The rewrite relation of the following TRS is considered.
zeros |
→ |
cons(0,n__zeros) |
(1) |
U11(tt) |
→ |
tt |
(2) |
U21(tt) |
→ |
tt |
(3) |
U31(tt) |
→ |
tt |
(4) |
U41(tt,V2) |
→ |
U42(isNatIList(activate(V2))) |
(5) |
U42(tt) |
→ |
tt |
(6) |
U51(tt,V2) |
→ |
U52(isNatList(activate(V2))) |
(7) |
U52(tt) |
→ |
tt |
(8) |
U61(tt,V2) |
→ |
U62(isNatIList(activate(V2))) |
(9) |
U62(tt) |
→ |
tt |
(10) |
U71(tt,L,N) |
→ |
U72(isNat(activate(N)),activate(L)) |
(11) |
U72(tt,L) |
→ |
s(length(activate(L))) |
(12) |
U81(tt) |
→ |
nil |
(13) |
U91(tt,IL,M,N) |
→ |
U92(isNat(activate(M)),activate(IL),activate(M),activate(N)) |
(14) |
U92(tt,IL,M,N) |
→ |
U93(isNat(activate(N)),activate(IL),activate(M),activate(N)) |
(15) |
U93(tt,IL,M,N) |
→ |
cons(activate(N),n__take(activate(M),activate(IL))) |
(16) |
isNat(n__0) |
→ |
tt |
(17) |
isNat(n__length(V1)) |
→ |
U11(isNatList(activate(V1))) |
(18) |
isNat(n__s(V1)) |
→ |
U21(isNat(activate(V1))) |
(19) |
isNatIList(V) |
→ |
U31(isNatList(activate(V))) |
(20) |
isNatIList(n__zeros) |
→ |
tt |
(21) |
isNatIList(n__cons(V1,V2)) |
→ |
U41(isNat(activate(V1)),activate(V2)) |
(22) |
isNatList(n__nil) |
→ |
tt |
(23) |
isNatList(n__cons(V1,V2)) |
→ |
U51(isNat(activate(V1)),activate(V2)) |
(24) |
isNatList(n__take(V1,V2)) |
→ |
U61(isNat(activate(V1)),activate(V2)) |
(25) |
length(nil) |
→ |
0 |
(26) |
length(cons(N,L)) |
→ |
U71(isNatList(activate(L)),activate(L),N) |
(27) |
take(0,IL) |
→ |
U81(isNatIList(IL)) |
(28) |
take(s(M),cons(N,IL)) |
→ |
U91(isNatIList(activate(IL)),activate(IL),M,N) |
(29) |
zeros |
→ |
n__zeros |
(30) |
take(X1,X2) |
→ |
n__take(X1,X2) |
(31) |
0 |
→ |
n__0 |
(32) |
length(X) |
→ |
n__length(X) |
(33) |
s(X) |
→ |
n__s(X) |
(34) |
cons(X1,X2) |
→ |
n__cons(X1,X2) |
(35) |
nil |
→ |
n__nil |
(36) |
activate(n__zeros) |
→ |
zeros |
(37) |
activate(n__take(X1,X2)) |
→ |
take(activate(X1),activate(X2)) |
(38) |
activate(n__0) |
→ |
0 |
(39) |
activate(n__length(X)) |
→ |
length(activate(X)) |
(40) |
activate(n__s(X)) |
→ |
s(activate(X)) |
(41) |
activate(n__cons(X1,X2)) |
→ |
cons(activate(X1),X2) |
(42) |
activate(n__nil) |
→ |
nil |
(43) |
activate(X) |
→ |
X |
(44) |
zeros# |
→ |
cons#(0,n__zeros) |
(45) |
zeros# |
→ |
0# |
(46) |
U41#(tt,V2) |
→ |
U42#(isNatIList(activate(V2))) |
(47) |
U41#(tt,V2) |
→ |
isNatIList#(activate(V2)) |
(48) |
U41#(tt,V2) |
→ |
activate#(V2) |
(49) |
U51#(tt,V2) |
→ |
U52#(isNatList(activate(V2))) |
(50) |
U51#(tt,V2) |
→ |
isNatList#(activate(V2)) |
(51) |
U51#(tt,V2) |
→ |
activate#(V2) |
(52) |
U61#(tt,V2) |
→ |
U62#(isNatIList(activate(V2))) |
(53) |
U61#(tt,V2) |
→ |
isNatIList#(activate(V2)) |
(54) |
U61#(tt,V2) |
→ |
activate#(V2) |
(55) |
U71#(tt,L,N) |
→ |
U72#(isNat(activate(N)),activate(L)) |
(56) |
U71#(tt,L,N) |
→ |
isNat#(activate(N)) |
(57) |
U71#(tt,L,N) |
→ |
activate#(N) |
(58) |
U71#(tt,L,N) |
→ |
activate#(L) |
(59) |
U72#(tt,L) |
→ |
s#(length(activate(L))) |
(60) |
U72#(tt,L) |
→ |
length#(activate(L)) |
(61) |
U72#(tt,L) |
→ |
activate#(L) |
(62) |
U81#(tt) |
→ |
nil# |
(63) |
U91#(tt,IL,M,N) |
→ |
U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) |
(64) |
U91#(tt,IL,M,N) |
→ |
isNat#(activate(M)) |
(65) |
U91#(tt,IL,M,N) |
→ |
activate#(M) |
(66) |
U91#(tt,IL,M,N) |
→ |
activate#(IL) |
(67) |
U91#(tt,IL,M,N) |
→ |
activate#(N) |
(68) |
U92#(tt,IL,M,N) |
→ |
U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) |
(69) |
U92#(tt,IL,M,N) |
→ |
isNat#(activate(N)) |
(70) |
U92#(tt,IL,M,N) |
→ |
activate#(N) |
(71) |
U92#(tt,IL,M,N) |
→ |
activate#(IL) |
(72) |
U92#(tt,IL,M,N) |
→ |
activate#(M) |
(73) |
U93#(tt,IL,M,N) |
→ |
cons#(activate(N),n__take(activate(M),activate(IL))) |
(74) |
U93#(tt,IL,M,N) |
→ |
activate#(N) |
(75) |
U93#(tt,IL,M,N) |
→ |
activate#(M) |
(76) |
U93#(tt,IL,M,N) |
→ |
activate#(IL) |
(77) |
isNat#(n__length(V1)) |
→ |
U11#(isNatList(activate(V1))) |
(78) |
isNat#(n__length(V1)) |
→ |
isNatList#(activate(V1)) |
(79) |
isNat#(n__length(V1)) |
→ |
activate#(V1) |
(80) |
isNat#(n__s(V1)) |
→ |
U21#(isNat(activate(V1))) |
(81) |
isNat#(n__s(V1)) |
→ |
isNat#(activate(V1)) |
(82) |
isNat#(n__s(V1)) |
→ |
activate#(V1) |
(83) |
isNatIList#(V) |
→ |
U31#(isNatList(activate(V))) |
(84) |
isNatIList#(V) |
→ |
isNatList#(activate(V)) |
(85) |
isNatIList#(V) |
→ |
activate#(V) |
(86) |
isNatIList#(n__cons(V1,V2)) |
→ |
U41#(isNat(activate(V1)),activate(V2)) |
(87) |
isNatIList#(n__cons(V1,V2)) |
→ |
isNat#(activate(V1)) |
(88) |
isNatIList#(n__cons(V1,V2)) |
→ |
activate#(V1) |
(89) |
isNatIList#(n__cons(V1,V2)) |
→ |
activate#(V2) |
(90) |
isNatList#(n__cons(V1,V2)) |
→ |
U51#(isNat(activate(V1)),activate(V2)) |
(91) |
isNatList#(n__cons(V1,V2)) |
→ |
isNat#(activate(V1)) |
(92) |
isNatList#(n__cons(V1,V2)) |
→ |
activate#(V1) |
(93) |
isNatList#(n__cons(V1,V2)) |
→ |
activate#(V2) |
(94) |
isNatList#(n__take(V1,V2)) |
→ |
U61#(isNat(activate(V1)),activate(V2)) |
(95) |
isNatList#(n__take(V1,V2)) |
→ |
isNat#(activate(V1)) |
(96) |
isNatList#(n__take(V1,V2)) |
→ |
activate#(V1) |
(97) |
isNatList#(n__take(V1,V2)) |
→ |
activate#(V2) |
(98) |
length#(nil) |
→ |
0# |
(99) |
length#(cons(N,L)) |
→ |
U71#(isNatList(activate(L)),activate(L),N) |
(100) |
length#(cons(N,L)) |
→ |
isNatList#(activate(L)) |
(101) |
length#(cons(N,L)) |
→ |
activate#(L) |
(102) |
take#(0,IL) |
→ |
U81#(isNatIList(IL)) |
(103) |
take#(0,IL) |
→ |
isNatIList#(IL) |
(104) |
take#(s(M),cons(N,IL)) |
→ |
U91#(isNatIList(activate(IL)),activate(IL),M,N) |
(105) |
take#(s(M),cons(N,IL)) |
→ |
isNatIList#(activate(IL)) |
(106) |
take#(s(M),cons(N,IL)) |
→ |
activate#(IL) |
(107) |
activate#(n__zeros) |
→ |
zeros# |
(108) |
activate#(n__take(X1,X2)) |
→ |
take#(activate(X1),activate(X2)) |
(109) |
activate#(n__take(X1,X2)) |
→ |
activate#(X1) |
(110) |
activate#(n__take(X1,X2)) |
→ |
activate#(X2) |
(111) |
activate#(n__0) |
→ |
0# |
(112) |
activate#(n__length(X)) |
→ |
length#(activate(X)) |
(113) |
activate#(n__length(X)) |
→ |
activate#(X) |
(114) |
activate#(n__s(X)) |
→ |
s#(activate(X)) |
(115) |
activate#(n__s(X)) |
→ |
activate#(X) |
(116) |
activate#(n__cons(X1,X2)) |
→ |
cons#(activate(X1),X2) |
(117) |
activate#(n__cons(X1,X2)) |
→ |
activate#(X1) |
(118) |
activate#(n__nil) |
→ |
nil# |
(119) |
It remains to prove infiniteness of the resulting DP problem.
activate#(n__take(X1,X2)) |
→ |
take#(activate(X1),activate(X2)) |
(109) |
isNatList#(n__cons(V1,V2)) |
→ |
isNat#(activate(V1)) |
(92) |
isNat#(n__length(V1)) |
→ |
isNatList#(activate(V1)) |
(79) |
isNatList#(n__cons(V1,V2)) |
→ |
activate#(V1) |
(93) |
activate#(n__take(X1,X2)) |
→ |
activate#(X1) |
(110) |
activate#(n__take(X1,X2)) |
→ |
activate#(X2) |
(111) |
length#(cons(N,L)) |
→ |
isNatList#(activate(L)) |
(101) |
isNatList#(n__cons(V1,V2)) |
→ |
activate#(V2) |
(94) |
activate#(n__length(X)) |
→ |
activate#(X) |
(114) |
isNatList#(n__take(V1,V2)) |
→ |
U61#(isNat(activate(V1)),activate(V2)) |
(95) |
isNatIList#(V) |
→ |
activate#(V) |
(86) |
isNatIList#(n__cons(V1,V2)) |
→ |
isNat#(activate(V1)) |
(88) |
isNat#(n__length(V1)) |
→ |
activate#(V1) |
(80) |
isNatIList#(n__cons(V1,V2)) |
→ |
activate#(V1) |
(89) |
isNatIList#(n__cons(V1,V2)) |
→ |
activate#(V2) |
(90) |
U41#(tt,V2) |
→ |
activate#(V2) |
(49) |
U61#(tt,V2) |
→ |
activate#(V2) |
(55) |
isNatList#(n__take(V1,V2)) |
→ |
isNat#(activate(V1)) |
(96) |
isNatList#(n__take(V1,V2)) |
→ |
activate#(V1) |
(97) |
isNatList#(n__take(V1,V2)) |
→ |
activate#(V2) |
(98) |
length#(cons(N,L)) |
→ |
activate#(L) |
(102) |
U72#(tt,L) |
→ |
activate#(L) |
(62) |
U71#(tt,L,N) |
→ |
isNat#(activate(N)) |
(57) |
U71#(tt,L,N) |
→ |
activate#(N) |
(58) |
U71#(tt,L,N) |
→ |
activate#(L) |
(59) |
U51#(tt,V2) |
→ |
activate#(V2) |
(52) |
U93#(tt,IL,M,N) |
→ |
activate#(N) |
(75) |
U93#(tt,IL,M,N) |
→ |
activate#(M) |
(76) |
U93#(tt,IL,M,N) |
→ |
activate#(IL) |
(77) |
U92#(tt,IL,M,N) |
→ |
isNat#(activate(N)) |
(70) |
U92#(tt,IL,M,N) |
→ |
activate#(N) |
(71) |
U92#(tt,IL,M,N) |
→ |
activate#(IL) |
(72) |
U92#(tt,IL,M,N) |
→ |
activate#(M) |
(73) |
U91#(tt,IL,M,N) |
→ |
isNat#(activate(M)) |
(65) |
U91#(tt,IL,M,N) |
→ |
activate#(M) |
(66) |
U91#(tt,IL,M,N) |
→ |
activate#(IL) |
(67) |
U91#(tt,IL,M,N) |
→ |
activate#(N) |
(68) |
take#(s(M),cons(N,IL)) |
→ |
activate#(IL) |
(107) |
and the following rules have been deleted.
take#(0,IL) |
→ |
isNatIList#(IL) |
(104) |
isNatIList#(V) |
→ |
isNatList#(activate(V)) |
(85) |
activate#(n__length(X)) |
→ |
length#(activate(X)) |
(113) |
length#(cons(N,L)) |
→ |
U71#(isNatList(activate(L)),activate(L),N) |
(100) |
U71#(tt,L,N) |
→ |
U72#(isNat(activate(N)),activate(L)) |
(56) |
U72#(tt,L) |
→ |
length#(activate(L)) |
(61) |
activate#(n__s(X)) |
→ |
activate#(X) |
(116) |
activate#(n__cons(X1,X2)) |
→ |
activate#(X1) |
(118) |
U61#(tt,V2) |
→ |
isNatIList#(activate(V2)) |
(54) |
isNatIList#(n__cons(V1,V2)) |
→ |
U41#(isNat(activate(V1)),activate(V2)) |
(87) |
U41#(tt,V2) |
→ |
isNatIList#(activate(V2)) |
(48) |
isNat#(n__s(V1)) |
→ |
isNat#(activate(V1)) |
(82) |
isNat#(n__s(V1)) |
→ |
activate#(V1) |
(83) |
take#(s(M),cons(N,IL)) |
→ |
U91#(isNatIList(activate(IL)),activate(IL),M,N) |
(105) |
U91#(tt,IL,M,N) |
→ |
U92#(isNat(activate(M)),activate(IL),activate(M),activate(N)) |
(64) |
U92#(tt,IL,M,N) |
→ |
U93#(isNat(activate(N)),activate(IL),activate(M),activate(N)) |
(69) |
take#(s(M),cons(N,IL)) |
→ |
isNatIList#(activate(IL)) |
(106) |
and the following rules have been deleted.
isNatList#(n__cons(n__zeros,y1)) |
→ |
U51#(isNat(zeros),activate(y1)) |
(128) |
isNatList#(n__cons(n__take(x0,x1),y1)) |
→ |
U51#(isNat(take(activate(x0),activate(x1))),activate(y1)) |
(129) |
isNatList#(n__cons(n__0,y1)) |
→ |
U51#(isNat(0),activate(y1)) |
(130) |
isNatList#(n__cons(n__length(x0),y1)) |
→ |
U51#(isNat(length(activate(x0))),activate(y1)) |
(131) |
isNatList#(n__cons(n__s(x0),y1)) |
→ |
U51#(isNat(s(activate(x0))),activate(y1)) |
(132) |
isNatList#(n__cons(n__cons(x0,x1),y1)) |
→ |
U51#(isNat(cons(activate(x0),x1)),activate(y1)) |
(133) |
isNatList#(n__cons(n__nil,y1)) |
→ |
U51#(isNat(nil),activate(y1)) |
(134) |
isNatList#(n__cons(x0,y1)) |
→ |
U51#(isNat(x0),activate(y1)) |
(135) |