The rewrite relation of the following TRS is considered.
and(tt,T) |
→ |
T |
(1) |
isNatIList(IL) |
→ |
isNatList(activate(IL)) |
(2) |
isNat(n__0) |
→ |
tt |
(3) |
isNat(n__s(N)) |
→ |
isNat(activate(N)) |
(4) |
isNat(n__length(L)) |
→ |
isNatList(activate(L)) |
(5) |
isNatIList(n__zeros) |
→ |
tt |
(6) |
isNatIList(n__cons(N,IL)) |
→ |
and(isNat(activate(N)),isNatIList(activate(IL))) |
(7) |
isNatList(n__nil) |
→ |
tt |
(8) |
isNatList(n__cons(N,L)) |
→ |
and(isNat(activate(N)),isNatList(activate(L))) |
(9) |
isNatList(n__take(N,IL)) |
→ |
and(isNat(activate(N)),isNatIList(activate(IL))) |
(10) |
zeros |
→ |
cons(0,n__zeros) |
(11) |
take(0,IL) |
→ |
uTake1(isNatIList(IL)) |
(12) |
uTake1(tt) |
→ |
nil |
(13) |
take(s(M),cons(N,IL)) |
→ |
uTake2(and(isNat(M),and(isNat(N),isNatIList(activate(IL)))),M,N,activate(IL)) |
(14) |
uTake2(tt,M,N,IL) |
→ |
cons(activate(N),n__take(activate(M),activate(IL))) |
(15) |
length(cons(N,L)) |
→ |
uLength(and(isNat(N),isNatList(activate(L))),activate(L)) |
(16) |
uLength(tt,L) |
→ |
s(length(activate(L))) |
(17) |
0 |
→ |
n__0 |
(18) |
s(X) |
→ |
n__s(X) |
(19) |
length(X) |
→ |
n__length(X) |
(20) |
zeros |
→ |
n__zeros |
(21) |
cons(X1,X2) |
→ |
n__cons(X1,X2) |
(22) |
nil |
→ |
n__nil |
(23) |
take(X1,X2) |
→ |
n__take(X1,X2) |
(24) |
activate(n__0) |
→ |
0 |
(25) |
activate(n__s(X)) |
→ |
s(X) |
(26) |
activate(n__length(X)) |
→ |
length(X) |
(27) |
activate(n__zeros) |
→ |
zeros |
(28) |
activate(n__cons(X1,X2)) |
→ |
cons(X1,X2) |
(29) |
activate(n__nil) |
→ |
nil |
(30) |
activate(n__take(X1,X2)) |
→ |
take(X1,X2) |
(31) |
activate(X) |
→ |
X |
(32) |
isNatIList#(IL) |
→ |
isNatList#(activate(IL)) |
(33) |
isNatIList#(IL) |
→ |
activate#(IL) |
(34) |
isNat#(n__s(N)) |
→ |
isNat#(activate(N)) |
(35) |
isNat#(n__s(N)) |
→ |
activate#(N) |
(36) |
isNat#(n__length(L)) |
→ |
isNatList#(activate(L)) |
(37) |
isNat#(n__length(L)) |
→ |
activate#(L) |
(38) |
isNatIList#(n__cons(N,IL)) |
→ |
and#(isNat(activate(N)),isNatIList(activate(IL))) |
(39) |
isNatIList#(n__cons(N,IL)) |
→ |
isNat#(activate(N)) |
(40) |
isNatIList#(n__cons(N,IL)) |
→ |
activate#(N) |
(41) |
isNatIList#(n__cons(N,IL)) |
→ |
isNatIList#(activate(IL)) |
(42) |
isNatIList#(n__cons(N,IL)) |
→ |
activate#(IL) |
(43) |
isNatList#(n__cons(N,L)) |
→ |
and#(isNat(activate(N)),isNatList(activate(L))) |
(44) |
isNatList#(n__cons(N,L)) |
→ |
isNat#(activate(N)) |
(45) |
isNatList#(n__cons(N,L)) |
→ |
activate#(N) |
(46) |
isNatList#(n__cons(N,L)) |
→ |
isNatList#(activate(L)) |
(47) |
isNatList#(n__cons(N,L)) |
→ |
activate#(L) |
(48) |
isNatList#(n__take(N,IL)) |
→ |
and#(isNat(activate(N)),isNatIList(activate(IL))) |
(49) |
isNatList#(n__take(N,IL)) |
→ |
isNat#(activate(N)) |
(50) |
isNatList#(n__take(N,IL)) |
→ |
activate#(N) |
(51) |
isNatList#(n__take(N,IL)) |
→ |
isNatIList#(activate(IL)) |
(52) |
isNatList#(n__take(N,IL)) |
→ |
activate#(IL) |
(53) |
zeros# |
→ |
cons#(0,n__zeros) |
(54) |
zeros# |
→ |
0# |
(55) |
take#(0,IL) |
→ |
uTake1#(isNatIList(IL)) |
(56) |
take#(0,IL) |
→ |
isNatIList#(IL) |
(57) |
uTake1#(tt) |
→ |
nil# |
(58) |
take#(s(M),cons(N,IL)) |
→ |
uTake2#(and(isNat(M),and(isNat(N),isNatIList(activate(IL)))),M,N,activate(IL)) |
(59) |
take#(s(M),cons(N,IL)) |
→ |
and#(isNat(M),and(isNat(N),isNatIList(activate(IL)))) |
(60) |
take#(s(M),cons(N,IL)) |
→ |
isNat#(M) |
(61) |
take#(s(M),cons(N,IL)) |
→ |
and#(isNat(N),isNatIList(activate(IL))) |
(62) |
take#(s(M),cons(N,IL)) |
→ |
isNat#(N) |
(63) |
take#(s(M),cons(N,IL)) |
→ |
isNatIList#(activate(IL)) |
(64) |
take#(s(M),cons(N,IL)) |
→ |
activate#(IL) |
(65) |
uTake2#(tt,M,N,IL) |
→ |
cons#(activate(N),n__take(activate(M),activate(IL))) |
(66) |
uTake2#(tt,M,N,IL) |
→ |
activate#(N) |
(67) |
uTake2#(tt,M,N,IL) |
→ |
activate#(M) |
(68) |
uTake2#(tt,M,N,IL) |
→ |
activate#(IL) |
(69) |
length#(cons(N,L)) |
→ |
uLength#(and(isNat(N),isNatList(activate(L))),activate(L)) |
(70) |
length#(cons(N,L)) |
→ |
and#(isNat(N),isNatList(activate(L))) |
(71) |
length#(cons(N,L)) |
→ |
isNat#(N) |
(72) |
length#(cons(N,L)) |
→ |
isNatList#(activate(L)) |
(73) |
length#(cons(N,L)) |
→ |
activate#(L) |
(74) |
uLength#(tt,L) |
→ |
s#(length(activate(L))) |
(75) |
uLength#(tt,L) |
→ |
length#(activate(L)) |
(76) |
uLength#(tt,L) |
→ |
activate#(L) |
(77) |
activate#(n__0) |
→ |
0# |
(78) |
activate#(n__s(X)) |
→ |
s#(X) |
(79) |
activate#(n__length(X)) |
→ |
length#(X) |
(80) |
activate#(n__zeros) |
→ |
zeros# |
(81) |
activate#(n__cons(X1,X2)) |
→ |
cons#(X1,X2) |
(82) |
activate#(n__nil) |
→ |
nil# |
(83) |
activate#(n__take(X1,X2)) |
→ |
take#(X1,X2) |
(84) |
It remains to prove infiniteness of the resulting DP problem.
isNatIList#(n__cons(N,IL)) |
→ |
and#(isNat(activate(N)),isNatIList(activate(IL))) |
(39) |
isNatList#(n__cons(N,L)) |
→ |
and#(isNat(activate(N)),isNatList(activate(L))) |
(44) |
isNatList#(n__take(N,IL)) |
→ |
and#(isNat(activate(N)),isNatIList(activate(IL))) |
(49) |
zeros# |
→ |
cons#(0,n__zeros) |
(54) |
zeros# |
→ |
0# |
(55) |
take#(0,IL) |
→ |
uTake1#(isNatIList(IL)) |
(56) |
uTake1#(tt) |
→ |
nil# |
(58) |
take#(s(M),cons(N,IL)) |
→ |
and#(isNat(M),and(isNat(N),isNatIList(activate(IL)))) |
(60) |
take#(s(M),cons(N,IL)) |
→ |
and#(isNat(N),isNatIList(activate(IL))) |
(62) |
uTake2#(tt,M,N,IL) |
→ |
cons#(activate(N),n__take(activate(M),activate(IL))) |
(66) |
length#(cons(N,L)) |
→ |
and#(isNat(N),isNatList(activate(L))) |
(71) |
uLength#(tt,L) |
→ |
s#(length(activate(L))) |
(75) |
activate#(n__0) |
→ |
0# |
(78) |
activate#(n__s(X)) |
→ |
s#(X) |
(79) |
activate#(n__zeros) |
→ |
zeros# |
(81) |
activate#(n__cons(X1,X2)) |
→ |
cons#(X1,X2) |
(82) |
activate#(n__nil) |
→ |
nil# |
(83) |
and the following rules have been deleted.
activate#(n__take(X1,X2)) |
→ |
take#(X1,X2) |
(84) |
isNatIList#(IL) |
→ |
isNatList#(activate(IL)) |
(33) |
isNatList#(n__take(N,IL)) |
→ |
isNat#(activate(N)) |
(50) |
isNatList#(n__take(N,IL)) |
→ |
activate#(N) |
(51) |
isNatList#(n__take(N,IL)) |
→ |
isNatIList#(activate(IL)) |
(52) |
isNatIList#(IL) |
→ |
activate#(IL) |
(34) |
isNatIList#(n__cons(N,IL)) |
→ |
isNat#(activate(N)) |
(40) |
isNatIList#(n__cons(N,IL)) |
→ |
activate#(N) |
(41) |
isNatIList#(n__cons(N,IL)) |
→ |
activate#(IL) |
(43) |
isNatList#(n__take(N,IL)) |
→ |
activate#(IL) |
(53) |
take#(s(M),cons(N,IL)) |
→ |
uTake2#(and(isNat(M),and(isNat(N),isNatIList(activate(IL)))),M,N,activate(IL)) |
(59) |
take#(s(M),cons(N,IL)) |
→ |
isNat#(M) |
(61) |
take#(s(M),cons(N,IL)) |
→ |
isNat#(N) |
(63) |
take#(s(M),cons(N,IL)) |
→ |
activate#(IL) |
(65) |
and the following rules have been deleted.
isNatList#(n__cons(y0,n__0)) |
→ |
isNatList#(0) |
(85) |
isNatList#(n__cons(y0,n__s(x0))) |
→ |
isNatList#(s(x0)) |
(86) |
isNatList#(n__cons(y0,n__length(x0))) |
→ |
isNatList#(length(x0)) |
(87) |
isNatList#(n__cons(y0,n__zeros)) |
→ |
isNatList#(zeros) |
(88) |
isNatList#(n__cons(y0,n__cons(x0,x1))) |
→ |
isNatList#(cons(x0,x1)) |
(89) |
isNatList#(n__cons(y0,n__nil)) |
→ |
isNatList#(nil) |
(90) |
isNatList#(n__cons(y0,n__take(x0,x1))) |
→ |
isNatList#(take(x0,x1)) |
(91) |
isNatList#(n__cons(y0,x0)) |
→ |
isNatList#(x0) |
(92) |
and(tt,T) |
→ |
T |
(1) |
isNatIList(IL) |
→ |
isNatList(activate(IL)) |
(2) |
isNat(n__0) |
→ |
tt |
(3) |
isNat(n__s(N)) |
→ |
isNat(activate(N)) |
(4) |
isNat(n__length(L)) |
→ |
isNatList(activate(L)) |
(5) |
isNatIList(n__zeros) |
→ |
tt |
(6) |
isNatIList(n__cons(N,IL)) |
→ |
and(isNat(activate(N)),isNatIList(activate(IL))) |
(7) |
isNatList(n__nil) |
→ |
tt |
(8) |
isNatList(n__cons(N,L)) |
→ |
and(isNat(activate(N)),isNatList(activate(L))) |
(9) |
isNatList(n__take(N,IL)) |
→ |
and(isNat(activate(N)),isNatIList(activate(IL))) |
(10) |
zeros |
→ |
cons(0,n__zeros) |
(11) |
take(0,IL) |
→ |
uTake1(isNatIList(IL)) |
(12) |
uTake1(tt) |
→ |
nil |
(13) |
take(s(M),cons(N,IL)) |
→ |
uTake2(and(isNat(M),and(isNat(N),isNatIList(activate(IL)))),M,N,activate(IL)) |
(14) |
uTake2(tt,M,N,IL) |
→ |
cons(activate(N),n__take(activate(M),activate(IL))) |
(15) |
length(cons(N,L)) |
→ |
uLength(and(isNat(N),isNatList(activate(L))),activate(L)) |
(16) |
uLength(tt,L) |
→ |
s(length(activate(L))) |
(17) |
s(X) |
→ |
n__s(X) |
(19) |
length(X) |
→ |
n__length(X) |
(20) |
zeros |
→ |
n__zeros |
(21) |
cons(X1,X2) |
→ |
n__cons(X1,X2) |
(22) |
nil |
→ |
n__nil |
(23) |
take(X1,X2) |
→ |
n__take(X1,X2) |
(24) |
activate(n__0) |
→ |
0 |
(25) |
activate(n__s(X)) |
→ |
s(X) |
(26) |
activate(n__length(X)) |
→ |
length(X) |
(27) |
activate(n__zeros) |
→ |
zeros |
(28) |
activate(n__cons(X1,X2)) |
→ |
cons(X1,X2) |
(29) |
activate(n__nil) |
→ |
nil |
(30) |
activate(n__take(X1,X2)) |
→ |
take(X1,X2) |
(31) |
activate(X) |
→ |
X |
(32) |
We restrict the innermost strategy to the following left hand sides.
There are no lhss.