The rewrite relation of the following TRS is considered.
|
le(0,Y) |
→ |
true |
(1) |
|
le(s(X),0) |
→ |
false |
(2) |
|
le(s(X),s(Y)) |
→ |
le(X,Y) |
(3) |
|
app(nil,Y) |
→ |
Y |
(4) |
|
app(cons(N,L),Y) |
→ |
cons(N,app(L,Y)) |
(5) |
|
low(N,nil) |
→ |
nil |
(6) |
|
low(N,cons(M,L)) |
→ |
iflow(le(M,N),N,cons(M,L)) |
(7) |
|
iflow(true,N,cons(M,L)) |
→ |
cons(M,low(N,L)) |
(8) |
|
iflow(false,N,cons(M,L)) |
→ |
low(N,L) |
(9) |
|
high(N,nil) |
→ |
nil |
(10) |
|
high(N,cons(M,L)) |
→ |
ifhigh(le(M,N),N,cons(M,L)) |
(11) |
|
ifhigh(true,N,cons(M,L)) |
→ |
high(N,L) |
(12) |
|
ifhigh(false,N,cons(M,L)) |
→ |
cons(M,high(N,L)) |
(13) |
|
quicksort(nil) |
→ |
nil |
(14) |
|
quicksort(cons(N,L)) |
→ |
app(quicksort(low(N,L)),cons(N,quicksort(high(N,L)))) |
(15) |
|
le#(s(X),s(Y)) |
→ |
le#(X,Y) |
(16) |
|
app#(cons(N,L),Y) |
→ |
app#(L,Y) |
(17) |
|
low#(N,cons(M,L)) |
→ |
le#(M,N) |
(18) |
|
low#(N,cons(M,L)) |
→ |
iflow#(le(M,N),N,cons(M,L)) |
(19) |
|
iflow#(true,N,cons(M,L)) |
→ |
low#(N,L) |
(20) |
|
iflow#(false,N,cons(M,L)) |
→ |
low#(N,L) |
(21) |
|
high#(N,cons(M,L)) |
→ |
le#(M,N) |
(22) |
|
high#(N,cons(M,L)) |
→ |
ifhigh#(le(M,N),N,cons(M,L)) |
(23) |
|
ifhigh#(true,N,cons(M,L)) |
→ |
high#(N,L) |
(24) |
|
ifhigh#(false,N,cons(M,L)) |
→ |
high#(N,L) |
(25) |
|
quicksort#(cons(N,L)) |
→ |
high#(N,L) |
(26) |
|
quicksort#(cons(N,L)) |
→ |
quicksort#(high(N,L)) |
(27) |
|
quicksort#(cons(N,L)) |
→ |
low#(N,L) |
(28) |
|
quicksort#(cons(N,L)) |
→ |
quicksort#(low(N,L)) |
(29) |
|
quicksort#(cons(N,L)) |
→ |
app#(quicksort(low(N,L)),cons(N,quicksort(high(N,L)))) |
(30) |
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
|
quicksort#(cons(N,L)) |
→ |
quicksort#(high(N,L)) |
(27) |
|
quicksort#(cons(N,L)) |
→ |
quicksort#(low(N,L)) |
(29) |
1.1.1 Subterm Criterion Processor
We use the projection to multisets
| π(quicksort#)
|
= |
{
1
}
|
| π(ifhigh)
|
= |
{
3
}
|
| π(high)
|
= |
{
2
}
|
| π(iflow)
|
= |
{
3
}
|
| π(low)
|
= |
{
2
}
|
| π(cons)
|
= |
{
1, 1, 1, 2
}
|
to remove the pairs:
|
quicksort#(cons(N,L)) |
→ |
quicksort#(high(N,L)) |
(27) |
|
quicksort#(cons(N,L)) |
→ |
quicksort#(low(N,L)) |
(29) |
1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
app#(cons(N,L),Y) |
→ |
app#(L,Y) |
(17) |
1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
app#(cons(N,L),Y) |
→ |
app#(L,Y) |
(17) |
|
|
| 2 |
≥ |
2 |
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
|
low#(N,cons(M,L)) |
→ |
iflow#(le(M,N),N,cons(M,L)) |
(19) |
|
iflow#(true,N,cons(M,L)) |
→ |
low#(N,L) |
(20) |
|
iflow#(false,N,cons(M,L)) |
→ |
low#(N,L) |
(21) |
1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
low#(N,cons(M,L)) |
→ |
iflow#(le(M,N),N,cons(M,L)) |
(19) |
|
|
| 2 |
≥ |
3 |
| 1 |
≥ |
2 |
|
iflow#(true,N,cons(M,L)) |
→ |
low#(N,L) |
(20) |
|
|
| 3 |
> |
2 |
| 2 |
≥ |
1 |
|
iflow#(false,N,cons(M,L)) |
→ |
low#(N,L) |
(21) |
|
|
| 3 |
> |
2 |
| 2 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
|
high#(N,cons(M,L)) |
→ |
ifhigh#(le(M,N),N,cons(M,L)) |
(23) |
|
ifhigh#(true,N,cons(M,L)) |
→ |
high#(N,L) |
(24) |
|
ifhigh#(false,N,cons(M,L)) |
→ |
high#(N,L) |
(25) |
1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
high#(N,cons(M,L)) |
→ |
ifhigh#(le(M,N),N,cons(M,L)) |
(23) |
|
|
| 2 |
≥ |
3 |
| 1 |
≥ |
2 |
|
ifhigh#(true,N,cons(M,L)) |
→ |
high#(N,L) |
(24) |
|
|
| 3 |
> |
2 |
| 2 |
≥ |
1 |
|
ifhigh#(false,N,cons(M,L)) |
→ |
high#(N,L) |
(25) |
|
|
| 3 |
> |
2 |
| 2 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
|
le#(s(X),s(Y)) |
→ |
le#(X,Y) |
(16) |
1.1.5 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
le#(s(X),s(Y)) |
→ |
le#(X,Y) |
(16) |
|
|
| 2 |
> |
2 |
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.