The rewrite relation of the following TRS is considered.
active(and(tt,T)) |
→ |
mark(T) |
(1) |
active(isNatIList(IL)) |
→ |
mark(isNatList(IL)) |
(2) |
active(isNat(0)) |
→ |
mark(tt) |
(3) |
active(isNat(s(N))) |
→ |
mark(isNat(N)) |
(4) |
active(isNat(length(L))) |
→ |
mark(isNatList(L)) |
(5) |
active(isNatIList(zeros)) |
→ |
mark(tt) |
(6) |
active(isNatIList(cons(N,IL))) |
→ |
mark(and(isNat(N),isNatIList(IL))) |
(7) |
active(isNatList(nil)) |
→ |
mark(tt) |
(8) |
active(isNatList(cons(N,L))) |
→ |
mark(and(isNat(N),isNatList(L))) |
(9) |
active(isNatList(take(N,IL))) |
→ |
mark(and(isNat(N),isNatIList(IL))) |
(10) |
active(zeros) |
→ |
mark(cons(0,zeros)) |
(11) |
active(take(0,IL)) |
→ |
mark(uTake1(isNatIList(IL))) |
(12) |
active(uTake1(tt)) |
→ |
mark(nil) |
(13) |
active(take(s(M),cons(N,IL))) |
→ |
mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) |
(14) |
active(uTake2(tt,M,N,IL)) |
→ |
mark(cons(N,take(M,IL))) |
(15) |
active(length(cons(N,L))) |
→ |
mark(uLength(and(isNat(N),isNatList(L)),L)) |
(16) |
active(uLength(tt,L)) |
→ |
mark(s(length(L))) |
(17) |
mark(and(X1,X2)) |
→ |
active(and(mark(X1),mark(X2))) |
(18) |
mark(tt) |
→ |
active(tt) |
(19) |
mark(isNatIList(X)) |
→ |
active(isNatIList(X)) |
(20) |
mark(isNatList(X)) |
→ |
active(isNatList(X)) |
(21) |
mark(isNat(X)) |
→ |
active(isNat(X)) |
(22) |
mark(0) |
→ |
active(0) |
(23) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(24) |
mark(length(X)) |
→ |
active(length(mark(X))) |
(25) |
mark(zeros) |
→ |
active(zeros) |
(26) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(27) |
mark(nil) |
→ |
active(nil) |
(28) |
mark(take(X1,X2)) |
→ |
active(take(mark(X1),mark(X2))) |
(29) |
mark(uTake1(X)) |
→ |
active(uTake1(mark(X))) |
(30) |
mark(uTake2(X1,X2,X3,X4)) |
→ |
active(uTake2(mark(X1),X2,X3,X4)) |
(31) |
mark(uLength(X1,X2)) |
→ |
active(uLength(mark(X1),X2)) |
(32) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(33) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(34) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(35) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(36) |
isNatIList(mark(X)) |
→ |
isNatIList(X) |
(37) |
isNatIList(active(X)) |
→ |
isNatIList(X) |
(38) |
isNatList(mark(X)) |
→ |
isNatList(X) |
(39) |
isNatList(active(X)) |
→ |
isNatList(X) |
(40) |
isNat(mark(X)) |
→ |
isNat(X) |
(41) |
isNat(active(X)) |
→ |
isNat(X) |
(42) |
s(mark(X)) |
→ |
s(X) |
(43) |
s(active(X)) |
→ |
s(X) |
(44) |
length(mark(X)) |
→ |
length(X) |
(45) |
length(active(X)) |
→ |
length(X) |
(46) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(47) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(48) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(49) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(50) |
take(mark(X1),X2) |
→ |
take(X1,X2) |
(51) |
take(X1,mark(X2)) |
→ |
take(X1,X2) |
(52) |
take(active(X1),X2) |
→ |
take(X1,X2) |
(53) |
take(X1,active(X2)) |
→ |
take(X1,X2) |
(54) |
uTake1(mark(X)) |
→ |
uTake1(X) |
(55) |
uTake1(active(X)) |
→ |
uTake1(X) |
(56) |
uTake2(mark(X1),X2,X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(57) |
uTake2(X1,mark(X2),X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(58) |
uTake2(X1,X2,mark(X3),X4) |
→ |
uTake2(X1,X2,X3,X4) |
(59) |
uTake2(X1,X2,X3,mark(X4)) |
→ |
uTake2(X1,X2,X3,X4) |
(60) |
uTake2(active(X1),X2,X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(61) |
uTake2(X1,active(X2),X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(62) |
uTake2(X1,X2,active(X3),X4) |
→ |
uTake2(X1,X2,X3,X4) |
(63) |
uTake2(X1,X2,X3,active(X4)) |
→ |
uTake2(X1,X2,X3,X4) |
(64) |
uLength(mark(X1),X2) |
→ |
uLength(X1,X2) |
(65) |
uLength(X1,mark(X2)) |
→ |
uLength(X1,X2) |
(66) |
uLength(active(X1),X2) |
→ |
uLength(X1,X2) |
(67) |
uLength(X1,active(X2)) |
→ |
uLength(X1,X2) |
(68) |
There are 115 ruless (increase limit for explicit display).
The dependency pairs are split into 12
components.
-
The
1st
component contains the
pair
mark#(and(X1,X2)) |
→ |
active#(and(mark(X1),mark(X2))) |
(115) |
active#(and(tt,T)) |
→ |
mark#(T) |
(69) |
mark#(and(X1,X2)) |
→ |
mark#(X1) |
(117) |
mark#(and(X1,X2)) |
→ |
mark#(X2) |
(118) |
mark#(isNatIList(X)) |
→ |
active#(isNatIList(X)) |
(120) |
active#(isNatIList(IL)) |
→ |
mark#(isNatList(IL)) |
(70) |
mark#(isNatList(X)) |
→ |
active#(isNatList(X)) |
(121) |
active#(isNat(s(N))) |
→ |
mark#(isNat(N)) |
(73) |
mark#(isNat(X)) |
→ |
active#(isNat(X)) |
(122) |
active#(isNat(length(L))) |
→ |
mark#(isNatList(L)) |
(75) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(124) |
active#(isNatIList(cons(N,IL))) |
→ |
mark#(and(isNat(N),isNatIList(IL))) |
(78) |
mark#(s(X)) |
→ |
mark#(X) |
(126) |
mark#(length(X)) |
→ |
active#(length(mark(X))) |
(127) |
active#(isNatList(cons(N,L))) |
→ |
mark#(and(isNat(N),isNatList(L))) |
(83) |
mark#(length(X)) |
→ |
mark#(X) |
(129) |
mark#(zeros) |
→ |
active#(zeros) |
(130) |
active#(zeros) |
→ |
mark#(cons(0,zeros)) |
(91) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(131) |
active#(isNatList(take(N,IL))) |
→ |
mark#(and(isNat(N),isNatIList(IL))) |
(87) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(133) |
mark#(take(X1,X2)) |
→ |
active#(take(mark(X1),mark(X2))) |
(135) |
active#(take(0,IL)) |
→ |
mark#(uTake1(isNatIList(IL))) |
(93) |
mark#(take(X1,X2)) |
→ |
mark#(X1) |
(137) |
mark#(take(X1,X2)) |
→ |
mark#(X2) |
(138) |
mark#(uTake1(X)) |
→ |
active#(uTake1(mark(X))) |
(139) |
active#(take(s(M),cons(N,IL))) |
→ |
mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) |
(97) |
mark#(uTake1(X)) |
→ |
mark#(X) |
(141) |
mark#(uTake2(X1,X2,X3,X4)) |
→ |
active#(uTake2(mark(X1),X2,X3,X4)) |
(142) |
active#(uTake2(tt,M,N,IL)) |
→ |
mark#(cons(N,take(M,IL))) |
(104) |
mark#(uTake2(X1,X2,X3,X4)) |
→ |
mark#(X1) |
(144) |
mark#(uLength(X1,X2)) |
→ |
active#(uLength(mark(X1),X2)) |
(145) |
active#(length(cons(N,L))) |
→ |
mark#(uLength(and(isNat(N),isNatList(L)),L)) |
(107) |
mark#(uLength(X1,X2)) |
→ |
mark#(X1) |
(147) |
active#(uLength(tt,L)) |
→ |
mark#(s(length(L))) |
(112) |
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 |
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
1 |
[isNatList(x1)] |
= |
1 |
[isNat(x1)] |
= |
1 |
[s(x1)] |
= |
0 |
[length(x1)] |
= |
1 |
[cons(x1, x2)] |
= |
0 |
[zeros] |
= |
1 |
[0] |
= |
0 |
[take(x1, x2)] |
= |
1 |
[uTake1(x1)] |
= |
0 |
[uTake2(x1,...,x4)] |
= |
1 |
[uLength(x1, x2)] |
= |
1 |
[active(x1)] |
= |
0 |
[nil] |
= |
0 |
together with the usable
rules
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(34) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(33) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(35) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(36) |
isNatIList(active(X)) |
→ |
isNatIList(X) |
(38) |
isNatIList(mark(X)) |
→ |
isNatIList(X) |
(37) |
isNatList(active(X)) |
→ |
isNatList(X) |
(40) |
isNatList(mark(X)) |
→ |
isNatList(X) |
(39) |
isNat(active(X)) |
→ |
isNat(X) |
(42) |
isNat(mark(X)) |
→ |
isNat(X) |
(41) |
s(active(X)) |
→ |
s(X) |
(44) |
s(mark(X)) |
→ |
s(X) |
(43) |
length(active(X)) |
→ |
length(X) |
(46) |
length(mark(X)) |
→ |
length(X) |
(45) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(48) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(47) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(49) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(50) |
take(X1,mark(X2)) |
→ |
take(X1,X2) |
(52) |
take(mark(X1),X2) |
→ |
take(X1,X2) |
(51) |
take(active(X1),X2) |
→ |
take(X1,X2) |
(53) |
take(X1,active(X2)) |
→ |
take(X1,X2) |
(54) |
uTake1(active(X)) |
→ |
uTake1(X) |
(56) |
uTake1(mark(X)) |
→ |
uTake1(X) |
(55) |
uTake2(X1,mark(X2),X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(58) |
uTake2(mark(X1),X2,X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(57) |
uTake2(X1,X2,mark(X3),X4) |
→ |
uTake2(X1,X2,X3,X4) |
(59) |
uTake2(X1,X2,X3,mark(X4)) |
→ |
uTake2(X1,X2,X3,X4) |
(60) |
uTake2(active(X1),X2,X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(61) |
uTake2(X1,active(X2),X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(62) |
uTake2(X1,X2,active(X3),X4) |
→ |
uTake2(X1,X2,X3,X4) |
(63) |
uTake2(X1,X2,X3,active(X4)) |
→ |
uTake2(X1,X2,X3,X4) |
(64) |
uLength(X1,mark(X2)) |
→ |
uLength(X1,X2) |
(66) |
uLength(mark(X1),X2) |
→ |
uLength(X1,X2) |
(65) |
uLength(active(X1),X2) |
→ |
uLength(X1,X2) |
(67) |
uLength(X1,active(X2)) |
→ |
uLength(X1,X2) |
(68) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(124) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(131) |
mark#(uTake1(X)) |
→ |
active#(uTake1(mark(X))) |
(139) |
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
|
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
0 |
[isNatList(x1)] |
= |
0 |
[isNat(x1)] |
= |
0 |
[s(x1)] |
= |
1 · x1
|
[length(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[zeros] |
= |
0 |
[0] |
= |
0 |
[take(x1, x2)] |
= |
1 + 1 · x1 + 1 · x2
|
[uTake1(x1)] |
= |
1 · x1
|
[uTake2(x1,...,x4)] |
= |
1 + 1 · x1 + 1 · x2 + 1 · x3 + 1 · x4
|
[uLength(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[active(x1)] |
= |
1 · x1
|
[nil] |
= |
0 |
the
pairs
active#(take(0,IL)) |
→ |
mark#(uTake1(isNatIList(IL))) |
(93) |
mark#(take(X1,X2)) |
→ |
mark#(X1) |
(137) |
mark#(take(X1,X2)) |
→ |
mark#(X2) |
(138) |
mark#(uTake2(X1,X2,X3,X4)) |
→ |
mark#(X1) |
(144) |
could be deleted.
1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
x1 |
[and(x1, x2)] |
= |
x1 + 2 · x2
|
[length(x1)] |
= |
2 + 2 · x1
|
[take(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[uLength(x1, x2)] |
= |
2 + 2 · x1 + 2 · x2
|
[uTake2(x1,...,x4)] |
= |
x1 + 2 · x2 + 2 · x3 + 2 · x4
|
[mark(x1)] |
= |
x1 |
[active(x1)] |
= |
x1 |
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
0 |
[isNatList(x1)] |
= |
0 |
[isNat(x1)] |
= |
0 |
[s(x1)] |
= |
x1 |
[cons(x1, x2)] |
= |
2 · x1 + x2
|
[zeros] |
= |
0 |
[0] |
= |
0 |
[uTake1(x1)] |
= |
x1 |
[nil] |
= |
0 |
[mark#(x1)] |
= |
x1 |
the
pairs
mark#(length(X)) |
→ |
mark#(X) |
(129) |
mark#(uLength(X1,X2)) |
→ |
mark#(X1) |
(147) |
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
2 · x1
|
[and(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[length(x1)] |
= |
-2 |
[take(x1, x2)] |
= |
2 + x2
|
[uLength(x1, x2)] |
= |
-2 |
[uTake2(x1,...,x4)] |
= |
x3 |
[mark(x1)] |
= |
x1 |
[active(x1)] |
= |
x1 |
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
0 |
[isNatList(x1)] |
= |
0 |
[isNat(x1)] |
= |
0 |
[s(x1)] |
= |
2 · x1
|
[cons(x1, x2)] |
= |
x1 |
[zeros] |
= |
2 |
[0] |
= |
1 |
[uTake1(x1)] |
= |
2 + x1
|
[nil] |
= |
1 |
[mark#(x1)] |
= |
2 · x1
|
the
pairs
active#(zeros) |
→ |
mark#(cons(0,zeros)) |
(91) |
active#(take(s(M),cons(N,IL))) |
→ |
mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) |
(97) |
mark#(uTake1(X)) |
→ |
mark#(X) |
(141) |
could be deleted.
1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
and#(X1,mark(X2)) |
→ |
and#(X1,X2) |
(149) |
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(148) |
and#(active(X1),X2) |
→ |
and#(X1,X2) |
(150) |
and#(X1,active(X2)) |
→ |
and#(X1,X2) |
(151) |
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) |
(149) |
|
1 |
≥ |
1 |
2 |
> |
2 |
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(148) |
|
1 |
> |
1 |
2 |
≥ |
2 |
and#(active(X1),X2) |
→ |
and#(X1,X2) |
(150) |
|
1 |
> |
1 |
2 |
≥ |
2 |
and#(X1,active(X2)) |
→ |
and#(X1,X2) |
(151) |
|
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
isNatIList#(active(X)) |
→ |
isNatIList#(X) |
(153) |
isNatIList#(mark(X)) |
→ |
isNatIList#(X) |
(152) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[isNatIList#(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.
isNatIList#(active(X)) |
→ |
isNatIList#(X) |
(153) |
|
1 |
> |
1 |
isNatIList#(mark(X)) |
→ |
isNatIList#(X) |
(152) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
isNatList#(active(X)) |
→ |
isNatList#(X) |
(155) |
isNatList#(mark(X)) |
→ |
isNatList#(X) |
(154) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[isNatList#(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.
isNatList#(active(X)) |
→ |
isNatList#(X) |
(155) |
|
1 |
> |
1 |
isNatList#(mark(X)) |
→ |
isNatList#(X) |
(154) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
isNat#(active(X)) |
→ |
isNat#(X) |
(157) |
isNat#(mark(X)) |
→ |
isNat#(X) |
(156) |
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
|
[isNat#(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.
isNat#(active(X)) |
→ |
isNat#(X) |
(157) |
|
1 |
> |
1 |
isNat#(mark(X)) |
→ |
isNat#(X) |
(156) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
s#(active(X)) |
→ |
s#(X) |
(159) |
s#(mark(X)) |
→ |
s#(X) |
(158) |
1.1.6 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.
1.1.6.1 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) |
(159) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(158) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
length#(active(X)) |
→ |
length#(X) |
(161) |
length#(mark(X)) |
→ |
length#(X) |
(160) |
1.1.7 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[length#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.7.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
length#(active(X)) |
→ |
length#(X) |
(161) |
|
1 |
> |
1 |
length#(mark(X)) |
→ |
length#(X) |
(160) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(163) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(162) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(164) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(165) |
1.1.8 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.
1.1.8.1 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) |
(163) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(162) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(164) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(165) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
9th
component contains the
pair
take#(X1,mark(X2)) |
→ |
take#(X1,X2) |
(167) |
take#(mark(X1),X2) |
→ |
take#(X1,X2) |
(166) |
take#(active(X1),X2) |
→ |
take#(X1,X2) |
(168) |
take#(X1,active(X2)) |
→ |
take#(X1,X2) |
(169) |
1.1.9 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[take#(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.
1.1.9.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
take#(X1,mark(X2)) |
→ |
take#(X1,X2) |
(167) |
|
1 |
≥ |
1 |
2 |
> |
2 |
take#(mark(X1),X2) |
→ |
take#(X1,X2) |
(166) |
|
1 |
> |
1 |
2 |
≥ |
2 |
take#(active(X1),X2) |
→ |
take#(X1,X2) |
(168) |
|
1 |
> |
1 |
2 |
≥ |
2 |
take#(X1,active(X2)) |
→ |
take#(X1,X2) |
(169) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
10th
component contains the
pair
uTake1#(active(X)) |
→ |
uTake1#(X) |
(171) |
uTake1#(mark(X)) |
→ |
uTake1#(X) |
(170) |
1.1.10 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[uTake1#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.10.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
uTake1#(active(X)) |
→ |
uTake1#(X) |
(171) |
|
1 |
> |
1 |
uTake1#(mark(X)) |
→ |
uTake1#(X) |
(170) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
11th
component contains the
pair
uTake2#(X1,mark(X2),X3,X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(173) |
uTake2#(mark(X1),X2,X3,X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(172) |
uTake2#(X1,X2,mark(X3),X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(174) |
uTake2#(X1,X2,X3,mark(X4)) |
→ |
uTake2#(X1,X2,X3,X4) |
(175) |
uTake2#(active(X1),X2,X3,X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(176) |
uTake2#(X1,active(X2),X3,X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(177) |
uTake2#(X1,X2,active(X3),X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(178) |
uTake2#(X1,X2,X3,active(X4)) |
→ |
uTake2#(X1,X2,X3,X4) |
(179) |
1.1.11 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[uTake2#(x1,...,x4)] |
= |
1 · x1 + 1 · x2 + 1 · x3 + 1 · x4
|
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.
uTake2#(X1,mark(X2),X3,X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(173) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
4 |
≥ |
4 |
uTake2#(mark(X1),X2,X3,X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(172) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
4 |
≥ |
4 |
uTake2#(X1,X2,mark(X3),X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(174) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
4 |
≥ |
4 |
uTake2#(X1,X2,X3,mark(X4)) |
→ |
uTake2#(X1,X2,X3,X4) |
(175) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
4 |
> |
4 |
uTake2#(active(X1),X2,X3,X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(176) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
4 |
≥ |
4 |
uTake2#(X1,active(X2),X3,X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(177) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
4 |
≥ |
4 |
uTake2#(X1,X2,active(X3),X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(178) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
4 |
≥ |
4 |
uTake2#(X1,X2,X3,active(X4)) |
→ |
uTake2#(X1,X2,X3,X4) |
(179) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
4 |
> |
4 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
12th
component contains the
pair
uLength#(X1,mark(X2)) |
→ |
uLength#(X1,X2) |
(181) |
uLength#(mark(X1),X2) |
→ |
uLength#(X1,X2) |
(180) |
uLength#(active(X1),X2) |
→ |
uLength#(X1,X2) |
(182) |
uLength#(X1,active(X2)) |
→ |
uLength#(X1,X2) |
(183) |
1.1.12 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[uLength#(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.
uLength#(X1,mark(X2)) |
→ |
uLength#(X1,X2) |
(181) |
|
1 |
≥ |
1 |
2 |
> |
2 |
uLength#(mark(X1),X2) |
→ |
uLength#(X1,X2) |
(180) |
|
1 |
> |
1 |
2 |
≥ |
2 |
uLength#(active(X1),X2) |
→ |
uLength#(X1,X2) |
(182) |
|
1 |
> |
1 |
2 |
≥ |
2 |
uLength#(X1,active(X2)) |
→ |
uLength#(X1,X2) |
(183) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.