The rewrite relation of the following TRS is considered.
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
activate#(n__terms(X)) |
→ |
activate#(X) |
(29) |
activate#(n__s(X)) |
→ |
activate#(X) |
(31) |
activate#(n__first(X1,X2)) |
→ |
first#(activate(X1),activate(X2)) |
(32) |
first#(s(X),cons(Y,Z)) |
→ |
activate#(Z) |
(27) |
activate#(n__first(X1,X2)) |
→ |
activate#(X1) |
(33) |
activate#(n__first(X1,X2)) |
→ |
activate#(X2) |
(34) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Knuth Bendix order with w0 = 2 and the following precedence and weight functions
prec(n__first) |
= |
0 |
|
weight(n__first) |
= |
2 |
|
|
|
prec(activate) |
= |
4 |
|
weight(activate) |
= |
0 |
|
|
|
prec(first) |
= |
2 |
|
weight(first) |
= |
2 |
|
|
|
prec(0) |
= |
1 |
|
weight(0) |
= |
2 |
|
|
|
prec(nil) |
= |
3 |
|
weight(nil) |
= |
4 |
|
|
|
in combination with the following argument filter
π(activate#) |
= |
1 |
π(n__terms) |
= |
1 |
π(n__s) |
= |
1 |
π(n__first) |
= |
[1,2] |
π(first#) |
= |
2 |
π(activate) |
= |
[1] |
π(cons) |
= |
2 |
π(terms) |
= |
1 |
π(s) |
= |
1 |
π(first) |
= |
[1,2] |
π(0) |
= |
[] |
π(nil) |
= |
[] |
together with the usable
rules
activate(n__terms(X)) |
→ |
terms(activate(X)) |
(13) |
activate(n__s(X)) |
→ |
s(activate(X)) |
(14) |
activate(n__first(X1,X2)) |
→ |
first(activate(X1),activate(X2)) |
(15) |
activate(X) |
→ |
X |
(16) |
first(s(X),cons(Y,Z)) |
→ |
cons(Y,n__first(X,activate(Z))) |
(9) |
s(X) |
→ |
n__s(X) |
(11) |
terms(N) |
→ |
cons(recip(sqr(N)),n__terms(n__s(N))) |
(1) |
terms(X) |
→ |
n__terms(X) |
(10) |
first(0,X) |
→ |
nil |
(8) |
first(X1,X2) |
→ |
n__first(X1,X2) |
(12) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
activate#(n__first(X1,X2)) |
→ |
first#(activate(X1),activate(X2)) |
(32) |
activate#(n__first(X1,X2)) |
→ |
activate#(X1) |
(33) |
activate#(n__first(X1,X2)) |
→ |
activate#(X2) |
(34) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
activate#(n__s(X)) |
→ |
activate#(X) |
(31) |
activate#(n__terms(X)) |
→ |
activate#(X) |
(29) |
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[n__s(x1)] |
= |
1 · x1
|
[n__terms(x1)] |
= |
1 · x1
|
[activate#(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.
activate#(n__s(X)) |
→ |
activate#(X) |
(31) |
|
1 |
> |
1 |
activate#(n__terms(X)) |
→ |
activate#(X) |
(29) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
sqr#(s(X)) |
→ |
sqr#(X) |
(20) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[s(x1)] |
= |
1 · x1
|
[sqr#(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.
sqr#(s(X)) |
→ |
sqr#(X) |
(20) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
dbl#(s(X)) |
→ |
dbl#(X) |
(24) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[s(x1)] |
= |
1 · x1
|
[dbl#(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.
dbl#(s(X)) |
→ |
dbl#(X) |
(24) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
add#(s(X),Y) |
→ |
add#(X,Y) |
(26) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[s(x1)] |
= |
1 · x1
|
[add#(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.
1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
add#(s(X),Y) |
→ |
add#(X,Y) |
(26) |
|
1 |
> |
1 |
2 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.