The rewrite relation of the following TRS is considered.
active(U11(tt,M,N)) |
→ |
mark(U12(tt,M,N)) |
(1) |
active(U12(tt,M,N)) |
→ |
mark(s(plus(N,M))) |
(2) |
active(plus(N,0)) |
→ |
mark(N) |
(3) |
active(plus(N,s(M))) |
→ |
mark(U11(tt,M,N)) |
(4) |
active(U11(X1,X2,X3)) |
→ |
U11(active(X1),X2,X3) |
(5) |
active(U12(X1,X2,X3)) |
→ |
U12(active(X1),X2,X3) |
(6) |
active(s(X)) |
→ |
s(active(X)) |
(7) |
active(plus(X1,X2)) |
→ |
plus(active(X1),X2) |
(8) |
active(plus(X1,X2)) |
→ |
plus(X1,active(X2)) |
(9) |
U11(mark(X1),X2,X3) |
→ |
mark(U11(X1,X2,X3)) |
(10) |
U12(mark(X1),X2,X3) |
→ |
mark(U12(X1,X2,X3)) |
(11) |
s(mark(X)) |
→ |
mark(s(X)) |
(12) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(13) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(14) |
proper(U11(X1,X2,X3)) |
→ |
U11(proper(X1),proper(X2),proper(X3)) |
(15) |
proper(tt) |
→ |
ok(tt) |
(16) |
proper(U12(X1,X2,X3)) |
→ |
U12(proper(X1),proper(X2),proper(X3)) |
(17) |
proper(s(X)) |
→ |
s(proper(X)) |
(18) |
proper(plus(X1,X2)) |
→ |
plus(proper(X1),proper(X2)) |
(19) |
proper(0) |
→ |
ok(0) |
(20) |
U11(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U11(X1,X2,X3)) |
(21) |
U12(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U12(X1,X2,X3)) |
(22) |
s(ok(X)) |
→ |
ok(s(X)) |
(23) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(24) |
top(mark(X)) |
→ |
top(proper(X)) |
(25) |
top(ok(X)) |
→ |
top(active(X)) |
(26) |
[mark(x1)] |
= |
· x1 +
|
[proper(x1)] |
= |
· x1 +
|
[U11(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
[U12(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
[plus(x1, x2)] |
= |
· x1 + · x2 +
|
[top(x1)] |
= |
· x1 +
|
[tt] |
= |
|
[0] |
= |
|
[ok(x1)] |
= |
· x1 +
|
[s(x1)] |
= |
· x1 +
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
[mark(x1)] |
= |
· x1 +
|
[proper(x1)] |
= |
· x1 +
|
[U11(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
[U12(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
[plus(x1, x2)] |
= |
· x1 + · x2 +
|
[top(x1)] |
= |
· x1 +
|
[tt] |
= |
|
[0] |
= |
|
[ok(x1)] |
= |
· x1 +
|
[s(x1)] |
= |
· x1 +
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
active#(U12(tt,M,N)) |
→ |
plus#(N,M) |
(27) |
active#(U12(tt,M,N)) |
→ |
s#(plus(N,M)) |
(28) |
active#(U11(X1,X2,X3)) |
→ |
active#(X1) |
(29) |
active#(U11(X1,X2,X3)) |
→ |
U11#(active(X1),X2,X3) |
(30) |
active#(U12(X1,X2,X3)) |
→ |
active#(X1) |
(31) |
active#(U12(X1,X2,X3)) |
→ |
U12#(active(X1),X2,X3) |
(32) |
active#(s(X)) |
→ |
active#(X) |
(33) |
active#(s(X)) |
→ |
s#(active(X)) |
(34) |
active#(plus(X1,X2)) |
→ |
active#(X1) |
(35) |
active#(plus(X1,X2)) |
→ |
plus#(active(X1),X2) |
(36) |
active#(plus(X1,X2)) |
→ |
active#(X2) |
(37) |
active#(plus(X1,X2)) |
→ |
plus#(X1,active(X2)) |
(38) |
U11#(mark(X1),X2,X3) |
→ |
U11#(X1,X2,X3) |
(39) |
U12#(mark(X1),X2,X3) |
→ |
U12#(X1,X2,X3) |
(40) |
s#(mark(X)) |
→ |
s#(X) |
(41) |
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(42) |
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(43) |
proper#(U11(X1,X2,X3)) |
→ |
proper#(X3) |
(44) |
proper#(U11(X1,X2,X3)) |
→ |
proper#(X2) |
(45) |
proper#(U11(X1,X2,X3)) |
→ |
proper#(X1) |
(46) |
proper#(U11(X1,X2,X3)) |
→ |
U11#(proper(X1),proper(X2),proper(X3)) |
(47) |
proper#(U12(X1,X2,X3)) |
→ |
proper#(X3) |
(48) |
proper#(U12(X1,X2,X3)) |
→ |
proper#(X2) |
(49) |
proper#(U12(X1,X2,X3)) |
→ |
proper#(X1) |
(50) |
proper#(U12(X1,X2,X3)) |
→ |
U12#(proper(X1),proper(X2),proper(X3)) |
(51) |
proper#(s(X)) |
→ |
proper#(X) |
(52) |
proper#(s(X)) |
→ |
s#(proper(X)) |
(53) |
proper#(plus(X1,X2)) |
→ |
proper#(X2) |
(54) |
proper#(plus(X1,X2)) |
→ |
proper#(X1) |
(55) |
proper#(plus(X1,X2)) |
→ |
plus#(proper(X1),proper(X2)) |
(56) |
U11#(ok(X1),ok(X2),ok(X3)) |
→ |
U11#(X1,X2,X3) |
(57) |
U12#(ok(X1),ok(X2),ok(X3)) |
→ |
U12#(X1,X2,X3) |
(58) |
s#(ok(X)) |
→ |
s#(X) |
(59) |
plus#(ok(X1),ok(X2)) |
→ |
plus#(X1,X2) |
(60) |
top#(mark(X)) |
→ |
proper#(X) |
(61) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(62) |
top#(ok(X)) |
→ |
active#(X) |
(63) |
top#(ok(X)) |
→ |
top#(active(X)) |
(64) |
The dependency pairs are split into 7
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(64) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(62) |
1.1.1.1.1 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(0) |
= |
0 |
|
stat(0) |
= |
lex
|
prec(s) |
= |
0 |
|
stat(s) |
= |
lex
|
prec(plus) |
= |
2 |
|
stat(plus) |
= |
lex
|
prec(mark) |
= |
0 |
|
stat(mark) |
= |
lex
|
prec(U12) |
= |
3 |
|
stat(U12) |
= |
lex
|
prec(active) |
= |
0 |
|
stat(active) |
= |
lex
|
prec(U11) |
= |
0 |
|
stat(U11) |
= |
lex
|
prec(tt) |
= |
0 |
|
stat(tt) |
= |
lex
|
π(top#) |
= |
1 |
π(ok) |
= |
1 |
π(proper) |
= |
1 |
π(0) |
= |
[] |
π(s) |
= |
1 |
π(plus) |
= |
[1,2] |
π(mark) |
= |
[1] |
π(U12) |
= |
[1,2,3] |
π(active) |
= |
1 |
π(U11) |
= |
1 |
π(tt) |
= |
[] |
together with the usable
rules
active(U12(tt,M,N)) |
→ |
mark(s(plus(N,M))) |
(2) |
active(U11(X1,X2,X3)) |
→ |
U11(active(X1),X2,X3) |
(5) |
active(U12(X1,X2,X3)) |
→ |
U12(active(X1),X2,X3) |
(6) |
active(s(X)) |
→ |
s(active(X)) |
(7) |
active(plus(X1,X2)) |
→ |
plus(active(X1),X2) |
(8) |
active(plus(X1,X2)) |
→ |
plus(X1,active(X2)) |
(9) |
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)) |
(24) |
s(mark(X)) |
→ |
mark(s(X)) |
(12) |
s(ok(X)) |
→ |
ok(s(X)) |
(23) |
U11(mark(X1),X2,X3) |
→ |
mark(U11(X1,X2,X3)) |
(10) |
U11(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U11(X1,X2,X3)) |
(21) |
U12(mark(X1),X2,X3) |
→ |
mark(U12(X1,X2,X3)) |
(11) |
U12(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U12(X1,X2,X3)) |
(22) |
proper(U11(X1,X2,X3)) |
→ |
U11(proper(X1),proper(X2),proper(X3)) |
(15) |
proper(tt) |
→ |
ok(tt) |
(16) |
proper(U12(X1,X2,X3)) |
→ |
U12(proper(X1),proper(X2),proper(X3)) |
(17) |
proper(s(X)) |
→ |
s(proper(X)) |
(18) |
proper(plus(X1,X2)) |
→ |
plus(proper(X1),proper(X2)) |
(19) |
proper(0) |
→ |
ok(0) |
(20) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(62) |
could be deleted.
1.1.1.1.1.1 Subterm Criterion Processor
We use the projection to multisets
π(top#)
|
= |
{
1
}
|
π(ok)
|
= |
{
1, 1, 1
}
|
π(s)
|
= |
{
1, 1
}
|
π(plus)
|
= |
{
1, 1
}
|
π(mark)
|
= |
{
1
}
|
π(U12)
|
= |
{
2, 2, 3, 3
}
|
π(active)
|
= |
{
1, 1
}
|
π(U11)
|
= |
{
2, 2
}
|
to remove the pairs:
top#(ok(X)) |
→ |
top#(active(X)) |
(64) |
1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
active#(U11(X1,X2,X3)) |
→ |
active#(X1) |
(29) |
active#(U12(X1,X2,X3)) |
→ |
active#(X1) |
(31) |
active#(s(X)) |
→ |
active#(X) |
(33) |
active#(plus(X1,X2)) |
→ |
active#(X1) |
(35) |
active#(plus(X1,X2)) |
→ |
active#(X2) |
(37) |
1.1.1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
active#(U11(X1,X2,X3)) |
→ |
active#(X1) |
(29) |
|
1 |
> |
1 |
active#(U12(X1,X2,X3)) |
→ |
active#(X1) |
(31) |
|
1 |
> |
1 |
active#(s(X)) |
→ |
active#(X) |
(33) |
|
1 |
> |
1 |
active#(plus(X1,X2)) |
→ |
active#(X1) |
(35) |
|
1 |
> |
1 |
active#(plus(X1,X2)) |
→ |
active#(X2) |
(37) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
proper#(U11(X1,X2,X3)) |
→ |
proper#(X3) |
(44) |
proper#(U11(X1,X2,X3)) |
→ |
proper#(X2) |
(45) |
proper#(U11(X1,X2,X3)) |
→ |
proper#(X1) |
(46) |
proper#(U12(X1,X2,X3)) |
→ |
proper#(X3) |
(48) |
proper#(U12(X1,X2,X3)) |
→ |
proper#(X2) |
(49) |
proper#(U12(X1,X2,X3)) |
→ |
proper#(X1) |
(50) |
proper#(s(X)) |
→ |
proper#(X) |
(52) |
proper#(plus(X1,X2)) |
→ |
proper#(X2) |
(54) |
proper#(plus(X1,X2)) |
→ |
proper#(X1) |
(55) |
1.1.1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
proper#(U11(X1,X2,X3)) |
→ |
proper#(X3) |
(44) |
|
1 |
> |
1 |
proper#(U11(X1,X2,X3)) |
→ |
proper#(X2) |
(45) |
|
1 |
> |
1 |
proper#(U11(X1,X2,X3)) |
→ |
proper#(X1) |
(46) |
|
1 |
> |
1 |
proper#(U12(X1,X2,X3)) |
→ |
proper#(X3) |
(48) |
|
1 |
> |
1 |
proper#(U12(X1,X2,X3)) |
→ |
proper#(X2) |
(49) |
|
1 |
> |
1 |
proper#(U12(X1,X2,X3)) |
→ |
proper#(X1) |
(50) |
|
1 |
> |
1 |
proper#(s(X)) |
→ |
proper#(X) |
(52) |
|
1 |
> |
1 |
proper#(plus(X1,X2)) |
→ |
proper#(X2) |
(54) |
|
1 |
> |
1 |
proper#(plus(X1,X2)) |
→ |
proper#(X1) |
(55) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(42) |
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(43) |
plus#(ok(X1),ok(X2)) |
→ |
plus#(X1,X2) |
(60) |
1.1.1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(42) |
|
2 |
≥ |
2 |
1 |
> |
1 |
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(43) |
|
2 |
> |
2 |
1 |
≥ |
1 |
plus#(ok(X1),ok(X2)) |
→ |
plus#(X1,X2) |
(60) |
|
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) |
(41) |
s#(ok(X)) |
→ |
s#(X) |
(59) |
1.1.1.1.5 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(mark(X)) |
→ |
s#(X) |
(41) |
|
1 |
> |
1 |
s#(ok(X)) |
→ |
s#(X) |
(59) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
U12#(mark(X1),X2,X3) |
→ |
U12#(X1,X2,X3) |
(40) |
U12#(ok(X1),ok(X2),ok(X3)) |
→ |
U12#(X1,X2,X3) |
(58) |
1.1.1.1.6 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U12#(mark(X1),X2,X3) |
→ |
U12#(X1,X2,X3) |
(40) |
|
3 |
≥ |
3 |
2 |
≥ |
2 |
1 |
> |
1 |
U12#(ok(X1),ok(X2),ok(X3)) |
→ |
U12#(X1,X2,X3) |
(58) |
|
3 |
> |
3 |
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
U11#(mark(X1),X2,X3) |
→ |
U11#(X1,X2,X3) |
(39) |
U11#(ok(X1),ok(X2),ok(X3)) |
→ |
U11#(X1,X2,X3) |
(57) |
1.1.1.1.7 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U11#(mark(X1),X2,X3) |
→ |
U11#(X1,X2,X3) |
(39) |
|
3 |
≥ |
3 |
2 |
≥ |
2 |
1 |
> |
1 |
U11#(ok(X1),ok(X2),ok(X3)) |
→ |
U11#(X1,X2,X3) |
(57) |
|
3 |
> |
3 |
2 |
> |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.