The rewrite relation of the following TRS is considered.
|
0(#) |
→ |
# |
(1) |
|
+(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) |
|
-(x,#) |
→ |
x |
(9) |
|
-(#,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) |
|
not(false) |
→ |
true |
(15) |
|
not(true) |
→ |
false |
(16) |
|
and(x,true) |
→ |
x |
(17) |
|
and(x,false) |
→ |
false |
(18) |
|
if(true,x,y) |
→ |
x |
(19) |
|
if(false,x,y) |
→ |
y |
(20) |
|
ge(0(x),0(y)) |
→ |
ge(x,y) |
(21) |
|
ge(0(x),1(y)) |
→ |
not(ge(y,x)) |
(22) |
|
ge(1(x),0(y)) |
→ |
ge(x,y) |
(23) |
|
ge(1(x),1(y)) |
→ |
ge(x,y) |
(24) |
|
ge(x,#) |
→ |
true |
(25) |
|
ge(#,1(x)) |
→ |
false |
(26) |
|
ge(#,0(x)) |
→ |
ge(#,x) |
(27) |
|
val(l(x)) |
→ |
x |
(28) |
|
val(n(x,y,z)) |
→ |
x |
(29) |
|
min(l(x)) |
→ |
x |
(30) |
|
min(n(x,y,z)) |
→ |
min(y) |
(31) |
|
max(l(x)) |
→ |
x |
(32) |
|
max(n(x,y,z)) |
→ |
max(z) |
(33) |
|
bs(l(x)) |
→ |
true |
(34) |
|
bs(n(x,y,z)) |
→ |
and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) |
(35) |
|
size(l(x)) |
→ |
1(#) |
(36) |
|
size(n(x,y,z)) |
→ |
+(+(size(x),size(y)),1(#)) |
(37) |
|
wb(l(x)) |
→ |
true |
(38) |
|
wb(n(x,y,z)) |
→ |
and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z))) |
(39) |
|
+#(0(x),0(y)) |
→ |
0#(+(x,y)) |
(40) |
|
+#(0(x),0(y)) |
→ |
+#(x,y) |
(41) |
|
+#(0(x),1(y)) |
→ |
+#(x,y) |
(42) |
|
+#(1(x),0(y)) |
→ |
+#(x,y) |
(43) |
|
+#(1(x),1(y)) |
→ |
0#(+(+(x,y),1(#))) |
(44) |
|
+#(1(x),1(y)) |
→ |
+#(+(x,y),1(#)) |
(45) |
|
+#(1(x),1(y)) |
→ |
+#(x,y) |
(46) |
|
+#(x,+(y,z)) |
→ |
+#(+(x,y),z) |
(47) |
|
+#(x,+(y,z)) |
→ |
+#(x,y) |
(48) |
|
-#(0(x),0(y)) |
→ |
0#(-(x,y)) |
(49) |
|
-#(0(x),0(y)) |
→ |
-#(x,y) |
(50) |
|
-#(0(x),1(y)) |
→ |
-#(-(x,y),1(#)) |
(51) |
|
-#(0(x),1(y)) |
→ |
-#(x,y) |
(52) |
|
-#(1(x),0(y)) |
→ |
-#(x,y) |
(53) |
|
-#(1(x),1(y)) |
→ |
0#(-(x,y)) |
(54) |
|
-#(1(x),1(y)) |
→ |
-#(x,y) |
(55) |
|
ge#(0(x),0(y)) |
→ |
ge#(x,y) |
(56) |
|
ge#(0(x),1(y)) |
→ |
not#(ge(y,x)) |
(57) |
|
ge#(0(x),1(y)) |
→ |
ge#(y,x) |
(58) |
|
ge#(1(x),0(y)) |
→ |
ge#(x,y) |
(59) |
|
ge#(1(x),1(y)) |
→ |
ge#(x,y) |
(60) |
|
ge#(#,0(x)) |
→ |
ge#(#,x) |
(61) |
|
min#(n(x,y,z)) |
→ |
min#(y) |
(62) |
|
max#(n(x,y,z)) |
→ |
max#(z) |
(63) |
|
bs#(n(x,y,z)) |
→ |
and#(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) |
(64) |
|
bs#(n(x,y,z)) |
→ |
and#(ge(x,max(y)),ge(min(z),x)) |
(65) |
|
bs#(n(x,y,z)) |
→ |
ge#(x,max(y)) |
(66) |
|
bs#(n(x,y,z)) |
→ |
max#(y) |
(67) |
|
bs#(n(x,y,z)) |
→ |
ge#(min(z),x) |
(68) |
|
bs#(n(x,y,z)) |
→ |
min#(z) |
(69) |
|
bs#(n(x,y,z)) |
→ |
and#(bs(y),bs(z)) |
(70) |
|
bs#(n(x,y,z)) |
→ |
bs#(y) |
(71) |
|
bs#(n(x,y,z)) |
→ |
bs#(z) |
(72) |
|
size#(n(x,y,z)) |
→ |
+#(+(size(x),size(y)),1(#)) |
(73) |
|
size#(n(x,y,z)) |
→ |
+#(size(x),size(y)) |
(74) |
|
size#(n(x,y,z)) |
→ |
size#(x) |
(75) |
|
size#(n(x,y,z)) |
→ |
size#(y) |
(76) |
|
wb#(n(x,y,z)) |
→ |
and#(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z))) |
(77) |
|
wb#(n(x,y,z)) |
→ |
if#(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))) |
(78) |
|
wb#(n(x,y,z)) |
→ |
ge#(size(y),size(z)) |
(79) |
|
wb#(n(x,y,z)) |
→ |
size#(y) |
(80) |
|
wb#(n(x,y,z)) |
→ |
size#(z) |
(81) |
|
wb#(n(x,y,z)) |
→ |
ge#(1(#),-(size(y),size(z))) |
(82) |
|
wb#(n(x,y,z)) |
→ |
-#(size(y),size(z)) |
(83) |
|
wb#(n(x,y,z)) |
→ |
ge#(1(#),-(size(z),size(y))) |
(84) |
|
wb#(n(x,y,z)) |
→ |
-#(size(z),size(y)) |
(85) |
|
wb#(n(x,y,z)) |
→ |
and#(wb(y),wb(z)) |
(86) |
|
wb#(n(x,y,z)) |
→ |
wb#(y) |
(87) |
|
wb#(n(x,y,z)) |
→ |
wb#(z) |
(88) |
The dependency pairs are split into 9
components.
-
The
1st
component contains the
pair
|
wb#(n(x,y,z)) |
→ |
wb#(z) |
(88) |
|
wb#(n(x,y,z)) |
→ |
wb#(y) |
(87) |
1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [n(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
| [wb#(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.
|
wb#(n(x,y,z)) |
→ |
wb#(z) |
(88) |
|
| 1 |
> |
1 |
|
wb#(n(x,y,z)) |
→ |
wb#(y) |
(87) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
|
bs#(n(x,y,z)) |
→ |
bs#(z) |
(72) |
|
bs#(n(x,y,z)) |
→ |
bs#(y) |
(71) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [n(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
| [bs#(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.
|
bs#(n(x,y,z)) |
→ |
bs#(z) |
(72) |
|
| 1 |
> |
1 |
|
bs#(n(x,y,z)) |
→ |
bs#(y) |
(71) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
|
size#(n(x,y,z)) |
→ |
size#(y) |
(76) |
|
size#(n(x,y,z)) |
→ |
size#(x) |
(75) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [n(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
| [size#(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.
|
size#(n(x,y,z)) |
→ |
size#(y) |
(76) |
|
| 1 |
> |
1 |
|
size#(n(x,y,z)) |
→ |
size#(x) |
(75) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
|
ge#(0(x),1(y)) |
→ |
ge#(y,x) |
(58) |
|
ge#(0(x),0(y)) |
→ |
ge#(x,y) |
(56) |
|
ge#(1(x),0(y)) |
→ |
ge#(x,y) |
(59) |
|
ge#(1(x),1(y)) |
→ |
ge#(x,y) |
(60) |
1.1.4 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.
1.1.4.1 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) |
(58) |
|
|
| 2 |
> |
1 |
| 1 |
> |
2 |
|
ge#(0(x),0(y)) |
→ |
ge#(x,y) |
(56) |
|
|
| 1 |
> |
1 |
| 2 |
> |
2 |
|
ge#(1(x),0(y)) |
→ |
ge#(x,y) |
(59) |
|
|
| 1 |
> |
1 |
| 2 |
> |
2 |
|
ge#(1(x),1(y)) |
→ |
ge#(x,y) |
(60) |
|
|
| 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
|
+#(0(x),1(y)) |
→ |
+#(x,y) |
(42) |
|
+#(0(x),0(y)) |
→ |
+#(x,y) |
(41) |
|
+#(1(x),0(y)) |
→ |
+#(x,y) |
(43) |
|
+#(1(x),1(y)) |
→ |
+#(+(x,y),1(#)) |
(45) |
|
+#(1(x),1(y)) |
→ |
+#(x,y) |
(46) |
|
+#(x,+(y,z)) |
→ |
+#(+(x,y),z) |
(47) |
|
+#(x,+(y,z)) |
→ |
+#(x,y) |
(48) |
1.1.5 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.
1.1.5.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [+(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [#] |
= |
0 |
| [0(x1)] |
= |
2 · x1
|
| [1(x1)] |
= |
1 + 2 · x1
|
| [+#(x1, x2)] |
= |
1 · x1 + 2 · x2
|
the
pairs
|
+#(0(x),1(y)) |
→ |
+#(x,y) |
(42) |
|
+#(1(x),0(y)) |
→ |
+#(x,y) |
(43) |
|
+#(1(x),1(y)) |
→ |
+#(+(x,y),1(#)) |
(45) |
|
+#(1(x),1(y)) |
→ |
+#(x,y) |
(46) |
and
no rules
could be deleted.
1.1.5.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
+#(0(x),0(y)) |
→ |
+#(x,y) |
(41) |
|
|
| 1 |
> |
1 |
| 2 |
> |
2 |
|
+#(x,+(y,z)) |
→ |
+#(+(x,y),z) |
(47) |
|
| 2 |
> |
2 |
|
+#(x,+(y,z)) |
→ |
+#(x,y) |
(48) |
|
|
| 1 |
≥ |
1 |
| 2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
|
-#(0(x),1(y)) |
→ |
-#(-(x,y),1(#)) |
(51) |
|
-#(0(x),1(y)) |
→ |
-#(x,y) |
(52) |
|
-#(0(x),0(y)) |
→ |
-#(x,y) |
(50) |
|
-#(1(x),0(y)) |
→ |
-#(x,y) |
(53) |
|
-#(1(x),1(y)) |
→ |
-#(x,y) |
(55) |
1.1.6 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 |
(9) |
|
-(#,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.
1.1.6.1 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(#)) |
(51) |
|
-#(0(x),1(y)) |
→ |
-#(x,y) |
(52) |
|
-#(0(x),0(y)) |
→ |
-#(x,y) |
(50) |
|
-#(1(x),0(y)) |
→ |
-#(x,y) |
(53) |
|
-#(1(x),1(y)) |
→ |
-#(x,y) |
(55) |
and
the
rules
|
-(x,#) |
→ |
x |
(9) |
|
-(#,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.
1.1.6.1.1 P is empty
There are no pairs anymore.
-
The
7th
component contains the
pair
|
ge#(#,0(x)) |
→ |
ge#(#,x) |
(61) |
1.1.7 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.
1.1.7.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
ge#(#,0(x)) |
→ |
ge#(#,x) |
(61) |
|
|
| 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
|
min#(n(x,y,z)) |
→ |
min#(y) |
(62) |
1.1.8 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [n(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
| [min#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.8.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
min#(n(x,y,z)) |
→ |
min#(y) |
(62) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
9th
component contains the
pair
|
max#(n(x,y,z)) |
→ |
max#(z) |
(63) |
1.1.9 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [n(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
| [max#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.9.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
max#(n(x,y,z)) |
→ |
max#(z) |
(63) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.