The rewrite relation of the following TRS is considered.
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
mark#(f(X1,X2)) |
→ |
active#(f(mark(X1),X2)) |
(12) |
active#(f(g(X),Y)) |
→ |
mark#(f(X,f(g(X),Y))) |
(10) |
mark#(f(X1,X2)) |
→ |
mark#(X1) |
(14) |
mark#(g(X)) |
→ |
active#(g(mark(X))) |
(15) |
mark#(g(X)) |
→ |
mark#(X) |
(17) |
1.1.1 Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(g) |
= |
1 |
|
weight(g) |
= |
1 |
|
|
|
in combination with the following argument filter
π(mark#) |
= |
1 |
π(f) |
= |
1 |
π(active#) |
= |
1 |
π(mark) |
= |
1 |
π(g) |
= |
[1] |
π(active) |
= |
1 |
the
pairs
active#(f(g(X),Y)) |
→ |
mark#(f(X,f(g(X),Y))) |
(10) |
mark#(g(X)) |
→ |
mark#(X) |
(17) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
mark#(f(X1,X2)) |
→ |
mark#(X1) |
(14) |
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[f(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[mark#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
mark#(f(X1,X2)) |
→ |
mark#(X1) |
(14) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
f#(X1,mark(X2)) |
→ |
f#(X1,X2) |
(19) |
f#(mark(X1),X2) |
→ |
f#(X1,X2) |
(18) |
f#(active(X1),X2) |
→ |
f#(X1,X2) |
(20) |
f#(X1,active(X2)) |
→ |
f#(X1,X2) |
(21) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[f#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
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.
f#(X1,mark(X2)) |
→ |
f#(X1,X2) |
(19) |
|
1 |
≥ |
1 |
2 |
> |
2 |
f#(mark(X1),X2) |
→ |
f#(X1,X2) |
(18) |
|
1 |
> |
1 |
2 |
≥ |
2 |
f#(active(X1),X2) |
→ |
f#(X1,X2) |
(20) |
|
1 |
> |
1 |
2 |
≥ |
2 |
f#(X1,active(X2)) |
→ |
f#(X1,X2) |
(21) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
g#(active(X)) |
→ |
g#(X) |
(23) |
g#(mark(X)) |
→ |
g#(X) |
(22) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(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.
1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
g#(active(X)) |
→ |
g#(X) |
(23) |
|
1 |
> |
1 |
g#(mark(X)) |
→ |
g#(X) |
(22) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.