The rewrite relation of the following TRS is considered.
The dependency pairs are split into 6
components.
-
The
1st
component contains the
pair
|
top#(mark(x)) |
→ |
top#(check(x)) |
(17) |
|
top#(found(x)) |
→ |
top#(active(x)) |
(31) |
|
top#(active(c)) |
→ |
top#(mark(c)) |
(16) |
1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [active(x1)] |
= |
1 · x1
|
| [f(x1)] |
= |
1 · x1
|
| [mark(x1)] |
= |
1 · x1
|
| [ok(x1)] |
= |
1 · x1
|
| [found(x1)] |
= |
1 · x1
|
| [check(x1)] |
= |
1 · x1
|
| [start(x1)] |
= |
1 · x1
|
| [match(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [X] |
= |
0 |
| [proper(x1)] |
= |
1 · x1
|
| [c] |
= |
0 |
| [top#(x1)] |
= |
1 · x1
|
together with the usable
rules
|
active(f(x)) |
→ |
mark(x) |
(1) |
|
active(f(x)) |
→ |
f(active(x)) |
(14) |
|
f(ok(x)) |
→ |
ok(f(x)) |
(10) |
|
f(found(x)) |
→ |
found(f(x)) |
(12) |
|
f(mark(x)) |
→ |
mark(f(x)) |
(15) |
|
check(f(x)) |
→ |
f(check(x)) |
(4) |
|
check(x) |
→ |
start(match(f(X),x)) |
(5) |
|
match(f(x),f(y)) |
→ |
f(match(x,y)) |
(6) |
|
start(ok(x)) |
→ |
found(x) |
(11) |
|
match(X,x) |
→ |
proper(x) |
(7) |
|
proper(c) |
→ |
ok(c) |
(8) |
|
proper(f(x)) |
→ |
f(proper(x)) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1 Reduction Pair Processor with Usable Rules
Using the
| prec(top#) |
= |
1 |
|
stat(top#) |
= |
lex
|
| prec(mark) |
= |
0 |
|
stat(mark) |
= |
lex
|
| prec(check) |
= |
0 |
|
stat(check) |
= |
lex
|
| prec(c) |
= |
1 |
|
stat(c) |
= |
lex
|
| prec(f) |
= |
0 |
|
stat(f) |
= |
lex
|
| prec(X) |
= |
1 |
|
stat(X) |
= |
lex
|
| prec(proper) |
= |
1 |
|
stat(proper) |
= |
lex
|
| π(top#) |
= |
[1] |
| π(mark) |
= |
[] |
| π(check) |
= |
[] |
| π(found) |
= |
1 |
| π(active) |
= |
1 |
| π(c) |
= |
[] |
| π(f) |
= |
[] |
| π(start) |
= |
1 |
| π(match) |
= |
1 |
| π(X) |
= |
[] |
| π(ok) |
= |
1 |
| π(proper) |
= |
[1] |
together with the usable
rules
|
check(f(x)) |
→ |
f(check(x)) |
(4) |
|
check(x) |
→ |
start(match(f(X),x)) |
(5) |
|
active(f(x)) |
→ |
mark(x) |
(1) |
|
active(f(x)) |
→ |
f(active(x)) |
(14) |
|
f(ok(x)) |
→ |
ok(f(x)) |
(10) |
|
f(found(x)) |
→ |
found(f(x)) |
(12) |
|
f(mark(x)) |
→ |
mark(f(x)) |
(15) |
|
match(f(x),f(y)) |
→ |
f(match(x,y)) |
(6) |
|
start(ok(x)) |
→ |
found(x) |
(11) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
top#(active(c)) |
→ |
top#(mark(c)) |
(16) |
could be deleted.
1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [active(x1)] |
= |
1 · x1
|
| [f(x1)] |
= |
2 · x1
|
| [mark(x1)] |
= |
2 · x1
|
| [ok(x1)] |
= |
1 · x1
|
| [found(x1)] |
= |
1 · x1
|
| [check(x1)] |
= |
2 · x1
|
| [start(x1)] |
= |
1 · x1
|
| [match(x1, x2)] |
= |
2 · x1 + 2 · x2
|
| [X] |
= |
0 |
| [proper(x1)] |
= |
2 · x1
|
| [c] |
= |
2 |
| [top#(x1)] |
= |
2 · x1
|
the
rule
could be deleted.
1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(X) |
= |
9 |
|
weight(X) |
= |
1 |
|
|
|
| prec(active) |
= |
7 |
|
weight(active) |
= |
6 |
|
|
|
| prec(f) |
= |
3 |
|
weight(f) |
= |
1 |
|
|
|
| prec(mark) |
= |
1 |
|
weight(mark) |
= |
6 |
|
|
|
| prec(ok) |
= |
2 |
|
weight(ok) |
= |
6 |
|
|
|
| prec(found) |
= |
0 |
|
weight(found) |
= |
7 |
|
|
|
| prec(check) |
= |
8 |
|
weight(check) |
= |
5 |
|
|
|
| prec(start) |
= |
6 |
|
weight(start) |
= |
1 |
|
|
|
| prec(proper) |
= |
4 |
|
weight(proper) |
= |
2 |
|
|
|
| prec(top#) |
= |
10 |
|
weight(top#) |
= |
1 |
|
|
|
| prec(match) |
= |
5 |
|
weight(match) |
= |
2 |
|
|
|
the
pairs
|
top#(mark(x)) |
→ |
top#(check(x)) |
(17) |
|
top#(found(x)) |
→ |
top#(active(x)) |
(31) |
and
the
rules
|
active(f(x)) |
→ |
mark(x) |
(1) |
|
active(f(x)) |
→ |
f(active(x)) |
(14) |
|
f(ok(x)) |
→ |
ok(f(x)) |
(10) |
|
f(found(x)) |
→ |
found(f(x)) |
(12) |
|
f(mark(x)) |
→ |
mark(f(x)) |
(15) |
|
check(f(x)) |
→ |
f(check(x)) |
(4) |
|
check(x) |
→ |
start(match(f(X),x)) |
(5) |
|
match(f(x),f(y)) |
→ |
f(match(x,y)) |
(6) |
|
start(ok(x)) |
→ |
found(x) |
(11) |
|
match(X,x) |
→ |
proper(x) |
(7) |
|
proper(f(x)) |
→ |
f(proper(x)) |
(9) |
could be deleted.
1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
check#(f(x)) |
→ |
check#(x) |
(20) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [f(x1)] |
= |
1 · x1
|
| [check#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
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.
|
check#(f(x)) |
→ |
check#(x) |
(20) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
|
match#(f(x),f(y)) |
→ |
match#(x,y) |
(25) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [f(x1)] |
= |
1 · x1
|
| [match#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
match#(f(x),f(y)) |
→ |
match#(x,y) |
(25) |
|
|
| 1 |
> |
1 |
| 2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
|
proper#(f(x)) |
→ |
proper#(x) |
(28) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [f(x1)] |
= |
1 · x1
|
| [proper#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
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.
|
proper#(f(x)) |
→ |
proper#(x) |
(28) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
|
active#(f(x)) |
→ |
active#(x) |
(34) |
1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [f(x1)] |
= |
1 · x1
|
| [active#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
active#(f(x)) |
→ |
active#(x) |
(34) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
|
f#(found(x)) |
→ |
f#(x) |
(30) |
|
f#(ok(x)) |
→ |
f#(x) |
(29) |
|
f#(mark(x)) |
→ |
f#(x) |
(35) |
1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [found(x1)] |
= |
1 · x1
|
| [ok(x1)] |
= |
1 · x1
|
| [mark(x1)] |
= |
1 · x1
|
| [f#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
f#(found(x)) |
→ |
f#(x) |
(30) |
|
| 1 |
> |
1 |
|
f#(ok(x)) |
→ |
f#(x) |
(29) |
|
| 1 |
> |
1 |
|
f#(mark(x)) |
→ |
f#(x) |
(35) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.