The rewrite relation of the following TRS is considered.
The dependency pairs are split into 7
components.
-
The
1st
component contains the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(28) |
top#(ok(X)) |
→ |
top#(active(X)) |
(46) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 0 |
[top(x1)] |
=
|
0 |
[2nd(x1)] |
=
|
x1 + 0 |
[top#(x1)] |
=
|
x1 + 0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 28103 |
[from(x1)] |
=
|
x1 + 0 |
[s#(x1)] |
=
|
0 |
[2nd#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
2 |
[proper#(x1)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 28102 |
[cons(x1, x2)] |
=
|
x1 + 0 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
s(ok(X)) |
→ |
ok(s(X)) |
(18) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(4) |
2nd(ok(X)) |
→ |
ok(2nd(X)) |
(15) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(8) |
active(2nd(cons(X,cons(Y,Z)))) |
→ |
mark(Y) |
(1) |
active(2nd(X)) |
→ |
2nd(active(X)) |
(3) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(16) |
from(ok(X)) |
→ |
ok(from(X)) |
(17) |
active(from(X)) |
→ |
from(active(X)) |
(5) |
s(mark(X)) |
→ |
mark(s(X)) |
(10) |
2nd(mark(X)) |
→ |
mark(2nd(X)) |
(7) |
proper(s(X)) |
→ |
s(proper(X)) |
(14) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(12) |
proper(2nd(X)) |
→ |
2nd(proper(X)) |
(11) |
from(mark(X)) |
→ |
mark(from(X)) |
(9) |
proper(from(X)) |
→ |
from(proper(X)) |
(13) |
active(s(X)) |
→ |
s(active(X)) |
(6) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
top#(mark(X)) |
→ |
top#(proper(X)) |
(28) |
top#(ok(X)) |
→ |
top#(active(X)) |
(46) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
active#(cons(X1,X2)) |
→ |
active#(X1) |
(30) |
active#(from(X)) |
→ |
active#(X) |
(45) |
active#(2nd(X)) |
→ |
active#(X) |
(40) |
active#(s(X)) |
→ |
active#(X) |
(37) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 0 |
[top(x1)] |
=
|
0 |
[2nd(x1)] |
=
|
x1 + 0 |
[top#(x1)] |
=
|
x1 + 0 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 29283 |
[from(x1)] |
=
|
x1 + 1 |
[s#(x1)] |
=
|
0 |
[2nd#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
1 |
[proper#(x1)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
x1 + 0 |
[active#(x1)] |
=
|
x1 + 0 |
together with the usable
rules
s(ok(X)) |
→ |
ok(s(X)) |
(18) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(4) |
2nd(ok(X)) |
→ |
ok(2nd(X)) |
(15) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(8) |
active(2nd(cons(X,cons(Y,Z)))) |
→ |
mark(Y) |
(1) |
active(2nd(X)) |
→ |
2nd(active(X)) |
(3) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(16) |
from(ok(X)) |
→ |
ok(from(X)) |
(17) |
active(from(X)) |
→ |
from(active(X)) |
(5) |
s(mark(X)) |
→ |
mark(s(X)) |
(10) |
2nd(mark(X)) |
→ |
mark(2nd(X)) |
(7) |
proper(s(X)) |
→ |
s(proper(X)) |
(14) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(12) |
proper(2nd(X)) |
→ |
2nd(proper(X)) |
(11) |
from(mark(X)) |
→ |
mark(from(X)) |
(9) |
proper(from(X)) |
→ |
from(proper(X)) |
(13) |
active(s(X)) |
→ |
s(active(X)) |
(6) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#(from(X)) |
→ |
active#(X) |
(45) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
3rd
component contains the
pair
proper#(s(X)) |
→ |
proper#(X) |
(50) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(29) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(24) |
proper#(2nd(X)) |
→ |
proper#(X) |
(36) |
proper#(from(X)) |
→ |
proper#(X) |
(35) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[2nd(x1)] |
=
|
x1 + 1 |
[top#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
x1 + 0 |
[ok(x1)] |
=
|
39825 |
[from(x1)] |
=
|
x1 + 1 |
[s#(x1)] |
=
|
0 |
[2nd#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
2 |
[proper#(x1)] |
=
|
x1 + 0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
2nd(ok(X)) |
→ |
ok(2nd(X)) |
(15) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(8) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(16) |
from(ok(X)) |
→ |
ok(from(X)) |
(17) |
s(mark(X)) |
→ |
mark(s(X)) |
(10) |
2nd(mark(X)) |
→ |
mark(2nd(X)) |
(7) |
from(mark(X)) |
→ |
mark(from(X)) |
(9) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
proper#(s(X)) |
→ |
proper#(X) |
(50) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(29) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(24) |
proper#(2nd(X)) |
→ |
proper#(X) |
(36) |
proper#(from(X)) |
→ |
proper#(X) |
(35) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(33) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(38) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
x2 + 0 |
[s(x1)] |
=
|
x1 + 0 |
[top(x1)] |
=
|
0 |
[2nd(x1)] |
=
|
x1 + 1 |
[top#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
x1 + 0 |
[ok(x1)] |
=
|
x1 + 1 |
[from(x1)] |
=
|
x1 + 2 |
[s#(x1)] |
=
|
0 |
[2nd#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
1 |
[proper#(x1)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
2nd(ok(X)) |
→ |
ok(2nd(X)) |
(15) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(8) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(16) |
from(ok(X)) |
→ |
ok(from(X)) |
(17) |
s(mark(X)) |
→ |
mark(s(X)) |
(10) |
2nd(mark(X)) |
→ |
mark(2nd(X)) |
(7) |
from(mark(X)) |
→ |
mark(from(X)) |
(9) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(33) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
5th
component contains the
pair
s#(ok(X)) |
→ |
s#(X) |
(42) |
s#(mark(X)) |
→ |
s#(X) |
(43) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 0 |
[top(x1)] |
=
|
0 |
[2nd(x1)] |
=
|
x1 + 1 |
[top#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
x1 + 0 |
[ok(x1)] |
=
|
x1 + 1 |
[from(x1)] |
=
|
x1 + 1 |
[s#(x1)] |
=
|
x1 + 0 |
[2nd#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[proper#(x1)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
2nd(ok(X)) |
→ |
ok(2nd(X)) |
(15) |
from(ok(X)) |
→ |
ok(from(X)) |
(17) |
2nd(mark(X)) |
→ |
mark(2nd(X)) |
(7) |
from(mark(X)) |
→ |
mark(from(X)) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
s#(ok(X)) |
→ |
s#(X) |
(42) |
s#(mark(X)) |
→ |
s#(X) |
(43) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
6th
component contains the
pair
2nd#(mark(X)) |
→ |
2nd#(X) |
(52) |
2nd#(ok(X)) |
→ |
2nd#(X) |
(22) |
1.1.6 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 0 |
[top(x1)] |
=
|
0 |
[2nd(x1)] |
=
|
x1 + 52194 |
[top#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
x1 + 0 |
[ok(x1)] |
=
|
x1 + 1 |
[from(x1)] |
=
|
x1 + 1 |
[s#(x1)] |
=
|
0 |
[2nd#(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
x1 + 1 |
[proper#(x1)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
2nd(ok(X)) |
→ |
ok(2nd(X)) |
(15) |
from(ok(X)) |
→ |
ok(from(X)) |
(17) |
2nd(mark(X)) |
→ |
mark(2nd(X)) |
(7) |
from(mark(X)) |
→ |
mark(from(X)) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
2nd#(mark(X)) |
→ |
2nd#(X) |
(52) |
2nd#(ok(X)) |
→ |
2nd#(X) |
(22) |
could be deleted.
1.1.6.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
7th
component contains the
pair
from#(mark(X)) |
→ |
from#(X) |
(49) |
from#(ok(X)) |
→ |
from#(X) |
(39) |
1.1.7 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 0 |
[top(x1)] |
=
|
0 |
[2nd(x1)] |
=
|
x1 + 17897 |
[top#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
x1 + 0 |
[ok(x1)] |
=
|
x1 + 1 |
[from(x1)] |
=
|
x1 + 1 |
[s#(x1)] |
=
|
0 |
[2nd#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[proper#(x1)] |
=
|
0 |
[from#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
2nd(ok(X)) |
→ |
ok(2nd(X)) |
(15) |
from(ok(X)) |
→ |
ok(from(X)) |
(17) |
2nd(mark(X)) |
→ |
mark(2nd(X)) |
(7) |
from(mark(X)) |
→ |
mark(from(X)) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
from#(mark(X)) |
→ |
from#(X) |
(49) |
from#(ok(X)) |
→ |
from#(X) |
(39) |
could be deleted.
1.1.7.1 Dependency Graph Processor
The dependency pairs are split into 0
components.