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) |
→ |
# |
(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) |
|
not(true) |
→ |
false |
(15) |
|
not(false) |
→ |
true |
(16) |
|
if(true,x,y) |
→ |
x |
(17) |
|
if(false,x,y) |
→ |
y |
(18) |
|
eq(#,#) |
→ |
true |
(19) |
|
eq(#,1(y)) |
→ |
false |
(20) |
|
eq(1(x),#) |
→ |
false |
(21) |
|
eq(#,0(y)) |
→ |
eq(#,y) |
(22) |
|
eq(0(x),#) |
→ |
eq(x,#) |
(23) |
|
eq(1(x),1(y)) |
→ |
eq(x,y) |
(24) |
|
eq(0(x),1(y)) |
→ |
false |
(25) |
|
eq(1(x),0(y)) |
→ |
false |
(26) |
|
eq(0(x),0(y)) |
→ |
eq(x,y) |
(27) |
|
ge(0(x),0(y)) |
→ |
ge(x,y) |
(28) |
|
ge(0(x),1(y)) |
→ |
not(ge(y,x)) |
(29) |
|
ge(1(x),0(y)) |
→ |
ge(x,y) |
(30) |
|
ge(1(x),1(y)) |
→ |
ge(x,y) |
(31) |
|
ge(x,#) |
→ |
true |
(32) |
|
ge(#,0(x)) |
→ |
ge(#,x) |
(33) |
|
ge(#,1(x)) |
→ |
false |
(34) |
|
log(x) |
→ |
-(log'(x),1(#)) |
(35) |
|
log'(#) |
→ |
# |
(36) |
|
log'(1(x)) |
→ |
+(log'(x),1(#)) |
(37) |
|
log'(0(x)) |
→ |
if(ge(x,1(#)),+(log'(x),1(#)),#) |
(38) |
|
*(#,x) |
→ |
# |
(39) |
|
*(0(x),y) |
→ |
0(*(x,y)) |
(40) |
|
*(1(x),y) |
→ |
+(0(*(x,y)),y) |
(41) |
|
*(*(x,y),z) |
→ |
*(x,*(y,z)) |
(42) |
|
*(x,+(y,z)) |
→ |
+(*(x,y),*(x,z)) |
(43) |
|
app(nil,l) |
→ |
l |
(44) |
|
app(cons(x,l1),l2) |
→ |
cons(x,app(l1,l2)) |
(45) |
|
sum(nil) |
→ |
0(#) |
(46) |
|
sum(cons(x,l)) |
→ |
+(x,sum(l)) |
(47) |
|
sum(app(l1,l2)) |
→ |
+(sum(l1),sum(l2)) |
(48) |
|
prod(nil) |
→ |
1(#) |
(49) |
|
prod(cons(x,l)) |
→ |
*(x,prod(l)) |
(50) |
|
prod(app(l1,l2)) |
→ |
*(prod(l1),prod(l2)) |
(51) |
|
mem(x,nil) |
→ |
false |
(52) |
|
mem(x,cons(y,l)) |
→ |
if(eq(x,y),true,mem(x,l)) |
(53) |
|
inter(x,nil) |
→ |
nil |
(54) |
|
inter(nil,x) |
→ |
nil |
(55) |
|
inter(app(l1,l2),l3) |
→ |
app(inter(l1,l3),inter(l2,l3)) |
(56) |
|
inter(l1,app(l2,l3)) |
→ |
app(inter(l1,l2),inter(l1,l3)) |
(57) |
|
inter(cons(x,l1),l2) |
→ |
ifinter(mem(x,l2),x,l1,l2) |
(58) |
|
inter(l1,cons(x,l2)) |
→ |
ifinter(mem(x,l1),x,l2,l1) |
(59) |
|
ifinter(true,x,l1,l2) |
→ |
cons(x,inter(l1,l2)) |
(60) |
|
ifinter(false,x,l1,l2) |
→ |
inter(l1,l2) |
(61) |
|
+#(0(x),0(y)) |
→ |
0#(+(x,y)) |
(62) |
|
+#(0(x),0(y)) |
→ |
+#(x,y) |
(63) |
|
+#(0(x),1(y)) |
→ |
+#(x,y) |
(64) |
|
+#(1(x),0(y)) |
→ |
+#(x,y) |
(65) |
|
+#(1(x),1(y)) |
→ |
0#(+(+(x,y),1(#))) |
(66) |
|
+#(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) |
|
-#(0(x),0(y)) |
→ |
0#(-(x,y)) |
(71) |
|
-#(0(x),0(y)) |
→ |
-#(x,y) |
(72) |
|
-#(0(x),1(y)) |
→ |
-#(-(x,y),1(#)) |
(73) |
|
-#(0(x),1(y)) |
→ |
-#(x,y) |
(74) |
|
-#(1(x),0(y)) |
→ |
-#(x,y) |
(75) |
|
-#(1(x),1(y)) |
→ |
0#(-(x,y)) |
(76) |
|
-#(1(x),1(y)) |
→ |
-#(x,y) |
(77) |
|
eq#(#,0(y)) |
→ |
eq#(#,y) |
(78) |
|
eq#(0(x),#) |
→ |
eq#(x,#) |
(79) |
|
eq#(1(x),1(y)) |
→ |
eq#(x,y) |
(80) |
|
eq#(0(x),0(y)) |
→ |
eq#(x,y) |
(81) |
|
ge#(0(x),0(y)) |
→ |
ge#(x,y) |
(82) |
|
ge#(0(x),1(y)) |
→ |
not#(ge(y,x)) |
(83) |
|
ge#(0(x),1(y)) |
→ |
ge#(y,x) |
(84) |
|
ge#(1(x),0(y)) |
→ |
ge#(x,y) |
(85) |
|
ge#(1(x),1(y)) |
→ |
ge#(x,y) |
(86) |
|
ge#(#,0(x)) |
→ |
ge#(#,x) |
(87) |
|
log#(x) |
→ |
-#(log'(x),1(#)) |
(88) |
|
log#(x) |
→ |
log'#(x) |
(89) |
|
log'#(1(x)) |
→ |
+#(log'(x),1(#)) |
(90) |
|
log'#(1(x)) |
→ |
log'#(x) |
(91) |
|
log'#(0(x)) |
→ |
if#(ge(x,1(#)),+(log'(x),1(#)),#) |
(92) |
|
log'#(0(x)) |
→ |
ge#(x,1(#)) |
(93) |
|
log'#(0(x)) |
→ |
+#(log'(x),1(#)) |
(94) |
|
log'#(0(x)) |
→ |
log'#(x) |
(95) |
|
*#(0(x),y) |
→ |
0#(*(x,y)) |
(96) |
|
*#(0(x),y) |
→ |
*#(x,y) |
(97) |
|
*#(1(x),y) |
→ |
+#(0(*(x,y)),y) |
(98) |
|
*#(1(x),y) |
→ |
0#(*(x,y)) |
(99) |
|
*#(1(x),y) |
→ |
*#(x,y) |
(100) |
|
*#(*(x,y),z) |
→ |
*#(x,*(y,z)) |
(101) |
|
*#(*(x,y),z) |
→ |
*#(y,z) |
(102) |
|
*#(x,+(y,z)) |
→ |
+#(*(x,y),*(x,z)) |
(103) |
|
*#(x,+(y,z)) |
→ |
*#(x,y) |
(104) |
|
*#(x,+(y,z)) |
→ |
*#(x,z) |
(105) |
|
app#(cons(x,l1),l2) |
→ |
app#(l1,l2) |
(106) |
|
sum#(nil) |
→ |
0#(#) |
(107) |
|
sum#(cons(x,l)) |
→ |
+#(x,sum(l)) |
(108) |
|
sum#(cons(x,l)) |
→ |
sum#(l) |
(109) |
|
sum#(app(l1,l2)) |
→ |
+#(sum(l1),sum(l2)) |
(110) |
|
sum#(app(l1,l2)) |
→ |
sum#(l1) |
(111) |
|
sum#(app(l1,l2)) |
→ |
sum#(l2) |
(112) |
|
prod#(cons(x,l)) |
→ |
*#(x,prod(l)) |
(113) |
|
prod#(cons(x,l)) |
→ |
prod#(l) |
(114) |
|
prod#(app(l1,l2)) |
→ |
*#(prod(l1),prod(l2)) |
(115) |
|
prod#(app(l1,l2)) |
→ |
prod#(l1) |
(116) |
|
prod#(app(l1,l2)) |
→ |
prod#(l2) |
(117) |
|
mem#(x,cons(y,l)) |
→ |
if#(eq(x,y),true,mem(x,l)) |
(118) |
|
mem#(x,cons(y,l)) |
→ |
eq#(x,y) |
(119) |
|
mem#(x,cons(y,l)) |
→ |
mem#(x,l) |
(120) |
|
inter#(app(l1,l2),l3) |
→ |
app#(inter(l1,l3),inter(l2,l3)) |
(121) |
|
inter#(app(l1,l2),l3) |
→ |
inter#(l1,l3) |
(122) |
|
inter#(app(l1,l2),l3) |
→ |
inter#(l2,l3) |
(123) |
|
inter#(l1,app(l2,l3)) |
→ |
app#(inter(l1,l2),inter(l1,l3)) |
(124) |
|
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) |
|
inter#(cons(x,l1),l2) |
→ |
mem#(x,l2) |
(128) |
|
inter#(l1,cons(x,l2)) |
→ |
ifinter#(mem(x,l1),x,l2,l1) |
(129) |
|
inter#(l1,cons(x,l2)) |
→ |
mem#(x,l1) |
(130) |
|
ifinter#(true,x,l1,l2) |
→ |
inter#(l1,l2) |
(131) |
|
ifinter#(false,x,l1,l2) |
→ |
inter#(l1,l2) |
(132) |
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.