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 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[proper(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[0] |
= |
0 |
[ok(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
|
[top#(x1)] |
= |
1 · x1
|
together with the usable
rules
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(0))) |
→ |
mark(0) |
(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) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
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 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[0] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[s(x1)] |
= |
1 · x1
|
[p(x1)] |
= |
1 · x1
|
[ok(x1)] |
= |
1 · x1
|
[top#(x1)] |
= |
1 · x1
|
together with the usable
rules
active(f(0)) |
→ |
mark(cons(0,f(s(0)))) |
(1) |
active(f(s(0))) |
→ |
mark(f(p(s(0)))) |
(2) |
active(p(s(0))) |
→ |
mark(0) |
(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) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
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(0))) |
→ |
mark(0) |
(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) |
could be deleted.
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 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[f(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[p(x1)] |
= |
1 · x1
|
[active#(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.
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 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[f(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[p(x1)] |
= |
1 · x1
|
[proper#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.3.1 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 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(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.
1.1.4.1 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 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[mark(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.
1.1.5.1 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 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(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.
1.1.6.1 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 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(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.
1.1.7.1 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.