
= |
5 |
|
status(cons) |
= |
[1] |
|
list-extension(cons) |
= |
Lex |
prec(active#) |
= |
0 |
|
status(active#) |
= |
[] |
|
list-extension(active#) |
= |
Lex |
and the following
Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
x2 + 1 |
[top(x1)] |
=
|
1 |
[top#(x1)] |
=
|
x1 + 1 |
[zeros] |
=
|
8946 |
[tail(x1)] |
=
|
x1 + 15922 |
[0] |
=
|
0 |
[tail#(x1)] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
[active#(x1)] |
=
|
1 |
together with the usable
rules
active(tail(X)) |
→ |
tail(active(X)) |
(4) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(8) |
active(zeros) |
→ |
mark(cons(0,zeros)) |
(1) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(3) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(5) |
proper(tail(X)) |
→ |
tail(proper(X)) |
(10) |
proper(zeros) |
→ |
ok(zeros) |
(7) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(12) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(11) |
proper(0) |
→ |
ok(0) |
(9) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(6) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(15) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(30) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(26) |
proper#(tail(X)) |
→ |
proper#(X) |
(18) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[zeros] |
=
|
1 |
[tail(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
0 |
[ok(x1)] |
=
|
25907 |
[0] |
=
|
20698 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
11050 |
[proper#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
11049 |
[cons(x1, x2)] |
=
|
x1 + x2 + 28882 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(5) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(11) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(30) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(26) |
proper#(tail(X)) |
→ |
proper#(X) |
(18) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
active#(tail(X)) |
→ |
active#(X) |
(32) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(20) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[zeros] |
=
|
1 |
[tail(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
0 |
[ok(x1)] |
=
|
2447 |
[0] |
=
|
19178 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
17749 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
17748 |
[cons(x1, x2)] |
=
|
x1 + x2 + 28882 |
[active#(x1)] |
=
|
x1 + 0 |
together with the usable
rules
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(5) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(11) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(tail(X)) |
→ |
active#(X) |
(32) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(20) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(31) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(23) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
x2 + 0 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[zeros] |
=
|
1 |
[tail(x1)] |
=
|
x1 + 29534 |
[proper(x1)] |
=
|
0 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
24766 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
2 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 28224 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(5) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(11) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(31) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
5th
component contains the
pair
tail#(mark(X)) |
→ |
tail#(X) |
(28) |
tail#(ok(X)) |
→ |
tail#(X) |
(24) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[zeros] |
=
|
1 |
[tail(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
1103 |
[ok(x1)] |
=
|
x1 + 2332 |
[0] |
=
|
40917 |
[tail#(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
x1 + 49044 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
49043 |
[cons(x1, x2)] |
=
|
x1 + 14137 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(5) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(11) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
tail#(mark(X)) |
→ |
tail#(X) |
(28) |
tail#(ok(X)) |
→ |
tail#(X) |
(24) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.