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)) |
(22) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(20) |
1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[proper(x1)] |
= |
1 · x1
|
[f(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
[a] |
= |
0 |
[ok(x1)] |
= |
1 · x1
|
[b] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[top#(x1)] |
= |
1 · x1
|
together with the usable
rules
proper(f(X1,X2,X3)) |
→ |
f(proper(X1),proper(X2),proper(X3)) |
(5) |
proper(a) |
→ |
ok(a) |
(6) |
proper(b) |
→ |
ok(b) |
(7) |
f(X1,mark(X2),X3) |
→ |
mark(f(X1,X2,X3)) |
(4) |
f(ok(X1),ok(X2),ok(X3)) |
→ |
ok(f(X1,X2,X3)) |
(8) |
active(f(a,X,X)) |
→ |
mark(f(X,b,b)) |
(1) |
active(b) |
→ |
mark(a) |
(2) |
active(f(X1,X2,X3)) |
→ |
f(X1,active(X2),X3) |
(3) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1 Narrowing Processor
We consider all narrowings of the pair
below position
1
to get the following set of pairs
top#(ok(f(a,x0,x0))) |
→ |
top#(mark(f(x0,b,b))) |
(24) |
top#(ok(b)) |
→ |
top#(mark(a)) |
(25) |
top#(ok(f(x0,x1,x2))) |
→ |
top#(f(x0,active(x1),x2)) |
(26) |
1.1.1.1.1 Narrowing Processor
We consider all narrowings of the pair
below position
1
to get the following set of pairs
top#(mark(f(x0,x1,x2))) |
→ |
top#(f(proper(x0),proper(x1),proper(x2))) |
(27) |
top#(mark(a)) |
→ |
top#(ok(a)) |
(28) |
top#(mark(b)) |
→ |
top#(ok(b)) |
(29) |
1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
active#(f(X1,X2,X3)) |
→ |
active#(X2) |
(13) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[f(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
[active#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.2.1 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) |
(13) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
proper#(f(X1,X2,X3)) |
→ |
proper#(X2) |
(17) |
proper#(f(X1,X2,X3)) |
→ |
proper#(X1) |
(16) |
proper#(f(X1,X2,X3)) |
→ |
proper#(X3) |
(18) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[f(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
[proper#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.3.1 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#(X2) |
(17) |
|
1 |
> |
1 |
proper#(f(X1,X2,X3)) |
→ |
proper#(X1) |
(16) |
|
1 |
> |
1 |
proper#(f(X1,X2,X3)) |
→ |
proper#(X3) |
(18) |
|
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 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[f#(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.4.1 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) |
|
1 |
> |
1 |
2 |
> |
2 |
3 |
> |
3 |
f#(X1,mark(X2),X3) |
→ |
f#(X1,X2,X3) |
(14) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
As there is no critical graph in the transitive closure, there are no infinite chains.