The rewrite relation of the following TRS is considered.
active#(from(X)) |
→ |
s#(X) |
(21) |
active#(from(X)) |
→ |
from#(s(X)) |
(22) |
active#(from(X)) |
→ |
cons#(X,from(s(X))) |
(23) |
active#(from(X)) |
→ |
mark#(cons(X,from(s(X)))) |
(24) |
active#(after(0,XS)) |
→ |
mark#(XS) |
(25) |
active#(after(s(N),cons(X,XS))) |
→ |
after#(N,XS) |
(26) |
active#(after(s(N),cons(X,XS))) |
→ |
mark#(after(N,XS)) |
(27) |
mark#(from(X)) |
→ |
mark#(X) |
(28) |
mark#(from(X)) |
→ |
from#(mark(X)) |
(29) |
mark#(from(X)) |
→ |
active#(from(mark(X))) |
(30) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(31) |
mark#(cons(X1,X2)) |
→ |
cons#(mark(X1),X2) |
(32) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(33) |
mark#(s(X)) |
→ |
mark#(X) |
(34) |
mark#(s(X)) |
→ |
s#(mark(X)) |
(35) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(36) |
mark#(after(X1,X2)) |
→ |
mark#(X2) |
(37) |
mark#(after(X1,X2)) |
→ |
mark#(X1) |
(38) |
mark#(after(X1,X2)) |
→ |
after#(mark(X1),mark(X2)) |
(39) |
mark#(after(X1,X2)) |
→ |
active#(after(mark(X1),mark(X2))) |
(40) |
mark#(0) |
→ |
active#(0) |
(41) |
from#(mark(X)) |
→ |
from#(X) |
(42) |
from#(active(X)) |
→ |
from#(X) |
(43) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(44) |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(45) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(46) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(47) |
s#(mark(X)) |
→ |
s#(X) |
(48) |
s#(active(X)) |
→ |
s#(X) |
(49) |
after#(mark(X1),X2) |
→ |
after#(X1,X2) |
(50) |
after#(X1,mark(X2)) |
→ |
after#(X1,X2) |
(51) |
after#(active(X1),X2) |
→ |
after#(X1,X2) |
(52) |
after#(X1,active(X2)) |
→ |
after#(X1,X2) |
(53) |
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
mark#(after(X1,X2)) |
→ |
mark#(X2) |
(37) |
mark#(after(X1,X2)) |
→ |
active#(after(mark(X1),mark(X2))) |
(40) |
active#(after(s(N),cons(X,XS))) |
→ |
mark#(after(N,XS)) |
(27) |
mark#(after(X1,X2)) |
→ |
mark#(X1) |
(38) |
mark#(s(X)) |
→ |
mark#(X) |
(34) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(31) |
mark#(from(X)) |
→ |
active#(from(mark(X))) |
(30) |
active#(from(X)) |
→ |
mark#(cons(X,from(s(X)))) |
(24) |
mark#(from(X)) |
→ |
mark#(X) |
(28) |
active#(after(0,XS)) |
→ |
mark#(XS) |
(25) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers
[mark(x1)] |
= |
0 · x1 +
-∞ |
[active(x1)] |
= |
0 · x1 +
-∞ |
[cons(x1, x2)] |
= |
0 · x1 + 0 · x2 + 4 |
[0] |
= |
0 |
[from(x1)] |
= |
1 · x1 + 7 |
[active#(x1)] |
= |
0 · x1 + 0 |
[mark#(x1)] |
= |
0 · x1 + 0 |
[after(x1, x2)] |
= |
0 · x1 + 3 · x2 + 0 |
[s(x1)] |
= |
0 · x1 +
-∞ |
together with the usable
rules
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(1) |
active(after(0,XS)) |
→ |
mark(XS) |
(2) |
active(after(s(N),cons(X,XS))) |
→ |
mark(after(N,XS)) |
(3) |
mark(from(X)) |
→ |
active(from(mark(X))) |
(4) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(5) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(6) |
mark(after(X1,X2)) |
→ |
active(after(mark(X1),mark(X2))) |
(7) |
mark(0) |
→ |
active(0) |
(8) |
from(mark(X)) |
→ |
from(X) |
(9) |
from(active(X)) |
→ |
from(X) |
(10) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(11) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(12) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(13) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(14) |
s(mark(X)) |
→ |
s(X) |
(15) |
s(active(X)) |
→ |
s(X) |
(16) |
after(mark(X1),X2) |
→ |
after(X1,X2) |
(17) |
after(X1,mark(X2)) |
→ |
after(X1,X2) |
(18) |
after(active(X1),X2) |
→ |
after(X1,X2) |
(19) |
after(X1,active(X2)) |
→ |
after(X1,X2) |
(20) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(from(X)) |
→ |
mark#(X) |
(28) |
could be deleted.
1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers
[mark(x1)] |
= |
0 · x1 +
-∞ |
[active(x1)] |
= |
0 · x1 + 0 |
[cons(x1, x2)] |
= |
0 · x1 + 0 · x2 + 0 |
[0] |
= |
0 |
[from(x1)] |
= |
0 · x1 + 2 |
[active#(x1)] |
= |
0 · x1 + 0 |
[mark#(x1)] |
= |
0 · x1 +
-∞ |
[after(x1, x2)] |
= |
7 · x1 + 0 · x2 + 7 |
[s(x1)] |
= |
0 · x1 + 0 |
together with the usable
rules
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(1) |
active(after(0,XS)) |
→ |
mark(XS) |
(2) |
active(after(s(N),cons(X,XS))) |
→ |
mark(after(N,XS)) |
(3) |
mark(from(X)) |
→ |
active(from(mark(X))) |
(4) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(5) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(6) |
mark(after(X1,X2)) |
→ |
active(after(mark(X1),mark(X2))) |
(7) |
mark(0) |
→ |
active(0) |
(8) |
from(mark(X)) |
→ |
from(X) |
(9) |
from(active(X)) |
→ |
from(X) |
(10) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(11) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(12) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(13) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(14) |
s(mark(X)) |
→ |
s(X) |
(15) |
s(active(X)) |
→ |
s(X) |
(16) |
after(mark(X1),X2) |
→ |
after(X1,X2) |
(17) |
after(X1,mark(X2)) |
→ |
after(X1,X2) |
(18) |
after(active(X1),X2) |
→ |
after(X1,X2) |
(19) |
after(X1,active(X2)) |
→ |
after(X1,X2) |
(20) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(after(X1,X2)) |
→ |
mark#(X1) |
(38) |
could be deleted.
1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers
[mark(x1)] |
= |
0 · x1 +
-∞ |
[active(x1)] |
= |
0 · x1 +
-∞ |
[cons(x1, x2)] |
= |
0 · x1 + 0 · x2 + 0 |
[0] |
= |
0 |
[from(x1)] |
= |
2 · x1 + 2 |
[active#(x1)] |
= |
0 · x1 +
-∞ |
[mark#(x1)] |
= |
0 · x1 + 0 |
[after(x1, x2)] |
= |
1 · x1 + 7 · x2 +
-∞ |
[s(x1)] |
= |
0 · x1 + 0 |
together with the usable
rules
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(1) |
active(after(0,XS)) |
→ |
mark(XS) |
(2) |
active(after(s(N),cons(X,XS))) |
→ |
mark(after(N,XS)) |
(3) |
mark(from(X)) |
→ |
active(from(mark(X))) |
(4) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(5) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(6) |
mark(after(X1,X2)) |
→ |
active(after(mark(X1),mark(X2))) |
(7) |
mark(0) |
→ |
active(0) |
(8) |
from(mark(X)) |
→ |
from(X) |
(9) |
from(active(X)) |
→ |
from(X) |
(10) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(11) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(12) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(13) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(14) |
s(mark(X)) |
→ |
s(X) |
(15) |
s(active(X)) |
→ |
s(X) |
(16) |
after(mark(X1),X2) |
→ |
after(X1,X2) |
(17) |
after(X1,mark(X2)) |
→ |
after(X1,X2) |
(18) |
after(active(X1),X2) |
→ |
after(X1,X2) |
(19) |
after(X1,active(X2)) |
→ |
after(X1,X2) |
(20) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#(after(0,XS)) |
→ |
mark#(XS) |
(25) |
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers
[mark(x1)] |
= |
0 · x1 +
-∞ |
[active(x1)] |
= |
0 · x1 +
-∞ |
[cons(x1, x2)] |
= |
0 · x1 + 0 · x2 +
-∞ |
[0] |
= |
5 |
[from(x1)] |
= |
0 · x1 +
-∞ |
[active#(x1)] |
= |
0 · x1 +
-∞ |
[mark#(x1)] |
= |
0 · x1 +
-∞ |
[after(x1, x2)] |
= |
1 · x1 + 4 · x2 + 1 |
[s(x1)] |
= |
0 · x1 +
-∞ |
together with the usable
rules
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(1) |
active(after(0,XS)) |
→ |
mark(XS) |
(2) |
active(after(s(N),cons(X,XS))) |
→ |
mark(after(N,XS)) |
(3) |
mark(from(X)) |
→ |
active(from(mark(X))) |
(4) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(5) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(6) |
mark(after(X1,X2)) |
→ |
active(after(mark(X1),mark(X2))) |
(7) |
mark(0) |
→ |
active(0) |
(8) |
from(mark(X)) |
→ |
from(X) |
(9) |
from(active(X)) |
→ |
from(X) |
(10) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(11) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(12) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(13) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(14) |
s(mark(X)) |
→ |
s(X) |
(15) |
s(active(X)) |
→ |
s(X) |
(16) |
after(mark(X1),X2) |
→ |
after(X1,X2) |
(17) |
after(X1,mark(X2)) |
→ |
after(X1,X2) |
(18) |
after(active(X1),X2) |
→ |
after(X1,X2) |
(19) |
after(X1,active(X2)) |
→ |
after(X1,X2) |
(20) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(after(X1,X2)) |
→ |
mark#(X2) |
(37) |
could be deleted.
1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
2nd
component contains the
pair
from#(active(X)) |
→ |
from#(X) |
(43) |
from#(mark(X)) |
→ |
from#(X) |
(42) |
1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
from#(active(X)) |
→ |
from#(X) |
(43) |
|
1 |
> |
1 |
from#(mark(X)) |
→ |
from#(X) |
(42) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(47) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(46) |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(45) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(44) |
1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(47) |
|
2 |
> |
2 |
1 |
≥ |
1 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(46) |
|
2 |
≥ |
2 |
1 |
> |
1 |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(45) |
|
2 |
> |
2 |
1 |
≥ |
1 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(44) |
|
2 |
≥ |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
s#(active(X)) |
→ |
s#(X) |
(49) |
s#(mark(X)) |
→ |
s#(X) |
(48) |
1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(active(X)) |
→ |
s#(X) |
(49) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(48) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
after#(mark(X1),X2) |
→ |
after#(X1,X2) |
(50) |
after#(X1,active(X2)) |
→ |
after#(X1,X2) |
(53) |
after#(active(X1),X2) |
→ |
after#(X1,X2) |
(52) |
after#(X1,mark(X2)) |
→ |
after#(X1,X2) |
(51) |
1.1.5 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
after#(mark(X1),X2) |
→ |
after#(X1,X2) |
(50) |
|
2 |
≥ |
2 |
1 |
> |
1 |
after#(X1,active(X2)) |
→ |
after#(X1,X2) |
(53) |
|
2 |
> |
2 |
1 |
≥ |
1 |
after#(active(X1),X2) |
→ |
after#(X1,X2) |
(52) |
|
2 |
≥ |
2 |
1 |
> |
1 |
after#(X1,mark(X2)) |
→ |
after#(X1,X2) |
(51) |
|
2 |
> |
2 |
1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.