The rewrite relation of the following TRS is considered.
[mark(x1)] |
= |
· x1 +
|
[add(x1, x2)] |
= |
· x1 + · x2 +
|
[fst(x1, x2)] |
= |
· x1 + · x2 +
|
[nil] |
= |
|
[s(x1)] |
= |
· x1 +
|
[proper(x1)] |
= |
· x1 +
|
[0] |
= |
|
[from(x1)] |
= |
· x1 +
|
[top(x1)] |
= |
· x1 +
|
[ok(x1)] |
= |
· x1 +
|
[len(x1)] |
= |
· x1 +
|
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
The dependency pairs are split into 9
components.
-
The
1st
component contains the
pair
proper#(len(X)) |
→ |
proper#(X) |
(81) |
proper#(add(X1,X2)) |
→ |
proper#(X1) |
(79) |
proper#(add(X1,X2)) |
→ |
proper#(X2) |
(78) |
proper#(from(X)) |
→ |
proper#(X) |
(76) |
proper#(fst(X1,X2)) |
→ |
proper#(X1) |
(74) |
proper#(fst(X1,X2)) |
→ |
proper#(X2) |
(73) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(71) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(70) |
proper#(s(X)) |
→ |
proper#(X) |
(68) |
1.1.1.1 Subterm Criterion Processor
We use the projection
and remove the pairs:
proper#(len(X)) |
→ |
proper#(X) |
(81) |
proper#(add(X1,X2)) |
→ |
proper#(X1) |
(79) |
proper#(add(X1,X2)) |
→ |
proper#(X2) |
(78) |
proper#(from(X)) |
→ |
proper#(X) |
(76) |
proper#(fst(X1,X2)) |
→ |
proper#(X1) |
(74) |
proper#(fst(X1,X2)) |
→ |
proper#(X2) |
(73) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(71) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(70) |
proper#(s(X)) |
→ |
proper#(X) |
(68) |
1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(90) |
1.1.1.2 Reduction Pair Processor with Usable Rules
Using the
prec(top#) |
= |
0 |
|
stat(top#) |
= |
lex
|
prec(ok) |
= |
0 |
|
stat(ok) |
= |
lex
|
prec(len) |
= |
10 |
|
stat(len) |
= |
lex
|
prec(add) |
= |
8 |
|
stat(add) |
= |
lex
|
prec(from) |
= |
0 |
|
stat(from) |
= |
lex
|
prec(cons) |
= |
0 |
|
stat(cons) |
= |
lex
|
prec(s) |
= |
0 |
|
stat(s) |
= |
lex
|
prec(mark) |
= |
0 |
|
stat(mark) |
= |
lex
|
prec(nil) |
= |
0 |
|
stat(nil) |
= |
lex
|
prec(active) |
= |
0 |
|
stat(active) |
= |
lex
|
prec(fst) |
= |
0 |
|
stat(fst) |
= |
lex
|
prec(0) |
= |
1 |
|
stat(0) |
= |
lex
|
π(top#) |
= |
1 |
π(ok) |
= |
[1] |
π(len) |
= |
[1] |
π(add) |
= |
[2] |
π(from) |
= |
1 |
π(cons) |
= |
2 |
π(s) |
= |
1 |
π(mark) |
= |
1 |
π(nil) |
= |
[] |
π(active) |
= |
1 |
π(fst) |
= |
1 |
π(0) |
= |
[] |
together with the usable
rules
active(fst(0,Z)) |
→ |
mark(nil) |
(1) |
active(fst(s(X),cons(Y,Z))) |
→ |
mark(cons(Y,fst(X,Z))) |
(2) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(3) |
active(add(0,X)) |
→ |
mark(X) |
(4) |
active(add(s(X),Y)) |
→ |
mark(s(add(X,Y))) |
(5) |
active(len(nil)) |
→ |
mark(0) |
(6) |
active(len(cons(X,Z))) |
→ |
mark(s(len(Z))) |
(7) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(8) |
active(fst(X1,X2)) |
→ |
fst(active(X1),X2) |
(9) |
active(fst(X1,X2)) |
→ |
fst(X1,active(X2)) |
(10) |
active(from(X)) |
→ |
from(active(X)) |
(11) |
active(add(X1,X2)) |
→ |
add(active(X1),X2) |
(12) |
active(add(X1,X2)) |
→ |
add(X1,active(X2)) |
(13) |
active(len(X)) |
→ |
len(active(X)) |
(14) |
fst(mark(X1),X2) |
→ |
mark(fst(X1,X2)) |
(16) |
fst(X1,mark(X2)) |
→ |
mark(fst(X1,X2)) |
(17) |
fst(ok(X1),ok(X2)) |
→ |
ok(fst(X1,X2)) |
(32) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(15) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(31) |
s(ok(X)) |
→ |
ok(s(X)) |
(30) |
from(mark(X)) |
→ |
mark(from(X)) |
(18) |
from(ok(X)) |
→ |
ok(from(X)) |
(33) |
add(mark(X1),X2) |
→ |
mark(add(X1,X2)) |
(19) |
add(X1,mark(X2)) |
→ |
mark(add(X1,X2)) |
(20) |
add(ok(X1),ok(X2)) |
→ |
ok(add(X1,X2)) |
(34) |
len(mark(X)) |
→ |
mark(len(X)) |
(21) |
len(ok(X)) |
→ |
ok(len(X)) |
(35) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(90) |
could be deleted.
1.1.1.2.1 P is empty
There are no pairs anymore.
-
The
3rd
component contains the
pair
active#(len(X)) |
→ |
active#(X) |
(59) |
active#(add(X1,X2)) |
→ |
active#(X2) |
(57) |
active#(add(X1,X2)) |
→ |
active#(X1) |
(55) |
active#(from(X)) |
→ |
active#(X) |
(53) |
active#(fst(X1,X2)) |
→ |
active#(X2) |
(51) |
active#(fst(X1,X2)) |
→ |
active#(X1) |
(49) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(47) |
1.1.1.3 Subterm Criterion Processor
We use the projection
and remove the pairs:
active#(len(X)) |
→ |
active#(X) |
(59) |
active#(add(X1,X2)) |
→ |
active#(X2) |
(57) |
active#(add(X1,X2)) |
→ |
active#(X1) |
(55) |
active#(from(X)) |
→ |
active#(X) |
(53) |
active#(fst(X1,X2)) |
→ |
active#(X2) |
(51) |
active#(fst(X1,X2)) |
→ |
active#(X1) |
(49) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(47) |
1.1.1.3.1 P is empty
There are no pairs anymore.
-
The
4th
component contains the
pair
1.1.1.4 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) |
(83) |
|
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) |
(84) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(61) |
1.1.1.5 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) |
(84) |
|
2 |
> |
2 |
1 |
> |
1 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(61) |
|
2 |
≥ |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
fst#(ok(X1),ok(X2)) |
→ |
fst#(X1,X2) |
(85) |
fst#(X1,mark(X2)) |
→ |
fst#(X1,X2) |
(63) |
fst#(mark(X1),X2) |
→ |
fst#(X1,X2) |
(62) |
1.1.1.6 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
fst#(ok(X1),ok(X2)) |
→ |
fst#(X1,X2) |
(85) |
|
2 |
> |
2 |
1 |
> |
1 |
fst#(X1,mark(X2)) |
→ |
fst#(X1,X2) |
(63) |
|
2 |
> |
2 |
1 |
≥ |
1 |
fst#(mark(X1),X2) |
→ |
fst#(X1,X2) |
(62) |
|
2 |
≥ |
2 |
1 |
> |
1 |
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) |
(86) |
from#(mark(X)) |
→ |
from#(X) |
(64) |
1.1.1.7 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) |
(86) |
|
1 |
> |
1 |
from#(mark(X)) |
→ |
from#(X) |
(64) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
add#(ok(X1),ok(X2)) |
→ |
add#(X1,X2) |
(87) |
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(66) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(65) |
1.1.1.8 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
add#(ok(X1),ok(X2)) |
→ |
add#(X1,X2) |
(87) |
|
2 |
> |
2 |
1 |
> |
1 |
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(66) |
|
2 |
> |
2 |
1 |
≥ |
1 |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(65) |
|
2 |
≥ |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
9th
component contains the
pair
len#(ok(X)) |
→ |
len#(X) |
(88) |
len#(mark(X)) |
→ |
len#(X) |
(67) |
1.1.1.9 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
len#(ok(X)) |
→ |
len#(X) |
(88) |
|
1 |
> |
1 |
len#(mark(X)) |
→ |
len#(X) |
(67) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.