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)) |
(58) |
top#(ok(X)) |
→ |
top#(active(X)) |
(29) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the argument filter
π(proper) |
= |
1 |
π(ok) |
= |
1 |
π(after#) |
= |
1 |
π(active) |
= |
1 |
π(active#) |
= |
1 |
in combination with the following Weighted Path Order with the following precedence and status
prec(cons#) |
= |
0 |
|
status(cons#) |
= |
[] |
|
list-extension(cons#) |
= |
Lex |
prec(s) |
= |
4 |
|
status(s) |
= |
[1] |
|
list-extension(s) |
= |
Lex |
prec(top) |
= |
0 |
|
status(top) |
= |
[] |
|
list-extension(top) |
= |
Lex |
prec(top#) |
= |
0 |
|
status(top#) |
= |
[1] |
|
list-extension(top#) |
= |
Lex |
prec(after) |
= |
5 |
|
status(after) |
= |
[1, 2] |
|
list-extension(after) |
= |
Lex |
prec(0) |
= |
0 |
|
status(0) |
= |
[] |
|
list-extension(0) |
= |
Lex |
prec(from) |
= |
4 |
|
status(from) |
= |
[1] |
|
list-extension(from) |
= |
Lex |
prec(s#) |
= |
0 |
|
status(s#) |
= |
[] |
|
list-extension(s#) |
= |
Lex |
prec(mark) |
= |
1 |
|
status(mark) |
= |
[1] |
|
list-extension(mark) |
= |
Lex |
prec(proper#) |
= |
0 |
|
status(proper#) |
= |
[] |
|
list-extension(proper#) |
= |
Lex |
prec(from#) |
= |
0 |
|
status(from#) |
= |
[] |
|
list-extension(from#) |
= |
Lex |
prec(cons) |
= |
3 |
|
status(cons) |
= |
[1] |
|
list-extension(cons) |
= |
Lex |
and the following
Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
max(x1 + 1, 0) |
[s(x1)] |
=
|
x1 + 0 |
[top(x1)] |
=
|
1 |
[top#(x1)] |
=
|
x1 + 1 |
[after(x1, x2)] |
=
|
x1 + x2 + 21238 |
[0] |
=
|
1 |
[from(x1)] |
=
|
x1 + 38147 |
[s#(x1)] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
1 |
[from#(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
max(x1 + 38146, x2 + 0, 0) |
together with the usable
rules
proper(0) |
→ |
ok(0) |
(18) |
active(from(X)) |
→ |
from(active(X)) |
(4) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(15) |
active(after(X1,X2)) |
→ |
after(X1,active(X2)) |
(8) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(1) |
active(after(s(N),cons(X,XS))) |
→ |
mark(after(N,XS)) |
(3) |
proper(s(X)) |
→ |
s(proper(X)) |
(16) |
s(ok(X)) |
→ |
ok(s(X)) |
(21) |
from(ok(X)) |
→ |
ok(from(X)) |
(19) |
proper(after(X1,X2)) |
→ |
after(proper(X1),proper(X2)) |
(17) |
after(ok(X1),ok(X2)) |
→ |
ok(after(X1,X2)) |
(22) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(5) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(10) |
active(after(X1,X2)) |
→ |
after(active(X1),X2) |
(7) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(20) |
proper(from(X)) |
→ |
from(proper(X)) |
(14) |
after(mark(X1),X2) |
→ |
mark(after(X1,X2)) |
(12) |
s(mark(X)) |
→ |
mark(s(X)) |
(11) |
from(mark(X)) |
→ |
mark(from(X)) |
(9) |
after(X1,mark(X2)) |
→ |
mark(after(X1,X2)) |
(13) |
active(s(X)) |
→ |
s(active(X)) |
(6) |
active(after(0,XS)) |
→ |
mark(XS) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(58) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
proper#(from(X)) |
→ |
proper#(X) |
(35) |
proper#(after(X1,X2)) |
→ |
proper#(X1) |
(54) |
proper#(s(X)) |
→ |
proper#(X) |
(52) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(47) |
proper#(after(X1,X2)) |
→ |
proper#(X2) |
(46) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(27) |
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 |
[after(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
5599 |
[ok(x1)] |
=
|
x1 + 0 |
[0] |
=
|
2016 |
[from(x1)] |
=
|
x1 + 14098 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
1 |
[after#(x1, x2)] |
=
|
0 |
[proper#(x1)] |
=
|
x1 + 0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
proper(0) |
→ |
ok(0) |
(18) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(1) |
active(after(s(N),cons(X,XS))) |
→ |
mark(after(N,XS)) |
(3) |
s(ok(X)) |
→ |
ok(s(X)) |
(21) |
from(ok(X)) |
→ |
ok(from(X)) |
(19) |
after(ok(X1),ok(X2)) |
→ |
ok(after(X1,X2)) |
(22) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(10) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(20) |
after(mark(X1),X2) |
→ |
mark(after(X1,X2)) |
(12) |
s(mark(X)) |
→ |
mark(s(X)) |
(11) |
from(mark(X)) |
→ |
mark(from(X)) |
(9) |
after(X1,mark(X2)) |
→ |
mark(after(X1,X2)) |
(13) |
active(after(0,XS)) |
→ |
mark(XS) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
proper#(from(X)) |
→ |
proper#(X) |
(35) |
proper#(after(X1,X2)) |
→ |
proper#(X1) |
(54) |
proper#(s(X)) |
→ |
proper#(X) |
(52) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(47) |
proper#(after(X1,X2)) |
→ |
proper#(X2) |
(46) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(27) |
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#(cons(X1,X2)) |
→ |
active#(X1) |
(61) |
active#(after(X1,X2)) |
→ |
active#(X1) |
(59) |
active#(after(X1,X2)) |
→ |
active#(X2) |
(50) |
active#(from(X)) |
→ |
active#(X) |
(28) |
active#(s(X)) |
→ |
active#(X) |
(25) |
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 |
[after(x1, x2)] |
=
|
x1 + x2 + 15261 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 0 |
[0] |
=
|
1 |
[from(x1)] |
=
|
x1 + 1 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
1 |
[after#(x1, x2)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 15943 |
[active#(x1)] |
=
|
x1 + 0 |
together with the usable
rules
proper(0) |
→ |
ok(0) |
(18) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(1) |
active(after(s(N),cons(X,XS))) |
→ |
mark(after(N,XS)) |
(3) |
s(ok(X)) |
→ |
ok(s(X)) |
(21) |
from(ok(X)) |
→ |
ok(from(X)) |
(19) |
after(ok(X1),ok(X2)) |
→ |
ok(after(X1,X2)) |
(22) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(10) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(20) |
after(mark(X1),X2) |
→ |
mark(after(X1,X2)) |
(12) |
s(mark(X)) |
→ |
mark(s(X)) |
(11) |
from(mark(X)) |
→ |
mark(from(X)) |
(9) |
after(X1,mark(X2)) |
→ |
mark(after(X1,X2)) |
(13) |
active(after(0,XS)) |
→ |
mark(XS) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(cons(X1,X2)) |
→ |
active#(X1) |
(61) |
active#(after(X1,X2)) |
→ |
active#(X1) |
(59) |
active#(after(X1,X2)) |
→ |
active#(X2) |
(50) |
active#(from(X)) |
→ |
active#(X) |
(28) |
active#(s(X)) |
→ |
active#(X) |
(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
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(55) |
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(30) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
x1 + 0 |
[s(x1)] |
=
|
x1 + 20534 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[after(x1, x2)] |
=
|
x1 + x2 + 31892 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 0 |
[0] |
=
|
1 |
[from(x1)] |
=
|
x1 + 1 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[after#(x1, x2)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
30574 |
[cons(x1, x2)] |
=
|
x1 + x2 + 10038 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
proper(0) |
→ |
ok(0) |
(18) |
s(ok(X)) |
→ |
ok(s(X)) |
(21) |
from(ok(X)) |
→ |
ok(from(X)) |
(19) |
after(ok(X1),ok(X2)) |
→ |
ok(after(X1,X2)) |
(22) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(10) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(20) |
after(mark(X1),X2) |
→ |
mark(after(X1,X2)) |
(12) |
s(mark(X)) |
→ |
mark(s(X)) |
(11) |
from(mark(X)) |
→ |
mark(from(X)) |
(9) |
after(X1,mark(X2)) |
→ |
mark(after(X1,X2)) |
(13) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(55) |
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) |
(33) |
s#(mark(X)) |
→ |
s#(X) |
(44) |
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 |
[after(x1, x2)] |
=
|
x1 + x2 + 2 |
[proper(x1)] |
=
|
0 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[from(x1)] |
=
|
3 |
[s#(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
x1 + 1 |
[after#(x1, x2)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
2 |
[cons(x1, x2)] |
=
|
1 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
after(ok(X1),ok(X2)) |
→ |
ok(after(X1,X2)) |
(22) |
after(mark(X1),X2) |
→ |
mark(after(X1,X2)) |
(12) |
after(X1,mark(X2)) |
→ |
mark(after(X1,X2)) |
(13) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
s#(ok(X)) |
→ |
s#(X) |
(33) |
s#(mark(X)) |
→ |
s#(X) |
(44) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
6th
component contains the
pair
from#(mark(X)) |
→ |
from#(X) |
(60) |
from#(ok(X)) |
→ |
from#(X) |
(34) |
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 |
[after(x1, x2)] |
=
|
x1 + x2 + 2 |
[proper(x1)] |
=
|
0 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[from(x1)] |
=
|
5971 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 5969 |
[after#(x1, x2)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[from#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
5970 |
[cons(x1, x2)] |
=
|
1 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
after(ok(X1),ok(X2)) |
→ |
ok(after(X1,X2)) |
(22) |
after(mark(X1),X2) |
→ |
mark(after(X1,X2)) |
(12) |
after(X1,mark(X2)) |
→ |
mark(after(X1,X2)) |
(13) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
from#(mark(X)) |
→ |
from#(X) |
(60) |
from#(ok(X)) |
→ |
from#(X) |
(34) |
could be deleted.
1.1.6.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
7th
component contains the
pair
after#(X1,mark(X2)) |
→ |
after#(X1,X2) |
(56) |
after#(ok(X1),ok(X2)) |
→ |
after#(X1,X2) |
(51) |
after#(mark(X1),X2) |
→ |
after#(X1,X2) |
(42) |
1.1.7 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 |
[after(x1, x2)] |
=
|
x1 + x2 + 2 |
[proper(x1)] |
=
|
0 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[from(x1)] |
=
|
25534 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 25532 |
[after#(x1, x2)] |
=
|
x1 + x2 + 0 |
[proper#(x1)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
25533 |
[cons(x1, x2)] |
=
|
1 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
after(ok(X1),ok(X2)) |
→ |
ok(after(X1,X2)) |
(22) |
after(mark(X1),X2) |
→ |
mark(after(X1,X2)) |
(12) |
after(X1,mark(X2)) |
→ |
mark(after(X1,X2)) |
(13) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
after#(X1,mark(X2)) |
→ |
after#(X1,X2) |
(56) |
after#(ok(X1),ok(X2)) |
→ |
after#(X1,X2) |
(51) |
after#(mark(X1),X2) |
→ |
after#(X1,X2) |
(42) |
could be deleted.
1.1.7.1 Dependency Graph Processor
The dependency pairs are split into 0
components.