The rewrite relation of the following TRS is considered.
active(zeros) |
→ |
mark(cons(0,zeros)) |
(1) |
active(U11(tt)) |
→ |
mark(tt) |
(2) |
active(U21(tt)) |
→ |
mark(tt) |
(3) |
active(U31(tt)) |
→ |
mark(tt) |
(4) |
active(U41(tt,V2)) |
→ |
mark(U42(isNatIList(V2))) |
(5) |
active(U42(tt)) |
→ |
mark(tt) |
(6) |
active(U51(tt,V2)) |
→ |
mark(U52(isNatList(V2))) |
(7) |
active(U52(tt)) |
→ |
mark(tt) |
(8) |
active(U61(tt,L,N)) |
→ |
mark(U62(isNat(N),L)) |
(9) |
active(U62(tt,L)) |
→ |
mark(s(length(L))) |
(10) |
active(isNat(0)) |
→ |
mark(tt) |
(11) |
active(isNat(length(V1))) |
→ |
mark(U11(isNatList(V1))) |
(12) |
active(isNat(s(V1))) |
→ |
mark(U21(isNat(V1))) |
(13) |
active(isNatIList(V)) |
→ |
mark(U31(isNatList(V))) |
(14) |
active(isNatIList(zeros)) |
→ |
mark(tt) |
(15) |
active(isNatIList(cons(V1,V2))) |
→ |
mark(U41(isNat(V1),V2)) |
(16) |
active(isNatList(nil)) |
→ |
mark(tt) |
(17) |
active(isNatList(cons(V1,V2))) |
→ |
mark(U51(isNat(V1),V2)) |
(18) |
active(length(nil)) |
→ |
mark(0) |
(19) |
active(length(cons(N,L))) |
→ |
mark(U61(isNatList(L),L,N)) |
(20) |
mark(zeros) |
→ |
active(zeros) |
(21) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(22) |
mark(0) |
→ |
active(0) |
(23) |
mark(U11(X)) |
→ |
active(U11(mark(X))) |
(24) |
mark(tt) |
→ |
active(tt) |
(25) |
mark(U21(X)) |
→ |
active(U21(mark(X))) |
(26) |
mark(U31(X)) |
→ |
active(U31(mark(X))) |
(27) |
mark(U41(X1,X2)) |
→ |
active(U41(mark(X1),X2)) |
(28) |
mark(U42(X)) |
→ |
active(U42(mark(X))) |
(29) |
mark(isNatIList(X)) |
→ |
active(isNatIList(X)) |
(30) |
mark(U51(X1,X2)) |
→ |
active(U51(mark(X1),X2)) |
(31) |
mark(U52(X)) |
→ |
active(U52(mark(X))) |
(32) |
mark(isNatList(X)) |
→ |
active(isNatList(X)) |
(33) |
mark(U61(X1,X2,X3)) |
→ |
active(U61(mark(X1),X2,X3)) |
(34) |
mark(U62(X1,X2)) |
→ |
active(U62(mark(X1),X2)) |
(35) |
mark(isNat(X)) |
→ |
active(isNat(X)) |
(36) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(37) |
mark(length(X)) |
→ |
active(length(mark(X))) |
(38) |
mark(nil) |
→ |
active(nil) |
(39) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(40) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(41) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(42) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(43) |
U11(mark(X)) |
→ |
U11(X) |
(44) |
U11(active(X)) |
→ |
U11(X) |
(45) |
U21(mark(X)) |
→ |
U21(X) |
(46) |
U21(active(X)) |
→ |
U21(X) |
(47) |
U31(mark(X)) |
→ |
U31(X) |
(48) |
U31(active(X)) |
→ |
U31(X) |
(49) |
U41(mark(X1),X2) |
→ |
U41(X1,X2) |
(50) |
U41(X1,mark(X2)) |
→ |
U41(X1,X2) |
(51) |
U41(active(X1),X2) |
→ |
U41(X1,X2) |
(52) |
U41(X1,active(X2)) |
→ |
U41(X1,X2) |
(53) |
U42(mark(X)) |
→ |
U42(X) |
(54) |
U42(active(X)) |
→ |
U42(X) |
(55) |
isNatIList(mark(X)) |
→ |
isNatIList(X) |
(56) |
isNatIList(active(X)) |
→ |
isNatIList(X) |
(57) |
U51(mark(X1),X2) |
→ |
U51(X1,X2) |
(58) |
U51(X1,mark(X2)) |
→ |
U51(X1,X2) |
(59) |
U51(active(X1),X2) |
→ |
U51(X1,X2) |
(60) |
U51(X1,active(X2)) |
→ |
U51(X1,X2) |
(61) |
U52(mark(X)) |
→ |
U52(X) |
(62) |
U52(active(X)) |
→ |
U52(X) |
(63) |
isNatList(mark(X)) |
→ |
isNatList(X) |
(64) |
isNatList(active(X)) |
→ |
isNatList(X) |
(65) |
U61(mark(X1),X2,X3) |
→ |
U61(X1,X2,X3) |
(66) |
U61(X1,mark(X2),X3) |
→ |
U61(X1,X2,X3) |
(67) |
U61(X1,X2,mark(X3)) |
→ |
U61(X1,X2,X3) |
(68) |
U61(active(X1),X2,X3) |
→ |
U61(X1,X2,X3) |
(69) |
U61(X1,active(X2),X3) |
→ |
U61(X1,X2,X3) |
(70) |
U61(X1,X2,active(X3)) |
→ |
U61(X1,X2,X3) |
(71) |
U62(mark(X1),X2) |
→ |
U62(X1,X2) |
(72) |
U62(X1,mark(X2)) |
→ |
U62(X1,X2) |
(73) |
U62(active(X1),X2) |
→ |
U62(X1,X2) |
(74) |
U62(X1,active(X2)) |
→ |
U62(X1,X2) |
(75) |
isNat(mark(X)) |
→ |
isNat(X) |
(76) |
isNat(active(X)) |
→ |
isNat(X) |
(77) |
s(mark(X)) |
→ |
s(X) |
(78) |
s(active(X)) |
→ |
s(X) |
(79) |
length(mark(X)) |
→ |
length(X) |
(80) |
length(active(X)) |
→ |
length(X) |
(81) |
There are 115 ruless (increase limit for explicit display).
The dependency pairs are split into 16
components.
-
The
1st
component contains the
pair
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(113) |
active#(U41(tt,V2)) |
→ |
mark#(U42(isNatIList(V2))) |
(85) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(115) |
mark#(zeros) |
→ |
active#(zeros) |
(112) |
active#(zeros) |
→ |
mark#(cons(0,zeros)) |
(82) |
mark#(U11(X)) |
→ |
active#(U11(mark(X))) |
(117) |
active#(U51(tt,V2)) |
→ |
mark#(U52(isNatList(V2))) |
(89) |
mark#(U11(X)) |
→ |
mark#(X) |
(119) |
mark#(U21(X)) |
→ |
active#(U21(mark(X))) |
(121) |
active#(U61(tt,L,N)) |
→ |
mark#(U62(isNat(N),L)) |
(93) |
mark#(U21(X)) |
→ |
mark#(X) |
(123) |
mark#(U31(X)) |
→ |
active#(U31(mark(X))) |
(124) |
active#(U62(tt,L)) |
→ |
mark#(s(length(L))) |
(96) |
mark#(U31(X)) |
→ |
mark#(X) |
(126) |
mark#(U41(X1,X2)) |
→ |
active#(U41(mark(X1),X2)) |
(127) |
active#(isNat(s(V1))) |
→ |
mark#(U21(isNat(V1))) |
(100) |
mark#(U41(X1,X2)) |
→ |
mark#(X1) |
(129) |
mark#(U42(X)) |
→ |
active#(U42(mark(X))) |
(130) |
active#(isNatIList(cons(V1,V2))) |
→ |
mark#(U41(isNat(V1),V2)) |
(103) |
mark#(U42(X)) |
→ |
mark#(X) |
(132) |
mark#(isNatIList(X)) |
→ |
active#(isNatIList(X)) |
(133) |
active#(isNatList(cons(V1,V2))) |
→ |
mark#(U51(isNat(V1),V2)) |
(106) |
mark#(U51(X1,X2)) |
→ |
active#(U51(mark(X1),X2)) |
(134) |
active#(length(cons(N,L))) |
→ |
mark#(U61(isNatList(L),L,N)) |
(109) |
mark#(U51(X1,X2)) |
→ |
mark#(X1) |
(136) |
mark#(U52(X)) |
→ |
active#(U52(mark(X))) |
(137) |
mark#(U52(X)) |
→ |
mark#(X) |
(139) |
mark#(isNatList(X)) |
→ |
active#(isNatList(X)) |
(140) |
mark#(U61(X1,X2,X3)) |
→ |
active#(U61(mark(X1),X2,X3)) |
(141) |
mark#(U61(X1,X2,X3)) |
→ |
mark#(X1) |
(143) |
mark#(U62(X1,X2)) |
→ |
active#(U62(mark(X1),X2)) |
(144) |
mark#(U62(X1,X2)) |
→ |
mark#(X1) |
(146) |
mark#(isNat(X)) |
→ |
active#(isNat(X)) |
(147) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(148) |
mark#(s(X)) |
→ |
mark#(X) |
(150) |
mark#(length(X)) |
→ |
active#(length(mark(X))) |
(151) |
mark#(length(X)) |
→ |
mark#(X) |
(153) |
1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[zeros] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[0] |
= |
0 |
[U21(x1)] |
= |
1 · x1
|
[tt] |
= |
0 |
[U41(x1, x2)] |
= |
2 · x1 + 1 · x2
|
[U42(x1)] |
= |
1 · x1
|
[isNatIList(x1)] |
= |
1 · x1
|
[U51(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[U52(x1)] |
= |
2 · x1
|
[isNatList(x1)] |
= |
1 · x1
|
[U61(x1, x2, x3)] |
= |
2 · x1 + 2 · x2 + 1 · x3
|
[U62(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[isNat(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[length(x1)] |
= |
2 · x1
|
[U11(x1)] |
= |
1 + 1 · x1
|
[U31(x1)] |
= |
1 · x1
|
[nil] |
= |
0 |
[mark#(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
2 · x1
|
the
pair
mark#(U11(X)) |
→ |
mark#(X) |
(119) |
and
no rules
could be deleted.
1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[zeros] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[0] |
= |
0 |
[U21(x1)] |
= |
1 · x1
|
[tt] |
= |
0 |
[U41(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[U42(x1)] |
= |
2 · x1
|
[isNatIList(x1)] |
= |
1 · x1
|
[U51(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[U52(x1)] |
= |
1 · x1
|
[isNatList(x1)] |
= |
1 · x1
|
[U61(x1, x2, x3)] |
= |
1 · x1 + 2 · x2 + 2 · x3
|
[U62(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[isNat(x1)] |
= |
2 · x1
|
[s(x1)] |
= |
1 · x1
|
[length(x1)] |
= |
2 · x1
|
[U11(x1)] |
= |
1 · x1
|
[U31(x1)] |
= |
2 + 2 · x1
|
[nil] |
= |
0 |
[mark#(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
2 · x1
|
the
pair
mark#(U31(X)) |
→ |
mark#(X) |
(126) |
and
no rules
could be deleted.
1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[zeros] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[0] |
= |
0 |
[U21(x1)] |
= |
1 · x1
|
[tt] |
= |
0 |
[U41(x1, x2)] |
= |
2 + 2 · x1 + 1 · x2
|
[U42(x1)] |
= |
1 · x1
|
[isNatIList(x1)] |
= |
2 + 1 · x1
|
[U51(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[U52(x1)] |
= |
2 · x1
|
[isNatList(x1)] |
= |
1 · x1
|
[U61(x1, x2, x3)] |
= |
2 · x1 + 2 · x2 + 2 · x3
|
[U62(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[isNat(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[length(x1)] |
= |
2 · x1
|
[U11(x1)] |
= |
1 · x1
|
[U31(x1)] |
= |
2 · x1
|
[nil] |
= |
0 |
[mark#(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
2 · x1
|
the
pair
mark#(U41(X1,X2)) |
→ |
mark#(X1) |
(129) |
and
no rules
could be deleted.
1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[zeros] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[0] |
= |
0 |
[U21(x1)] |
= |
1 · x1
|
[tt] |
= |
0 |
[U41(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[U42(x1)] |
= |
2 · x1
|
[isNatIList(x1)] |
= |
1 · x1
|
[U51(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[U52(x1)] |
= |
1 · x1
|
[isNatList(x1)] |
= |
1 · x1
|
[U61(x1, x2, x3)] |
= |
1 + 2 · x1 + 2 · x2 + 2 · x3
|
[U62(x1, x2)] |
= |
1 + 2 · x1 + 2 · x2
|
[isNat(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[length(x1)] |
= |
1 + 2 · x1
|
[U11(x1)] |
= |
2 · x1
|
[U31(x1)] |
= |
1 · x1
|
[nil] |
= |
0 |
[mark#(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
2 · x1
|
the
pairs
mark#(U61(X1,X2,X3)) |
→ |
mark#(X1) |
(143) |
mark#(U62(X1,X2)) |
→ |
mark#(X1) |
(146) |
mark#(length(X)) |
→ |
mark#(X) |
(153) |
and
no rules
could be deleted.
1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
x1 |
[U11(x1)] |
= |
-2 |
[U21(x1)] |
= |
-2 |
[U31(x1)] |
= |
1 |
[U41(x1, x2)] |
= |
2 |
[U42(x1)] |
= |
2 |
[U51(x1, x2)] |
= |
2 |
[U52(x1)] |
= |
0 |
[U61(x1, x2, x3)] |
= |
2 |
[U62(x1, x2)] |
= |
2 |
[cons(x1, x2)] |
= |
-2 |
[length(x1)] |
= |
2 |
[s(x1)] |
= |
0 |
[mark(x1)] |
= |
-2 |
[active(x1)] |
= |
0 |
[zeros] |
= |
2 |
[0] |
= |
0 |
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
2 |
[isNatList(x1)] |
= |
2 |
[isNat(x1)] |
= |
2 |
[nil] |
= |
0 |
[mark#(x1)] |
= |
2 |
together with the usable
rules
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(41) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(40) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(42) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(43) |
isNatIList(active(X)) |
→ |
isNatIList(X) |
(57) |
isNatIList(mark(X)) |
→ |
isNatIList(X) |
(56) |
U42(active(X)) |
→ |
U42(X) |
(55) |
U42(mark(X)) |
→ |
U42(X) |
(54) |
U11(active(X)) |
→ |
U11(X) |
(45) |
U11(mark(X)) |
→ |
U11(X) |
(44) |
isNatList(active(X)) |
→ |
isNatList(X) |
(65) |
isNatList(mark(X)) |
→ |
isNatList(X) |
(64) |
U52(active(X)) |
→ |
U52(X) |
(63) |
U52(mark(X)) |
→ |
U52(X) |
(62) |
U21(active(X)) |
→ |
U21(X) |
(47) |
U21(mark(X)) |
→ |
U21(X) |
(46) |
isNat(active(X)) |
→ |
isNat(X) |
(77) |
isNat(mark(X)) |
→ |
isNat(X) |
(76) |
U62(X1,mark(X2)) |
→ |
U62(X1,X2) |
(73) |
U62(mark(X1),X2) |
→ |
U62(X1,X2) |
(72) |
U62(active(X1),X2) |
→ |
U62(X1,X2) |
(74) |
U62(X1,active(X2)) |
→ |
U62(X1,X2) |
(75) |
U31(active(X)) |
→ |
U31(X) |
(49) |
U31(mark(X)) |
→ |
U31(X) |
(48) |
length(active(X)) |
→ |
length(X) |
(81) |
length(mark(X)) |
→ |
length(X) |
(80) |
s(active(X)) |
→ |
s(X) |
(79) |
s(mark(X)) |
→ |
s(X) |
(78) |
U41(X1,mark(X2)) |
→ |
U41(X1,X2) |
(51) |
U41(mark(X1),X2) |
→ |
U41(X1,X2) |
(50) |
U41(active(X1),X2) |
→ |
U41(X1,X2) |
(52) |
U41(X1,active(X2)) |
→ |
U41(X1,X2) |
(53) |
U51(X1,mark(X2)) |
→ |
U51(X1,X2) |
(59) |
U51(mark(X1),X2) |
→ |
U51(X1,X2) |
(58) |
U51(active(X1),X2) |
→ |
U51(X1,X2) |
(60) |
U51(X1,active(X2)) |
→ |
U51(X1,X2) |
(61) |
U61(X1,mark(X2),X3) |
→ |
U61(X1,X2,X3) |
(67) |
U61(mark(X1),X2,X3) |
→ |
U61(X1,X2,X3) |
(66) |
U61(X1,X2,mark(X3)) |
→ |
U61(X1,X2,X3) |
(68) |
U61(active(X1),X2,X3) |
→ |
U61(X1,X2,X3) |
(69) |
U61(X1,active(X2),X3) |
→ |
U61(X1,X2,X3) |
(70) |
U61(X1,X2,active(X3)) |
→ |
U61(X1,X2,X3) |
(71) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(113) |
mark#(U11(X)) |
→ |
active#(U11(mark(X))) |
(117) |
mark#(U21(X)) |
→ |
active#(U21(mark(X))) |
(121) |
mark#(U31(X)) |
→ |
active#(U31(mark(X))) |
(124) |
mark#(U52(X)) |
→ |
active#(U52(mark(X))) |
(137) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(148) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
x1 |
[mark#(x1)] |
= |
1 |
[U42(x1)] |
= |
-2 |
[isNatIList(x1)] |
= |
1 |
[active(x1)] |
= |
-2 |
[mark(x1)] |
= |
-2 |
[U52(x1)] |
= |
-2 |
[U61(x1, x2, x3)] |
= |
1 |
[isNatList(x1)] |
= |
1 |
[U41(x1, x2)] |
= |
1 |
[U51(x1, x2)] |
= |
1 |
[U62(x1, x2)] |
= |
1 |
[length(x1)] |
= |
1 |
[U21(x1)] |
= |
2 |
[isNat(x1)] |
= |
1 |
[s(x1)] |
= |
-2 |
[cons(x1, x2)] |
= |
-2 + x2
|
[zeros] |
= |
1 |
[0] |
= |
0 |
[tt] |
= |
0 |
[U11(x1)] |
= |
2 |
[U31(x1)] |
= |
0 |
[nil] |
= |
0 |
together with the usable
rules
isNatIList(active(X)) |
→ |
isNatIList(X) |
(57) |
isNatIList(mark(X)) |
→ |
isNatIList(X) |
(56) |
U42(active(X)) |
→ |
U42(X) |
(55) |
U42(mark(X)) |
→ |
U42(X) |
(54) |
isNatList(active(X)) |
→ |
isNatList(X) |
(65) |
isNatList(mark(X)) |
→ |
isNatList(X) |
(64) |
isNat(active(X)) |
→ |
isNat(X) |
(77) |
isNat(mark(X)) |
→ |
isNat(X) |
(76) |
U62(X1,mark(X2)) |
→ |
U62(X1,X2) |
(73) |
U62(mark(X1),X2) |
→ |
U62(X1,X2) |
(72) |
U62(active(X1),X2) |
→ |
U62(X1,X2) |
(74) |
U62(X1,active(X2)) |
→ |
U62(X1,X2) |
(75) |
length(active(X)) |
→ |
length(X) |
(81) |
length(mark(X)) |
→ |
length(X) |
(80) |
U41(X1,mark(X2)) |
→ |
U41(X1,X2) |
(51) |
U41(mark(X1),X2) |
→ |
U41(X1,X2) |
(50) |
U41(active(X1),X2) |
→ |
U41(X1,X2) |
(52) |
U41(X1,active(X2)) |
→ |
U41(X1,X2) |
(53) |
U51(X1,mark(X2)) |
→ |
U51(X1,X2) |
(59) |
U51(mark(X1),X2) |
→ |
U51(X1,X2) |
(58) |
U51(active(X1),X2) |
→ |
U51(X1,X2) |
(60) |
U51(X1,active(X2)) |
→ |
U51(X1,X2) |
(61) |
U61(X1,mark(X2),X3) |
→ |
U61(X1,X2,X3) |
(67) |
U61(mark(X1),X2,X3) |
→ |
U61(X1,X2,X3) |
(66) |
U61(X1,X2,mark(X3)) |
→ |
U61(X1,X2,X3) |
(68) |
U61(active(X1),X2,X3) |
→ |
U61(X1,X2,X3) |
(69) |
U61(X1,active(X2),X3) |
→ |
U61(X1,X2,X3) |
(70) |
U61(X1,X2,active(X3)) |
→ |
U61(X1,X2,X3) |
(71) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(U42(X)) |
→ |
active#(U42(mark(X))) |
(130) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-2 |
[mark#(x1)] |
= |
-2 + 2 · x1
|
[U42(x1)] |
= |
1 + 2 · x1
|
[isNatIList(x1)] |
= |
0 |
[active(x1)] |
= |
2 · x1
|
[mark(x1)] |
= |
x1 |
[U41(x1, x2)] |
= |
1 + x1
|
[U51(x1, x2)] |
= |
1 + 2 · x1
|
[U61(x1, x2, x3)] |
= |
-2 |
[U62(x1, x2)] |
= |
-2 |
[length(x1)] |
= |
-2 |
[U52(x1)] |
= |
1 + 2 · x1
|
[isNatList(x1)] |
= |
0 |
[U21(x1)] |
= |
1 + 2 · x1
|
[isNat(x1)] |
= |
0 |
[s(x1)] |
= |
1 + 2 · x1
|
[cons(x1, x2)] |
= |
1 + 2 · x1
|
[zeros] |
= |
2 |
[0] |
= |
0 |
[tt] |
= |
1 |
[U11(x1)] |
= |
1 + x1
|
[U31(x1)] |
= |
2 + x1
|
[nil] |
= |
0 |
together with the usable
rules
isNatIList(active(X)) |
→ |
isNatIList(X) |
(57) |
isNatIList(mark(X)) |
→ |
isNatIList(X) |
(56) |
U42(active(X)) |
→ |
U42(X) |
(55) |
U42(mark(X)) |
→ |
U42(X) |
(54) |
isNatList(active(X)) |
→ |
isNatList(X) |
(65) |
isNatList(mark(X)) |
→ |
isNatList(X) |
(64) |
U52(active(X)) |
→ |
U52(X) |
(63) |
U52(mark(X)) |
→ |
U52(X) |
(62) |
isNat(active(X)) |
→ |
isNat(X) |
(77) |
isNat(mark(X)) |
→ |
isNat(X) |
(76) |
U62(X1,mark(X2)) |
→ |
U62(X1,X2) |
(73) |
U62(mark(X1),X2) |
→ |
U62(X1,X2) |
(72) |
U62(active(X1),X2) |
→ |
U62(X1,X2) |
(74) |
U62(X1,active(X2)) |
→ |
U62(X1,X2) |
(75) |
length(active(X)) |
→ |
length(X) |
(81) |
length(mark(X)) |
→ |
length(X) |
(80) |
s(active(X)) |
→ |
s(X) |
(79) |
s(mark(X)) |
→ |
s(X) |
(78) |
U41(X1,mark(X2)) |
→ |
U41(X1,X2) |
(51) |
U41(mark(X1),X2) |
→ |
U41(X1,X2) |
(50) |
U41(active(X1),X2) |
→ |
U41(X1,X2) |
(52) |
U41(X1,active(X2)) |
→ |
U41(X1,X2) |
(53) |
U21(active(X)) |
→ |
U21(X) |
(47) |
U21(mark(X)) |
→ |
U21(X) |
(46) |
U51(X1,mark(X2)) |
→ |
U51(X1,X2) |
(59) |
U51(mark(X1),X2) |
→ |
U51(X1,X2) |
(58) |
U51(active(X1),X2) |
→ |
U51(X1,X2) |
(60) |
U51(X1,active(X2)) |
→ |
U51(X1,X2) |
(61) |
U61(X1,mark(X2),X3) |
→ |
U61(X1,X2,X3) |
(67) |
U61(mark(X1),X2,X3) |
→ |
U61(X1,X2,X3) |
(66) |
U61(X1,X2,mark(X3)) |
→ |
U61(X1,X2,X3) |
(68) |
U61(active(X1),X2,X3) |
→ |
U61(X1,X2,X3) |
(69) |
U61(X1,active(X2),X3) |
→ |
U61(X1,X2,X3) |
(70) |
U61(X1,X2,active(X3)) |
→ |
U61(X1,X2,X3) |
(71) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(zeros) |
→ |
active#(zeros) |
(112) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(156) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(155) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(157) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(158) |
1.1.1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[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.
1.1.1.1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(156) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(155) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(157) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(158) |
|
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
U11#(active(X)) |
→ |
U11#(X) |
(160) |
U11#(mark(X)) |
→ |
U11#(X) |
(159) |
1.1.1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[U11#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U11#(active(X)) |
→ |
U11#(X) |
(160) |
|
1 |
> |
1 |
U11#(mark(X)) |
→ |
U11#(X) |
(159) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
U21#(active(X)) |
→ |
U21#(X) |
(162) |
U21#(mark(X)) |
→ |
U21#(X) |
(161) |
1.1.1.1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[U21#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U21#(active(X)) |
→ |
U21#(X) |
(162) |
|
1 |
> |
1 |
U21#(mark(X)) |
→ |
U21#(X) |
(161) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
U31#(active(X)) |
→ |
U31#(X) |
(164) |
U31#(mark(X)) |
→ |
U31#(X) |
(163) |
1.1.1.1.1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[U31#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U31#(active(X)) |
→ |
U31#(X) |
(164) |
|
1 |
> |
1 |
U31#(mark(X)) |
→ |
U31#(X) |
(163) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
U41#(X1,mark(X2)) |
→ |
U41#(X1,X2) |
(166) |
U41#(mark(X1),X2) |
→ |
U41#(X1,X2) |
(165) |
U41#(active(X1),X2) |
→ |
U41#(X1,X2) |
(167) |
U41#(X1,active(X2)) |
→ |
U41#(X1,X2) |
(168) |
1.1.1.1.1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[U41#(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.
1.1.1.1.1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U41#(X1,mark(X2)) |
→ |
U41#(X1,X2) |
(166) |
|
1 |
≥ |
1 |
2 |
> |
2 |
U41#(mark(X1),X2) |
→ |
U41#(X1,X2) |
(165) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U41#(active(X1),X2) |
→ |
U41#(X1,X2) |
(167) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U41#(X1,active(X2)) |
→ |
U41#(X1,X2) |
(168) |
|
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
U42#(active(X)) |
→ |
U42#(X) |
(170) |
U42#(mark(X)) |
→ |
U42#(X) |
(169) |
1.1.1.1.1.1.7 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[U42#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.7.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U42#(active(X)) |
→ |
U42#(X) |
(170) |
|
1 |
> |
1 |
U42#(mark(X)) |
→ |
U42#(X) |
(169) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
isNatIList#(active(X)) |
→ |
isNatIList#(X) |
(172) |
isNatIList#(mark(X)) |
→ |
isNatIList#(X) |
(171) |
1.1.1.1.1.1.8 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[isNatIList#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.8.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
isNatIList#(active(X)) |
→ |
isNatIList#(X) |
(172) |
|
1 |
> |
1 |
isNatIList#(mark(X)) |
→ |
isNatIList#(X) |
(171) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
9th
component contains the
pair
U51#(X1,mark(X2)) |
→ |
U51#(X1,X2) |
(174) |
U51#(mark(X1),X2) |
→ |
U51#(X1,X2) |
(173) |
U51#(active(X1),X2) |
→ |
U51#(X1,X2) |
(175) |
U51#(X1,active(X2)) |
→ |
U51#(X1,X2) |
(176) |
1.1.1.1.1.1.9 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[U51#(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.
1.1.1.1.1.1.9.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U51#(X1,mark(X2)) |
→ |
U51#(X1,X2) |
(174) |
|
1 |
≥ |
1 |
2 |
> |
2 |
U51#(mark(X1),X2) |
→ |
U51#(X1,X2) |
(173) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U51#(active(X1),X2) |
→ |
U51#(X1,X2) |
(175) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U51#(X1,active(X2)) |
→ |
U51#(X1,X2) |
(176) |
|
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
U52#(active(X)) |
→ |
U52#(X) |
(178) |
U52#(mark(X)) |
→ |
U52#(X) |
(177) |
1.1.1.1.1.1.10 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[U52#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.10.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U52#(active(X)) |
→ |
U52#(X) |
(178) |
|
1 |
> |
1 |
U52#(mark(X)) |
→ |
U52#(X) |
(177) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
11th
component contains the
pair
isNatList#(active(X)) |
→ |
isNatList#(X) |
(180) |
isNatList#(mark(X)) |
→ |
isNatList#(X) |
(179) |
1.1.1.1.1.1.11 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.
1.1.1.1.1.1.11.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
isNatList#(active(X)) |
→ |
isNatList#(X) |
(180) |
|
1 |
> |
1 |
isNatList#(mark(X)) |
→ |
isNatList#(X) |
(179) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
12th
component contains the
pair
U61#(X1,mark(X2),X3) |
→ |
U61#(X1,X2,X3) |
(182) |
U61#(mark(X1),X2,X3) |
→ |
U61#(X1,X2,X3) |
(181) |
U61#(X1,X2,mark(X3)) |
→ |
U61#(X1,X2,X3) |
(183) |
U61#(active(X1),X2,X3) |
→ |
U61#(X1,X2,X3) |
(184) |
U61#(X1,active(X2),X3) |
→ |
U61#(X1,X2,X3) |
(185) |
U61#(X1,X2,active(X3)) |
→ |
U61#(X1,X2,X3) |
(186) |
1.1.1.1.1.1.12 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[U61#(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.
1.1.1.1.1.1.12.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U61#(X1,mark(X2),X3) |
→ |
U61#(X1,X2,X3) |
(182) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
U61#(mark(X1),X2,X3) |
→ |
U61#(X1,X2,X3) |
(181) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
U61#(X1,X2,mark(X3)) |
→ |
U61#(X1,X2,X3) |
(183) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
U61#(active(X1),X2,X3) |
→ |
U61#(X1,X2,X3) |
(184) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
U61#(X1,active(X2),X3) |
→ |
U61#(X1,X2,X3) |
(185) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
U61#(X1,X2,active(X3)) |
→ |
U61#(X1,X2,X3) |
(186) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
13th
component contains the
pair
U62#(X1,mark(X2)) |
→ |
U62#(X1,X2) |
(188) |
U62#(mark(X1),X2) |
→ |
U62#(X1,X2) |
(187) |
U62#(active(X1),X2) |
→ |
U62#(X1,X2) |
(189) |
U62#(X1,active(X2)) |
→ |
U62#(X1,X2) |
(190) |
1.1.1.1.1.1.13 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[U62#(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.
1.1.1.1.1.1.13.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U62#(X1,mark(X2)) |
→ |
U62#(X1,X2) |
(188) |
|
1 |
≥ |
1 |
2 |
> |
2 |
U62#(mark(X1),X2) |
→ |
U62#(X1,X2) |
(187) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U62#(active(X1),X2) |
→ |
U62#(X1,X2) |
(189) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U62#(X1,active(X2)) |
→ |
U62#(X1,X2) |
(190) |
|
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
isNat#(active(X)) |
→ |
isNat#(X) |
(192) |
isNat#(mark(X)) |
→ |
isNat#(X) |
(191) |
1.1.1.1.1.1.14 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.
1.1.1.1.1.1.14.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
isNat#(active(X)) |
→ |
isNat#(X) |
(192) |
|
1 |
> |
1 |
isNat#(mark(X)) |
→ |
isNat#(X) |
(191) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
15th
component contains the
pair
s#(active(X)) |
→ |
s#(X) |
(194) |
s#(mark(X)) |
→ |
s#(X) |
(193) |
1.1.1.1.1.1.15 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.
1.1.1.1.1.1.15.1 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) |
(194) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(193) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
16th
component contains the
pair
length#(active(X)) |
→ |
length#(X) |
(196) |
length#(mark(X)) |
→ |
length#(X) |
(195) |
1.1.1.1.1.1.16 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.
1.1.1.1.1.1.16.1 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) |
(196) |
|
1 |
> |
1 |
length#(mark(X)) |
→ |
length#(X) |
(195) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.