The rewrite relation of the following TRS is considered.
The dependency pairs are split into 8
components.
-
The
1st
component contains the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(31) |
top#(ok(X)) |
→ |
top#(active(X)) |
(40) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
x1 + 1 |
[d(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[c(x1)] |
=
|
x1 + 48638 |
[f(x1)] |
=
|
x1 + 1424 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 48642 |
[h#(x1)] |
=
|
0 |
[d#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 2 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 48641 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 1 |
together with the usable
rules
active(f(X)) |
→ |
f(active(X)) |
(4) |
g(ok(X)) |
→ |
ok(g(X)) |
(15) |
proper(f(X)) |
→ |
f(proper(X)) |
(8) |
active(f(f(X))) |
→ |
mark(c(f(g(f(X))))) |
(1) |
active(h(X)) |
→ |
mark(c(d(X))) |
(3) |
d(ok(X)) |
→ |
ok(d(X)) |
(16) |
h(ok(X)) |
→ |
ok(h(X)) |
(17) |
active(h(X)) |
→ |
h(active(X)) |
(5) |
proper(g(X)) |
→ |
g(proper(X)) |
(10) |
h(mark(X)) |
→ |
mark(h(X)) |
(7) |
c(ok(X)) |
→ |
ok(c(X)) |
(14) |
proper(h(X)) |
→ |
h(proper(X)) |
(12) |
proper(d(X)) |
→ |
d(proper(X)) |
(11) |
proper(c(X)) |
→ |
c(proper(X)) |
(9) |
f(ok(X)) |
→ |
ok(f(X)) |
(13) |
f(mark(X)) |
→ |
mark(f(X)) |
(6) |
active(c(X)) |
→ |
mark(d(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
top#(mark(X)) |
→ |
top#(proper(X)) |
(31) |
top#(ok(X)) |
→ |
top#(active(X)) |
(40) |
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#(f(X)) |
→ |
proper#(X) |
(42) |
proper#(h(X)) |
→ |
proper#(X) |
(24) |
proper#(g(X)) |
→ |
proper#(X) |
(23) |
proper#(c(X)) |
→ |
proper#(X) |
(36) |
proper#(d(X)) |
→ |
proper#(X) |
(34) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
x1 + 1 |
[d(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[c(x1)] |
=
|
x1 + 1 |
[f(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 17678 |
[ok(x1)] |
=
|
x1 + 1 |
[h#(x1)] |
=
|
0 |
[d#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
x1 + 0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 3 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 1 |
together with the usable
rules
active(f(X)) |
→ |
f(active(X)) |
(4) |
g(ok(X)) |
→ |
ok(g(X)) |
(15) |
proper(f(X)) |
→ |
f(proper(X)) |
(8) |
active(f(f(X))) |
→ |
mark(c(f(g(f(X))))) |
(1) |
active(h(X)) |
→ |
mark(c(d(X))) |
(3) |
d(ok(X)) |
→ |
ok(d(X)) |
(16) |
h(ok(X)) |
→ |
ok(h(X)) |
(17) |
active(h(X)) |
→ |
h(active(X)) |
(5) |
proper(g(X)) |
→ |
g(proper(X)) |
(10) |
h(mark(X)) |
→ |
mark(h(X)) |
(7) |
c(ok(X)) |
→ |
ok(c(X)) |
(14) |
proper(h(X)) |
→ |
h(proper(X)) |
(12) |
proper(d(X)) |
→ |
d(proper(X)) |
(11) |
proper(c(X)) |
→ |
c(proper(X)) |
(9) |
f(ok(X)) |
→ |
ok(f(X)) |
(13) |
f(mark(X)) |
→ |
mark(f(X)) |
(6) |
active(c(X)) |
→ |
mark(d(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
proper#(f(X)) |
→ |
proper#(X) |
(42) |
proper#(h(X)) |
→ |
proper#(X) |
(24) |
proper#(g(X)) |
→ |
proper#(X) |
(23) |
proper#(c(X)) |
→ |
proper#(X) |
(36) |
proper#(d(X)) |
→ |
proper#(X) |
(34) |
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#(h(X)) |
→ |
active#(X) |
(48) |
active#(f(X)) |
→ |
active#(X) |
(21) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
x1 + 1 |
[d(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[c(x1)] |
=
|
x1 + 1 |
[f(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 34519 |
[ok(x1)] |
=
|
x1 + 1 |
[h#(x1)] |
=
|
0 |
[d#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 12174 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
x1 + 12172 |
together with the usable
rules
active(f(X)) |
→ |
f(active(X)) |
(4) |
g(ok(X)) |
→ |
ok(g(X)) |
(15) |
proper(f(X)) |
→ |
f(proper(X)) |
(8) |
active(f(f(X))) |
→ |
mark(c(f(g(f(X))))) |
(1) |
active(h(X)) |
→ |
mark(c(d(X))) |
(3) |
d(ok(X)) |
→ |
ok(d(X)) |
(16) |
h(ok(X)) |
→ |
ok(h(X)) |
(17) |
active(h(X)) |
→ |
h(active(X)) |
(5) |
proper(g(X)) |
→ |
g(proper(X)) |
(10) |
h(mark(X)) |
→ |
mark(h(X)) |
(7) |
c(ok(X)) |
→ |
ok(c(X)) |
(14) |
proper(h(X)) |
→ |
h(proper(X)) |
(12) |
proper(d(X)) |
→ |
d(proper(X)) |
(11) |
proper(c(X)) |
→ |
c(proper(X)) |
(9) |
f(ok(X)) |
→ |
ok(f(X)) |
(13) |
f(mark(X)) |
→ |
mark(f(X)) |
(6) |
active(c(X)) |
→ |
mark(d(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(h(X)) |
→ |
active#(X) |
(48) |
active#(f(X)) |
→ |
active#(X) |
(21) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
h#(ok(X)) |
→ |
h#(X) |
(44) |
h#(mark(X)) |
→ |
h#(X) |
(25) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
x1 + 1 |
[d(x1)] |
=
|
x1 + 2 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[c(x1)] |
=
|
x1 + 1 |
[f(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 1 |
[h#(x1)] |
=
|
x1 + 0 |
[d#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 3 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 1 |
together with the usable
rules
active(f(X)) |
→ |
f(active(X)) |
(4) |
g(ok(X)) |
→ |
ok(g(X)) |
(15) |
proper(f(X)) |
→ |
f(proper(X)) |
(8) |
active(f(f(X))) |
→ |
mark(c(f(g(f(X))))) |
(1) |
active(h(X)) |
→ |
mark(c(d(X))) |
(3) |
d(ok(X)) |
→ |
ok(d(X)) |
(16) |
h(ok(X)) |
→ |
ok(h(X)) |
(17) |
active(h(X)) |
→ |
h(active(X)) |
(5) |
proper(g(X)) |
→ |
g(proper(X)) |
(10) |
h(mark(X)) |
→ |
mark(h(X)) |
(7) |
c(ok(X)) |
→ |
ok(c(X)) |
(14) |
proper(h(X)) |
→ |
h(proper(X)) |
(12) |
proper(d(X)) |
→ |
d(proper(X)) |
(11) |
proper(c(X)) |
→ |
c(proper(X)) |
(9) |
f(ok(X)) |
→ |
ok(f(X)) |
(13) |
f(mark(X)) |
→ |
mark(f(X)) |
(6) |
active(c(X)) |
→ |
mark(d(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
h#(ok(X)) |
→ |
h#(X) |
(44) |
h#(mark(X)) |
→ |
h#(X) |
(25) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
5th
component contains the
pair
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
x1 + 1 |
[d(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[c(x1)] |
=
|
x1 + 1 |
[f(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 29405 |
[h#(x1)] |
=
|
0 |
[d#(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 3 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 1 |
together with the usable
rules
active(f(X)) |
→ |
f(active(X)) |
(4) |
g(ok(X)) |
→ |
ok(g(X)) |
(15) |
proper(f(X)) |
→ |
f(proper(X)) |
(8) |
active(f(f(X))) |
→ |
mark(c(f(g(f(X))))) |
(1) |
active(h(X)) |
→ |
mark(c(d(X))) |
(3) |
d(ok(X)) |
→ |
ok(d(X)) |
(16) |
h(ok(X)) |
→ |
ok(h(X)) |
(17) |
active(h(X)) |
→ |
h(active(X)) |
(5) |
proper(g(X)) |
→ |
g(proper(X)) |
(10) |
h(mark(X)) |
→ |
mark(h(X)) |
(7) |
c(ok(X)) |
→ |
ok(c(X)) |
(14) |
proper(h(X)) |
→ |
h(proper(X)) |
(12) |
proper(d(X)) |
→ |
d(proper(X)) |
(11) |
proper(c(X)) |
→ |
c(proper(X)) |
(9) |
f(ok(X)) |
→ |
ok(f(X)) |
(13) |
f(mark(X)) |
→ |
mark(f(X)) |
(6) |
active(c(X)) |
→ |
mark(d(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
6th
component contains the
pair
1.1.6 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
x1 + 1 |
[d(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[c(x1)] |
=
|
x1 + 1 |
[f(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 1 |
[h#(x1)] |
=
|
0 |
[d#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 3 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 1 |
together with the usable
rules
active(f(X)) |
→ |
f(active(X)) |
(4) |
g(ok(X)) |
→ |
ok(g(X)) |
(15) |
proper(f(X)) |
→ |
f(proper(X)) |
(8) |
active(f(f(X))) |
→ |
mark(c(f(g(f(X))))) |
(1) |
active(h(X)) |
→ |
mark(c(d(X))) |
(3) |
d(ok(X)) |
→ |
ok(d(X)) |
(16) |
h(ok(X)) |
→ |
ok(h(X)) |
(17) |
active(h(X)) |
→ |
h(active(X)) |
(5) |
proper(g(X)) |
→ |
g(proper(X)) |
(10) |
h(mark(X)) |
→ |
mark(h(X)) |
(7) |
c(ok(X)) |
→ |
ok(c(X)) |
(14) |
proper(h(X)) |
→ |
h(proper(X)) |
(12) |
proper(d(X)) |
→ |
d(proper(X)) |
(11) |
proper(c(X)) |
→ |
c(proper(X)) |
(9) |
f(ok(X)) |
→ |
ok(f(X)) |
(13) |
f(mark(X)) |
→ |
mark(f(X)) |
(6) |
active(c(X)) |
→ |
mark(d(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
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#(ok(X)) |
→ |
f#(X) |
(45) |
f#(mark(X)) |
→ |
f#(X) |
(20) |
1.1.7 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
x1 + 1 |
[d(x1)] |
=
|
1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[c(x1)] |
=
|
x1 + 1 |
[f(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 0 |
[h#(x1)] |
=
|
0 |
[d#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
x1 + 0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 3 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 1 |
together with the usable
rules
active(f(X)) |
→ |
f(active(X)) |
(4) |
g(ok(X)) |
→ |
ok(g(X)) |
(15) |
proper(f(X)) |
→ |
f(proper(X)) |
(8) |
active(f(f(X))) |
→ |
mark(c(f(g(f(X))))) |
(1) |
active(h(X)) |
→ |
mark(c(d(X))) |
(3) |
d(ok(X)) |
→ |
ok(d(X)) |
(16) |
h(ok(X)) |
→ |
ok(h(X)) |
(17) |
active(h(X)) |
→ |
h(active(X)) |
(5) |
proper(g(X)) |
→ |
g(proper(X)) |
(10) |
h(mark(X)) |
→ |
mark(h(X)) |
(7) |
c(ok(X)) |
→ |
ok(c(X)) |
(14) |
proper(h(X)) |
→ |
h(proper(X)) |
(12) |
proper(d(X)) |
→ |
d(proper(X)) |
(11) |
proper(c(X)) |
→ |
c(proper(X)) |
(9) |
f(ok(X)) |
→ |
ok(f(X)) |
(13) |
f(mark(X)) |
→ |
mark(f(X)) |
(6) |
active(c(X)) |
→ |
mark(d(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
could be deleted.
1.1.7.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
8th
component contains the
pair
1.1.8 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
x1 + 2 |
[d(x1)] |
=
|
x1 + 3 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[c(x1)] |
=
|
x1 + 1 |
[f(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 1 |
[h#(x1)] |
=
|
0 |
[d#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 31742 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
x1 + 31744 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 1 |
together with the usable
rules
active(f(X)) |
→ |
f(active(X)) |
(4) |
g(ok(X)) |
→ |
ok(g(X)) |
(15) |
proper(f(X)) |
→ |
f(proper(X)) |
(8) |
active(f(f(X))) |
→ |
mark(c(f(g(f(X))))) |
(1) |
active(h(X)) |
→ |
mark(c(d(X))) |
(3) |
d(ok(X)) |
→ |
ok(d(X)) |
(16) |
h(ok(X)) |
→ |
ok(h(X)) |
(17) |
active(h(X)) |
→ |
h(active(X)) |
(5) |
proper(g(X)) |
→ |
g(proper(X)) |
(10) |
h(mark(X)) |
→ |
mark(h(X)) |
(7) |
c(ok(X)) |
→ |
ok(c(X)) |
(14) |
proper(h(X)) |
→ |
h(proper(X)) |
(12) |
proper(d(X)) |
→ |
d(proper(X)) |
(11) |
proper(c(X)) |
→ |
c(proper(X)) |
(9) |
f(ok(X)) |
→ |
ok(f(X)) |
(13) |
f(mark(X)) |
→ |
mark(f(X)) |
(6) |
active(c(X)) |
→ |
mark(d(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
could be deleted.
1.1.8.1 Dependency Graph Processor
The dependency pairs are split into 0
components.