The rewrite relation of the following TRS is considered.
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(31) |
active#(f(0)) |
→ |
mark#(cons(0,f(s(0)))) |
(19) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(33) |
mark#(f(X)) |
→ |
active#(f(mark(X))) |
(27) |
active#(f(s(0))) |
→ |
mark#(f(p(s(0)))) |
(23) |
mark#(f(X)) |
→ |
mark#(X) |
(29) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(34) |
active#(p(s(X))) |
→ |
mark#(X) |
(26) |
mark#(s(X)) |
→ |
mark#(X) |
(36) |
mark#(p(X)) |
→ |
active#(p(mark(X))) |
(37) |
mark#(p(X)) |
→ |
mark#(X) |
(39) |
1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 + 2 · x1
|
[0] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
2 · x1 + 1 · x2
|
[s(x1)] |
= |
2 · x1
|
[p(x1)] |
= |
2 · x1
|
[mark#(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
2 · x1
|
the
pair
mark#(f(X)) |
→ |
mark#(X) |
(29) |
and
no rules
could be deleted.
1.1.1.1 Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(f) |
= |
1 |
|
weight(f) |
= |
1 |
|
|
|
prec(0) |
= |
0 |
|
weight(0) |
= |
1 |
|
|
|
in combination with the following argument filter
π(mark#) |
= |
1 |
π(cons) |
= |
1 |
π(active#) |
= |
1 |
π(mark) |
= |
1 |
π(f) |
= |
[] |
π(0) |
= |
[] |
π(s) |
= |
1 |
π(p) |
= |
1 |
π(active) |
= |
1 |
the
pair
active#(f(0)) |
→ |
mark#(cons(0,f(s(0)))) |
(19) |
could be deleted.
1.1.1.1.1 Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(f) |
= |
2 |
|
weight(f) |
= |
1 |
|
|
|
prec(p) |
= |
0 |
|
weight(p) |
= |
1 |
|
|
|
prec(0) |
= |
1 |
|
weight(0) |
= |
1 |
|
|
|
in combination with the following argument filter
π(mark#) |
= |
1 |
π(cons) |
= |
1 |
π(active#) |
= |
1 |
π(mark) |
= |
1 |
π(f) |
= |
[] |
π(s) |
= |
1 |
π(p) |
= |
[1] |
π(active) |
= |
1 |
π(0) |
= |
[] |
the
pairs
active#(p(s(X))) |
→ |
mark#(X) |
(26) |
mark#(p(X)) |
→ |
mark#(X) |
(39) |
could be deleted.
1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
mark#(s(X)) |
→ |
mark#(X) |
(36) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(33) |
1.1.1.1.1.1.1 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
cons(mark(x0),x1) |
cons(x0,mark(x1)) |
cons(active(x0),x1) |
cons(x0,active(x1)) |
s(mark(x0)) |
s(active(x0)) |
1.1.1.1.1.1.1.1.1 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) |
(36) |
|
1 |
> |
1 |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(33) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
active#(f(s(0))) |
→ |
mark#(f(p(s(0)))) |
(23) |
mark#(f(X)) |
→ |
active#(f(mark(X))) |
(27) |
1.1.1.1.1.1.2 Instantiation Processor
We instantiate the pair
to the following set of pairs
mark#(f(p(s(0)))) |
→ |
active#(f(mark(p(s(0))))) |
(50) |
1.1.1.1.1.1.2.1 Reduction Pair Processor
Using the linear polynomial interpretation over the rationals with delta = 1/64
[active#(x1)] |
= |
0 + 1/4 · x1
|
[f(x1)] |
= |
0 + 1/4 · x1
|
[s(x1)] |
= |
0 + 2 · x1
|
[0] |
= |
1/4 |
[mark#(x1)] |
= |
0 + 1/2 · x1
|
[p(x1)] |
= |
0 + 1/2 · x1
|
[mark(x1)] |
= |
0 + 1 · x1
|
[cons(x1, x2)] |
= |
0 + 0 · x1 + 1/4 · x2
|
[active(x1)] |
= |
0 + 1 · x1
|
the
pair
mark#(f(p(s(0)))) |
→ |
active#(f(mark(p(s(0))))) |
(50) |
could be deleted.
1.1.1.1.1.1.2.1.1 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.2.1.1.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
f(mark(x0)) |
f(active(x0)) |
s(mark(x0)) |
s(active(x0)) |
p(mark(x0)) |
p(active(x0)) |
1.1.1.1.1.1.2.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
2 · x1
|
[f(x1)] |
= |
2 + 1 · x1
|
[s(x1)] |
= |
2 + 1 · x1
|
[0] |
= |
1 |
[mark#(x1)] |
= |
1 · x1
|
[p(x1)] |
= |
2 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
active#(f(s(0))) |
→ |
mark#(f(p(s(0)))) |
(23) |
and
no rules
could be deleted.
1.1.1.1.1.1.2.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
f#(active(X)) |
→ |
f#(X) |
(41) |
f#(mark(X)) |
→ |
f#(X) |
(40) |
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.
active(f(0)) |
active(f(s(0))) |
active(p(s(x0))) |
mark(f(x0)) |
mark(0) |
mark(cons(x0,x1)) |
mark(s(x0)) |
mark(p(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.
f#(active(X)) |
→ |
f#(X) |
(41) |
|
1 |
> |
1 |
f#(mark(X)) |
→ |
f#(X) |
(40) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(43) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(42) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(44) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(45) |
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.
active(f(0)) |
active(f(s(0))) |
active(p(s(x0))) |
mark(f(x0)) |
mark(0) |
mark(cons(x0,x1)) |
mark(s(x0)) |
mark(p(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.
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(43) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(42) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(44) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(45) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
s#(active(X)) |
→ |
s#(X) |
(47) |
s#(mark(X)) |
→ |
s#(X) |
(46) |
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.
active(f(0)) |
active(f(s(0))) |
active(p(s(x0))) |
mark(f(x0)) |
mark(0) |
mark(cons(x0,x1)) |
mark(s(x0)) |
mark(p(x0)) |
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.
s#(active(X)) |
→ |
s#(X) |
(47) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(46) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
p#(active(X)) |
→ |
p#(X) |
(49) |
p#(mark(X)) |
→ |
p#(X) |
(48) |
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.
active(f(0)) |
active(f(s(0))) |
active(p(s(x0))) |
mark(f(x0)) |
mark(0) |
mark(cons(x0,x1)) |
mark(s(x0)) |
mark(p(x0)) |
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.
p#(active(X)) |
→ |
p#(X) |
(49) |
|
1 |
> |
1 |
p#(mark(X)) |
→ |
p#(X) |
(48) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.