(STRATEGY INNERMOST) (VAR IL L M N V V1 V2 X X1 X2 X3 X4) (DATATYPES A = µX.< zeros, mark(X), 0, tt, nil, ok(X) >) (SIGNATURES active :: [A] -> A cons :: [A x A] -> A U101 :: [A x A x A] -> A U102 :: [A x A x A] -> A U103 :: [A x A x A] -> A U104 :: [A x A x A] -> A U105 :: [A x A] -> A U106 :: [A] -> A U11 :: [A x A] -> A U12 :: [A x A] -> A U111 :: [A x A x A] -> A U112 :: [A x A x A] -> A U113 :: [A x A x A] -> A U114 :: [A x A] -> A s :: [A] -> A length :: [A] -> A U13 :: [A] -> A U121 :: [A x A] -> A U122 :: [A] -> A U131 :: [A x A x A x A] -> A U132 :: [A x A x A x A] -> A U133 :: [A x A x A x A] -> A U134 :: [A x A x A x A] -> A U135 :: [A x A x A x A] -> A U136 :: [A x A x A x A] -> A take :: [A x A] -> A U21 :: [A x A] -> A U22 :: [A x A] -> A U23 :: [A] -> A U31 :: [A x A] -> A U32 :: [A x A] -> A U33 :: [A] -> A U41 :: [A x A x A] -> A U42 :: [A x A x A] -> A U43 :: [A x A x A] -> A U44 :: [A x A x A] -> A U45 :: [A x A] -> A U46 :: [A] -> A U51 :: [A x A] -> A U52 :: [A] -> A U61 :: [A x A] -> A U62 :: [A] -> A U71 :: [A] -> A U81 :: [A] -> A U91 :: [A x A x A] -> A U92 :: [A x A x A] -> A U93 :: [A x A x A] -> A U94 :: [A x A x A] -> A U95 :: [A x A] -> A U96 :: [A] -> A proper :: [A] -> A isNatKind :: [A] -> A isNatIListKind :: [A] -> A isNat :: [A] -> A isNatIList :: [A] -> A isNatList :: [A] -> A top :: [A] -> A) (RULES active(zeros()) -> mark(cons(0() ,zeros())) active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2)) active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2) ,V1 ,V2)) active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2) ,V1 ,V2)) active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2)) active(U105(tt(),V2)) -> mark(U106(isNatIList(V2))) active(U106(tt())) -> mark(tt()) active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1)) active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L) ,L ,N)) active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N)) active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L)) active(U114(tt(),L)) -> mark(s(length(L))) active(U12(tt(),V1)) -> mark(U13(isNatList(V1))) active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL))) active(U122(tt())) -> mark(nil()) active(U13(tt())) -> mark(tt()) active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL) ,IL ,M ,N)) active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N)) active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N)) active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N)) active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N)) active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt(),V1)) -> mark(U23(isNat(V1))) active(U23(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V)) active(U32(tt(),V)) -> mark(U33(isNatList(V))) active(U33(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2)) active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2) ,V1 ,V2)) active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2) ,V1 ,V2)) active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2)) active(U45(tt(),V2)) -> mark(U46(isNatIList(V2))) active(U46(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2))) active(U62(tt())) -> mark(tt()) active(U71(tt())) -> mark(tt()) active(U81(tt())) -> mark(tt()) active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2)) active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2) ,V1 ,V2)) active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2) ,V1 ,V2)) active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2)) active(U95(tt(),V2)) -> mark(U96(isNatList(V2))) active(U96(tt())) -> mark(tt()) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1 ,V2))) -> mark(U41(isNatKind(V1) ,V1 ,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1 ,V2))) -> mark(U51(isNatKind(V1) ,V2)) active(isNatIListKind(take(V1 ,V2))) -> mark(U61(isNatKind(V1) ,V2)) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1))) active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1 ,V2))) -> mark(U91(isNatKind(V1) ,V1 ,V2)) active(isNatList(take(V1 ,V2))) -> mark(U101(isNatKind(V1),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N)) active(take(0(),IL)) -> mark(U121(isNatIList(IL),IL)) active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL) ,IL ,M ,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(U102(X1,X2,X3)) -> U102(active(X1),X2,X3) active(U103(X1,X2,X3)) -> U103(active(X1),X2,X3) active(U104(X1,X2,X3)) -> U104(active(X1),X2,X3) active(U105(X1,X2)) -> U105(active(X1),X2) active(U106(X)) -> U106(active(X)) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X1,X2)) -> U12(active(X1),X2) active(U111(X1,X2,X3)) -> U111(active(X1),X2,X3) active(U112(X1,X2,X3)) -> U112(active(X1),X2,X3) active(U113(X1,X2,X3)) -> U113(active(X1),X2,X3) active(U114(X1,X2)) -> U114(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U13(X)) -> U13(active(X)) active(U121(X1,X2)) -> U121(active(X1),X2) active(U122(X)) -> U122(active(X)) active(U131(X1,X2,X3,X4)) -> U131(active(X1),X2,X3,X4) active(U132(X1,X2,X3,X4)) -> U132(active(X1),X2,X3,X4) active(U133(X1,X2,X3,X4)) -> U133(active(X1),X2,X3,X4) active(U134(X1,X2,X3,X4)) -> U134(active(X1),X2,X3,X4) active(U135(X1,X2,X3,X4)) -> U135(active(X1),X2,X3,X4) active(U136(X1,X2,X3,X4)) -> U136(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1 ,active(X2)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X1,X2)) -> U22(active(X1),X2) active(U23(X)) -> U23(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X1,X2)) -> U32(active(X1),X2) active(U33(X)) -> U33(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2,X3)) -> U42(active(X1),X2,X3) active(U43(X1,X2,X3)) -> U43(active(X1),X2,X3) active(U44(X1,X2,X3)) -> U44(active(X1),X2,X3) active(U45(X1,X2)) -> U45(active(X1),X2) active(U46(X)) -> U46(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U62(X)) -> U62(active(X)) active(U71(X)) -> U71(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3)) -> U91(active(X1),X2,X3) active(U92(X1,X2,X3)) -> U92(active(X1),X2,X3) active(U93(X1,X2,X3)) -> U93(active(X1),X2,X3) active(U94(X1,X2,X3)) -> U94(active(X1),X2,X3) active(U95(X1,X2)) -> U95(active(X1),X2) active(U96(X)) -> U96(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3)) U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3)) U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3)) U105(mark(X1),X2) -> mark(U105(X1,X2)) U106(mark(X)) -> mark(U106(X)) U11(mark(X1),X2) -> mark(U11(X1 ,X2)) U12(mark(X1),X2) -> mark(U12(X1 ,X2)) U111(mark(X1),X2,X3) -> mark(U111(X1,X2,X3)) U112(mark(X1),X2,X3) -> mark(U112(X1,X2,X3)) U113(mark(X1),X2,X3) -> mark(U113(X1,X2,X3)) U114(mark(X1),X2) -> mark(U114(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U13(mark(X)) -> mark(U13(X)) U121(mark(X1),X2) -> mark(U121(X1,X2)) U122(mark(X)) -> mark(U122(X)) U131(mark(X1),X2,X3,X4) -> mark(U131(X1,X2,X3,X4)) U132(mark(X1),X2,X3,X4) -> mark(U132(X1,X2,X3,X4)) U133(mark(X1),X2,X3,X4) -> mark(U133(X1,X2,X3,X4)) U134(mark(X1),X2,X3,X4) -> mark(U134(X1,X2,X3,X4)) U135(mark(X1),X2,X3,X4) -> mark(U135(X1,X2,X3,X4)) U136(mark(X1),X2,X3,X4) -> mark(U136(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) U21(mark(X1),X2) -> mark(U21(X1 ,X2)) U22(mark(X1),X2) -> mark(U22(X1 ,X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1),X2) -> mark(U31(X1 ,X2)) U32(mark(X1),X2) -> mark(U32(X1 ,X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2,X3) -> mark(U42(X1,X2,X3)) U43(mark(X1),X2,X3) -> mark(U43(X1,X2,X3)) U44(mark(X1),X2,X3) -> mark(U44(X1,X2,X3)) U45(mark(X1),X2) -> mark(U45(X1 ,X2)) U46(mark(X)) -> mark(U46(X)) U51(mark(X1),X2) -> mark(U51(X1 ,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2) -> mark(U61(X1 ,X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X)) -> mark(U71(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3) -> mark(U91(X1,X2,X3)) U92(mark(X1),X2,X3) -> mark(U92(X1,X2,X3)) U93(mark(X1),X2,X3) -> mark(U93(X1,X2,X3)) U94(mark(X1),X2,X3) -> mark(U94(X1,X2,X3)) U95(mark(X1),X2) -> mark(U95(X1 ,X2)) U96(mark(X)) -> mark(U96(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U101(X1,X2,X3)) -> U101(proper(X1) ,proper(X2) ,proper(X3)) proper(tt()) -> ok(tt()) proper(U102(X1,X2,X3)) -> U102(proper(X1) ,proper(X2) ,proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1,X2,X3)) -> U103(proper(X1) ,proper(X2) ,proper(X3)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(U104(X1,X2,X3)) -> U104(proper(X1) ,proper(X2) ,proper(X3)) proper(U105(X1,X2)) -> U105(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U106(X)) -> U106(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(U111(X1,X2,X3)) -> U111(proper(X1) ,proper(X2) ,proper(X3)) proper(U112(X1,X2,X3)) -> U112(proper(X1) ,proper(X2) ,proper(X3)) proper(U113(X1,X2,X3)) -> U113(proper(X1) ,proper(X2) ,proper(X3)) proper(U114(X1,X2)) -> U114(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U121(X1,X2)) -> U121(proper(X1),proper(X2)) proper(U122(X)) -> U122(proper(X)) proper(nil()) -> ok(nil()) proper(U131(X1,X2,X3,X4)) -> U131(proper(X1) ,proper(X2) ,proper(X3) ,proper(X4)) proper(U132(X1,X2,X3,X4)) -> U132(proper(X1) ,proper(X2) ,proper(X3) ,proper(X4)) proper(U133(X1,X2,X3,X4)) -> U133(proper(X1) ,proper(X2) ,proper(X3) ,proper(X4)) proper(U134(X1,X2,X3,X4)) -> U134(proper(X1) ,proper(X2) ,proper(X3) ,proper(X4)) proper(U135(X1,X2,X3,X4)) -> U135(proper(X1) ,proper(X2) ,proper(X3) ,proper(X4)) proper(U136(X1,X2,X3,X4)) -> U136(proper(X1) ,proper(X2) ,proper(X3) ,proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X1,X2)) -> U32(proper(X1),proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1) ,proper(X2) ,proper(X3)) proper(U42(X1,X2,X3)) -> U42(proper(X1) ,proper(X2) ,proper(X3)) proper(U43(X1,X2,X3)) -> U43(proper(X1) ,proper(X2) ,proper(X3)) proper(U44(X1,X2,X3)) -> U44(proper(X1) ,proper(X2) ,proper(X3)) proper(U45(X1,X2)) -> U45(proper(X1),proper(X2)) proper(U46(X)) -> U46(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X)) -> U71(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(U91(X1,X2,X3)) -> U91(proper(X1) ,proper(X2) ,proper(X3)) proper(U92(X1,X2,X3)) -> U92(proper(X1) ,proper(X2) ,proper(X3)) proper(U93(X1,X2,X3)) -> U93(proper(X1) ,proper(X2) ,proper(X3)) proper(U94(X1,X2,X3)) -> U94(proper(X1) ,proper(X2) ,proper(X3)) proper(U95(X1,X2)) -> U95(proper(X1),proper(X2)) proper(U96(X)) -> U96(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3)) U105(ok(X1),ok(X2)) -> ok(U105(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) U106(ok(X)) -> ok(U106(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U11(ok(X1),ok(X2)) -> ok(U11(X1 ,X2)) U12(ok(X1),ok(X2)) -> ok(U12(X1 ,X2)) U111(ok(X1),ok(X2),ok(X3)) -> ok(U111(X1,X2,X3)) U112(ok(X1),ok(X2),ok(X3)) -> ok(U112(X1,X2,X3)) U113(ok(X1),ok(X2),ok(X3)) -> ok(U113(X1,X2,X3)) U114(ok(X1),ok(X2)) -> ok(U114(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U13(ok(X)) -> ok(U13(X)) isNatList(ok(X)) -> ok(isNatList(X)) U121(ok(X1),ok(X2)) -> ok(U121(X1,X2)) U122(ok(X)) -> ok(U122(X)) U131(ok(X1) ,ok(X2) ,ok(X3) ,ok(X4)) -> ok(U131(X1 ,X2 ,X3 ,X4)) U132(ok(X1) ,ok(X2) ,ok(X3) ,ok(X4)) -> ok(U132(X1 ,X2 ,X3 ,X4)) U133(ok(X1) ,ok(X2) ,ok(X3) ,ok(X4)) -> ok(U133(X1 ,X2 ,X3 ,X4)) U134(ok(X1) ,ok(X2) ,ok(X3) ,ok(X4)) -> ok(U134(X1 ,X2 ,X3 ,X4)) U135(ok(X1) ,ok(X2) ,ok(X3) ,ok(X4)) -> ok(U135(X1 ,X2 ,X3 ,X4)) U136(ok(X1) ,ok(X2) ,ok(X3) ,ok(X4)) -> ok(U136(X1 ,X2 ,X3 ,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) U21(ok(X1),ok(X2)) -> ok(U21(X1 ,X2)) U22(ok(X1),ok(X2)) -> ok(U22(X1 ,X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1 ,X2)) U32(ok(X1),ok(X2)) -> ok(U32(X1 ,X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3)) U43(ok(X1),ok(X2),ok(X3)) -> ok(U43(X1,X2,X3)) U44(ok(X1),ok(X2),ok(X3)) -> ok(U44(X1,X2,X3)) U45(ok(X1),ok(X2)) -> ok(U45(X1 ,X2)) U46(ok(X)) -> ok(U46(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1 ,X2)) U52(ok(X)) -> ok(U52(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1 ,X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X)) -> ok(U71(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3)) -> ok(U91(X1,X2,X3)) U92(ok(X1),ok(X2),ok(X3)) -> ok(U92(X1,X2,X3)) U93(ok(X1),ok(X2),ok(X3)) -> ok(U93(X1,X2,X3)) U94(ok(X1),ok(X2),ok(X3)) -> ok(U94(X1,X2,X3)) U95(ok(X1),ok(X2)) -> ok(U95(X1 ,X2)) U96(ok(X)) -> ok(U96(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)))