The rewrite relation of the following TRS is considered.
The dependency pairs are split into 8
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(56) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(41) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals
[cons#(x1, x2)] |
= |
|
[s(x1)] |
= |
· x1 +
|
[length1#(x1)] |
= |
|
[top(x1)] |
= |
|
[length1(x1)] |
= |
· x1 +
|
[top#(x1)] |
= |
· x1 +
|
[proper(x1)] |
= |
x1 +
|
[ok(x1)] |
= |
x1 +
|
[0] |
= |
|
[from(x1)] |
= |
x1 +
|
[s#(x1)] |
= |
|
[nil] |
= |
|
[mark(x1)] |
= |
· x1 +
|
[proper#(x1)] |
= |
|
[from#(x1)] |
= |
|
[active(x1)] |
= |
x1 +
|
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[active#(x1)] |
= |
|
[length(x1)] |
= |
· x1 +
|
[length#(x1)] |
= |
|
together with the usable
rules
from(ok(X)) |
→ |
ok(from(X)) |
(18) |
active(length1(X)) |
→ |
mark(length(X)) |
(4) |
proper(nil) |
→ |
ok(nil) |
(15) |
from(mark(X)) |
→ |
mark(from(X)) |
(8) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(1) |
active(length(cons(X,Y))) |
→ |
mark(s(length1(Y))) |
(3) |
proper(0) |
→ |
ok(0) |
(16) |
length(ok(X)) |
→ |
ok(length(X)) |
(21) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(19) |
proper(length1(X)) |
→ |
length1(proper(X)) |
(17) |
length1(ok(X)) |
→ |
ok(length1(X)) |
(22) |
active(from(X)) |
→ |
from(active(X)) |
(5) |
s(mark(X)) |
→ |
mark(s(X)) |
(10) |
active(s(X)) |
→ |
s(active(X)) |
(7) |
s(ok(X)) |
→ |
ok(s(X)) |
(20) |
proper(length(X)) |
→ |
length(proper(X)) |
(14) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(12) |
proper(from(X)) |
→ |
from(proper(X)) |
(11) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(9) |
proper(s(X)) |
→ |
s(proper(X)) |
(13) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(6) |
active(length(nil)) |
→ |
mark(0) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
top#(ok(X)) |
→ |
top#(active(X)) |
(56) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(41) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
proper#(length1(X)) |
→ |
proper#(X) |
(38) |
proper#(s(X)) |
→ |
proper#(X) |
(58) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(36) |
proper#(length(X)) |
→ |
proper#(X) |
(53) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(29) |
proper#(from(X)) |
→ |
proper#(X) |
(40) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[length1#(x1)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[length1(x1)] |
=
|
x1 + 16704 |
[top#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
33265 |
[ok(x1)] |
=
|
33266 |
[0] |
=
|
1 |
[from(x1)] |
=
|
x1 + 28872 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
19095 |
[mark(x1)] |
=
|
46797 |
[proper#(x1)] |
=
|
x1 + 0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
44049 |
[cons(x1, x2)] |
=
|
x1 + x2 + 2 |
[active#(x1)] |
=
|
0 |
[length(x1)] |
=
|
x1 + 1 |
[length#(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
proper#(length1(X)) |
→ |
proper#(X) |
(38) |
proper#(s(X)) |
→ |
proper#(X) |
(58) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(36) |
proper#(length(X)) |
→ |
proper#(X) |
(53) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(29) |
proper#(from(X)) |
→ |
proper#(X) |
(40) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
active#(s(X)) |
→ |
active#(X) |
(50) |
active#(from(X)) |
→ |
active#(X) |
(45) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(25) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[length1#(x1)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[length1(x1)] |
=
|
x1 + 27691 |
[top#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
33266 |
[0] |
=
|
1 |
[from(x1)] |
=
|
x1 + 28872 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
240 |
[mark(x1)] |
=
|
2 |
[proper#(x1)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 2 |
[active#(x1)] |
=
|
x1 + 0 |
[length(x1)] |
=
|
x1 + 1 |
[length#(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
active#(s(X)) |
→ |
active#(X) |
(50) |
active#(from(X)) |
→ |
active#(X) |
(45) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(25) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
length1#(ok(X)) |
→ |
length1#(X) |
(33) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[length1#(x1)] |
=
|
x1 + 0 |
[top(x1)] |
=
|
0 |
[length1(x1)] |
=
|
x1 + 1 |
[top#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 2 |
[0] |
=
|
34495 |
[from(x1)] |
=
|
x1 + 21394 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
0 |
[mark(x1)] |
=
|
34661 |
[proper#(x1)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
18640 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[length(x1)] |
=
|
x1 + 26608 |
[length#(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
length1#(ok(X)) |
→ |
length1#(X) |
(33) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
5th
component contains the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(51) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(43) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
x2 + 0 |
[s(x1)] |
=
|
x1 + 1 |
[length1#(x1)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[length1(x1)] |
=
|
x1 + 1 |
[top#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 2 |
[0] |
=
|
36560 |
[from(x1)] |
=
|
x1 + 56977 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
0 |
[mark(x1)] |
=
|
2 |
[proper#(x1)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 36337 |
[active#(x1)] |
=
|
0 |
[length(x1)] |
=
|
x1 + 15620 |
[length#(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(51) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
6th
component contains the
pair
from#(mark(X)) |
→ |
from#(X) |
(42) |
from#(ok(X)) |
→ |
from#(X) |
(27) |
1.1.6 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 0 |
[length1#(x1)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[length1(x1)] |
=
|
x1 + 0 |
[top#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 2 |
[0] |
=
|
0 |
[from(x1)] |
=
|
x1 + 17525 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
29109 |
[mark(x1)] |
=
|
x1 + 15048 |
[proper#(x1)] |
=
|
0 |
[from#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[length(x1)] |
=
|
x1 + 0 |
[length#(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
from#(mark(X)) |
→ |
from#(X) |
(42) |
from#(ok(X)) |
→ |
from#(X) |
(27) |
could be deleted.
1.1.6.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
7th
component contains the
pair
length#(ok(X)) |
→ |
length#(X) |
(47) |
1.1.7 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 0 |
[length1#(x1)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[length1(x1)] |
=
|
x1 + 0 |
[top#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 2 |
[0] |
=
|
0 |
[from(x1)] |
=
|
x1 + 32135 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 48064 |
[proper#(x1)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
33017 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[length(x1)] |
=
|
x1 + 0 |
[length#(x1)] |
=
|
x1 + 0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
length#(ok(X)) |
→ |
length#(X) |
(47) |
could be deleted.
1.1.7.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
8th
component contains the
pair
s#(ok(X)) |
→ |
s#(X) |
(48) |
s#(mark(X)) |
→ |
s#(X) |
(39) |
1.1.8 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 0 |
[length1#(x1)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[length1(x1)] |
=
|
x1 + 0 |
[top#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 2 |
[0] |
=
|
0 |
[from(x1)] |
=
|
x1 + 26815 |
[s#(x1)] |
=
|
x1 + 0 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 15048 |
[proper#(x1)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[length(x1)] |
=
|
x1 + 0 |
[length#(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
s#(ok(X)) |
→ |
s#(X) |
(48) |
s#(mark(X)) |
→ |
s#(X) |
(39) |
could be deleted.
1.1.8.1 Dependency Graph Processor
The dependency pairs are split into 0
components.