The rewrite relation of the following TRS is considered.
active#(eq(0,0)) |
→ |
mark#(true) |
(37) |
active#(eq(s(X),s(Y))) |
→ |
mark#(eq(X,Y)) |
(38) |
active#(eq(s(X),s(Y))) |
→ |
eq#(X,Y) |
(39) |
active#(eq(X,Y)) |
→ |
mark#(false) |
(40) |
active#(inf(X)) |
→ |
mark#(cons(X,inf(s(X)))) |
(41) |
active#(inf(X)) |
→ |
cons#(X,inf(s(X))) |
(42) |
active#(inf(X)) |
→ |
inf#(s(X)) |
(43) |
active#(inf(X)) |
→ |
s#(X) |
(44) |
active#(take(0,X)) |
→ |
mark#(nil) |
(45) |
active#(take(s(X),cons(Y,L))) |
→ |
mark#(cons(Y,take(X,L))) |
(46) |
active#(take(s(X),cons(Y,L))) |
→ |
cons#(Y,take(X,L)) |
(47) |
active#(take(s(X),cons(Y,L))) |
→ |
take#(X,L) |
(48) |
active#(length(nil)) |
→ |
mark#(0) |
(49) |
active#(length(cons(X,L))) |
→ |
mark#(s(length(L))) |
(50) |
active#(length(cons(X,L))) |
→ |
s#(length(L)) |
(51) |
active#(length(cons(X,L))) |
→ |
length#(L) |
(52) |
mark#(eq(X1,X2)) |
→ |
active#(eq(X1,X2)) |
(53) |
mark#(0) |
→ |
active#(0) |
(54) |
mark#(true) |
→ |
active#(true) |
(55) |
mark#(s(X)) |
→ |
active#(s(X)) |
(56) |
mark#(false) |
→ |
active#(false) |
(57) |
mark#(inf(X)) |
→ |
active#(inf(mark(X))) |
(58) |
mark#(inf(X)) |
→ |
inf#(mark(X)) |
(59) |
mark#(inf(X)) |
→ |
mark#(X) |
(60) |
mark#(cons(X1,X2)) |
→ |
active#(cons(X1,X2)) |
(61) |
mark#(take(X1,X2)) |
→ |
active#(take(mark(X1),mark(X2))) |
(62) |
mark#(take(X1,X2)) |
→ |
take#(mark(X1),mark(X2)) |
(63) |
mark#(take(X1,X2)) |
→ |
mark#(X1) |
(64) |
mark#(take(X1,X2)) |
→ |
mark#(X2) |
(65) |
mark#(nil) |
→ |
active#(nil) |
(66) |
mark#(length(X)) |
→ |
active#(length(mark(X))) |
(67) |
mark#(length(X)) |
→ |
length#(mark(X)) |
(68) |
mark#(length(X)) |
→ |
mark#(X) |
(69) |
eq#(mark(X1),X2) |
→ |
eq#(X1,X2) |
(70) |
eq#(X1,mark(X2)) |
→ |
eq#(X1,X2) |
(71) |
eq#(active(X1),X2) |
→ |
eq#(X1,X2) |
(72) |
eq#(X1,active(X2)) |
→ |
eq#(X1,X2) |
(73) |
s#(mark(X)) |
→ |
s#(X) |
(74) |
s#(active(X)) |
→ |
s#(X) |
(75) |
inf#(mark(X)) |
→ |
inf#(X) |
(76) |
inf#(active(X)) |
→ |
inf#(X) |
(77) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(78) |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(79) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(80) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(81) |
take#(mark(X1),X2) |
→ |
take#(X1,X2) |
(82) |
take#(X1,mark(X2)) |
→ |
take#(X1,X2) |
(83) |
take#(active(X1),X2) |
→ |
take#(X1,X2) |
(84) |
take#(X1,active(X2)) |
→ |
take#(X1,X2) |
(85) |
length#(mark(X)) |
→ |
length#(X) |
(86) |
length#(active(X)) |
→ |
length#(X) |
(87) |
The dependency pairs are split into 7
components.
-
The
1st
component contains the
pair
active#(eq(s(X),s(Y))) |
→ |
mark#(eq(X,Y)) |
(38) |
mark#(eq(X1,X2)) |
→ |
active#(eq(X1,X2)) |
(53) |
active#(inf(X)) |
→ |
mark#(cons(X,inf(s(X)))) |
(41) |
mark#(s(X)) |
→ |
active#(s(X)) |
(56) |
active#(take(s(X),cons(Y,L))) |
→ |
mark#(cons(Y,take(X,L))) |
(46) |
mark#(inf(X)) |
→ |
active#(inf(mark(X))) |
(58) |
active#(length(cons(X,L))) |
→ |
mark#(s(length(L))) |
(50) |
mark#(inf(X)) |
→ |
mark#(X) |
(60) |
mark#(cons(X1,X2)) |
→ |
active#(cons(X1,X2)) |
(61) |
mark#(take(X1,X2)) |
→ |
active#(take(mark(X1),mark(X2))) |
(62) |
mark#(take(X1,X2)) |
→ |
mark#(X1) |
(64) |
mark#(take(X1,X2)) |
→ |
mark#(X2) |
(65) |
mark#(length(X)) |
→ |
active#(length(mark(X))) |
(67) |
mark#(length(X)) |
→ |
mark#(X) |
(69) |
1.1.1 Reduction Pair Processor
Using the Knuth Bendix order with w0 = 3 and the following precedence and weight functions
prec(eq) |
= |
2 |
|
weight(eq) |
= |
7 |
|
|
|
prec(cons) |
= |
6 |
|
weight(cons) |
= |
3 |
|
|
|
prec(s) |
= |
0 |
|
weight(s) |
= |
5 |
|
|
|
prec(take) |
= |
4 |
|
weight(take) |
= |
3 |
|
|
|
prec(length) |
= |
3 |
|
weight(length) |
= |
4 |
|
|
|
prec(0) |
= |
5 |
|
weight(0) |
= |
8 |
|
|
|
prec(true) |
= |
1 |
|
weight(true) |
= |
5 |
|
|
|
prec(false) |
= |
7 |
|
weight(false) |
= |
5 |
|
|
|
prec(nil) |
= |
8 |
|
weight(nil) |
= |
8 |
|
|
|
in combination with the following argument filter
π(active#) |
= |
1 |
π(eq) |
= |
[] |
π(mark#) |
= |
1 |
π(inf) |
= |
1 |
π(cons) |
= |
[] |
π(s) |
= |
[] |
π(take) |
= |
[1,2] |
π(mark) |
= |
1 |
π(length) |
= |
[1] |
π(active) |
= |
1 |
π(0) |
= |
[] |
π(true) |
= |
[] |
π(false) |
= |
[] |
π(nil) |
= |
[] |
the
pairs
active#(take(s(X),cons(Y,L))) |
→ |
mark#(cons(Y,take(X,L))) |
(46) |
active#(length(cons(X,L))) |
→ |
mark#(s(length(L))) |
(50) |
mark#(take(X1,X2)) |
→ |
mark#(X1) |
(64) |
mark#(take(X1,X2)) |
→ |
mark#(X2) |
(65) |
mark#(length(X)) |
→ |
mark#(X) |
(69) |
could be deleted.
1.1.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(active#) |
= |
2 |
|
stat(active#) |
= |
mul
|
prec(eq) |
= |
2 |
|
stat(eq) |
= |
mul
|
prec(s) |
= |
2 |
|
stat(s) |
= |
mul
|
prec(inf) |
= |
3 |
|
stat(inf) |
= |
mul
|
prec(cons) |
= |
2 |
|
stat(cons) |
= |
mul
|
prec(mark) |
= |
0 |
|
stat(mark) |
= |
mul
|
prec(take) |
= |
4 |
|
stat(take) |
= |
mul
|
prec(length) |
= |
2 |
|
stat(length) |
= |
mul
|
prec(active) |
= |
1 |
|
stat(active) |
= |
mul
|
prec(0) |
= |
4 |
|
stat(0) |
= |
mul
|
prec(true) |
= |
1 |
|
stat(true) |
= |
mul
|
prec(false) |
= |
5 |
|
stat(false) |
= |
mul
|
prec(nil) |
= |
4 |
|
stat(nil) |
= |
mul
|
π(active#) |
= |
[] |
π(eq) |
= |
[] |
π(s) |
= |
[1] |
π(mark#) |
= |
1 |
π(inf) |
= |
[1] |
π(cons) |
= |
[] |
π(mark) |
= |
[1] |
π(take) |
= |
[1,2] |
π(length) |
= |
[] |
π(active) |
= |
[] |
π(0) |
= |
[] |
π(true) |
= |
[] |
π(false) |
= |
[] |
π(nil) |
= |
[] |
together with the usable
rules
eq(X1,mark(X2)) |
→ |
eq(X1,X2) |
(20) |
eq(mark(X1),X2) |
→ |
eq(X1,X2) |
(19) |
eq(active(X1),X2) |
→ |
eq(X1,X2) |
(21) |
eq(X1,active(X2)) |
→ |
eq(X1,X2) |
(22) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(28) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(27) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(29) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(30) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(s(X)) |
→ |
active#(s(X)) |
(56) |
mark#(inf(X)) |
→ |
active#(inf(mark(X))) |
(58) |
mark#(inf(X)) |
→ |
mark#(X) |
(60) |
mark#(take(X1,X2)) |
→ |
active#(take(mark(X1),mark(X2))) |
(62) |
could be deleted.
1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(eq) |
= |
0 |
|
weight(eq) |
= |
1 |
|
|
|
prec(inf) |
= |
2 |
|
weight(inf) |
= |
3 |
|
|
|
prec(cons) |
= |
3 |
|
weight(cons) |
= |
2 |
|
|
|
prec(length) |
= |
1 |
|
weight(length) |
= |
2 |
|
|
|
in combination with the following argument filter
π(active#) |
= |
1 |
π(eq) |
= |
[] |
π(mark#) |
= |
1 |
π(inf) |
= |
[] |
π(cons) |
= |
[] |
π(length) |
= |
[] |
together with the usable
rules
eq(X1,mark(X2)) |
→ |
eq(X1,X2) |
(20) |
eq(mark(X1),X2) |
→ |
eq(X1,X2) |
(19) |
eq(active(X1),X2) |
→ |
eq(X1,X2) |
(21) |
eq(X1,active(X2)) |
→ |
eq(X1,X2) |
(22) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(28) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(27) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(29) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(30) |
length(active(X)) |
→ |
length(X) |
(36) |
length(mark(X)) |
→ |
length(X) |
(35) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#(inf(X)) |
→ |
mark#(cons(X,inf(s(X)))) |
(41) |
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(s) |
= |
0 |
|
weight(s) |
= |
1 |
|
|
|
prec(cons) |
= |
2 |
|
weight(cons) |
= |
2 |
|
|
|
prec(length) |
= |
1 |
|
weight(length) |
= |
1 |
|
|
|
in combination with the following argument filter
π(active#) |
= |
1 |
π(eq) |
= |
2 |
π(s) |
= |
[1] |
π(mark#) |
= |
1 |
π(cons) |
= |
[] |
π(length) |
= |
[] |
π(mark) |
= |
1 |
π(active) |
= |
1 |
together with the usable
rules
eq(X1,mark(X2)) |
→ |
eq(X1,X2) |
(20) |
eq(mark(X1),X2) |
→ |
eq(X1,X2) |
(19) |
eq(active(X1),X2) |
→ |
eq(X1,X2) |
(21) |
eq(X1,active(X2)) |
→ |
eq(X1,X2) |
(22) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(28) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(27) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(29) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(30) |
length(active(X)) |
→ |
length(X) |
(36) |
length(mark(X)) |
→ |
length(X) |
(35) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#(eq(s(X),s(Y))) |
→ |
mark#(eq(X,Y)) |
(38) |
could be deleted.
1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
eq#(X1,mark(X2)) |
→ |
eq#(X1,X2) |
(71) |
eq#(mark(X1),X2) |
→ |
eq#(X1,X2) |
(70) |
eq#(active(X1),X2) |
→ |
eq#(X1,X2) |
(72) |
eq#(X1,active(X2)) |
→ |
eq#(X1,X2) |
(73) |
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
|
[eq#(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.
eq#(X1,mark(X2)) |
→ |
eq#(X1,X2) |
(71) |
|
1 |
≥ |
1 |
2 |
> |
2 |
eq#(mark(X1),X2) |
→ |
eq#(X1,X2) |
(70) |
|
1 |
> |
1 |
2 |
≥ |
2 |
eq#(active(X1),X2) |
→ |
eq#(X1,X2) |
(72) |
|
1 |
> |
1 |
2 |
≥ |
2 |
eq#(X1,active(X2)) |
→ |
eq#(X1,X2) |
(73) |
|
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
s#(active(X)) |
→ |
s#(X) |
(75) |
s#(mark(X)) |
→ |
s#(X) |
(74) |
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
|
[s#(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.
s#(active(X)) |
→ |
s#(X) |
(75) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(74) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
inf#(active(X)) |
→ |
inf#(X) |
(77) |
inf#(mark(X)) |
→ |
inf#(X) |
(76) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[inf#(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.
inf#(active(X)) |
→ |
inf#(X) |
(77) |
|
1 |
> |
1 |
inf#(mark(X)) |
→ |
inf#(X) |
(76) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(79) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(78) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(80) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(81) |
1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[cons#(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.
1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(79) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(78) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(80) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(81) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
take#(X1,mark(X2)) |
→ |
take#(X1,X2) |
(83) |
take#(mark(X1),X2) |
→ |
take#(X1,X2) |
(82) |
take#(active(X1),X2) |
→ |
take#(X1,X2) |
(84) |
take#(X1,active(X2)) |
→ |
take#(X1,X2) |
(85) |
1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[take#(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.
1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
take#(X1,mark(X2)) |
→ |
take#(X1,X2) |
(83) |
|
1 |
≥ |
1 |
2 |
> |
2 |
take#(mark(X1),X2) |
→ |
take#(X1,X2) |
(82) |
|
1 |
> |
1 |
2 |
≥ |
2 |
take#(active(X1),X2) |
→ |
take#(X1,X2) |
(84) |
|
1 |
> |
1 |
2 |
≥ |
2 |
take#(X1,active(X2)) |
→ |
take#(X1,X2) |
(85) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
length#(active(X)) |
→ |
length#(X) |
(87) |
length#(mark(X)) |
→ |
length#(X) |
(86) |
1.1.7 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[length#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.7.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
length#(active(X)) |
→ |
length#(X) |
(87) |
|
1 |
> |
1 |
length#(mark(X)) |
→ |
length#(X) |
(86) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.