The rewrite relation of the following TRS is considered.
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
d'#(h(active(x))) |
→ |
c'#(g(mark(x))) |
(31) |
c'#(active(x)) |
→ |
d'#(mark(x)) |
(29) |
d'#(proper(x)) |
→ |
d'#(ok(x)) |
(38) |
c'#(proper(x)) |
→ |
c'#(ok(x)) |
(36) |
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[g(x1)] |
= |
1 · x1
|
[h(x1)] |
= |
1 · x1
|
[top(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[proper(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[c'#(x1)] |
= |
1 · x1
|
[d'#(x1)] |
= |
1 · x1
|
together with the usable
rules
ok(g(X)) |
→ |
g(ok(X)) |
(23) |
ok(h(X)) |
→ |
h(ok(X)) |
(24) |
ok(top(X)) |
→ |
active(top(X)) |
(26) |
h(proper(X)) |
→ |
proper(h(X)) |
(20) |
g(active(X)) |
→ |
h(mark(X)) |
(16) |
g(proper(X)) |
→ |
proper(g(X)) |
(19) |
mark(top(X)) |
→ |
proper(top(X)) |
(25) |
(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
[d'#(x1)] |
= |
0 |
[h(x1)] |
= |
0 |
[active(x1)] |
= |
1 |
[c'#(x1)] |
= |
1 · x1
|
[g(x1)] |
= |
0 |
[mark(x1)] |
= |
0 |
[proper(x1)] |
= |
1 · x1
|
[ok(x1)] |
= |
1 · x1
|
[top(x1)] |
= |
1 + 1 · x1
|
together with the usable
rules
g(active(X)) |
→ |
h(mark(X)) |
(16) |
g(proper(X)) |
→ |
proper(g(X)) |
(19) |
ok(g(X)) |
→ |
g(ok(X)) |
(23) |
ok(h(X)) |
→ |
h(ok(X)) |
(24) |
ok(top(X)) |
→ |
active(top(X)) |
(26) |
h(proper(X)) |
→ |
proper(h(X)) |
(20) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
c'#(active(x)) |
→ |
d'#(mark(x)) |
(29) |
could be deleted.
1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
2nd
component contains the
pair
ok#(h(X)) |
→ |
ok#(X) |
(43) |
ok#(g(X)) |
→ |
ok#(X) |
(41) |
1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[h(x1)] |
= |
1 · x1
|
[g(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#(h(X)) |
→ |
ok#(X) |
(43) |
|
1 |
> |
1 |
ok#(g(X)) |
→ |
ok#(X) |
(41) |
|
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) |
(34) |
1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[proper(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.
1.1.1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
g#(proper(X)) |
→ |
g#(X) |
(34) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
h#(proper(X)) |
→ |
h#(X) |
(35) |
1.1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[proper(x1)] |
= |
1 · x1
|
[h#(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.
h#(proper(X)) |
→ |
h#(X) |
(35) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.