The rewrite relation of the following TRS is considered.
|
eq(0,0) |
→ |
true |
(1) |
|
eq(0,s(x)) |
→ |
false |
(2) |
|
eq(s(x),0) |
→ |
false |
(3) |
|
eq(s(x),s(y)) |
→ |
eq(x,y) |
(4) |
|
le(0,y) |
→ |
true |
(5) |
|
le(s(x),0) |
→ |
false |
(6) |
|
le(s(x),s(y)) |
→ |
le(x,y) |
(7) |
|
app(nil,y) |
→ |
y |
(8) |
|
app(add(n,x),y) |
→ |
add(n,app(x,y)) |
(9) |
|
min(add(n,nil)) |
→ |
n |
(10) |
|
min(add(n,add(m,x))) |
→ |
if_min(le(n,m),add(n,add(m,x))) |
(11) |
|
if_min(true,add(n,add(m,x))) |
→ |
min(add(n,x)) |
(12) |
|
if_min(false,add(n,add(m,x))) |
→ |
min(add(m,x)) |
(13) |
|
rm(n,nil) |
→ |
nil |
(14) |
|
rm(n,add(m,x)) |
→ |
if_rm(eq(n,m),n,add(m,x)) |
(15) |
|
if_rm(true,n,add(m,x)) |
→ |
rm(n,x) |
(16) |
|
if_rm(false,n,add(m,x)) |
→ |
add(m,rm(n,x)) |
(17) |
|
minsort(nil,nil) |
→ |
nil |
(18) |
|
minsort(add(n,x),y) |
→ |
if_minsort(eq(n,min(add(n,x))),add(n,x),y) |
(19) |
|
if_minsort(true,add(n,x),y) |
→ |
add(n,minsort(app(rm(n,x),y),nil)) |
(20) |
|
if_minsort(false,add(n,x),y) |
→ |
minsort(x,add(n,y)) |
(21) |
|
eq#(s(x),s(y)) |
→ |
eq#(x,y) |
(22) |
|
le#(s(x),s(y)) |
→ |
le#(x,y) |
(23) |
|
app#(add(n,x),y) |
→ |
app#(x,y) |
(24) |
|
min#(add(n,add(m,x))) |
→ |
le#(n,m) |
(25) |
|
min#(add(n,add(m,x))) |
→ |
if_min#(le(n,m),add(n,add(m,x))) |
(26) |
|
if_min#(true,add(n,add(m,x))) |
→ |
min#(add(n,x)) |
(27) |
|
if_min#(false,add(n,add(m,x))) |
→ |
min#(add(m,x)) |
(28) |
|
rm#(n,add(m,x)) |
→ |
eq#(n,m) |
(29) |
|
rm#(n,add(m,x)) |
→ |
if_rm#(eq(n,m),n,add(m,x)) |
(30) |
|
if_rm#(true,n,add(m,x)) |
→ |
rm#(n,x) |
(31) |
|
if_rm#(false,n,add(m,x)) |
→ |
rm#(n,x) |
(32) |
|
minsort#(add(n,x),y) |
→ |
min#(add(n,x)) |
(33) |
|
minsort#(add(n,x),y) |
→ |
eq#(n,min(add(n,x))) |
(34) |
|
minsort#(add(n,x),y) |
→ |
if_minsort#(eq(n,min(add(n,x))),add(n,x),y) |
(35) |
|
if_minsort#(true,add(n,x),y) |
→ |
rm#(n,x) |
(36) |
|
if_minsort#(true,add(n,x),y) |
→ |
app#(rm(n,x),y) |
(37) |
|
if_minsort#(true,add(n,x),y) |
→ |
minsort#(app(rm(n,x),y),nil) |
(38) |
|
if_minsort#(false,add(n,x),y) |
→ |
minsort#(x,add(n,y)) |
(39) |
The dependency pairs are split into 6
components.
-
The
1st
component contains the
pair
|
if_minsort#(false,add(n,x),y) |
→ |
minsort#(x,add(n,y)) |
(39) |
|
minsort#(add(n,x),y) |
→ |
if_minsort#(eq(n,min(add(n,x))),add(n,x),y) |
(35) |
|
if_minsort#(true,add(n,x),y) |
→ |
minsort#(app(rm(n,x),y),nil) |
(38) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [false] |
= |
0 |
| [add(x1, x2)] |
= |
3 · x1 + 1 · x2 + 1 |
| [eq(x1, x2)] |
= |
1 · x1 + 2 · x2 + 0 |
| [s(x1)] |
= |
0 · x1 + 0 |
| [minsort#(x1, x2)] |
= |
4 · x1 + 4 · x2 + 0 |
| [le(x1, x2)] |
= |
2 · x1 + 0 · x2 + 1 |
| [if_min(x1, x2)] |
= |
7 · x1 + 0 · x2 + 2 |
| [0] |
= |
0 |
| [app(x1, x2)] |
= |
1 · x1 + 1 · x2 + 0 |
| [if_rm(x1, x2, x3)] |
= |
0 · x1 + 0 · x2 + 1 · x3 + 0 |
| [if_minsort#(x1, x2, x3)] |
= |
0 · x1 + 4 · x2 + 4 · x3 + 0 |
| [rm(x1, x2)] |
= |
0 · x1 + 1 · x2 + 0 |
| [min(x1)] |
= |
5 · x1 + 3 |
| [nil] |
= |
0 |
| [true] |
= |
0 |
together with the usable
rules
|
rm(n,nil) |
→ |
nil |
(14) |
|
rm(n,add(m,x)) |
→ |
if_rm(eq(n,m),n,add(m,x)) |
(15) |
|
if_rm(true,n,add(m,x)) |
→ |
rm(n,x) |
(16) |
|
if_rm(false,n,add(m,x)) |
→ |
add(m,rm(n,x)) |
(17) |
|
app(nil,y) |
→ |
y |
(8) |
|
app(add(n,x),y) |
→ |
add(n,app(x,y)) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
if_minsort#(true,add(n,x),y) |
→ |
minsort#(app(rm(n,x),y),nil) |
(38) |
could be deleted.
1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
if_minsort#(false,add(n,x),y) |
→ |
minsort#(x,add(n,y)) |
(39) |
|
| 2 |
> |
1 |
|
minsort#(add(n,x),y) |
→ |
if_minsort#(eq(n,min(add(n,x))),add(n,x),y) |
(35) |
|
|
| 2 |
≥ |
3 |
| 1 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
|
app#(add(n,x),y) |
→ |
app#(x,y) |
(24) |
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#(add(n,x),y) |
→ |
app#(x,y) |
(24) |
|
|
| 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
|
rm#(n,add(m,x)) |
→ |
if_rm#(eq(n,m),n,add(m,x)) |
(30) |
|
if_rm#(true,n,add(m,x)) |
→ |
rm#(n,x) |
(31) |
|
if_rm#(false,n,add(m,x)) |
→ |
rm#(n,x) |
(32) |
1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
rm#(n,add(m,x)) |
→ |
if_rm#(eq(n,m),n,add(m,x)) |
(30) |
|
|
| 2 |
≥ |
3 |
| 1 |
≥ |
2 |
|
if_rm#(true,n,add(m,x)) |
→ |
rm#(n,x) |
(31) |
|
|
| 3 |
> |
2 |
| 2 |
≥ |
1 |
|
if_rm#(false,n,add(m,x)) |
→ |
rm#(n,x) |
(32) |
|
|
| 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
|
eq#(s(x),s(y)) |
→ |
eq#(x,y) |
(22) |
1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
eq#(s(x),s(y)) |
→ |
eq#(x,y) |
(22) |
|
|
| 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
|
min#(add(n,add(m,x))) |
→ |
if_min#(le(n,m),add(n,add(m,x))) |
(26) |
|
if_min#(true,add(n,add(m,x))) |
→ |
min#(add(n,x)) |
(27) |
|
if_min#(false,add(n,add(m,x))) |
→ |
min#(add(m,x)) |
(28) |
1.1.5 Subterm Criterion Processor
We use the projection to multisets
| π(if_min#)
|
= |
{
2
}
|
| π(min#)
|
= |
{
1
}
|
| π(add)
|
= |
{
1, 1, 1, 2
}
|
to remove the pairs:
|
if_min#(true,add(n,add(m,x))) |
→ |
min#(add(n,x)) |
(27) |
|
if_min#(false,add(n,add(m,x))) |
→ |
min#(add(m,x)) |
(28) |
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
6th
component contains the
pair
|
le#(s(x),s(y)) |
→ |
le#(x,y) |
(23) |
1.1.6 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) |
(23) |
|
|
| 2 |
> |
2 |
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.