The rewrite relation of the following TRS is considered.
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(31) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(29) |
1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[proper(x1)] |
= |
1 · x1
|
[zeros] |
= |
0 |
[ok(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[0] |
= |
0 |
[tail(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[top#(x1)] |
= |
1 · x1
|
together with the usable
rules
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) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(6) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(12) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(5) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(11) |
active(zeros) |
→ |
mark(cons(0,zeros)) |
(1) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(3) |
active(tail(X)) |
→ |
tail(active(X)) |
(4) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[top#(x1)] |
= |
-1 + x1
|
[active(x1)] |
= |
1 + x1
|
[zeros] |
= |
1 |
[mark(x1)] |
= |
2 + 2 · x1
|
[cons(x1, x2)] |
= |
x1 |
[0] |
= |
0 |
[tail(x1)] |
= |
x1 |
[proper(x1)] |
= |
2 + 2 · x1
|
[ok(x1)] |
= |
2 + 2 · x1
|
the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(31) |
could be deleted.
1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[proper(x1)] |
= |
1 · x1
|
[zeros] |
= |
0 |
[ok(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[0] |
= |
0 |
[tail(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[top#(x1)] |
= |
1 · x1
|
together with the usable
rules
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) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(6) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(12) |
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
rule
could be deleted.
1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[proper(x1)] |
= |
1 · x1
|
[zeros] |
= |
0 |
[ok(x1)] |
= |
2 · x1
|
[cons(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[0] |
= |
0 |
[tail(x1)] |
= |
2 · x1
|
[mark(x1)] |
= |
1 + 1 · x1
|
[top#(x1)] |
= |
1 · x1
|
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(29) |
and
the
rules
tail(mark(X)) |
→ |
mark(tail(X)) |
(6) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(5) |
could be deleted.
1.1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
active#(tail(X)) |
→ |
active#(X) |
(19) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(17) |
1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[tail(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[active#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
active#(tail(X)) |
→ |
active#(X) |
(19) |
|
1 |
> |
1 |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(17) |
|
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#(X2) |
(24) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(23) |
proper#(tail(X)) |
→ |
proper#(X) |
(26) |
1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[tail(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.
1.1.1.3.1 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#(X2) |
(24) |
|
1 |
> |
1 |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(23) |
|
1 |
> |
1 |
proper#(tail(X)) |
→ |
proper#(X) |
(26) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(27) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(20) |
1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[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.
1.1.1.4.1 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) |
(27) |
|
1 |
> |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(20) |
|
1 |
> |
1 |
2 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
tail#(ok(X)) |
→ |
tail#(X) |
(28) |
tail#(mark(X)) |
→ |
tail#(X) |
(21) |
1.1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[tail#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
tail#(ok(X)) |
→ |
tail#(X) |
(28) |
|
1 |
> |
1 |
tail#(mark(X)) |
→ |
tail#(X) |
(21) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.