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)) |
(70) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(68) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(top#) |
= |
0 |
|
stat(top#) |
= |
lex
|
prec(ok) |
= |
0 |
|
stat(ok) |
= |
lex
|
prec(proper) |
= |
0 |
|
stat(proper) |
= |
lex
|
prec(x) |
= |
5 |
|
stat(x) |
= |
lex
|
prec(s) |
= |
1 |
|
stat(s) |
= |
lex
|
prec(plus) |
= |
3 |
|
stat(plus) |
= |
lex
|
prec(0) |
= |
0 |
|
stat(0) |
= |
lex
|
prec(mark) |
= |
0 |
|
stat(mark) |
= |
lex
|
prec(active) |
= |
0 |
|
stat(active) |
= |
lex
|
prec(and) |
= |
1 |
|
stat(and) |
= |
lex
|
prec(tt) |
= |
0 |
|
stat(tt) |
= |
lex
|
π(top#) |
= |
1 |
π(ok) |
= |
1 |
π(proper) |
= |
1 |
π(x) |
= |
[1,2] |
π(s) |
= |
[1] |
π(plus) |
= |
[1,2] |
π(0) |
= |
[] |
π(mark) |
= |
[1] |
π(active) |
= |
1 |
π(and) |
= |
[1,2] |
π(tt) |
= |
[] |
together with the usable
rules
active(and(tt,X)) |
→ |
mark(X) |
(1) |
active(plus(N,0)) |
→ |
mark(N) |
(2) |
active(plus(N,s(M))) |
→ |
mark(s(plus(N,M))) |
(3) |
active(x(N,0)) |
→ |
mark(0) |
(4) |
active(x(N,s(M))) |
→ |
mark(plus(x(N,M),N)) |
(5) |
active(and(X1,X2)) |
→ |
and(active(X1),X2) |
(6) |
active(plus(X1,X2)) |
→ |
plus(active(X1),X2) |
(7) |
active(plus(X1,X2)) |
→ |
plus(X1,active(X2)) |
(8) |
active(s(X)) |
→ |
s(active(X)) |
(9) |
active(x(X1,X2)) |
→ |
x(active(X1),X2) |
(10) |
active(x(X1,X2)) |
→ |
x(X1,active(X2)) |
(11) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(13) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(14) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(25) |
s(mark(X)) |
→ |
mark(s(X)) |
(15) |
s(ok(X)) |
→ |
ok(s(X)) |
(26) |
x(mark(X1),X2) |
→ |
mark(x(X1,X2)) |
(16) |
x(X1,mark(X2)) |
→ |
mark(x(X1,X2)) |
(17) |
x(ok(X1),ok(X2)) |
→ |
ok(x(X1,X2)) |
(27) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(12) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(24) |
proper(and(X1,X2)) |
→ |
and(proper(X1),proper(X2)) |
(18) |
proper(tt) |
→ |
ok(tt) |
(19) |
proper(plus(X1,X2)) |
→ |
plus(proper(X1),proper(X2)) |
(20) |
proper(0) |
→ |
ok(0) |
(21) |
proper(s(X)) |
→ |
s(proper(X)) |
(22) |
proper(x(X1,X2)) |
→ |
x(proper(X1),proper(X2)) |
(23) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(68) |
could be deleted.
1.1.1.1 Subterm Criterion Processor
We use the projection to multisets
π(top#)
|
= |
{
1
}
|
π(ok)
|
= |
{
1, 1, 1
}
|
π(x)
|
= |
{
2, 2
}
|
π(s)
|
= |
{
1, 1
}
|
π(plus)
|
= |
{
1, 1
}
|
π(mark)
|
= |
{
1
}
|
π(active)
|
= |
{
1, 1
}
|
π(and)
|
= |
{
2
}
|
to remove the pairs:
top#(ok(X)) |
→ |
top#(active(X)) |
(70) |
1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
active#(and(X1,X2)) |
→ |
active#(X1) |
(34) |
active#(plus(X1,X2)) |
→ |
active#(X1) |
(36) |
active#(plus(X1,X2)) |
→ |
active#(X2) |
(38) |
active#(s(X)) |
→ |
active#(X) |
(40) |
active#(x(X1,X2)) |
→ |
active#(X1) |
(42) |
active#(x(X1,X2)) |
→ |
active#(X2) |
(44) |
1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
active#(and(X1,X2)) |
→ |
active#(X1) |
(34) |
|
1 |
> |
1 |
active#(plus(X1,X2)) |
→ |
active#(X1) |
(36) |
|
1 |
> |
1 |
active#(plus(X1,X2)) |
→ |
active#(X2) |
(38) |
|
1 |
> |
1 |
active#(s(X)) |
→ |
active#(X) |
(40) |
|
1 |
> |
1 |
active#(x(X1,X2)) |
→ |
active#(X1) |
(42) |
|
1 |
> |
1 |
active#(x(X1,X2)) |
→ |
active#(X2) |
(44) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
proper#(and(X1,X2)) |
→ |
proper#(X2) |
(52) |
proper#(and(X1,X2)) |
→ |
proper#(X1) |
(53) |
proper#(plus(X1,X2)) |
→ |
proper#(X2) |
(55) |
proper#(plus(X1,X2)) |
→ |
proper#(X1) |
(56) |
proper#(s(X)) |
→ |
proper#(X) |
(58) |
proper#(x(X1,X2)) |
→ |
proper#(X2) |
(60) |
proper#(x(X1,X2)) |
→ |
proper#(X1) |
(61) |
1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
proper#(and(X1,X2)) |
→ |
proper#(X2) |
(52) |
|
1 |
> |
1 |
proper#(and(X1,X2)) |
→ |
proper#(X1) |
(53) |
|
1 |
> |
1 |
proper#(plus(X1,X2)) |
→ |
proper#(X2) |
(55) |
|
1 |
> |
1 |
proper#(plus(X1,X2)) |
→ |
proper#(X1) |
(56) |
|
1 |
> |
1 |
proper#(s(X)) |
→ |
proper#(X) |
(58) |
|
1 |
> |
1 |
proper#(x(X1,X2)) |
→ |
proper#(X2) |
(60) |
|
1 |
> |
1 |
proper#(x(X1,X2)) |
→ |
proper#(X1) |
(61) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
x#(mark(X1),X2) |
→ |
x#(X1,X2) |
(50) |
x#(X1,mark(X2)) |
→ |
x#(X1,X2) |
(51) |
x#(ok(X1),ok(X2)) |
→ |
x#(X1,X2) |
(66) |
1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
x#(mark(X1),X2) |
→ |
x#(X1,X2) |
(50) |
|
2 |
≥ |
2 |
1 |
> |
1 |
x#(X1,mark(X2)) |
→ |
x#(X1,X2) |
(51) |
|
2 |
> |
2 |
1 |
≥ |
1 |
x#(ok(X1),ok(X2)) |
→ |
x#(X1,X2) |
(66) |
|
2 |
> |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
s#(mark(X)) |
→ |
s#(X) |
(49) |
s#(ok(X)) |
→ |
s#(X) |
(65) |
1.1.5 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(mark(X)) |
→ |
s#(X) |
(49) |
|
1 |
> |
1 |
s#(ok(X)) |
→ |
s#(X) |
(65) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(47) |
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(48) |
plus#(ok(X1),ok(X2)) |
→ |
plus#(X1,X2) |
(64) |
1.1.6 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(47) |
|
2 |
≥ |
2 |
1 |
> |
1 |
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(48) |
|
2 |
> |
2 |
1 |
≥ |
1 |
plus#(ok(X1),ok(X2)) |
→ |
plus#(X1,X2) |
(64) |
|
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
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(46) |
and#(ok(X1),ok(X2)) |
→ |
and#(X1,X2) |
(63) |
1.1.7 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(46) |
|
2 |
≥ |
2 |
1 |
> |
1 |
and#(ok(X1),ok(X2)) |
→ |
and#(X1,X2) |
(63) |
|
2 |
> |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.