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) |
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)] |
= |
2 + 2 · x1
|
[0] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[s(x1)] |
= |
1 · x1
|
[p(x1)] |
= |
1 · x1
|
[mark#(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
the
pair
mark#(f(X)) |
→ |
mark#(X) |
(29) |
and
no rules
could be deleted.
1.1.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(mark#) |
= |
1 |
|
stat(mark#) |
= |
lex
|
prec(cons) |
= |
1 |
|
stat(cons) |
= |
lex
|
prec(mark) |
= |
0 |
|
stat(mark) |
= |
lex
|
prec(f) |
= |
1 |
|
stat(f) |
= |
lex
|
prec(0) |
= |
1 |
|
stat(0) |
= |
lex
|
prec(s) |
= |
0 |
|
stat(s) |
= |
lex
|
prec(p) |
= |
1 |
|
stat(p) |
= |
lex
|
π(mark#) |
= |
[] |
π(cons) |
= |
[] |
π(active#) |
= |
1 |
π(mark) |
= |
[1] |
π(f) |
= |
[] |
π(0) |
= |
[] |
π(s) |
= |
[] |
π(p) |
= |
[] |
π(active) |
= |
1 |
together with the usable
rules
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(12) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(11) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(13) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(14) |
f(active(X)) |
→ |
f(X) |
(10) |
f(mark(X)) |
→ |
f(X) |
(9) |
s(active(X)) |
→ |
s(X) |
(16) |
s(mark(X)) |
→ |
s(X) |
(15) |
p(active(X)) |
→ |
p(X) |
(18) |
p(mark(X)) |
→ |
p(X) |
(17) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(34) |
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) |
= |
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.
1.1.1.1.1.1 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) |
mark#(p(X)) |
→ |
mark#(X) |
(39) |
1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[s(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[p(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.
1.1.1.1.1.1.1.1 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 |
mark#(p(X)) |
→ |
mark#(X) |
(39) |
|
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 Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(s) |
= |
4 |
|
weight(s) |
= |
1 |
|
|
|
prec(mark#) |
= |
3 |
|
weight(mark#) |
= |
1 |
|
|
|
prec(p) |
= |
2 |
|
weight(p) |
= |
1 |
|
|
|
prec(cons) |
= |
0 |
|
weight(cons) |
= |
1 |
|
|
|
prec(0) |
= |
1 |
|
weight(0) |
= |
1 |
|
|
|
in combination with the following argument filter
π(active#) |
= |
1 |
π(f) |
= |
1 |
π(s) |
= |
[] |
π(mark#) |
= |
[] |
π(p) |
= |
[] |
π(mark) |
= |
1 |
π(active) |
= |
1 |
π(cons) |
= |
[] |
π(0) |
= |
[] |
the
pairs
active#(f(s(0))) |
→ |
mark#(f(p(s(0)))) |
(23) |
mark#(f(p(s(0)))) |
→ |
active#(f(mark(p(s(0))))) |
(50) |
could be deleted.
1.1.1.1.1.1.2.1.1 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 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[f#(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.
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 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[cons#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.3.1 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 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[s#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.4.1 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 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[p#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.5.1 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.