The rewrite relation of the following TRS is considered.
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
mark#(x(X1,X2)) |
→ |
mark#(X2) |
(47) |
mark#(x(X1,X2)) |
→ |
active#(x(mark(X1),mark(X2))) |
(50) |
active#(x(N,s(M))) |
→ |
mark#(plus(x(N,M),N)) |
(34) |
mark#(plus(X1,X2)) |
→ |
active#(plus(mark(X1),mark(X2))) |
(42) |
active#(plus(N,s(M))) |
→ |
mark#(s(plus(N,M))) |
(30) |
mark#(s(X)) |
→ |
mark#(X) |
(44) |
mark#(x(X1,X2)) |
→ |
mark#(X1) |
(48) |
mark#(plus(X1,X2)) |
→ |
mark#(X1) |
(40) |
mark#(plus(X1,X2)) |
→ |
mark#(X2) |
(39) |
mark#(and(X1,X2)) |
→ |
active#(and(mark(X1),X2)) |
(37) |
active#(and(tt,X)) |
→ |
mark#(X) |
(26) |
mark#(and(X1,X2)) |
→ |
mark#(X1) |
(35) |
active#(plus(N,0)) |
→ |
mark#(N) |
(27) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(mark#) |
= |
0 |
|
stat(mark#) |
= |
lex
|
prec(active#) |
= |
0 |
|
stat(active#) |
= |
lex
|
prec(x) |
= |
2 |
|
stat(x) |
= |
lex
|
prec(s) |
= |
0 |
|
stat(s) |
= |
lex
|
prec(plus) |
= |
1 |
|
stat(plus) |
= |
lex
|
prec(0) |
= |
0 |
|
stat(0) |
= |
lex
|
prec(mark) |
= |
0 |
|
stat(mark) |
= |
lex
|
prec(active) |
= |
0 |
|
stat(active) |
= |
lex
|
prec(and) |
= |
0 |
|
stat(and) |
= |
lex
|
prec(tt) |
= |
0 |
|
stat(tt) |
= |
lex
|
π(mark#) |
= |
1 |
π(active#) |
= |
1 |
π(x) |
= |
[1,2] |
π(s) |
= |
[1] |
π(plus) |
= |
[1,2] |
π(0) |
= |
[] |
π(mark) |
= |
1 |
π(active) |
= |
1 |
π(and) |
= |
[1,2] |
π(tt) |
= |
[] |
together with the usable
rules
active(and(tt,X)) |
→ |
mark(X) |
(1) |
active(plus(N,0)) |
→ |
mark(N) |
(2) |
active(plus(N,s(M))) |
→ |
mark(s(plus(N,M))) |
(3) |
active(x(N,0)) |
→ |
mark(0) |
(4) |
active(x(N,s(M))) |
→ |
mark(plus(x(N,M),N)) |
(5) |
mark(and(X1,X2)) |
→ |
active(and(mark(X1),X2)) |
(6) |
mark(tt) |
→ |
active(tt) |
(7) |
mark(plus(X1,X2)) |
→ |
active(plus(mark(X1),mark(X2))) |
(8) |
mark(0) |
→ |
active(0) |
(9) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(10) |
mark(x(X1,X2)) |
→ |
active(x(mark(X1),mark(X2))) |
(11) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(12) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(13) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(14) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(15) |
plus(mark(X1),X2) |
→ |
plus(X1,X2) |
(16) |
plus(X1,mark(X2)) |
→ |
plus(X1,X2) |
(17) |
plus(active(X1),X2) |
→ |
plus(X1,X2) |
(18) |
plus(X1,active(X2)) |
→ |
plus(X1,X2) |
(19) |
s(mark(X)) |
→ |
s(X) |
(20) |
s(active(X)) |
→ |
s(X) |
(21) |
x(mark(X1),X2) |
→ |
x(X1,X2) |
(22) |
x(X1,mark(X2)) |
→ |
x(X1,X2) |
(23) |
x(active(X1),X2) |
→ |
x(X1,X2) |
(24) |
x(X1,active(X2)) |
→ |
x(X1,X2) |
(25) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(x(X1,X2)) |
→ |
mark#(X2) |
(47) |
active#(x(N,s(M))) |
→ |
mark#(plus(x(N,M),N)) |
(34) |
active#(plus(N,s(M))) |
→ |
mark#(s(plus(N,M))) |
(30) |
mark#(s(X)) |
→ |
mark#(X) |
(44) |
mark#(x(X1,X2)) |
→ |
mark#(X1) |
(48) |
mark#(plus(X1,X2)) |
→ |
mark#(X1) |
(40) |
mark#(plus(X1,X2)) |
→ |
mark#(X2) |
(39) |
active#(and(tt,X)) |
→ |
mark#(X) |
(26) |
mark#(and(X1,X2)) |
→ |
mark#(X1) |
(35) |
active#(plus(N,0)) |
→ |
mark#(N) |
(27) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(55) |
plus#(X1,active(X2)) |
→ |
plus#(X1,X2) |
(58) |
plus#(active(X1),X2) |
→ |
plus#(X1,X2) |
(57) |
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(56) |
1.1.2 Subterm Criterion Processor
We use the projection
and remove the pairs:
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(55) |
plus#(active(X1),X2) |
→ |
plus#(X1,X2) |
(57) |
1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
plus#(X1,active(X2)) |
→ |
plus#(X1,X2) |
(58) |
|
2 |
> |
2 |
1 |
≥ |
1 |
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(56) |
|
2 |
> |
2 |
1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
s#(mark(X)) |
→ |
s#(X) |
(59) |
s#(active(X)) |
→ |
s#(X) |
(60) |
1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(mark(X)) |
→ |
s#(X) |
(59) |
|
1 |
> |
1 |
s#(active(X)) |
→ |
s#(X) |
(60) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
x#(mark(X1),X2) |
→ |
x#(X1,X2) |
(61) |
x#(X1,active(X2)) |
→ |
x#(X1,X2) |
(64) |
x#(active(X1),X2) |
→ |
x#(X1,X2) |
(63) |
x#(X1,mark(X2)) |
→ |
x#(X1,X2) |
(62) |
1.1.4 Subterm Criterion Processor
We use the projection
and remove the pairs:
x#(mark(X1),X2) |
→ |
x#(X1,X2) |
(61) |
x#(active(X1),X2) |
→ |
x#(X1,X2) |
(63) |
1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
x#(X1,active(X2)) |
→ |
x#(X1,X2) |
(64) |
|
2 |
> |
2 |
1 |
≥ |
1 |
x#(X1,mark(X2)) |
→ |
x#(X1,X2) |
(62) |
|
2 |
> |
2 |
1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(51) |
and#(X1,active(X2)) |
→ |
and#(X1,X2) |
(54) |
and#(active(X1),X2) |
→ |
and#(X1,X2) |
(53) |
and#(X1,mark(X2)) |
→ |
and#(X1,X2) |
(52) |
1.1.5 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(51) |
|
2 |
≥ |
2 |
1 |
> |
1 |
and#(X1,active(X2)) |
→ |
and#(X1,X2) |
(54) |
|
2 |
> |
2 |
1 |
≥ |
1 |
and#(active(X1),X2) |
→ |
and#(X1,X2) |
(53) |
|
2 |
≥ |
2 |
1 |
> |
1 |
and#(X1,mark(X2)) |
→ |
and#(X1,X2) |
(52) |
|
2 |
> |
2 |
1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.