The rewrite relation of the following TRS is considered.
zeros |
→ |
cons(0,n__zeros) |
(1) |
U101(tt,V1,V2) |
→ |
U102(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(2) |
U102(tt,V1,V2) |
→ |
U103(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(3) |
U103(tt,V1,V2) |
→ |
U104(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(4) |
U104(tt,V1,V2) |
→ |
U105(isNat(activate(V1)),activate(V2)) |
(5) |
U105(tt,V2) |
→ |
U106(isNatIList(activate(V2))) |
(6) |
U106(tt) |
→ |
tt |
(7) |
U11(tt,V1) |
→ |
U12(isNatIListKind(activate(V1)),activate(V1)) |
(8) |
U111(tt,L,N) |
→ |
U112(isNatIListKind(activate(L)),activate(L),activate(N)) |
(9) |
U112(tt,L,N) |
→ |
U113(isNat(activate(N)),activate(L),activate(N)) |
(10) |
U113(tt,L,N) |
→ |
U114(isNatKind(activate(N)),activate(L)) |
(11) |
U114(tt,L) |
→ |
s(length(activate(L))) |
(12) |
U12(tt,V1) |
→ |
U13(isNatList(activate(V1))) |
(13) |
U121(tt,IL) |
→ |
U122(isNatIListKind(activate(IL))) |
(14) |
U122(tt) |
→ |
nil |
(15) |
U13(tt) |
→ |
tt |
(16) |
U131(tt,IL,M,N) |
→ |
U132(isNatIListKind(activate(IL)),activate(IL),activate(M),activate(N)) |
(17) |
U132(tt,IL,M,N) |
→ |
U133(isNat(activate(M)),activate(IL),activate(M),activate(N)) |
(18) |
U133(tt,IL,M,N) |
→ |
U134(isNatKind(activate(M)),activate(IL),activate(M),activate(N)) |
(19) |
U134(tt,IL,M,N) |
→ |
U135(isNat(activate(N)),activate(IL),activate(M),activate(N)) |
(20) |
U135(tt,IL,M,N) |
→ |
U136(isNatKind(activate(N)),activate(IL),activate(M),activate(N)) |
(21) |
U136(tt,IL,M,N) |
→ |
cons(activate(N),n__take(activate(M),activate(IL))) |
(22) |
U21(tt,V1) |
→ |
U22(isNatKind(activate(V1)),activate(V1)) |
(23) |
U22(tt,V1) |
→ |
U23(isNat(activate(V1))) |
(24) |
U23(tt) |
→ |
tt |
(25) |
U31(tt,V) |
→ |
U32(isNatIListKind(activate(V)),activate(V)) |
(26) |
U32(tt,V) |
→ |
U33(isNatList(activate(V))) |
(27) |
U33(tt) |
→ |
tt |
(28) |
U41(tt,V1,V2) |
→ |
U42(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(29) |
U42(tt,V1,V2) |
→ |
U43(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(30) |
U43(tt,V1,V2) |
→ |
U44(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(31) |
U44(tt,V1,V2) |
→ |
U45(isNat(activate(V1)),activate(V2)) |
(32) |
U45(tt,V2) |
→ |
U46(isNatIList(activate(V2))) |
(33) |
U46(tt) |
→ |
tt |
(34) |
U51(tt,V2) |
→ |
U52(isNatIListKind(activate(V2))) |
(35) |
U52(tt) |
→ |
tt |
(36) |
U61(tt,V2) |
→ |
U62(isNatIListKind(activate(V2))) |
(37) |
U62(tt) |
→ |
tt |
(38) |
U71(tt) |
→ |
tt |
(39) |
U81(tt) |
→ |
tt |
(40) |
U91(tt,V1,V2) |
→ |
U92(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(41) |
U92(tt,V1,V2) |
→ |
U93(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(42) |
U93(tt,V1,V2) |
→ |
U94(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(43) |
U94(tt,V1,V2) |
→ |
U95(isNat(activate(V1)),activate(V2)) |
(44) |
U95(tt,V2) |
→ |
U96(isNatList(activate(V2))) |
(45) |
U96(tt) |
→ |
tt |
(46) |
isNat(n__0) |
→ |
tt |
(47) |
isNat(n__length(V1)) |
→ |
U11(isNatIListKind(activate(V1)),activate(V1)) |
(48) |
isNat(n__s(V1)) |
→ |
U21(isNatKind(activate(V1)),activate(V1)) |
(49) |
isNatIList(V) |
→ |
U31(isNatIListKind(activate(V)),activate(V)) |
(50) |
isNatIList(n__zeros) |
→ |
tt |
(51) |
isNatIList(n__cons(V1,V2)) |
→ |
U41(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(52) |
isNatIListKind(n__nil) |
→ |
tt |
(53) |
isNatIListKind(n__zeros) |
→ |
tt |
(54) |
isNatIListKind(n__cons(V1,V2)) |
→ |
U51(isNatKind(activate(V1)),activate(V2)) |
(55) |
isNatIListKind(n__take(V1,V2)) |
→ |
U61(isNatKind(activate(V1)),activate(V2)) |
(56) |
isNatKind(n__0) |
→ |
tt |
(57) |
isNatKind(n__length(V1)) |
→ |
U71(isNatIListKind(activate(V1))) |
(58) |
isNatKind(n__s(V1)) |
→ |
U81(isNatKind(activate(V1))) |
(59) |
isNatList(n__nil) |
→ |
tt |
(60) |
isNatList(n__cons(V1,V2)) |
→ |
U91(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(61) |
isNatList(n__take(V1,V2)) |
→ |
U101(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(62) |
length(nil) |
→ |
0 |
(63) |
length(cons(N,L)) |
→ |
U111(isNatList(activate(L)),activate(L),N) |
(64) |
take(0,IL) |
→ |
U121(isNatIList(IL),IL) |
(65) |
take(s(M),cons(N,IL)) |
→ |
U131(isNatIList(activate(IL)),activate(IL),M,N) |
(66) |
zeros |
→ |
n__zeros |
(67) |
take(X1,X2) |
→ |
n__take(X1,X2) |
(68) |
0 |
→ |
n__0 |
(69) |
length(X) |
→ |
n__length(X) |
(70) |
s(X) |
→ |
n__s(X) |
(71) |
cons(X1,X2) |
→ |
n__cons(X1,X2) |
(72) |
nil |
→ |
n__nil |
(73) |
activate(n__zeros) |
→ |
zeros |
(74) |
activate(n__take(X1,X2)) |
→ |
take(X1,X2) |
(75) |
activate(n__0) |
→ |
0 |
(76) |
activate(n__length(X)) |
→ |
length(X) |
(77) |
activate(n__s(X)) |
→ |
s(X) |
(78) |
activate(n__cons(X1,X2)) |
→ |
cons(X1,X2) |
(79) |
activate(n__nil) |
→ |
nil |
(80) |
activate(X) |
→ |
X |
(81) |
There are 182 ruless (increase limit for explicit display).
It remains to prove infiniteness of the resulting DP problem.
activate#(n__take(X1,X2)) |
→ |
take#(X1,X2) |
(258) |
isNatList#(n__take(V1,V2)) |
→ |
U101#(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(244) |
isNatList#(n__take(V1,V2)) |
→ |
isNatKind#(activate(V1)) |
(245) |
isNatList#(n__take(V1,V2)) |
→ |
activate#(V1) |
(246) |
isNatList#(n__take(V1,V2)) |
→ |
activate#(V2) |
(247) |
isNatIListKind#(n__take(V1,V2)) |
→ |
U61#(isNatKind(activate(V1)),activate(V2)) |
(230) |
isNatIListKind#(n__take(V1,V2)) |
→ |
isNatKind#(activate(V1)) |
(231) |
isNatIListKind#(n__take(V1,V2)) |
→ |
activate#(V1) |
(232) |
isNatIListKind#(n__take(V1,V2)) |
→ |
activate#(V2) |
(233) |
and the following rules have been deleted.
take#(0,IL) |
→ |
U121#(isNatIList(IL),IL) |
(252) |
U121#(tt,IL) |
→ |
isNatIListKind#(activate(IL)) |
(125) |
U101#(tt,V1,V2) |
→ |
U102#(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(84) |
U102#(tt,V1,V2) |
→ |
U103#(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(88) |
U103#(tt,V1,V2) |
→ |
U104#(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(92) |
U104#(tt,V1,V2) |
→ |
U105#(isNat(activate(V1)),activate(V2)) |
(96) |
U105#(tt,V2) |
→ |
isNatIList#(activate(V2)) |
(101) |
isNatIList#(V) |
→ |
U31#(isNatIListKind(activate(V)),activate(V)) |
(219) |
U31#(tt,V) |
→ |
U32#(isNatIListKind(activate(V)),activate(V)) |
(163) |
U32#(tt,V) |
→ |
isNatList#(activate(V)) |
(167) |
U32#(tt,V) |
→ |
activate#(V) |
(168) |
U31#(tt,V) |
→ |
isNatIListKind#(activate(V)) |
(164) |
U61#(tt,V2) |
→ |
isNatIListKind#(activate(V2)) |
(192) |
U61#(tt,V2) |
→ |
activate#(V2) |
(193) |
U31#(tt,V) |
→ |
activate#(V) |
(165) |
isNatIList#(V) |
→ |
isNatIListKind#(activate(V)) |
(220) |
isNatIList#(V) |
→ |
activate#(V) |
(221) |
isNatIList#(n__cons(V1,V2)) |
→ |
U41#(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(222) |
U41#(tt,V1,V2) |
→ |
U42#(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(169) |
U42#(tt,V1,V2) |
→ |
U43#(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(173) |
U43#(tt,V1,V2) |
→ |
U44#(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(177) |
U44#(tt,V1,V2) |
→ |
U45#(isNat(activate(V1)),activate(V2)) |
(181) |
U45#(tt,V2) |
→ |
isNatIList#(activate(V2)) |
(186) |
isNatIList#(n__cons(V1,V2)) |
→ |
isNatKind#(activate(V1)) |
(223) |
isNatIList#(n__cons(V1,V2)) |
→ |
activate#(V1) |
(224) |
isNatIList#(n__cons(V1,V2)) |
→ |
activate#(V2) |
(225) |
U45#(tt,V2) |
→ |
activate#(V2) |
(187) |
U44#(tt,V1,V2) |
→ |
isNat#(activate(V1)) |
(182) |
U44#(tt,V1,V2) |
→ |
activate#(V1) |
(183) |
U44#(tt,V1,V2) |
→ |
activate#(V2) |
(184) |
U43#(tt,V1,V2) |
→ |
isNatIListKind#(activate(V2)) |
(178) |
U43#(tt,V1,V2) |
→ |
activate#(V2) |
(179) |
U43#(tt,V1,V2) |
→ |
activate#(V1) |
(180) |
U42#(tt,V1,V2) |
→ |
isNatIListKind#(activate(V2)) |
(174) |
U42#(tt,V1,V2) |
→ |
activate#(V2) |
(175) |
U42#(tt,V1,V2) |
→ |
activate#(V1) |
(176) |
U41#(tt,V1,V2) |
→ |
isNatKind#(activate(V1)) |
(170) |
U41#(tt,V1,V2) |
→ |
activate#(V1) |
(171) |
U41#(tt,V1,V2) |
→ |
activate#(V2) |
(172) |
U105#(tt,V2) |
→ |
activate#(V2) |
(102) |
U104#(tt,V1,V2) |
→ |
isNat#(activate(V1)) |
(97) |
U104#(tt,V1,V2) |
→ |
activate#(V1) |
(98) |
U104#(tt,V1,V2) |
→ |
activate#(V2) |
(99) |
U103#(tt,V1,V2) |
→ |
isNatIListKind#(activate(V2)) |
(93) |
U103#(tt,V1,V2) |
→ |
activate#(V2) |
(94) |
U103#(tt,V1,V2) |
→ |
activate#(V1) |
(95) |
U102#(tt,V1,V2) |
→ |
isNatIListKind#(activate(V2)) |
(89) |
U102#(tt,V1,V2) |
→ |
activate#(V2) |
(90) |
U102#(tt,V1,V2) |
→ |
activate#(V1) |
(91) |
U101#(tt,V1,V2) |
→ |
isNatKind#(activate(V1)) |
(85) |
U101#(tt,V1,V2) |
→ |
activate#(V1) |
(86) |
U101#(tt,V1,V2) |
→ |
activate#(V2) |
(87) |
U121#(tt,IL) |
→ |
activate#(IL) |
(126) |
take#(0,IL) |
→ |
isNatIList#(IL) |
(253) |
take#(s(M),cons(N,IL)) |
→ |
U131#(isNatIList(activate(IL)),activate(IL),M,N) |
(254) |
U131#(tt,IL,M,N) |
→ |
U132#(isNatIListKind(activate(IL)),activate(IL),activate(M),activate(N)) |
(128) |
U132#(tt,IL,M,N) |
→ |
U133#(isNat(activate(M)),activate(IL),activate(M),activate(N)) |
(133) |
U133#(tt,IL,M,N) |
→ |
U134#(isNatKind(activate(M)),activate(IL),activate(M),activate(N)) |
(138) |
U134#(tt,IL,M,N) |
→ |
U135#(isNat(activate(N)),activate(IL),activate(M),activate(N)) |
(143) |
U135#(tt,IL,M,N) |
→ |
U136#(isNatKind(activate(N)),activate(IL),activate(M),activate(N)) |
(148) |
U136#(tt,IL,M,N) |
→ |
activate#(N) |
(154) |
U136#(tt,IL,M,N) |
→ |
activate#(M) |
(155) |
U136#(tt,IL,M,N) |
→ |
activate#(IL) |
(156) |
U135#(tt,IL,M,N) |
→ |
isNatKind#(activate(N)) |
(149) |
U135#(tt,IL,M,N) |
→ |
activate#(N) |
(150) |
U135#(tt,IL,M,N) |
→ |
activate#(IL) |
(151) |
U135#(tt,IL,M,N) |
→ |
activate#(M) |
(152) |
U134#(tt,IL,M,N) |
→ |
isNat#(activate(N)) |
(144) |
U134#(tt,IL,M,N) |
→ |
activate#(N) |
(145) |
U134#(tt,IL,M,N) |
→ |
activate#(IL) |
(146) |
U134#(tt,IL,M,N) |
→ |
activate#(M) |
(147) |
U133#(tt,IL,M,N) |
→ |
isNatKind#(activate(M)) |
(139) |
U133#(tt,IL,M,N) |
→ |
activate#(M) |
(140) |
U133#(tt,IL,M,N) |
→ |
activate#(IL) |
(141) |
U133#(tt,IL,M,N) |
→ |
activate#(N) |
(142) |
U132#(tt,IL,M,N) |
→ |
isNat#(activate(M)) |
(134) |
U132#(tt,IL,M,N) |
→ |
activate#(M) |
(135) |
U132#(tt,IL,M,N) |
→ |
activate#(IL) |
(136) |
U132#(tt,IL,M,N) |
→ |
activate#(N) |
(137) |
U131#(tt,IL,M,N) |
→ |
isNatIListKind#(activate(IL)) |
(129) |
U131#(tt,IL,M,N) |
→ |
activate#(IL) |
(130) |
U131#(tt,IL,M,N) |
→ |
activate#(M) |
(131) |
U131#(tt,IL,M,N) |
→ |
activate#(N) |
(132) |
take#(s(M),cons(N,IL)) |
→ |
isNatIList#(activate(IL)) |
(255) |
take#(s(M),cons(N,IL)) |
→ |
activate#(IL) |
(256) |
and the following rules have been deleted.
U51#(tt,V2) |
→ |
activate#(V2) |
(190) |
length#(cons(N,L)) |
→ |
U111#(isNatList(activate(L)),activate(L),N) |
(249) |
U111#(tt,L,N) |
→ |
U112#(isNatIListKind(activate(L)),activate(L),activate(N)) |
(106) |
U112#(tt,L,N) |
→ |
U113#(isNat(activate(N)),activate(L),activate(N)) |
(110) |
U113#(tt,L,N) |
→ |
U114#(isNatKind(activate(N)),activate(L)) |
(114) |
U114#(tt,L) |
→ |
length#(activate(L)) |
(119) |
length#(cons(N,L)) |
→ |
isNatList#(activate(L)) |
(250) |
isNatList#(n__cons(V1,V2)) |
→ |
U91#(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(240) |
U91#(tt,V1,V2) |
→ |
U92#(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(194) |
U92#(tt,V1,V2) |
→ |
U93#(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(198) |
U93#(tt,V1,V2) |
→ |
U94#(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(202) |
U94#(tt,V1,V2) |
→ |
U95#(isNat(activate(V1)),activate(V2)) |
(206) |
U95#(tt,V2) |
→ |
isNatList#(activate(V2)) |
(211) |
isNatList#(n__cons(V1,V2)) |
→ |
isNatKind#(activate(V1)) |
(241) |
isNatIListKind#(n__cons(V1,V2)) |
→ |
isNatKind#(activate(V1)) |
(227) |
isNatKind#(n__s(V1)) |
→ |
isNatKind#(activate(V1)) |
(238) |
isNatKind#(n__s(V1)) |
→ |
activate#(V1) |
(239) |
isNatIListKind#(n__cons(V1,V2)) |
→ |
activate#(V1) |
(228) |
isNatIListKind#(n__cons(V1,V2)) |
→ |
activate#(V2) |
(229) |
isNatList#(n__cons(V1,V2)) |
→ |
activate#(V1) |
(242) |
isNatList#(n__cons(V1,V2)) |
→ |
activate#(V2) |
(243) |
U95#(tt,V2) |
→ |
activate#(V2) |
(212) |
U94#(tt,V1,V2) |
→ |
isNat#(activate(V1)) |
(207) |
U11#(tt,V1) |
→ |
U12#(isNatIListKind(activate(V1)),activate(V1)) |
(103) |
U12#(tt,V1) |
→ |
isNatList#(activate(V1)) |
(122) |
U12#(tt,V1) |
→ |
activate#(V1) |
(123) |
U11#(tt,V1) |
→ |
isNatIListKind#(activate(V1)) |
(104) |
U11#(tt,V1) |
→ |
activate#(V1) |
(105) |
isNat#(n__s(V1)) |
→ |
U21#(isNatKind(activate(V1)),activate(V1)) |
(216) |
U21#(tt,V1) |
→ |
U22#(isNatKind(activate(V1)),activate(V1)) |
(157) |
U22#(tt,V1) |
→ |
isNat#(activate(V1)) |
(161) |
isNat#(n__s(V1)) |
→ |
isNatKind#(activate(V1)) |
(217) |
isNat#(n__s(V1)) |
→ |
activate#(V1) |
(218) |
U22#(tt,V1) |
→ |
activate#(V1) |
(162) |
U21#(tt,V1) |
→ |
isNatKind#(activate(V1)) |
(158) |
U21#(tt,V1) |
→ |
activate#(V1) |
(159) |
U94#(tt,V1,V2) |
→ |
activate#(V1) |
(208) |
U94#(tt,V1,V2) |
→ |
activate#(V2) |
(209) |
U93#(tt,V1,V2) |
→ |
isNatIListKind#(activate(V2)) |
(203) |
U93#(tt,V1,V2) |
→ |
activate#(V2) |
(204) |
U93#(tt,V1,V2) |
→ |
activate#(V1) |
(205) |
U92#(tt,V1,V2) |
→ |
isNatIListKind#(activate(V2)) |
(199) |
U92#(tt,V1,V2) |
→ |
activate#(V2) |
(200) |
U92#(tt,V1,V2) |
→ |
activate#(V1) |
(201) |
U91#(tt,V1,V2) |
→ |
isNatKind#(activate(V1)) |
(195) |
U91#(tt,V1,V2) |
→ |
activate#(V1) |
(196) |
U91#(tt,V1,V2) |
→ |
activate#(V2) |
(197) |
length#(cons(N,L)) |
→ |
activate#(L) |
(251) |
U114#(tt,L) |
→ |
activate#(L) |
(120) |
U113#(tt,L,N) |
→ |
isNatKind#(activate(N)) |
(115) |
U113#(tt,L,N) |
→ |
activate#(N) |
(116) |
U113#(tt,L,N) |
→ |
activate#(L) |
(117) |
U112#(tt,L,N) |
→ |
isNat#(activate(N)) |
(111) |
U112#(tt,L,N) |
→ |
activate#(N) |
(112) |
U112#(tt,L,N) |
→ |
activate#(L) |
(113) |
U111#(tt,L,N) |
→ |
isNatIListKind#(activate(L)) |
(107) |
U111#(tt,L,N) |
→ |
activate#(L) |
(108) |
U111#(tt,L,N) |
→ |
activate#(N) |
(109) |
and the following rules have been deleted.
isNatIListKind#(n__cons(n__zeros,y1)) |
→ |
U51#(isNatKind(zeros),activate(y1)) |
(264) |
isNatIListKind#(n__cons(n__take(x0,x1),y1)) |
→ |
U51#(isNatKind(take(x0,x1)),activate(y1)) |
(265) |
isNatIListKind#(n__cons(n__0,y1)) |
→ |
U51#(isNatKind(0),activate(y1)) |
(266) |
isNatIListKind#(n__cons(n__length(x0),y1)) |
→ |
U51#(isNatKind(length(x0)),activate(y1)) |
(267) |
isNatIListKind#(n__cons(n__s(x0),y1)) |
→ |
U51#(isNatKind(s(x0)),activate(y1)) |
(268) |
isNatIListKind#(n__cons(n__cons(x0,x1),y1)) |
→ |
U51#(isNatKind(cons(x0,x1)),activate(y1)) |
(269) |
isNatIListKind#(n__cons(n__nil,y1)) |
→ |
U51#(isNatKind(nil),activate(y1)) |
(270) |
isNatIListKind#(n__cons(x0,y1)) |
→ |
U51#(isNatKind(x0),activate(y1)) |
(271) |