The rewrite relation of the following TRS is considered.
The dependency pairs are split into 7
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(56) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(42) |
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 +
|
[top(x1)] |
= |
|
[top#(x1)] |
= |
· x1 +
|
[p#(x1)] |
= |
|
[f(x1)] |
= |
· x1 +
|
[p(x1)] |
= |
· x1 +
|
[proper(x1)] |
= |
x1 +
|
[ok(x1)] |
= |
x1 +
|
[0] |
= |
|
[s#(x1)] |
= |
|
[mark(x1)] |
= |
x1 +
|
[f#(x1)] |
= |
|
[proper#(x1)] |
= |
|
[active(x1)] |
= |
x1 +
|
[cons(x1, x2)] |
= |
· x1 +
|
[active#(x1)] |
= |
|
together with the usable
rules
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(18) |
active(f(X)) |
→ |
f(active(X)) |
(4) |
proper(s(X)) |
→ |
s(proper(X)) |
(15) |
f(mark(X)) |
→ |
mark(f(X)) |
(8) |
active(f(0)) |
→ |
mark(cons(0,f(s(0)))) |
(1) |
active(p(s(X))) |
→ |
mark(X) |
(3) |
proper(p(X)) |
→ |
p(proper(X)) |
(16) |
s(ok(X)) |
→ |
ok(s(X)) |
(19) |
f(ok(X)) |
→ |
ok(f(X)) |
(17) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(5) |
s(mark(X)) |
→ |
mark(s(X)) |
(10) |
active(p(X)) |
→ |
p(active(X)) |
(7) |
p(ok(X)) |
→ |
ok(p(X)) |
(20) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(14) |
proper(f(X)) |
→ |
f(proper(X)) |
(12) |
p(mark(X)) |
→ |
mark(p(X)) |
(11) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(9) |
proper(0) |
→ |
ok(0) |
(13) |
active(s(X)) |
→ |
s(active(X)) |
(6) |
active(f(s(0))) |
→ |
mark(f(p(s(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)) |
(42) |
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#(s(X)) |
→ |
active#(X) |
(55) |
active#(p(X)) |
→ |
active#(X) |
(33) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(45) |
active#(f(X)) |
→ |
active#(X) |
(39) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[p#(x1)] |
=
|
0 |
[f(x1)] |
=
|
x1 + 1 |
[p(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
19456 |
[ok(x1)] |
=
|
19457 |
[0] |
=
|
13506 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
48368 |
[f#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
23979 |
[cons(x1, x2)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
together with the usable
rules
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(18) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(s(X)) |
→ |
active#(X) |
(55) |
active#(p(X)) |
→ |
active#(X) |
(33) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(45) |
active#(f(X)) |
→ |
active#(X) |
(39) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
proper#(p(X)) |
→ |
proper#(X) |
(35) |
proper#(s(X)) |
→ |
proper#(X) |
(32) |
proper#(f(X)) |
→ |
proper#(X) |
(53) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(28) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(38) |
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 |
[top#(x1)] |
=
|
0 |
[p#(x1)] |
=
|
0 |
[f(x1)] |
=
|
x1 + 1 |
[p(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
2 |
[0] |
=
|
49110 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
47553 |
[f#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
19696 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(18) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
proper#(p(X)) |
→ |
proper#(X) |
(35) |
proper#(s(X)) |
→ |
proper#(X) |
(32) |
proper#(f(X)) |
→ |
proper#(X) |
(53) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(28) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(38) |
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) |
(43) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(40) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
x2 + 0 |
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[p#(x1)] |
=
|
0 |
[f(x1)] |
=
|
x1 + 1 |
[p(x1)] |
=
|
x1 + 15943 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
30124 |
[f#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
30123 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(18) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(43) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
5th
component contains the
pair
p#(ok(X)) |
→ |
p#(X) |
(31) |
p#(mark(X)) |
→ |
p#(X) |
(25) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[p#(x1)] |
=
|
x1 + 0 |
[f(x1)] |
=
|
x1 + 1 |
[p(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 2 |
[f#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
p#(ok(X)) |
→ |
p#(X) |
(31) |
p#(mark(X)) |
→ |
p#(X) |
(25) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
6th
component contains the
pair
s#(mark(X)) |
→ |
s#(X) |
(54) |
s#(ok(X)) |
→ |
s#(X) |
(49) |
1.1.6 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[p#(x1)] |
=
|
0 |
[f(x1)] |
=
|
x1 + 1 |
[p(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
x1 + 2 |
[f#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
s#(mark(X)) |
→ |
s#(X) |
(54) |
s#(ok(X)) |
→ |
s#(X) |
(49) |
could be deleted.
1.1.6.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
7th
component contains the
pair
f#(mark(X)) |
→ |
f#(X) |
(24) |
f#(ok(X)) |
→ |
f#(X) |
(36) |
1.1.7 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 27570 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[p#(x1)] |
=
|
0 |
[f(x1)] |
=
|
x1 + 1 |
[p(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 19897 |
[f#(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
19896 |
[cons(x1, x2)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
f#(mark(X)) |
→ |
f#(X) |
(24) |
f#(ok(X)) |
→ |
f#(X) |
(36) |
could be deleted.
1.1.7.1 Dependency Graph Processor
The dependency pairs are split into 0
components.