The rewrite relation of the following TRS is considered.
The dependency pairs are split into 14
components.
-
The
1st
component contains the
pair
|
prod#(app(l1,l2)) |
→ |
prod#(l1) |
(116) |
|
prod#(cons(x,l)) |
→ |
prod#(l) |
(114) |
|
prod#(app(l1,l2)) |
→ |
prod#(l2) |
(117) |
1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [app(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [prod#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
prod#(app(l1,l2)) |
→ |
prod#(l1) |
(116) |
|
| 1 |
> |
1 |
|
prod#(cons(x,l)) |
→ |
prod#(l) |
(114) |
|
| 1 |
> |
1 |
|
prod#(app(l1,l2)) |
→ |
prod#(l2) |
(117) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
|
inter#(app(l1,l2),l3) |
→ |
inter#(l2,l3) |
(123) |
|
inter#(app(l1,l2),l3) |
→ |
inter#(l1,l3) |
(122) |
|
inter#(l1,app(l2,l3)) |
→ |
inter#(l1,l2) |
(125) |
|
inter#(l1,app(l2,l3)) |
→ |
inter#(l1,l3) |
(126) |
|
inter#(cons(x,l1),l2) |
→ |
ifinter#(mem(x,l2),x,l1,l2) |
(127) |
|
ifinter#(true,x,l1,l2) |
→ |
inter#(l1,l2) |
(131) |
|
inter#(l1,cons(x,l2)) |
→ |
ifinter#(mem(x,l1),x,l2,l1) |
(129) |
|
ifinter#(false,x,l1,l2) |
→ |
inter#(l1,l2) |
(132) |
1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
inter#(app(l1,l2),l3) |
→ |
inter#(l2,l3) |
(123) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
|
inter#(app(l1,l2),l3) |
→ |
inter#(l1,l3) |
(122) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
|
inter#(l1,app(l2,l3)) |
→ |
inter#(l1,l2) |
(125) |
|
|
| 1 |
≥ |
1 |
| 2 |
> |
2 |
|
inter#(l1,app(l2,l3)) |
→ |
inter#(l1,l3) |
(126) |
|
|
| 1 |
≥ |
1 |
| 2 |
> |
2 |
|
inter#(cons(x,l1),l2) |
→ |
ifinter#(mem(x,l2),x,l1,l2) |
(127) |
|
|
| 1 |
> |
2 |
| 1 |
> |
3 |
| 2 |
≥ |
4 |
|
inter#(l1,cons(x,l2)) |
→ |
ifinter#(mem(x,l1),x,l2,l1) |
(129) |
|
|
| 2 |
> |
2 |
| 2 |
> |
3 |
| 1 |
≥ |
4 |
|
ifinter#(true,x,l1,l2) |
→ |
inter#(l1,l2) |
(131) |
|
|
| 3 |
≥ |
1 |
| 4 |
≥ |
2 |
|
ifinter#(false,x,l1,l2) |
→ |
inter#(l1,l2) |
(132) |
|
|
| 3 |
≥ |
1 |
| 4 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
|
log'#(0(x)) |
→ |
log'#(x) |
(95) |
|
log'#(1(x)) |
→ |
log'#(x) |
(91) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [0(x1)] |
= |
1 · x1
|
| [1(x1)] |
= |
1 · x1
|
| [log'#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
log'#(0(x)) |
→ |
log'#(x) |
(95) |
|
| 1 |
> |
1 |
|
log'#(1(x)) |
→ |
log'#(x) |
(91) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
|
*#(1(x),y) |
→ |
*#(x,y) |
(100) |
|
*#(0(x),y) |
→ |
*#(x,y) |
(97) |
|
*#(*(x,y),z) |
→ |
*#(x,*(y,z)) |
(101) |
|
*#(*(x,y),z) |
→ |
*#(y,z) |
(102) |
|
*#(x,+(y,z)) |
→ |
*#(x,y) |
(104) |
|
*#(x,+(y,z)) |
→ |
*#(x,z) |
(105) |
1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
*#(1(x),y) |
→ |
*#(x,y) |
(100) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
|
*#(0(x),y) |
→ |
*#(x,y) |
(97) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
|
*#(*(x,y),z) |
→ |
*#(x,*(y,z)) |
(101) |
|
| 1 |
> |
1 |
|
*#(*(x,y),z) |
→ |
*#(y,z) |
(102) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
|
*#(x,+(y,z)) |
→ |
*#(x,y) |
(104) |
|
|
| 1 |
≥ |
1 |
| 2 |
> |
2 |
|
*#(x,+(y,z)) |
→ |
*#(x,z) |
(105) |
|
|
| 1 |
≥ |
1 |
| 2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
|
sum#(app(l1,l2)) |
→ |
sum#(l1) |
(111) |
|
sum#(cons(x,l)) |
→ |
sum#(l) |
(109) |
|
sum#(app(l1,l2)) |
→ |
sum#(l2) |
(112) |
1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [app(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [sum#(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.
|
sum#(app(l1,l2)) |
→ |
sum#(l1) |
(111) |
|
| 1 |
> |
1 |
|
sum#(cons(x,l)) |
→ |
sum#(l) |
(109) |
|
| 1 |
> |
1 |
|
sum#(app(l1,l2)) |
→ |
sum#(l2) |
(112) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
|
mem#(x,cons(y,l)) |
→ |
mem#(x,l) |
(120) |
1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [mem#(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.
1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
mem#(x,cons(y,l)) |
→ |
mem#(x,l) |
(120) |
|
|
| 1 |
≥ |
1 |
| 2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
|
ge#(0(x),1(y)) |
→ |
ge#(y,x) |
(84) |
|
ge#(0(x),0(y)) |
→ |
ge#(x,y) |
(82) |
|
ge#(1(x),0(y)) |
→ |
ge#(x,y) |
(85) |
|
ge#(1(x),1(y)) |
→ |
ge#(x,y) |
(86) |
1.1.7 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [0(x1)] |
= |
1 · x1
|
| [1(x1)] |
= |
1 · x1
|
| [ge#(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.
1.1.7.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
ge#(0(x),1(y)) |
→ |
ge#(y,x) |
(84) |
|
|
| 2 |
> |
1 |
| 1 |
> |
2 |
|
ge#(0(x),0(y)) |
→ |
ge#(x,y) |
(82) |
|
|
| 1 |
> |
1 |
| 2 |
> |
2 |
|
ge#(1(x),0(y)) |
→ |
ge#(x,y) |
(85) |
|
|
| 1 |
> |
1 |
| 2 |
> |
2 |
|
ge#(1(x),1(y)) |
→ |
ge#(x,y) |
(86) |
|
|
| 1 |
> |
1 |
| 2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
|
+#(0(x),1(y)) |
→ |
+#(x,y) |
(64) |
|
+#(0(x),0(y)) |
→ |
+#(x,y) |
(63) |
|
+#(1(x),0(y)) |
→ |
+#(x,y) |
(65) |
|
+#(1(x),1(y)) |
→ |
+#(+(x,y),1(#)) |
(67) |
|
+#(1(x),1(y)) |
→ |
+#(x,y) |
(68) |
|
+#(+(x,y),z) |
→ |
+#(x,+(y,z)) |
(69) |
|
+#(+(x,y),z) |
→ |
+#(y,z) |
(70) |
1.1.8 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [+(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [#] |
= |
0 |
| [0(x1)] |
= |
1 · x1
|
| [1(x1)] |
= |
1 · x1
|
| [+#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
together with the usable
rules
|
+(x,#) |
→ |
x |
(2) |
|
+(#,x) |
→ |
x |
(3) |
|
+(0(x),0(y)) |
→ |
0(+(x,y)) |
(4) |
|
+(0(x),1(y)) |
→ |
1(+(x,y)) |
(5) |
|
+(1(x),0(y)) |
→ |
1(+(x,y)) |
(6) |
|
+(1(x),1(y)) |
→ |
0(+(+(x,y),1(#))) |
(7) |
|
+(+(x,y),z) |
→ |
+(x,+(y,z)) |
(8) |
|
0(#) |
→ |
# |
(1) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.8.1 Monotonic Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(#) |
= |
4 |
|
weight(#) |
= |
2 |
|
|
|
| prec(0) |
= |
1 |
|
weight(0) |
= |
1 |
|
|
|
| prec(1) |
= |
2 |
|
weight(1) |
= |
3 |
|
|
|
| prec(+) |
= |
3 |
|
weight(+) |
= |
0 |
|
|
|
| prec(+#) |
= |
0 |
|
weight(+#) |
= |
0 |
|
|
|
the
pairs
|
+#(0(x),1(y)) |
→ |
+#(x,y) |
(64) |
|
+#(0(x),0(y)) |
→ |
+#(x,y) |
(63) |
|
+#(1(x),0(y)) |
→ |
+#(x,y) |
(65) |
|
+#(1(x),1(y)) |
→ |
+#(+(x,y),1(#)) |
(67) |
|
+#(1(x),1(y)) |
→ |
+#(x,y) |
(68) |
|
+#(+(x,y),z) |
→ |
+#(x,+(y,z)) |
(69) |
|
+#(+(x,y),z) |
→ |
+#(y,z) |
(70) |
and
the
rules
|
+(x,#) |
→ |
x |
(2) |
|
+(#,x) |
→ |
x |
(3) |
|
+(0(x),0(y)) |
→ |
0(+(x,y)) |
(4) |
|
+(0(x),1(y)) |
→ |
1(+(x,y)) |
(5) |
|
+(1(x),0(y)) |
→ |
1(+(x,y)) |
(6) |
|
+(1(x),1(y)) |
→ |
0(+(+(x,y),1(#))) |
(7) |
|
+(+(x,y),z) |
→ |
+(x,+(y,z)) |
(8) |
|
0(#) |
→ |
# |
(1) |
could be deleted.
1.1.8.1.1 P is empty
There are no pairs anymore.
-
The
9th
component contains the
pair
|
-#(0(x),1(y)) |
→ |
-#(-(x,y),1(#)) |
(73) |
|
-#(0(x),1(y)) |
→ |
-#(x,y) |
(74) |
|
-#(0(x),0(y)) |
→ |
-#(x,y) |
(72) |
|
-#(1(x),0(y)) |
→ |
-#(x,y) |
(75) |
|
-#(1(x),1(y)) |
→ |
-#(x,y) |
(77) |
1.1.9 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [-(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [#] |
= |
0 |
| [0(x1)] |
= |
1 · x1
|
| [1(x1)] |
= |
1 · x1
|
| [-#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
together with the usable
rules
|
-(#,x) |
→ |
# |
(9) |
|
-(x,#) |
→ |
x |
(10) |
|
-(0(x),0(y)) |
→ |
0(-(x,y)) |
(11) |
|
-(0(x),1(y)) |
→ |
1(-(-(x,y),1(#))) |
(12) |
|
-(1(x),0(y)) |
→ |
1(-(x,y)) |
(13) |
|
-(1(x),1(y)) |
→ |
0(-(x,y)) |
(14) |
|
0(#) |
→ |
# |
(1) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.9.1 Monotonic Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(#) |
= |
0 |
|
weight(#) |
= |
2 |
|
|
|
| prec(0) |
= |
2 |
|
weight(0) |
= |
5 |
|
|
|
| prec(1) |
= |
1 |
|
weight(1) |
= |
3 |
|
|
|
| prec(-) |
= |
4 |
|
weight(-) |
= |
0 |
|
|
|
| prec(-#) |
= |
3 |
|
weight(-#) |
= |
0 |
|
|
|
the
pairs
|
-#(0(x),1(y)) |
→ |
-#(-(x,y),1(#)) |
(73) |
|
-#(0(x),1(y)) |
→ |
-#(x,y) |
(74) |
|
-#(0(x),0(y)) |
→ |
-#(x,y) |
(72) |
|
-#(1(x),0(y)) |
→ |
-#(x,y) |
(75) |
|
-#(1(x),1(y)) |
→ |
-#(x,y) |
(77) |
and
the
rules
|
-(#,x) |
→ |
# |
(9) |
|
-(x,#) |
→ |
x |
(10) |
|
-(0(x),0(y)) |
→ |
0(-(x,y)) |
(11) |
|
-(0(x),1(y)) |
→ |
1(-(-(x,y),1(#))) |
(12) |
|
-(1(x),0(y)) |
→ |
1(-(x,y)) |
(13) |
|
-(1(x),1(y)) |
→ |
0(-(x,y)) |
(14) |
|
0(#) |
→ |
# |
(1) |
could be deleted.
1.1.9.1.1 P is empty
There are no pairs anymore.
-
The
10th
component contains the
pair
|
eq#(0(x),0(y)) |
→ |
eq#(x,y) |
(81) |
|
eq#(1(x),1(y)) |
→ |
eq#(x,y) |
(80) |
1.1.10 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [0(x1)] |
= |
1 · x1
|
| [1(x1)] |
= |
1 · x1
|
| [eq#(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.
1.1.10.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
eq#(0(x),0(y)) |
→ |
eq#(x,y) |
(81) |
|
|
| 1 |
> |
1 |
| 2 |
> |
2 |
|
eq#(1(x),1(y)) |
→ |
eq#(x,y) |
(80) |
|
|
| 1 |
> |
1 |
| 2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
11th
component contains the
pair
|
ge#(#,0(x)) |
→ |
ge#(#,x) |
(87) |
1.1.11 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [#] |
= |
0 |
| [0(x1)] |
= |
1 · x1
|
| [ge#(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.
1.1.11.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
ge#(#,0(x)) |
→ |
ge#(#,x) |
(87) |
|
|
| 1 |
≥ |
1 |
| 2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
12th
component contains the
pair
|
eq#(#,0(y)) |
→ |
eq#(#,y) |
(78) |
1.1.12 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [#] |
= |
0 |
| [0(x1)] |
= |
1 · x1
|
| [eq#(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.
1.1.12.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
eq#(#,0(y)) |
→ |
eq#(#,y) |
(78) |
|
|
| 1 |
≥ |
1 |
| 2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
13th
component contains the
pair
|
eq#(0(x),#) |
→ |
eq#(x,#) |
(79) |
1.1.13 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [0(x1)] |
= |
1 · x1
|
| [#] |
= |
0 |
| [eq#(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.
1.1.13.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
eq#(0(x),#) |
→ |
eq#(x,#) |
(79) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
14th
component contains the
pair
|
app#(cons(x,l1),l2) |
→ |
app#(l1,l2) |
(106) |
1.1.14 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [app#(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.
1.1.14.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
app#(cons(x,l1),l2) |
→ |
app#(l1,l2) |
(106) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.