The rewrite relation of the following TRS is considered.
active#(and(true,X)) |
→ |
mark#(X) |
(47) |
active#(and(false,Y)) |
→ |
mark#(false) |
(48) |
active#(if(true,X,Y)) |
→ |
mark#(X) |
(49) |
active#(if(false,X,Y)) |
→ |
mark#(Y) |
(50) |
active#(add(0,X)) |
→ |
mark#(X) |
(51) |
active#(add(s(X),Y)) |
→ |
mark#(s(add(X,Y))) |
(52) |
active#(add(s(X),Y)) |
→ |
s#(add(X,Y)) |
(53) |
active#(add(s(X),Y)) |
→ |
add#(X,Y) |
(54) |
active#(first(0,X)) |
→ |
mark#(nil) |
(55) |
active#(first(s(X),cons(Y,Z))) |
→ |
mark#(cons(Y,first(X,Z))) |
(56) |
active#(first(s(X),cons(Y,Z))) |
→ |
cons#(Y,first(X,Z)) |
(57) |
active#(first(s(X),cons(Y,Z))) |
→ |
first#(X,Z) |
(58) |
active#(from(X)) |
→ |
mark#(cons(X,from(s(X)))) |
(59) |
active#(from(X)) |
→ |
cons#(X,from(s(X))) |
(60) |
active#(from(X)) |
→ |
from#(s(X)) |
(61) |
active#(from(X)) |
→ |
s#(X) |
(62) |
mark#(and(X1,X2)) |
→ |
active#(and(mark(X1),X2)) |
(63) |
mark#(and(X1,X2)) |
→ |
and#(mark(X1),X2) |
(64) |
mark#(and(X1,X2)) |
→ |
mark#(X1) |
(65) |
mark#(true) |
→ |
active#(true) |
(66) |
mark#(false) |
→ |
active#(false) |
(67) |
mark#(if(X1,X2,X3)) |
→ |
active#(if(mark(X1),X2,X3)) |
(68) |
mark#(if(X1,X2,X3)) |
→ |
if#(mark(X1),X2,X3) |
(69) |
mark#(if(X1,X2,X3)) |
→ |
mark#(X1) |
(70) |
mark#(add(X1,X2)) |
→ |
active#(add(mark(X1),X2)) |
(71) |
mark#(add(X1,X2)) |
→ |
add#(mark(X1),X2) |
(72) |
mark#(add(X1,X2)) |
→ |
mark#(X1) |
(73) |
mark#(0) |
→ |
active#(0) |
(74) |
mark#(s(X)) |
→ |
active#(s(X)) |
(75) |
mark#(first(X1,X2)) |
→ |
active#(first(mark(X1),mark(X2))) |
(76) |
mark#(first(X1,X2)) |
→ |
first#(mark(X1),mark(X2)) |
(77) |
mark#(first(X1,X2)) |
→ |
mark#(X1) |
(78) |
mark#(first(X1,X2)) |
→ |
mark#(X2) |
(79) |
mark#(nil) |
→ |
active#(nil) |
(80) |
mark#(cons(X1,X2)) |
→ |
active#(cons(X1,X2)) |
(81) |
mark#(from(X)) |
→ |
active#(from(X)) |
(82) |
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(83) |
and#(X1,mark(X2)) |
→ |
and#(X1,X2) |
(84) |
and#(active(X1),X2) |
→ |
and#(X1,X2) |
(85) |
and#(X1,active(X2)) |
→ |
and#(X1,X2) |
(86) |
if#(mark(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(87) |
if#(X1,mark(X2),X3) |
→ |
if#(X1,X2,X3) |
(88) |
if#(X1,X2,mark(X3)) |
→ |
if#(X1,X2,X3) |
(89) |
if#(active(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(90) |
if#(X1,active(X2),X3) |
→ |
if#(X1,X2,X3) |
(91) |
if#(X1,X2,active(X3)) |
→ |
if#(X1,X2,X3) |
(92) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(93) |
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(94) |
add#(active(X1),X2) |
→ |
add#(X1,X2) |
(95) |
add#(X1,active(X2)) |
→ |
add#(X1,X2) |
(96) |
s#(mark(X)) |
→ |
s#(X) |
(97) |
s#(active(X)) |
→ |
s#(X) |
(98) |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(99) |
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(100) |
first#(active(X1),X2) |
→ |
first#(X1,X2) |
(101) |
first#(X1,active(X2)) |
→ |
first#(X1,X2) |
(102) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(103) |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(104) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(105) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(106) |
from#(mark(X)) |
→ |
from#(X) |
(107) |
from#(active(X)) |
→ |
from#(X) |
(108) |
The dependency pairs are split into 8
components.
-
The
1st
component contains the
pair
mark#(and(X1,X2)) |
→ |
active#(and(mark(X1),X2)) |
(63) |
active#(and(true,X)) |
→ |
mark#(X) |
(47) |
mark#(and(X1,X2)) |
→ |
mark#(X1) |
(65) |
mark#(if(X1,X2,X3)) |
→ |
active#(if(mark(X1),X2,X3)) |
(68) |
active#(if(true,X,Y)) |
→ |
mark#(X) |
(49) |
mark#(if(X1,X2,X3)) |
→ |
mark#(X1) |
(70) |
mark#(add(X1,X2)) |
→ |
active#(add(mark(X1),X2)) |
(71) |
active#(if(false,X,Y)) |
→ |
mark#(Y) |
(50) |
mark#(add(X1,X2)) |
→ |
mark#(X1) |
(73) |
mark#(s(X)) |
→ |
active#(s(X)) |
(75) |
active#(add(0,X)) |
→ |
mark#(X) |
(51) |
mark#(first(X1,X2)) |
→ |
active#(first(mark(X1),mark(X2))) |
(76) |
active#(add(s(X),Y)) |
→ |
mark#(s(add(X,Y))) |
(52) |
mark#(first(X1,X2)) |
→ |
mark#(X1) |
(78) |
mark#(first(X1,X2)) |
→ |
mark#(X2) |
(79) |
mark#(cons(X1,X2)) |
→ |
active#(cons(X1,X2)) |
(81) |
active#(first(s(X),cons(Y,Z))) |
→ |
mark#(cons(Y,first(X,Z))) |
(56) |
mark#(from(X)) |
→ |
active#(from(X)) |
(82) |
active#(from(X)) |
→ |
mark#(cons(X,from(s(X)))) |
(59) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark#(x1)] |
= |
1 |
[and(x1, x2)] |
= |
1 |
[active#(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
0 |
[true] |
= |
0 |
[if(x1, x2, x3)] |
= |
1 |
[add(x1, x2)] |
= |
1 |
[false] |
= |
0 |
[s(x1)] |
= |
0 |
[0] |
= |
0 |
[first(x1, x2)] |
= |
1 |
[cons(x1, x2)] |
= |
0 |
[from(x1)] |
= |
1 |
[active(x1)] |
= |
0 |
[nil] |
= |
0 |
together with the usable
rules
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(22) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(21) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(23) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(24) |
if(X1,mark(X2),X3) |
→ |
if(X1,X2,X3) |
(26) |
if(mark(X1),X2,X3) |
→ |
if(X1,X2,X3) |
(25) |
if(X1,X2,mark(X3)) |
→ |
if(X1,X2,X3) |
(27) |
if(active(X1),X2,X3) |
→ |
if(X1,X2,X3) |
(28) |
if(X1,active(X2),X3) |
→ |
if(X1,X2,X3) |
(29) |
if(X1,X2,active(X3)) |
→ |
if(X1,X2,X3) |
(30) |
add(X1,mark(X2)) |
→ |
add(X1,X2) |
(32) |
add(mark(X1),X2) |
→ |
add(X1,X2) |
(31) |
add(active(X1),X2) |
→ |
add(X1,X2) |
(33) |
add(X1,active(X2)) |
→ |
add(X1,X2) |
(34) |
s(active(X)) |
→ |
s(X) |
(36) |
s(mark(X)) |
→ |
s(X) |
(35) |
first(X1,mark(X2)) |
→ |
first(X1,X2) |
(38) |
first(mark(X1),X2) |
→ |
first(X1,X2) |
(37) |
first(active(X1),X2) |
→ |
first(X1,X2) |
(39) |
first(X1,active(X2)) |
→ |
first(X1,X2) |
(40) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(42) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(41) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(43) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(44) |
from(active(X)) |
→ |
from(X) |
(46) |
from(mark(X)) |
→ |
from(X) |
(45) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(s(X)) |
→ |
active#(s(X)) |
(75) |
mark#(cons(X1,X2)) |
→ |
active#(cons(X1,X2)) |
(81) |
could be deleted.
1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[mark#(x1)] |
= |
1 · x1
|
[and(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[active#(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[true] |
= |
1 |
[if(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
[add(x1, x2)] |
= |
1 + 1 · x1 + 1 · x2
|
[false] |
= |
0 |
[0] |
= |
1 |
[first(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[s(x1)] |
= |
0 |
[cons(x1, x2)] |
= |
0 |
[from(x1)] |
= |
0 |
[active(x1)] |
= |
1 · x1
|
[nil] |
= |
0 |
the
pairs
active#(and(true,X)) |
→ |
mark#(X) |
(47) |
active#(if(true,X,Y)) |
→ |
mark#(X) |
(49) |
mark#(add(X1,X2)) |
→ |
mark#(X1) |
(73) |
active#(add(0,X)) |
→ |
mark#(X) |
(51) |
active#(add(s(X),Y)) |
→ |
mark#(s(add(X,Y))) |
(52) |
could be deleted.
1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark#(x1)] |
= |
1 |
[and(x1, x2)] |
= |
0 |
[active#(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
0 |
[if(x1, x2, x3)] |
= |
1 |
[add(x1, x2)] |
= |
0 |
[false] |
= |
0 |
[first(x1, x2)] |
= |
1 |
[s(x1)] |
= |
0 |
[cons(x1, x2)] |
= |
1 · x1
|
[from(x1)] |
= |
1 |
[active(x1)] |
= |
0 |
[true] |
= |
0 |
[0] |
= |
0 |
[nil] |
= |
0 |
together with the usable
rules
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(22) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(21) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(23) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(24) |
if(X1,mark(X2),X3) |
→ |
if(X1,X2,X3) |
(26) |
if(mark(X1),X2,X3) |
→ |
if(X1,X2,X3) |
(25) |
if(X1,X2,mark(X3)) |
→ |
if(X1,X2,X3) |
(27) |
if(active(X1),X2,X3) |
→ |
if(X1,X2,X3) |
(28) |
if(X1,active(X2),X3) |
→ |
if(X1,X2,X3) |
(29) |
if(X1,X2,active(X3)) |
→ |
if(X1,X2,X3) |
(30) |
add(X1,mark(X2)) |
→ |
add(X1,X2) |
(32) |
add(mark(X1),X2) |
→ |
add(X1,X2) |
(31) |
add(active(X1),X2) |
→ |
add(X1,X2) |
(33) |
add(X1,active(X2)) |
→ |
add(X1,X2) |
(34) |
first(X1,mark(X2)) |
→ |
first(X1,X2) |
(38) |
first(mark(X1),X2) |
→ |
first(X1,X2) |
(37) |
first(active(X1),X2) |
→ |
first(X1,X2) |
(39) |
first(X1,active(X2)) |
→ |
first(X1,X2) |
(40) |
from(active(X)) |
→ |
from(X) |
(46) |
from(mark(X)) |
→ |
from(X) |
(45) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(and(X1,X2)) |
→ |
active#(and(mark(X1),X2)) |
(63) |
mark#(add(X1,X2)) |
→ |
active#(add(mark(X1),X2)) |
(71) |
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[mark#(x1)] |
= |
1 · x1
|
[and(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[if(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
[active#(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[false] |
= |
1 |
[first(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[s(x1)] |
= |
0 |
[cons(x1, x2)] |
= |
0 |
[from(x1)] |
= |
0 |
[active(x1)] |
= |
1 · x1
|
[true] |
= |
0 |
[add(x1, x2)] |
= |
1 · x2
|
[0] |
= |
0 |
[nil] |
= |
0 |
the
pair
active#(if(false,X,Y)) |
→ |
mark#(Y) |
(50) |
could be deleted.
1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(if) |
= |
0 |
|
weight(if) |
= |
1 |
|
|
|
prec(active#) |
= |
2 |
|
weight(active#) |
= |
1 |
|
|
|
prec(first) |
= |
3 |
|
weight(first) |
= |
1 |
|
|
|
prec(cons) |
= |
1 |
|
weight(cons) |
= |
1 |
|
|
|
prec(from) |
= |
4 |
|
weight(from) |
= |
2 |
|
|
|
in combination with the following argument filter
π(mark#) |
= |
1 |
π(and) |
= |
1 |
π(if) |
= |
[1] |
π(active#) |
= |
[] |
π(first) |
= |
[1,2] |
π(cons) |
= |
[] |
π(from) |
= |
[] |
together with the usable
rules
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(42) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(41) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(43) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(44) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(if(X1,X2,X3)) |
→ |
active#(if(mark(X1),X2,X3)) |
(68) |
mark#(if(X1,X2,X3)) |
→ |
mark#(X1) |
(70) |
mark#(first(X1,X2)) |
→ |
active#(first(mark(X1),mark(X2))) |
(76) |
mark#(first(X1,X2)) |
→ |
mark#(X1) |
(78) |
mark#(first(X1,X2)) |
→ |
mark#(X2) |
(79) |
active#(first(s(X),cons(Y,Z))) |
→ |
mark#(cons(Y,first(X,Z))) |
(56) |
mark#(from(X)) |
→ |
active#(from(X)) |
(82) |
active#(from(X)) |
→ |
mark#(cons(X,from(s(X)))) |
(59) |
could be deleted.
1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[and(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[mark#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
mark#(and(X1,X2)) |
→ |
mark#(X1) |
(65) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
and#(X1,mark(X2)) |
→ |
and#(X1,X2) |
(84) |
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(83) |
and#(active(X1),X2) |
→ |
and#(X1,X2) |
(85) |
and#(X1,active(X2)) |
→ |
and#(X1,X2) |
(86) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[and#(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.
1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
and#(X1,mark(X2)) |
→ |
and#(X1,X2) |
(84) |
|
1 |
≥ |
1 |
2 |
> |
2 |
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(83) |
|
1 |
> |
1 |
2 |
≥ |
2 |
and#(active(X1),X2) |
→ |
and#(X1,X2) |
(85) |
|
1 |
> |
1 |
2 |
≥ |
2 |
and#(X1,active(X2)) |
→ |
and#(X1,X2) |
(86) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
if#(X1,mark(X2),X3) |
→ |
if#(X1,X2,X3) |
(88) |
if#(mark(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(87) |
if#(X1,X2,mark(X3)) |
→ |
if#(X1,X2,X3) |
(89) |
if#(active(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(90) |
if#(X1,active(X2),X3) |
→ |
if#(X1,X2,X3) |
(91) |
if#(X1,X2,active(X3)) |
→ |
if#(X1,X2,X3) |
(92) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[if#(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
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.
if#(X1,mark(X2),X3) |
→ |
if#(X1,X2,X3) |
(88) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
if#(mark(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(87) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
if#(X1,X2,mark(X3)) |
→ |
if#(X1,X2,X3) |
(89) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
if#(active(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(90) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
if#(X1,active(X2),X3) |
→ |
if#(X1,X2,X3) |
(91) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
if#(X1,X2,active(X3)) |
→ |
if#(X1,X2,X3) |
(92) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(94) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(93) |
add#(active(X1),X2) |
→ |
add#(X1,X2) |
(95) |
add#(X1,active(X2)) |
→ |
add#(X1,X2) |
(96) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[add#(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.
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(94) |
|
1 |
≥ |
1 |
2 |
> |
2 |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(93) |
|
1 |
> |
1 |
2 |
≥ |
2 |
add#(active(X1),X2) |
→ |
add#(X1,X2) |
(95) |
|
1 |
> |
1 |
2 |
≥ |
2 |
add#(X1,active(X2)) |
→ |
add#(X1,X2) |
(96) |
|
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
s#(active(X)) |
→ |
s#(X) |
(98) |
s#(mark(X)) |
→ |
s#(X) |
(97) |
1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[s#(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.
s#(active(X)) |
→ |
s#(X) |
(98) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(97) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(100) |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(99) |
first#(active(X1),X2) |
→ |
first#(X1,X2) |
(101) |
first#(X1,active(X2)) |
→ |
first#(X1,X2) |
(102) |
1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[first#(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.
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(100) |
|
1 |
≥ |
1 |
2 |
> |
2 |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(99) |
|
1 |
> |
1 |
2 |
≥ |
2 |
first#(active(X1),X2) |
→ |
first#(X1,X2) |
(101) |
|
1 |
> |
1 |
2 |
≥ |
2 |
first#(X1,active(X2)) |
→ |
first#(X1,X2) |
(102) |
|
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
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(104) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(103) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(105) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(106) |
1.1.7 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[cons#(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.
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(104) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(103) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(105) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(106) |
|
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
from#(active(X)) |
→ |
from#(X) |
(108) |
from#(mark(X)) |
→ |
from#(X) |
(107) |
1.1.8 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[from#(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.
from#(active(X)) |
→ |
from#(X) |
(108) |
|
1 |
> |
1 |
from#(mark(X)) |
→ |
from#(X) |
(107) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.