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)) |
(62) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(60) |
1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[proper(x1)] |
= |
1 · x1
|
[first(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[0] |
= |
0 |
[ok(x1)] |
= |
2 · x1
|
[nil] |
= |
0 |
[s(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[from(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
2 · x1
|
[top#(x1)] |
= |
1 · x1
|
together with the usable
rules
proper(first(X1,X2)) |
→ |
first(proper(X1),proper(X2)) |
(14) |
proper(0) |
→ |
ok(0) |
(15) |
proper(nil) |
→ |
ok(nil) |
(16) |
proper(s(X)) |
→ |
s(proper(X)) |
(17) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(18) |
proper(from(X)) |
→ |
from(proper(X)) |
(19) |
from(mark(X)) |
→ |
mark(from(X)) |
(13) |
from(ok(X)) |
→ |
ok(from(X)) |
(23) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(12) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(22) |
s(mark(X)) |
→ |
mark(s(X)) |
(11) |
s(ok(X)) |
→ |
ok(s(X)) |
(21) |
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(9) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(10) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(20) |
active(first(0,X)) |
→ |
mark(nil) |
(1) |
active(first(s(X),cons(Y,Z))) |
→ |
mark(cons(Y,first(X,Z))) |
(2) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(3) |
active(first(X1,X2)) |
→ |
first(active(X1),X2) |
(4) |
active(first(X1,X2)) |
→ |
first(X1,active(X2)) |
(5) |
active(s(X)) |
→ |
s(active(X)) |
(6) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(7) |
active(from(X)) |
→ |
from(active(X)) |
(8) |
(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 linear polynomial interpretation over the naturals
[top#(x1)] |
= |
x1 |
[active(x1)] |
= |
x1 |
[first(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[0] |
= |
2 |
[mark(x1)] |
= |
2 + x1
|
[nil] |
= |
0 |
[s(x1)] |
= |
1 + x1
|
[cons(x1, x2)] |
= |
2 · x1
|
[from(x1)] |
= |
2 + 2 · x1
|
[proper(x1)] |
= |
x1 |
[ok(x1)] |
= |
x1 |
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(60) |
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)] |
= |
2 · x1
|
[first(x1, x2)] |
= |
2 · x1 + 1 · x2
|
[0] |
= |
1 |
[mark(x1)] |
= |
1 · x1
|
[nil] |
= |
2 |
[s(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[from(x1)] |
= |
1 · x1
|
[ok(x1)] |
= |
2 · x1
|
[top#(x1)] |
= |
2 · x1
|
together with the usable
rules
active(first(0,X)) |
→ |
mark(nil) |
(1) |
active(first(s(X),cons(Y,Z))) |
→ |
mark(cons(Y,first(X,Z))) |
(2) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(3) |
active(first(X1,X2)) |
→ |
first(active(X1),X2) |
(4) |
active(first(X1,X2)) |
→ |
first(X1,active(X2)) |
(5) |
active(s(X)) |
→ |
s(active(X)) |
(6) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(7) |
active(from(X)) |
→ |
from(active(X)) |
(8) |
from(mark(X)) |
→ |
mark(from(X)) |
(13) |
from(ok(X)) |
→ |
ok(from(X)) |
(23) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(12) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(22) |
s(mark(X)) |
→ |
mark(s(X)) |
(11) |
s(ok(X)) |
→ |
ok(s(X)) |
(21) |
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(9) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(10) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(20) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
active(first(0,X)) |
→ |
mark(nil) |
(1) |
could be deleted.
1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
2 · x1
|
[first(x1, x2)] |
= |
2 · x1 + 1 · x2
|
[s(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[mark(x1)] |
= |
1 · x1
|
[from(x1)] |
= |
1 · x1
|
[ok(x1)] |
= |
1 + 2 · x1
|
[top#(x1)] |
= |
1 · x1
|
the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(62) |
and
the
rules
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(22) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(20) |
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#(first(X1,X2)) |
→ |
active#(X2) |
(34) |
active#(first(X1,X2)) |
→ |
active#(X1) |
(32) |
active#(s(X)) |
→ |
active#(X) |
(36) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(38) |
active#(from(X)) |
→ |
active#(X) |
(40) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[first(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[s(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[from(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#(first(X1,X2)) |
→ |
active#(X2) |
(34) |
|
1 |
> |
1 |
active#(first(X1,X2)) |
→ |
active#(X1) |
(32) |
|
1 |
> |
1 |
active#(s(X)) |
→ |
active#(X) |
(36) |
|
1 |
> |
1 |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(38) |
|
1 |
> |
1 |
active#(from(X)) |
→ |
active#(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
proper#(first(X1,X2)) |
→ |
proper#(X2) |
(48) |
proper#(first(X1,X2)) |
→ |
proper#(X1) |
(47) |
proper#(s(X)) |
→ |
proper#(X) |
(50) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(52) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(53) |
proper#(from(X)) |
→ |
proper#(X) |
(55) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[first(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[s(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[from(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#(first(X1,X2)) |
→ |
proper#(X2) |
(48) |
|
1 |
> |
1 |
proper#(first(X1,X2)) |
→ |
proper#(X1) |
(47) |
|
1 |
> |
1 |
proper#(s(X)) |
→ |
proper#(X) |
(50) |
|
1 |
> |
1 |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(52) |
|
1 |
> |
1 |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(53) |
|
1 |
> |
1 |
proper#(from(X)) |
→ |
proper#(X) |
(55) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(42) |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(41) |
first#(ok(X1),ok(X2)) |
→ |
first#(X1,X2) |
(56) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[ok(x1)] |
= |
1 · x1
|
[first#(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.
1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(42) |
|
1 |
≥ |
1 |
2 |
> |
2 |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(41) |
|
1 |
> |
1 |
2 |
≥ |
2 |
first#(ok(X1),ok(X2)) |
→ |
first#(X1,X2) |
(56) |
|
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
s#(ok(X)) |
→ |
s#(X) |
(57) |
s#(mark(X)) |
→ |
s#(X) |
(43) |
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
|
[s#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.5.1 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) |
(57) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(43) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(58) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(44) |
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
|
[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.
1.1.6.1 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) |
(58) |
|
1 |
> |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(44) |
|
1 |
> |
1 |
2 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
from#(ok(X)) |
→ |
from#(X) |
(59) |
from#(mark(X)) |
→ |
from#(X) |
(45) |
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
|
[from#(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.
from#(ok(X)) |
→ |
from#(X) |
(59) |
|
1 |
> |
1 |
from#(mark(X)) |
→ |
from#(X) |
(45) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.