The rewrite relation of the following TRS is considered.
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(23) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(21) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
[mark(x1)] |
= |
· x1 +
|
[c] |
= |
|
[active(x1)] |
= |
· x1 +
|
[proper(x1)] |
= |
· x1 +
|
[b] |
= |
|
[top#(x1)] |
= |
· x1 +
|
[ok(x1)] |
= |
· x1 +
|
[f(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
together with the usable
rules
active(f(b,X,c)) |
→ |
mark(f(X,c,X)) |
(1) |
active(c) |
→ |
mark(b) |
(2) |
active(f(X1,X2,X3)) |
→ |
f(X1,active(X2),X3) |
(3) |
f(X1,mark(X2),X3) |
→ |
mark(f(X1,X2,X3)) |
(4) |
f(ok(X1),ok(X2),ok(X3)) |
→ |
ok(f(X1,X2,X3)) |
(8) |
proper(f(X1,X2,X3)) |
→ |
f(proper(X1),proper(X2),proper(X3)) |
(5) |
proper(b) |
→ |
ok(b) |
(6) |
proper(c) |
→ |
ok(c) |
(7) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(21) |
could be deleted.
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(mark) |
= |
0 |
|
stat(mark) |
= |
lex
|
prec(active) |
= |
0 |
|
stat(active) |
= |
lex
|
prec(f) |
= |
1 |
|
stat(f) |
= |
lex
|
prec(c) |
= |
1 |
|
stat(c) |
= |
lex
|
prec(b) |
= |
0 |
|
stat(b) |
= |
lex
|
π(top#) |
= |
1 |
π(ok) |
= |
[1] |
π(mark) |
= |
[] |
π(active) |
= |
1 |
π(f) |
= |
[2] |
π(c) |
= |
[] |
π(b) |
= |
[] |
together with the usable
rules
active(f(b,X,c)) |
→ |
mark(f(X,c,X)) |
(1) |
active(c) |
→ |
mark(b) |
(2) |
active(f(X1,X2,X3)) |
→ |
f(X1,active(X2),X3) |
(3) |
f(X1,mark(X2),X3) |
→ |
mark(f(X1,X2,X3)) |
(4) |
f(ok(X1),ok(X2),ok(X3)) |
→ |
ok(f(X1,X2,X3)) |
(8) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(23) |
could be deleted.
1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
proper#(f(X1,X2,X3)) |
→ |
proper#(X1) |
(17) |
proper#(f(X1,X2,X3)) |
→ |
proper#(X2) |
(16) |
proper#(f(X1,X2,X3)) |
→ |
proper#(X3) |
(15) |
1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
proper#(f(X1,X2,X3)) |
→ |
proper#(X1) |
(17) |
|
1 |
> |
1 |
proper#(f(X1,X2,X3)) |
→ |
proper#(X2) |
(16) |
|
1 |
> |
1 |
proper#(f(X1,X2,X3)) |
→ |
proper#(X3) |
(15) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
active#(f(X1,X2,X3)) |
→ |
active#(X2) |
(12) |
1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
active#(f(X1,X2,X3)) |
→ |
active#(X2) |
(12) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
f#(ok(X1),ok(X2),ok(X3)) |
→ |
f#(X1,X2,X3) |
(19) |
f#(X1,mark(X2),X3) |
→ |
f#(X1,X2,X3) |
(14) |
1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
f#(ok(X1),ok(X2),ok(X3)) |
→ |
f#(X1,X2,X3) |
(19) |
|
3 |
> |
3 |
2 |
> |
2 |
1 |
> |
1 |
f#(X1,mark(X2),X3) |
→ |
f#(X1,X2,X3) |
(14) |
|
3 |
≥ |
3 |
2 |
> |
2 |
1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.