The rewrite relation of the following TRS is considered.
The dependency pairs are split into 7
components.
-
The
1st
component contains the
pair
mark#(tail(X)) |
→ |
mark#(X) |
(63) |
mark#(head(X)) |
→ |
mark#(X) |
(60) |
mark#(zeros) |
→ |
active#(zeros) |
(58) |
active#(zeros) |
→ |
mark#(cons(0,zeros)) |
(43) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(48) |
mark#(adx(X)) |
→ |
active#(adx(mark(X))) |
(56) |
active#(adx(cons(X,L))) |
→ |
mark#(incr(cons(X,adx(L)))) |
(41) |
mark#(incr(X)) |
→ |
active#(incr(mark(X))) |
(46) |
active#(incr(cons(X,L))) |
→ |
mark#(cons(s(X),incr(L))) |
(37) |
mark#(incr(X)) |
→ |
mark#(X) |
(44) |
mark#(adx(X)) |
→ |
mark#(X) |
(54) |
mark#(s(X)) |
→ |
mark#(X) |
(51) |
1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(mark#) |
= |
0 |
|
stat(mark#) |
= |
lex
|
prec(active#) |
= |
0 |
|
stat(active#) |
= |
lex
|
prec(tail) |
= |
0 |
|
stat(tail) |
= |
lex
|
prec(head) |
= |
0 |
|
stat(head) |
= |
lex
|
prec(0) |
= |
0 |
|
stat(0) |
= |
lex
|
prec(zeros) |
= |
1 |
|
stat(zeros) |
= |
lex
|
prec(nats) |
= |
0 |
|
stat(nats) |
= |
lex
|
prec(adx) |
= |
3 |
|
stat(adx) |
= |
lex
|
prec(s) |
= |
0 |
|
stat(s) |
= |
lex
|
prec(cons) |
= |
0 |
|
stat(cons) |
= |
lex
|
prec(mark) |
= |
0 |
|
stat(mark) |
= |
lex
|
prec(active) |
= |
0 |
|
stat(active) |
= |
lex
|
prec(incr) |
= |
2 |
|
stat(incr) |
= |
lex
|
prec(nil) |
= |
0 |
|
stat(nil) |
= |
lex
|
π(mark#) |
= |
1 |
π(active#) |
= |
1 |
π(tail) |
= |
1 |
π(head) |
= |
1 |
π(0) |
= |
[] |
π(zeros) |
= |
[] |
π(nats) |
= |
[] |
π(adx) |
= |
[1] |
π(s) |
= |
1 |
π(cons) |
= |
1 |
π(mark) |
= |
1 |
π(active) |
= |
1 |
π(incr) |
= |
[1] |
π(nil) |
= |
[] |
together with the usable
rules
active(incr(nil)) |
→ |
mark(nil) |
(1) |
active(incr(cons(X,L))) |
→ |
mark(cons(s(X),incr(L))) |
(2) |
active(adx(cons(X,L))) |
→ |
mark(incr(cons(X,adx(L)))) |
(4) |
active(zeros) |
→ |
mark(cons(0,zeros)) |
(6) |
mark(incr(X)) |
→ |
active(incr(mark(X))) |
(9) |
mark(nil) |
→ |
active(nil) |
(10) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(11) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(12) |
mark(adx(X)) |
→ |
active(adx(mark(X))) |
(13) |
mark(nats) |
→ |
active(nats) |
(14) |
mark(zeros) |
→ |
active(zeros) |
(15) |
mark(0) |
→ |
active(0) |
(16) |
mark(head(X)) |
→ |
active(head(mark(X))) |
(17) |
mark(tail(X)) |
→ |
active(tail(mark(X))) |
(18) |
incr(mark(X)) |
→ |
incr(X) |
(19) |
incr(active(X)) |
→ |
incr(X) |
(20) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(21) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(22) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(23) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(24) |
s(mark(X)) |
→ |
s(X) |
(25) |
s(active(X)) |
→ |
s(X) |
(26) |
adx(mark(X)) |
→ |
adx(X) |
(27) |
adx(active(X)) |
→ |
adx(X) |
(28) |
head(mark(X)) |
→ |
head(X) |
(29) |
head(active(X)) |
→ |
head(X) |
(30) |
tail(mark(X)) |
→ |
tail(X) |
(31) |
tail(active(X)) |
→ |
tail(X) |
(32) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(zeros) |
→ |
mark#(cons(0,zeros)) |
(43) |
active#(adx(cons(X,L))) |
→ |
mark#(incr(cons(X,adx(L)))) |
(41) |
active#(incr(cons(X,L))) |
→ |
mark#(cons(s(X),incr(L))) |
(37) |
mark#(incr(X)) |
→ |
mark#(X) |
(44) |
mark#(adx(X)) |
→ |
mark#(X) |
(54) |
could be deleted.
1.1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
mark#(tail(X)) |
→ |
mark#(X) |
(63) |
mark#(head(X)) |
→ |
mark#(X) |
(60) |
mark#(s(X)) |
→ |
mark#(X) |
(51) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(48) |
1.1.1.1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
mark#(tail(X)) |
→ |
mark#(X) |
(63) |
|
1 |
> |
1 |
mark#(head(X)) |
→ |
mark#(X) |
(60) |
|
1 |
> |
1 |
mark#(s(X)) |
→ |
mark#(X) |
(51) |
|
1 |
> |
1 |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(48) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
incr#(mark(X)) |
→ |
incr#(X) |
(66) |
incr#(active(X)) |
→ |
incr#(X) |
(67) |
1.1.1.1.1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
incr#(mark(X)) |
→ |
incr#(X) |
(66) |
|
1 |
> |
1 |
incr#(active(X)) |
→ |
incr#(X) |
(67) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
s#(mark(X)) |
→ |
s#(X) |
(72) |
s#(active(X)) |
→ |
s#(X) |
(73) |
1.1.1.1.1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(mark(X)) |
→ |
s#(X) |
(72) |
|
1 |
> |
1 |
s#(active(X)) |
→ |
s#(X) |
(73) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(68) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(71) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(70) |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(69) |
1.1.1.1.1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(68) |
|
2 |
≥ |
2 |
1 |
> |
1 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(71) |
|
2 |
> |
2 |
1 |
≥ |
1 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(70) |
|
2 |
≥ |
2 |
1 |
> |
1 |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(69) |
|
2 |
> |
2 |
1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
adx#(mark(X)) |
→ |
adx#(X) |
(74) |
adx#(active(X)) |
→ |
adx#(X) |
(75) |
1.1.1.1.1.1.5 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
adx#(mark(X)) |
→ |
adx#(X) |
(74) |
|
1 |
> |
1 |
adx#(active(X)) |
→ |
adx#(X) |
(75) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
head#(mark(X)) |
→ |
head#(X) |
(76) |
head#(active(X)) |
→ |
head#(X) |
(77) |
1.1.1.1.1.1.6 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
head#(mark(X)) |
→ |
head#(X) |
(76) |
|
1 |
> |
1 |
head#(active(X)) |
→ |
head#(X) |
(77) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
tail#(mark(X)) |
→ |
tail#(X) |
(78) |
tail#(active(X)) |
→ |
tail#(X) |
(79) |
1.1.1.1.1.1.7 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
tail#(mark(X)) |
→ |
tail#(X) |
(78) |
|
1 |
> |
1 |
tail#(active(X)) |
→ |
tail#(X) |
(79) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.