The rewrite relation of the following TRS is considered.
zeros |
→ |
cons(0,n__zeros) |
(1) |
U11(tt,V1) |
→ |
U12(isNatIListKind(activate(V1)),activate(V1)) |
(2) |
U12(tt,V1) |
→ |
U13(isNatList(activate(V1))) |
(3) |
U13(tt) |
→ |
tt |
(4) |
U21(tt,V1) |
→ |
U22(isNatKind(activate(V1)),activate(V1)) |
(5) |
U22(tt,V1) |
→ |
U23(isNat(activate(V1))) |
(6) |
U23(tt) |
→ |
tt |
(7) |
U31(tt,V) |
→ |
U32(isNatIListKind(activate(V)),activate(V)) |
(8) |
U32(tt,V) |
→ |
U33(isNatList(activate(V))) |
(9) |
U33(tt) |
→ |
tt |
(10) |
U41(tt,V1,V2) |
→ |
U42(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(11) |
U42(tt,V1,V2) |
→ |
U43(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(12) |
U43(tt,V1,V2) |
→ |
U44(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(13) |
U44(tt,V1,V2) |
→ |
U45(isNat(activate(V1)),activate(V2)) |
(14) |
U45(tt,V2) |
→ |
U46(isNatIList(activate(V2))) |
(15) |
U46(tt) |
→ |
tt |
(16) |
U51(tt,V2) |
→ |
U52(isNatIListKind(activate(V2))) |
(17) |
U52(tt) |
→ |
tt |
(18) |
U61(tt) |
→ |
tt |
(19) |
U71(tt) |
→ |
tt |
(20) |
U81(tt,V1,V2) |
→ |
U82(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(21) |
U82(tt,V1,V2) |
→ |
U83(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(22) |
U83(tt,V1,V2) |
→ |
U84(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(23) |
U84(tt,V1,V2) |
→ |
U85(isNat(activate(V1)),activate(V2)) |
(24) |
U85(tt,V2) |
→ |
U86(isNatList(activate(V2))) |
(25) |
U86(tt) |
→ |
tt |
(26) |
U91(tt,L,N) |
→ |
U92(isNatIListKind(activate(L)),activate(L),activate(N)) |
(27) |
U92(tt,L,N) |
→ |
U93(isNat(activate(N)),activate(L),activate(N)) |
(28) |
U93(tt,L,N) |
→ |
U94(isNatKind(activate(N)),activate(L)) |
(29) |
U94(tt,L) |
→ |
s(length(activate(L))) |
(30) |
isNat(n__0) |
→ |
tt |
(31) |
isNat(n__length(V1)) |
→ |
U11(isNatIListKind(activate(V1)),activate(V1)) |
(32) |
isNat(n__s(V1)) |
→ |
U21(isNatKind(activate(V1)),activate(V1)) |
(33) |
isNatIList(V) |
→ |
U31(isNatIListKind(activate(V)),activate(V)) |
(34) |
isNatIList(n__zeros) |
→ |
tt |
(35) |
isNatIList(n__cons(V1,V2)) |
→ |
U41(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(36) |
isNatIListKind(n__nil) |
→ |
tt |
(37) |
isNatIListKind(n__zeros) |
→ |
tt |
(38) |
isNatIListKind(n__cons(V1,V2)) |
→ |
U51(isNatKind(activate(V1)),activate(V2)) |
(39) |
isNatKind(n__0) |
→ |
tt |
(40) |
isNatKind(n__length(V1)) |
→ |
U61(isNatIListKind(activate(V1))) |
(41) |
isNatKind(n__s(V1)) |
→ |
U71(isNatKind(activate(V1))) |
(42) |
isNatList(n__nil) |
→ |
tt |
(43) |
isNatList(n__cons(V1,V2)) |
→ |
U81(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(44) |
length(nil) |
→ |
0 |
(45) |
length(cons(N,L)) |
→ |
U91(isNatList(activate(L)),activate(L),N) |
(46) |
zeros |
→ |
n__zeros |
(47) |
0 |
→ |
n__0 |
(48) |
length(X) |
→ |
n__length(X) |
(49) |
s(X) |
→ |
n__s(X) |
(50) |
cons(X1,X2) |
→ |
n__cons(X1,X2) |
(51) |
nil |
→ |
n__nil |
(52) |
activate(n__zeros) |
→ |
zeros |
(53) |
activate(n__0) |
→ |
0 |
(54) |
activate(n__length(X)) |
→ |
length(X) |
(55) |
activate(n__s(X)) |
→ |
s(X) |
(56) |
activate(n__cons(X1,X2)) |
→ |
cons(X1,X2) |
(57) |
activate(n__nil) |
→ |
nil |
(58) |
activate(X) |
→ |
X |
(59) |
There are 113 ruless (increase limit for explicit display).
It remains to prove infiniteness of the resulting DP problem.
zeros# |
→ |
cons#(0,n__zeros) |
(60) |
zeros# |
→ |
0# |
(61) |
U12#(tt,V1) |
→ |
U13#(isNatList(activate(V1))) |
(65) |
U22#(tt,V1) |
→ |
U23#(isNat(activate(V1))) |
(71) |
U31#(tt,V) |
→ |
U32#(isNatIListKind(activate(V)),activate(V)) |
(74) |
U31#(tt,V) |
→ |
isNatIListKind#(activate(V)) |
(75) |
U31#(tt,V) |
→ |
activate#(V) |
(76) |
U32#(tt,V) |
→ |
U33#(isNatList(activate(V))) |
(77) |
U32#(tt,V) |
→ |
isNatList#(activate(V)) |
(78) |
U32#(tt,V) |
→ |
activate#(V) |
(79) |
U41#(tt,V1,V2) |
→ |
U42#(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(80) |
U41#(tt,V1,V2) |
→ |
isNatKind#(activate(V1)) |
(81) |
U41#(tt,V1,V2) |
→ |
activate#(V1) |
(82) |
U41#(tt,V1,V2) |
→ |
activate#(V2) |
(83) |
U42#(tt,V1,V2) |
→ |
U43#(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(84) |
U42#(tt,V1,V2) |
→ |
isNatIListKind#(activate(V2)) |
(85) |
U42#(tt,V1,V2) |
→ |
activate#(V2) |
(86) |
U42#(tt,V1,V2) |
→ |
activate#(V1) |
(87) |
U43#(tt,V1,V2) |
→ |
U44#(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(88) |
U43#(tt,V1,V2) |
→ |
isNatIListKind#(activate(V2)) |
(89) |
U43#(tt,V1,V2) |
→ |
activate#(V2) |
(90) |
U43#(tt,V1,V2) |
→ |
activate#(V1) |
(91) |
U44#(tt,V1,V2) |
→ |
U45#(isNat(activate(V1)),activate(V2)) |
(92) |
U44#(tt,V1,V2) |
→ |
isNat#(activate(V1)) |
(93) |
U44#(tt,V1,V2) |
→ |
activate#(V1) |
(94) |
U44#(tt,V1,V2) |
→ |
activate#(V2) |
(95) |
U45#(tt,V2) |
→ |
U46#(isNatIList(activate(V2))) |
(96) |
U45#(tt,V2) |
→ |
isNatIList#(activate(V2)) |
(97) |
U45#(tt,V2) |
→ |
activate#(V2) |
(98) |
U51#(tt,V2) |
→ |
U52#(isNatIListKind(activate(V2))) |
(99) |
U85#(tt,V2) |
→ |
U86#(isNatList(activate(V2))) |
(118) |
U94#(tt,L) |
→ |
s#(length(activate(L))) |
(133) |
isNatIList#(V) |
→ |
U31#(isNatIListKind(activate(V)),activate(V)) |
(142) |
isNatIList#(V) |
→ |
isNatIListKind#(activate(V)) |
(143) |
isNatIList#(V) |
→ |
activate#(V) |
(144) |
isNatIList#(n__cons(V1,V2)) |
→ |
U41#(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(145) |
isNatIList#(n__cons(V1,V2)) |
→ |
isNatKind#(activate(V1)) |
(146) |
isNatIList#(n__cons(V1,V2)) |
→ |
activate#(V1) |
(147) |
isNatIList#(n__cons(V1,V2)) |
→ |
activate#(V2) |
(148) |
isNatKind#(n__length(V1)) |
→ |
U61#(isNatIListKind(activate(V1))) |
(153) |
isNatKind#(n__s(V1)) |
→ |
U71#(isNatKind(activate(V1))) |
(156) |
length#(nil) |
→ |
0# |
(163) |
activate#(n__zeros) |
→ |
zeros# |
(167) |
activate#(n__0) |
→ |
0# |
(168) |
activate#(n__s(X)) |
→ |
s#(X) |
(170) |
activate#(n__cons(X1,X2)) |
→ |
cons#(X1,X2) |
(171) |
activate#(n__nil) |
→ |
nil# |
(172) |
and the following rules have been deleted.
isNatList#(n__cons(V1,V2)) |
→ |
isNatKind#(activate(V1)) |
(160) |
isNatKind#(n__length(V1)) |
→ |
isNatIListKind#(activate(V1)) |
(154) |
isNatIListKind#(n__cons(V1,V2)) |
→ |
isNatKind#(activate(V1)) |
(150) |
isNatKind#(n__length(V1)) |
→ |
activate#(V1) |
(155) |
isNatIListKind#(n__cons(V1,V2)) |
→ |
activate#(V1) |
(151) |
isNatIListKind#(n__cons(V1,V2)) |
→ |
activate#(V2) |
(152) |
U51#(tt,V2) |
→ |
activate#(V2) |
(101) |
isNatList#(n__cons(V1,V2)) |
→ |
activate#(V1) |
(161) |
isNatList#(n__cons(V1,V2)) |
→ |
activate#(V2) |
(162) |
U85#(tt,V2) |
→ |
activate#(V2) |
(120) |
U84#(tt,V1,V2) |
→ |
isNat#(activate(V1)) |
(115) |
U12#(tt,V1) |
→ |
activate#(V1) |
(67) |
U11#(tt,V1) |
→ |
isNatIListKind#(activate(V1)) |
(63) |
U11#(tt,V1) |
→ |
activate#(V1) |
(64) |
isNat#(n__length(V1)) |
→ |
isNatIListKind#(activate(V1)) |
(137) |
isNat#(n__length(V1)) |
→ |
activate#(V1) |
(138) |
U84#(tt,V1,V2) |
→ |
activate#(V1) |
(116) |
U84#(tt,V1,V2) |
→ |
activate#(V2) |
(117) |
U83#(tt,V1,V2) |
→ |
isNatIListKind#(activate(V2)) |
(111) |
U83#(tt,V1,V2) |
→ |
activate#(V2) |
(112) |
U83#(tt,V1,V2) |
→ |
activate#(V1) |
(113) |
U82#(tt,V1,V2) |
→ |
isNatIListKind#(activate(V2)) |
(107) |
U82#(tt,V1,V2) |
→ |
activate#(V2) |
(108) |
U82#(tt,V1,V2) |
→ |
activate#(V1) |
(109) |
U81#(tt,V1,V2) |
→ |
isNatKind#(activate(V1)) |
(103) |
U81#(tt,V1,V2) |
→ |
activate#(V1) |
(104) |
U81#(tt,V1,V2) |
→ |
activate#(V2) |
(105) |
length#(cons(N,L)) |
→ |
activate#(L) |
(166) |
U94#(tt,L) |
→ |
activate#(L) |
(135) |
U93#(tt,L,N) |
→ |
isNatKind#(activate(N)) |
(130) |
U93#(tt,L,N) |
→ |
activate#(N) |
(131) |
U93#(tt,L,N) |
→ |
activate#(L) |
(132) |
U92#(tt,L,N) |
→ |
isNat#(activate(N)) |
(126) |
U92#(tt,L,N) |
→ |
activate#(N) |
(127) |
U92#(tt,L,N) |
→ |
activate#(L) |
(128) |
U91#(tt,L,N) |
→ |
isNatIListKind#(activate(L)) |
(122) |
U91#(tt,L,N) |
→ |
activate#(L) |
(123) |
U91#(tt,L,N) |
→ |
activate#(N) |
(124) |
and the following rules have been deleted.
activate#(n__length(X)) |
→ |
length#(X) |
(169) |
length#(cons(N,L)) |
→ |
U91#(isNatList(activate(L)),activate(L),N) |
(164) |
U91#(tt,L,N) |
→ |
U92#(isNatIListKind(activate(L)),activate(L),activate(N)) |
(121) |
U92#(tt,L,N) |
→ |
U93#(isNat(activate(N)),activate(L),activate(N)) |
(125) |
U93#(tt,L,N) |
→ |
U94#(isNatKind(activate(N)),activate(L)) |
(129) |
U94#(tt,L) |
→ |
length#(activate(L)) |
(134) |
length#(cons(N,L)) |
→ |
isNatList#(activate(L)) |
(165) |
isNatList#(n__cons(V1,V2)) |
→ |
U81#(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(159) |
U81#(tt,V1,V2) |
→ |
U82#(isNatKind(activate(V1)),activate(V1),activate(V2)) |
(102) |
U82#(tt,V1,V2) |
→ |
U83#(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(106) |
U83#(tt,V1,V2) |
→ |
U84#(isNatIListKind(activate(V2)),activate(V1),activate(V2)) |
(110) |
U84#(tt,V1,V2) |
→ |
U85#(isNat(activate(V1)),activate(V2)) |
(114) |
U85#(tt,V2) |
→ |
isNatList#(activate(V2)) |
(119) |
isNatKind#(n__s(V1)) |
→ |
isNatKind#(activate(V1)) |
(157) |
isNatKind#(n__s(V1)) |
→ |
activate#(V1) |
(158) |
isNat#(n__length(V1)) |
→ |
U11#(isNatIListKind(activate(V1)),activate(V1)) |
(136) |
U11#(tt,V1) |
→ |
U12#(isNatIListKind(activate(V1)),activate(V1)) |
(62) |
U12#(tt,V1) |
→ |
isNatList#(activate(V1)) |
(66) |
isNat#(n__s(V1)) |
→ |
U21#(isNatKind(activate(V1)),activate(V1)) |
(139) |
U21#(tt,V1) |
→ |
U22#(isNatKind(activate(V1)),activate(V1)) |
(68) |
U22#(tt,V1) |
→ |
isNat#(activate(V1)) |
(72) |
isNat#(n__s(V1)) |
→ |
isNatKind#(activate(V1)) |
(140) |
isNat#(n__s(V1)) |
→ |
activate#(V1) |
(141) |
U22#(tt,V1) |
→ |
activate#(V1) |
(73) |
U21#(tt,V1) |
→ |
isNatKind#(activate(V1)) |
(69) |
U21#(tt,V1) |
→ |
activate#(V1) |
(70) |
and the following rules have been deleted.
isNatIListKind#(n__cons(n__zeros,y1)) |
→ |
U51#(isNatKind(zeros),activate(y1)) |
(180) |
isNatIListKind#(n__cons(n__0,y1)) |
→ |
U51#(isNatKind(0),activate(y1)) |
(181) |
isNatIListKind#(n__cons(n__length(x0),y1)) |
→ |
U51#(isNatKind(length(x0)),activate(y1)) |
(182) |
isNatIListKind#(n__cons(n__s(x0),y1)) |
→ |
U51#(isNatKind(s(x0)),activate(y1)) |
(183) |
isNatIListKind#(n__cons(n__cons(x0,x1),y1)) |
→ |
U51#(isNatKind(cons(x0,x1)),activate(y1)) |
(184) |
isNatIListKind#(n__cons(n__nil,y1)) |
→ |
U51#(isNatKind(nil),activate(y1)) |
(185) |
isNatIListKind#(n__cons(x0,y1)) |
→ |
U51#(isNatKind(x0),activate(y1)) |
(186) |