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)) |
(66) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(61) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the argument filter
π(top) |
= |
1 |
π(plus#) |
= |
2 |
π(top#) |
= |
1 |
π(proper) |
= |
1 |
π(ok) |
= |
1 |
π(active) |
= |
1 |
π(active#) |
= |
1 |
in combination with the following Weighted Path Order with the following precedence and status
prec(s) |
= |
1 |
|
status(s) |
= |
[1] |
|
list-extension(s) |
= |
Lex |
prec(and) |
= |
2 |
|
status(and) |
= |
[2, 1] |
|
list-extension(and) |
= |
Lex |
prec(x) |
= |
4 |
|
status(x) |
= |
[2, 1] |
|
list-extension(x) |
= |
Lex |
prec(0) |
= |
0 |
|
status(0) |
= |
[] |
|
list-extension(0) |
= |
Lex |
prec(x#) |
= |
0 |
|
status(x#) |
= |
[1, 2] |
|
list-extension(x#) |
= |
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(plus) |
= |
3 |
|
status(plus) |
= |
[2, 1] |
|
list-extension(plus) |
= |
Lex |
prec(tt) |
= |
3 |
|
status(tt) |
= |
[] |
|
list-extension(tt) |
= |
Lex |
prec(and#) |
= |
0 |
|
status(and#) |
= |
[1, 2] |
|
list-extension(and#) |
= |
Lex |
and the following
Max-polynomial interpretation
[s(x1)] |
=
|
x1 + 0 |
[and(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
[x(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
[0] |
=
|
0 |
[x#(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
[tt] |
=
|
0 |
[and#(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
together with the usable
rules
proper(and(X1,X2)) |
→ |
and(proper(X1),proper(X2)) |
(18) |
active(x(N,0)) |
→ |
mark(0) |
(4) |
s(mark(X)) |
→ |
mark(s(X)) |
(15) |
active(plus(X1,X2)) |
→ |
plus(X1,active(X2)) |
(8) |
active(and(tt,X)) |
→ |
mark(X) |
(1) |
active(plus(N,s(M))) |
→ |
mark(s(plus(N,M))) |
(3) |
x(mark(X1),X2) |
→ |
mark(x(X1,X2)) |
(16) |
proper(0) |
→ |
ok(0) |
(21) |
s(ok(X)) |
→ |
ok(s(X)) |
(26) |
proper(tt) |
→ |
ok(tt) |
(19) |
x(X1,mark(X2)) |
→ |
mark(x(X1,X2)) |
(17) |
x(ok(X1),ok(X2)) |
→ |
ok(x(X1,X2)) |
(27) |
proper(s(X)) |
→ |
s(proper(X)) |
(22) |
active(x(N,s(M))) |
→ |
mark(plus(x(N,M),N)) |
(5) |
active(x(X1,X2)) |
→ |
x(active(X1),X2) |
(10) |
active(plus(X1,X2)) |
→ |
plus(active(X1),X2) |
(7) |
proper(plus(X1,X2)) |
→ |
plus(proper(X1),proper(X2)) |
(20) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(25) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(14) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(12) |
proper(x(X1,X2)) |
→ |
x(proper(X1),proper(X2)) |
(23) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(24) |
active(x(X1,X2)) |
→ |
x(X1,active(X2)) |
(11) |
active(s(X)) |
→ |
s(active(X)) |
(9) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(13) |
active(and(X1,X2)) |
→ |
and(active(X1),X2) |
(6) |
active(plus(N,0)) |
→ |
mark(N) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(61) |
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#(plus(X1,X2)) |
→ |
proper#(X1) |
(70) |
proper#(x(X1,X2)) |
→ |
proper#(X1) |
(43) |
proper#(x(X1,X2)) |
→ |
proper#(X2) |
(38) |
proper#(plus(X1,X2)) |
→ |
proper#(X2) |
(60) |
proper#(and(X1,X2)) |
→ |
proper#(X1) |
(37) |
proper#(and(X1,X2)) |
→ |
proper#(X2) |
(33) |
proper#(s(X)) |
→ |
proper#(X) |
(31) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x1 + x2 + 1 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[x(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
2 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[x#(x1, x2)] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
x1 + 0 |
[plus(x1, x2)] |
=
|
x1 + x2 + 1 |
[active(x1)] |
=
|
27858 |
[active#(x1)] |
=
|
0 |
[tt] |
=
|
1 |
[and#(x1, x2)] |
=
|
0 |
together with the usable
rules
active(x(N,0)) |
→ |
mark(0) |
(4) |
s(mark(X)) |
→ |
mark(s(X)) |
(15) |
x(mark(X1),X2) |
→ |
mark(x(X1,X2)) |
(16) |
proper(0) |
→ |
ok(0) |
(21) |
s(ok(X)) |
→ |
ok(s(X)) |
(26) |
proper(tt) |
→ |
ok(tt) |
(19) |
x(X1,mark(X2)) |
→ |
mark(x(X1,X2)) |
(17) |
x(ok(X1),ok(X2)) |
→ |
ok(x(X1,X2)) |
(27) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(25) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(14) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(12) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(24) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(13) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
proper#(plus(X1,X2)) |
→ |
proper#(X1) |
(70) |
proper#(x(X1,X2)) |
→ |
proper#(X1) |
(43) |
proper#(x(X1,X2)) |
→ |
proper#(X2) |
(38) |
proper#(plus(X1,X2)) |
→ |
proper#(X2) |
(60) |
proper#(and(X1,X2)) |
→ |
proper#(X1) |
(37) |
proper#(and(X1,X2)) |
→ |
proper#(X2) |
(33) |
proper#(s(X)) |
→ |
proper#(X) |
(31) |
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#(and(X1,X2)) |
→ |
active#(X1) |
(69) |
active#(x(X1,X2)) |
→ |
active#(X1) |
(42) |
active#(plus(X1,X2)) |
→ |
active#(X1) |
(63) |
active#(x(X1,X2)) |
→ |
active#(X2) |
(39) |
active#(plus(X1,X2)) |
→ |
active#(X2) |
(34) |
active#(s(X)) |
→ |
active#(X) |
(49) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x1 + x2 + 1 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[x(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
2 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[x#(x1, x2)] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + x2 + 1 |
[active(x1)] |
=
|
2 |
[active#(x1)] |
=
|
x1 + 0 |
[tt] |
=
|
1 |
[and#(x1, x2)] |
=
|
0 |
together with the usable
rules
active(x(N,0)) |
→ |
mark(0) |
(4) |
s(mark(X)) |
→ |
mark(s(X)) |
(15) |
x(mark(X1),X2) |
→ |
mark(x(X1,X2)) |
(16) |
proper(0) |
→ |
ok(0) |
(21) |
s(ok(X)) |
→ |
ok(s(X)) |
(26) |
proper(tt) |
→ |
ok(tt) |
(19) |
x(X1,mark(X2)) |
→ |
mark(x(X1,X2)) |
(17) |
x(ok(X1),ok(X2)) |
→ |
ok(x(X1,X2)) |
(27) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(12) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(24) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(and(X1,X2)) |
→ |
active#(X1) |
(69) |
active#(x(X1,X2)) |
→ |
active#(X1) |
(42) |
active#(plus(X1,X2)) |
→ |
active#(X1) |
(63) |
active#(x(X1,X2)) |
→ |
active#(X2) |
(39) |
active#(plus(X1,X2)) |
→ |
active#(X2) |
(34) |
active#(s(X)) |
→ |
active#(X) |
(49) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(45) |
and#(ok(X1),ok(X2)) |
→ |
and#(X1,X2) |
(51) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x1 + x2 + 3153 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[x(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
2 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[x#(x1, x2)] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + x2 + 1 |
[active(x1)] |
=
|
2 |
[active#(x1)] |
=
|
0 |
[tt] |
=
|
1 |
[and#(x1, x2)] |
=
|
x1 + x2 + 0 |
together with the usable
rules
active(x(N,0)) |
→ |
mark(0) |
(4) |
s(mark(X)) |
→ |
mark(s(X)) |
(15) |
x(mark(X1),X2) |
→ |
mark(x(X1,X2)) |
(16) |
proper(0) |
→ |
ok(0) |
(21) |
s(ok(X)) |
→ |
ok(s(X)) |
(26) |
proper(tt) |
→ |
ok(tt) |
(19) |
x(X1,mark(X2)) |
→ |
mark(x(X1,X2)) |
(17) |
x(ok(X1),ok(X2)) |
→ |
ok(x(X1,X2)) |
(27) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(12) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(24) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
and#(ok(X1),ok(X2)) |
→ |
and#(X1,X2) |
(51) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
5th
component contains the
pair
x#(ok(X1),ok(X2)) |
→ |
x#(X1,X2) |
(46) |
x#(X1,mark(X2)) |
→ |
x#(X1,X2) |
(44) |
x#(mark(X1),X2) |
→ |
x#(X1,X2) |
(52) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x1 + x2 + 1 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[x(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
2 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[x#(x1, x2)] |
=
|
x2 + 0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + x2 + 1 |
[active(x1)] |
=
|
3 |
[active#(x1)] |
=
|
0 |
[tt] |
=
|
1 |
[and#(x1, x2)] |
=
|
0 |
together with the usable
rules
active(x(N,0)) |
→ |
mark(0) |
(4) |
s(mark(X)) |
→ |
mark(s(X)) |
(15) |
x(mark(X1),X2) |
→ |
mark(x(X1,X2)) |
(16) |
proper(0) |
→ |
ok(0) |
(21) |
s(ok(X)) |
→ |
ok(s(X)) |
(26) |
proper(tt) |
→ |
ok(tt) |
(19) |
x(X1,mark(X2)) |
→ |
mark(x(X1,X2)) |
(17) |
x(ok(X1),ok(X2)) |
→ |
ok(x(X1,X2)) |
(27) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(12) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(24) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
x#(ok(X1),ok(X2)) |
→ |
x#(X1,X2) |
(46) |
x#(X1,mark(X2)) |
→ |
x#(X1,X2) |
(44) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
6th
component contains the
pair
s#(ok(X)) |
→ |
s#(X) |
(59) |
s#(mark(X)) |
→ |
s#(X) |
(35) |
1.1.6 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[s(x1)] |
=
|
x1 + 18748 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x2 + 18751 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[x(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
2 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[x#(x1, x2)] |
=
|
0 |
[s#(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
x1 + 1 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + x2 + 1 |
[active(x1)] |
=
|
18750 |
[active#(x1)] |
=
|
0 |
[tt] |
=
|
1 |
[and#(x1, x2)] |
=
|
0 |
together with the usable
rules
active(x(N,0)) |
→ |
mark(0) |
(4) |
s(mark(X)) |
→ |
mark(s(X)) |
(15) |
x(mark(X1),X2) |
→ |
mark(x(X1,X2)) |
(16) |
proper(0) |
→ |
ok(0) |
(21) |
s(ok(X)) |
→ |
ok(s(X)) |
(26) |
proper(tt) |
→ |
ok(tt) |
(19) |
x(X1,mark(X2)) |
→ |
mark(x(X1,X2)) |
(17) |
x(ok(X1),ok(X2)) |
→ |
ok(x(X1,X2)) |
(27) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
s#(ok(X)) |
→ |
s#(X) |
(59) |
s#(mark(X)) |
→ |
s#(X) |
(35) |
could be deleted.
1.1.6.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
7th
component contains the
pair
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(68) |
plus#(ok(X1),ok(X2)) |
→ |
plus#(X1,X2) |
(64) |
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(53) |
1.1.7 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x2 + 4 |
[plus#(x1, x2)] |
=
|
x1 + x2 + 0 |
[top#(x1)] |
=
|
0 |
[x(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
2 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[x#(x1, x2)] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + x2 + 1 |
[active(x1)] |
=
|
3 |
[active#(x1)] |
=
|
0 |
[tt] |
=
|
1 |
[and#(x1, x2)] |
=
|
0 |
together with the usable
rules
active(x(N,0)) |
→ |
mark(0) |
(4) |
s(mark(X)) |
→ |
mark(s(X)) |
(15) |
x(mark(X1),X2) |
→ |
mark(x(X1,X2)) |
(16) |
proper(0) |
→ |
ok(0) |
(21) |
s(ok(X)) |
→ |
ok(s(X)) |
(26) |
proper(tt) |
→ |
ok(tt) |
(19) |
x(X1,mark(X2)) |
→ |
mark(x(X1,X2)) |
(17) |
x(ok(X1),ok(X2)) |
→ |
ok(x(X1,X2)) |
(27) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(68) |
plus#(ok(X1),ok(X2)) |
→ |
plus#(X1,X2) |
(64) |
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(53) |
could be deleted.
1.1.7.1 Dependency Graph Processor
The dependency pairs are split into 0
components.