The rewrite relation of the following TRS is considered.
The dependency pairs are split into 6
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(35) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(33) |
1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[proper(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[a] |
= |
0 |
[ok(x1)] |
= |
1 · x1
|
[c(x1)] |
= |
1 · x1
|
[g(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[top#(x1)] |
= |
1 · x1
|
together with the usable
rules
proper(f(X)) |
→ |
f(proper(X)) |
(6) |
proper(a) |
→ |
ok(a) |
(7) |
proper(c(X)) |
→ |
c(proper(X)) |
(8) |
proper(g(X)) |
→ |
g(proper(X)) |
(9) |
g(mark(X)) |
→ |
mark(g(X)) |
(5) |
g(ok(X)) |
→ |
ok(g(X)) |
(12) |
c(ok(X)) |
→ |
ok(c(X)) |
(11) |
f(mark(X)) |
→ |
mark(f(X)) |
(4) |
f(ok(X)) |
→ |
ok(f(X)) |
(10) |
active(f(f(a))) |
→ |
mark(c(f(g(f(a))))) |
(1) |
active(f(X)) |
→ |
f(active(X)) |
(2) |
active(g(X)) |
→ |
g(active(X)) |
(3) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[top#(x1)] |
= |
1 · x1
|
[ok(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 + 1 · x1
|
[proper(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[a] |
= |
1 |
[c(x1)] |
= |
0 |
[g(x1)] |
= |
1 · x1
|
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(33) |
could be deleted.
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[a] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[c(x1)] |
= |
1 · x1
|
[g(x1)] |
= |
1 · x1
|
[ok(x1)] |
= |
1 · x1
|
[top#(x1)] |
= |
1 · x1
|
together with the usable
rules
active(f(f(a))) |
→ |
mark(c(f(g(f(a))))) |
(1) |
active(f(X)) |
→ |
f(active(X)) |
(2) |
active(g(X)) |
→ |
g(active(X)) |
(3) |
g(mark(X)) |
→ |
mark(g(X)) |
(5) |
g(ok(X)) |
→ |
ok(g(X)) |
(12) |
f(mark(X)) |
→ |
mark(f(X)) |
(4) |
f(ok(X)) |
→ |
ok(f(X)) |
(10) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[top#(x1)] |
= |
1 · x1
|
[ok(x1)] |
= |
1 + 1 · x1
|
[active(x1)] |
= |
0 |
[f(x1)] |
= |
1 · x1
|
[a] |
= |
0 |
[mark(x1)] |
= |
0 |
[c(x1)] |
= |
1 |
[g(x1)] |
= |
1 · x1
|
the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(35) |
could be deleted.
1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
active#(g(X)) |
→ |
active#(X) |
(21) |
active#(f(X)) |
→ |
active#(X) |
(19) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[g(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[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#(g(X)) |
→ |
active#(X) |
(21) |
|
1 |
> |
1 |
active#(f(X)) |
→ |
active#(X) |
(19) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
proper#(c(X)) |
→ |
proper#(X) |
(27) |
proper#(f(X)) |
→ |
proper#(X) |
(25) |
proper#(g(X)) |
→ |
proper#(X) |
(29) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[c(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[g(x1)] |
= |
1 · x1
|
[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#(c(X)) |
→ |
proper#(X) |
(27) |
|
1 |
> |
1 |
proper#(f(X)) |
→ |
proper#(X) |
(25) |
|
1 |
> |
1 |
proper#(g(X)) |
→ |
proper#(X) |
(29) |
|
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(X)) |
→ |
f#(X) |
(30) |
f#(mark(X)) |
→ |
f#(X) |
(22) |
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)] |
= |
1 · x1
|
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(X)) |
→ |
f#(X) |
(30) |
|
1 |
> |
1 |
f#(mark(X)) |
→ |
f#(X) |
(22) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
g#(ok(X)) |
→ |
g#(X) |
(32) |
g#(mark(X)) |
→ |
g#(X) |
(23) |
1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[g#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
g#(ok(X)) |
→ |
g#(X) |
(32) |
|
1 |
> |
1 |
g#(mark(X)) |
→ |
g#(X) |
(23) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[c#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
c#(ok(X)) |
→ |
c#(X) |
(31) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.