The rewrite relation of the following TRS is considered.
The dependency pairs are split into 7
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(55) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(53) |
1.1.1 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
proper(f(X)) |
→ |
f(proper(X)) |
(12) |
proper(0) |
→ |
ok(0) |
(13) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(14) |
proper(s(X)) |
→ |
s(proper(X)) |
(15) |
proper(p(X)) |
→ |
p(proper(X)) |
(16) |
p(mark(X)) |
→ |
mark(p(X)) |
(11) |
p(ok(X)) |
→ |
ok(p(X)) |
(20) |
s(mark(X)) |
→ |
mark(s(X)) |
(10) |
s(ok(X)) |
→ |
ok(s(X)) |
(19) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(9) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(18) |
f(mark(X)) |
→ |
mark(f(X)) |
(8) |
f(ok(X)) |
→ |
ok(f(X)) |
(17) |
active(f(0)) |
→ |
mark(cons(0,f(s(0)))) |
(1) |
active(f(s(0))) |
→ |
mark(f(p(s(0)))) |
(2) |
active(p(s(X))) |
→ |
mark(X) |
(3) |
active(f(X)) |
→ |
f(active(X)) |
(4) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(5) |
active(s(X)) |
→ |
s(active(X)) |
(6) |
active(p(X)) |
→ |
p(active(X)) |
(7) |
1.1.1.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(f(x0)) |
active(cons(x0,x1)) |
active(s(x0)) |
active(p(x0)) |
f(mark(x0)) |
cons(mark(x0),x1) |
s(mark(x0)) |
p(mark(x0)) |
proper(f(x0)) |
proper(0) |
proper(cons(x0,x1)) |
proper(s(x0)) |
proper(p(x0)) |
f(ok(x0)) |
cons(ok(x0),ok(x1)) |
s(ok(x0)) |
p(ok(x0)) |
1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
[top#(x1)] |
= |
+ · x1
|
[ok(x1)] |
= |
+ · x1
|
[active(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[proper(x1)] |
= |
+ · x1
|
[f(x1)] |
= |
+ · x1
|
[0] |
= |
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[s(x1)] |
= |
+ · x1
|
[p(x1)] |
= |
+ · x1
|
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(53) |
could be deleted.
1.1.1.1.1.1 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
active(f(0)) |
→ |
mark(cons(0,f(s(0)))) |
(1) |
active(f(s(0))) |
→ |
mark(f(p(s(0)))) |
(2) |
active(p(s(X))) |
→ |
mark(X) |
(3) |
active(f(X)) |
→ |
f(active(X)) |
(4) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(5) |
active(s(X)) |
→ |
s(active(X)) |
(6) |
active(p(X)) |
→ |
p(active(X)) |
(7) |
p(mark(X)) |
→ |
mark(p(X)) |
(11) |
p(ok(X)) |
→ |
ok(p(X)) |
(20) |
s(mark(X)) |
→ |
mark(s(X)) |
(10) |
s(ok(X)) |
→ |
ok(s(X)) |
(19) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(9) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(18) |
f(mark(X)) |
→ |
mark(f(X)) |
(8) |
f(ok(X)) |
→ |
ok(f(X)) |
(17) |
1.1.1.1.1.1.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(f(x0)) |
active(cons(x0,x1)) |
active(s(x0)) |
active(p(x0)) |
f(mark(x0)) |
cons(mark(x0),x1) |
s(mark(x0)) |
p(mark(x0)) |
f(ok(x0)) |
cons(ok(x0),ok(x1)) |
s(ok(x0)) |
p(ok(x0)) |
1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(0) |
= |
0 |
|
weight(0) |
= |
1 |
|
|
|
prec(active) |
= |
8 |
|
weight(active) |
= |
4 |
|
|
|
prec(f) |
= |
4 |
|
weight(f) |
= |
6 |
|
|
|
prec(mark) |
= |
1 |
|
weight(mark) |
= |
1 |
|
|
|
prec(s) |
= |
6 |
|
weight(s) |
= |
2 |
|
|
|
prec(p) |
= |
5 |
|
weight(p) |
= |
3 |
|
|
|
prec(ok) |
= |
3 |
|
weight(ok) |
= |
5 |
|
|
|
prec(top#) |
= |
7 |
|
weight(top#) |
= |
1 |
|
|
|
prec(cons) |
= |
2 |
|
weight(cons) |
= |
0 |
|
|
|
the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(55) |
and
the
rules
active(f(0)) |
→ |
mark(cons(0,f(s(0)))) |
(1) |
active(f(s(0))) |
→ |
mark(f(p(s(0)))) |
(2) |
active(p(s(X))) |
→ |
mark(X) |
(3) |
active(f(X)) |
→ |
f(active(X)) |
(4) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(5) |
active(s(X)) |
→ |
s(active(X)) |
(6) |
active(p(X)) |
→ |
p(active(X)) |
(7) |
f(mark(X)) |
→ |
mark(f(X)) |
(8) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(9) |
s(mark(X)) |
→ |
mark(s(X)) |
(10) |
p(mark(X)) |
→ |
mark(p(X)) |
(11) |
proper(f(X)) |
→ |
f(proper(X)) |
(12) |
proper(0) |
→ |
ok(0) |
(13) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(14) |
proper(s(X)) |
→ |
s(proper(X)) |
(15) |
proper(p(X)) |
→ |
p(proper(X)) |
(16) |
f(ok(X)) |
→ |
ok(f(X)) |
(17) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(18) |
s(ok(X)) |
→ |
ok(s(X)) |
(19) |
p(ok(X)) |
→ |
ok(p(X)) |
(20) |
top(mark(X)) |
→ |
top(proper(X)) |
(21) |
top(ok(X)) |
→ |
top(active(X)) |
(22) |
could be deleted.
1.1.1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
active#(cons(X1,X2)) |
→ |
active#(X1) |
(31) |
active#(f(X)) |
→ |
active#(X) |
(29) |
active#(s(X)) |
→ |
active#(X) |
(33) |
active#(p(X)) |
→ |
active#(X) |
(35) |
1.1.2 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.2.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
f(mark(x0)) |
cons(mark(x0),x1) |
s(mark(x0)) |
p(mark(x0)) |
f(ok(x0)) |
cons(ok(x0),ok(x1)) |
s(ok(x0)) |
p(ok(x0)) |
1.1.2.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
active#(cons(X1,X2)) |
→ |
active#(X1) |
(31) |
|
1 |
> |
1 |
active#(f(X)) |
→ |
active#(X) |
(29) |
|
1 |
> |
1 |
active#(s(X)) |
→ |
active#(X) |
(33) |
|
1 |
> |
1 |
active#(p(X)) |
→ |
active#(X) |
(35) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(43) |
proper#(f(X)) |
→ |
proper#(X) |
(41) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(44) |
proper#(s(X)) |
→ |
proper#(X) |
(46) |
proper#(p(X)) |
→ |
proper#(X) |
(48) |
1.1.3 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.3.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
f(mark(x0)) |
cons(mark(x0),x1) |
s(mark(x0)) |
p(mark(x0)) |
f(ok(x0)) |
cons(ok(x0),ok(x1)) |
s(ok(x0)) |
p(ok(x0)) |
1.1.3.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(43) |
|
1 |
> |
1 |
proper#(f(X)) |
→ |
proper#(X) |
(41) |
|
1 |
> |
1 |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(44) |
|
1 |
> |
1 |
proper#(s(X)) |
→ |
proper#(X) |
(46) |
|
1 |
> |
1 |
proper#(p(X)) |
→ |
proper#(X) |
(48) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
f#(ok(X)) |
→ |
f#(X) |
(49) |
f#(mark(X)) |
→ |
f#(X) |
(36) |
1.1.4 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.4.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
There are no lhss.
1.1.4.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
f#(ok(X)) |
→ |
f#(X) |
(49) |
|
1 |
> |
1 |
f#(mark(X)) |
→ |
f#(X) |
(36) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(50) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(37) |
1.1.5 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.5.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
There are no lhss.
1.1.5.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(50) |
|
1 |
> |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(37) |
|
1 |
> |
1 |
2 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
s#(ok(X)) |
→ |
s#(X) |
(51) |
s#(mark(X)) |
→ |
s#(X) |
(38) |
1.1.6 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.6.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
There are no lhss.
1.1.6.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(ok(X)) |
→ |
s#(X) |
(51) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(38) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
p#(ok(X)) |
→ |
p#(X) |
(52) |
p#(mark(X)) |
→ |
p#(X) |
(39) |
1.1.7 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.7.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
There are no lhss.
1.1.7.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
p#(ok(X)) |
→ |
p#(X) |
(52) |
|
1 |
> |
1 |
p#(mark(X)) |
→ |
p#(X) |
(39) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.