The rewrite relation of the following TRS is considered.
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
c'#(proper(x)) |
→ |
c'#(ok(x)) |
(27) |
c'#(active(x)) |
→ |
c'#(g(f(mark(x)))) |
(21) |
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[top(x1)] |
= |
1 · x1
|
[proper(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[g(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[ok(x1)] |
= |
1 · x1
|
[c'#(x1)] |
= |
1 · x1
|
together with the usable
rules
mark(top(X)) |
→ |
proper(top(X)) |
(19) |
f(proper(X)) |
→ |
proper(f(X)) |
(15) |
g(f(active(X))) |
→ |
g(mark(X)) |
(13) |
g(proper(X)) |
→ |
proper(g(X)) |
(16) |
ok(f(X)) |
→ |
f(ok(X)) |
(17) |
ok(g(X)) |
→ |
g(ok(X)) |
(18) |
ok(top(X)) |
→ |
active(top(X)) |
(20) |
(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 with Usable Rules
Using the linear polynomial interpretation over the naturals
[c'#(x1)] |
= |
1 · x1
|
[proper(x1)] |
= |
1 · x1
|
[ok(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 |
[g(x1)] |
= |
0 |
[f(x1)] |
= |
0 |
[mark(x1)] |
= |
0 |
[top(x1)] |
= |
1 + 1 · x1
|
together with the usable
rules
ok(f(X)) |
→ |
f(ok(X)) |
(17) |
ok(g(X)) |
→ |
g(ok(X)) |
(18) |
ok(top(X)) |
→ |
active(top(X)) |
(20) |
f(proper(X)) |
→ |
proper(f(X)) |
(15) |
g(f(active(X))) |
→ |
g(mark(X)) |
(13) |
g(proper(X)) |
→ |
proper(g(X)) |
(16) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
c'#(active(x)) |
→ |
c'#(g(f(mark(x)))) |
(21) |
could be deleted.
1.1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[c'#(x1)] |
= |
1 · x1
|
[proper(x1)] |
= |
1 + 1 · x1
|
[ok(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 + 1 · x1
|
[g(x1)] |
= |
1 + 1 · x1
|
[top(x1)] |
= |
1 + 1 · x1
|
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 + 1 · x1
|
the
pair
c'#(proper(x)) |
→ |
c'#(ok(x)) |
(27) |
could be deleted.
1.1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
ok#(g(X)) |
→ |
ok#(X) |
(34) |
ok#(f(X)) |
→ |
ok#(X) |
(32) |
1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[g(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[ok#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
ok#(g(X)) |
→ |
ok#(X) |
(34) |
|
1 |
> |
1 |
ok#(f(X)) |
→ |
ok#(X) |
(32) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
g#(proper(X)) |
→ |
g#(X) |
(30) |
g#(f(active(X))) |
→ |
g#(mark(X)) |
(25) |
1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[top(x1)] |
= |
1 · x1
|
[proper(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[g#(x1)] |
= |
1 · x1
|
together with the usable
rule
mark(top(X)) |
→ |
proper(top(X)) |
(19) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.3.1 Switch to Innermost Termination
The TRS does not have overlaps with the pairs and is locally confluent:
20
Hence, it suffices to show innermost termination in the following.
1.1.1.1.3.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
2 · x1
|
[top(x1)] |
= |
2 + 2 · x1
|
[proper(x1)] |
= |
1 + 1 · x1
|
[g#(x1)] |
= |
3 · x1
|
[f(x1)] |
= |
3 + 1 · x1
|
[active(x1)] |
= |
2 + 2 · x1
|
the
pairs
g#(proper(X)) |
→ |
g#(X) |
(30) |
g#(f(active(X))) |
→ |
g#(mark(X)) |
(25) |
and
the
rule
mark(top(X)) |
→ |
proper(top(X)) |
(19) |
could be deleted.
1.1.1.1.3.1.1.1 P is empty
There are no pairs anymore.
-
The
4th
component contains the
pair
f#(proper(X)) |
→ |
f#(X) |
(29) |
1.1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[proper(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.
1.1.1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
f#(proper(X)) |
→ |
f#(X) |
(29) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.