(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ZEROS → CONS(0, n__zeros)
ZEROS → 01
U1011(tt, V1, V2) → U1021(isNatKind(activate(V1)), activate(V1), activate(V2))
U1011(tt, V1, V2) → ISNATKIND(activate(V1))
U1011(tt, V1, V2) → ACTIVATE(V1)
U1011(tt, V1, V2) → ACTIVATE(V2)
U1021(tt, V1, V2) → U1031(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1021(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1021(tt, V1, V2) → ACTIVATE(V2)
U1021(tt, V1, V2) → ACTIVATE(V1)
U1031(tt, V1, V2) → U1041(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1031(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1031(tt, V1, V2) → ACTIVATE(V2)
U1031(tt, V1, V2) → ACTIVATE(V1)
U1041(tt, V1, V2) → U1051(isNat(activate(V1)), activate(V2))
U1041(tt, V1, V2) → ISNAT(activate(V1))
U1041(tt, V1, V2) → ACTIVATE(V1)
U1041(tt, V1, V2) → ACTIVATE(V2)
U1051(tt, V2) → U1061(isNatIList(activate(V2)))
U1051(tt, V2) → ISNATILIST(activate(V2))
U1051(tt, V2) → ACTIVATE(V2)
U111(tt, V1) → U121(isNatIListKind(activate(V1)), activate(V1))
U111(tt, V1) → ISNATILISTKIND(activate(V1))
U111(tt, V1) → ACTIVATE(V1)
U1111(tt, L, N) → U1121(isNatIListKind(activate(L)), activate(L), activate(N))
U1111(tt, L, N) → ISNATILISTKIND(activate(L))
U1111(tt, L, N) → ACTIVATE(L)
U1111(tt, L, N) → ACTIVATE(N)
U1121(tt, L, N) → U1131(isNat(activate(N)), activate(L), activate(N))
U1121(tt, L, N) → ISNAT(activate(N))
U1121(tt, L, N) → ACTIVATE(N)
U1121(tt, L, N) → ACTIVATE(L)
U1131(tt, L, N) → U1141(isNatKind(activate(N)), activate(L))
U1131(tt, L, N) → ISNATKIND(activate(N))
U1131(tt, L, N) → ACTIVATE(N)
U1131(tt, L, N) → ACTIVATE(L)
U1141(tt, L) → S(length(activate(L)))
U1141(tt, L) → LENGTH(activate(L))
U1141(tt, L) → ACTIVATE(L)
U121(tt, V1) → U131(isNatList(activate(V1)))
U121(tt, V1) → ISNATLIST(activate(V1))
U121(tt, V1) → ACTIVATE(V1)
U1211(tt, IL) → U1221(isNatIListKind(activate(IL)))
U1211(tt, IL) → ISNATILISTKIND(activate(IL))
U1211(tt, IL) → ACTIVATE(IL)
U1221(tt) → NIL
U1311(tt, IL, M, N) → U1321(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U1311(tt, IL, M, N) → ISNATILISTKIND(activate(IL))
U1311(tt, IL, M, N) → ACTIVATE(IL)
U1311(tt, IL, M, N) → ACTIVATE(M)
U1311(tt, IL, M, N) → ACTIVATE(N)
U1321(tt, IL, M, N) → U1331(isNat(activate(M)), activate(IL), activate(M), activate(N))
U1321(tt, IL, M, N) → ISNAT(activate(M))
U1321(tt, IL, M, N) → ACTIVATE(M)
U1321(tt, IL, M, N) → ACTIVATE(IL)
U1321(tt, IL, M, N) → ACTIVATE(N)
U1331(tt, IL, M, N) → U1341(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U1331(tt, IL, M, N) → ISNATKIND(activate(M))
U1331(tt, IL, M, N) → ACTIVATE(M)
U1331(tt, IL, M, N) → ACTIVATE(IL)
U1331(tt, IL, M, N) → ACTIVATE(N)
U1341(tt, IL, M, N) → U1351(isNat(activate(N)), activate(IL), activate(M), activate(N))
U1341(tt, IL, M, N) → ISNAT(activate(N))
U1341(tt, IL, M, N) → ACTIVATE(N)
U1341(tt, IL, M, N) → ACTIVATE(IL)
U1341(tt, IL, M, N) → ACTIVATE(M)
U1351(tt, IL, M, N) → U1361(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U1351(tt, IL, M, N) → ISNATKIND(activate(N))
U1351(tt, IL, M, N) → ACTIVATE(N)
U1351(tt, IL, M, N) → ACTIVATE(IL)
U1351(tt, IL, M, N) → ACTIVATE(M)
U1361(tt, IL, M, N) → CONS(activate(N), n__take(activate(M), activate(IL)))
U1361(tt, IL, M, N) → ACTIVATE(N)
U1361(tt, IL, M, N) → ACTIVATE(M)
U1361(tt, IL, M, N) → ACTIVATE(IL)
U211(tt, V1) → U221(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNATKIND(activate(V1))
U211(tt, V1) → ACTIVATE(V1)
U221(tt, V1) → U231(isNat(activate(V1)))
U221(tt, V1) → ISNAT(activate(V1))
U221(tt, V1) → ACTIVATE(V1)
U311(tt, V) → U321(isNatIListKind(activate(V)), activate(V))
U311(tt, V) → ISNATILISTKIND(activate(V))
U311(tt, V) → ACTIVATE(V)
U321(tt, V) → U331(isNatList(activate(V)))
U321(tt, V) → ISNATLIST(activate(V))
U321(tt, V) → ACTIVATE(V)
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U411(tt, V1, V2) → ISNATKIND(activate(V1))
U411(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ACTIVATE(V2)
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U421(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U421(tt, V1, V2) → ACTIVATE(V2)
U421(tt, V1, V2) → ACTIVATE(V1)
U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U431(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U431(tt, V1, V2) → ACTIVATE(V2)
U431(tt, V1, V2) → ACTIVATE(V1)
U441(tt, V1, V2) → U451(isNat(activate(V1)), activate(V2))
U441(tt, V1, V2) → ISNAT(activate(V1))
U441(tt, V1, V2) → ACTIVATE(V1)
U441(tt, V1, V2) → ACTIVATE(V2)
U451(tt, V2) → U461(isNatIList(activate(V2)))
U451(tt, V2) → ISNATILIST(activate(V2))
U451(tt, V2) → ACTIVATE(V2)
U511(tt, V2) → U521(isNatIListKind(activate(V2)))
U511(tt, V2) → ISNATILISTKIND(activate(V2))
U511(tt, V2) → ACTIVATE(V2)
U611(tt, V2) → U621(isNatIListKind(activate(V2)))
U611(tt, V2) → ISNATILISTKIND(activate(V2))
U611(tt, V2) → ACTIVATE(V2)
U911(tt, V1, V2) → U921(isNatKind(activate(V1)), activate(V1), activate(V2))
U911(tt, V1, V2) → ISNATKIND(activate(V1))
U911(tt, V1, V2) → ACTIVATE(V1)
U911(tt, V1, V2) → ACTIVATE(V2)
U921(tt, V1, V2) → U931(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U921(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U921(tt, V1, V2) → ACTIVATE(V2)
U921(tt, V1, V2) → ACTIVATE(V1)
U931(tt, V1, V2) → U941(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U931(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U931(tt, V1, V2) → ACTIVATE(V2)
U931(tt, V1, V2) → ACTIVATE(V1)
U941(tt, V1, V2) → U951(isNat(activate(V1)), activate(V2))
U941(tt, V1, V2) → ISNAT(activate(V1))
U941(tt, V1, V2) → ACTIVATE(V1)
U941(tt, V1, V2) → ACTIVATE(V2)
U951(tt, V2) → U961(isNatList(activate(V2)))
U951(tt, V2) → ISNATLIST(activate(V2))
U951(tt, V2) → ACTIVATE(V2)
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__cons(V1, V2)) → U511(isNatKind(activate(V1)), activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__take(V1, V2)) → U611(isNatKind(activate(V1)), activate(V2))
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__length(V1)) → U711(isNatIListKind(activate(V1)))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → U811(isNatKind(activate(V1)))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → U911(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → U1011(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
LENGTH(nil) → 01
LENGTH(cons(N, L)) → U1111(isNatList(activate(L)), activate(L), N)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
TAKE(0, IL) → U1211(isNatIList(IL), IL)
TAKE(0, IL) → ISNATILIST(IL)
TAKE(s(M), cons(N, IL)) → U1311(isNatIList(activate(IL)), activate(IL), M, N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ACTIVATE(n__zeros) → ZEROS
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__0) → 01
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__nil) → NIL
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 22 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
TAKE(0, IL) → U1211(isNatIList(IL), IL)
U1211(tt, IL) → ISNATILISTKIND(activate(IL))
ISNATILISTKIND(n__cons(V1, V2)) → U511(isNatKind(activate(V1)), activate(V2))
U511(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
LENGTH(cons(N, L)) → U1111(isNatList(activate(L)), activate(L), N)
U1111(tt, L, N) → U1121(isNatIListKind(activate(L)), activate(L), activate(N))
U1121(tt, L, N) → U1131(isNat(activate(N)), activate(L), activate(N))
U1131(tt, L, N) → U1141(isNatKind(activate(N)), activate(L))
U1141(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → U911(isNatKind(activate(V1)), activate(V1), activate(V2))
U911(tt, V1, V2) → U921(isNatKind(activate(V1)), activate(V1), activate(V2))
U921(tt, V1, V2) → U931(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U931(tt, V1, V2) → U941(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U941(tt, V1, V2) → U951(isNat(activate(V1)), activate(V2))
U951(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → U1011(isNatKind(activate(V1)), activate(V1), activate(V2))
U1011(tt, V1, V2) → U1021(isNatKind(activate(V1)), activate(V1), activate(V2))
U1021(tt, V1, V2) → U1031(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1031(tt, V1, V2) → U1041(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1041(tt, V1, V2) → U1051(isNat(activate(V1)), activate(V2))
U1051(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
U311(tt, V) → U321(isNatIListKind(activate(V)), activate(V))
U321(tt, V) → ISNATLIST(activate(V))
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
U321(tt, V) → ACTIVATE(V)
U311(tt, V) → ISNATILISTKIND(activate(V))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__take(V1, V2)) → U611(isNatKind(activate(V1)), activate(V2))
U611(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
U611(tt, V2) → ACTIVATE(V2)
U311(tt, V) → ACTIVATE(V)
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, V1, V2) → U451(isNat(activate(V1)), activate(V2))
U451(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U451(tt, V2) → ACTIVATE(V2)
U441(tt, V1, V2) → ISNAT(activate(V1))
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
U111(tt, V1) → U121(isNatIListKind(activate(V1)), activate(V1))
U121(tt, V1) → ISNATLIST(activate(V1))
U121(tt, V1) → ACTIVATE(V1)
U111(tt, V1) → ISNATILISTKIND(activate(V1))
U111(tt, V1) → ACTIVATE(V1)
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → U221(isNatKind(activate(V1)), activate(V1))
U221(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U221(tt, V1) → ACTIVATE(V1)
U211(tt, V1) → ISNATKIND(activate(V1))
U211(tt, V1) → ACTIVATE(V1)
U441(tt, V1, V2) → ACTIVATE(V1)
U441(tt, V1, V2) → ACTIVATE(V2)
U431(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U431(tt, V1, V2) → ACTIVATE(V2)
U431(tt, V1, V2) → ACTIVATE(V1)
U421(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U421(tt, V1, V2) → ACTIVATE(V2)
U421(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ISNATKIND(activate(V1))
U411(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ACTIVATE(V2)
U1051(tt, V2) → ACTIVATE(V2)
U1041(tt, V1, V2) → ISNAT(activate(V1))
U1041(tt, V1, V2) → ACTIVATE(V1)
U1041(tt, V1, V2) → ACTIVATE(V2)
U1031(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1031(tt, V1, V2) → ACTIVATE(V2)
U1031(tt, V1, V2) → ACTIVATE(V1)
U1021(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1021(tt, V1, V2) → ACTIVATE(V2)
U1021(tt, V1, V2) → ACTIVATE(V1)
U1011(tt, V1, V2) → ISNATKIND(activate(V1))
U1011(tt, V1, V2) → ACTIVATE(V1)
U1011(tt, V1, V2) → ACTIVATE(V2)
U951(tt, V2) → ACTIVATE(V2)
U941(tt, V1, V2) → ISNAT(activate(V1))
U941(tt, V1, V2) → ACTIVATE(V1)
U941(tt, V1, V2) → ACTIVATE(V2)
U931(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U931(tt, V1, V2) → ACTIVATE(V2)
U931(tt, V1, V2) → ACTIVATE(V1)
U921(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U921(tt, V1, V2) → ACTIVATE(V2)
U921(tt, V1, V2) → ACTIVATE(V1)
U911(tt, V1, V2) → ISNATKIND(activate(V1))
U911(tt, V1, V2) → ACTIVATE(V1)
U911(tt, V1, V2) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U1141(tt, L) → ACTIVATE(L)
U1131(tt, L, N) → ISNATKIND(activate(N))
U1131(tt, L, N) → ACTIVATE(N)
U1131(tt, L, N) → ACTIVATE(L)
U1121(tt, L, N) → ISNAT(activate(N))
U1121(tt, L, N) → ACTIVATE(N)
U1121(tt, L, N) → ACTIVATE(L)
U1111(tt, L, N) → ISNATILISTKIND(activate(L))
U1111(tt, L, N) → ACTIVATE(L)
U1111(tt, L, N) → ACTIVATE(N)
U511(tt, V2) → ACTIVATE(V2)
U1211(tt, IL) → ACTIVATE(IL)
TAKE(0, IL) → ISNATILIST(IL)
TAKE(s(M), cons(N, IL)) → U1311(isNatIList(activate(IL)), activate(IL), M, N)
U1311(tt, IL, M, N) → U1321(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U1321(tt, IL, M, N) → U1331(isNat(activate(M)), activate(IL), activate(M), activate(N))
U1331(tt, IL, M, N) → U1341(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U1341(tt, IL, M, N) → U1351(isNat(activate(N)), activate(IL), activate(M), activate(N))
U1351(tt, IL, M, N) → U1361(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U1361(tt, IL, M, N) → ACTIVATE(N)
U1361(tt, IL, M, N) → ACTIVATE(M)
U1361(tt, IL, M, N) → ACTIVATE(IL)
U1351(tt, IL, M, N) → ISNATKIND(activate(N))
U1351(tt, IL, M, N) → ACTIVATE(N)
U1351(tt, IL, M, N) → ACTIVATE(IL)
U1351(tt, IL, M, N) → ACTIVATE(M)
U1341(tt, IL, M, N) → ISNAT(activate(N))
U1341(tt, IL, M, N) → ACTIVATE(N)
U1341(tt, IL, M, N) → ACTIVATE(IL)
U1341(tt, IL, M, N) → ACTIVATE(M)
U1331(tt, IL, M, N) → ISNATKIND(activate(M))
U1331(tt, IL, M, N) → ACTIVATE(M)
U1331(tt, IL, M, N) → ACTIVATE(IL)
U1331(tt, IL, M, N) → ACTIVATE(N)
U1321(tt, IL, M, N) → ISNAT(activate(M))
U1321(tt, IL, M, N) → ACTIVATE(M)
U1321(tt, IL, M, N) → ACTIVATE(IL)
U1321(tt, IL, M, N) → ACTIVATE(N)
U1311(tt, IL, M, N) → ISNATILISTKIND(activate(IL))
U1311(tt, IL, M, N) → ACTIVATE(IL)
U1311(tt, IL, M, N) → ACTIVATE(M)
U1311(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
U111(tt, V1) → ISNATILISTKIND(activate(V1))
U111(tt, V1) → ACTIVATE(V1)
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNAT(n__length(V1)) → ACTIVATE(V1)
U1011(tt, V1, V2) → ACTIVATE(V1)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
POL(ACTIVATE(x1)) = | -I | + | 0A | · | x1 |
POL(n__take(x1, x2)) = | 1A | + | 1A | · | x1 | + | 5A | · | x2 |
POL(TAKE(x1, x2)) = | 1A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(activate(x1)) = | -I | + | 0A | · | x1 |
POL(U1211(x1, x2)) = | -I | + | -I | · | x1 | + | 0A | · | x2 |
POL(isNatIList(x1)) = | -I | + | 0A | · | x1 |
POL(ISNATILISTKIND(x1)) = | -I | + | 0A | · | x1 |
POL(n__cons(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U511(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(isNatKind(x1)) = | 0A | + | 0A | · | x1 |
POL(ISNATKIND(x1)) = | 0A | + | 0A | · | x1 |
POL(n__length(x1)) = | -I | + | 2A | · | x1 |
POL(LENGTH(x1)) = | -I | + | 0A | · | x1 |
POL(cons(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U1111(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(isNatList(x1)) = | -I | + | 0A | · | x1 |
POL(U1121(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(isNatIListKind(x1)) = | 0A | + | 0A | · | x1 |
POL(U1131(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(isNat(x1)) = | 0A | + | 0A | · | x1 |
POL(U1141(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(ISNATLIST(x1)) = | 0A | + | 0A | · | x1 |
POL(U911(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U921(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U931(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U941(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U951(x1, x2)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 |
POL(n__s(x1)) = | -I | + | 0A | · | x1 |
POL(U1011(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 1A | · | x2 | + | 0A | · | x3 |
POL(U1021(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U1031(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U1041(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U1051(x1, x2)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 |
POL(ISNATILIST(x1)) = | 0A | + | 0A | · | x1 |
POL(U311(x1, x2)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 |
POL(U321(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U611(x1, x2)) = | 0A | + | 1A | · | x1 | + | 0A | · | x2 |
POL(U411(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U421(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U431(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U441(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U451(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(ISNAT(x1)) = | 0A | + | 0A | · | x1 |
POL(U111(x1, x2)) = | -I | + | 0A | · | x1 | + | 2A | · | x2 |
POL(U121(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U211(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U221(x1, x2)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 |
POL(U1311(x1, x2, x3, x4)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 | + | 0A | · | x4 |
POL(U1321(x1, x2, x3, x4)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 | + | 0A | · | x4 |
POL(U1331(x1, x2, x3, x4)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 | + | 0A | · | x4 |
POL(U1341(x1, x2, x3, x4)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 | + | 0A | · | x4 |
POL(U1351(x1, x2, x3, x4)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 | + | 0A | · | x4 |
POL(U1361(x1, x2, x3, x4)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 | + | 0A | · | x4 |
POL(U102(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 1A | · | x3 |
POL(U103(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U104(x1, x2, x3)) = | -I | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U101(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 5A | · | x3 |
POL(U11(x1, x2)) = | 0A | + | -I | · | x1 | + | -I | · | x2 |
POL(U12(x1, x2)) = | 0A | + | -I | · | x1 | + | -I | · | x2 |
POL(U106(x1)) = | -I | + | 0A | · | x1 |
POL(U105(x1, x2)) = | -I | + | -I | · | x1 | + | 0A | · | x2 |
POL(U114(x1, x2)) = | 0A | + | 0A | · | x1 | + | 2A | · | x2 |
POL(length(x1)) = | -I | + | 2A | · | x1 |
POL(U113(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 2A | · | x2 | + | 0A | · | x3 |
POL(U112(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 2A | · | x2 | + | 0A | · | x3 |
POL(U111(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 2A | · | x2 | + | 1A | · | x3 |
POL(U13(x1)) = | 0A | + | -I | · | x1 |
POL(U122(x1)) = | 0A | + | -I | · | x1 |
POL(U121(x1, x2)) = | 0A | + | -I | · | x1 | + | -I | · | x2 |
POL(U134(x1, x2, x3, x4)) = | 2A | + | 0A | · | x1 | + | 5A | · | x2 | + | 1A | · | x3 | + | 5A | · | x4 |
POL(U135(x1, x2, x3, x4)) = | 2A | + | -I | · | x1 | + | 5A | · | x2 | + | 1A | · | x3 | + | 0A | · | x4 |
POL(U133(x1, x2, x3, x4)) = | 2A | + | -I | · | x1 | + | 5A | · | x2 | + | 1A | · | x3 | + | 5A | · | x4 |
POL(U132(x1, x2, x3, x4)) = | 2A | + | -I | · | x1 | + | 5A | · | x2 | + | 1A | · | x3 | + | 5A | · | x4 |
POL(U131(x1, x2, x3, x4)) = | 5A | + | 0A | · | x1 | + | 5A | · | x2 | + | 1A | · | x3 | + | 5A | · | x4 |
POL(U136(x1, x2, x3, x4)) = | 2A | + | -I | · | x1 | + | 5A | · | x2 | + | 1A | · | x3 | + | 0A | · | x4 |
POL(U21(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U22(x1, x2)) = | -I | + | 0A | · | x1 | + | -I | · | x2 |
POL(U23(x1)) = | 0A | + | -I | · | x1 |
POL(U31(x1, x2)) = | -I | + | -I | · | x1 | + | 0A | · | x2 |
POL(U32(x1, x2)) = | -I | + | -I | · | x1 | + | 0A | · | x2 |
POL(U33(x1)) = | -I | + | 0A | · | x1 |
POL(U41(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | -I | · | x2 | + | 0A | · | x3 |
POL(U42(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | -I | · | x2 | + | 0A | · | x3 |
POL(U43(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | -I | · | x2 | + | 0A | · | x3 |
POL(U44(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | -I | · | x2 | + | -I | · | x3 |
POL(U45(x1, x2)) = | 0A | + | -I | · | x1 | + | -I | · | x2 |
POL(U46(x1)) = | 0A | + | -I | · | x1 |
POL(U51(x1, x2)) = | 0A | + | -I | · | x1 | + | -I | · | x2 |
POL(U52(x1)) = | 0A | + | -I | · | x1 |
POL(U62(x1)) = | -I | + | 0A | · | x1 |
POL(U61(x1, x2)) = | 1A | + | -I | · | x1 | + | 0A | · | x2 |
POL(U81(x1)) = | 0A | + | -I | · | x1 |
POL(U71(x1)) = | 0A | + | 0A | · | x1 |
POL(U92(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U93(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U91(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U94(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | -I | · | x2 | + | -I | · | x3 |
POL(U95(x1, x2)) = | 0A | + | -I | · | x1 | + | -I | · | x2 |
POL(U96(x1)) = | 0A | + | -I | · | x1 |
POL(take(x1, x2)) = | 1A | + | 1A | · | x1 | + | 5A | · | x2 |
The following usable rules [FROCOS05] were oriented:
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U106(tt) → tt
U105(tt, V2) → U106(isNatIList(activate(V2)))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U114(tt, L) → s(length(activate(L)))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U13(tt) → tt
U122(tt) → nil
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U62(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U81(tt) → tt
U71(tt) → tt
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U96(tt) → tt
U95(tt, V2) → U96(isNatList(activate(V2)))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__0) → tt
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIList(n__zeros) → tt
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
TAKE(0, IL) → U1211(isNatIList(IL), IL)
U1211(tt, IL) → ISNATILISTKIND(activate(IL))
ISNATILISTKIND(n__cons(V1, V2)) → U511(isNatKind(activate(V1)), activate(V2))
U511(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → U1111(isNatList(activate(L)), activate(L), N)
U1111(tt, L, N) → U1121(isNatIListKind(activate(L)), activate(L), activate(N))
U1121(tt, L, N) → U1131(isNat(activate(N)), activate(L), activate(N))
U1131(tt, L, N) → U1141(isNatKind(activate(N)), activate(L))
U1141(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → U911(isNatKind(activate(V1)), activate(V1), activate(V2))
U911(tt, V1, V2) → U921(isNatKind(activate(V1)), activate(V1), activate(V2))
U921(tt, V1, V2) → U931(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U931(tt, V1, V2) → U941(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U941(tt, V1, V2) → U951(isNat(activate(V1)), activate(V2))
U951(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → U1011(isNatKind(activate(V1)), activate(V1), activate(V2))
U1011(tt, V1, V2) → U1021(isNatKind(activate(V1)), activate(V1), activate(V2))
U1021(tt, V1, V2) → U1031(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1031(tt, V1, V2) → U1041(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1041(tt, V1, V2) → U1051(isNat(activate(V1)), activate(V2))
U1051(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
U311(tt, V) → U321(isNatIListKind(activate(V)), activate(V))
U321(tt, V) → ISNATLIST(activate(V))
U321(tt, V) → ACTIVATE(V)
U311(tt, V) → ISNATILISTKIND(activate(V))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__take(V1, V2)) → U611(isNatKind(activate(V1)), activate(V2))
U611(tt, V2) → ISNATILISTKIND(activate(V2))
U611(tt, V2) → ACTIVATE(V2)
U311(tt, V) → ACTIVATE(V)
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, V1, V2) → U451(isNat(activate(V1)), activate(V2))
U451(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U451(tt, V2) → ACTIVATE(V2)
U441(tt, V1, V2) → ISNAT(activate(V1))
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
U111(tt, V1) → U121(isNatIListKind(activate(V1)), activate(V1))
U121(tt, V1) → ISNATLIST(activate(V1))
U121(tt, V1) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → U221(isNatKind(activate(V1)), activate(V1))
U221(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U221(tt, V1) → ACTIVATE(V1)
U211(tt, V1) → ISNATKIND(activate(V1))
U211(tt, V1) → ACTIVATE(V1)
U441(tt, V1, V2) → ACTIVATE(V1)
U441(tt, V1, V2) → ACTIVATE(V2)
U431(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U431(tt, V1, V2) → ACTIVATE(V2)
U431(tt, V1, V2) → ACTIVATE(V1)
U421(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U421(tt, V1, V2) → ACTIVATE(V2)
U421(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ISNATKIND(activate(V1))
U411(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ACTIVATE(V2)
U1051(tt, V2) → ACTIVATE(V2)
U1041(tt, V1, V2) → ISNAT(activate(V1))
U1041(tt, V1, V2) → ACTIVATE(V1)
U1041(tt, V1, V2) → ACTIVATE(V2)
U1031(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1031(tt, V1, V2) → ACTIVATE(V2)
U1031(tt, V1, V2) → ACTIVATE(V1)
U1021(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1021(tt, V1, V2) → ACTIVATE(V2)
U1021(tt, V1, V2) → ACTIVATE(V1)
U1011(tt, V1, V2) → ISNATKIND(activate(V1))
U1011(tt, V1, V2) → ACTIVATE(V2)
U951(tt, V2) → ACTIVATE(V2)
U941(tt, V1, V2) → ISNAT(activate(V1))
U941(tt, V1, V2) → ACTIVATE(V1)
U941(tt, V1, V2) → ACTIVATE(V2)
U931(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U931(tt, V1, V2) → ACTIVATE(V2)
U931(tt, V1, V2) → ACTIVATE(V1)
U921(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U921(tt, V1, V2) → ACTIVATE(V2)
U921(tt, V1, V2) → ACTIVATE(V1)
U911(tt, V1, V2) → ISNATKIND(activate(V1))
U911(tt, V1, V2) → ACTIVATE(V1)
U911(tt, V1, V2) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U1141(tt, L) → ACTIVATE(L)
U1131(tt, L, N) → ISNATKIND(activate(N))
U1131(tt, L, N) → ACTIVATE(N)
U1131(tt, L, N) → ACTIVATE(L)
U1121(tt, L, N) → ISNAT(activate(N))
U1121(tt, L, N) → ACTIVATE(N)
U1121(tt, L, N) → ACTIVATE(L)
U1111(tt, L, N) → ISNATILISTKIND(activate(L))
U1111(tt, L, N) → ACTIVATE(L)
U1111(tt, L, N) → ACTIVATE(N)
U511(tt, V2) → ACTIVATE(V2)
U1211(tt, IL) → ACTIVATE(IL)
TAKE(0, IL) → ISNATILIST(IL)
TAKE(s(M), cons(N, IL)) → U1311(isNatIList(activate(IL)), activate(IL), M, N)
U1311(tt, IL, M, N) → U1321(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U1321(tt, IL, M, N) → U1331(isNat(activate(M)), activate(IL), activate(M), activate(N))
U1331(tt, IL, M, N) → U1341(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U1341(tt, IL, M, N) → U1351(isNat(activate(N)), activate(IL), activate(M), activate(N))
U1351(tt, IL, M, N) → U1361(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U1361(tt, IL, M, N) → ACTIVATE(N)
U1361(tt, IL, M, N) → ACTIVATE(M)
U1361(tt, IL, M, N) → ACTIVATE(IL)
U1351(tt, IL, M, N) → ISNATKIND(activate(N))
U1351(tt, IL, M, N) → ACTIVATE(N)
U1351(tt, IL, M, N) → ACTIVATE(IL)
U1351(tt, IL, M, N) → ACTIVATE(M)
U1341(tt, IL, M, N) → ISNAT(activate(N))
U1341(tt, IL, M, N) → ACTIVATE(N)
U1341(tt, IL, M, N) → ACTIVATE(IL)
U1341(tt, IL, M, N) → ACTIVATE(M)
U1331(tt, IL, M, N) → ISNATKIND(activate(M))
U1331(tt, IL, M, N) → ACTIVATE(M)
U1331(tt, IL, M, N) → ACTIVATE(IL)
U1331(tt, IL, M, N) → ACTIVATE(N)
U1321(tt, IL, M, N) → ISNAT(activate(M))
U1321(tt, IL, M, N) → ACTIVATE(M)
U1321(tt, IL, M, N) → ACTIVATE(IL)
U1321(tt, IL, M, N) → ACTIVATE(N)
U1311(tt, IL, M, N) → ISNATILISTKIND(activate(IL))
U1311(tt, IL, M, N) → ACTIVATE(IL)
U1311(tt, IL, M, N) → ACTIVATE(M)
U1311(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 12 less nodes.
(8) Complex Obligation (AND)
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TAKE(0, IL) → U1211(isNatIList(IL), IL)
U1211(tt, IL) → ISNATILISTKIND(activate(IL))
ISNATILISTKIND(n__cons(V1, V2)) → U511(isNatKind(activate(V1)), activate(V2))
U511(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
U311(tt, V) → U321(isNatIListKind(activate(V)), activate(V))
U321(tt, V) → ISNATLIST(activate(V))
ISNATLIST(n__cons(V1, V2)) → U911(isNatKind(activate(V1)), activate(V1), activate(V2))
U911(tt, V1, V2) → U921(isNatKind(activate(V1)), activate(V1), activate(V2))
U921(tt, V1, V2) → U931(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U931(tt, V1, V2) → U941(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U941(tt, V1, V2) → U951(isNat(activate(V1)), activate(V2))
U951(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → U1011(isNatKind(activate(V1)), activate(V1), activate(V2))
U1011(tt, V1, V2) → U1021(isNatKind(activate(V1)), activate(V1), activate(V2))
U1021(tt, V1, V2) → U1031(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1031(tt, V1, V2) → U1041(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1041(tt, V1, V2) → U1051(isNat(activate(V1)), activate(V2))
U1051(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__take(V1, V2)) → U611(isNatKind(activate(V1)), activate(V2))
U611(tt, V2) → ISNATILISTKIND(activate(V2))
U611(tt, V2) → ACTIVATE(V2)
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, V1, V2) → U451(isNat(activate(V1)), activate(V2))
U451(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U451(tt, V2) → ACTIVATE(V2)
U441(tt, V1, V2) → ISNAT(activate(V1))
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
U111(tt, V1) → U121(isNatIListKind(activate(V1)), activate(V1))
U121(tt, V1) → ISNATLIST(activate(V1))
U121(tt, V1) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → U221(isNatKind(activate(V1)), activate(V1))
U221(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U221(tt, V1) → ACTIVATE(V1)
U211(tt, V1) → ISNATKIND(activate(V1))
U211(tt, V1) → ACTIVATE(V1)
U441(tt, V1, V2) → ACTIVATE(V1)
U441(tt, V1, V2) → ACTIVATE(V2)
U431(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U431(tt, V1, V2) → ACTIVATE(V2)
U431(tt, V1, V2) → ACTIVATE(V1)
U421(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U421(tt, V1, V2) → ACTIVATE(V2)
U421(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ISNATKIND(activate(V1))
U411(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ACTIVATE(V2)
U1051(tt, V2) → ACTIVATE(V2)
U1041(tt, V1, V2) → ISNAT(activate(V1))
U1041(tt, V1, V2) → ACTIVATE(V1)
U1041(tt, V1, V2) → ACTIVATE(V2)
U1031(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1031(tt, V1, V2) → ACTIVATE(V2)
U1031(tt, V1, V2) → ACTIVATE(V1)
U1021(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1021(tt, V1, V2) → ACTIVATE(V2)
U1021(tt, V1, V2) → ACTIVATE(V1)
U1011(tt, V1, V2) → ISNATKIND(activate(V1))
U1011(tt, V1, V2) → ACTIVATE(V2)
U951(tt, V2) → ACTIVATE(V2)
U941(tt, V1, V2) → ISNAT(activate(V1))
U941(tt, V1, V2) → ACTIVATE(V1)
U941(tt, V1, V2) → ACTIVATE(V2)
U931(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U931(tt, V1, V2) → ACTIVATE(V2)
U931(tt, V1, V2) → ACTIVATE(V1)
U921(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U921(tt, V1, V2) → ACTIVATE(V2)
U921(tt, V1, V2) → ACTIVATE(V1)
U911(tt, V1, V2) → ISNATKIND(activate(V1))
U911(tt, V1, V2) → ACTIVATE(V1)
U911(tt, V1, V2) → ACTIVATE(V2)
U321(tt, V) → ACTIVATE(V)
U311(tt, V) → ISNATILISTKIND(activate(V))
U311(tt, V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → U1311(isNatIList(activate(IL)), activate(IL), M, N)
U1311(tt, IL, M, N) → U1321(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U1321(tt, IL, M, N) → U1331(isNat(activate(M)), activate(IL), activate(M), activate(N))
U1331(tt, IL, M, N) → U1341(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U1341(tt, IL, M, N) → U1351(isNat(activate(N)), activate(IL), activate(M), activate(N))
U1351(tt, IL, M, N) → U1361(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U1361(tt, IL, M, N) → ACTIVATE(N)
U1361(tt, IL, M, N) → ACTIVATE(M)
U1361(tt, IL, M, N) → ACTIVATE(IL)
U1351(tt, IL, M, N) → ISNATKIND(activate(N))
U1351(tt, IL, M, N) → ACTIVATE(N)
U1351(tt, IL, M, N) → ACTIVATE(IL)
U1351(tt, IL, M, N) → ACTIVATE(M)
U1341(tt, IL, M, N) → ISNAT(activate(N))
U1341(tt, IL, M, N) → ACTIVATE(N)
U1341(tt, IL, M, N) → ACTIVATE(IL)
U1341(tt, IL, M, N) → ACTIVATE(M)
U1331(tt, IL, M, N) → ISNATKIND(activate(M))
U1331(tt, IL, M, N) → ACTIVATE(M)
U1331(tt, IL, M, N) → ACTIVATE(IL)
U1331(tt, IL, M, N) → ACTIVATE(N)
U1321(tt, IL, M, N) → ISNAT(activate(M))
U1321(tt, IL, M, N) → ACTIVATE(M)
U1321(tt, IL, M, N) → ACTIVATE(IL)
U1321(tt, IL, M, N) → ACTIVATE(N)
U1311(tt, IL, M, N) → ISNATILISTKIND(activate(IL))
U1311(tt, IL, M, N) → ACTIVATE(IL)
U1311(tt, IL, M, N) → ACTIVATE(M)
U1311(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U511(tt, V2) → ACTIVATE(V2)
U1211(tt, IL) → ACTIVATE(IL)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(10) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
U111(tt, V1) → U121(isNatIListKind(activate(V1)), activate(V1))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
POL(TAKE(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U1211(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatIList(x1)) = | | + | | · | x1 |
POL(ISNATILISTKIND(x1)) = | | + | | · | x1 |
POL(activate(x1)) = | | + | | · | x1 |
POL(n__cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U511(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatKind(x1)) = | | + | | · | x1 |
POL(ISNATKIND(x1)) = | | + | | · | x1 |
POL(ACTIVATE(x1)) = | | + | | · | x1 |
POL(n__take(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(ISNATILIST(x1)) = | | + | | · | x1 |
POL(U311(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatIListKind(x1)) = | | + | | · | x1 |
POL(U321(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(ISNATLIST(x1)) = | | + | | · | x1 |
POL(U911(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U921(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U931(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U941(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U951(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U1011(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U1021(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U1031(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U1041(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U1051(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U611(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U411(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U421(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U431(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U441(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U451(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(n__length(x1)) = | | + | | · | x1 |
POL(U111(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U121(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U211(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U221(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U1311(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U1321(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U1331(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U1341(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U1351(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U1361(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U102(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U103(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U104(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U101(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U11(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U12(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U105(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U114(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U113(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U112(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U111(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U121(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatList(x1)) = | | + | | · | x1 |
POL(U134(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U135(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U133(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U132(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U131(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U136(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U21(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U22(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U31(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U32(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U41(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U42(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U43(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U44(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U45(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U51(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U61(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U92(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U93(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U91(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U94(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U95(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(take(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U106(tt) → tt
U105(tt, V2) → U106(isNatIList(activate(V2)))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U114(tt, L) → s(length(activate(L)))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U13(tt) → tt
U122(tt) → nil
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U62(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U81(tt) → tt
U71(tt) → tt
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U96(tt) → tt
U95(tt, V2) → U96(isNatList(activate(V2)))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__0) → tt
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TAKE(0, IL) → U1211(isNatIList(IL), IL)
U1211(tt, IL) → ISNATILISTKIND(activate(IL))
ISNATILISTKIND(n__cons(V1, V2)) → U511(isNatKind(activate(V1)), activate(V2))
U511(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
U311(tt, V) → U321(isNatIListKind(activate(V)), activate(V))
U321(tt, V) → ISNATLIST(activate(V))
ISNATLIST(n__cons(V1, V2)) → U911(isNatKind(activate(V1)), activate(V1), activate(V2))
U911(tt, V1, V2) → U921(isNatKind(activate(V1)), activate(V1), activate(V2))
U921(tt, V1, V2) → U931(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U931(tt, V1, V2) → U941(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U941(tt, V1, V2) → U951(isNat(activate(V1)), activate(V2))
U951(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → U1011(isNatKind(activate(V1)), activate(V1), activate(V2))
U1011(tt, V1, V2) → U1021(isNatKind(activate(V1)), activate(V1), activate(V2))
U1021(tt, V1, V2) → U1031(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1031(tt, V1, V2) → U1041(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1041(tt, V1, V2) → U1051(isNat(activate(V1)), activate(V2))
U1051(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__take(V1, V2)) → U611(isNatKind(activate(V1)), activate(V2))
U611(tt, V2) → ISNATILISTKIND(activate(V2))
U611(tt, V2) → ACTIVATE(V2)
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, V1, V2) → U451(isNat(activate(V1)), activate(V2))
U451(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U451(tt, V2) → ACTIVATE(V2)
U441(tt, V1, V2) → ISNAT(activate(V1))
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
U121(tt, V1) → ISNATLIST(activate(V1))
U121(tt, V1) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → U221(isNatKind(activate(V1)), activate(V1))
U221(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U221(tt, V1) → ACTIVATE(V1)
U211(tt, V1) → ISNATKIND(activate(V1))
U211(tt, V1) → ACTIVATE(V1)
U441(tt, V1, V2) → ACTIVATE(V1)
U441(tt, V1, V2) → ACTIVATE(V2)
U431(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U431(tt, V1, V2) → ACTIVATE(V2)
U431(tt, V1, V2) → ACTIVATE(V1)
U421(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U421(tt, V1, V2) → ACTIVATE(V2)
U421(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ISNATKIND(activate(V1))
U411(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ACTIVATE(V2)
U1051(tt, V2) → ACTIVATE(V2)
U1041(tt, V1, V2) → ISNAT(activate(V1))
U1041(tt, V1, V2) → ACTIVATE(V1)
U1041(tt, V1, V2) → ACTIVATE(V2)
U1031(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1031(tt, V1, V2) → ACTIVATE(V2)
U1031(tt, V1, V2) → ACTIVATE(V1)
U1021(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1021(tt, V1, V2) → ACTIVATE(V2)
U1021(tt, V1, V2) → ACTIVATE(V1)
U1011(tt, V1, V2) → ISNATKIND(activate(V1))
U1011(tt, V1, V2) → ACTIVATE(V2)
U951(tt, V2) → ACTIVATE(V2)
U941(tt, V1, V2) → ISNAT(activate(V1))
U941(tt, V1, V2) → ACTIVATE(V1)
U941(tt, V1, V2) → ACTIVATE(V2)
U931(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U931(tt, V1, V2) → ACTIVATE(V2)
U931(tt, V1, V2) → ACTIVATE(V1)
U921(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U921(tt, V1, V2) → ACTIVATE(V2)
U921(tt, V1, V2) → ACTIVATE(V1)
U911(tt, V1, V2) → ISNATKIND(activate(V1))
U911(tt, V1, V2) → ACTIVATE(V1)
U911(tt, V1, V2) → ACTIVATE(V2)
U321(tt, V) → ACTIVATE(V)
U311(tt, V) → ISNATILISTKIND(activate(V))
U311(tt, V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → U1311(isNatIList(activate(IL)), activate(IL), M, N)
U1311(tt, IL, M, N) → U1321(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U1321(tt, IL, M, N) → U1331(isNat(activate(M)), activate(IL), activate(M), activate(N))
U1331(tt, IL, M, N) → U1341(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U1341(tt, IL, M, N) → U1351(isNat(activate(N)), activate(IL), activate(M), activate(N))
U1351(tt, IL, M, N) → U1361(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U1361(tt, IL, M, N) → ACTIVATE(N)
U1361(tt, IL, M, N) → ACTIVATE(M)
U1361(tt, IL, M, N) → ACTIVATE(IL)
U1351(tt, IL, M, N) → ISNATKIND(activate(N))
U1351(tt, IL, M, N) → ACTIVATE(N)
U1351(tt, IL, M, N) → ACTIVATE(IL)
U1351(tt, IL, M, N) → ACTIVATE(M)
U1341(tt, IL, M, N) → ISNAT(activate(N))
U1341(tt, IL, M, N) → ACTIVATE(N)
U1341(tt, IL, M, N) → ACTIVATE(IL)
U1341(tt, IL, M, N) → ACTIVATE(M)
U1331(tt, IL, M, N) → ISNATKIND(activate(M))
U1331(tt, IL, M, N) → ACTIVATE(M)
U1331(tt, IL, M, N) → ACTIVATE(IL)
U1331(tt, IL, M, N) → ACTIVATE(N)
U1321(tt, IL, M, N) → ISNAT(activate(M))
U1321(tt, IL, M, N) → ACTIVATE(M)
U1321(tt, IL, M, N) → ACTIVATE(IL)
U1321(tt, IL, M, N) → ACTIVATE(N)
U1311(tt, IL, M, N) → ISNATILISTKIND(activate(IL))
U1311(tt, IL, M, N) → ACTIVATE(IL)
U1311(tt, IL, M, N) → ACTIVATE(M)
U1311(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U511(tt, V2) → ACTIVATE(V2)
U1211(tt, IL) → ACTIVATE(IL)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(12) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
(13) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1211(tt, IL) → ISNATILISTKIND(activate(IL))
ISNATILISTKIND(n__cons(V1, V2)) → U511(isNatKind(activate(V1)), activate(V2))
U511(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
TAKE(0, IL) → U1211(isNatIList(IL), IL)
U1211(tt, IL) → ACTIVATE(IL)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
U311(tt, V) → U321(isNatIListKind(activate(V)), activate(V))
U321(tt, V) → ISNATLIST(activate(V))
ISNATLIST(n__cons(V1, V2)) → U911(isNatKind(activate(V1)), activate(V1), activate(V2))
U911(tt, V1, V2) → U921(isNatKind(activate(V1)), activate(V1), activate(V2))
U921(tt, V1, V2) → U931(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U931(tt, V1, V2) → U941(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U941(tt, V1, V2) → U951(isNat(activate(V1)), activate(V2))
U951(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → U1011(isNatKind(activate(V1)), activate(V1), activate(V2))
U1011(tt, V1, V2) → U1021(isNatKind(activate(V1)), activate(V1), activate(V2))
U1021(tt, V1, V2) → U1031(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1031(tt, V1, V2) → U1041(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1041(tt, V1, V2) → U1051(isNat(activate(V1)), activate(V2))
U1051(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__take(V1, V2)) → U611(isNatKind(activate(V1)), activate(V2))
U611(tt, V2) → ISNATILISTKIND(activate(V2))
U611(tt, V2) → ACTIVATE(V2)
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, V1, V2) → U451(isNat(activate(V1)), activate(V2))
U451(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U451(tt, V2) → ACTIVATE(V2)
U441(tt, V1, V2) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → U221(isNatKind(activate(V1)), activate(V1))
U221(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U221(tt, V1) → ACTIVATE(V1)
U211(tt, V1) → ISNATKIND(activate(V1))
U211(tt, V1) → ACTIVATE(V1)
U441(tt, V1, V2) → ACTIVATE(V1)
U441(tt, V1, V2) → ACTIVATE(V2)
U431(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U431(tt, V1, V2) → ACTIVATE(V2)
U431(tt, V1, V2) → ACTIVATE(V1)
U421(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U421(tt, V1, V2) → ACTIVATE(V2)
U421(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ISNATKIND(activate(V1))
U411(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ACTIVATE(V2)
U1051(tt, V2) → ACTIVATE(V2)
U1041(tt, V1, V2) → ISNAT(activate(V1))
U1041(tt, V1, V2) → ACTIVATE(V1)
U1041(tt, V1, V2) → ACTIVATE(V2)
U1031(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1031(tt, V1, V2) → ACTIVATE(V2)
U1031(tt, V1, V2) → ACTIVATE(V1)
U1021(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1021(tt, V1, V2) → ACTIVATE(V2)
U1021(tt, V1, V2) → ACTIVATE(V1)
U1011(tt, V1, V2) → ISNATKIND(activate(V1))
U1011(tt, V1, V2) → ACTIVATE(V2)
U951(tt, V2) → ACTIVATE(V2)
U941(tt, V1, V2) → ISNAT(activate(V1))
U941(tt, V1, V2) → ACTIVATE(V1)
U941(tt, V1, V2) → ACTIVATE(V2)
U931(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U931(tt, V1, V2) → ACTIVATE(V2)
U931(tt, V1, V2) → ACTIVATE(V1)
U921(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U921(tt, V1, V2) → ACTIVATE(V2)
U921(tt, V1, V2) → ACTIVATE(V1)
U911(tt, V1, V2) → ISNATKIND(activate(V1))
U911(tt, V1, V2) → ACTIVATE(V1)
U911(tt, V1, V2) → ACTIVATE(V2)
U321(tt, V) → ACTIVATE(V)
U311(tt, V) → ISNATILISTKIND(activate(V))
U311(tt, V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → U1311(isNatIList(activate(IL)), activate(IL), M, N)
U1311(tt, IL, M, N) → U1321(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U1321(tt, IL, M, N) → U1331(isNat(activate(M)), activate(IL), activate(M), activate(N))
U1331(tt, IL, M, N) → U1341(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U1341(tt, IL, M, N) → U1351(isNat(activate(N)), activate(IL), activate(M), activate(N))
U1351(tt, IL, M, N) → U1361(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U1361(tt, IL, M, N) → ACTIVATE(N)
U1361(tt, IL, M, N) → ACTIVATE(M)
U1361(tt, IL, M, N) → ACTIVATE(IL)
U1351(tt, IL, M, N) → ISNATKIND(activate(N))
U1351(tt, IL, M, N) → ACTIVATE(N)
U1351(tt, IL, M, N) → ACTIVATE(IL)
U1351(tt, IL, M, N) → ACTIVATE(M)
U1341(tt, IL, M, N) → ISNAT(activate(N))
U1341(tt, IL, M, N) → ACTIVATE(N)
U1341(tt, IL, M, N) → ACTIVATE(IL)
U1341(tt, IL, M, N) → ACTIVATE(M)
U1331(tt, IL, M, N) → ISNATKIND(activate(M))
U1331(tt, IL, M, N) → ACTIVATE(M)
U1331(tt, IL, M, N) → ACTIVATE(IL)
U1331(tt, IL, M, N) → ACTIVATE(N)
U1321(tt, IL, M, N) → ISNAT(activate(M))
U1321(tt, IL, M, N) → ACTIVATE(M)
U1321(tt, IL, M, N) → ACTIVATE(IL)
U1321(tt, IL, M, N) → ACTIVATE(N)
U1311(tt, IL, M, N) → ISNATILISTKIND(activate(IL))
U1311(tt, IL, M, N) → ACTIVATE(IL)
U1311(tt, IL, M, N) → ACTIVATE(M)
U1311(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U511(tt, V2) → ACTIVATE(V2)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(14) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ISNATLIST(n__take(V1, V2)) → U1011(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATILISTKIND(n__take(V1, V2)) → U611(isNatKind(activate(V1)), activate(V2))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
POL(U1211(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(ISNATILISTKIND(x1)) = | | + | | · | x1 |
POL(activate(x1)) = | | + | | · | x1 |
POL(n__cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U511(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatKind(x1)) = | | + | | · | x1 |
POL(ISNATKIND(x1)) = | | + | | · | x1 |
POL(ACTIVATE(x1)) = | | + | | · | x1 |
POL(n__take(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(TAKE(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatIList(x1)) = | | + | | · | x1 |
POL(ISNATILIST(x1)) = | | + | | · | x1 |
POL(U311(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatIListKind(x1)) = | | + | | · | x1 |
POL(U321(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(ISNATLIST(x1)) = | | + | | · | x1 |
POL(U911(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U921(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U931(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U941(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U951(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U1011(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U1021(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U1031(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U1041(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U1051(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U611(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U411(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U421(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U431(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U441(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U451(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U211(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U221(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U1311(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U1321(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U1331(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U1341(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U1351(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U1361(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U102(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U103(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U104(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U101(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U11(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U12(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U105(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U114(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U113(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U112(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U111(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U121(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatList(x1)) = | | + | | · | x1 |
POL(U134(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U135(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U133(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U132(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U131(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U136(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U21(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U22(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U31(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U32(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U41(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U42(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U43(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U44(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U45(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U51(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U61(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U92(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U93(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U91(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U94(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U95(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(n__length(x1)) = | | + | | · | x1 |
POL(take(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U106(tt) → tt
U105(tt, V2) → U106(isNatIList(activate(V2)))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U114(tt, L) → s(length(activate(L)))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U13(tt) → tt
U122(tt) → nil
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U62(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U81(tt) → tt
U71(tt) → tt
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U96(tt) → tt
U95(tt, V2) → U96(isNatList(activate(V2)))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
(15) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1211(tt, IL) → ISNATILISTKIND(activate(IL))
ISNATILISTKIND(n__cons(V1, V2)) → U511(isNatKind(activate(V1)), activate(V2))
U511(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
TAKE(0, IL) → U1211(isNatIList(IL), IL)
U1211(tt, IL) → ACTIVATE(IL)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
U311(tt, V) → U321(isNatIListKind(activate(V)), activate(V))
U321(tt, V) → ISNATLIST(activate(V))
ISNATLIST(n__cons(V1, V2)) → U911(isNatKind(activate(V1)), activate(V1), activate(V2))
U911(tt, V1, V2) → U921(isNatKind(activate(V1)), activate(V1), activate(V2))
U921(tt, V1, V2) → U931(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U931(tt, V1, V2) → U941(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U941(tt, V1, V2) → U951(isNat(activate(V1)), activate(V2))
U951(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U1011(tt, V1, V2) → U1021(isNatKind(activate(V1)), activate(V1), activate(V2))
U1021(tt, V1, V2) → U1031(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1031(tt, V1, V2) → U1041(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1041(tt, V1, V2) → U1051(isNat(activate(V1)), activate(V2))
U1051(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
U611(tt, V2) → ISNATILISTKIND(activate(V2))
U611(tt, V2) → ACTIVATE(V2)
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, V1, V2) → U451(isNat(activate(V1)), activate(V2))
U451(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U451(tt, V2) → ACTIVATE(V2)
U441(tt, V1, V2) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → U221(isNatKind(activate(V1)), activate(V1))
U221(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U221(tt, V1) → ACTIVATE(V1)
U211(tt, V1) → ISNATKIND(activate(V1))
U211(tt, V1) → ACTIVATE(V1)
U441(tt, V1, V2) → ACTIVATE(V1)
U441(tt, V1, V2) → ACTIVATE(V2)
U431(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U431(tt, V1, V2) → ACTIVATE(V2)
U431(tt, V1, V2) → ACTIVATE(V1)
U421(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U421(tt, V1, V2) → ACTIVATE(V2)
U421(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ISNATKIND(activate(V1))
U411(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ACTIVATE(V2)
U1051(tt, V2) → ACTIVATE(V2)
U1041(tt, V1, V2) → ISNAT(activate(V1))
U1041(tt, V1, V2) → ACTIVATE(V1)
U1041(tt, V1, V2) → ACTIVATE(V2)
U1031(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1031(tt, V1, V2) → ACTIVATE(V2)
U1031(tt, V1, V2) → ACTIVATE(V1)
U1021(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1021(tt, V1, V2) → ACTIVATE(V2)
U1021(tt, V1, V2) → ACTIVATE(V1)
U1011(tt, V1, V2) → ISNATKIND(activate(V1))
U1011(tt, V1, V2) → ACTIVATE(V2)
U951(tt, V2) → ACTIVATE(V2)
U941(tt, V1, V2) → ISNAT(activate(V1))
U941(tt, V1, V2) → ACTIVATE(V1)
U941(tt, V1, V2) → ACTIVATE(V2)
U931(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U931(tt, V1, V2) → ACTIVATE(V2)
U931(tt, V1, V2) → ACTIVATE(V1)
U921(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U921(tt, V1, V2) → ACTIVATE(V2)
U921(tt, V1, V2) → ACTIVATE(V1)
U911(tt, V1, V2) → ISNATKIND(activate(V1))
U911(tt, V1, V2) → ACTIVATE(V1)
U911(tt, V1, V2) → ACTIVATE(V2)
U321(tt, V) → ACTIVATE(V)
U311(tt, V) → ISNATILISTKIND(activate(V))
U311(tt, V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → U1311(isNatIList(activate(IL)), activate(IL), M, N)
U1311(tt, IL, M, N) → U1321(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U1321(tt, IL, M, N) → U1331(isNat(activate(M)), activate(IL), activate(M), activate(N))
U1331(tt, IL, M, N) → U1341(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U1341(tt, IL, M, N) → U1351(isNat(activate(N)), activate(IL), activate(M), activate(N))
U1351(tt, IL, M, N) → U1361(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U1361(tt, IL, M, N) → ACTIVATE(N)
U1361(tt, IL, M, N) → ACTIVATE(M)
U1361(tt, IL, M, N) → ACTIVATE(IL)
U1351(tt, IL, M, N) → ISNATKIND(activate(N))
U1351(tt, IL, M, N) → ACTIVATE(N)
U1351(tt, IL, M, N) → ACTIVATE(IL)
U1351(tt, IL, M, N) → ACTIVATE(M)
U1341(tt, IL, M, N) → ISNAT(activate(N))
U1341(tt, IL, M, N) → ACTIVATE(N)
U1341(tt, IL, M, N) → ACTIVATE(IL)
U1341(tt, IL, M, N) → ACTIVATE(M)
U1331(tt, IL, M, N) → ISNATKIND(activate(M))
U1331(tt, IL, M, N) → ACTIVATE(M)
U1331(tt, IL, M, N) → ACTIVATE(IL)
U1331(tt, IL, M, N) → ACTIVATE(N)
U1321(tt, IL, M, N) → ISNAT(activate(M))
U1321(tt, IL, M, N) → ACTIVATE(M)
U1321(tt, IL, M, N) → ACTIVATE(IL)
U1321(tt, IL, M, N) → ACTIVATE(N)
U1311(tt, IL, M, N) → ISNATILISTKIND(activate(IL))
U1311(tt, IL, M, N) → ACTIVATE(IL)
U1311(tt, IL, M, N) → ACTIVATE(M)
U1311(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U511(tt, V2) → ACTIVATE(V2)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(16) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 6 SCCs with 104 less nodes.
(17) Complex Obligation (AND)
(18) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(19) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(20) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(21) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
The graph contains the following edges 1 > 1
- ACTIVATE(n__s(X)) → ACTIVATE(X)
The graph contains the following edges 1 > 1
(22) TRUE
(23) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(24) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
POL(ISNATKIND(x1)) = | | + | | · | x1 |
POL(activate(x1)) = | | + | | · | x1 |
POL(U102(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U103(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(isNatIListKind(x1)) = | | + | | · | x1 |
POL(U104(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U101(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(isNatKind(x1)) = | | + | | · | x1 |
POL(U11(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U12(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U105(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatIList(x1)) = | | + | | · | x1 |
POL(U114(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U113(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U112(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U111(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U121(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatList(x1)) = | | + | | · | x1 |
POL(U134(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U135(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U133(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U132(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U131(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U136(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(n__take(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U21(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U22(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U31(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U32(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U41(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U42(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U43(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U44(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U45(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U51(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U61(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U92(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U93(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U91(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U94(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U95(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(n__length(x1)) = | | + | | · | x1 |
POL(n__cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(take(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U106(tt) → tt
U105(tt, V2) → U106(isNatIList(activate(V2)))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U114(tt, L) → s(length(activate(L)))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U13(tt) → tt
U122(tt) → nil
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U62(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U81(tt) → tt
U71(tt) → tt
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U96(tt) → tt
U95(tt, V2) → U96(isNatList(activate(V2)))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIList(n__zeros) → tt
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
(25) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(26) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(27) TRUE
(28) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → U221(isNatKind(activate(V1)), activate(V1))
U221(tt, V1) → ISNAT(activate(V1))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(29) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
POL(U211(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatKind(x1)) = | | + | | · | x1 |
POL(activate(x1)) = | | + | | · | x1 |
POL(U221(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U102(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U103(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(isNatIListKind(x1)) = | | + | | · | x1 |
POL(U104(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U101(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U11(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U12(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U105(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatIList(x1)) = | | + | | · | x1 |
POL(U114(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U113(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U112(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U111(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U121(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatList(x1)) = | | + | | · | x1 |
POL(U134(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U135(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U133(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U132(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U131(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U136(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(n__take(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U21(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U22(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U31(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U32(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U41(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U42(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U43(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U44(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U45(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U51(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U61(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U92(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U93(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U91(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U94(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U95(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(n__length(x1)) = | | + | | · | x1 |
POL(n__cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(take(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U106(tt) → tt
U105(tt, V2) → U106(isNatIList(activate(V2)))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U114(tt, L) → s(length(activate(L)))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U13(tt) → tt
U122(tt) → nil
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U62(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U81(tt) → tt
U71(tt) → tt
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U96(tt) → tt
U95(tt, V2) → U96(isNatList(activate(V2)))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIList(n__zeros) → tt
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
(30) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U211(tt, V1) → U221(isNatKind(activate(V1)), activate(V1))
U221(tt, V1) → ISNAT(activate(V1))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(31) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
(32) TRUE
(33) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → U511(isNatKind(activate(V1)), activate(V2))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(34) Narrowing (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
U511(
tt,
V2) →
ISNATILISTKIND(
activate(
V2)) at position [0] we obtained the following new rules [LPAR04]:
U511(tt, n__zeros) → ISNATILISTKIND(zeros)
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U511(tt, n__0) → ISNATILISTKIND(0)
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
U511(tt, n__nil) → ISNATILISTKIND(nil)
U511(tt, x0) → ISNATILISTKIND(x0)
(35) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ISNATILISTKIND(n__cons(V1, V2)) → U511(isNatKind(activate(V1)), activate(V2))
U511(tt, n__zeros) → ISNATILISTKIND(zeros)
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U511(tt, n__0) → ISNATILISTKIND(0)
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
U511(tt, n__nil) → ISNATILISTKIND(nil)
U511(tt, x0) → ISNATILISTKIND(x0)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(36) Narrowing (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
ISNATILISTKIND(
n__cons(
V1,
V2)) →
U511(
isNatKind(
activate(
V1)),
activate(
V2)) at position [0] we obtained the following new rules [LPAR04]:
ISNATILISTKIND(n__cons(n__zeros, y1)) → U511(isNatKind(zeros), activate(y1))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
ISNATILISTKIND(n__cons(n__0, y1)) → U511(isNatKind(0), activate(y1))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
ISNATILISTKIND(n__cons(n__nil, y1)) → U511(isNatKind(nil), activate(y1))
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
(37) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, n__zeros) → ISNATILISTKIND(zeros)
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U511(tt, n__0) → ISNATILISTKIND(0)
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
U511(tt, n__nil) → ISNATILISTKIND(nil)
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__zeros, y1)) → U511(isNatKind(zeros), activate(y1))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
ISNATILISTKIND(n__cons(n__0, y1)) → U511(isNatKind(0), activate(y1))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
ISNATILISTKIND(n__cons(n__nil, y1)) → U511(isNatKind(nil), activate(y1))
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(38) Narrowing (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
U511(
tt,
n__zeros) →
ISNATILISTKIND(
zeros) at position [0] we obtained the following new rules [LPAR04]:
U511(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
U511(tt, n__zeros) → ISNATILISTKIND(n__zeros)
(39) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U511(tt, n__0) → ISNATILISTKIND(0)
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
U511(tt, n__nil) → ISNATILISTKIND(nil)
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__zeros, y1)) → U511(isNatKind(zeros), activate(y1))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
ISNATILISTKIND(n__cons(n__0, y1)) → U511(isNatKind(0), activate(y1))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
ISNATILISTKIND(n__cons(n__nil, y1)) → U511(isNatKind(nil), activate(y1))
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
U511(tt, n__zeros) → ISNATILISTKIND(n__zeros)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(40) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(41) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ISNATILISTKIND(n__cons(n__zeros, y1)) → U511(isNatKind(zeros), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__0) → ISNATILISTKIND(0)
ISNATILISTKIND(n__cons(n__0, y1)) → U511(isNatKind(0), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(n__nil, y1)) → U511(isNatKind(nil), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(42) Narrowing (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
ISNATILISTKIND(
n__cons(
n__zeros,
y1)) →
U511(
isNatKind(
zeros),
activate(
y1)) at position [0] we obtained the following new rules [LPAR04]:
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(n__zeros), activate(y0))
(43) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__0) → ISNATILISTKIND(0)
ISNATILISTKIND(n__cons(n__0, y1)) → U511(isNatKind(0), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(n__nil, y1)) → U511(isNatKind(nil), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(n__zeros), activate(y0))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(44) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(45) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__0, y1)) → U511(isNatKind(0), activate(y1))
U511(tt, n__0) → ISNATILISTKIND(0)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__nil, y1)) → U511(isNatKind(nil), activate(y1))
U511(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(cons(0, n__zeros)), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(46) Narrowing (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
ISNATILISTKIND(
n__cons(
n__0,
y1)) →
U511(
isNatKind(
0),
activate(
y1)) at position [0] we obtained the following new rules [LPAR04]:
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
(47) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U511(tt, n__0) → ISNATILISTKIND(0)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__nil, y1)) → U511(isNatKind(nil), activate(y1))
U511(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(cons(0, n__zeros)), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(48) Narrowing (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
U511(
tt,
n__0) →
ISNATILISTKIND(
0) at position [0] we obtained the following new rules [LPAR04]:
U511(tt, n__0) → ISNATILISTKIND(n__0)
(49) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__nil, y1)) → U511(isNatKind(nil), activate(y1))
U511(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(cons(0, n__zeros)), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__0) → ISNATILISTKIND(n__0)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(50) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(51) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(n__nil, y1)) → U511(isNatKind(nil), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(52) Narrowing (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
U511(
tt,
n__nil) →
ISNATILISTKIND(
nil) at position [0] we obtained the following new rules [LPAR04]:
U511(tt, n__nil) → ISNATILISTKIND(n__nil)
(53) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
ISNATILISTKIND(n__cons(n__nil, y1)) → U511(isNatKind(nil), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__nil) → ISNATILISTKIND(n__nil)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(54) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(55) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__nil, y1)) → U511(isNatKind(nil), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(56) Narrowing (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
ISNATILISTKIND(
n__cons(
n__nil,
y1)) →
U511(
isNatKind(
nil),
activate(
y1)) at position [0] we obtained the following new rules [LPAR04]:
ISNATILISTKIND(n__cons(n__nil, y0)) → U511(isNatKind(n__nil), activate(y0))
(57) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
ISNATILISTKIND(n__cons(n__nil, y0)) → U511(isNatKind(n__nil), activate(y0))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(58) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(59) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(60) Narrowing (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
U511(
tt,
n__zeros) →
ISNATILISTKIND(
cons(
0,
n__zeros)) at position [0] we obtained the following new rules [LPAR04]:
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U511(tt, n__zeros) → ISNATILISTKIND(cons(n__0, n__zeros))
(61) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U511(tt, n__zeros) → ISNATILISTKIND(cons(n__0, n__zeros))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(62) Narrowing (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
ISNATILISTKIND(
n__cons(
n__zeros,
y0)) →
U511(
isNatKind(
cons(
0,
n__zeros)),
activate(
y0)) at position [0] we obtained the following new rules [LPAR04]:
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(n__cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(cons(n__0, n__zeros)), activate(y0))
(63) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U511(tt, n__zeros) → ISNATILISTKIND(cons(n__0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(n__cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(cons(n__0, n__zeros)), activate(y0))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(64) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(65) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(cons(n__0, n__zeros)), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(cons(n__0, n__zeros))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(66) Narrowing (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
ISNATILISTKIND(
n__cons(
n__zeros,
y0)) →
U511(
isNatKind(
cons(
n__0,
n__zeros)),
activate(
y0)) at position [0] we obtained the following new rules [LPAR04]:
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(n__cons(n__0, n__zeros)), activate(y0))
(67) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U511(tt, n__zeros) → ISNATILISTKIND(cons(n__0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(n__cons(n__0, n__zeros)), activate(y0))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(68) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(69) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(cons(n__0, n__zeros))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(70) Narrowing (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
U511(
tt,
n__zeros) →
ISNATILISTKIND(
cons(
n__0,
n__zeros)) at position [0] we obtained the following new rules [LPAR04]:
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))
(71) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(72) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U511(isNatKind(take(activate(x0), activate(x1))), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(ISNATILISTKIND(x1)) = x1
POL(U101(x1, x2, x3)) = 1
POL(U102(x1, x2, x3)) = 1
POL(U103(x1, x2, x3)) = 1
POL(U104(x1, x2, x3)) = 1
POL(U105(x1, x2)) = 0
POL(U106(x1)) = 0
POL(U11(x1, x2)) = 0
POL(U111(x1, x2, x3)) = 0
POL(U112(x1, x2, x3)) = 0
POL(U113(x1, x2, x3)) = 0
POL(U114(x1, x2)) = 0
POL(U12(x1, x2)) = 0
POL(U121(x1, x2)) = 0
POL(U122(x1)) = 0
POL(U13(x1)) = 0
POL(U131(x1, x2, x3, x4)) = 1 + x2 + x4
POL(U132(x1, x2, x3, x4)) = 1 + x2 + x4
POL(U133(x1, x2, x3, x4)) = 1 + x2 + x4
POL(U134(x1, x2, x3, x4)) = 1 + x2 + x4
POL(U135(x1, x2, x3, x4)) = 1 + x2 + x4
POL(U136(x1, x2, x3, x4)) = 1 + x2 + x4
POL(U21(x1, x2)) = 0
POL(U22(x1, x2)) = 0
POL(U23(x1)) = 0
POL(U31(x1, x2)) = 1
POL(U32(x1, x2)) = 1
POL(U33(x1)) = 1
POL(U41(x1, x2, x3)) = 0
POL(U42(x1, x2, x3)) = 0
POL(U43(x1, x2, x3)) = 0
POL(U44(x1, x2, x3)) = 0
POL(U45(x1, x2)) = 0
POL(U46(x1)) = 0
POL(U51(x1, x2)) = 0
POL(U511(x1, x2)) = x2
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3)) = 0
POL(U92(x1, x2, x3)) = 0
POL(U93(x1, x2, x3)) = 0
POL(U94(x1, x2, x3)) = 0
POL(U95(x1, x2)) = 0
POL(U96(x1)) = 0
POL(activate(x1)) = x1
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 1
POL(isNatIListKind(x1)) = 1
POL(isNatKind(x1)) = 0
POL(isNatList(x1)) = 1 + x1
POL(length(x1)) = 0
POL(n__0) = 0
POL(n__cons(x1, x2)) = x1 + x2
POL(n__length(x1)) = 0
POL(n__nil) = 0
POL(n__s(x1)) = 0
POL(n__take(x1, x2)) = 1 + x2
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x1)) = 0
POL(take(x1, x2)) = 1 + x2
POL(tt) = 0
POL(zeros) = 0
The following usable rules [FROCOS05] were oriented:
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U106(tt) → tt
U105(tt, V2) → U106(isNatIList(activate(V2)))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U114(tt, L) → s(length(activate(L)))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U13(tt) → tt
U122(tt) → nil
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U62(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U81(tt) → tt
U71(tt) → tt
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U96(tt) → tt
U95(tt, V2) → U96(isNatList(activate(V2)))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
(73) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(74) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(ISNATILISTKIND(x1)) = x1
POL(U101(x1, x2, x3)) = 0
POL(U102(x1, x2, x3)) = 0
POL(U103(x1, x2, x3)) = 0
POL(U104(x1, x2, x3)) = 0
POL(U105(x1, x2)) = 0
POL(U106(x1)) = 0
POL(U11(x1, x2)) = 0
POL(U111(x1, x2, x3)) = 1 + x2
POL(U112(x1, x2, x3)) = 1 + x2
POL(U113(x1, x2, x3)) = 1 + x2
POL(U114(x1, x2)) = 1 + x2
POL(U12(x1, x2)) = 0
POL(U121(x1, x2)) = x2
POL(U122(x1)) = 0
POL(U13(x1)) = 0
POL(U131(x1, x2, x3, x4)) = x2 + x3 + x4
POL(U132(x1, x2, x3, x4)) = x2 + x3 + x4
POL(U133(x1, x2, x3, x4)) = x2 + x3 + x4
POL(U134(x1, x2, x3, x4)) = x2 + x3 + x4
POL(U135(x1, x2, x3, x4)) = x2 + x3 + x4
POL(U136(x1, x2, x3, x4)) = x2 + x3 + x4
POL(U21(x1, x2)) = 0
POL(U22(x1, x2)) = 0
POL(U23(x1)) = 0
POL(U31(x1, x2)) = 0
POL(U32(x1, x2)) = 0
POL(U33(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U42(x1, x2, x3)) = 0
POL(U43(x1, x2, x3)) = 0
POL(U44(x1, x2, x3)) = 0
POL(U45(x1, x2)) = 0
POL(U46(x1)) = 0
POL(U51(x1, x2)) = 0
POL(U511(x1, x2)) = x2
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3)) = 0
POL(U92(x1, x2, x3)) = 0
POL(U93(x1, x2, x3)) = 0
POL(U94(x1, x2, x3)) = 0
POL(U95(x1, x2)) = 0
POL(U96(x1)) = 0
POL(activate(x1)) = x1
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 0
POL(isNatKind(x1)) = 0
POL(isNatList(x1)) = 1
POL(length(x1)) = 1 + x1
POL(n__0) = 0
POL(n__cons(x1, x2)) = x1 + x2
POL(n__length(x1)) = 1 + x1
POL(n__nil) = 0
POL(n__s(x1)) = x1
POL(n__take(x1, x2)) = x1 + x2
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = x1 + x2
POL(tt) = 0
POL(zeros) = 0
The following usable rules [FROCOS05] were oriented:
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U106(tt) → tt
U105(tt, V2) → U106(isNatIList(activate(V2)))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U114(tt, L) → s(length(activate(L)))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U13(tt) → tt
U122(tt) → nil
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U62(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U81(tt) → tt
U71(tt) → tt
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U96(tt) → tt
U95(tt, V2) → U96(isNatList(activate(V2)))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
(75) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(76) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(ISNATILISTKIND(x1)) = x1
POL(U101(x1, x2, x3)) = 0
POL(U102(x1, x2, x3)) = 0
POL(U103(x1, x2, x3)) = 0
POL(U104(x1, x2, x3)) = 0
POL(U105(x1, x2)) = 0
POL(U106(x1)) = 0
POL(U11(x1, x2)) = 0
POL(U111(x1, x2, x3)) = 1
POL(U112(x1, x2, x3)) = 1
POL(U113(x1, x2, x3)) = 1
POL(U114(x1, x2)) = 1
POL(U12(x1, x2)) = 0
POL(U121(x1, x2)) = 0
POL(U122(x1)) = 0
POL(U13(x1)) = 0
POL(U131(x1, x2, x3, x4)) = x2 + x4
POL(U132(x1, x2, x3, x4)) = x2 + x4
POL(U133(x1, x2, x3, x4)) = x2 + x4
POL(U134(x1, x2, x3, x4)) = x2 + x4
POL(U135(x1, x2, x3, x4)) = x2 + x4
POL(U136(x1, x2, x3, x4)) = x2 + x4
POL(U21(x1, x2)) = 0
POL(U22(x1, x2)) = 0
POL(U23(x1)) = 0
POL(U31(x1, x2)) = 0
POL(U32(x1, x2)) = 0
POL(U33(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U42(x1, x2, x3)) = 0
POL(U43(x1, x2, x3)) = 0
POL(U44(x1, x2, x3)) = 0
POL(U45(x1, x2)) = 0
POL(U46(x1)) = 0
POL(U51(x1, x2)) = 0
POL(U511(x1, x2)) = x2
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1)) = 1
POL(U81(x1)) = 0
POL(U91(x1, x2, x3)) = 0
POL(U92(x1, x2, x3)) = 0
POL(U93(x1, x2, x3)) = 0
POL(U94(x1, x2, x3)) = 0
POL(U95(x1, x2)) = 0
POL(U96(x1)) = 0
POL(activate(x1)) = x1
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = 1
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 0
POL(isNatKind(x1)) = x1
POL(isNatList(x1)) = 0
POL(length(x1)) = 1
POL(n__0) = 0
POL(n__cons(x1, x2)) = x1 + x2
POL(n__length(x1)) = 1
POL(n__nil) = 0
POL(n__s(x1)) = 1
POL(n__take(x1, x2)) = x2
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x1)) = 1
POL(take(x1, x2)) = x2
POL(tt) = 0
POL(zeros) = 0
The following usable rules [FROCOS05] were oriented:
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U106(tt) → tt
U105(tt, V2) → U106(isNatIList(activate(V2)))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U114(tt, L) → s(length(activate(L)))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U13(tt) → tt
U122(tt) → nil
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U62(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U81(tt) → tt
U71(tt) → tt
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U96(tt) → tt
U95(tt, V2) → U96(isNatList(activate(V2)))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
(77) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(78) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
POL(U511(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(n__take(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(ISNATILISTKIND(x1)) = | | + | | · | x1 |
POL(take(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(activate(x1)) = | | + | | · | x1 |
POL(n__length(x1)) = | | + | | · | x1 |
POL(n__cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatKind(x1)) = | | + | | · | x1 |
POL(U102(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U103(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(isNatIListKind(x1)) = | | + | | · | x1 |
POL(U104(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U101(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U11(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U12(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U105(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatIList(x1)) = | | + | | · | x1 |
POL(U114(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U113(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U112(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U111(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U121(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatList(x1)) = | | + | | · | x1 |
POL(U134(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U135(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U133(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U132(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U131(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U136(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U21(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U22(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U31(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U32(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U41(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U42(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U43(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U44(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U45(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U51(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U61(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U92(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U93(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U91(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U94(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U95(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U106(tt) → tt
U105(tt, V2) → U106(isNatIList(activate(V2)))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U114(tt, L) → s(length(activate(L)))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U13(tt) → tt
U122(tt) → nil
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U62(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U81(tt) → tt
U71(tt) → tt
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U96(tt) → tt
U95(tt, V2) → U96(isNatList(activate(V2)))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIList(n__zeros) → tt
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
(79) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(80) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
POL(U511(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(n__take(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(ISNATILISTKIND(x1)) = | | + | | · | x1 |
POL(take(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(activate(x1)) = | | + | | · | x1 |
POL(n__length(x1)) = | | + | | · | x1 |
POL(n__cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatKind(x1)) = | | + | | · | x1 |
POL(U102(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U103(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(isNatIListKind(x1)) = | | + | | · | x1 |
POL(U104(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U101(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U11(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U12(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U105(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatIList(x1)) = | | + | | · | x1 |
POL(U114(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U113(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U112(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U111(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U121(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatList(x1)) = | | + | | · | x1 |
POL(U134(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U135(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U133(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U132(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U131(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U136(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U21(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U22(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U31(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U32(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U41(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U42(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U43(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U44(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U45(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U51(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U61(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U92(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U93(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U91(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U94(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U95(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U106(tt) → tt
U105(tt, V2) → U106(isNatIList(activate(V2)))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U114(tt, L) → s(length(activate(L)))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U13(tt) → tt
U122(tt) → nil
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U62(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U81(tt) → tt
U71(tt) → tt
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U96(tt) → tt
U95(tt, V2) → U96(isNatList(activate(V2)))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
(81) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(82) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
POL(U511(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(n__take(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(ISNATILISTKIND(x1)) = | | + | | · | x1 |
POL(take(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(activate(x1)) = | | + | | · | x1 |
POL(n__cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatKind(x1)) = | | + | | · | x1 |
POL(U102(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U103(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(isNatIListKind(x1)) = | | + | | · | x1 |
POL(U104(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U101(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U11(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U12(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U105(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatIList(x1)) = | | + | | · | x1 |
POL(U114(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U113(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U112(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U111(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U121(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatList(x1)) = | | + | | · | x1 |
POL(U134(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U135(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U133(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U132(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U131(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U136(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U21(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U22(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U31(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U32(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U41(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U42(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U43(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U44(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U45(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U51(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U61(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U92(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U93(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U91(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U94(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U95(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(n__length(x1)) = | | + | | · | x1 |
The following usable rules [FROCOS05] were oriented:
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U106(tt) → tt
U105(tt, V2) → U106(isNatIList(activate(V2)))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U114(tt, L) → s(length(activate(L)))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U13(tt) → tt
U122(tt) → nil
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U62(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U81(tt) → tt
U71(tt) → tt
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U96(tt) → tt
U95(tt, V2) → U96(isNatList(activate(V2)))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
(83) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(84) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
U511(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
POL(U511(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(n__take(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(ISNATILISTKIND(x1)) = | | + | | · | x1 |
POL(take(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(activate(x1)) = | | + | | · | x1 |
POL(n__cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatKind(x1)) = | | + | | · | x1 |
POL(U102(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U103(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(isNatIListKind(x1)) = | | + | | · | x1 |
POL(U104(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U101(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U11(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U12(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U105(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatIList(x1)) = | | + | | · | x1 |
POL(U114(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U113(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U112(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U111(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U121(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatList(x1)) = | | + | | · | x1 |
POL(U134(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U135(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U133(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U132(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U131(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U136(x1, x2, x3, x4)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
POL(U21(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U22(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U31(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U32(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U41(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U42(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U43(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U44(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U45(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U51(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U61(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U92(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U93(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U91(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U94(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U95(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(n__length(x1)) = | | + | | · | x1 |
The following usable rules [FROCOS05] were oriented:
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U106(tt) → tt
U105(tt, V2) → U106(isNatIList(activate(V2)))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U114(tt, L) → s(length(activate(L)))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U13(tt) → tt
U122(tt) → nil
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U62(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U81(tt) → tt
U71(tt) → tt
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U96(tt) → tt
U95(tt, V2) → U96(isNatList(activate(V2)))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIList(n__zeros) → tt
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
(85) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(86) NonTerminationProof (EQUIVALENT transformation)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:
s =
U511(
isNatKind(
n__0),
activate(
n__zeros)) evaluates to t =
U511(
isNatKind(
n__0),
activate(
n__zeros))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequenceU511(isNatKind(n__0), activate(n__zeros)) →
U511(
isNatKind(
n__0),
n__zeros)
with rule
activate(
X) →
X at position [1] and matcher [
X /
n__zeros]
U511(isNatKind(n__0), n__zeros) →
U511(
tt,
n__zeros)
with rule
isNatKind(
n__0) →
tt at position [0] and matcher [ ]
U511(tt, n__zeros) →
ISNATILISTKIND(
n__cons(
n__0,
n__zeros))
with rule
U511(
tt,
n__zeros) →
ISNATILISTKIND(
n__cons(
n__0,
n__zeros)) at position [] and matcher [ ]
ISNATILISTKIND(n__cons(n__0, n__zeros)) →
U511(
isNatKind(
n__0),
activate(
n__zeros))
with rule
ISNATILISTKIND(
n__cons(
x0,
y1)) →
U511(
isNatKind(
x0),
activate(
y1))
Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
All these steps are and every following step will be a correct step w.r.t to Q.
(87) FALSE
(88) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U931(tt, V1, V2) → U941(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U941(tt, V1, V2) → U951(isNat(activate(V1)), activate(V2))
U951(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → U911(isNatKind(activate(V1)), activate(V1), activate(V2))
U911(tt, V1, V2) → U921(isNatKind(activate(V1)), activate(V1), activate(V2))
U921(tt, V1, V2) → U931(isNatIListKind(activate(V2)), activate(V1), activate(V2))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(89) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, V1, V2) → U451(isNat(activate(V1)), activate(V2))
U451(tt, V2) → ISNATILIST(activate(V2))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(90) Obligation:
Q DP problem:
The TRS P consists of the following rules:
LENGTH(cons(N, L)) → U1111(isNatList(activate(L)), activate(L), N)
U1111(tt, L, N) → U1121(isNatIListKind(activate(L)), activate(L), activate(N))
U1121(tt, L, N) → U1131(isNat(activate(N)), activate(L), activate(N))
U1131(tt, L, N) → U1141(isNatKind(activate(N)), activate(L))
U1141(tt, L) → LENGTH(activate(L))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.